Science-Based Multiscale Modeling of Fatigue Damage for ... · American Institute of Aeronautics and Astronautics 1 Science-Based Multiscale Modeling of Fatigue Damage for Structural

American Institute of Aeronautics and Astronautics

1

Science-Based Multiscale Modeling of Fatigue Damage for Structural Prognosis

Elias L. Anagnostou1, Stephen J. Engel2, Daniel Fridline3, David H. Hoitsma4, John S. Madsen5, Jerrell A. Nardiello6, and John M. Papazian7

Northrop Grumman Corporation, Aerospace Systems, Bethpage, N.Y. 11714

Probabilistic microstructurally-based models for fatigue are being developed as part of a DARPA/Northrop Grumman Structural Integrity Prognosis System (SIPS). An integrated framework has been developed to provide probabilistic fatigue life prognosis of individual air vehicles by leveraging research across a large multi-disciplinary team. Development of the framework was made possible by advancements made by the SIPS Modeling and Simulation multidisciplinary research team in the statistical modeling of polycrystalline metal microstructures, finite element meshing, multiscale analysis, modeling of constitutive behavior of polycrystals, and materials testing technology for property measurement and validation procedures. The fatigue models are based on a fundamental understanding of the fatigue process, and trace the structural degradation caused by fatigue back to its physical origins in the microstructure of the metallic component. The objectives are to discover and link all the important damage mechanisms leading to a macroscopically observable crack and to allow defects to emerge naturally from statistically meaningful ensembles of material representations, subject to more accurate, scale-specific, damage inducing fields. An example is modeling the interplay between the grain crystallography and crack incubation at a constituent particle. A microstructurally-based model allows estimation of the total fatigue life from incubation and nucleation at a constituent second phase particle to propagation of micro-cracks to emergence of macro-cracks all in a statistical sense to permit accurate estimation of reliability indices.

Dedication

This paper is dedicated in memory of Dr. John M. Papazian, principal investigator on the SIPS program, our technical leader and friend, who worked tirelessly to bring his vision of the program to fruition. John’s wisdom,

leadership and love of life have been a true inspiration.

Nomenclature N = cycles ΔK = stress intensity factor range ΔKeff = effective stress intensity factor range a = half-surface crack along thickness c = crack along width R = stress ratio (Smin/Smax) Smax = maximum applied gross stress Smin = minimum applied gross stress

1 Associate Technical Fellow, Advanced Programs & Technology, 925 S. Oyster Bay Rd., Mail Stop U01-26 2 Associate Technical Fellow, Advanced Programs & Technology, 925 S. Oyster Bay Rd., Mail Stop U01-26 3 Research Engineer, Advanced Programs & Technology, 925 S. Oyster Bay Rd., Mail Stop U01-26 4 Research Engineer, Technical Services, 925 S. Oyster Bay Rd., Mail Stop U01-26 5 Associate Technical Fellow, Advanced Programs & Technology, 925 S. Oyster Bay Rd., Mail Stop U01-26 6 Principal Engineer, Advanced Programs & Technology, 925 S. Oyster Bay Rd., Mail Stop U01-26 7 Technical Fellow, Advanced Programs & Technology, 925 S. Oyster Bay Rd., Mail Stop U01-26

Approved for Public Release, Distribution Unlimited: DISTAR Case 14849 Dated 20 January 2010

51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 18th12 - 15 April 2010, Orlando, Florida

AIAA 2010-2971

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc.The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.All other rights are reserved by the copyright owner.


2

β = nonlocal micronotch-root maximum plastic shear strain amplitude γ = shear strain α = ductility exponent in modified microscale Coffin-Manson relation Cnucl = ductility coefficient in modified microscale Coffin-Manson relation Δ = increment of a quantity G = linear proportional constant between crack opening displacement and crack growth rate KIC = fracture toughness ap = constituent particle size af = inherent flaw size C = proportional constant between af and ap σcr = constituent particle strength σp = constituent particle tensile stress Kt = stress concentration factor

I. Introduction urrent fatigue life models are empirical and deterministic. They represent statistically and geometrically average behaviors at the continuum scale, thereby resulting in simple surrogates of smaller length scale

mechanisms. Advantages of current empirical methods are that they are mathematically and computationally less exacting, i.e. they require fewer measurements of properties and relationships. Current methods work well when these properties and relationships are well known empirically. An example is using Linear Elastic Fracture Mechanics (LEFM) to predict fatigue crack growth through a power law relationship. Disadvantages are that they are subject to the effects of size, shape, rate, aging, etc. thereby limiting their applicability and resulting in decay of accuracy. They do not account for the important physical/chemical events acting on the system being modeled. Most importantly, current methods mask inherent variability because they represent average behaviors at the continuum scale. However, it has been shown that fatigue life is strongly influenced by the tail ends of the defect distributions which are often related to the microstructural-feature distributions, such as constituent particle size, crystallography, grain boundaries, grain size and shape distributions.

A microstructurally-based probabilistic framework has been developed as part of the SIPS1 program for more accurate predictions of the structural viability of individual air vehicles. The SIPS Modeling and Simulation (M&S) research team comprises a consortium of leading universities and high-tech companies whose primary objective has been to develop science-based modeling and simulation to faithfully capture details of material microstructure and degradation process due to fatigue. The main objectives of the M&S team have been to develop realistic physics of failure models that will estimate the total fatigue life of a structure and address the stochastic and probabilistic effects in materials utilizing knowledge of the uncertain nature of the material’s microstructure. Rules for modeling damage accumulation and evolution in materials were developed for each critical damage mechanism. Critical damage mechanisms were established through experimental observations, and rules have been ascertained by exploiting experimental truth data in conjunction with micromechanical simulations of the actual observations at the various length scales/damage mechanisms. The final goal is to replace macroscale, empirical, deterministic based fatigue models with mechanistic, microstructural, probabilistic based models for more accurate assessment of reliability indices. Figure 1 shows the overall M&S group integration and Table 1 summarizes team member’s primary roles.

C

Cornell

Lehigh

ALCOA

RPI

CMU

MS State

OSU

UVa

NorthropGrumman

CorrosionScience

MechanicalProperties

ExperimentalCharacterization

Atoms StructuralScale

VEXTEC

Figure 1. Modeling and Simulation group integration.


3

The M&S approach has been to: 1) Develop a digital material description that will provide statistically accurate 3-D representations of the

material’s real microstructure. 2) Develop physics-based models for fatigue for adaptive real-time prognosis. 3) Develop a crystal elastic-plastic material model that will accurately capture the heterogeneous stress and

strain states at the grain scale. 4) Develop a multiscale fatigue modeling environment to explicitly capture damage evolution at the grain

scale. 5) Develop microstructurally-based corrosion and corrosion fatigue models. Experimental observations of fatigued aluminum alloy 7075-T651 notched specimens have shown that in a high

stress area such as a notch or hole, fatigue cracks initiate and nucleate into the matrix from cracked constituent particles. Not all constituent particles crack in the high stress area of a notch or hole. The process that leads to the event of crack nucleation has been defined as the incubation stage of fatigue. It has also been observed that not all cracked particles spawn a crack in the matrix. The nucleation stage is thereby defined as the appearance of new surface area in the matrix adjacent to a cracked constituent particle. Nucleation is followed by growth of small cracks which tend to be sensitive to their surroundings. It has been observed that most small cracks are crystallographic i.e., they follow a crystallographic orientation relative to the loading axis. Multiple cracked particles cause multiple microstructurally small cracks. Some arrest, some grow then link and form the dominant crack that leads to failure. Based on experimental observations of aluminum alloy 7075-T651, our approach separates the total fatigue life (Ntotal) into the four stages as shown in Fig. 2. Table 2 summarizes the fatigue modeling approaches that have been employed to address each stage of fatigue.

