FINITE ELEMENT IMPLEMENTATION OF THE PRESTON-TONKS …

The Pennsylvania State University

The Graduate School

College of Engineering

FINITE ELEMENT IMPLEMENTATION OF THE PRESTON-TONKS-WALLACE

PLASTICITY MODEL AND ENERGY BASED BONDING PARAMETER FOR THE

COLD SPRAY PROCESS

A Dissertation in

Engineering Science and Mechanics

by

Jeremy M. Schreiber

© 2016 Jeremy M. Schreiber

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

December 2016

The dissertation of Jeremy M. Schreiber was reviewed and approved* by the following:

Ivica Smid

Associate Professor of Engineering Science and Mechanics

Co-Chair of Committee

Dissertation Co-Advisor

Albert E. Segall

Professor of Engineering Science and Mechanics

Co-Chair of Committee


Timothy J. Eden

Associate Professor of Engineering Science and Mechanics


Special Member

Michael Lanagan

Professor of Engineering Science and Mechanics


Allison Beese

Assistant Professor of Materials Science and Engineering

Norris B. McFarlane Faculty Professor

Victor K. Champagne

U.S. Army Research Laboratory

Weapons and Materials Directorate

Special Member

Judith A. Todd

P.B. Breneman Department Head

Department of Engineering Science and Mechanics

*Signatures are on file in the Graduate School

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Abstract

In the Cold Spray process, solid powder particles are heated and accelerated to

approximately 800 m/s by injecting them into a pressurized gas stream that is expanded

through a converging-diverging nozzle. The high velocity particles impact a substrate and

undergo severe plastic deformation at very-high-strain-rates. The impact generates a bond

between the particles and the substrate, or on particles already deposited on the substrate. The

process parameters for each material system are developed experimentally. One of the most

important and least understood phenomena is the particle bonding mechanism. Single particle

experiments are being conducted to provide insight into the bonding process and to provide

data for model validation. The most common method of predicting particle deformation was

to use a constitutive equation such as the Johnson-Cook material model that is a curve fit of

measured high-strain-rate properties. However, the constant for these types of models have to

be determined experimentally for each material of interest and this model does not incorporate

important impact events such as bonding or rebound of the particle. To better predict particle

deformation and to reduce the need for experimental data, a thermodynamics based material

model developed by Preston et.al was implemented through a user developed subroutine in

finite element analysis. An energy based bonding parameter was also included in this

model. The model was used to predict the deformation of single particles at different impact

velocities. The model showed excellent agreement with experimental single particle impact

data provided by other universities. No adjustment to the material parameters was

required. The development of the model will be explained and model results for deformation

and bonding will be presented.

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Table of Contents List of Figures ......................................................................................................................... vii

List of Tables ............................................................................................................................ x

Acknowledgements .................................................................................................................. xi

1. Introduction ....................................................................................................................... 1

1.1. Current State-of-the-Art ................................................................................................. 2

1.2. Research Objective ........................................................................................................ 2

2. Background ........................................................................................................................ 4

2.1. History of the Cold Spray Process ............................................................................. 4

2.2. The Cold Spray Process ............................................................................................. 4

2.3. Finite Element Analysis ............................................................................................. 6

2.4. Material Constitutive Models ..................................................................................... 6

2.4.1. Johnson-Cook Constitutive Material Model ....................................................... 6

2.4.2. Zerilli-Armstrong Constitutive Material Model ................................................. 8

2.4.3. Mechanical Threshold Stress Constitutive Material Model (MTS) .................. 10

2.4.4. Preston-Tonks-Wallace Material Model (PTW) ............................................... 11

2.5. Bonding in Cold Spray ............................................................................................. 13

2.6. Hypothesis ................................................................................................................ 13

3 PTW Model Development, Calibration and Validation .................................................. 15

3.1 Initial Model Considerations .................................................................................... 15

3.2 Development and validation of a Johnson-Cook subroutine ................................... 16

3.3 PTW Model Development ....................................................................................... 22

3.3.1 Preston-Tonks-Wallace Model Derivation ....................................................... 23

3.3.2 PTW Subroutine Implementation via VUHARD ............................................. 25

3.4 PTW Parameter Sensitivity Analysis ....................................................................... 31

3.5 PTW model comparison to Johnson-Cook .............................................................. 45

3.6 Discussion ................................................................................................................ 49

Chapter 4 ................................................................................................................................. 52

4.1. Instituting a Bonding Mechanism for Cold Spray Impact ....................................... 52

4.1.1. Executive Summary .......................................................................................... 52

4.1.2. Objective ........................................................................................................... 53

4.1.3. Background and Current State-of-the-Art ........................................................ 53

4.2. Traction-Separation Behavior .................................................................................. 59

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4.2.1. Background Information ................................................................................... 59

4.2.2. Traction-Separation Option .............................................................................. 60

4.2.3. Damage Model .................................................................................................. 61

4.2.4. Damage Evolution ............................................................................................ 64

4.2.5. Damage Stabilization ........................................................................................ 67

4.3. Implementation into Abaqus .................................................................................... 67

4.3.1. Block Impact Model ......................................................................................... 68

4.3.2. Parameter Investigation for Traction-Separation .............................................. 72

4.3.3. Model Refinement ............................................................................................ 75

4.4. Conclusions .............................................................................................................. 85

4.5.1. Subroutine Development- VUINTER .................................................................. 85

5. Implementing PTW VUHARD and Bonding Parameter to Cold Spray Impact ............. 89

5.1. Chapter Summary and Introduction ......................................................................... 89

5.2. Single Particle Impact Model Setup ......................................................................... 90

5.2.1. Dimensioning and Material Properties ................................................................. 90

5.2.2. Mie-Gruneisen EOS .......................................................................................... 92

5.4. Single Particle Impact Results and Discussion ........................................................ 97

5.5. Al6061 on Al6061 Impact Results ......................................................................... 103

5.5.1. Model Setup .................................................................................................... 103

5.5.2. Interactions and Boundary Conditions............................................................ 105

5.5.3. Meshing........................................................................................................... 105

5.5.4. Results and Discussion ................................................................................... 106

5.6. Conclusions ............................................................................................................ 108

6. Conclusions ................................................................................................................... 109

7. Future Work ................................................................................................................... 114

References ............................................................................................................................. 115

Appendix A: FE Model Inputs .............................................................................................. 119

Traction-Separation Model Testing Input ......................................................................... 119

PTW Deformation Model Input ........................................................................................ 121

Al6061-Al6061 Bonding Input ......................................................................................... 126

Appendix B Johnson-Cook VUMAT Subroutine ................................................................. 130

Appendix C Bonding Results................................................................................................ 134

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Traction Separation Results at Failure (von Mises Stress) ............................................... 134

Traction Separation Results at Failure (Equivalent Plastic Strain, PEEQ) ....................... 140

Damage Parameter Investigation (von Mises stress) ........................................................ 146

Maximum Displacement Based Damage Evolution Results (von Mises Stress) .............. 150

Fracture Energy Based Results for Damage Model (von Mises stress) ............................ 153

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List of Figures Figure 2-1 Schematic of the Cold Spray Process...................................................................... 5 Figure 2-2 Strain-rate material dependence predicted by the Johnson-Cook model. ............... 8 Figure 3-1 Typical strain rate dependence predicted using the Johnson-Cook material model.................................................................................................................................................. 17 Figure 3-2 Axisymmetric cylinder model used to develop the Johnson-Cook user subroutine.................................................................................................................................................. 20 Figure 3-3 Comparison between built-in JC (left) and the JC subroutine (right) for copper. 21 Figure 3-4 Stress vs. strain rate comparison between subroutine (left) and published data (right)for copper [27]. ............................................................................................................. 26 Figure 3-5 Comparison of the saturation and yield stress for global structure. Subroutine (left) and published data (right) for copper [27]. .................................................................... 27 Figure 3-6 Flowchart of the processes that occur in the PTW VUHARD subroutine. ........... 28 Figure 3-7 Graphical flowchart of the equation breakdown in the PTW subroutine. ............. 30 Figure 3-8 Sensitivity analysis of the effect of hardening parameter p on predicted von Mises flow stress for aluminum. ....................................................................................................... 32 Figure 3-9 Sensitivity analysis of the effect of yield stress constant Y0 on the predicted von Mises flow stress for aluminum. ............................................................................................. 33 Figure 3-10 Magnified scale showing the behavior of the yield stress constant Y0 with respect to the predicted von Mises flow stress for aluminum. ............................................................ 34 Figure 3-11 Sensitivity analysis of the saturation stress S0 versus predicted von Mises flow stress for aluminum. ................................................................................................................ 35 Figure 3-12 Sensitivity analysis of temperature dependent flow stress, S∞ versus predicted von Mises flow stress for aluminum. ...................................................................................... 36 Figure 3-13 Sensitivity analysis of density ρ versus predicted von Mises flow stress in the PTW material model for aluminum. ....................................................................................... 37 Figure 3-14 Magnified scale of predicted von Mises flow stress versus density. Note that material density will not change enough during heating to significantly change the flow stress for aluminum. .......................................................................................................................... 38 Figure 3-15 Sensitivity analysis of the alpha parameter versus the predicted von Mises flow stress in the PTW material model for aluminum. ................................................................... 39 Figure 3-16 Sensitivity analysis of the effect of β on the predicted von Mises flow stress in the PTW material model for aluminum. ................................................................................. 40 Figure 3-17 Sensitivity analysis of the effect of kappa on the predicted von Mises flow stress in the PTW material model for aluminum. ............................................................................. 41 Figure 3-18 Predicted von Mises flow stress with respect to strain in the PTW material model for aluminum. .......................................................................................................................... 42 Figure 3-19 Effect of strain-rate on the predicted von Mises flow stress in the PTW material model for aluminum. ............................................................................................................... 42 Figure 3-20 Magnified view of the von Mises flow stress prediction with respect to strain-rate. The dashed curve is a logarithmic approximation for aluminum. .................................. 43 Figure 3-21 Effect of temperature on the predicted von Mises flow stress in the PTW material model. Note, the melting point of aluminum is 660 °C ............................................ 44

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Figure 3-22 Effect of changing the melting point of the material with respect to the predicted von Mises flow stress in the PTW material model for aluminum. ......................................... 45 Figure 3-23 Axisymmetric JC/PTW test model setup ............................................................ 46 Figure 3-24 1000 N comparison between PTW subroutine (left) and built-in JC (right) for copper. ..................................................................................................................................... 47 Figure 3-25 Von Mises stress comparison between the PTW subroutine (left) and the built-in JC model (right) for copper..................................................................................................... 48 Figure 3-26 Von Mises comparison between the PTW subroutine (left) and the built-in JC model (right) for copper. ......................................................................................................... 49 Figure 4-1 Schematic diagram of mechanical mixing found in cold spray bonding [40] ...... 54 Figure 4-2 Results of Northeastern Universities cold spray particle impact bonding model. Notice that the bonded area varies over time [5]. ................................................................... 56 Figure 4-3 Non-normal cold spray impact analysis conducted by Wang. et.al. Note the prediction of the gap and jetting in the model [43]. ................................................................ 57 Figure 4-4 Example of the linear progression and initiation of the damage parameter in the traction-separation model in Abaqus [45]............................................................................... 62 Figure 4-5 Exponential description of fracture energy base damage parameter used in the traction-separation model in Abaqus. [45].............................................................................. 66 Figure 4-6 Three-dimensional model setup for traction-separation parameter investigation. 69 Figure 4-7 Meshed three-dimensional model for traction-separation parameter investigation.................................................................................................................................................. 71 Figure 4-8 Trends found in the analysis of the traction-separation stiffness coefficients. ..... 72 Figure 4-9 Results of the parameter investigation for the maximum displacement based damage evolution. Units are in meters. ................................................................................... 74 Figure 4-10 Results of the parameter investigation for the fracture energy based damage evolution. ................................................................................................................................ 75 Figure 4-11 Effect of orientating element geometry to increase deformation tolerance in dynamic impact models. ......................................................................................................... 77 Figure 4-12 Refined traction-separation impact model. Note that the model now contains over 3 million elements. .......................................................................................................... 78 Figure 4-13 Results of the von Mises stress prediction in the fracture energy based damage evolution model. Note that it is difficult to see the stress distribution due to the extremely small elements. ........................................................................................................................ 79 Figure 4-14 Results of the von Mises stress prediction in the fracture energy based damage evolution model. ..................................................................................................................... 80 Figure 4-15 Meshed results of the PEEQ for the fracture energy based damage evolution. .. 81 Figure 4-16 Hidden mesh results of PEEQ for the fracture energy based damage evolution model. It is apparent that there is nodal immersion at the impacting interface. ..................... 82 Figure 4-17 Saw-tooth pattern caused by nodal immersion at the interface........................... 82 Figure 4-18 Plastic strain results for the refined fracture energy based damage evolution model. Note, it is very difficult to identify the strain due to the small mesh size. ................. 83

ix

Figure 4-19 Hidden mesh results for the plastic strain in the refined fracture energy based damage evolution model. Note, that the largest amount of strain is found at the corners where bonding has occurred. ............................................................................................................. 84 Figure 5-1 Single Particle Impact Setup. Notice that the particle has been heavily partitioned for enhanced mesh refinement and interaction parameter. ..................................................... 90 Figure 5-2 Relationship between pressure and density known as a Hugoniot curve [55]. ..... 93 Figure 5-3 Example of the meshed single particle impact model before impact. ................... 96 Figure 5-4 Comparison between the experimental 175 m/sec impact and the PTW modeling result. Deformation appears to be very similar. Experiments conducted by W. Xie and J. Lee at UMASS. .............................................................................................................................. 97 Figure 5-5 Comparison between the experimental 286 m/sec impact and the PTW modeling result. Deformation appears to be quite similar. Experiments conducted by W. Xie and J. Lee at UMASS. .............................................................................................................................. 98 Figure 5-6 Comparison between the experimental 416 m/sec impact and the PTW modeling result. The general deformation appears to be comparible. Note the bottom of the particle appears to be rounded due to viewing angle. Experiments conducted by W. Xie and J. Lee at UMASS. .................................................................................................................................. 99 Figure 5-7 Comparison between the experimental 530 m/sec impact and the PTW modeling result. Deformation appears to be similar. Experiments conducted by W. Xie and J. Lee at UMASS. .................................................................................................................................. 99 Figure 5-8 Comparison between experimental 663 m/sec impact and the PW modeling result. Both the experimental and finite element model show very similar deformation. Experimental results conducted by W. Xie and J. Lee at UMASS....................................... 100 Figure 5-9 Comparison between experimental 699 m/sec impact and the PW modeling result. Both the experimental and finite element model show similar deformation. Experimental results conducted by W. Xie and J. Lee at UMASS. ............................................................ 101 Figure 5-10 Particle outline overlays for both the experimental results (blue), and the PTW model (red). Note that deformation is very similar in each case. ......................................... 102 Figure 5-11 Model setup for Al6061-Al6061 impact. Note, there is much less partitioning in the particle compared to the initial PTW model. .................................................................. 104 Figure 5-12 Meshed Al6061-Al6061 impact model. ............................................................ 105 Figure 5-13 Experimental results from WPI. Note the numbered lines were used in the analysis of the sample, but not needed for the finite element analysis. ................................ 107 Figure 5-14 Finite element result of Al6061-Al6061 impact and bonding model. .............. 107

x

List of Tables Table 3-1 Copper material properties used in the Johnson-Cook user subroutine development [2]. ........................................................................................................................................... 18 Table 3-2 PTW material properties for copper shown in comparison to the simple Johnson-Cook parameters [27]. ............................................................................................................. 18 Table 4-1 Part dimensions for traction-separation parameter investigation model. ............... 68 Table 4-2 Quasi-steady-state material properties used in the traction-separation parameter analysis (AISI 4340) [22]........................................................................................................ 70 Table 4-3 Johnson-Cook material properties for AISI 4340 steel used in the traction-separation behavior analysis [22]. ........................................................................................... 70 Table 5-1 Non-PTW material parameters used in the single particle impact model. Data from Northeastern University [52]. ................................................................................................. 91 Table 5-2 High-strain-rate PTW parameters used in the single particle impact model for the Al6061 particle [27]. ............................................................................................................... 91 Table 5-3 Mie-Gruneisen EOS properties for Aluminum 6061-T6 [56]. ............................... 94 Table 5-4 Substrate dimensions used in Al6061-Al6061 impact model. ............................. 104

xi

Acknowledgements

There are a few people that I would like to thank for helping me along during the writing of

this thesis.

First, I would like to thank my family for being very supportive during my time as a

graduate student. I could not have finished this work without them behind me. Thanks also go

to my graduate school roommates who all were working towards the same goal as me, and

were a great sounding board for my ideas during my candidacy and comprehensive exams. I

would most likely still be writing this thesis if it were not for the help of my advisor Ivi Smid,

who would always remind me how long I have been in graduate school during our meetings.

He is also one of the best cooks in State College. I also owe thanks to Vic Champagne at the

Army Research Lab, who provided me with great resources for my modeling efforts in the

Cold Spray Modeling Team. I would like to thank Mike Lanagan, the graduate officer of the

Engineering Science and Mechanics Department for stepping in at the last minute due to a

paperwork issue, reading my thesis, and allowing me to defend. I also appreciated the

thoughtful questions that Dr. Beese provided at my defense, which I added to my thesis

corrections. Finally, I would like to thank Tim Eden for his support. I have worked for Tim for

nearly seven years at the Applied Research Lab and was exposed to many opportunities that

typical graduate students never have. His advice and funding throughout the years has made

my graduate career possible, not to mention the liberal use of red ink on any of my manuscripts.

Thanks again.

1

1. Introduction

The objective of this study was to develop a comprehensive fracture energy based

approach to particle adhesion in cold spray using finite element analysis (FEA). FEA has been

used in engineering applications for decades to solve complex problems that cannot be done

by hand. In the past, applications for FEA were mainly in the areas of structural analysis, heat

transfer, vibration analysis, and fluid flow. With the advances in computational performance

and software in recent years, dynamic models have become commonplace.

Exceedingly complicated topics such as forging, rolling, and fragmentation are now

modeled on a daily basis with personal computers. One area that finite element analysts have

had trouble with are models that require dynamic bonding of two materials. Many researchers

have attempted to model adhesive bonding phenomena with mixed results [1]–[8].

Currently, there are a number of finite element bonding models based mainly on the

use of cohesive elements [5], [9]. These models rely on a traction-separation law that is derived

from a potential energy function of a non-linear spring to determine bonding criteria. While

this method is generally used in automobile crash simulations, it does not work in every case.

Other researchers have investigated energy based models [4], [8], [10], that have shown

that there are attractive forces between two materials in lightly loaded contact [8], [11]. The

force to separate these two materials can be easily measured. An energy based model may

prove to be ideal for the cold spray process. During deformation, there is an immense amount

of strain imparted into the system. Strain energy can be calculated from these impacts, leading

to a parameter in a bonding model. Many of the variables that control the amount of energy

that a particle has, are directly controlled via the cold spray parameters. Utilizing both

2

experimental results and finite elements, this study investigated the effect these parameters had

on particle bonding. This effort will provide a basis for a finite element bonding subroutine, as

well as an alternative method for bonding prediction.

1.1. Current State-of-the-Art

The current state-of-the-art in finite element modeling is to use either a cohesive

element based impact model or an energy based model. Yildirim, et.al., have proposed the use

of a cohesive strength value to initiate bonding [5]. This method relies on a lower bound which

correlates to the critical impact velocity to bond a particle to a substrate [5]. This model has

shown that different materials have varying cohesive strength values. The other method

currently used to simulate particle adhesion in finite elements is an energy based model. Using

the JKR formalism, the surface energy of a particle contacting a substrate can be expressed [8],

[11]. Quesnel and Rimai have investigated energy contributions to particle adhesion such as

the van der Waals forces and stresses and strains in the system [8]. Currently, there is no

universally agreed upon method in finite element modeling to determine when particle bonding

has occurred in cold spray.

1.2. Research Objective

This effort provides a practical basis for a finite element bonding subroutine, as well as an

alternative method for bonding prediction. This effort will add to the body of modeling

knowledge by developing an approach for predicting material parameters that are inaccessible

by any other means (e.g. strain energy required to bond, localized strain-rate deformation). A

fracture strain based approach for fragmentation was recently developed by Schreiber et.al.

[12]that provided insight into the bonding mechanisms in the cold spray process. In addition

to providing a new bonding model for the cold spray process, the groundwork for experimental

3

mechanical property prediction will be developed for cold spray impact in conjunction with

collaborating universities.

4

2. Background

2.1.History of the Cold Spray Process

Cold spray, also known as cold gas dynamic spray or high velocity particle

consolidation, was developed in the 1980s by researchers at the Institute of Theoretical and

Applied Mechanics of the Siberian Division of the Russian Academy of Science in Novosibirsk

[13]. Soviet researchers were studying the effect of small particle impact on aircraft engines in

supersonic conditions. They found that in certain conditions, instead of causing erosion, the

particles would adhere to the substrate [3].The researchers were able to deposit many different

materials under these conditions, ranging from metals, to polymers, and ceramics.

2.2.The Cold Spray Process

In the cold spray process, particles are injected into an inert carrier gas that is

accelerated to supersonic velocities through a converging-diverging de Laval-type nozzle.

Particles are generally accelerated to approximately 500-1000 m/s. Once accelerated, the

particles are directed towards a substrate using a robot or manually. Upon impact with the

substrate, the particles deform and create a mechanical bond with the substrate, forming a

coating. Further passes over the original impacts can be made to increase the coating thickness.

A schematic of the cold spray system is shown in Figure 2-1.

5

Figure 2-1 Schematic of the Cold Spray Process

The process was named Cold Spray because of the relatively low pre-impact

temperatures used in the process, typically ambient to 700°C. This is in comparison to other

thermal spray techniques such as flame spray, arc, and plasma spray, typically operating above

the melting point of the materials being deposited. However, these other thermal spray

processes rely on feedstock melting to develop a coating. Cold Spray utilizes the kinetic energy

of the particles to initiate bonding to the substrate [3]. Using only kinetic energy is one of the

main advantages to cold spray, and gives several benefits [14]:

1) Low oxide content in coatings due to low temperature.

