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i
A Comparative Finite Element Stress Analysis of Isotropic and Fusion Deposited 3D Printed Polymer
by
Robert Sayre III
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
In Partial Fulfillment of the
Requirements for the degree of
MASTERS OF ENGINEERING
Major Subject: Mechanical Engineering
Approved:
_________________________________________ Ernesto GutierrezMiravete, Project Adviser
Rensselaer Polytechnic Institute Hartford, Connecticut
December, 2014
ii
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
ACKNOWLEDGMENT .................................................................................................. vi
ABSTRACT .................................................................................................................... vii
Keywords ........................................................................................................................ viii
Nomenclature .................................................................................................................... ix
Acronyms .......................................................................................................................... xi
1. Introduction .................................................................................................................. 1
1.1 Additive Manufacturing ..................................................................................... 1
1.2 Classic Laminate Theory.................................................................................... 4
1.3 Modifications to CLT ......................................................................................... 7
1.4 Experimental Testing of FDM3D Printed Materials ...................................... 10
2. Theory/Methodology ................................................................................................. 11
2.1 Approach .......................................................................................................... 11
2.2 Finite Element Modeling of Isotropic Materials .............................................. 11
2.2.1 Material Properties ............................................................................... 11
2.2.2 Tensile Test .......................................................................................... 12
2.2.3 Compression Test ................................................................................. 13
2.2.4 Three Point Bend Test .......................................................................... 14
2.3 Finite Element Model of 3D Printed Materials ................................................ 16
2.3.1 Assumptions ......................................................................................... 16
2.3.2 Material Properties ............................................................................... 16
2.3.3 Tensile Test .......................................................................................... 17
2.3.4 Compression Test ................................................................................. 18
2.3.5 Three Point Bending Test .................................................................... 19
3. Results and Discussion .............................................................................................. 21
iii
3.1 Isotropic Material ............................................................................................. 21
3.1.1 Tensile Test Results ............................................................................. 21
3.1.2 Compression Test Results .................................................................... 22
3.1.3 Three Point Bending Test Results ........................................................ 23
3.2 3D Printed Material .......................................................................................... 24
3.2.1 Tensile Test Results ............................................................................. 24
3.2.2 Compression Test Results .................................................................... 26
3.2.3 Three Point Bending Test Results ........................................................ 27
4. Conclusion ................................................................................................................. 32
5. References .................................................................................................................. 33
iv
LIST OF TABLES
Table 1: Printers used for FDM3D Printing .................................................................. 10
Table 2: Tensile Lab Test for 3D printed ABS Specimens ............................................ 10
Table 3: Isotropic Material Properties ............................................................................ 12
Table 4: Filament Material Properties ............................................................................ 17
Table 5: Tensile Test Configuration ................................................................................ 17
Table 6: Composite Compression Laminate Configuration ............................................ 19
Table 7: Composite Tensile Analysis Results ................................................................. 25
Table 8: Composite Compression Analysis Results ........................................................ 26
Table 9: Composite Bend Analysis Results..................................................................... 28
v
LIST OF FIGURES
Figure 1: 3D Object Slicing and Path Generation ............................................................. 2
Figure 2: FDM3D Printing Process .................................................................................. 3
Figure 3: Coordinate Axis ................................................................................................. 5
Figure 4: Forces Acting on a Lamina ............................................................................... 6
Figure 5: Gap Representation ............................................................................................ 9
Figure 6: Tensile Test Specimen ..................................................................................... 12
Figure 7: Tensile Test Mesh ............................................................................................ 13
Figure 8: Compression Test Specimen and Mesh ........................................................... 14
Figure 9: Three Point Bending Test Mesh ....................................................................... 15
Figure 10: Ply Stack Up ................................................................................................... 18
Figure 11: Composite Tensile Test Mesh ........................................................................ 18
Figure 12: Composite Compression 1/4Symmetry Mesh .............................................. 19
Figure 13: Isotropic Tensile Test Model ......................................................................... 21
Figure 14: Isotropic Tensile Test Plastic Strain ............................................................... 21
Figure 15: Isotropic Compression Test von Mises Stress ............................................... 22
Figure 16: Isotopic Compression Test Plastic Strain ....................................................... 22
Figure 17: Isotropic Bend Test von Mises Stresses ......................................................... 23
Figure 18: Isotropic Bend Test Plastic Strain .................................................................. 24
Figure 19: Tensile Test von Mises Stress (16ply [0/0]) ................................................. 25
Figure 20: Composite Tensile Test TsaiWu Failure (32ply [45/45]) .......................... 26
Figure 21: Composite Compression Test von Mises Stress (64ply [0/0]) ..................... 27
Figure 22: Composite Compression Test TsaiHill Criteria (128ply [45/45]) ............... 27
Figure 23: Composite Bend Test von Mises Stress (16 ply [0/0]) .................................. 28
Figure 24: Composite Bend Analysis (16 ply [0/90]) von Mises Stress ......................... 29
Figure 25: Composite Bend Analysis (16 ply [45/45]) von Mises Stress ...................... 30
Figure 26: Composite Bend Test (16 ply [0/0]) TsaiHill Failure ................................... 31
vi
ACKNOWLEDGMENT
I would like to thank my project advisor, Ernesto GutierrezMiravete for his continuing
assistance on the development and guidance of this project.