Table 1. Modeling and Simulation group member’s primary roles.

SIPS M&S Research Team RoleALCOAMaterial Characterization, Structure-Property Characterization.

Microstructure characterization & creation of data for statistical representation of microstructure,characterize damage progression.

Cornell University (CU)Multi-Scale Modeling, Mesoscale Models, Web-Based Simulation.

Multiscale modeling environment, crack incubation, nucleation & growth models, probabilistic crack growth modeling.

Mississippi State University (MS State)Atomistic Simulation, Microstructural Models, Multi-Scale Modeling.

Multiscale analysis using internal state variables,define matrix for microstructure-property model development, develop multi-stage fatigue model.

Lehigh University (LU)Corrosion and Corrosion Fatigue Models.

Mechanistic modeling of corrosion & corrosion fatigue, stochastic & probabilistic effects, corrosion-fatigue testing.

Rensselaer Polytechnic Institute (RPI)Crystallographic Deformation

Development of 3-D mesoscale (polycrystalline) finite element models of crystallographic deformation.

Carnegie Mellon University (CMU)Microstructure Builder

Methods for constructing 3-D representations of materials microstructures, statistically representative materials microstructures.

Ohio State University (OSU)Corrosion

Characterization of corrosion parameters and development of mechanistic models for localized corrosion.

University of Virginia (UVa)Fatigue Damage Characterization

Experimental characterization of nucleation, small crack growth, crack crystallography and environment effects.

VEXTECFatigue Models

Probabilistic microstructural-based fatigue life prediction model.


4

A summary of the technologies developed under the M&S thrust area of the SIPS program is provided below for

the key tasks involved in the approach of developing microstructurally-based models for fatigue life prognosis.

Ntotal = Nincubation + Nnucleation+ Nsmall crack + Nlarge crack

FASTRAN

FASTRAN (& Crack Coalescence)(Models for small crack growth & link-up)

Multi-Stage Fatigue/VPS-MICRO(Models for nucleation & small crack growth)

Geometric Approach(Explicit simulation of multiscale mechanics for crack growth - models for incubation, nucleation, & small

crack growth stages)

incubation

nucleation

small crack growth

Figure 2. Microstructurally-based models for fatigue life prognosis.

Table 2. Fatigue modeling approaches.

Model Approach Microstructural Element (Al Alloys)

FASTRAN 4.43 (MS State)

non-linear fracture mechanics constituent particle

Microcrack Coalescence Model (ALCOA)

non-linear fracture mechanics + crack coalescence

array of constituent particles

Multi-Stage Fatigue (MS State)

independent treatment of crack nucleation and small crack growth

constituent particles, grain size & texture

VPS-MICRO Probabilistic Model (VEXTEC)

probabilistic treatment of nucleation, crystallographic propagation and LEFM

distribution of constituent particles, grain size, texture

Geometric Approach (Cornell, Carnegie Mellon, Rensselaer Polytechnic)

explicit multiscale mechanism-based calculations, incubation model, nucleation model, etc.

statistically representative polycrystal, constituent particle distributions, grain size & texture


5

II. Statistically Representative 3-D Digital Materials Models Most polycrystalline materials are self-evidently three dimensional with respect to microstructure, and yet

characterization is generally confined to two dimensional cross-sections. Clearly, in order to progress with establishing microstructure-behavior relationships, we need to be able to describe microstructures in all three dimensions. This in turn, presupposes that we can make the descriptions suitable for use by numerical systems such as finite element models, which entails detailed geometrical descriptions. Many material behaviors such as plasticity are strongly anisotropic and this means that it is desirable to include crystallographic orientation (texture) in the descriptions. Our approach exploited statistical stereological approaches, combined with reasonable assumptions about the microstructural features of a particular material in developing a useful representation of the material’s microstructure2.

Carnegie Mellon University (CMU) and ALCOA have developed a “Microstructure Builder” for constructing three-dimensional digital representations of material microstructure based on two-dimensional orientation maps. The Microstructure Builder allows readily obtained cross-sectional information to be used to generate appropriate initial conditions for simulation of elastic and plastic deformation. Crystallographic orientation (texture) is included as one of the primary features of the method. Thus the input to the model requires statistical characterization of the orientation maps for the three orthogonal planes that can be obtained from electron back-scatter diffraction (EBSD) in a scanning electron microscope (SEM). Statistical characterization of microstructural features of interest includes size and shape of grains, constituent particle3 size and shape, misorientation distribution, and orientation distribution. “Microstructure Builder” can create numerous instances of a material microstructure all of which can be statistically representative of the real material. The generation process is divided into four independent components. First, a voxel based rolled aluminum polycrystalline grain structure is generated which matches the shape and size of the grains of interest. Second, a crystallographic orientation is assigned to each of the grains in the digital structure such that the orientation distribution and misorientation distribution of the structure is matched. Third, a 3D array of particles is generated so that they can be inserted into the grain structure. Fourth, these models are then post processed via surface meshing and mesh smoothing so that they can be used by SIPS team collaborators for finite element crystal-plasticity computations. Generating synthetic 3D polycrystalline grain structures that match the statistics for AA7075 is accomplished through a multistep process which includes; generating representative ellipsoids (in terms of size and shape distributions), placing those ellipsoids into a volume, allowing that volume to be filled with voxels that are grouped as grains, and then modifying the grains through isotropic Monte Carlo grain growth. An example of a statistically representative 3-D digital microstructure is shown in Fig. 3 which illustrates a microstructure with 51 grains generated via cellular automation, a method used to generate the voxelized structure.

RD TD

ND

Color Coded Map Type:Inverse Pole Figure [001]

Each face is 1mm by 1mm

RD, x

TD, y

ND, z

RD, x

TD, y

ND, z

Digital Replication of Polycrystalline Grain Structure

First-Phase RealizationSEM scans of 7075-T651 along 3

orthogonal planes

μstructure Builder

Figure 3. “Microstructure Builder” generated, statistically representative 3-D digital microstructure with 51 grains. The size of the volume element is approximately 1000 (RD) x 500 (TD) x 500 (ND) µm.


6

III. Non-linear Fracture Mechanics Model – FASTRAN Mississippi State University (MS State) has developed an analytical crack growth life model (FASTRAN4) that

is based on the Dugdale strip yield model. Dugdale derived an expression that relates the size of the plastic zone ahead of a crack to the applied stress intensity. FASTRAN modifies this model, originally for thin sheet (plane stress), to account for three dimensional effects, (plane strain). Additionally, FASTRAN accounts for the presence of plastically strained material in the wake of the crack which tends to cause the faces to contact and the crack to close before the applied stress is equal to zero (plasticity induced crack closure). It is assumed that the crack will not propagate while its faces are in contact, thus this factor is used to modify the applied ΔK to obtain ΔKeff, which is assumed to be the actual crack extension driving force. The crack growth rate is then obtained from experimentally measured da/dN vs. ΔK curves. FASTRAN has been developed to include short crack effects, and has been successfully applied to predictions of short crack growth in commercial aluminum alloys.5,6 FASTRAN requires an initial flaw size to initiate a prediction. As an initial approximation for commercial aluminum alloys, this flaw size has been taken as the size of the constituent particle that is most likely to start a crack. This approach has the advantage that the initial particle size distribution can be used as an input to probabilistic simulations. FASTRAN has been integrated into the SIPS probabilistic and reasoning framework,7 and has been successfully applied to many problems in the SIPS program. Thus, a probabilistic answer can be obtained for the predicted life to a given crack size.