2) Polymers can be coated without melting [15].

3) The duration of interaction between the feedstock and the substrate is reduced.

6

2.3.Finite Element Analysis

Finite element usage has increased significantly over the past twenty years due to

higher performing computers, improved constitutive material models, and better software.

FEM allows engineers to solve complex systems that were previously impossible to solve. The

degradation and failure of a material under extremely high strain rates could only be estimated

from dangerous experiments and empirical predictions that are at times impractical and

inaccurate [16]. Today, there are numerous fracture and failure models, some even pre-loaded

in the commercial FEM software. One can simply choose the model that best represents the

material system. These models can readily be modified for other material systems and can be

translated to many practical applications.

2.4.Material Constitutive Models

There has been a large amount of research on high-strain-rate deformation of ductile materials

[17]–[20]. These models are complex and are usually confirmed using experimental data.

These models are used in many finite element codes to predict material behavior at high-strain-

rates.

2.4.1. Johnson-Cook Constitutive Material Model

One of the most well-known and most widely implemented constitutive material model

is the Johnson-Cook plasticity model [21], [22]. Johnson and Cook developed an empirical

7

model for ductile materials at elevated strain rates. They found that fracture mainly depends

on hydrostatic pressure rather than strain rate and temperature [21], [22]. The Johnson-Cook

constitutive material model is a purely empirical model that is used to represent the strength

behavior of materials subjected to large strain rates, such as when a structure is exposed to

intense impulsive loading during the detonation of explosives. This model is commonly used

in finite element simulations of fracture and failure of materials at high strain rates. The model

defines the yield stress, σy, of the material shown in Equation 2-1.

𝜎𝜎𝑦𝑦 = �𝐴𝐴 + 𝐵𝐵𝜀𝜀𝑝𝑝𝑛𝑛��1 + 𝐶𝐶𝐶𝐶𝐶𝐶𝜀𝜀�̇�𝑝�[1− 𝑇𝑇𝐻𝐻𝑚𝑚] [21], [22] Equation 2-1

where 𝜀𝜀𝑝𝑝 is the amount of effective plastic strain in the system, 𝜀𝜀�̇�𝑝 is the normalized effective

plastic strain rate, and 𝑇𝑇𝐻𝐻 is a normalized temperature, shown in Equation 2-2.

𝑇𝑇𝐻𝐻 = (𝑇𝑇−𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟)(𝑇𝑇𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚−𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟)

[21], [22] Equation 2-2

Constants A, B, and C are experimentally determined material constants, where A is the quasi-

steady-state yield stress, B is the power law pre-exponential factor, n is the strain hardening

exponent, C is the strain rate pre-exponential factor, and m is the thermal softening exponent.

An example of the strain-rate dependence predicted by the Johnson-Cook model is shown in

Figure 2-2.

8

Figure 2-2 Strain-rate material dependence predicted by the Johnson-Cook model.

2.4.2. Zerilli-Armstrong Constitutive Material Model

The Zerilli-Armstrong constitutive material model is a dislocation based model that built off

of the work of Johnson and Cook. This model recognizes the dependence of crystal structure

on the plastic deformation of the material [23], [24]. A number of new aspects are introduced

into this model over the Johnson-Cook approach. These aspects are thermal activation, the

influence of the solute and grain size in face centered cubic (FCC) and body centered cubic

(BCC) metals. This model accounts for dislocation generation and motion using the thermal

activation relationship found in Equation 2-3 [23], [24].

�̇�𝛾 = 𝑚𝑚′𝑏𝑏𝑏𝑏𝜈𝜈 [23], [24] Equation 2-3

9

where �̇�𝛾 is the plastic shear strain-rate, 𝑚𝑚′ is a tensor orientation factor, 𝑏𝑏 is the Burgers vector,

N is the dislocation density, and 𝜈𝜈 is the average dislocation velocity. The average dislocation

velocity, 𝜈𝜈, is given as a function of Gibbs free energy at lower velocities. This relationship is

shown in Equation 2-4.

𝜈𝜈 = 𝜈𝜈0exp (−𝐺𝐺/𝑘𝑘𝑇𝑇) [23], [24] Equation 2-4

where G is the Gibbs free energy of activation, k is Boltzmann’s constant, and T is temperature.

This equation is converted to integral form and is shown in Equation 2-5.

𝐺𝐺 = 𝐺𝐺0 − ∫ 𝐴𝐴∗𝑏𝑏𝑏𝑏𝜏𝜏𝑡𝑡ℎ′𝜏𝜏𝑚𝑚ℎ0 [23], [24] Equation 2-5

where 𝐺𝐺0 is the reference Gibbs energy at T=0, 𝐴𝐴∗ is the area of activation, and 𝜏𝜏𝑡𝑡ℎ′ is the

thermal component of the shear stress. Equations 2-3, 2-4, and 2-5 are used to develop the

basis for the constitutive model. Both FCC and BCC materials are addressed by this model,

however, each crystal structure behaves differently. BCC metals show a greater dependence of

the yield stress with respect to temperature and strain-rate. This is opposed to FCC metals,

which have a stronger thermal activation energy behavior that depends on mainly strain. This

means that each predicted yield strength will be dependent on different parameters depending

on crystal structure [23], [24]. In FCC metals, the plastic flow stress is shown in Equation 2-6.

𝜃𝜃 = Δ𝜃𝜃𝑔𝑔′ + 𝑐𝑐2𝜖𝜖0.5 exp(−𝑐𝑐3𝑇𝑇 + 𝑐𝑐4𝑇𝑇𝐶𝐶𝐶𝐶𝜖𝜖̇) + 𝑘𝑘l−0.5[23], [24] Equation 2-6

where Δ𝜃𝜃𝑔𝑔′ is an additional component of stress that can occur between the solute and the

dislocation density on the yield stress, k is the microstructural stress intensity, and l is the

inverse square root of the average grain size diameter. Variables c2, c3, and c4, are effects

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from strain, temperature, and strain-rate, respectively. In BCC metals, the plastic flow stress

is shown in Equation 2-7.

𝜃𝜃 = Δ𝜃𝜃𝑔𝑔′ + 𝑐𝑐1 exp(−𝑐𝑐3𝑇𝑇 + 𝑐𝑐4𝑇𝑇𝐶𝐶𝐶𝐶𝜖𝜖̇) + 𝑐𝑐5𝜖𝜖𝑛𝑛 + 𝑘𝑘𝐶𝐶−0.5 [23], [24] Equation 2-7

Variable c1 can be written without a strain factor due to the lack of strain rate dependence on

strain in BCC metals. Variables c5 and n are properties of the power law relationship, where

c5 is the pre-exponential factor, and n is the strain hardening exponent.

2.4.3. Mechanical Threshold Stress Constitutive Material Model (MTS)

The MTS model was one of the first constitutive material models that did not compare

flow stress directly at constant strain such as the Johnson-Cook model [21], [22], [25]. Instead,

the mechanical threshold stress, the flow stress at 0 K, is used as a proper state variable [25].

This model is based on dislocation motion and dislocation interaction, similar to the Zerilli-

Armstrong model, but the MTS model relies solely on the mechanical threshold stress and an

Arrhenius relationship with respect to thermal activation through Voce behavior, shown in

Equation 2-8 [25].

𝜃𝜃 = 𝜃𝜃0 �1 −𝜃𝜃�−𝜃𝜃�𝑎𝑎

𝜃𝜃�𝑠𝑠(𝑇𝑇,�̇�𝜖)−𝜃𝜃�𝑎𝑎�[25] Equation 2-8

where 𝜃𝜃0 is dislocation accumulation hardening, 𝜃𝜃�𝑠𝑠 is a saturation stress defined by MTS, 𝜃𝜃�𝑎𝑎

is the yield stress, T is temperature, and 𝜖𝜖̇ is the strain rate. Taking this relationship into

consideration, the MTS model is typically expressed as Equation 2-9.

𝜃𝜃�

𝜇𝜇= 𝜃𝜃�𝑎𝑎

𝜇𝜇+ ∑𝜃𝜃�𝑗𝑗

𝜇𝜇 [26] Equation 2-9

11

where 𝜃𝜃� is the mechanical threshold stress, 𝜃𝜃�𝑗𝑗 is the mechanical threshold stress with thermally

activated effects such as Hall-Petch strengthening and dislocation motion, and 𝜇𝜇 is the

temperature dependent shear modulus [26].

2.4.4. Preston-Tonks-Wallace Material Model (PTW)

Adding to the work of Follansbee and Kocks with MTS, Preston, Tonks, and Wallace

developed a physically based materials model that is capable of predicting yield stresses at

strain-rates ranging from 10-3 to 1012 1/sec [27], [28]. This model relies on three dimensionless

variables which are based on flow stress, temperature, and strain-rate. The first dimensionless

stress parameter is defined as the flow stress divided by the shear modulus as a function of

density and temperature. Flow stress in this case is considered to be one half of the equivalent

von Mises deviatoric stress [27]. The stress parameter is shown in Equation 2-10.

�̂�𝜏 = 𝜏𝜏𝐺𝐺(𝜌𝜌,𝑇𝑇)

[27] Equation 2-10

where �̂�𝜏 is the dimensionless stress parameter, 𝜏𝜏 is the flow stress, G is the shear modulus, 𝜌𝜌 is

mass density, and T is temperature. The second dimensionless parameter is the temperature

dependence. This is expressed as a scaled temperature variable, shown in Equation 2-11.

𝑇𝑇� = 𝑇𝑇𝑇𝑇𝑟𝑟

[27] Equation 2-11

where Tm is the melting temperature. The last dimensionless variable is the strain-rate

dependence. This was chosen to be the equivalent plastic strain rate, �̇�𝜓, divided by an

equivalent scaling factor, �̇�𝜉. The approximation for �̇�𝜉 is shown in Equation 2-12.

12

𝑐𝑐𝑇𝑇2𝑎𝑎

= �̇�𝜉 [27] Equation 2-12

where cT is the transverse speed of sound, and a is the distance between atoms. This model

integrates these dimensionless parameters with thermal activation work hardening saturation

stress and yield stress through Equations 2-13 and 2-14, shown below.

�̂�𝜏𝑠𝑠 = 𝑠𝑠0 − (𝑠𝑠0 − 𝑠𝑠∞)erf [𝜅𝜅𝑇𝑇� ln (𝛾𝛾�̇�𝜉/�̇�𝜓)][27] Equation 2-13

�̂�𝜏𝑦𝑦 = 𝑦𝑦0 − (𝑦𝑦0 − 𝑦𝑦∞)erf [𝜅𝜅𝑇𝑇� ln (𝛾𝛾�̇�𝜉/�̇�𝜓)] [27] Equation 2-14

where 𝑠𝑠0,𝑦𝑦0, 𝑠𝑠∞,𝑎𝑎𝐶𝐶𝑏𝑏 𝑦𝑦∞ are material constants, and 𝜅𝜅 and 𝛾𝛾 are dimensionless material

constants. Adding Equations 2-10 through 2-14 to the Voce relationship and integrating along

a constant strain-rate, gives the PTW model shown in Equation 2-15.

�̂�𝜏 = �̂�𝜏𝑠𝑠 +1𝑝𝑝�𝑠𝑠0 − �̂�𝜏𝑦𝑦�𝐶𝐶𝐶𝐶

⎣⎢⎢⎡1 − �1 − 𝑒𝑒𝑒𝑒𝑝𝑝 �−𝑝𝑝

�̂�𝜏𝑠𝑠 − �̂�𝜏𝑦𝑦𝑠𝑠0 − �̂�𝜏𝑦𝑦

��

∗ 𝑒𝑒𝑒𝑒𝑝𝑝

⎩⎨

⎧−

𝑝𝑝𝜃𝜃𝜓𝜓

(𝑠𝑠0 − �̂�𝜏𝑦𝑦) �𝑒𝑒𝑒𝑒𝑝𝑝 �𝑝𝑝�̂�𝜏𝑠𝑠 − �̂�𝜏𝑦𝑦𝑠𝑠0 − �̂�𝜏𝑦𝑦

� − 1�⎭⎬

⎫

⎦⎥⎥⎤

[27] Equation 2-15

This is more complex than previous flow stress predictions, but the authors have shown that

the overall prediction of the flow stress is much closer than that of strictly empirical material

models such as Johnson-Cook.

13

2.5.Bonding in Cold Spray

The bonding mechanism in the cold spray process is not well understood. It is known that

the particle and substrate undergo extreme plastic deformation upon impact. This causes the

thin oxide layer on the particle and substrate to break down allowing contact between pristine

metallic surfaces. Coupled with localized heating due to deformation, these extreme conditions

cause the particles to bond with the substrate. Some theorize that there is intermetallic

formation at the interface between the particle and the substrate which causes bonding [29].

Others believe that there occurs an adiabatic shear instability at the particle/substrate interface

which causes bonding of the particle [30]. Experiments have shown that bonding occurs above

a critical particle velocity, which suggests that the adiabatic shear instability only occurs above

this velocity during cold spray due to localized shear [7], [31], [32].

2.6. Hypothesis

Prediction of bonding behavior in cold spray is a very difficult task. Two main bonding

regimes exist in the cold spray bond, mechanical interlocking and metallurgical bonding due

to the adiabatic shear instability effect. Previous work using cohesive type elements have

shown great promise in predicting bonding and rebound in cold spray, but cohesive elements

were not originally designed to model this behavior. A strain energy based cohesive interface

model should be able to predict bonding better than a cohesive element based model. Cohesive

element models are based on estimated surface energy values that can vary from particle to

particle, making any bonding prediction model dependent rather than material dependent. A

strain energy based cohesive interface model should be able to:

14

• Use experimental data to measure the strain energy in the system

• Correlate experimental data to the finite element prediction

• Validate any changes in particle sizes, velocities, temperature, etc. to predictions

• Relate processing parameters to bonding

15

3 PTW Model Development, Calibration and Validation

3.1 Initial Model Considerations The objective of this work is to develop a Preston-Tonks-Wallace (PTW) subroutine in

Abaqus that is capable of predicting plasticity in both two and three dimensions. The effort

started with the development of a user defined Johnson-Cook (JC) model. In previous

unpublished work, J. Schreiber at Penn State University developed an explicit user subroutine

for the Johnson-Cook material model using the Abaqus VUMAT subroutine. The subroutine

was written following FORTRAN77 conventions. This subroutine showed excellent

agreement with the software’s built-in Johnson-Cook model as shown later in this chapter. The

success of this effort verified that external variables were passed in and out of the software

correctly, and were being updated accordingly during each step.

This subroutine was used to assist in the validation and troubleshooting of the Preston-

Tonks-Wallace material model. The subroutine was modified for the PTW model by adding

the relevant PTW mathematics in place of the simple JC equation. The PTW material model

is a very complex thermodynamics based model that utilizes many more material properties

than the Johnson-Cook model. Initially, the PTW model was to be developed in both Abaqus

and LS-DYNA. However, due to the success of the JC subroutine in Abaqus, LS-DYNA was

not needed. This saved a large amount of extra effort since another subroutine would need to

be written for the PTW subroutine in LS-DYNA.

However, even though there are many more parameters required, the advantage of the

PTW material model is that the material properties are easy to measure without the high-strain-

rate testing required in the JC model. The PTW model was originally planned to be introduced

into Abaqus CAE (Abaqus) using the VUMAT user subroutine, but it was found that the PTW

model does not operate in the same manner as the JC model. The issues found in the subroutine

16

development are discussed further in this chapter. In the initial modeling stages, comparisons

were made between the results of Johnson-Cook and PTW at similar strain rates to validate the

PTW model.

3.2 Development and validation of a Johnson-Cook subroutine The Johnson-Cook (JC) material model is a constitutive material model that takes into

account power law hardening, strain rate dependence, and thermal softening effects[21], [22].

It is most often expressed as shown in Equation 3-16. Parameters A, B, and n are the quasi-

steady-state yield strength, power law pre-exponential factor, and strain hardening exponent,

respectively. Parameters C and m are the strain-rate exponential factor and thermal softening

exponent, respectively. These values have been shown to be strain-rate dependent [33] and are

typically obtained using the Split Hopkinson Pressure Bar (SHPB) test.

Equation 3-16 [21]

𝜀𝜀𝑝𝑝 = 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑐𝑐𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝐶𝐶𝑎𝑎𝑠𝑠𝑒𝑒𝑒𝑒𝑐𝑐 𝑠𝑠𝑒𝑒𝑠𝑠𝑎𝑎𝑒𝑒𝐶𝐶

𝜀𝜀𝑝𝑝∗ = 𝐶𝐶𝑛𝑛𝑠𝑠𝑚𝑚𝑎𝑎𝐶𝐶𝑒𝑒𝑛𝑛𝑒𝑒𝑏𝑏 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑐𝑐𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝐶𝐶𝑎𝑎𝑠𝑠𝑒𝑒𝑒𝑒𝑐𝑐 𝑠𝑠𝑒𝑒𝑠𝑠𝑎𝑎𝑒𝑒𝐶𝐶 𝑠𝑠𝑎𝑎𝑒𝑒𝑒𝑒

𝑇𝑇𝐻𝐻 = ℎ𝑛𝑛𝑚𝑚𝑛𝑛𝐶𝐶𝑛𝑛𝑜𝑜𝑛𝑛𝑜𝑜𝑠𝑠 𝑒𝑒𝑒𝑒𝑚𝑚𝑝𝑝𝑒𝑒𝑠𝑠𝑎𝑎𝑒𝑒𝑜𝑜𝑠𝑠𝑒𝑒 = (𝑇𝑇 − 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑚𝑚)

(𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡 − 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑚𝑚)

The results of the SHPB test provides a stress-strain curve of the material at strain-rates

that are not obtainable using a standard tensile test. This stress-strain curve is used to create a

curve fit of the Equation 3-16. The JC material model has been the standard high-strain-rate

17

model for the past few decades. However, materials data is not easily accessible for the JC

model and the SHPB test is only done by a few universities and research institutes. Figure 3-1

shows the results of plotting the JC equation at various strain-rates, which shows the typical

rate dependence of materials. It can be shown that as the strain rate increases, the flow stress

increases. However, this is not always the case, especially if temperature is taken into account.

Figure 3-1 Typical strain rate dependence predicted using the Johnson-Cook material model.

The main issue with the JC material model is that predictions typically break down at

strain rates exceeding 104 (1/sec). This is not sufficient for cold spray modeling, even if

corrections are made at higher strain-rates [33]. The strain rate during particle impact in the

cold spray process is normally in the range of 107 (1/sec). The PTW model allows for accurate

stress predictions for strain rates ranging from 10-3 to 1012 (1/sec) [27].

18

Once the JC model was defined in a user defined VUMAT subroutine, it was initially

used to ensure that the variables were being passed into the software correctly. However, the

VUMAT user subroutine substitutes all material property relationships in Abaqus and uses

only what is found in the subroutine. This is not an issue with the JC model since it predicts

properties in the elastic regime due to the inclusion of the quasi-steady-state yield strength in

the JC equation used in the subroutine, and the elastic modulus and Poisson’s ratio that is input

into Abaqus.

To validate the JC VUMAT against the built-in JC model, a simple axisymmetric

cylinder loaded from the top and fixed on the bottom, was used. Both models were run under

the same conditions using copper material properties shown in Table 3-1. For comparison, the

PTW material properties for copper are shown in Table 3-2 [27].

Table 3-1 Copper material properties used in the Johnson-Cook user subroutine development [2].

Young’s Modulus

Poisson’s Ratio

A B C n m Tm Tref

110 GPa 0.34 90 MPa 292 MPa 0.025 0.31 1.09 1356 K 294 K

Table 3-2 PTW material properties for copper shown in comparison to the simple Johnson-Cook parameters [27].

θ p S0 s∞ κ γ Y0 y∞

0.025 2.0 0.0085 0.00055 0.11 0.00001 0.0001 0.0001 Y1 Y2 β G0 α αp M (mu) C

(cm/µs) 0.094 0.575 0.25 510 0.20 0.43 63.54 0.3933

s ρa γa G/B ρ0

(g/cm3) Γ (10-4

cal/mol K2) g k (cal/s

cm K) 1.50 8.933 2.0 0.35 9.02 1.6 2/3 1.8

19

A load of 1x107 N, corresponding to a pressure of 100 MPa was applied to the top of

the axisymmetric cylinder under steady state conditions shown in Figure 3-2. The model was

meshed with 5000 4-node bilinear axisymmetric quadrilateral, reduced integration elements,

CAX4R. These elements were chosen due to the quick calculation time of a reduced integration

element and the fact that quadrilateral type elements reduce the possibility of encountering

over-stiffening of elements. Since these elements are reduced integration, there is only one

Gauss point, which is a linear interpolation between nodes. This may mean that additional

elements may be required to address any mesh sensitivity issues. A pinned boundary condition

was chosen for the bottom of the model to account for any displacements due to loading. A

fully welded boundary condition was considered, but was determined to be over constraining

the model. The difference between a pinned boundary condition and an encastred boundary

condition in Abaqus is that only the X, Y, and Z displacements are fixed in a pinned condition,

whereas in an encastred condition, any nodal rotation is fixed in addition to nodal

displacements.

20

Figure 3-2 Axisymmetric cylinder model used to develop the Johnson-Cook user subroutine.

This makes a difference in axisymmetric models due to constraining higher order

displacements like nodal rotation. This can cause convergence issues and was avoided in this

modeling by using only pinned boundary conditions. Figure 3-3 shows the results of the JC

subroutine development compared to the built-in JC model. There was less than 3% error in

the von Mises stress between the two models, which was determined to be more than

satisfactory for this type of dynamic modeling. Results of the JC comparison were very

promising, and were used in the development of the PTW model user subroutine.

21

Figure 3-3 Comparison between built-in JC (left) and the JC subroutine (right) for copper.

22

3.3 PTW Model Development The PTW model is loosely based on the mechanical threshold stress model that was

discussed in the background and has three main dimensionless variables that are dependent on

flow stress, temperature, and strain-rate [2]. This is similar to the JC model, with the exception

that it is significantly more complex. The PTW model has 19 parameters that must be addressed

in both the VUMAT and internally in Abaqus, compared to the 7 parameters in the JC

VUMAT. PTW is inherently more complex than JC due to the fact it is based on a

thermodynamic basis rather than the curve fit physical constitutive model as found in JC.