A constant-amplitude fatigue crack growth test program was conducted on 7075-T651 aluminum alloy to generate small- and large-crack rate data over a wide range of stress ratios (R = 0.1 to 0.7) from threshold to fracture. The baseline fatigue crack-growth-rate relation (da/dN vs. ΔKeff) was established in both the thickness (a) and width (c) directions.8 Crack-growth behavior was found to be significantly different in the thickness and in the in-plane directions. Using inclusion-particle-cluster sizes and shapes, like those that are observed in tests, the fatigue and crack-growth lives are predicted on various SIPS fatigue specimens from coupon to component to system level configurations under constant-amplitude and flight-by-flight spectrum loading. The predicted deterministic results compare well with test data.8

A test program was then conducted to demonstrate the validity of the SIPS probabilistic framework. The overall objective of the test was to validate SIPS probabilistic fatigue life model predictions against laboratory generated crack size distributions at various damage states and under two representative flight spectra. A description of the experimental procedure, results and analysis is provided in a companion paper.9 Figure 4 shows a comparison of the FASTRAN probabilistic predictions as compared to the experimentally obtained crack size distributions for two different damage states at 100% and 200% Fatigue Life Expended (FLE10) for the SIPS two-hole specimen under spectrum loading. Figure 4(a) shows results along the thickness (2a) and Fig. 4(b) shows results along the width (c). In both cases, the experimentally generated distributions and the predictions are fit to a lognormal distribution. The tails of the distributions in the plots are close for small and large crack sizes. Because the experimental data at 200%

(a) At 100% FLE damage state (b) At 200% FLE damage state Figure 4. Probabilistic FASTRAN predictions vs. experiment for 7075-T651 under spectrum loading.


7

FLE only includes crack depths of size c = 0.000275 inches and larger9, the predicted cumulative probability (Fig. 4(b)) is a conditional distribution that only considers crack sizes (c) greater than 0.000275 inches (7 µm). The probabilistic predictions are in good agreement with the test data.

IV. Microcrack Coalescence Model The ALCOA Technical Center, in collaboration with Carnegie Mellon University (CMU) and Cornell University

(CU), has developed a fatigue crack growth life model that addresses the problem of multiple microcrack growth and coalescence. Current fatigue crack growth life methodologies require an initial damage state to carry-out predictions of damage evolution to failure of a component. The initial damage state is defined by a single initial flaw that grows independently to failure. However, experimental observations have indicated that in the high stress area in the bore of a hole multiple cracked constituent particles cause multiple microstructurally small cracks as shown in Fig. 5. The random formation of these microcracks forms a cluster of cracks whose growth is affected by interacting mechanisms through crack shielding and coalescence. Some microcracks arrest and some grow then link and form the dominant life-limiting crack that leads to failure of the component. Microcrack formation and coalescence is strongly influenced by the material’s microstructure and can lead to a significant variability and reduction in fatigue life. A formal approach has been developed to explicitly model the physics of microcrack growth and coalescence including the effects of crack shielding.

The Multi-Site Crack Growth (MSCG) computational method is designed to enable fatigue crack growth life predictions for structural hot-spots potentially containing many crack nucleating features. The model deals directly with a set of a large number of cracks in a defined geometrical region and can start with an initial uncracked state, where cracks may nucleate from cracked particles or other features, or from an initial cracked state such as might be expected at a percentage of Fatigue Life Expended (FLE). The software then tracks the simultaneous growth and interaction of these cracks, including link-up, shielding, and final failure. The physical mechanisms employed to determine whether the cracks extend, link-up or arrest are based on an approximations of each crack’s strain field and a non-linear elastoplastic finite element formulation. The code is designed to work from pre-computed crack growth solutions. The method uses FASTRAN as the driver to generate crack growth life solutions under spectrum loading; however, any other crack growth simulation code can be used as the driver. The use of efficient computational algorithms enables rapid calculations; a crack growth simulation involving 1000 interacting cracks runs in about a minute on a typical personal computer.

A set of experimental data has been obtained to validate SIPS probabilistic models for fatigue.9 Crack size distributions from optical and scanning electron microscope (SEM) observations were obtained experimentally at various damage states at 100%, 150%, 200% and 300% FLE. For the SEM measured cracks, a crack mapping method was established whereby the location, number and size of each crack in the high stress area of the hole were catalogued. Four two-hole specimens were fatigued using a flight-by-flight spectrum loading to 100% FLE. Sixteen SEM high resolution configurations were generated and 1106 cracks in total were measured using the crack mapping technique. These measured configurations at 100% FLE are ideal in defining the initial crack configuration for MSCG model predictions to 200% FLE and to failure. Validation of MSCG model can be carried-out by comparing the MSCG model predicted crack size distribution at 200% FLE to the measured crack size distribution at 200% FLE.

7075-T651

Figure 5. Multiple fatigue cracks in the high stress region of a typical fastener hole.


8

Figure 6 shows one of these configurations with cracks of varying lengths of size 2a for a strip representing the high stress area in one of the hole sides with dimensions of 0.225” (5715 µm) in width and 0.0215” (546 µm) in height. This configuration represents the crack population as measured by the crack mapping technique at 15,129 cycles corresponding to 100% FLE. The dimensions of the fatigued specimen were: hole diameter = 0.190”, width = 1.0”, and thickness = 0.225”. As shown, longer cracks have larger plastic zones as expected. Figure 7 shows an intermediate configuration. Figure 8 shows the computed final configuration with a single long crack extending across the section that has an irregular path as a consequence of multiple linkup events. Many of the initial cracks have grown to some extent and then ceased to propagate as they become shielded by the fatal crack. The link-up ligatures (coalescence events) are shown in red.

Multi-Site Crack Growth predictions were repeated for fifteen more initial crack configurations. Each initial configuration represents damage in the high stress area of a side of a hole at 15,129 cycles corresponding to 100% FLE. Figure 9 shows a comparison between MSCG predictions to experimental measured data at 200% FLE for the flight-by-flight spectrum loading. Figure 9 shows a higher frequency of crack lengths (2a) in the 10 to 30 µm range for the MSCG predictions, and a larger frequency of cracks for 30+ µm size cracks for the experimental data. However it should be noted that the comparison shown in Fig. 9 of experiments to predictions is not directly equivalent. The predictions are for a computational domain corresponding to the high stress area in a hole and the experimental data are obtained from optical measurements on the fracture surface. The predictions include small cracks that will eventually arrest and will not be part of the fracture surface. It is expected that if these small cracks are removed from the MSCG predictions the results will be more comparable.

Figure 6. Initial measured configuration at 100% FLE. Each crack is represented by a set of line segments, at each end of which is a rectangle that represents the plastic zone size associated with each crack tip.

Figure 8. Final configuration. One crack has extended across the entire width (0.225”) and has several changes in direction as a result of link-up events. The link-up ligatures are shown in red.

Figure 7. Intermediate configuration. Some cracks have grown as indicated by the lines that are longer than in the initial configuration.


9

Figure 10 shows the predicted variation of cycles required for a crack to span the area element, which can be taken as representative of failure, and is compared to experimental results to failure. Note the considerable range in effective lifetime that is suggested by this set of calculations, which differ only by the instantiation with different measured microstructures and therefore illustrates the strong influence of microstructural variation on lifetime.

V. VPS-MICRO Probabilistic Model The VEXTEC VPS-MICRO model includes a probabilistic fatigue life prediction algorithm that explicitly

models fatigue crack initiation and growth starting at the microstructural scale.11 The overall fatigue damage model uses Monte Carlo simulation to proceed through three different levels of fatigue damage accumulation that are commonly observed in the process of fatigue, each with its separate modeling regimes: (1) crack nucleation, (2) short crack growth and (3) long crack growth. The probabilistic micromechanics model is based on dislocation theory, crystal plasticity and fracture mechanics. It considers the variability in grain size, microstress, inclusion density and grain orientation. This enables “total life” analysis from damage nucleation to final fracture. Crack models incorporate randomness using Monte Carlo probabilistic techniques. Computer simulation is set up with built-in material libraries and appropriate modeling linkages are established to predict the scatter in fatigue life. VEXTEC has modified their physics-based models for SIPS aluminum alloy 7075-T651 and 7050-T7451 materials. The models have been used for fatigue life predictions of SIPS smooth round and double edge notch (DEN) specimens under constant amplitude loading and SIPS two-hole fatigue specimen under spectrum loading.

The crack nucleation model, which is based on cracking due to the build-up of dislocation density is incorporated into the probabilistic fatigue algorithm. This model accounts for grain orientation and size along with inclusion size and density. The crack nucleation model determines the number of cycles to create a crack equal to the grain size. The output from the crack nucleation model provides the input to the small crack growth model. The

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10 30 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490

Experimental 200% FLEMulti-Site Predicted 200% FLE

Freq

uenc

y of

cra

cks

per h

ole

Crack length, 2a (μm)

Figure 9. Multi-Site Crack Growth predictions vs. experiment at 200% FLE under spectrum loading

20000 25000 30000 35000 40000 45000 50000

Multi-Site Crack Growth PredictionsExperimental

Cycles to Failure Figure 10. Multi-Site Crack Growth predictions vs. experiment cycles to failure


10

microstructural variations are explicitly considered in this module. The small crack growth model determines the number of cycles and the size of a crack for which the long crack model is applicable. The long crack growth rate is modeled using linear elastic fracture mechanics. All variation in long crack growth is modeled by allowing the fracture mechanics coefficients to be random variables. The long crack growth model determines the number of cycles to exceed a critical crack size.

The overall framework of the micromechanics model for aluminum materials is shown in Fig. 11 along with the elements of the technology that are integrated together. A finite element analysis of the actual specimen configuration provides stresses and stressed volumes. The material microstructure modeled as a statistical volume element (SVE) provides the distribution of stresses at the grain scale. These inputs are used within the microstructural damage algorithm to predict fatigue failure.

Model predictions were validated by comparing probabilistic simulations to SIPS experimental data. One such comparison is shown in Fig. 12 for a 7075-T651 smooth round coupon under constant amplitude loading (R = -1). Six components were simulated at each stress level, as shown with red symbols. The predicted fatigue lives are in good agreement with the experimental data. At lower stress levels (≤ 23 ksi), the model predicted run-outs as observed in the experiments. All fatigue failures were initiated by constituent particles, similar to the experimental observations. Statistical analysis at a confidence level of 95% was performed to determine whether the experimental data could reflect the simulation data. Figure 12 shows that the vast majority of predictions are within a 95% confidence bound of the actual test data. The predicted results compare well with test data.

Figure 11. VPS-MICRO probabilistic micromechanics model framework.

Figure 4.5.5: Probabilistic micromechanics model framework

180

160

140

120

100

80

60

40

20

01.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Fatigue Life, Cycles

Max

imum

App

lied

Stre

ss, K

si

AI7075-T651: SIPS1006 Fatigue Smooth Round Coupon; R = -1

VEXTEC simulation data (6 components)Experimental dataVEXTEC – 95% lower confidence boundVEXTEC – 95% upper confidence bound

VEXTEC simulation data (6 components)Experimental dataVEXTEC – 95% lower confidence boundVEXTEC – 95% upper confidence bound

Figure 12. VPS-MICRO predictions vs. experiment for constant amplitude loading


11

VI. Multi-Stage Fatigue Models Mississippi State University (MS State) has developed a Multi-Stage Fatigue (MSF) model that incorporates the

microstructural features to the fatigue life predictions for nucleation, microstructurally/physically small crack growth, and long crack growth stages in both high cycle and low cycle regimes. The MSF model specifically addresses the role of local constrained cyclic microplasticity at fractured inclusions in fatigue crack nucleation and small crack growth regimes, including the effect of crystallographic orientation on crack tip displacement as the driving force. In addition to being grounded in micromechanics theories of fatigue crack propagation, this approach has the advantage of providing a set of equations that can be readily evaluated under a variety of conditions. As such, it can be employed in the SIPS probabilistic reasoning and prognosis framework which utilizes statistical distributions of the input parameters to provide probabilistic predictions of fatigue crack evolution.

In the MSF approach,12 the total fatigue life (Ntotal) is separated into three stages: nucleation (Nnucl), microstructurally small/physically small crack growth (Nmsc/psc) and long crack growth (Nlc) as follows:

lcmsc/pscnucltotal NNNN ++= (1)

For 7075-T651 the nucleation life, Nnucl, is the number of cycles to nucleate a crack at a micronotch and is assumed to consist of two stages, the cracking of a constituent particle and the early crack propagation through the zone of the micronotch root influence. A modified Coffin-Manson law for 7075-T651 has been developed13 linking the microplasticity to the nucleation life, Nnucl according to

nuclnuclmax

2CN

pαγβ =

Δ= (2)

where β is the local maximum plastic shear strain around the constituent particle and α, Cnucl are the linear and exponential coefficients. The model was developed by combining micromechanics finite element analyses of single cracked particles and experimental observations of damage. The micromechanics simulations were performed to obtain relationships between the far-field strain, the nonlocal (notch root volume averaged) micronotch root maximum plastic shear strain amplitude around the cracked particle, the plastic zone size and the particle diameter. Experiments were performed on smooth cylindrical shaped specimens for various levels of maximum applied strain amplitudes with a loading ratio, R = -1.0. The experimental observations of damage initiation provide relationship between far-field strain and number of cycles to initiation. A statistical analysis was performed on interrupted damage accumulation specimens14 to determine what percent of initiation life (life to a 0.01 in. crack) is attributed to nucleation. The experimental data provided a relationship between far-field strain and cycles to nucleation life (Nnucl).

The microstructurally small/physically small portion of fatigue life, Nmsc/psc, encompasses crack sizes, α, between the incubated size (αi ≈ .56D where D is the average constituent particle diameter) and a crack size that is treatable by linear elastic fracture mechanics, approximately 0.01 in. (250 µm) in this case. Crack growth in the MSC/PSC regime is governed by the range of the crack tip displacement (ΔCTD) according to

( )thNa CTDCTDG

dd

Δ−Δ= (2)

where G is a material constant that reflects the crack tip irreversibility, and ΔCTDth is the threshold value of the crack tip displacement and is assigned a value corresponding to the Burger’s vector for the Al rich matrix. A ΔCTD expression for 7075-T651 has been developed13 in terms of the grain orientation, the modulus, the grain size, the mean stress, the applied stress range and the plastic shear strain range.

The MSF model has been validated by comparing predictions to SIPS experimental data. One such comparison is shown in Fig. 13 for a SIPS two-hole specimen under spectrum loading. Two predictions are shown each with an initial flaw size corresponding to the size of a constituent particle. The open and solid circles denote surface crack lengths at a particular number of cycles or at failure. The open and solid squares denote through crack lengths from the hole at a particular number of cycles or at failure. The diamond symbols show surface crack lengths measured during a test using a photographic method. A JENTEK meandering-winding-magnetometer (MWM) array sensor15


12

was used to monitor the crack lengths along the bore of the hole in some other tests. Results of one of these test is shown as curves in Fig. 13. The predicted results compare well with test data.