Numerous changes had to be made to the JC subroutine to accommodate the PTW

mathematics. It was found that a standard VUMAT subroutine was no longer valid, since the

PTW model only predicts results in the plastic regime due to the lack of elastic properties. The

JC model defines the elastic parameters as part of the curve fit, so there is no need to define an

elasticity model. However, PTW does not prescribe any elasticity definition so another type of

subroutine needed to be utilized. Another issue with implementing the PTW model with a

VUMAT is updating the internal stresses and von Mises stress back into Abaqus. Abaqus is

expecting a vector input for stress and energy at the end of each step. However, PTW only will

supply a scalar value of energy and von Mises stress. This was another reason to investigate

another subroutine. It was determined that to effectively implement the PTW material model,

another subroutine called VUHARD had to be utilized. VUHARD is only activated when the

material is being plastically deformed. This is ideal for the PTW model since there is no elastic

response considered. Another advantage of the VUHARD subroutine is that it will not override

any Abaqus material parameters other than plastic properties. This is ideal for cases where

multiple subroutines may be required.

23

3.3.1 Preston-Tonks-Wallace Model Derivation The Preston-Tonks-Wallace material model is based partially on the Mechanical

Threshold Stress (MTS) derivation. The MTS model was one of the first constitutive material

models that did not compare flow stress at constant strain such as the Johnson-Cook model

[21], [22], [25]. Instead, in the mechanical threshold stress the flow stress at 0 K is used as a

proper state variable [25]. This model is also based on dislocation motion and dislocation

interaction, similar to the Zerilli-Armstrong model. However, the MTS model relies solely on

the mechanical threshold stress and an Arrhenius relationship with respect to thermal activation

through Voce behavior, shown in Equation 3-2 [25]. The Voce Law is an isotropic work

hardening model that is based on a saturation stress,

𝜃𝜃 = 𝜃𝜃0 �1 −𝜃𝜃�−𝜃𝜃�𝑎𝑎

𝜃𝜃�𝑠𝑠(𝑇𝑇,�̇�𝜖)−𝜃𝜃�𝑎𝑎�[25] Equation 3-2

where 𝜃𝜃0 is dislocation accumulation hardening, 𝜃𝜃�𝑠𝑠 is the saturation stress, 𝜃𝜃�𝑎𝑎 is the yield stress,

T is temperature, and 𝜖𝜖̇ is the strain rate. Taking this relationship into consideration, the MTS

model is typically expressed as Equation 3-3.

𝜃𝜃�

𝜇𝜇= 𝜃𝜃�𝑎𝑎

𝜇𝜇+ ∑𝜃𝜃�𝑗𝑗

𝜇𝜇 [26] Equation 3-3

Where 𝜃𝜃� is the mechanical threshold stress, 𝜃𝜃�𝑗𝑗 is the mechanical threshold stress with thermally

activated effects such as Hall-Petch strengthening and dislocation motion, and 𝜇𝜇 is the

temperature dependent shear modulus [26].

Adding to the MTS model, Preston, Tonks, and Wallace developed a physically based

material model that is capable of predicting yield stresses at strain-rates ranging from 10-3 to

1012 1/sec [27], [28]. This model relies on three dimensionless variables which are based on

24

flow stress, temperature, and strain-rate. These variables give the three main parameters in the

equation. The three parameters are the stress parameter �̂�𝜏, the temperature dependence 𝑇𝑇� , and

the strain-rate dependence �̇�𝜉. The stress parameter is defined as the flow stress divided by the

shear modulus as a function of density and temperature. Flow stress in this case is considered

to be one half of the equivalent von Mises deviatoric stress [27]. The stress parameter is shown

in Equation 3-4.

�̂�𝜏 = 𝜏𝜏𝐺𝐺(𝜌𝜌,𝑇𝑇)

[27] Equation 3-4

where �̂�𝜏 is the dimensionless stress parameter, 𝜏𝜏 is the flow stress, G is the shear modulus, 𝜌𝜌 is

mass density, and T is temperature. The second dimensionless parameter is the temperature

dependence. This is expressed as a scaled temperature variable, shown in Equation 3-5.

𝑇𝑇� = 𝑇𝑇𝑇𝑇𝑟𝑟

[27] Equation 3-5

where Tm is the melting temperature. The last dimensionless variable is the strain-rate

dependence. This was chosen to be the equivalent plastic strain rate, �̇�𝜓, divided by an

equivalent scaling factor, �̇�𝜉. The approximation for �̇�𝜉 is shown in Equation 3-6.

𝑐𝑐𝑇𝑇2𝑎𝑎

= �̇�𝜉 [27] Equation 3-6

where cT is the transverse speed of sound, and a is the distance between atoms. PTW combines

these dimensionless parameters with thermal activation work hardening saturation stress and

yield stress through Equations 3-7 and 3-8, shown below.

�̂�𝜏𝑠𝑠 = 𝑠𝑠0 − (𝑠𝑠0 − 𝑠𝑠∞)erf [𝜅𝜅𝑇𝑇� ln (𝛾𝛾�̇�𝜉/�̇�𝜓)][27] Equation 3-7

�̂�𝜏𝑦𝑦 = 𝑦𝑦0 − (𝑦𝑦0 − 𝑦𝑦∞)erf [𝜅𝜅𝑇𝑇� ln (𝛾𝛾�̇�𝜉/�̇�𝜓)] [27] Equation 3-8

25

where 𝑠𝑠0,𝑦𝑦0, 𝑠𝑠∞,𝑎𝑎𝐶𝐶𝑏𝑏 𝑦𝑦∞ are material constants, and 𝜅𝜅 and 𝛾𝛾 are dimensionless material

constants. Adding Equations 3-4 through 3-8 to the Voce relationship and integrating along a

constant strain-rate, gives the PTW model shown in Equation 3-9.

�̂�𝜏 = �̂�𝜏𝑠𝑠 +1𝑝𝑝�𝑠𝑠0 − �̂�𝜏𝑦𝑦�𝐶𝐶𝐶𝐶

⎣⎢⎢⎡1 − �1 − 𝑒𝑒𝑒𝑒𝑝𝑝 �−𝑝𝑝

�̂�𝜏𝑠𝑠 − �̂�𝜏𝑦𝑦𝑠𝑠0 − �̂�𝜏𝑦𝑦

��

∗ 𝑒𝑒𝑒𝑒𝑝𝑝

⎩⎨

⎧−

𝑝𝑝𝜃𝜃𝜓𝜓

(𝑠𝑠0 − �̂�𝜏𝑦𝑦) �𝑒𝑒𝑒𝑒𝑝𝑝 �𝑝𝑝�̂�𝜏𝑠𝑠 − �̂�𝜏𝑦𝑦𝑠𝑠0 − �̂�𝜏𝑦𝑦

� − 1�⎭⎬

⎫

⎦⎥⎥⎤

[27] Equation 3-9

This model is more complex than previous high-strain-rate models, but the authors have shown

that the overall prediction of the flow stress compared against experimental data is much more

consistent than that of strictly empirical material models such as Johnson-Cook. Prediction of

the flow stress is important to understand since the flow stress is the initiation of plastic

deformation in the model.

3.3.2 PTW Subroutine Implementation via VUHARD The first step in the development of PTW model was to program the mathematics and

compare to published data to ensure that the subroutine iterations were correct [2]. The

mathematical relationships were written into FORTRAN77 format and iterated through

numerous steps. Figures 3-4 and 3-5 show good correlation between the VUHARD and

published data for changes in strain-rate and thermal effects, respectively [27]. These results

validated that the code is consistent with published data, and allowed further development of

the code.

26

Figure 3-4 Stress vs. strain rate comparison between subroutine (left) and published data

(right)for copper [27].

27

Figure 3-5 Comparison of the saturation and yield stress for global structure. Subroutine (left)

and published data (right) for copper [27].

Incorporating the mathematics in the subroutine proved to be fairly difficult, due to the

addition of integration and error functions into the subroutine with the proper numerical

precision. The next step after the mathematics were validated, was to utilize the subroutine in

the iteration process to predict the stress. This is not a simple task since Abaqus may run

millions of iterations of a subroutine in one simulation step. Rounding errors or imprecise

command steps in the subroutine may result in calculations that will not converge or produce

results that may appear to be correct but are false. To help with programming the PTW model,

a flowchart of the subroutine steps was needed. These steps include, declaring universal

variables that are used by Abaqus, declaring the PTW variables, starting the PTW mathematics,

and updating strains and energies to be transferred back to Abaqus. The initial flowchart for

the PTW model is shown in Figure 3-6. Each step is color coordinated.

28

Figure 3-6 Flowchart of the processes that occur in the PTW VUHARD subroutine.

This flowchart only describes the events that occur inside of the subroutine and what

variables are sent from the subroutine. This does not include any internal Abaqus operations

such as the calculation of the von Mises stress for each step increment. The flowchart first

starts with defining the user input variables and any other constants that are input through the

Abaqus user interface. Pre-defined values such as temperature, initial particle velocity, and any

other boundary condition is defined outside of the subroutine in Abaqus.

These values are read into the model via the universal subroutine callout before any

user coding is read. Once the variables are read by the FORTRAN code, the subroutine will

read what material is specified and start the PTW mathematics with the specified material

properties at each increment. From there, strain-rate and temperature are checked at every

29

iteration and the PTW mathematics will be run with each updated variable. At the end of each

iteration, the stresses, internal energy, and any external energy values, such as heat generation

are updated. There were several areas in this subroutine that were very difficult to implement.

The most difficult part of this implementation was calculating a von Mises stress from

the output data in the PTW model. This step is mandatory when using the VUMAT subroutine,

and was one of the driving factors in making the change from VUMAT to VUHARD.

VUHARD does not require the von Mises stress to be calculated in the subroutine unlike

VUMAT, which replaces nearly all Abaqus calculations with the subroutine. All stress

calculations in VUHARD are handled internally by Abaqus, which greatly reduces the amount

of subroutine coding required. The original VUMAT subroutine for the Johnson-Cook model

that was developed did not require a definition of the von Mises stress due to the Johnson-

Cook model defining a flow stress value in the elastic regime.

To fully implement the PTW model using a VUHARD subroutine, each equation that

feeds into the model must be translated and input into the FORTRAN code. To do this,

Equation 3-9 was broken down into simple iterable steps that can utilize all of the user input

parameters defined in the model and Abaqus. These steps are shown graphically as a flowchart

in Figure 3-7. Step 1 in the flowchart calculates a time keeping factor related to the Debye

frequency, which gives an idea of what time stepping is required for convergence. The Debye

frequency is the maximum natural resonance frequency for atoms in a crystal, and is used in

other calculations such as heat capacity and diffusion. Step 2 normalizes the temperatures in

the model to the melting point of the material, similar to what happens in JC model. Step 3 is

an error function relationship that assigns the saturation stress to the appropriate temperature

and strain-rate relationship. Step 4 is introduced to ensure that the proper relationship has been

30

chosen by the subroutine. Step 5 is the second error function relationship that chooses the

appropriate internal stress depending on the strain-rate conditions. In the PTW model, there

are two strain-rate regimes, the lower strain-rate thermal regime, and the higher strain-rate

overdriven shock regime. These regimes are discussed further below and in Chapter 2. Step 6

calculates the shear stress in the final PTW equation. Step 7 simply converts the Step 6 results

into the flow stress used in the PTW VUHARD subroutine.

Figure 3-7 Graphical flowchart of the equation breakdown in the PTW subroutine.

The PTW equation has two main regimes that change the way the parameters are modified that

is completely different than the JC model formulation. There is a strain rate dependence in the

equation which has two distinct regions, the first region allows slip dislocations, grain

boundary movements, and other types of deformation to occur at lower strain rates. The second

31

region occurs at higher strain rates, does not allow for standard deformation methods such as

crystal dislocations. This region is called the overdriven shock regime. This has an effect that

can modify the PTW equation at Step 3 and Step 5 in Figure 3-7 Graphical flowchart of the

equation breakdown in the PTW subroutine. This defining region is both temperature and

strain-rate dependent, and relies on multiple material properties and non-dimensional constants

to define the transition. Other research groups have utilized a bi-linear Johnson-Cook model

instead of developing a more complex model such as PTW. The results of this work have been

promising, with good correlation with PTW [34]. However, the bi-linear Johnson-Cook model

requires high-strain-rate data that is difficult to measure. Even then, the material properties

must be “tuned” to achieve acceptable results, where the PTW model does not.

3.4 PTW Parameter Sensitivity Analysis Since there are so many parameters required in the PTW material model, it is necessary

to conduct a sensitivity analysis on the variables in the model to better understand what controls

the prediction of the flow stress. Using the formulas developed for the subroutine and the

material properties for pure aluminum, some of the non-dimensional variables were varied

versus the von Mises flow stress. Figure 3-8 shows the relationship between the predicted von

Mises stress and the hardening parameter p. p is a dimensionless material property modifier

that is used in the Voce hardening law found in PTW.

32

Figure 3-8 Sensitivity analysis of the effect of hardening parameter p on predicted von Mises flow stress for aluminum.

It can be shown that as p increases, the predicted von Mises stress decreases from 400

MPa down to 300 MPa. This decrease is fairly significant. The default value for aluminum is

3.

The next variable to be analyzed is Y0. This non-dimensional parameter is referred to

as the yield stress constant at zero Kelvin. This is a material constant that is part of the error

function description of the yield saturation stress in the thermal activation, also known as the

low-strain-rate regime. Figure 3-9 shows the dependence of von Mises flow stress on the value

of Y0.

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6

Von

Mise

s Str

ess (

MPa

)

p

von Mises Flow Stress vs. p

33

Figure 3-9 Sensitivity analysis of the effect of yield stress constant Y0 on the predicted von Mises flow stress for aluminum.

Similar to what was shown in in the previous figure, there is not a large change as the

variable is increased. Note that the yield strength increases as Y0 is increased. This make sense

since this coefficient is essentially increasing the yield strength of the material in the thermal

regime as it is increased. The default value for pure aluminum is 0.00942.

This value can be estimated using thermodynamic software such as JMatPro or

Thermocalc. As Y0 is increased, there appears to be a point where the yield stress reaches a

plateau where, Y0 makes no significant change. This appears to coincide with the pure

aluminum value of 0.00942. This region may be the reason why Preston et.al concluded that

0

50

100

150

200

250

300

350

400

450

500

0 0.002 0.004 0.006 0.008 0.01

Von

Mise

s Str

ess (

MPa

)

Y0

von Mises Flow Stress vs Y0

34

materials with similar crystal structures have similar PTW variables. This trend is shown in

Figure 3-10 with a magnified scale.

Figure 3-10 Magnified scale showing the behavior of the yield stress constant Y0 with respect to the predicted von Mises flow stress for aluminum.

The next parameter to be analyzed is very similar to Y0. S0 is the value of the saturation

stress at zero Kelvin. This value can be determined similarly using thermodynamic database

software, as well as curve fitting of high-strain-rate data. This value controls the level of work

hardening that can occur in the model and is predicted using the same error function as Y0.

Plotting S0 against the von Mises stress shows a very different prediction in total stress.

S0 increases significantly in nearly a logarithmic relationship. This correlation between von

Mises flow stress makes sense due to the fact that S0 controls the amount of work hardening in

the material. As S0 increases, more work hardening can occur, increasing the required von

350

370

390

410

430

450

470

490

0 0.002 0.004 0.006 0.008 0.01

Von

Mise

s Str

ess (

MPa

)

Y0

Stress vs Y0

35

Mises flow stress to yield. The published value for S0 in aluminum is 0.032. This trend is

shown in Figure 3-11.

Figure 3-11 Sensitivity analysis of the saturation stress S0 versus predicted von Mises flow stress for aluminum.

S0 is only used at very low temperatures. However, at higher temperatures, another

parameter is used, S∞. At higher temperatures, thermal softening is controlling the von Mises

flow stress more than work hardening, so the effect of S∞ is less significant than the low

temperature saturation stress value. This trend is shown in Figure 3-12.

y = 136.87ln(x) + 1140.4R² = 0.8717

0

200

400

600

800

1000

1200

0 0.2 0.4 0.6 0.8 1 1.2

Von

Mise

s Str

ess (

MPa

)

S0

Stress vs S0

Saturation Stress

Log. (Saturation Stress)

36

Figure 3-12 Sensitivity analysis of temperature dependent flow stress, S∞ versus predicted von Mises flow stress for aluminum.

Another important parameter in the PTW model is the material density. As shown in

Figure 3-13, as the density of the material increases, there is very little change in the von Mises

flow stress. Even though there is only negligible change in the von Mises flow stress, this is an

important piece of information to know in the development of the model and the determination

of material properties. Abaqus is capable of inputting temperature dependent data for a number

of material properties, including density. However, this is not necessary, which should

decrease calculation time slightly.

0

100

200

300

400

500

600

0 0.01 0.02 0.03 0.04 0.05 0.06

Von

Mise

s Str

ess (

MPa

)

S∞

Stress vs S∞

37

Figure 3-13 Sensitivity analysis of density ρ versus predicted von Mises flow stress in the PTW material model for aluminum.

The overall dependence of flow stress on density is shown in Figure 3-14. Focusing on

a range of densities from 2,000 kg/m3 to 15,000 kg/m3, there is an overall change in predicted

von Mises flow stress of 2.5 MPa. It can also be seen that the von Mises flow stress follows a

logarithmic function of density.

350355360365370375380385390395400

0 2000 4000 6000 8000 10000 12000 14000 16000

Von

Mise

s Str

ess (

MPa

)

ρ

Stress vs ρ

38

Figure 3-14 Magnified scale of predicted von Mises flow stress versus density. Note that material density will not change enough during heating to significantly change the flow stress for aluminum.

Another non-dimensional constant in the PTW model is alpha. Alpha is used in the

calculation of the shear modulus of the material at different strain-rates. The authors of the

PTW model describes alpha as a scaling variable with a dependence on density, temperature,

and strain-rate. Alpha is used when the shear modulus is to be calculated at very high pressures

using the Burakovsky-Preston melting relationship [35]. Notice that there is a slight decrease

in the predicted von Mises flow stress as the value of alpha is increased as shown in Figure 3-

15. In pure aluminum, the value of alpha was found to be 0.475. It appears that near 0.475,

there is a small change in slope for the predicted flow stress value.

y = 1.2193ln(x) + 363.78R² = 1

372.5

373

373.5

374

374.5

375

375.5

376

0 2000 4000 6000 8000 10000 12000 14000 16000

Von

Mise

s Str

ess (

MPa

)

ρ

Stress vs ρ

39

Figure 3-15 Sensitivity analysis of the alpha parameter versus the predicted von Mises flow stress in the PTW material model for aluminum.

At large strain-rates, there is another non-dimensional constant used in the PTW

material model called beta (β); β is an exponent that is used to determine the saturation stress

as well as the yield stress of a material at very high strain-rates. At these strain-rates, any work

hardening effects are neglected completely. Also known as the overdriven shock regime, is

typically observed between 1x109 – 1x1012 s-1 strain-rate. This is well above the strain-rates

found in the cold spray process, ~1x107 s-1, but must be addressed for the model to work

correctly. β is used in Equation 3-10.

𝜏𝜏𝑦𝑦� = 𝜏𝜏𝑠𝑠� = 𝐶𝐶 ∗ ��̇�𝜓/𝜀𝜀̇�𝛽𝛽

Equation 3-10

Where C is a constant, �̇�𝜓 is the equivalent plastic strain, and 𝜀𝜀̇ is the strain rate. Plotting

the predicted von Mises flow stress against β is shown in Figure 3-16. It can be seen that at

low values of β, there is a strong dependence on the flow stress. However, the value of β was

250270290310330350370390410430450

0 0.2 0.4 0.6 0.8 1 1.2

Von

Mise

s Str

ess (

MPa

)

alpha

Stress vs alpha

40

found to be 0.23 in pure aluminum, which is well into the regime where there is no dependence

on flow stress.

Figure 3-16 Sensitivity analysis of the effect of β on the predicted von Mises flow stress in the PTW material model for aluminum.

The last non-dimensional variable in the PTW model that will be discussed is kappa

(κ). κ is a constant of temperature dependence that is used in the prediction of the yield and

saturation stresses. The equation for yield and saturation stresses follows an Arrhenius

relationship and is shown in Equation 3-11 in a generalized form.

�̂�𝜏𝑥𝑥 = 𝑒𝑒0 − (𝑒𝑒0 − 𝑒𝑒∞)𝑒𝑒𝑠𝑠𝑒𝑒 �𝜅𝜅𝑇𝑇�𝐶𝐶𝐶𝐶 �𝛾𝛾𝛾𝛾�̇�𝜀�� Equation 3-11

κ is used as an analog to the Boltzmann constant in the Arrhenius relationship. This

stress prediction is used through the entire PTW model and the dependence of the predicted

von Mises flow stress versus κ is shown in Figure 3-17. At lower values, there is a very large

350

370

390

410

430

450

470

490

510

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Von

Mise

s Str

ess (

MPa

)

Beta

Stress vs Beta

41

dependence between flow stress and κ. For pure aluminum, Preston et.al. found that the value

for κ is 0.494, which is where the function starts to lose its flow stress dependence.

Figure 3-17 Sensitivity analysis of the effect of kappa on the predicted von Mises flow stress in the PTW material model for aluminum.

The final five figures in this section are plotted to show the dependence of the PTW

material to the strain, strain-rate, temperature, and melting point of the material. The first

dependence of interest is the strain. Figure 3-18 plots the dependence of the predicted von

Mises flow stress to the strain in the system. Values below ~0.1 were considered to be in the

elastic regime. The predicted von Mises flow stress increases as the strain increases in the

material as expected. This dependence can be correlated to the strain-rate dependence shown

in Figures 3-19 and 3-20. Note that there is little change in the predicted von Mises flow stress

as the strain-rate is increased. This was not expected at first due to prior experiences with the

Johnson-Cook material model typically increasing the flow stress of the material as the strain-

rate is increased. However, if the strain-rate plot is magnified, as shown in Figure 20, it can be

050

100150200250300350400450500

0 0.2 0.4 0.6 0.8 1

Von

Mise

s Str

ess (

MPa

)

κ

Stress vs κ

42

seen that as strain-rate increases, the predicted von Mises flow stress also increases following

a logarithmic function.