VII. 3-D Crystal Plasticity and Damage Evolution Models Rensselaer Polytechnic Institute (RPI) has developed a crystal elastic-plastic material model at the grain scale.

The overall response of polycrystalline metals is strongly tied to grain-scale phenomena. Material heterogeneities, particularly constituent particles in Al 7075 and grain boundaries in Al 7050, are closely related to the initiation of localized plastic deformation, damage accumulation, and crack nucleation. The local grain orientations and structure surrounding constituent particles strongly affect the heterogeneous stress and strain states that develop, which, in turn, affect crack nucleation and microstructurally short crack propagation. The formulation and implementation of crystal plasticity models and damage metrics for the SIPS program are centered on the relationship between heterogeneous response at the grain scale and the accumulation of fatigue damage leading to crack nucleation.

The response of the AA7075-T651 matrix material is governed by a crystal elasto-viscoplastic formulation that captures the relevant microstructural mechanisms.16 The constitutive model provides information about heterogeneous stress, strain, and plasticity states at the grain scale, aiding fatigue modeling and the prediction of crack nucleation. Key aspects of the work include defining an appropriate constitutive model and calibrating and validating both constitutive and numerical models. The RPI crystal elastic-plastic constitutive model has been developed using a programming language and style that allow for direct integration into the multiscale modeling environment at Cornell University. The constitutive model has also been implemented in a commercial finite element analysis code, as an ABAQUS UMAT.

The crystal plasticity constitutive model was calibrated and validated against experimental data provided by collaborators at Carnegie Mellon University, Mississippi State University17, and Northrop Grumman Corporation. Simulation data were obtained by analyzing a 299-grain polycrystal model containing approximately 42,000 elements, shown in Fig. 14. The grains were constructed with an aspect ratio of RD:TD:ND = 25:4:1, which approximates the observed aspect ratio of 50:4:1. The rolling dimension was

Figure 13. Multi-Stage fatigue model predictions vs. experiment for spectrum loading.

Figure 14. Idealized grain structure used for calibration against experimental data.


13

reduced to facilitate the creation of a finite element model with a sufficiently large number of grains. Lattice orientations were assigned based on scanning electron microscope back scatter diffraction measurements. The test specimen was prestrained in compression then uniaxially reloaded in tension. Results from a reversed-loading test validation are shown in Fig. 15.

Additionally, RPI focused on the nucleation process by defining a simulation protocol to identify key metrics related to spawning of matrix cracks from cracked constitutive particles. A set of five different damage metrics have been defined to relate plastic slip and stress to fatigue damage. A number of series of simulations has been carried out by Cornell University and RPI to evaluate and validate the crystal plasticity model and its implementation. An example of a simulated experiment is presented in the case of one of the damage metrics that looks at a combination of accumulated slip on a slip plane and normal stress on that slip plane. The accumulated damage was computed in the neighborhood of an idealized cracked particle in a single grain. The finite element model is presented in Fig. 16. The results in Fig. 17 compare two simulations which differed only by the orientation of the grain. The results presented in Fig. 17 demonstrate the effect of microstructure on damage evolution and also the distribution of damage in the neighborhood of a cracked constituent particle.

5499-421

Stre

ss (M

Pa)

Strain

-0.06 -0.02 0 0.04 0.06

-600

-400

-200

0

200

400

600

-0.04 0.025499-421

Stre

ss (M

Pa)

Strain

-0.06 -0.02 0 0.04 0.06

-600

-400

-200

0

200

400

600

-0.04 0.025499-422

Stre

ss (M

Pa)

Strain

-0.06 -0.02 0 0.04 0.06

-600

-400

-200

0

200

400

600

-0.04 0.025499-422

Stre

ss (M

Pa)

Strain

-0.06 -0.02 0 0.04 0.06

-600

-400

-200

0

200

400

600

-0.04 0.02

(a) Validation in the rolling direction (RD) (b) Validation in the transverse direction (TD) Figure 15. Predictions vs. experiment for a compression-tension reversed-loading test.

5499-4255499-425

Figure 16. Model containing a single cracked particle.

5499-4265499-426 Figure 17. Crystal plasticity simulations of damage accumulation in a model containing a single cracked particle in a grain with two different orientations; detail near tip of crack in a particle.


14

VIII. Multiscale Fatigue Modeling Environment - Geometric Approach Cornell University (CU) has developed a computational framework for explicitly simulating the multiscale

mechanics of crack growth, from incubation at the microscale to failure at the component-scale. Development of the framework was made possible by advancements made by the SIPS M&S multidisciplinary research team18; in the statistical modeling of polycrystalline metal microstructures (ALCOA, CMU and LU), finite element meshing (CU), multiscale analysis (CU and MS State), modeling of constitutive behavior of polycrystals (RPI and MS State), and materials testing technology for property measurement and validation procedures (Northrop Grumman, MS State, UVa, and OSU). The framework is plug-compatible, allowing for future improvements in the physics-based models to be hypothesized, implemented, analyzed and validated, and applied to other material systems.

The current framework shown in Fig. 18 contains 3 levels, each of which has a unique degree of fidelity and, consequently, computational demand.19,20 In this approach, an explicit geometric representation of the aluminum alloy microstructure and damage evolution caused by fatigue loading are being modeled. At various stages of the modeling process, length scales from meters, (the aircraft), to millimeters, (the fastener holes), to micrometers, (the grains and constituent particles), are modeled. This multiscale modeling approach would be intractable if one were to attempt to include the microstructural scale throughout the model, thus we have developed a hierarchical sequence of models at the various scales. At the macroscale, the global state of stress on the component is determined. This highlights the hot spots. A smaller (mesoscale) region, typically a 1 mm cube, containing the critical damage site is then extracted from the larger model along with the boundary conditions determined from the macroscale analysis. This region is then populated with the microstructural features that control the damage process, in this case, the grain structure and the constituent particles. The microstructural features are determined by quantitative 3-dimensional metallography. The results of the metallographic characterization are captured as a statistically representative digital microstructure, which is used as input to Microstructure Builder to generate the microstructure used in the mesoscale models. The mesoscale model is then interrogated to determine its response to the applied fatigue loads using a crystal plasticity constitutive model. The damage at the mesoscale is then inserted back into the macroscale model by adjusting the local stiffness at each of the integration points in the macroscale model to reflect the damage calculated at the mesoscale. In this manner, explicit calculations of the physical basis for fatigue damage could be performed and used for predictions of the expected fatigue life of an aircraft component. The overall philosophy of the Geometric Approach is depicted in Fig. 18.

Figure 18. Multiscale fatigue modeling environment – Geometric Approach


15

Experiment observations of early damage were combined with micromechanics simulations to develop criteria for particle cracking, crack initiation into grains from cracked particles, and small crack growth. Micromechanics simulations at the microscale include modeling single particles in single grains and performing crystal plasticity simulations. These models are used to calculate the effect of local plasticity on the cracking and debonding of the constituent particle (incubation stage), the subsequent initiation of a microcrack in the matrix adjacent to the cracked particle (nucleation stage), and the propagation of the crack through the parent grain and into the adjacent grains (microstructurally small crack growth stage). In all cases, the crystallographic nature of plastic deformation in each grain is explicitly accounted for.