Figure 3-18 Predicted von Mises flow stress with respect to strain in the PTW material model for aluminum.

Figure 3-19 Effect of strain-rate on the predicted von Mises flow stress in the PTW material model for aluminum.

050

100150200250300350400450

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Von

Mise

s Str

ess (

MPa

)

ɛ (mm/mm)

Stress vs ɛ

050

100150200250300350400

0 0.5 1 1.5 2 2.5

Von

Mise

s Str

ess (

MPa

)

𝜀𝜀 ̇ s-1

Stress vs strain-rate (𝜀𝜀 ̇s-1)

43

Figure 3-20 Magnified view of the von Mises flow stress prediction with respect to strain-rate. The dashed curve is a logarithmic approximation for aluminum.

The last two figures shown in this section show the dependence of the von Mises flow

stress on temperature changes and material melting point. Figure 3-21 shows what happens as

the temperature of the system is changed. It is evident that the predicted von Mises flow stress

decreases as temperature increases, as expected.

y = 7.557ln(x) + 374.81R² = 0.9997

350

355

360

365

370

375

380

385

0 0.5 1 1.5 2 2.5

Von

Mise

s Str

ess (

MPa

)

𝜀𝜀 ̇ s-1

Stress vs strain-rate (𝜀𝜀 ̇s-1)

44

Figure 3-21 Effect of temperature on the predicted von Mises flow stress in the PTW material model. Note, the melting point of aluminum is 660 °C

Figure 3-22 shows the dependence of the material melting point on the predicted von

Mises flow stress. It is shown that as the melting point of the material increases, the higher the

predicted von Mises flow stress will be. This may be important to the model, since material

melting point may be significantly changed locally during cold spray impact or other

applications such as friction welding. This parameter gives a better understanding into what

parameters matter in the PTW model.

0

50

100

150

200

250

300

350

400

450

200 300 400 500 600 700 800

Von

Mise

s Str

ess (

MPa

)

T (°C)

Stress vs T

45

Figure 3-22 Effect of changing the melting point of the material with respect to the predicted von Mises flow stress in the PTW material model for aluminum.

3.5 PTW model comparison to Johnson-Cook An axisymmetric model of a copper cylinder was built in Abaqus with a diameter of 0.2 m and

a height of 0.2 m to compare PTW against built-in JC. For both models, 5000 4-node bilinear

axisymmetric quadrilateral, reduced integration elements, CAX4R, were used for the

modeling. Three different loading conditions were used, 1000 N, 1x107 N, and 1x108 N. These

correlate to pressures of 10 kPa, 100 MPa, and 1 GPa, respectively. Both JC and PTW models

were compared at each loading condition. The model is shown in Figure 3-23. PTW does not

predict elastic properties, so linear elastic material properties were defined in Abaqus for the

PTW model.

050

100150200250300350400450

0 500 1000 1500 2000 2500

Von

Mise

s Str

ess (

MPa

)

Tm (K)

Stress vs Tm

46

Figure 3-23 Axisymmetric JC/PTW test model setup

Loading case 1 shows the elastic response of the model at low loads as shown in Figure 3-24.

Results of the elastic case show very good agreement in the 1000 N load case. These results

are comparing the built-in JC properties against the PTW elastic properties. The results of this

loading condition give an error of less than 1%. The stress distribution between the two models

is nearly identical, which is to be expected.

47

Figure 3-24 1000 N comparison between PTW subroutine (left) and built-in JC (right) for

copper.

Loading case 2 is a near-yield response case with a load of 1x107 N. The results of this

modeling are shown in Figure 3-25. There is also good agreement in the von Mises stress of

the two models. This is to be expected since they are still in the elastic regime. The stress

distribution is nearly identical in both models. There is a slightly larger area of higher stress in

the top left of the PTW model, but the trend is similar between the two. The results of this

modeling also produced an error of less than 1%.

48

Figure 3-25 Von Mises stress comparison between the PTW subroutine (left) and the built-in

JC model (right) for copper.

The final case in the preliminary model testing was a model that would see plastic

deformation. Loading case 3 increases the loading by an order of magnitude, to 1x108 N. At

this loading condition, there is a change in the stress predictions due to the activation of the

PTW VUHARD subroutine. The results of this model are shown in Figure 3-26. The stress

distribution between the two models is completely different. In terms of stress prediction, the

PTW model predicts an order of magnitude lower von Mises stress at the bottom of the model.

This may be due to a strain-rate difference through the model since the load is being applied

instantaneously with no loading curve. On the upper end of the stress prediction, both models

predict a very similar maximum stress at the top of the model. These models were not expected

to be the same once plasticity occurred, but the stress distribution is markedly different.

49

Figure 3-26 Von Mises comparison between the PTW subroutine (left) and the built-in JC

model (right) for copper.

3.6 Discussion

The PTW model is a thermodynamically based model that does not rely on curve fitting

of experimental data. This is one of the reasons that the PTW model can accurately predict

deformation and stress values over wide ranges of strain-rates. However, this model is much

more complex and calculation intensive than the Johnson-Cook model, for example.

Complexity and increased calculation times may be the reason that PTW is not currently found

built into commercial software. It may also be that it was only recently developed. Other

authors have investigated the PTW model previously, for use in the cold spray process, but

50

they did not implement the model to its fullest extent in three dimensions or institute a bonding

criterion as was conducted in this work [36].

Overall, the development of the PTW VUHARD user subroutine was a straightforward

effort. The PTW mathematics were converted to a FORTRAN77 specification and checked for

accuracy using published data. Once the mathematics were checked, a Johnson-Cook user

subroutine was developed to ensure that material parameters were being passed through to

Abaqus successfully and to ensure that a subroutine would predict similar values to a built-in

Abaqus material option. Results of the Johnson-Cook user subroutine correlated well with the

built-in Johnson-Cook material model. Results were within 3% of each other. An extensive

sensitivity analysis was conducted on the PTW parameters to identify which variables had the

greatest effect on the model. The PTW model was then compared to the Johnson-Cook

subroutine and was subsequently debugged. The success of the JC subroutine in Abaqus made

the development of any work in LS-DYNA unnecessary.

There were however, some issues with the development of this subroutine. Initially, it

was thought that a VUMAT user subroutine would be a viable solution to the PTW

implementation, but there was an issue of not defining the elastic regime in the PTW model.

This was overcome by using a VUHARD subroutine, which only modifies materials that have

plastically deformed. This made the development of the final PTW user subroutine much

easier. Some other issues with the PTW subroutine were the extensive use of error functions

and numerous material properties that are not found in the Johnson-Cook model.

Development of the PTW user subroutine was a significant portion of this work, and

its development was used to predict the deformation behavior of cold spray particles. The PTW

51

user subroutine will be coupled with a bonding criterion, which is discussed in the next few

chapters.

52

Chapter 4 4.1. Instituting a Bonding Mechanism for Cold Spray Impact

4.1.1. Executive Summary

This chapter is dedicated to the development and implementation of bonding criteria

for the cold spray process using fracture energy. Abaqus has many built-in options for

describing the surface interaction between two parts. One of these methods is known as

traction-separation behavior. In addition to these built-in options, Abaqus allows for

subroutines to be developed to describe surface interactions.

The advantage of the traction-separation behavior option is that it already supports

fracture energy based damage evolution. After an extensive literature review, it appears that

there has been little work done with this option for modeling dynamic systems. The traction-

separation behavior option was originally developed for adhesive contacts such as glue or

carbon-fiber composites. The subsequent sections discuss each parameter in detail and a finite

element model is developed to evaluate each parameter. Two damage evolution models are

investigated; maximum displacement or fracture energy based. It appears that the displacement

based model shows good bonding characteristics without any mesh refinement, as opposed to

the fracture energy based model. After refining the mesh, the fracture energy based damage

evolution model initiated bonding. These parameters are evaluated over many orders of

magnitude to determine trends in bonding behavior. The bonding parameters are then used in

subsequent chapters to institute bonding during cold spray impact.

53

4.1.2. Objective

The objective of this chapter is to implement bonding criteria for the cold spray process.

Bonding in the cold spray process has been an area of debate for well over two decades,

modeling of which has been attempted since Assadi et.al. [30]. Most cold spray impact finite

element models do not include a bonding model in the input, but typically show the particle in

its most deformed state correlating to experimental results, before any rebound can occur. This

has been the most common method of predicting the deformation in cold spray impact, but this

type of modeling is neglecting any rebound or de-bonding that may or may not occur in real

life.

To account for effects due to bonding and de-bonding, this chapter investigated various

methods that have been tried in the past, as well as proposing new methods that may show

promise in cold spray modeling [5]. This bonding model has three main requirements to be

considered successful. First, the bonding criteria must be easy to obtain, via experimental

results, experimental correlation, or estimated using thermodynamic methods. Second, the

bonding model must be able to be applied to particle/substrate impacts, and particle/cold spray

deposit impacts without changing governing parameters for each condition. And finally, the

bonding model must be compatible with any outside user defined subroutines in Abaqus, such

as the VUMAT and the VUHARD user subroutines.

4.1.3. Background and Current State-of-the-Art

Modeling bonding or de-bonding in finite elements is a fairly new concept. Some of

the first attempts for modeling bonding and de-bonding were for adhesively bonded structures

such as glue joints and composites [37], [38]. There have been numerous studies conducted in

this field, as well as a thorough understanding of the bonding and failure mechanisms that

54

occur in adhesive bonds and composites [39]. However, this is not the case in the cold spray

process. Since the development of the cold spray process in the mid 1980’s, there have been

numerous explanations for the processes that occur during impact [7], [30], [32]. Many authors

have identified two bonding phenomena that can occur during cold spray deposition. The first,

perhaps the simplest bonding mechanism that can occur in cold spray is mechanical mixing of

the particle and the substrate. This process is typically found in impacts between soft particles

and substrates. A schematic of mechanical mixing is shown in Figure 4-1 [40].

Figure 4-1 Schematic diagram of mechanical mixing found in cold spray bonding [40]

This bonding mechanism is very simple and straightforward to understand, however

finite element modeling of this process is nowhere near as straightforward. Finite element

modeling is very good at modeling structural loading and deformation, but standard

Lagrangian finite element models are not designed for predicting mechanical mixing. Due to

the extreme deformations found in the cold spray process, Lagrangian models experience

severe mesh distortion, which causes the model to not converge. Other methods such as

Eulerian models are very good at modeling mechanical mixing due to the material not being

assigned to a certain meshed element as found in the Lagrangian models. However, it is very

difficult to define contact parameters and other boundary conditions in Eulerian models. The

55

model proposed in this chapter will address mechanical mixing by allowing small amounts of

nodal immersion between the particle and substrate.

The next bonding mechanism that is discussed in the cold spray process is referred to

as bonding due to adiabatic shear instability. Adiabatic shear instability occurs when a particle

impacts a substrate at sufficient speed to cause significant softening and shear localization.

This is correlated to bonding in cold spray, especially in hard particle/hard substrate impacts,

where little mechanical mixing occurs, and there is significant contact pressure.

When a particle impacts a substrate, it is thought that the oxide layers in both the

substrate and particle are destroyed, leaving pristine material in contact and bonding [32]. This

is similar to what occurs in cold welding [41]. The characteristic jetting effect found in cold

spray is believed to carry away the broken down oxide layer, whose removal allows the

bonding to occur [32]. The criteria in this work will address the bonding behavior in cold spray

using a fracture energy approach, similar to work done by Schreiber et.al. for fragmentation of

high-strength steel rings [42].

The state-of-the-art in cold spray bonding has been discussed by numerous research

groups, from a numerical explanation, to a cohesive element bonding criterion [4], [5], [30],

[43]. One of the first finite element investigations of bonding in cold spray was conducted by

Schmidt et.al at Helmut Schmidt University [44]. Their work concluded that the

particle/substrate bond is reliant on a critical impact velocity, which is generally agreed upon

today as a threshold that must be met to have bonding occur [3]. Their finite element model

mainly identified that extreme deformation occurred near the outer edges of the particle, which

is attributed to shear instability. The model does not include a bonding criteria; it was only

used to define the threshold for bonding via the critical impact velocity. However, this

56

pioneering work is very important to help in the understanding of the critical impact velocity,

which may indicate that there is an amount of internal energy required to initiate bonding.

The next important work on cold spray bonding is from Yildirim et.al from

Northeastern University [5]. This research attempted to identify and correlate an interfacial

bonding energy to the critical velocity of an impacting cold spray particle. This research

showed that the effective cohesive strength of the particle/substrate bond changes as the

particle velocity is increased above the critical impact velocity [1], [11]. It was also shown

which areas were bonded using the interfacial bonding criterion. Figure 4-2 shows the results

of the interfacial bonding energy model with respect to time conducted by Northeastern

University. The particle impact velocity was 750 m/sec and the interfacial bonding energy was

set to 400 MPa.

Figure 4-2 Results of Northeastern Universities cold spray particle impact bonding model. Notice that the bonded area varies over time [5]. The dark areas show the bonded region. Notice that the bonded region covers the entire particle

until approximately 33 nanoseconds, and then begins to de-bond. De-bonding and re-bonding

at later time steps may occur due to the elastic rebound of the particle. This correlates well to

experimental results, which show areas of de-bonding near the center of the particle.

An attempt to identify a lower threshold value to keep the particle bonded to the

substrate was conducted. It was found that the prediction of the lower bound was less than

57

ideal. Also, a significant effect on the bonding parameters was found to be due to the coefficient

of friction between the particle and substrate as well as the overall mesh size [5]. This work

was very important in the development of this chapters bonding criteria due to the overall

similarities in the general design.

Current work on cold spray bonding is mainly based on phenomenological aspects of

cold spray impact. Wang et.al. at United Technologies Research Center have done considerable

research on the effects of cold spray bonding for non-normal impacts. The effect of angular

impact on the cold spray particle is important information, since many particles do not impact

at a perfect 90° angle [43]. This work does not include any bonding parameters in the model,

but gives an idea of what an off-normal impact should look like. Figure 4-3 shows the results

of their study of non-normal cold spray impacts and their bonding characteristics.

Figure 4-3 Non-normal cold spray impact analysis conducted by Wang. et.al. Note the prediction of the gap and jetting in the model [43].

The cold spray impact model shows very good agreement with the experimental results,

regarding the formation of a gap on the left hand side of the particle. Jetting is also predicted

in the model, which correlates well with experimental results.

58

Overall, bonding in finite element analysis in regards to the cold spray process is not fully

implemented. This may be due to issues with the definition of the contact interactions, the

mechanical mixing aspect, or the effect on particle rebound in the model. Contact interactions

such as friction coefficient have been shown to significantly change the bonding between the

particle and the substrate [1], [11]. This has crucial implications in the development of the

bonding model.

It is possible that interfacial friction coefficient parameters will need to be estimated, since

there is currently no feasible way to measure the dynamic coefficient of friction between a

substrate and a micron size particle at strain-rates reaching 1x107 sec-1. However, there are

ways around this issue. Currently, the best method for tuning properties in a high-strain-rate

dynamic finite element model is to correlate finite element results with experimental data.

Experimental data in cold spray is difficult to obtain, and is typically in the form of a cross

section of a single particle impact. Subsequently, the finite element model properties can be

tuned to better represent experimental data.

Mechanical mixing is another area where a standard Lagrangian finite element model

falls short. In reality, mixing does occur, especially in soft particles and substrates. Methods

such as allowing some nodal immersion can simulate mechanical mixing, but as of today, there

are no established methods for addressing mechanical mixing without moving to an Eulerian

type finite element model. Rebound is an area that has had some investigation. Northeastern

University has done an excellent job of investigating the coefficient of restitution for impacting

cold spray particles, and attempted to implement that in a bonding model [4]. The work in this

chapter will attempt a similar style of bonding parameter, but will use fracture energy as a basis

for debonding.

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4.2. Traction-Separation Behavior

4.2.1. Background Information

Traction-separation is a type of surface based cohesive behavior that is built into

Abaqus for a number of applications. As mentioned previously, this type of cohesive behavior

was used for predicting the behavior of adhesively bonded structures. There are two main types

of cohesive models available in Abaqus, cohesive elements and surfaces. Cohesive elements

are used mainly in the modeling of glues and other adhesives where the thickness of the

adhesive can be considered as an element. It is possible to use a ‘zero thickness’ cohesive

element, but this may cause some issues when modeling the cold spray process, due to

deformation across interfaces. The interaction formulation between cohesive elements and

cohesive surfaces are very similar, but thickness effects are not considered in cohesive surface

models. A cohesive surface model is most likely the best option for modeling the cold spray

particle bonding for a number of reasons. First, since there is no adhesive material on top of

the substrate or on the particle, utilizing a cohesive element would not make sense for cold

spray.

In addition, adding another element type in the model could cause convergence issues

due to the tie constraints that would be required to bond the two element types together, as well

as adding to calculation time. Currently, the cold spray impact model is subjected to extreme

deformations and very high strain-rates, which the cohesive element was not designed for. This

may cause more issues in further modeling if features such as adaptive re-meshing are

implemented. The final reason why a cohesive surface model is preferred is that future

modeling may include the thin oxide layer on the substrate and particles, and a “fake” cohesive

layer may skew the results.

60

There are a few reasons that traction-separation behavior was chosen to be used over

the development of a user subroutine. After a thorough background literature survey, it was

found that there has not been much work done on investigating the usefulness of this contact

interaction in dynamic modeling, especially for cold spray. A search through the Abaqus

manual, provided insight into both the theory behind traction-separation behavior and the

implementation possibilities. According to Abaqus 2016 Section 37.1.10, surface based

cohesive behavior can be used to model what is referred to as a ‘sticky’ contact. This is

referring to parts that may not be in contact initially, but may come into contact throughout the

modeling step and bond with each other, as well as debond. This appears to be exactly what is

required for the cold spray bonding model. Literature has shown no use of this for cold spray

applications, except for a paper by Northeastern University which does not fully describe how

they implemented bonding [5].

4.2.2. Traction-Separation Option

For this work, a simple linear elastic traction-separation model was utilized in Abaqus.

This is accomplished in Abaqus by utilizing Equation 4-1 [45]. This equation relates stress to

the separation in the model

𝑒𝑒 = �𝑒𝑒𝑛𝑛𝑒𝑒𝑠𝑠𝑒𝑒𝑡𝑡� = �

𝐾𝐾𝑛𝑛𝑛𝑛 𝐾𝐾𝑛𝑛𝑠𝑠 𝐾𝐾𝑛𝑛𝑡𝑡𝐾𝐾𝑛𝑛𝑠𝑠 𝐾𝐾𝑠𝑠𝑠𝑠 𝐾𝐾𝑠𝑠𝑡𝑡𝐾𝐾𝑛𝑛𝑡𝑡 𝐾𝐾𝑠𝑠𝑡𝑡 𝐾𝐾𝑡𝑡𝑡𝑡

� �𝛿𝛿𝑛𝑛𝛿𝛿𝑠𝑠𝛿𝛿𝑡𝑡� = 𝐾𝐾𝛿𝛿

Equation 4-1 [45]

where 𝐾𝐾𝑥𝑥𝑥𝑥 refers to the corresponding stiffness coefficients, 𝑒𝑒𝑥𝑥 refers to the traction

stress vector in each direction, and 𝛿𝛿𝑥𝑥 refers to the corresponding separation values in each

61

direction. When utilizing the linear elastic separation parameters in Abaqus, only the principle

stiffness coefficients are specified.

It is possible to utilize a coupled relationship for the stiffness coefficients, but for this

initial work, only linear elastic will be analyzed. Since the cold spray process relies on extreme

plastic deformation of the impacting particle to create a bond, it may be necessary to investigate

the stiffness parameters in more than only the normal direction. In the linear elastic model, 𝐾𝐾𝑠𝑠𝑠𝑠

and 𝐾𝐾𝑡𝑡𝑡𝑡 correspond to the tangential cohesive stiffness components. These values will be set

to the same value as the normal stiffness component. Some estimating must be done at this

point to determine the proper stiffness components. Since the majority of the work on traction-

separation mechanics has been done on adhesives, typically the stiffness values are related to

the adhesive strength. Using this knowledge, a fair estimate of the stiffness component

strengths would be the yield strength, the ultimate tensile strength, and possibly the shear

strength of the materials being impacted. However, this may pose an issue with defining the

model. At high-strain-rates, the yield strength of a material increases dramatically, as shown

in the last chapter. This is the reason why a high-strain-rate plasticity model is needed in the

cold spray model such as the Johnson-Cook and the Preston-Tonks-Wallace model. On the

other hand, just because the yield strength increases as strain-rate increases, this does not mean

that the bond strength will not be on the order of the quasi-steady-state yield and shear strength.

More study of this area is required.

4.2.3. Damage Model

The damage approach in the linear elastic traction-separation is just as important as

the bonding model itself. A damage parameter is needed to define the point at which the newly

bonded interface will degrade and de-bond. Without a damage model, the bonding parameter

62

will only assume that the material has bonded perfectly at all points, which is not correct, as

has been shown in numerous studies by other researchers [46]–[48]. As shown in Figure 4-4,

the Abaqus technical manual section describes the damage parameter to be a linear response,

similar to what is found with the bonding criteria. As the bonded area reaches a threshold value,

the bonds will start to break, leading to de-bonding of the particle.

Figure 4-4 Example of the linear progression and initiation of the damage parameter in the traction-separation model in Abaqus [45].

There are a number of ways that damage can be instituted via the traction-separation

law in Abaqus. Perhaps the simplest method to threshold a damage parameter would be to

specify a maximum nominal stress that will define de-bonding. In Abaqus, both normal and

shear stresses can be defined to cause failure, as well as the ability to add temperature

dependent stress values to the model. This may be beneficial to the cold spray process, since

many materials must be sprayed at elevated temperatures to ensure bonding. Another option

in Abaqus to cause failure in the bond would be a maximum separation value for damage

63

initiation. This parameter may be difficult to implement in the cold spray model since the

model experiences severe deformation and elastic rebound.

Moreover, it may be possible that if the maximum separation value is set too low or

high, false or excessive bonding would occur. Both of these models follow the same equation

to initiate failure. Failure is identified as the ratio of stress (or separation) to initial stress (or

separation) being greater than one. Once this ratio is exceeded, damage will occur, breaking

bonds. This relationship is shown as Equation 4-2 [45] for the maximum stress criterion and

as Equation 4-3 for the maximum separation criterion [45].

𝑚𝑚𝑎𝑎𝑒𝑒 �⟨𝑡𝑡𝑛𝑛⟩𝑡𝑡𝑛𝑛0

, 𝑡𝑡𝑠𝑠𝑡𝑡𝑠𝑠0

, 𝑡𝑡𝑚𝑚𝑡𝑡𝑚𝑚0� = 1 Equation 4-2 [45]

𝑚𝑚𝑎𝑎𝑒𝑒 �⟨𝛿𝛿𝑛𝑛⟩𝛿𝛿𝑛𝑛0

, 𝛿𝛿𝑠𝑠𝛿𝛿𝑠𝑠0

, 𝛿𝛿𝑚𝑚𝛿𝛿𝑚𝑚0� = 1 Equation 4-3 [45]

Damage initiation criteria can be also expressed as quadratic functions, provided that fits the

type of failure found in the model. The only way to determine if this is the case, would be to

compare against experimental results. The quadratic forms are shown for maximum stress and

maximum separation in Equations 4-4 and 4-5, respectively [45].

�⟨𝑡𝑡𝑛𝑛⟩𝑡𝑡𝑛𝑛0�2

+ �𝑡𝑡𝑠𝑠𝑡𝑡𝑠𝑠0�2

+ �𝑡𝑡𝑚𝑚𝑡𝑡𝑚𝑚0�2

= 1 Equation 4-4 [45]

�⟨𝛿𝛿𝑛𝑛⟩𝛿𝛿𝑛𝑛0�2

+ �𝛿𝛿𝑠𝑠𝛿𝛿𝑠𝑠0�2

+ �𝛿𝛿𝑚𝑚𝛿𝛿𝑚𝑚0�2

= 1 Equation 4-5 [45]

64

For this work, the linear damage equations will be investigated further, since it is very difficult

to determine what type of failure actually occurs in the cold spray process due to the very high

strain-rates. The damage model is further defined by a few more damage options, mainly

damage evolution and damage stabilization. These two topics are discussed in the next section.

4.2.4. Damage Evolution

A maximum displacement based method was chosen to define damage evolution in

Abaqus. As input, the software requires a softening definition and a total displacement at

failure. The softening options in Abaqus can be defined through three methods: linear,

exponential, and tabular. The linear definition for displacement based softening follows

Equation 4-6 [45]

𝐷𝐷 = 𝛿𝛿𝑟𝑟𝑓𝑓 (𝛿𝛿𝑟𝑟𝑟𝑟𝑎𝑎𝑚𝑚−𝛿𝛿𝑟𝑟0 )

𝛿𝛿𝑟𝑟𝑟𝑟𝑎𝑎𝑚𝑚(𝛿𝛿𝑟𝑟𝑓𝑓 −𝛿𝛿𝑟𝑟0 )

Equation 4-6 [45]

where D is the damage variable, 𝛿𝛿𝑚𝑚𝑚𝑚𝑎𝑎𝑥𝑥 is the maximum separation reached during loading, 𝛿𝛿𝑚𝑚𝑓𝑓

is the failure separation, and 𝛿𝛿𝑚𝑚0 is the initial separation. This is a relatively simple method for

determining failure. This method is used in subsequent sections where the damage parameter

is investigated in a finite element model. Damage evolution can also be described as an

exponential function. This is a more complex model than the linear damage evolution. It also

may not be required for the cold spray bonding parameter. A tabular description is also

available for damage evolution with respect to displacement. A simple chart can be developed

to describe the damage evolution with respect to the effective separation at the start of the

model. This option may not work for this modeling effort due to the lack of bonding and de-

bonding information available for the cold spray process.

65

It is also possible to describe the damage evolution of the model using a fracture energy

based solution. At the beginning of this thesis, it was proposed that a fracture energy value

could be used to describe the bonding characteristics of the cold spray process. Work by J.

Schreiber showed that the fragmentation of a high strength steel ring could be predicted using

values of fracture strain [10], [20]. Reversing this idea for a bonding parameter may be possible

using the built-in options found in Abaqus. Abaqus has an option that allowed for fracture

energy to be described as a means to initiate damage for bonding which may be applicable to

the cold spray process. Fracture energy is described as the area under the traction-separation

curve. This is slightly different than the fracture energy used in previous work by Schreiber,

but still may be useful for bonding criteria [42]. An example of the fracture energy based curve

is shown in Figure 4-5 [45]. This curve assumes that the loading section will be linear,

however, this may not be the case in real life. Due to the extreme plastic deformation associated

with the cold spray impact, this loading curve may not be an accurate description of the

traction-separation behavior upon impact. It is possible that a subroutine could be written to

describe this curve, but obtaining experimental confirmation may be impractical.

66

Figure 4-5 Exponential description of fracture energy base damage parameter used in the traction-separation model in Abaqus. [45]

Fracture energy is implemented in the same manner as the displacement based

evolution model, but instead of requiring the total displacement at failure, the fracture energy

based model requires a fracture energy value. The tabular softening option is not available for

the fracture energy based damage evolution, but both the linear and exponential forms of

evolution can be used. They are shown in Equations 4-7 and 4-8 [45].

𝐷𝐷 =2𝐺𝐺𝐶𝐶

𝑇𝑇𝑚𝑚𝑓𝑓𝑓𝑓0� (𝛿𝛿𝑟𝑟𝑟𝑟𝑎𝑎𝑚𝑚−𝛿𝛿𝑟𝑟0 )

𝛿𝛿𝑟𝑟𝑟𝑟𝑎𝑎𝑚𝑚(2𝐺𝐺𝐶𝐶 𝑇𝑇𝑚𝑚𝑓𝑓𝑓𝑓0� −𝛿𝛿𝑟𝑟0 )

Equation 4-7 [45]

where 𝐺𝐺𝐶𝐶 is the fracture energy, and 𝑇𝑇𝑚𝑚𝑓𝑓𝑓𝑓0 is the effective traction at separation

𝐷𝐷 = ∫𝑇𝑇𝑚𝑚𝑓𝑓𝑓𝑓𝑑𝑑𝛿𝛿𝐺𝐺𝐶𝐶−𝐺𝐺0

𝛿𝛿𝑟𝑟𝑓𝑓

𝛿𝛿𝑟𝑟0 Equation 4-8 [45]

67

where 𝑇𝑇𝑚𝑚𝑓𝑓𝑓𝑓 is the effective traction, 𝛿𝛿 is the effective separation, and 𝐺𝐺0 is the elastic energy

at damage initiation. Equation 4-7 is almost identical to Equation 4-6 for maximum separation

definition. The only change is the replacement of maximum separation at failure with the

fracture energy divided by the effective traction at separation. This option will be investigated

further for this work to predict bonding behavior in the subsequent sections.

4.2.5. Damage Stabilization

The last value available for defining the damage parameter in the traction-separation law

is the damage stabilization. This option is used mainly for models that exhibit softening

behavior convergence issues, typically used with cohesive elements, fasteners, and concrete

damage [45]. These convergence issues may arise from severe plastic deformation causing

excessive distortion in the elements. Abaqus provides damage stabilization to overcome these

issues. In the software, a viscosity coefficient can be prescribed that will cause a small amount

of damping in the system. The Abaqus technical manual describes this as causing the tangent

stiffness matrix to define contact stresses as positive values during very small time increments.

This may become more important in the cold spray impact model due to the extremely small

time increments that are required to find convergence. In some cases, the cold spray impact

model is incrementing at 1x10-12 sec to find convergence.

4.3. Implementation into Abaqus

Implementing the traction-separation behavior into Abaqus was a fairly straightforward

process. First a model needed to be developed that was easy to modify, fast to compute, and

did not have a complex geometry. Once this model was developed, the traction-separation

parameters, the damage initiation parameters, and the damage evolution parameters were

investigated.

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4.3.1. Block Impact Model

A model to conduct a quick parameterization study was required to identify the most

critical parameters in bonding. To do this, a three-dimensional Lagrangian finite element

model was developed using Abaqus CAE. This model consisted of two rectangular blocks that

would impact at high velocity. Dimensions were kept very simple to ensure that no issues

would arise from dimensions or conversion factors used in the particle impact model. Standard

SI base units were used where possible. Dimensions for each block are shown in Table 4-1.

Table 4-1 Part dimensions for traction-separation parameter investigation model.

Bottom Block Top Block

Length, X (meters) 3 1

Width, Z (meters) 3 1

Thickness, Y (meters) 2 1

The bottom block was fixed in space on the bottom surface to simulate being placed on a

table, and the upper block impacted the lower block at 400 m/sec. The setup of the model is

shown in Figure 4-6.

69

Figure 4-6 Three-dimensional model setup for traction-separation parameter investigation.

The material properties used for this analysis were for AISI 4340 steel. This alloy was

chosen for this modeling due to the enormous amount of material property information

available. AISI 4340 is used in a variety of applications, including ordinance applications, so

high-strain-rate properties were readily available for this modeling. A Johnson-Cook plasticity

model was implemented for this study, due to the available property data, and that a user

subroutine is not required. Material properties used for this analysis are shown in Tables 4-2

and 4-3 for quasi-steady-state data and Johnson-Cook data, respectively.

70

Table 4-2 Quasi-steady-state material properties used in the traction-separation parameter

analysis (AISI 4340) [22].

Density 7830 kg/m3

Young’s Modulus 200x109 Pa

Poisson’s Ratio 0.29

Table 4-3 Johnson-Cook material properties for AISI 4340 steel used in the traction-separation

behavior analysis [22].

A 792x106 Pa

B 510x106 Pa

C 0.014

n 0.26

m 1.03

Tm 1793 K

Tt 255 K

𝜀𝜀0̇ 1

71

Meshing of the blocks was conducted using the automatic mesh generator in Abaqus. Since

the blocks are very simple in composition and geometry, a standard brick mesh was used. For

the bottom block, 144,000 C3D8R, 8-node linear brick elements with reduced integration and

hourglass control were used. For the upper block, 35,937 C3D8R, 8-node linear brick elements

with reduced integration and hourglass control were used. These elements were chosen over

tetrahedral elements due to the geometry being rectangular and hexahedral elements typically

provide the fastest convergence with the least amount of elements. Only linear brick elements

are available in Abaqus/Explicit as opposed to tetrahedral elements, which can be linear or

quadratic in nature. The meshed model is shown in Figure 4-7.

Figure 4-7 Meshed three-dimensional model for traction-separation parameter investigation.

72

4.3.2. Parameter Investigation for Traction-Separation

An investigation of the controlling parameters in the traction-separation behavior was

conducted. Since traction-separation behavior is phenomologically based, it is worthwhile

investigating how the model behaves as the values are varied. During this first analysis, the

stiffness coefficients were varied across many orders of magnitude to identify any changes in

behavior. Starting at a stiffness of 1, and continuing to over 1x1011, a trend started to appear.

As discussed in Chapter 3, as the traction-separation stiffness coefficients reached the yield

strength values, the distorted elements started to disappear. The trend is shown in Figure 4-8.

Figure 4-8 Trends found in the analysis of the traction-separation stiffness coefficients.

73

Notice as the stiffness coefficient reaches 1,000 MPa, the extreme deformation and

distortion of the elements appears to subside. This may be an indication that the bonding

parameters are material property dependent, and further modeling uses a traction-separation

variable of 1,000 MPa. The published yield strength for AISI 4340 steel is 710 MPa, and the

published ultimate tensile strength is 1,110 MPa [49].

Once the traction-separation stiffness coefficient was determined, the next step in the

development of the bonding parameter was to decide which damage evolution model best

described the bonding characteristics and allowed for de-bonding to occur. This bonding model

is heavily reliant on the de-bonding parameter since it considers any contact with another body

to be a perfect contact. Only when the particle rebounds from the substrate will it start to

activate the damage parameter. This behavior can be modified if needed.

The first damage parameter investigated was the maximum displacement damage

model. Five separate models were used to identify any bonding characteristics for

displacements ranging from 0.01 m to 1 m. Figure 4-9 shows the results of the modeling effort.

It can be seen that in the model with the maximum displacement of 0.01 m, complete de-

bonding occurred after impact. This may be representative of real life since it is not known

whether these blocks will bond or not, but the goal of this modeling effort is to determine what

is the best method for instituting bonding and whether or not it can be controlled.

Also in Figure 4-9, the results of the 0.1 m maximum displacement model showed

evidence of jetting and no de-bonding of the upper block. This indicates that a successful bond

has occurred between the upper and bottom blocks. As the maximum displacement is increased

further, extreme deformation and convergence issue began to start, making it impossible to

investigate a displacement above 1 m. It is thought that the optimal displacement value lies

74

between 0.01 m and 0.1 m, but it may not be feasible to use a displacement based bonding

parameter due to the issues discussed in the previous sections.

Figure 4-9 Results of the parameter investigation for the maximum displacement based damage evolution. Units are in meters.

The final damage evolution parameter that was explored was the fracture energy based

traction-separation behavior. Similar to what was done to the displacement based damage

evolution, the fracture energy was varied across many orders of magnitude to visualize any

trends that may occur. Figure 4-10 shows the trends in rebound and deformation for fracture

energy values ranging from 1 to 1x109 Pascal, based on Equation 4-4. Starting with the value

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of 1 in Figure 4-10, it can be seen that no bonding has occurred in either the upper or bottom

block. This value was increased by an order of magnitude until a noticeable change had

occurred. Once the fracture energy value reached 100,000, a significant amount of distortion

occurred. Above this value, there appears to be regions of bonding, but due to the immense

amounts of distortion in the elements, no real conclusions can be made. To alleviate the

distortion issues, mesh refinement and some meshing tricks were required.

Figure 4-10 Results of the parameter investigation for the fracture energy based damage evolution.

4.3.3. Model Refinement Due to the excessive element distortion found in the fracture energy based damage

evolution model, some modifications to the mesh were required. Typically, larger elements

can withstand large deformations due to the greater distance between nodes. This is important

in dynamic analyses since the speed of sound of the material is considered. It is possible that

76

some elements can be distorted so quickly, that the rate of deformation exceeds that of the

speed of sound. This is a very common error found in impact models as well as other types of

dynamic finite element models such as fragmentation [42]. In most cases, this type of failure

can be alleviated by either increasing the mesh size, instituting a dynamic remeshing algorithm,

or simply tailoring the shape of the elements to accommodate deformation in an expected

direction. For this work, each option was considered, but there are drawbacks. First, increasing

the element size will cause significant issues with the contact parameters due to the increased

spacing between nodes. This will allow a much larger amount of nodal immersion between the

two impacting bodies. Nodal immersion can be reduced by aligning the impacting nodes with

the stationary nodes of the other part. However, this method will not work for cold spray

impact, since the deformation of the particle is mostly perpendicular to the axis of travel. The

second method, dynamic remeshing is also a challenge to implement in dynamic models. In

cases where contact is made with another object, dynamic remeshing may not be achievable

due to the contact interface continually requiring remeshing, which could change or remove

the contact parameters. Dynamic remeshing also increases calculation time significantly. The

last method to be discussed is simply modifying the orientation of the mesh to accommodate

the impact. There are a number of ways to accomplish this. Instead of using a square hexahedral

element for impact, a rectangular element can be chosen. The advantage of the rectangular

hexahedral element is that the “narrow” end can be oriented towards the impacting particle.

This will cause the element to distort in a manner that drives it from a rectangle, into a square,

and finally into another rectangle oriented in the opposite direction. Doing this modification

increases the amount of displacement an element can absorb before severely distorting and

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failing. Figure 4-11 shows an example of stages of distortion the element will see before

failure.

Figure 4-11 Effect of orientating element geometry to increase deformation tolerance in dynamic impact models.

In this refined model, the elements were simply meshed in a manner that allowed them

to distort while not allowing increased nodal immersion. Using the same particle geometry as

in the previous section, the model mesh was significantly refined. The upper block mesh was

increased from ~35,000 elements, to 1,000,000 elements, effectively tripling the 3D mesh

density. This decreased nodal spacing from approximately 3 cm, down to 1 cm. In the lower

78

block, the mesh refinement was conducted with the same 1/3 scaling factor as the upper block.

Mesh density was increased from 144,000 elements to 2,250,000 elements. This change

decreased the nodal spacing from approximately 5.7 mm, down to 2.2 mm. The refined model

is shown in Figure 4-12.

Figure 4-12 Refined traction-separation impact model. Note that the model now contains over 3 million elements. The refined model was run on a Dell Workstation T7600 with 256 GB of RAM, as well as 14

processor cores operating at up to 3.1 GHz. Calculation time for one run increased from ~20

minutes to well over 3 hours.

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This refined model was run with 1x106 as the fracture energy value to see if the distortion and

strain decreased in the model. It appears that both distortion and strain issues have significantly

decreased. Figure 4-13 shows the results of the von Mises stress prediction at maximum

deformation. It is difficult to identify the stresses in the model due to the small element size,

so Figure 4-14 is also shown without the element outlines.

Figure 4-13 Results of the von Mises stress prediction in the fracture energy based damage evolution model. Note that it is difficult to see the stress distribution due to the extremely small elements.

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Figure 4-14 Results of the von Mises stress prediction in the fracture energy based damage evolution model. It is necessary to plot the strain values for this model to compare against stress. The first set of

plots show the equivalent plastic strain (PEEQ) found in the model. PEEQ is the strain analog

to the von Mises stress. It is a scalar value of strain defined by integration of plastic strain rate.

This is shown in Equation 4-8 [50].

𝜀𝜀𝑝𝑝�� = ∫ 𝜀𝜀𝑝𝑝𝑏𝑏𝑒𝑒̇𝑡𝑡0 Equation 4-8 [50]

Where 𝜀𝜀𝑝𝑝�� is the equivalent plastic strain, 𝜀𝜀�̇�𝑝is the plastic strain-rate, and t is time.

Results of the PEEQ are shown in Figures 4-15 and 4-16. Both the meshed and hidden mesh

results are shown.

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Figure 4-15 Meshed results of the PEEQ for the fracture energy based damage evolution.

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Figure 4-16 Hidden mesh results of PEEQ for the fracture energy based damage evolution model. It is apparent that there is nodal immersion at the impacting interface.

As shown in Figure 4-16, there appears to be a “saw-tooth” interface between the two

blocks. A magnified image of the saw-tooth pattern is shown in Figure 4-17.

Figure 4-17 Saw-tooth pattern caused by nodal immersion at the interface. This is most likely due to nodal immersion between the two parts. There are no good

options to resolve this issue due to the nature of finite elements. There may be some meshing

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tricks that could reduce the amount of immersion, but overall this is a problem with every finite

element model that analyzes contact. There is little PEEQ in the model. This is surprising, as

there is a large amount of plastic deformation occurring at the interface. It may be beneficial

to investigate the plastic strain results in addition to the PEEQ results. Figures 4-18 and 4-19

show the results of the plastic strain output in the model.

Figure 4-18 Plastic strain results for the refined fracture energy based damage evolution model.

Note, it is very difficult to identify the strain due to the small mesh size.

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Figure 4-19 Hidden mesh results for the plastic strain in the refined fracture energy based

damage evolution model. Note, that the largest amount of strain is found at the corners where

bonding has occurred.

It is difficult to see any strain output in Figure 4-18 due to the extremely small mesh size.

Figure 4-19 is included without the mesh overlay to help identify the strain output. As opposed

to the PEEQ, the plastic strain output shows increased plastic strain at the corners of the block.

Since there is bonding, the plastic strain is continued into the bottom block as well. This is a

good sign that the bonding parameter is working. In further particle/substrate bonding models,

these values will be tailored to the model and implemented.

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4.4. Conclusions

A traction-separation method for implementing a bonding model for cold spray particle

impact was investigated. Using the built-in options in the Abaqus software for traction-

separation behavior, a reverse method for bonding using fracture energy was developed. The

bonding parameters are based on phenomologically based assumptions and certain parameters,

such as the stiffness coefficients, which show a trend that follows yield and ultimate tensile

strength of the material. This method of bonding implementation is easy to use and requires

little experimental data to use. Overall, the traction-separation bonding criterion may not be

the best overall method to implement bonding, but it gives the researcher a simple model to

add bonding information into otherwise a very complex dynamic model. A user subroutine,

VUINTER would be the preferred method of implementation of an advanced interfacial

bonding model. However, due to the overwhelming issues with developing, writing,

debugging, and validating a code like that without the ability to compare against experimental

results would be unachievable within the next 1.5 years.

4.5. Suggested Future Work

4.5.1. Subroutine Development- VUINTER

There are two schools of thought when it comes to developing a new model or parameter

in finite elements. The first method is to use what is currently available in the software, and

the second is to develop a user defined subroutine. The previous chapter dealt extensively with

the development and implementation of a VUHARD user subroutine for the use of the PTW

material model in Abaqus CAE. Subroutines are very powerful codes that give the researcher

the ability to define nearly any parameter or incorporate any material model that they see fit.

However, this control comes at a cost. It is very time consuming to develop and evaluate a new

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subroutine, as well as significantly increasing calculation times in explicit analyses due to the

millions of iterations that could occur in a single analysis run.

During this work, a significant background search was conducted to identify the best

solution for implementing a bonding parameter for cold spray. It was found that there were

two acceptable solutions. The first solution being the development of a user subroutine called

VUINTER. The second solution is to utilize the built-in capabilities of the Abaqus traction-

separation behavior. This section will cover the background research conducted for the

VUINTER subroutine.