It has been experimentally observed that in AA 7075-T651 fatigue cracks initiate and nucleate into the matrix from cracked Al7Cu2Fe constituent particles. However, it has been observed that only a small percentage of the Fe-bearing particles crack. Consequently, inserting all particles into a microstructural model is physically unnecessary and computationally intractable due to the resulting increased finite element model size. Therefore, a model for particle cracking21 has been developed and implemented to select only particles most likely to crack and nucleate a crack into the matrix. In this approach, shown in Fig. 19, three-dimensional crystal plasticity finite element simulations were performed of a single particle embedded in a grain to develop a response surface for the tensile stress (σp) in the particle as a function of the strain level surrounding the particle, parent grain orientation, and particle aspect ratio. A technique for estimating particle strength (σcr) from fracture toughness, particle size, and intrinsic flaw size was developed. Particle cracking was then determined by comparing particle stress and strength.

A series of experiments were performed on double edge notch (DEN) specimens8 to develop a detailed understanding of the sequence of events involved in the creation of a microstructurally long fatigue crack and to provide information for calibrating SIPS models pertaining to early stages of fatigue. These experiments involved constant amplitude fatigue loading on notched DEN specimens with periodic interruptions of the fatigue cycling in order to make damage observations in an SEM. The material parameter, C, the average ratio of intrinsic crack size to particle size, which is used to estimate particle strength (σcr), was calibrated for selected microstructure samples as observed in the high stress area of the DEN specimen. Three configurations were selected with single crack particles where no matrix crack had developed and three configurations were selected with single intact particles. The sample observed local texture, microstructure geometry, and applied strain at the time of particle cracking were also recorded. The resulting image of each sample was traced to create a computational replication of the microstructural

Estimated CriticalFracture Stress

Response Surface To Interpolate For Particle Tensile Stress, σp

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10 11

High Stress OrientationIntermediate Stress OrientationLow Stress Orientation

Particle Aspect Ratio (RD:ND, RD:TD)

σ xx

(MPa

)

Switching from low stress orientation to high stress orientation results in “curve

shifting” for all particle aspect ratios

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10 11



σ xx

(MPa

)

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10 11



σ xx

(MPa

)





400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10 11



σ xx

(MPa

)



400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10 11



σ xx

(MPa

)

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8 9 10 11



σ xx

(MPa

)





Random Input Parameters

• Grain Orientation• Particle Aspect Ratio• Strain Level

yx

yx

yx

Micromechanics Crystal Plasticity Simulations Random Input Parameters

• Particle Size• Fracture Toughness• Model Calibration, C

Particles Likely to Crack

Figure 19. Fatigue incubation model for particle cracking


16

geometry, representing a 2-D model of the geometry, shown as physical data in Fig. 20. The 2-D geometry was then extruded to create a 3-D computational model. The grain was extruded based on an observed average grain aspect ratio and the particle was extruded based on fitting a circle to the particle boundary and utilizing the radius as the extrusion distance. After the 3-D geometry is generated, a mesh is generated, in which elements are assigned experimentally observed material orientations. In this study, the particle is modeled as linear elastic, isotropic solid and the grains as rate-dependent, face-centered cubic (FCC) crystal plastic solids of varying orientation. Details of the incubation model development work can be found in Ref. 21. Similarly, a model has also been developed for the crack nucleation stage22 and additional research is currently underway at CU to develop models for microstructurally small crack growth.

The incubation model has been validated by computing the frequency of particle cracking for a statistical realization of particles and comparing to an experimentally obtained frequency of particle cracking. The frequency can be predicted by sampling the respective strain, grain orientation, and particle size distributions to find the stress and strain in each particle. A histogram of the difference between particle stress and strength for an example realization is shown in Fig. 21. For this realization of 20,000 sampled particles, the predicted frequency of particle cracking is 2.9%. The predicted particle crack frequency was in good agreement with the experimentally observed average particle cracking frequency of 2.3%.8

A computational framework has been developed for explicitly simulating the multiscale mechanics of crack growth, from incubation at the microscale to failure at the component-scale. The geometric modeling approach holds the promise of providing a detailed, physically realistic, high-fidelity model of the fatigue process.

Figure 20. Replication experiment to determine proportionality constant, C

Figure 21. Histogram of the difference between particle stress and strength, Δσ = σp - σcr. The subset of the histogram where Δσ ≥ 0 corresponds to the cracked particles.


17

IX. SIPS Integrated Modeling Approach Under the DARPA Structural Integrity Prognosis Program, Northrop Grumman researchers, in collaboration

with other team members, have developed a methodology to produce probability distributions of initial crack sizes from the probabilities that cracks have formed in constituent particles and grown into the surrounding matrix for a given aircraft usage history. These distributions are used with fatigue crack growth models for prediction of fatigue crack evolution in aerospace structural components. Since the probability distribution of the stochastic process for crack size is a function of the initial crack size distribution, the improved representation and accuracy of the initial distribution can produce greater fidelity in the resulting distributions for crack size over time.

A microstructurally-based probabilistic framework has been developed that incorporates SIPS developed models and methods into a single integrated modeling approach. The objective was to link the various models that have been developed to address each critical fatigue stage into one coherent model. The integrated model illustrated in Fig. 22, comprises the following components: 1) Microstructure builder (CMU) to define the bulk material’s initial microstructure statistics, 2) Meso-scale tool box including models for fatigue incubation and nucleation (CU) to define a statistically representative physics-based initial state of damage, 3) Crystal plasticity model (RPI) to accurately capture the state of stress and strain at the grain scale, 4) Fatigue crack growth models (MS State) to capture damage evolution, 5) Northrop Grumman developed methods to rapidly calculate inelastic responses at a critical hot spot based on specific individual aircraft usage, and 6) Northrop Grumman developed probabilistic methods.

The framework tailors probability distributions of initial flaw sizes to individual aircraft. Prior microstructural methods used either distributions of constituent particle sizes derived from bulk material statistics (pristine) or distributions of cracked particles found in interrupted fatigued specimens as initial flaws for failure prediction. The new method utilizes the usage history of each individual aircraft to create initial flaw size distributions tailored for that aircraft. Although these distributions begin with a common set of bulk material statistics, the new methods account for the probability that a given particle may lead to a fatal crack given one vehicle usage history and may not lead to a fatal crack given a different usage history. Tailored distributions of initial flaw sizes enable better predictions of fatigue crack growth life.

Fatigue damage predictions of structural components rely on the accurate calculation of elastic and inelastic material response at the critical locations of interest. Since fatigue calculations are performed over very large numbers of cycles, it is even more crucial to carry out these calculations in an expeditious manner. Conventional 1-D uniaxial methods that relate local stresses and strains to nominal values are fast but do not offer the necessary detail required by multiaxial based fatigue models. Detailed elastic-plastic finite element calculations offer the required accuracy however these methods are computationally expensive and impractical for variable amplitude

Calculate Notch StrainFor Maximum Load in

Flight History(Neuber Model)

Calculate Notch StrainFor Maximum Load in

Flight History(Neuber Model)

Bulk Material μStructure Statistics

Grain OrientationParticle Aspect Ratio

Particle Size

Response Surface to Select Particles

Most Likely to Crack(Incubation Filter)

Crack Size, a

P(a)

Crack Size, a

P(a)Aircraft Specific Geometry, Material

& Flight History

Probabilistic Predictions

FASTRAN/MSF Predictionsµstructurally Small

& Large Crack Growth to failure

FASTRAN/MSF Predictionsµstructurally Small

& Large Crack Growth to failure

Physics-Based Initial Crack Size DistributionPhysics-Based Initial

Crack Size Distribution

Particle Size, a

P(a)P(a)

Particle Size, a

P(a)P(a)

Spectrum LoadSpectrum LoadSpectrum Load

Figure 22. Integrated Prognosis Model Framework


18

loading histories as are often encountered in aerospace structures. To this end, several multiaxial approximate approaches have been examined in the literature.