The VUINTER/VUINTERACTION user subroutine is used to define interactions between

two contacting surfaces for both thermal and mechanical properties. It allows the researcher to

have nearly full control of the contact properties in Abaqus. The advantages of this subroutine

are that nearly any user defined variable can be utilized and updated as needed. This type of

subroutine is used extensively in tool cutting applications, where there are no built-in models

to predict wear, heat generation due to friction, friction, and heat flux [51]. Mathematical

models can be implemented through the VUINTER subroutine in the same manner as the

VUHARD subroutine that was used in Chapter 3. Another advantage of this subroutine is that

it is fully compatible with other subroutines that may be used in the model, such as the

VUHARD subroutine. Utilizing this subroutine would be a substantial undertaking due to the

lack of bonding criteria in the cold spray process. There are some example subroutine codes in

the Abaqus help files, but a better understanding of the contact interactions will be required.

This subroutine overrides all contact interactions that are defined in Abaqus, so an entire

section of the subroutine would be simply defining the contact interactions, and not the bonding

criteria. Another area that may detract from the use of this subroutine is the limitation on the

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contact definition. VUINTER only works with penalty contact. This may become an issue

since all of the contact parameters will be overridden. It will be fairly difficult to debug the

subroutine, since no contact information can be enabled internally. There is however, a

universal format for this subroutine, which is similar to the VUHARD subroutine. The format

is shown below.

subroutine vuinter( C Write only 1 sfd, scd, spd, svd, C Read/Write - 2 stress, fluxSlv, fluxMst, sed, statev, C Read only - 3 kStep, kInc, nFacNod, nSlvNod, nMstNod, nSurfDir, 4 nDir, nStateVar, nProps, nTemp, nPred, numDefTfv, 5 jSlvUid, jMstUid, jConMstid, timStep, timGlb, 6 dTimCur, surfInt, surfSlv, surfMst, 7 rdisp, drdisp, drot, stiffDflt, condDflt, 8 shape, coordSlv, coordMst, alocaldir, props, 9 areaSlv, tempSlv, dtempSlv, preDefSlv, dpreDefSlv, 1 tempMst, dtempMst, preDefMst, dpreDefMst) C include `vaba_param.inc' C character*80 surfInt, surfSlv, surfMst C dimension props(nProps), statev(nStateVar,nSlvNod), 1 drot(2,2,nSlvNod), sed(nSlvNod), sfd(nSlvNod), 2 scd(nSlvNod), spd(nSlvNod), svd(nSlvNod), 3 rdisp(nDir,nSlvNod), drdisp(nDir,nSlvNod), 4 stress(nDir,nSlvNod), fluxSlv(nSlvNod), 5 fluxMst(nSlvNod), areaSlv(nSlvNod), 6 stiffDflt(nSlvNod), condDflt(nSlvNod), 7 alocaldir(nDir,nDir,nSlvNod), shape(nFacNod,nSlvNod), 8 coordSlv(nDir,nSlvNod), coordMst(nDir,nMstNod), 9 jSlvUid(nSlvNod), jMstUid(nMstNod), 1 jConMstid(nFacNod,nSlvNod), tempSlv(nSlvNod), 2 dtempSlv(nSlvNod), preDefSlv(nPred,nSlvNod), 3 dpreDefSlv(nPred,nSlvNod), tempMst(numDefTfv), 4 dtempMst(numDefTfv), preDefMst(nPred,numDefTfv), 5 dpreDefMst(nPred,numDefTfv) user coding to define stress,

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and, optionally, fluxSlv, fluxMst, statev, sed, sfd, scd, spd, and svd return end

Overall, VUINTER gives the researcher full control of the contact parameters and

interaction properties between two impacting surfaces. This method may be the preferred

method to describe what happens during bonding in cold spray, but the validation of this

subroutine would be a daunting task for a single person.

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5. Implementing PTW VUHARD and Bonding Parameter to Cold Spray Impact

5.1. Chapter Summary and Introduction

This chapter is dedicated to the implementation of the PTW VUHARD user subroutine

developed in Chapter 3 and the addition of the fracture energy bonding parameter discussed in

Chapter 4. Moving from the axisymmetric preliminary PTW comparison model in Chapter 3

to a full three dimensional impact model was the next task to be completed for the modeling

effort. An advantage of using the VUHARD user subroutine it is not dependent on the type of

model being analyzed, unlike other user subroutines such as VUMAT. From previous

discussions and unpublished work, a comparison was to be made between the single particle

impact experiments conducted by the University of Massachusetts at Amherst (UMASS) and

the PTW VUHARD Abaqus user subroutine. Initial single particle experiments were

conducted using aluminum Al6061 particles impacting a sapphire substrate at multiple

velocities. This work was conducted at UMASS by W. Xie and J. Lee using a laser based

particle acceleration apparatus of their own design. After impact, the particles were analyzed

using a Scanning Electron Microscope. Photomicrographs of the impacted particles were used

to compare the results of the computer modeling.

During the initial stages of modeling, a standard Hookean elastic response was used.

However, it became evident that the elastic modulus required “tuning” as the strain-rate

changed. The Mie-Gruneisen Equation of State (EOS) was implemented into the model to

account for this. Once all parameters were found and implemented, finite element models were

developed for each impact condition, as shown in the next few sections. Once the Al6061-

sapphire impact model was finished and compared to experimental results, the model was

adapted to analyze the impact of an Al6061 particle onto an Al6061 substrate. This modeling

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is where the bonding parameter values determined in Chapter 4 were used. Good agreement

was found between the experimental results and the finite element models in both cases.

5.2. Single Particle Impact Model Setup 5.2.1. Dimensioning and Material Properties

A three dimensional impact model was developed in Abaqus Explicit 6.14-2 using a

rectangular sapphire substrate and varied spherical particles. The substrate dimensions are

100µm x 100µm x 50µm. Particle sizes are varied from approximately 19 µm to 25 µm, to be

consistent with experimental results. The model setup is shown in Figure 5-1.

Figure 5-1 Single Particle Impact Setup. Notice that the particle has been heavily partitioned for enhanced mesh refinement and interaction parameter.

Note the partitioning lines in the particle in Figure 5-1. Partitioning of the particle was

required to allow a hexahedral type mesh to be used. The substrate is fixed on the bottom face

to keep model runaway from occurring. All dimensions in this model are converted from base

SI units as used in previous chapters, to follow the unit consistent tonne/mm/s format. This

was done due to the very small size of the particle and substrate and size limitations in Abaqus.

Material parameters are also converted to match the dimensional changes due to Abaqus

requiring consistent units throughout. Multiple parameters are required not only for the elastic

91

properties in the 6061 aluminum and the sapphire, but high-strain-rate properties are needed

for the PTW VUHARD subroutine. Material properties are given in Tables 5-1 and 5-2.

Table 5-1 Non-PTW material parameters used in the single particle impact model. Data from Northeastern University [52].

Al-6061 Sapphire

Elastic Modulus (GPa) 68.9 345 Poisson’s Ratio 0.33 0.29 Density (kg/m3) 2700 3980

Thermal Conductivity (W/m*K) 167 23 Thermal Expansion (µm/m*K) 23.6 6.95

Specific Heat (J/kg*K) --- 896 Inelastic Heat Fraction 0.9 ---

Table 5-2 High-strain-rate PTW parameters used in the single particle impact model for the Al6061 particle [27].

Parameter Al6061 Unit Description

S0 0.032 --- The value of saturation shear stress (Ts) at 0 Kelvin S∞ 0.00791 --- The value of saturation shear stress (Ts) near the melt

temperature (Tm) κ 0.494 --- Constant of temperature dependence

Tm 932 K Melting Temperature γ 0.0001522 --- Constant of strain rate dependence β 0.23 --- Exponential material constant for stress at large strain

rates Y0 0.00942 --- Yield Stress constant at 0 Kelvin Y1 0.0142 --- Medium strain rate constant Y2 0.4 --- Medium strain rate exponent p 3 --- Dimensionless material property modifier for the Voce

hardening law θ 0.0529 --- Strain hardening constant in the Voce Hardening Law

G0 29900000000 Pa Initial Shear Modulus M 4.48E-26 kg/atom Molecular Mass

alpha 0.475 --- Constant used to calculate shear modulus hard 1 --- Hardness Constant

E 69 GPa Young's modulus μ 0.334 --- Poisson's Ratio ρ 2712 kg/m3 Density 𝜀𝜀̇ Variable cm/(cm*s) Strain rate ε Variable cm/cm Strain T Variable K Initial Temperature

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All non-PTW properties are input directly into the standard Abaqus material module. User

subroutine values are currently coded directly into the user subroutine. However, work is being

done to parameterize the PTW inputs in a manner that will allow for direct entry for any

material in the Abaqus material module.

5.2.2. Mie-Gruneisen EOS

At very high-strain-rates, it becomes necessary to define the elastic response of a

material in a different manner than at quasi-static-strain-rates. Instead of defining elasticity in

a Hookean type manner, the response of materials at high-strain-rates can be defined in terms

of internal energy and density. This is known as a shock loading response. The derivation of

the response originally comes from the Euler equations, which are a set of hyperbolic functions

governing fluid dynamics for adiabatic and inviscid flow. Inviscid flow refers to a fluid

exhibiting no viscosity, which simplifies the derivation of the Euler equations [53]. The Mie-

Gruneisen EOS is based on a small subset of these equations, called the Rankine-Hugoniot

relation. This relation describes the behavior of a shock wave propagating through a material

[54]. The relationship between these equations is expressed as a function of internal energy

versus density. Figure 5-2 shows this relationship known as a Hugoniot curve [55]

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Figure 5-2 Relationship between pressure and density known as a Hugoniot curve [55].

where pH is the Hugoniot pressure, pH0 is the beginning Hugoinot pressure, pH1 is the

final Hugoniot pressure, and ρ0 and ρ1 are beginning and final densities, respectively. This

relationship is expressed as the Mie-Gruneisen EOS in Abaqus. This EOS can be expressed in

its most common form, shown in Equation 5-1[55].

𝑝𝑝 − 𝑝𝑝𝐻𝐻 = Γ𝜌𝜌(𝐸𝐸𝑚𝑚 − 𝐸𝐸𝐻𝐻) Equation 5-1 [55]

Where p is pressure, 𝑝𝑝𝐻𝐻 is the Hugoniot pressure, 𝐸𝐸𝑚𝑚 is the internal energy, 𝐸𝐸𝐻𝐻is the

specific energy per unit mass, and Γ is the Gruneisen ratio. This common form of the Mie-

Gruneisen EOS is not used by Abaqus directly. Abaqus requires a function written in the linear

𝑈𝑈𝑠𝑠 − 𝑈𝑈𝑝𝑝 format. This format is shown in Equation 5-2 [55].

𝑝𝑝 = 𝜌𝜌0𝑐𝑐02𝜂𝜂(1−𝑠𝑠𝜂𝜂)2 �1 − Γ0𝜂𝜂

2� + Γ0𝜌𝜌0𝐸𝐸𝑚𝑚 Equation 5-2 [55]

94

Where 𝜌𝜌0𝑐𝑐02 is the elastic bulk modulus at small strains, 𝜂𝜂 is the nominal volumetric

compressive strain, and s is the relationship to the shock velocity [55]. The values required for

the Mie-Gruneisen input in Abaqus for Aluminum 6061-T6 are shown in Table 5-3 [56].

Table 5-3 Mie-Gruneisen EOS properties for Aluminum 6061-T6 [56].

Density (g/cm3) Gruneisen coefficient, Γ0

c0 (cm/µs) s

2.703 1.97 0.524 1.40

5.3. Interactions, Boundary Conditions and Meshing

One of the most difficult aspects of modeling the cold spray impact is properly defining

the contact interactions between particle and substrate, as well as the boundary conditions and

overall meshing of the model. The first issue is to define how the particle interacts with the

substrate. During this preliminary modeling, several built-in Abaqus contact parameters were

utilized. The first major decision in the interaction was whether or not a general contact

definition would be possible, or a surface-to-surface contact was required. In terms of

specifying contact, general contact is typically easier to implement, but surface-to-surface is

more tailorable to the model. A surface-to-surface contact was chosen with the penalty contact

method for the mechanical constraint formulation. The penalty contact method has a number

of advantages that are defined by the Abaqus help manual. First, there are no problems due to

conflicts with other types of constraints, which may become a problem during subsequent

modeling with multiple contact definitions. Second, the advantage is that penalty contact

automatically chooses the penalty stiffness, which greatly decreases outside input during

modeling, but increases model run time.

95

Once the interaction format was chosen, the contact interaction properties can be

determined. During this initial stage of modeling, only two options have been considered.

These two options are the normal and tangential behavior of the particle-substrate contact. The

first contact interaction parameter to be discussed is the normal behavior. Normal behavior

allows for pressure-overclosure, constraint enforcement, and separation after contact to be

selected. For this model, the pressure-overclosure option was “Hard” contact. Overclosure is

the penetration of a node of one part into another part. This hard contact does not allow for any

overclosure during impact. There are some other models that may be interesting to investigate

such as an exponential or tabular description of overclosure, but for this initial modeling, a

hard contact should suffice. There are no other built-in choices for the constraint enforcement

method in Abaqus Explicit other than default, so the only other choice for the normal behavior

is the separation after contact. During this initial modeling, separation was not allowed after

contact, linking the particle to the substrate. This option will be investigated later during the

bonding parameter development.

Tangential behavior between the particle and substrate was defined as a penalty contact

friction formulation, where only the friction coefficient must be specified. The friction

coefficient was varied from 0.1-1.0 to investigate any changes in the model. It was found that

values above 0.3 did not significantly change the results. This correlates well with the findings

of Northeastern University [5].

Boundary conditions for the particle and substrate were set in a very simplistic manner.

The bottom face of the substrate was fixed in all three directions to simulate being rigidly

connected to a hard surface. The only boundary condition for the particle was velocity. The

particle velocity was defined using a pre-defined field in Abaqus. The velocity was varied

96

according to the impact velocity specified in the single particle impact tests conducted by

UMASS. The particle was placed directly above the substrate, to minimize the run time. The

model can become unstable if the particle travels through free space for too long of a distance

without interacting with a surface. The step time for this modeling was set to be 10

microseconds with data output every nanosecond. This provides enough resolution to properly

examine the impact, deformation, and rebound of the particle.

Meshing was conducted using all brick elements, in an attempt to eliminate any over-

stiffening effects that can be found when deforming tetrahedral elements. A significant amount

of partitioning was required to allow brick elements to be meshed in the particle. Both the

particle and the substrate were meshed with Abaqus designation C3D8R, 8-node linear brick,

reduced integration, hourglass control elements. The substrate was meshed with 32,000

elements, while the particle mesh varied between 2,500 and 8,000 elements, depending on the

particle size. An example of the meshed model is shown in Figure 5-3.

Figure 5-3 Example of the meshed single particle impact model before impact.

97

5.4. Single Particle Impact Results and Discussion

Six models were developed using the PTW VUHARD user subroutine, comparing the

results against experimental data. The results of the first model were compared against a 20.75

µm particle with an impact velocity of 175 m/sec. Figure 5-4 shows the comparison between

the experimental and modeling results.

Figure 5-4 Comparison between the experimental 175 m/sec impact and the PTW modeling result. Deformation appears to be very similar. Experiments conducted by W. Xie and J. Lee at UMASS.

The deformation predicted using the model correlates well with the experimental

results. The majority of the deformation is localized at the bottom of the particle where it

impacted the substrate.

Increasing the particle impact velocity to 286 m/sec was investigated in the next model.

The particle diameter was also increased to 24.40 µm to agree with the experiment. Figure 5-

5 shows good agreement between both the particle and the substrate. During the initial stages

of modeling, the elastic modulus had to be modified at low strain rates to correlate the elastic

98

response of the model to the experiment. After changing to the Mie-Gruneisen EOS, the elastic

response no longer required “tuning” with respect to strain-rate changes.

Figure 5-5 Comparison between the experimental 286 m/sec impact and the PTW modeling result. Deformation appears to be quite similar. Experiments conducted by W. Xie and J. Lee at UMASS.

The next model further increased the particle impact velocity to 416 m/sec. Particle

diameter did not change significantly, but the model was modified to reflect the change.

Deformation in the finite element model is slightly different than the experimental results, but

the similarities are very close, as shown in Figure 5-6. Other models change the high-strain-

rate parameters for each condition to tune the deformation of the model to meet the

experimental results. The PTW model is a thermodynamics based model, and no changes are

required at any particle velocity.

99

Figure 5-6 Comparison between the experimental 416 m/sec impact and the PTW modeling result. The general deformation appears to be comparible. Note the bottom of the particle appears to be rounded due to viewing angle. Experiments conducted by W. Xie and J. Lee at UMASS.

The next model increased the impact velocity to 530 m/sec and decreased the particle diameter

slightly. As shown in Figure 5-7, there is good agreement between the model and experimental

results.

Figure 5-7 Comparison between the experimental 530 m/sec impact and the PTW modeling result. Deformation appears to be similar. Experiments conducted by W. Xie and J. Lee at UMASS.

100

The next model was developed to predict the deformation of a 19.75 µm Al6061

particle that impacted a sapphire substrate at 663 m/sec. This model was developed in the same

manner as the previous simulations, with the exception of changing the particle size and

velocity. Results of this modeling effort and an image of the deformed particle are shown in

Figure 5-8.

Figure 5-8 Comparison between experimental 663 m/sec impact and the PW modeling result. Both the experimental and finite element model show very similar deformation.

Experimental results conducted by W. Xie and J. Lee at UMASS.

Results of this model show a much greater amount of deformation compared to the first

few models. Non the less, the model still closely predicts the deformation. There is no evidence

of jetting in this particle, which is correlated in the model.

The last model that was developed for this initial set of experiments was for a 23.40

µm diameter Al6061 particle that had an impact speed of 699 m/sec into a sapphire substrate.

This model was developed in the same manner as the first two simulations, with the exception

of changing the particle size and velocity. Results of this final modeling effort and the

deformed particle are shown in Figure 5-9.

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Figure 5-9 Comparison between experimental 699 m/sec impact and the PW modeling result. Both the experimental and finite element model show similar deformation. Experimental results conducted by W. Xie and J. Lee at UMASS.

Results of this simulation effort show similar results to the previous two models. There

is some evidence that jetting appears on the edges of the particles. Jetting is not very evident

in the model, however, the material properties used in the model may not fully reflect those of

the particles. To compare the results of all the particle impacts in one figure, Figure 5-10 was

developed to show the outlines of each experimental particle with a PTW overlay.

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Figure 5-10 Particle outline overlays for both the experimental results (blue), and the PTW model (red). Note that deformation is very similar in each case.

As can be seen, there is relatively good agreement between the experimental results

and the finite element model. Any discrepancies are most likely due to the use of the available

material properties. It would be a worthwhile endeavor to develop PTW properties for the

Al6061 particles. It may be possible that the grain structure of the aluminum particles are

oriented in a certain direction as well, changing the overall yield strength of the alloy. This

effect has been shown by numerous authors over the years [57], [58]. This could change the

103

overall shape of the deformed particle. The current properties are for a wrought Al6061 sheet,

which may be significantly different than that of a particle. Another reason why the simulation

results may differ is through the elastic property definition in Abaqus. At high-strain-rates, the

Hooke’s law description of elastic properties breaks down due to a number of factors such as

having too short of a time scale for dislocation motion.

5.5. Al6061 on Al6061 Impact Results

The final research on the PTW and bonding parameter implementation culminated in the

development of a finite element model for Al6061 impacting Al6061. While there is almost no

published experimental data for cold spray impact of Al6061 on Al6061, experimental results

for this work was provided by Worcester Polytechnic Institute.

5.5.1. Model Setup

A model similar to what was developed for the initial PTW investigation was used for

the Al6061-Al6061 impact. The initial particle diameter was 18.9 µm. The model setup is

shown in Figure 5-11. Note that the particle in this model was not partitioned as much as in

the initial PTW implementation; it was realized that excessive partitioning did not change the

results. The substrate was fixed at the bottom to prevent runaway, similar to the substrate used

in Chapter 4. Substrate dimensions are shown in Table 5-5.

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Figure 5-11 Model setup for Al6061-Al6061 impact. Note, there is much less partitioning in the particle compared to the initial PTW model.

Table 5-4 Substrate dimensions used in Al6061-Al6061 impact model.

Length (mm) 0.1

Width (mm) 0.1

Thickness (mm) 0.05

105

5.5.2. Interactions and Boundary Conditions

This model follows the previous formats of fixing the bottom of the substrate in all

three directions. The only particle boundary condition to set was the initial velocity condition.

Particle impact velocity was set to 927 m/sec, to match experimental particle conditions In

terms of interaction parameters, the same values were specified for the normal and tangential

impact conditions. The only difference is that this model incorporates the fracture energy based

damage evolution model for bonding that was developed in Chapter 4.

5.5.3. Meshing

Meshing in this finite element model was conducted with C3D8R elements in a similar manner

to the initial PTW Al-sapphire model. The substrate was meshed with 32,000 elements and the

particle was meshed with 2,560 elements. The meshed assembly is shown in Figure 5-12.

Figure 5-12 Meshed Al6061-Al6061 impact model.

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5.5.4. Results and Discussion

The results of this modeling effort are shown in the next two figures. Figure 5-13 shows the

experimental results supplied by WPI. This is a secondary electron image of a focused ion

beam milled single particle splat sample. Notice that the particle is located on the right hand

side of Figure 5-13. The three numbered lines in the figure were used in sample analysis, but

were not required for the finite element analysis. Comparing these experimental results to the

finite element results in Figure 5-14, it is easy to see that there is very good correlation in the

deformation of the particle. However, there is not as much of an indent in the substrate as

would be expected. This may be due to some issues with the material parameters, but overall

it is not a bad prediction of the deformation of the particle. Due to rolling of the substrate, the

grain structure may have a preferred orientation. Orienting of the grains increases material

performance and also increases hardness, which may change the deformation behavior of the

substrate. It is also possible that the focused ion beam mill did not prepare the sample

perpendicular to the impact direction. This would account for the apparent increase in

deformation in the substrate in the experimental results. Material parameters used in this

analysis are Al6061-T6. This may not accurately represent the actual material properties of the

cold sprayed powder that was deposited. Currently, a large amount of work is being conducted

by various universities to measure the mechanical properties of powders. In terms of bonding,

the finite element model shows a good bond across the entire substrate, which correlates well

with the bond found in the experimental results.