Approximate methods such as multi-axial versions of Neuber's rule23 and Glinka's ESED24 method have gained widespread use in the fatigue community. The new method expands upon previous implementations by generalizing the 1-D Neuber Rule for multiaxial stress states combined with material constitutive laws capable of capturing the cyclic response of the material (such as Von Mises kinematic or combined hardening and finite strain crystal plasticity). The decomposition of stress into a stress magnitude and stress direction tensor allows a more general formulation so that the method can be used at free surfaces of notches as well as at internal material points in the structure. To account for the progression of stress redistribution that occurs with increasing amounts of plastic strain, a correction factor can be applied to the formulation provided some a priori knowledge of the stress direction tensor is established. This allows the use of the method in situations where the nominal loading results in significant local plastic strain.

To verify the approach, several test cases were developed and results of the new approximate method were compared with detailed 3D finite element analyses (FEA). One of verification test consisted of applying a remote tensile stress field to a finite element model of a single-hole specimen, performing an elastic-plastic analysis, and probing the local notch stresses and inelastic strains. Very good agreement is shown between the approximate method and finite element analysis (FEA) results for both local notch inelastic strain as well as stress. Figure 23 shows the results using the new method with modifications to both the Neuber Method as well as to the Glinka ESED method.

The current implementation is in the form of a module within the integrated modeling framework. It takes as input the fatigue spectrum, an elastic stress concentration factor Kt at the critical location (notch), the applied gross stress level, and material properties for the constitutive behavior chosen and provides as output the maximum spectrum strain at the critical location (notch). This is in turn is passed to the incubation model as one of the inputs to determine particles most likely to crack.

The overall process shown in Fig. 24 begins with initial probability distributions for particle dimensions: a (particle half-width), c (particle depth), and h (particle height) that are determined from bulk statistics obtained from SEM scans of the three orthogonal planes. A range of particle dimensions are computed and subjected to an aspect ratio (AR) filter that eliminates any particles whose dimensions do not have the appropriate aspect ratios. These particle dimensions with additional inputs such as fracture toughness distribution, notch strain, and grain orientation distribution are then passed into the incubation filter that determines the probability of the particle fracturing

Figure 23. Elasto-plastic finite element model results vs. incremental Nueber/Glinka models


19

P(fr|a,c,h). This conditional probability is the output of the incubation filter, and it represents the probability of fracture given particle dimensions (a, c, h). The next step in Fig. 24 uses the Theorem of Total Probability to obtain the P(fr|a,c) from each of the probabilities P(fr|a,c,h) by summing each probability multiplied by the probability that the corresponding depth h occurs.

Crack growth models produce curves of crack size as a function of time for each initial crack value, however, these physics models are deterministic and do not capture the uncertainty in the inputs and its effects on the crack growth curve. This uncertainty is represented by the probability density f(a, c|fr) that defines the probability of the occurrence of an initial crack with half-length a and depth of the crack c. The randomness of the input crack values produces a random output in the crack growth model that is defined as a stochastic process. By assigning the probabilities defined by f(a, c|fr) to each crack growth curve, the probability distribution of the stochastic process of the crack at any time can be computed.9 Therefore, it is necessary to accurately compute the joint probability density f(a, c|fr) of the initial crack sizes. Using Bayes Theorem we can obtain the density f(a, c|fr) from the conditional probability of fracture P(fr|a,c) and the original probability density f(a, c) of particle dimensions for the half-length and depth obtained from bulk statistics. As shown in Fig. 24 this joint density f(a,c | fr) represents the probability of the occurrence of the two random inputs to a crack growth model.

Ongoing work focuses on verifying these distributions and utilizing them with small crack growth simulations to produce new predictions and make comparisons to experimental data. Figure 25(a) compares two probability density function distributions of fractured particles that represent the initial crack size distributions for small crack growth simulations: 1) experimentally obtained from one interrupted fatigued specimen25, and 2) physics-based derived from the integrated modeling approach. The bulk material probability density function is only shown for comparison. Figure 25(b) compares the results of small crack growth analyses to an experimentally9 obtained crack size distribution for a damage state at 100% FLE for the two-hole specimen using a flight-by-flight spectrum loading. FASTRAN probabilistic predictions were carried-out using the experimentally obtained and the physics-based initial crack size distributions. Both input distributions represent the crack depth (c) along the specimen width direction with an assumed initial aspect ratio of a/c of 1.0. The probabilistic prediction based on the physics-based derived initial crack size distribution is in good agreement with the test data.

IncubationNucleation

Filter

Bulk Statistics:F(a,h)F(c,h)F(a,c)

aicihi

P(fr|a, c, h) P(fr|a, c)f(a, c|fr)

Notch Strain

f(a, c)

Aircraft specific

Load history

Fracture toughness

AR filt

er

Choose n instances

Particle hemi-ellipsoid

cycles

c

2D Crack Growth Modeling

Probability that a particle of size (a, c, h) will fracture for

this vehicle history

Probability reduced to (a, c) dimensions

Probability that a particle of size (a, c) exists in critical

area

Each crack trajectory modeled is assigned a probability f(a, c|fr)

3-D Neuber/Glinka Model

2a

c 2h

Figure 24. Probabilistic Framework for the Integrated Modeling Prognosis System


20

X. Conclusion Over the course of the SIPS Program, technologies were developed and validated in a graduated progression

from simple-representative laboratory coupon tests to full-scale system tests using active military aircraft vehicles. Testing and analyses of fatigue mechanisms on over 400 laboratory specimens with varying degrees of damage/usage, teardown inspections of two retired outer wing panels, and full-scale tests of three retired outer wing panels have yielded a measurement database of over 10,000 cracks. Microstructural characterization of old and new samples of AA 7075-T651 quantified the critical distributions for the random variables used by the SIPS fatigue models. Development of detailed modeling of fatigue mechanisms at various scales based on experimentally observed damage accumulation and evolution produced an integrated model that captures probabilistically the crack progression from micrometers to meters. Advances in understanding of the microstructural processes that give rise to fatigue crack initiation and propagation, along with dramatic increases in computing resources, have combined to create a situation where it may be feasible to create detailed, physically rigorous models of the basic processes of fatigue cracking in metals. When these models are combined with information about the history and the current state of the vehicle, it may be possible to achieve a realistic picture of the state of structural heath of the vehicle. It is true that much work remains to be done in this area, but the framework of a modular modeling system has been demonstrated here. Models ranging from the micron-scale up to macro-scale can now be brought to bear on a host of problems, and the procedures necessary to expand their utility to other materials have also been developed. The current work was focused on two alloys, 7075-T651 and 7050-T7451, primarily because the former has broad application in a variety of existing platforms with known fatigue issues, and the latter is being employed in newer platforms which we wish to understand and model.

Acknowledgments This work is partially sponsored by the Defense Advanced Research Projects Agency under contract HR0011-

04-C-0003. Dr. Leo Christodoulou and Dr. William Scheuren served as the DARPA Program Managers. Many subcontractors have contributed to this paper; their participation is gratefully acknowledged.

The views, opinions, and/or findings contained in this article are those of the author/presenter and should not be interpreted as representing the official views or policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the Department of Defense.

References 1Madsen, J. S., Anagnostou, E. L., Engel, S. J., Hoitsma, D. H., Jr., Nardiello, J. A., and Papazian, J. M., “Overview of the

DARPA Structural Integrity Prognosis System (SIPS),” Proceedings of 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 2010 (to be published).

(a). Initial crack size distributions (b) Probabilistic FASTRAN vs. experiment Figure 25. (a) Input distributions for small crack growth life predictions, and (b) predictions at 100% FLE for 7075-T651 under spectrum loading.


21

2Rollett, A.D., Lee, S.B., Campman, R., and Robrer, G.S., “Three-Dimensional Characterization of Microstructure by Electron Back-Scatter Diffraction,” Annual Review of Materials Research, Vol. 37, 2007, pp. 627-658.

3Rollett, A. D., Campman, R., and Saylor, D., “Three Dimensional Microstructures: Statistical Analysis of Second Phase Particles in AA7075-T651,” Materials Science Forum, Vol. 519-521, 2006, pp. 1-10.

4Newman, J. C., Jr., Brot, A., and Matias, C., “Crack-Growth Calculations in 7075-T7351 Aluminum Alloy Under Various Load Spectra Using an Improved Crack-Closure Model,” Engineering Fracture Mechanics, Vol. 71, 2004, pp. 2347-2363.

5Edwards, R. P., and Newman, J.C., Jr., “Short-Crack Growth Behavior in Various Aircraft Materials”, AGARD Report, Vol. 767, 1990.

6Newman, J. C., Jr., and Edwards, R. P., “Short-Crack Growth Behavior in Aluminum Alloys - An AGARD Cooperative Test Program”, AGRD Report, Vol. R-732, 1988.

7Hoitsma, D. H., Jr., Anagnostou, E. L., Engel, S. J., , Madsen, J. S., Papazian, J. M., and Tang, L., “Structural Integrity Prognosis System Reasoning,” Proceedings of 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 2010 (to be published).

8Newman, J. C., Jr., Jordon, B., Anagnostou, E. L., Fridline, D., and Rusk, D., “Fatigue and Crack-Growth Analyses on Specimens Simulating Details in Wing Panels of Naval Aircraft,” Proceedings of Aging Aircraft Conference, 2008, Phoenix, AZ.

9Nardiello, J. A., Anagnostou, E. L., Christ, R. J., Jr. , Engel, S. J., Madsen, J. S., and Papazian, J. M., “Review of SIPS Experimental Results for System Calibration and Validation,” Proceedings of 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 2010 (to be published).

10Engel, S. J., Anagnostou, E. L., Fridline, D., Madsen, J. S., Nardiello, J. A., and Papazian, J. M., “Structural Integrity Prognosis System Demonstration and Transition,” Proceedings of 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 2010 (to be published).

11Tryon, R., “Using Probabilistic Microstructural Methods to Predict the Fatigue Response of a Simple Laboratory Specimen,” Engineering Design Reliability Handbook, edited by E. Nikolaidis, D. Ghiocel and S. Singhal, CRC Press, New York, 2005, Chap. 42.

12McDowell, D. L., Gall, K., Horstemeyer, M. F., and Fan, J., “Microstructure-Based Fatigue Modeling of Cast A356-T6 Alloy,” Engineering Fracture Mechanics, Vol. 70, 2003, pp. 49-80.

13Xue Y., McDowell D. L., Horstemeyer M. F., Dale M. H., and Jordon B., “Microstructure-Based Multistage Fatigue Modeling of a 7075-T651 Aluminum Alloy,” Engineering Fracture Mechanics, Vol. 74, 2007, pp. 2810-2823.

14Weiland, H., Nardiello, J., Zaefferer, S., Cheong, S., Papazian, J., Raabe, D., “Microstructural Aspects of Crack Nucleation During Cyclic Loading of AA7075-T651,” Engineering Fracture Mechanics, Vol. 76, 2009, pp.709-714.

15Zilberstein, V., Walrath, K., Grundy, D., Schlicker, D., Goldfine, N., and Abramovici, E., “MWM Eddy-Current Arrays for Crack Initiation and Growth Monitoring,” International Journal of Fatigue, Vol. 25, 2003, pp. 1147-1155.

16Matous, K., and Maniatty, A. M., “Finite Element Formulation for Modeling Large Deformations in Elasto-Viscoplastic Polycrystals,” International Journal for Numerical Methods in Engineering, Vol. 60, 2004, pp. 2313-2333.

17Jordon, J. B., Horstemeyer, M. F., Solanki, K., and Xue, Y., 2007, “Damage and Stress State Influence on the Baushinger Effect in Aluminum Alloys,” Mechanics of Materials, Vol. 39, 2007, pp. 920-931.

18Anagnostou, E., Brahme, A., Cornwell, C., El-Dasher, B. S., Fridy, J., Horstemeyer, M. F., Ingraffea, A. R., Lee, S. B., Maniatty, A., Noack, R., Papazian, J., Rollet, A. D., Saylor, D., and Weiland, H., “Simulation of Fatigue Crack Initiation and Propagation in Aluminum Alloys Using Realistic Microstructures”, Proceedings of 11th International Conference on Fracture, edited by A. Carpinteri et al., Paper 5612, Politecnico de Torino, Turin, Italy, 2005.

19Emery, J. M., Hochhalter, J. D., Wawrzynek, P. A., Heber, G., and Ingraffea, A. R., “DDSim: A Hierarchical, Probabilistic, Multiscale Damage and Durability Simulation System - Part I: Methodology and Level I,” Engineering Fracture Mechanics, Vol. 76, 2009, pp. 1500-1530.

20Emery J. M., “DDSIM: A Hierarchical, Probabilistic, Multiscale Damage and Durability Simulation Methodology,” Ph.D. Dissertation, Civil Engineering Dept., Cornell Univ., Ithaca, NY, 2007.

21Bozek, J. E., Hochhalter, J. D., Veilleux, M. G., Liu M., Heber, G., Sintay, S. D., Rollett, A. D., Littlewood, D. J., Maniatty, A. M., Weiland, H., Christ, R. J., Jr., Payne, J., Welsh, G., Harlow, D.G., Wawrzynek, P. A., and Ingraffea, A.R., “A Geometric Approach to Modeling Microstructurally Small Fatigue Crack Formation- Part I: Probabilistic Simulation of Constituent Particle Cracking in AA 7075-T651,” Modeling and Simulation In Materials Science and Engineering, Vol. 16, 2008, pp. 28.

22Hochhalter, J. D., Littlewood, D. J., Christ, R. J., Jr., Veilleux, M. G., Bozek, J. E., Ingraffea, A.R., and Maniatty, A. M., “A Geometric Approach to Modeling Microstructurally Small Fatigue Crack Formation- Part II: Simulation and Prediction of Crack Nucleation in AA 7075-T651,” Modeling and Simulation In Materials Science and Engineering (to be published).

23Neuber, H., “Theory of Stress Concentration of Shear Strained Prismatic Bodies with Arbitrary Non Linear Stress-Strain Law”, ASME Journal of Applied Mechanics, Vol. 28, 1961, pp. 544-550.

24Buczynski, A., and Glinka, G., “Multiaxial Stress-Strain Notch Analysis,” Symposium on Multiaxial Fatigue and Deformation: Testing and Prediction, Seattle, WA, May 1999.

25Harlow, D. G., Nardiello, J., and Payne, J., “The Effect of Constituent Particles in Aluminum Alloys on Fatigue Damage Evolution: Statistical Observations,” International Journal of Fatigue, Vol. 23, 2010 pp. 505-511.

Documents

Science-Based Multiscale Modeling of Fatigue Damage for ... · American Institute of Aeronautics and Astronautics 1 Science-Based Multiscale Modeling of Fatigue Damage for Structural