107

Figure 5-13 Experimental results from WPI. Note the numbered lines were used in the analysis of the sample, but not needed for the finite element analysis.

Figure 5-14 Finite element result of Al6061-Al6061 impact and bonding model.

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5.6. Conclusions Two single particle impact models were developed for Al6061 particles impacting a

sapphire substrate and an Al6061 substrate. These models utilized the PTW VUHARD user

subroutine to define the high-strain-rate behavior of Al6061 during impact. Overall, the

Al6061-Sapphire impact model produced results that closely matched the experimental data.

There were a few minor differences in the deformation especially in the substrate. This may

be attributed to the Al6061 particles not having exactly the same properties as the PTW

properties for Al6061, as well as the strain-rate dependency in the elastic regime. The fracture

energy based bonding model is a significant improvement over the current state-of-the-art,

since most other cold spray finite element models do not include bonding criteria at all. Other

finite element models only show the particle impact until the maximum deformation of the

particle, and do not include any rebound effects. Overall, it has been shown that a bonding

model can be developed that utilizes fracture energy as a means to initiate de-bonding.

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6. Conclusions

6.1. Preston-Tonks-Wallace Subroutine Development

Two topics were studied and analyzed in this work to predict the deformation and bonding

of a cold spray particle. The first topic was the development and implementation of the

complex thermodynamics based Preston-Tonks-Wallace plasticity model using a VUHARD

user subroutine in Abaqus. The PTW model has not been fully implemented in commercial

software, and was used for the first time in this work to compare experimental single particle

impacts with success. The second topic was implementing the strain-energy based bonding

criteria. Implementing these models was not a straightforward process since the PTW model

is not built-in to commercial finite element codes, and no simple validation method exists for

either. Validation of the models was conducted using a few different methods.

6.1.1. Johnson-Cook Subroutine

A Johnson-Cook plasticity user subroutine was written following FORTRAN77

specifications and was compared against the built-in Johnson-Cook plasticity model in

Abaqus. This was done to verify how Abaqus interpreted and transferred data through a

subroutine. These results were compared to each other and was less than a 3% error between

built-in and subroutine. Once this comparison was completed, the Preston-Tonks-Wallace

mathematics were written in place of the Johnson-Cook subroutine and compared using an

axisymmetric model.

6.1.2. Axisymmetric PTW Validation

To ensure that the PTW mathematics were correct, the FORTRAN code was run at

different parameters externally from Abaqus to compare against experimental data published

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by Preston et.al. The FORTRAN mathematics plotted values of strain vs. strain-rate, and were

shown to follow the same trend and order of magnitude as the published data. The same

axisymmetric model that was used in the Johnson-Cook comparison was used in the Johnson-

Cook and PTW comparison. Below the estimated yield strength of the material, both the

Johnson-Cook and the PTW models predicted identical stresses, as expected. Above the yield

strength, the PTW code was activated and predicted a different stress distribution than the

Johnson-Cook. This change in stress distribution was expected due to the overall formulation

of the PTW compared to the Johnson-Cook model. The J-C model is a simple curve fit of high-

strain-rate data, and results are somewhat suspect when strain-rates in the model are different

than experimental result strain-rate due to interpolation of the data in a non-linear function.

Once the Preston-Tonks-Wallace model was validated, it was used to simulate the deformation

behavior of an Al6061 particle impacting a single crystal sapphire substrate. This subroutine

correlated well with experimental results without having to change any material parameters,

as found in other deformation models.

6.2. Strain-Energy Based Bonding Parameter

After the deformation modeling was completed, work began on developing a bonding

criterion for the cold spray process. It is known that there are multiple bonding phenomena

occurring during cold spray impact, and this bonding model attempted to address each

phenomena. The chosen bonding parameter has changed slightly in implementation since the

original hypothesis. Instead of a “strain-energy” based model, a “fracture-energy” based

damage evolution bonding model was developed that addressed the adiabatic shear instability

type bonding that is attributed to the cold spray process. This is a new concept for bonding that

is described for the first time in this work. Traction-separation bonding works by allowing

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perfect contact at the beginning of the impact, but during particle rebound, a “fracture-energy”

threshold value must be met for de-bonding to occur. The traction-separation still requires

more development, but due to the lack of experimental data available for a pure single particle

impact (i.e. not a simple impact analysis), some estimation was required. Bonding in cold spray

is not limited to adiabatic shear instability. For mechanical mixing and interlocking, which is

typically found in soft materials, nodal immersion between the particle and the substrate must

be allowed to occur. If this work were to be developed into an Eulerian based model, or

evaluated using smoothed particle hydrodynamics, a new type of mechanical mixing parameter

would need to be developed.

6.3. Combining PTW and Bonding to a Three Dimensional Model

Once both models were developed, they were applied to an impact of an Al6061 particle onto

an Al6061 substrate. It was found that there was very good correlation between the

experimental results and the finite element model of the deformation of the particle. There was

less elastic substrate deformation found in the finite element model than what was found in the

experimental results. There are a few explanations for this phenomenon. First, the properties

of the substrate were most likely not identical to the particle. The substrate was Al6061-T6,

which is harder than the particles, thus decreasing deformation. Second, the orientation of the

grains in the substrate can also affect the deformation behavior. The most common method for

producing aluminum sheets is rolling. Rolling orients the grain structure in a preferred

direction, which increases material performance. Another reason for the discrepancy is the grit

blasting of the substrate that is used to enhance bonding. This is not taken into account in the

model, and may change the results. These reasons may indicate why the substrate did not

deform the same. Finally, the way the sample was prepared could have affected the results. It

112

is possible that the particle did not hit normal to the surface or the sample was not focused ion

beam milled perpendicular to the angle of impact.

6.4. Issues and Shortcomings

There were a few issues that arose during this work that lead to an increase in model

complexity and error. The first issue was that the finite element model was originally intended

to be developed in LS-DYNA in conjunction with Abaqus. It was determined that LS-DYNA

would require a subroutine for PTW, similar to Abaqus, but could not support the proposed

bonding criteria. Although LS-DYNA is very good at dynamic modeling, these issues gave it

no advantage over Abaqus.

The second issue that arose during this work is that using a VUHARD subroutine in

Abaqus requires a manual elastic property re-definition. Previously, the Johnson-Cook

material model was used to model high-strain-rates, but this included a definition for the elastic

response. The Preston-Tonks-Wallace material model does not describe elasticity. Therefore,

a baseline elastic definition was required. At first, the standard Hookean description of

elasticity was used, but the elastic modulus required “tuning” at higher strain-rates to provide

accurate results. This was unsatisfactory since months of effort were spent on developing the

Preston-Tonks-Wallace model which was supposed to eliminate this issue. To remedy this

situation, a hydro-shock definition of elasticity was used, which came in the form of an

equation called the Mie-Gruneisen Equation of State. This equation of state relates material

density to pressure and predicts the shock wave propagation through what is called the

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Hugoniot. However, this increases the complexity of the model, but provides good results

without modifying parameters during runs.

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7. Future Work Some additional work required to better understand the bonding parameter and the

deformation of cold spray particle impact would be the following:

• Residual stress measurements at the particle/substrate interface

• Transmission electron microscopy at the particle/substrate interface to identify any

secondary phases or regions of localized melting

• Attempt to measure Preston-Tonks-Wallace (PTW) material parameters directly or

indirectly for cold spray particles

• Identify what PTW parameters are easiest to measure, and which can be estimated for

future material property investigations

• Refine meshing in all models, beyond one million elements to conduct a thorough

sensitivity analysis

• Conduct more experiments for the Al6061-Al6061 impact analysis to make better

deformation and bonding correlations

• This research also has a high potential to describe high velocity forging and explosive

forming better than other theories. More research in this area is needed.

115

References [1] S. Gu and S. Kamnis, “Bonding Mechanism from the Impact of Thermally Sprayed

Solid Particles,” Metall. Mater. Trans. A, vol. 40, no. 11, pp. 2664–2674, Aug. 2009.

[2] K. Kim, M. Watanabe, and S. Kuroda, “Bonding mechanisms of thermally softened metallic powder particles and substrates impacted at high velocity,” Surf. Coatings Technol., vol. 204, no. 14, pp. 2175–2180, Apr. 2010.

[3] T. Schmidt, H. Assadi, F. Gärtner, H. Richter, T. Stoltenhoff, H. Kreye, and T. Klassen, “From Particle Acceleration to Impact and Bonding in Cold Spraying,” J. Therm. Spray Technol., vol. 18, no. 5–6, pp. 794–808, Aug. 2009.

[4] B. Yildirim, A. Hulton, S. A. Alavian, T. Ando, A. Gouldstone, and S. Muftu, “On Simulation of Multi-Particle Impact Interactions in the Cold Spray Process,” in Proceedings of the ASME/STLE 2012 International Joint Tribology Conference, 2012, pp. 2–4.

[5] B. Yildirim, H. Fukanuma, T. Ando, A. Gouldstone, and S. Müftü, “A Numerical Investigation Into Cold Spray Bonding Processes,” J. Tribol., vol. 137, no. 1, p. 011102, Oct. 2014.

[6] A. Hulton, “Investigation of the effects of particle temperature and spacing on multi-particle impacts in cold spray,” Northeastern University, 2013.

[7] M. Grujicic, C. L. Zhao, W. S. DeRosset, and D. Helfritch, “Adiabatic Shear Instability Based Mechanism for Particles/Substrate Bonding in the Cold Gas Dynamic Spray Process,” Jan. 2004.

[8] D. J. Quesnel and D. S. Rimai, “Finite Element Modeling of Particle Adhesion: A Surface Energy Formalism,” J. Adhes., vol. 74, no. 1–4, pp. 177–194, Dec. 2000.

[9] D. Morin, B. Bourel, B. Bennani, F. Lauro, and D. Lesueur, “A new cohesive element for structural bonding modelling under dynamic loading,” Int. J. Impact Eng., vol. 53, pp. 94–105, Mar. 2013.

[10] P. C. King, G. Bae, S. H. Zahiri, M. Jahedi, and C. Lee, “An Experimental and Finite Element Study of Cold Spray Copper Impact onto Two Aluminum Substrates,” J. Therm. Spray Technol., vol. 19, no. 3, pp. 620–634, Dec. 2009.

[11] K. L. Johnson, K. Kendall, and A. D. Roberts, “Surface Energy and the Contact of Elastic Solids,” Proc. R. Soc. London, vol. 324, no. 1558, pp. 301–313, 1971.

[12] J. M. Schreiber, I. Smid, and T. J. Eden, “Fragmentation of a Steel Ring under Explosive Loading,” in 1 International Conference on 3D Materials Science, 2012, pp. 43–48.

[13] A. P. Alkhimov, A. N. Papyrin, V. F. Kosarev, N. I. Nesterovich, and M. M. Shushpanov, “Gas-dynamic spraying method for applying a coating,” US 5302414 A, 1994.

[14] P. L. Fauchais, J. V. R. Heberlein, and M. I. Boulos, Thermal Spray Fundamentals. Boston, MA: Springer US, 2014.

116

[15] R. Lupoi and W. O’Neill, “Deposition of metallic coatings on polymer surfaces using cold spray,” Surf. Coatings Technol., vol. 205, no. 7, pp. 2167–2173, Dec. 2010.

[16] N. Rushton, G. K. Schleyer, A. M. Clayton, and S. Thompson, “Internal explosive loading of steel pipes,” Thin-Walled Struct., vol. 46, no. 7–9, pp. 870–877, Jul. 2008.

[17] L. Campagne, L. Daridon, and S. Ahzi, “A physically based model for dynamic failure in ductile metals,” Mech. Mater., vol. 37, no. 8, pp. 869–886, Aug. 2005.

[18] G. V. Stepanov and A. V. Shirokov, “Modeling of crack propagation kinetics,” Strength Mater., vol. 42, no. 4, pp. 426–431, Aug. 2010.

[19] W. Altenhof, A. Raczy, M. Laframboise, J. Loscher, and A. Alpas, “Numerical simulation of AM50A magnesium alloy under large deformation,” Int. J. Impact Eng., vol. 30, no. 2, pp. 117–142, Feb. 2004.

[20] A. Mishra, M. Martin, N. N. Thadhani, B. K. Kad, E. A. Kenik, and M. A. Meyers, “High-strain-rate response of ultra-fine-grained copper,” Acta Mater., vol. 56, no. 12, pp. 2770–2783, Jul. 2008.

[21] G. R. Johnson and W. H. Cook, “A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures,” Proceedings of the 7th International Symposium on Ballistics, vol. 547. pp. 541–547, 1983.

[22] G. R. Johnson and W. H. Cook, “Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures,” Eng. Fract. Mech., vol. 21, no. 1, pp. 31–48, Jan. 1985.

[23] F. J. Zerilli and R. W. Armstrong, “Dislocation-mechanics-based constitutive relations for material dynamics calculations,” J. Appl. Phys., vol. 61, no. 5, p. 1816, Mar. 1987.

[24] F. J. Zerilli and R. W. Armstrong, “Constitutive relations for the plastic deformation of metals,” in AIP Conference Proceedings, 1994, vol. 309, no. 1, pp. 989–992.

[25] P. S. Follansbee and U. F. Kocks, “A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable,” Acta Metall., vol. 36, no. 1, pp. 81–93, Jan. 1988.

[26] D. M. Goto, R. K. Garrett, J. Bingert, S.-R. Chen, and G. T. Gray, “Mechanical Threshold Stress Constitutive Strength Model Description of HY-100 Steel,” 1999.

[27] D. L. Preston, D. L. Tonks, and D. C. Wallace, “Model of plastic deformation for extreme loading conditions,” J. Appl. Phys., vol. 93, no. 1, p. 211, Dec. 2003.

[28] B. Banerjee, “The Mechanical Threshold Stress model for various tempers of AISI 4340 steel,” Int. J. Solids Struct., vol. 44, no. 3–4, pp. 834–859, Feb. 2007.

[29] A. V. Bolesta, V. M. Fomin, M. R. Sharafutdinov, and B. P. Tolochko, “Investigation of interface boundary occurring during cold gas-dynamic spraying of metallic particles,” Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers, Detect. Assoc. Equip., vol. 470, no. 1–2, pp. 249–252, Sep. 2001.

[30] H. Assadi, F. Gärtner, T. Stoltenhoff, and H. Kreye, “Bonding mechanism in cold gas

117

spraying,” Acta Mater., vol. 51, no. 15, pp. 4379–4394, Sep. 2003.

[31] M. Grujicic, J. R. Saylor, D. E. Beasley, W. S. DeRosset, and D. Helfritch, “Computational analysis of the interfacial bonding between feed-powder particles and the substrate in the cold-gas dynamic-spray process,” Appl. Surf. Sci., vol. 219, no. 3–4, pp. 211–227, Dec. 2003.

[32] M. Grujicic, C. . Zhao, C. Tong, W. . DeRosset, and D. Helfritch, “Analysis of the impact velocity of powder particles in the cold-gas dynamic-spray process,” Mater. Sci. Eng. A, vol. 368, no. 1–2, pp. 222–230, Mar. 2004.

[33] A. Manes, L. Peroni, M. Scapin, and M. Giglio, “Analysis of strain rate behavior of an Al 6061 T6 alloy,” Procedia Eng., vol. 10, pp. 3477–3482, 2011.

[34] “Private Communication,” 2016.

[35] L. Burakovsky and D. L. Preston, “Analysis of dislocation mechanism for melting of elements,” Solid State Commun., vol. 115, no. 7, pp. 341–345, 2000.

[36] S. Rahmati and A. Ghaei, “The Use of Particle/Substrate Material Models in Simulation of Cold-Gas Dynamic-Spray Process,” J. Therm. Spray Technol., vol. 23, no. 3, pp. 530–540, Feb. 2014.

[37] M. Alfano, F. Furgiuele, A. Leonardi, C. Maletta, and G. H. Paulino, “RACTURE ANALYSIS OF ADHESIVE JOINTS USING INTRINSIC COHESIVE ZONE MODELS.”

[38] A. G. Tsamos, L. Margetts, and A. P. Jivkov, “IMPLEMENTATION OF A COHESIVE ZONE MODEL INTO THE OPEN SOURCE FINITE ELEMENT SOFTWARE ParaFEM.”

[39] T. Diehl, “Modeling Surface-Bonded Structures with ABAQUS Cohesive Elements: Beam-Type Solutions.”

[40] V. K. Champagne, The cold spray materials deposition process : fundamentals and applications. Woodhead, 2007.

[41] N. Bay, “COLD WELDING. PART 1: CHARACTERISTICS, BONDING MECHANISMS, BOND STRENGTH.” 1986.

[42] J. M. Schreiber, I. Smid, T. J. Eden, and D. Jann, “Prediction of fragmentation and experimentally inaccessible material properties of steel using finite element analysis,” Finite Elem. Anal. Des., vol. 104, pp. 72–79, Oct. 2015.

[43] X. Wang, F. Feng, M. A. Klecka, M. D. Mordasky, J. K. Garofano, T. El-Wardany, A. Nardi, and V. K. Champagne, “Characterization and modeling of the bonding process in cold spray additive manufacturing,” Addit. Manuf., vol. 8, pp. 149–162, Oct. 2015.

[44] T. Schmidt, H. Assadi, F. Gärtner, H. Richter, T. Stoltenhoff, H. Kreye, and T. Klassen, “From Particle Acceleration to Impact and Bonding in Cold Spraying,” J. Therm. Spray Technol., vol. 18, no. 5–6, pp. 794–808, Dec. 2009.

[45] “Abaqus 2016 Help,” in Abaqus Analysis User’s Guide, .

118

[46] M. Ridha, V. B. C. Tan, and T. E. Tay, “Traction–separation laws for progressive failure of bonded scarf repair of composite panel,” Compos. Struct., vol. 93, no. 4, pp. 1239–1245, 2011.

[47] Q. D. Yang and M. D. Thouless, “Mixed-mode fracture analyses of plastically-deforming adhesive joints,” Int. J. Fract., vol. 110, no. 2, pp. 175–187, 2001.

[48] T. Siegmund and W. Brocks, “Prediction of the Work of Separation and Implications to Modeling,” Int. J. Fract., vol. 99, no. 1/2, pp. 97–116, 1999.

[49] “MatWeb Material Property Data.” [Online]. Available: www.matweb.com.

[50] H. Tan, “Flow Theory of Plasticity – Handout 4,” 2009.

[51] S. Atlati, B. Haddag, M. Nouari, and M. Zenasni, “Thermomechanical modelling of the tool–workmaterial interface in machining and its implementation using the ABAQUS VUINTER subroutine,” Int. J. Mech. Sci., vol. 87, pp. 102–117, Oct. 2014.

[52] N. University, “Unpublished Work,” 2016.

[53] ©. M. Ragheb, “FLUID MECHANICS, EULER AND BERNOULLI EQUATIONS,” 2013.

[54] B. Militzer, “Shock Hugoniot.” [Online]. Available: https://legacy.gl.ciw.edu/static/users/bmilitzer/diss/node53.html.

[55] “Equation of State,” in Abaqus Analysis User’s Guide, .

[56] “Simulation of the ballistic perforation of aluminum plates with Abaqus/Explicit.”

[57] R. R. Cervay, “MECHANICAL PROPERTY EVALUATION OF ALUMINUM ALLOY 7010-T73651 Interim Technical Report for Period,” 1980.

[58] ASM International, Aluminum and Aluminum Alloys, 6th ed. 1993.

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Appendix A: FE Model Inputs Traction-Separation Model Testing Input *Heading ** Job name: TS_1e9_D_100e4_DE_0_01 Model name: Model-1 ** Generated by: Abaqus/CAE 2016 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=One_Meter_Block *Node *Element, type=C3D8R *Nset, nset=Set-1, generate 1, 39304, 1 *Elset, elset=Set-1, generate 1, 35937, 1 ** Section: Steel *Solid Section, elset=Set-1, material=Steel , *End Part ** *Part, name=Ten_Meter_Block *Node *Element, type=C3D8R *Nset, nset=Set-1, generate 1, 152561, 1 *Elset, elset=Set-1, generate 1, 144000, 1 ** Section: Steel *Solid Section, elset=Set-1, material=Steel , *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=One_Meter_Block-1, part=One_Meter_Block 0., 1., -0.5 *End Instance ** *Instance, name=Ten_Meter_Block-1, part=Ten_Meter_Block 0., -0.5, -1.5 *End Instance ** *Nset, nset=Set-1, instance=Ten_Meter_Block-1, generate

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1, 152521, 41 *Elset, elset=Set-1, instance=Ten_Meter_Block-1, generate 1, 143961, 40 *Nset, nset=Set-2, instance=One_Meter_Block-1, generate 1, 39304, 1 *Elset, elset=Set-2, instance=One_Meter_Block-1, generate 1, 35937, 1 *Elset, elset=_m_Surf-1_S6, internal, instance=One_Meter_Block-1, generate 1, 35905, 33 *Surface, type=ELEMENT, name=m_Surf-1 _m_Surf-1_S6, S6 *Elset, elset=_m_Surf-3_S4, internal, instance=Ten_Meter_Block-1, generate 40, 144000, 40 *Surface, type=ELEMENT, name=m_Surf-3 _m_Surf-3_S4, S4 *Elset, elset=_s_Surf-1_S4, internal, instance=Ten_Meter_Block-1, generate 40, 144000, 40 *Surface, type=ELEMENT, name=s_Surf-1 _s_Surf-1_S4, S4 *Elset, elset=_s_Surf-3_S6, internal, instance=One_Meter_Block-1, generate 1, 35905, 33 *Surface, type=ELEMENT, name=s_Surf-3 _s_Surf-3_S6, S6 *End Assembly ** ** MATERIALS ** *Material, name=Steel *Density 7810., *Elastic 2e+11, 0.33 *Plastic, hardening=JOHNSON COOK 7.92e+08, 5.1e+08, 0.26, 1.03, 1793., 255. *Rate Dependent, type=JOHNSON COOK 0.014,1. ** ** INTERACTION PROPERTIES ** *Surface Interaction, name=Global_Contact_Def *Surface Interaction, name=Traction_Sep *Cohesive Behavior, REPEATED CONTACTS 1e+09, 1e+09, 1e+09 *Damage Initiation, criterion=MAXS 1e+06, 1e+06, 1e+06 *Damage Evolution, type=DISPLACEMENT 0.01, ** ** PREDEFINED FIELDS ** ** Name: Small_Block_Velocity Type: Velocity *Initial Conditions, type=VELOCITY

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Set-2, 1, 0. Set-2, 2, -400. Set-2, 3, 0. ** ---------------------------------------------------------------- ** ** STEP: Impact ** *Step, name=Impact, nlgeom=YES *Dynamic, Explicit , 0.01 *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: Fixed_Bottom_Block Type: Symmetry/Antisymmetry/Encastre *Boundary Set-1, PINNED ** ** INTERACTIONS ** ** Interaction: Contact *Contact, op=NEW *Contact Inclusions, ALL EXTERIOR *Contact Property Assignment , , Global_Contact_Def m_Surf-1 , s_Surf-1 , Traction_Sep ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT, time interval=0.0001 ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

PTW Deformation Model Input *Heading ** Job name: PTW_530_compare Model name: Model-1 ** Generated by: Abaqus/CAE 6.14-2 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Particle_2440um_286msec *Node

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*Nset, nset=Set-1, generate 1, 2937, 1 *Elset, elset=Set-1, generate 1, 2560, 1 ** Section: Particle *Solid Section, elset=Set-1, material=Aluminum , *End Part ** *Part, name=Substrate_mm *Node *Element, type=C3D8R *Nset, nset=Set-1, generate 1, 35301, 1 *Elset, elset=Set-1, generate 1, 32000, 1 ** Section: Substrate *Solid Section, elset=Set-1, material=Sapphire , *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Substrate_mm-1, part=Substrate_mm -0.035, 0., -0.05 *End Instance ** *Instance, name=Particle_2440um_286msec-1, part=Particle_2440um_286msec 0., -0.1278, 0. *End Instance ** *Nset, nset=Set-1, instance=Substrate_mm-1, generate 1, 35281, 21 *Elset, elset=Set-1, instance=Substrate_mm-1, generate 1, 31981, 20 *Nset, nset=Set-5, instance=Substrate_mm-1, generate 1, 35301, 1 *Elset, elset=Set-5, instance=Substrate_mm-1, generate 1, 32000, 1 *Nset, nset=Set-13, instance=Particle_2440um_286msec-1, generate 1, 2937, 1 *Elset, elset=Set-13, instance=Particle_2440um_286msec-1, generate 1, 2560, 1 *Elset, elset=_m_Surf-1_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-1

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_m_Surf-1_S4, S4 *Elset, elset=_m_Surf-3_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-3 _m_Surf-3_S4, S4 *Elset, elset=_m_Surf-5_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-5 _m_Surf-5_S4, S4 *Elset, elset=_m_Surf-7_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-7 _m_Surf-7_S4, S4 *Elset, elset=_m_Surf-25_S3, internal, instance=Particle_2440um_286msec-1 *Elset, elset=_m_Surf-25_S4, internal, instance=Particle_2440um_286msec-1 *Elset, elset=_m_Surf-25_S2, internal, instance=Particle_2440um_286msec-1 *Surface, type=ELEMENT, name=m_Surf-25 _m_Surf-25_S3, S3 _m_Surf-25_S4, S4 _m_Surf-25_S2, S2 *Elset, elset=_m_Surf-26_S2, internal, instance=Particle_2440um_286msec-1 *Elset, elset=_m_Surf-26_S3, internal, instance=Particle_2440um_286msec-1 *Elset, elset=_m_Surf-26_S4, internal, instance=Particle_2440um_286msec-1, generate 1162, 1220, 2 *Surface, type=ELEMENT, name=m_Surf-26 _m_Surf-26_S2, S2 _m_Surf-26_S3, S3 _m_Surf-26_S4, S4 *Elset, elset=_m_Surf-30_S2, internal, instance=Particle_2440um_286msec-1 *Elset, elset=_m_Surf-30_S3, internal, instance=Particle_2440um_286msec-1 *Elset, elset=_m_Surf-30_S4, internal, instance=Particle_2440um_286msec-1, generate 1162, 1220, 2 *Surface, type=ELEMENT, name=m_Surf-30 _m_Surf-30_S2, S2 _m_Surf-30_S3, S3 _m_Surf-30_S4, S4 *Elset, elset=_s_Surf-9_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-9 _s_Surf-9_S4, S4 *Elset, elset=_s_Surf-11_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-11 _s_Surf-11_S4, S4 *Elset, elset=_s_Surf-13_S4, internal, instance=Substrate_mm-1, generate

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20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-13 _s_Surf-13_S4, S4 *Elset, elset=_s_Surf-15_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-15 _s_Surf-15_S4, S4 *Elset, elset=_s_Surf-17_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-17 _s_Surf-17_S4, S4 *Elset, elset=_s_Surf-19_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-19 _s_Surf-19_S4, S4 *Elset, elset=_s_Surf-21_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-21 _s_Surf-21_S4, S4 *Elset, elset=_s_Surf-23_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-23 _s_Surf-23_S4, S4 *Elset, elset=_s_Surf-28_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-28 _s_Surf-28_S4, S4 *End Assembly ** ** MATERIALS ** *Material, name=Aluminum *Density 2.7e-09, *Depvar, delete=3 19, *Elastic 6.8e+10, 0.3 *Plastic, hardening=USER, properties=2 0.5, 0.5 *Material, name=Sapphire *Density 3.98e-09, *Elastic 3.45e+07, 0.29 ** ** INTERACTION PROPERTIES ** *Surface Interaction, name=Contact *Friction 1., *Surface Behavior, no separation, pressure-overclosure=HARD

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** ** PREDEFINED FIELDS ** ** Name: PArticle_Vel Type: Velocity *Initial Conditions, type=VELOCITY Set-13, 1, 0. Set-13, 2, -530000. Set-13, 3, 0. ** ---------------------------------------------------------------- ** ** STEP: Impact ** *Step, name=Impact, nlgeom=YES *Dynamic, Explicit , 6e-07 *Bulk Viscosity 0.06, 1.2 ** Mass Scaling: Semi-Automatic ** Whole Model *Fixed Mass Scaling, dt=1e-12, type=below min, factor=2. ** ** BOUNDARY CONDITIONS ** ** Name: Fixed_Sub Type: Displacement/Rotation *Boundary Set-1, 1, 1 Set-1, 2, 2 Set-1, 3, 3 ** ** INTERACTIONS ** ** Interaction: Int-1 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY, cpset=Int-1 m_Surf-30, s_Surf-23 ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT, time interval=1e-09 ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

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Al6061-Al6061 Bonding Input *Heading ** Job name: PTW_18_9um_927_msec Model name: Model-1 ** Generated by: Abaqus/CAE 6.14-2 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Particle_18_9UM *Node *Element, type=C3D8R *Nset, nset=Set-1, generate 1, 2937, 1 *Elset, elset=Set-1, generate 1, 2560, 1 ** Section: Particle *Solid Section, elset=Set-1, material=Aluminum , *End Part ** *Part, name=Substrate_mm *Node *Element, type=C3D8R *Nset, nset=Set-1, generate 1, 35301, 1 *Elset, elset=Set-1, generate 1, 32000, 1 ** Section: Particle *Solid Section, elset=Set-1, material=Aluminum , *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Substrate_mm-1, part=Substrate_mm -0.035, 0., -0.05 *End Instance ** *Instance, name=Particle_18_9UM-1, part=Particle_18_9UM 0., -0.11055, 0. *End Instance ** *Nset, nset=Set-1, instance=Substrate_mm-1, generate 1, 35281, 21 *Elset, elset=Set-1, instance=Substrate_mm-1, generate 1, 31981, 20 *Nset, nset=Set-5, instance=Substrate_mm-1, generate 1, 35301, 1 *Elset, elset=Set-5, instance=Substrate_mm-1, generate 1, 32000, 1

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*Nset, nset=Set-14, instance=Particle_18_9UM-1, generate 1, 2937, 1 *Elset, elset=Set-14, instance=Particle_18_9UM-1, generate 1, 2560, 1 *Nset, nset=Set-15, instance=Particle_18_9UM-1, generate 1, 2937, 1 *Elset, elset=Set-15, instance=Particle_18_9UM-1, generate 1, 2560, 1 *Nset, nset=Set-16, instance=Substrate_mm-1, generate 1, 35301, 1 *Elset, elset=Set-16, instance=Substrate_mm-1, generate 1, 32000, 1 *Elset, elset=_m_Surf-1_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-1 _m_Surf-1_S4, S4 *Elset, elset=_m_Surf-3_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-3 _m_Surf-3_S4, S4 *Elset, elset=_m_Surf-5_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-5 _m_Surf-5_S4, S4 *Elset, elset=_m_Surf-7_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=m_Surf-7 _m_Surf-7_S4, S4 *Elset, elset=_m_Surf-27_S3, internal, instance=Particle_18_9UM-1 *Elset, elset=_m_Surf-27_S4, internal, instance=Particle_18_9UM-1 *Elset, elset=_m_Surf-27_S2, internal, instance=Particle_18_9UM-1 3 *Surface, type=ELEMENT, name=m_Surf-27 _m_Surf-27_S3, S3 _m_Surf-27_S4, S4 _m_Surf-27_S2, S2 *Elset, elset=_s_Surf-9_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-9 _s_Surf-9_S4, S4 *Elset, elset=_s_Surf-11_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-11 _s_Surf-11_S4, S4 *Elset, elset=_s_Surf-13_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-13 _s_Surf-13_S4, S4 *Elset, elset=_s_Surf-15_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-15 _s_Surf-15_S4, S4 *Elset, elset=_s_Surf-17_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20

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*Surface, type=ELEMENT, name=s_Surf-17 _s_Surf-17_S4, S4 *Elset, elset=_s_Surf-19_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-19 _s_Surf-19_S4, S4 *Elset, elset=_s_Surf-21_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-21 _s_Surf-21_S4, S4 *Elset, elset=_s_Surf-23_S4, internal, instance=Substrate_mm-1, generate 20, 32000, 20 *Surface, type=ELEMENT, name=s_Surf-23 _s_Surf-23_S4, S4 *End Assembly ** ** MATERIALS ** *Material, name=Aluminum *Conductivity 0.167, *Density 2.7e-09, *Depvar, delete=3 19, *Elastic 1e+08, 0.3 *Inelastic Heat Fraction 0.9, *Plastic, hardening=USER, properties=2 0.5, 0.5 *Specific Heat 0.9, *Material, name=Sapphire *Density 3.98e-09, *Elastic 3.45e+07, 0.29 ** ** INTERACTION PROPERTIES ** *Surface Interaction, name=Contact *Friction 1., *Surface Behavior, no separation, pressure-overclosure=HARD ** ** PHYSICAL CONSTANTS ** *Physical Constants, absolute zero=0. ** ** PREDEFINED FIELDS ** ** Name: PArt_Vel Type: Velocity *Initial Conditions, type=VELOCITY Set-14, 1, 0. Set-14, 2, -927000.

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Set-14, 3, 0. ** Name: Part_Temp Type: Temperature *Initial Conditions, type=TEMPERATURE Set-15, 300. ** Name: Sub_Temp Type: Temperature *Initial Conditions, type=TEMPERATURE Set-16, 300. ** ---------------------------------------------------------------- ** ** STEP: Impact ** *Step, name=Impact, nlgeom=YES *Dynamic, Explicit , 6e-07 *Bulk Viscosity 0.06, 1.2 ** Mass Scaling: Semi-Automatic ** Whole Model *Fixed Mass Scaling, dt=1e-12, type=below min, factor=2. ** ** BOUNDARY CONDITIONS ** ** Name: Fixed_Sub Type: Displacement/Rotation *Boundary Set-1, 1, 1 Set-1, 2, 2 Set-1, 3, 3 ** ** INTERACTIONS ** ** Interaction: Int-1 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY, cpset=Int-1 m_Surf-27, s_Surf-23 ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, time interval=1e-09 *Node Output A, RF, U, V *Element Output, directions=YES EVF, LE, PE, PEEQ, PEEQVAVG, PEVAVG, S, SVAVG, TEMP, TEMPMAVG *Contact Output CSTRESS, ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

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Appendix B Johnson-Cook VUMAT Subroutine subroutine vumat( C Read only - 1 nblock, ndir, nshr, nstatev, nfieldv, nprops, lanneal, 2 stepTime, totalTime, dt, cmname, coordMp, charLength, 3 props, density, strainInc, relSpinInc, 4 tempOld, stretchOld, defgradOld, fieldOld, 5 stressOld, stateOld, enerInternOld, enerInelasOld, 6 tempNew, stretchNew, defgradNew, fieldNew, C Write only - 7 stressNew, stateNew, enerInternNew, enerInelasNew ) C include 'vaba_param.inc' C C All arrays dimensioned by (*) are not used in this algorithm dimension props(nprops), density(nblock), coordMp(nblock,*), 1 charLength(nblock), strainInc(nblock,ndir+nshr), 2 relSpinInc(nblock,nshr), tempOld(nblock), 3 stretchOld(nblock,ndir+nshr), defgradOld(nblock,ndir+nshr), 4 fieldOld(nblock,nfieldv), stressOld(nblock,ndir+nshr), 5 stateOld(nblock,nstatev), enerInternOld(nblock), 6 enerInelasOld(nblock), tempNew(nblock), 7 stretchNew(nblock,ndir+nshr), defgradNew(nblock,ndir+nshr), 8 fieldNew(nblock,nfieldv),stressNew(nblock,ndir+nshr), 9 stateNew(nblock,nstatev),enerInternNew(nblock), 1 enerInelasNew(nblock) C character*80 cmname C C C Parameter that may be used parameter( zero = 0., one = 1., two = 2., three = 3., 1 third = one/three, half = 0.5, threeHalfs = 1.5 ) C The state variables are stored as: C STATE(*,1) = equivalent plastic strain C STATE(*,2) = plastic strain rate C C user-defined material C C props(1) Young's modulus C props(2) Poisson's ratio C props(3) A C props(4) B

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C props(5) n C props(6) C C props(7) m C props(8) melting temperature C props(9) reference temperature C C C if state allows JC model applied to two different material C C e = props(1) xnu = props(2) A = props(3) B = props(4) EN = props(5) C = props(6) EM = props(7) C C constants and modulus C twomu = e / ( one + xnu ) thremu = threeHalfs * twomu sixmu = three * twomu alamda = twomu * ( e - twomu ) / ( sixmu - two * e ) if(cmname=='AL6061_T6') then do i = 1, nblock C C calculate Johnson-Cook yield stress C C plastic strain peeqOld=stateOld(i,1) C if(peeqOld .eq. zero) then yieldOld = A hard=0 else hard=EN*B*peeqOld**(EN-1) yieldOld = A+B*peeqOld**EN end if C C Trial stress trace = strainInc(i,1) + strainInc(i,2) + strainInc(i,3) s11 = stressOld(i,1) + alamda*trace + twomu*strainInc(i,1)

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s22 = stressOld(i,2) + alamda*trace + twomu*strainInc(i,2) s33 = stressOld(i,3) + alamda*trace + twomu*strainInc(i,3) s12 = stressOld(i,4) + twomu*strainInc(i,4) C C C Deviatoric part of trial stress measured from the back stress smean = third * ( s11 + s22 + s33 ) s11 = s11 - smean s22 = s22 - smean s33 = s33 - smean C C Magnitude of the deviatoric trial stress difference vmises = sqrt(threeHalfs*( s11**2 + s22**2 + s33**2 + two*s12**2)) C C if(stateOld(i,2) .eq. zero) then tvp=one hard1=hard else tvp=one+C*log(stateOld(i,2)/1000) hard1=hard*tvp+yieldOld*1000*C/stateOld(i,2) end if yieldOld=yieldOld*tvp C C C Check for yield by determining the factor for plasticity, C zero for elastic, one for yield sigdif=vmises-yieldOld facyld = zero if( sigdif .gt. zero ) facyld = one deqps = facyld * sigdif / (thremu + hard1) C C Update the state variable StateNew(i,1)=stateOld(i,1)+deqps StateNew(i,2)=deqps/dt C C Update back stress yieldNew = yieldOld + hard1 * deqps factor = yieldNew / ( yieldNew + thremu * deqps ) stressNew(i,1) = s11 * factor + smean stressNew(i,2) = s22 * factor + smean stressNew(i,3) = s33 * factor + smean stressNew(i,4) = s12 * factor

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C C C Update the specific internal energy - stressPower = half * ( 1 ( stressOld(i,1)+stressNew(i,1) )*strainInc(i,1) 1 + ( stressOld(i,2)+stressNew(i,2) )*strainInc(i,2) 1 + ( stressOld(i,3)+stressNew(i,3) )*strainInc(i,3) 1 + two*( stressOld(i,4)+stressNew(i,4) )*strainInc(i,4) ) C enerInternNew(i) = enerInternOld(i) 1 + stressPower / density(i) C C Update the dissipated inelastic specific energy - plasticWorkInc = yieldNew * deqps enerInelasNew(i) = enerInelasOld(i) 1 + plasticWorkInc / density(i) C end do end if C return end

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Appendix C Bonding Results Traction Separation Results at Failure (von Mises Stress)

C- 1 von Mises stress output for a traction-separation parameter of 1.


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136

C- 5 von Mises stress output for a traction-separation parameter of 10,000.

C- 6 von Mises stress output for a traction-separation parameter of 100,000.

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C- 7 von Mises stress output for a traction-separation parameter of 1x106.


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139



140

Traction Separation Results at Failure (Equivalent Plastic Strain, PEEQ)

C- 13 PEEQ output for a traction-separation parameter of 1.


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142

C- 17 PEEQ output for a traction-separation parameter of 10,000.

C- 18 PEEQ output for a traction-separation parameter of 100,000.

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C- 19 PEEQ output for a traction-separation parameter of 1x106.


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145



146

Damage Parameter Investigation (von Mises stress)

C- 25 von Mises stress output for a damage parameter of 1x104.


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148



149



150

Maximum Displacement Based Damage Evolution Results (von Mises Stress)

C- 33 von Mises stress output for a maximum displacement value of 0.01.


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152

C- 37 von Mises stress output for a maximum displacement value of 1.

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Fracture Energy Based Results for Damage Model (von Mises stress)

C- 38 von Mises stress output for a fracture energy value of 1.


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155

C- 42 von Mises stress output for a fracture energy value of 10,000.

C- 43 von Mises stress output for a fracture energy value of 100,000.

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C- 44 von Mises stress output for a fracture energy value of 1x106.


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Vita – Jeremy Schreiber

PUBLICATIONS

Schreiber, J. M., Smid, I., Eden, T. J., & Jann, D. (2015). Prediction of fragmentation and experimentally inaccessible material properties of steel using finite element analysis. Finite Elements in Analysis and Design, 104, 72–79. doi:10.1016/j.finel.2015.06.001

Schreiber, J. M., Omcikus, Z. R., Eden, T. J., Sharma, M. M., Champagne, V., & Patankar, S. N. (2014). Combined effect of hot extrusion and heat treatment on the mechanical behavior of 7055 AA processed via spray metal forming. Journal of Alloys and Compounds, 617, 135–139. doi:10.1016/j.jallcom.2014.07.184

J. Schreiber, I. Smid, T. Eden (2015). A Finite Element Based Approach to Deformation of Encapsulated Particles in Cold Spray. Materials Science and Technology (MS&T 2015 Columbus, OH)

A. Wiltner, B. Kloden, F. Dittrich, Th. Weissgarber, B. Kieback, J.M. Schreiber, T.J. Eden, J.K. Potter, I. Smid, “Aluminum Deposition onto molybdenum and tungsten applying cold spray technology: study of morphology, cold spray parameters, diffusion kinetics and alloy formation,” 18th Plansee Seminar 2013.

J. Schreiber, I. Smid, and T. Eden, “Determination of High Strain Rate Behavior of Steel Using Finite Element Analysis and High Strain Rate Experimentation,” Proceedings of TMS 2013, TMS (The Minerals, Metals, and Materials Society, March 2013.

J. Schreiber, I. Smid, and T. Eden, “Fragmentation of a Steel Ring Under Explosive Loading,” Proceedings of the 1st International Conference on 3D Materials Science, TMS (The Minerals, Metals, and Materials Society, July 2012.

PRESENTATIONS/POSTERS

Presentation: High-Strain-Rate Tensile Property Determination Using Finite Element Analysis and Experimental Methods. MS&T 2012 August 2012 Pittsburgh, PA

Presentation: High-Strain-Rate Property Determination using Finite Element Analysis and Split-Hopkinson MS&T 2013 Montreal, Quebec, Canada

Presentation: High-Strain-Rate Property Determination of High-Strength Steel Using Finite Element Analysis and Experimental Data TMS 2013 San Antonio, TX

Presentation: J. Schreiber, I. Smid, T. Eden, “Coupling Experimental and Numerical Methods to Predict Fragmentation Under Extremely High Strain Rates”. MS&T 2014 Pittsburgh Pennsylvania. October 2014

Presentation: Modeling of Operational Parameters for Cold Spray Deposition. Jeremy Schreiber, Ivi Smid, Timothy Eden TMS 2015 Orlando, FL

Presentation: T. Eden, J. Schreiber, “Finite Element Analysis of Cold Spray Particle Impact”. Cold Spray Action Team Meeting, Worcester Massachusetts, June 2015

Presentation: J. Schreiber, I. Smid, T. Eden, “A Finite Element Based Approach to Deformation of Encapsulated Particles in Cold Spray”. MS&T 2015 Columbus Ohio, October 2015

Poster: J. Schreiber, T. Eden, I. Smid, M. Sharma. “Finite Element Modeling of Hard Particle Impact During Cold Spray Deposition.” North American Cold Spray Conference. Montreal, Quebec, Canada 2014

Poster: J. Schreiber, I. Smid, T. Eden, “Characterization and Constitutive Material Model Implementation for High-strain-rate Deformation Modeling with Finite Elements”. TMS 2014 San Diego California, March 2014.

Documents

FINITE ELEMENT IMPLEMENTATION OF THE PRESTON-TONKS …