vii
ABSTRACT
This project investigates modeling fusion deposition modeling (FDM) 3D printed ABS
parts using composite methods in FEA software. Prior research indicates the assumption
of perfect bonding utilized in composite theory is inaccurate and modifications to the
theory have been proposed. These modifications were included in the material
properties in Abaqus FEA software. These results were compared to an isotropic model
consisting of samples simulating a milled ABS plate. The results were also compared to
experimental values existing on FDM3D printed parts. The analysis revealed the FDM
3D printed parts indeed yield at a lower load than their comparable isotropic
counterparts. Treating a FDM3D printed part as a composite using simple
modifications to the material properties appears to agree better with experimental data
than the isotropic models. The number of layers within a given thickness does not
change the results for a uniaxial loaded sample, only changing the orientation of the
laminate results in changes. Thus, modeling a FDM3D printed component as a layered
composite with modified properties is an appropriate tool to simulate real world
conditions.
viii
Keywords
Fused Deposition Modeling
Finite Element Analysis
Additive Manufacturing
3D Printing
Material Properties
Computer Modeling
ASTM D638
ASTM D695
ASTM D790
ix
Nomenclature
Nx Normal force in the Xdirection Lbf
Ny Normal force in the Ydirection Lbf
Nxy Shear Force Lbf
Mx Moment about Xaxis Inchlbf
My Moment about Yaxis Inchlbf
Mxy Torsional Moment Inchlbf
��� Strain in Xdirection Inch/inch
���º Inplane normal strain, Xdirection Inch/inch
��� Bending curvature, Xdirection inch/inch2
��� Strain in Ydirection Inch/inch
���º Inplane normal strain, Ydirection Inch/inch
��� Bending Curvature, Ydirection inch/inch2
��� Shear strain inch/inch
���º Inplane shear strain Inch/inch
��� Inplane torsion inch/inch2
z Layer thickness Inch
�� Normal Stress in Xdirection Psi
�� Normal Stress in Ydirection Psi
��� Shear Stress Psi
[��]� Stiffness matrix for generally orthotropic
lamina

[A] Laminate extensional stiffness matrix 
[B] Laminate crosscoupling stiffness matrix 
[D] Laminate bending thickness matrix 
[T] Transformation matrix 
�� Stress in lamina local coordinate 1
direction (axial or fiber direction)
Psi
x
�� Stress in lamina local coordinate 2
driection (transverse direction)
Psi
��� Stress in lamina local coordinate 12
direction (shear direction)
Psi
��� E11 is the elastic modulus in the 1direction Psi
�� Area void density 
��� Elastic modulus of the plastic filament Psi
��� Elastic modulus in the 2direction Psi
�� Linear void ratio, along the direction of the
filament in the 2direction

��� Shear modulus of the laminate in the 12
direction
Psi
� Shear Modulus of the plastic filament Psi
� Empiral factor ranging from 0 to 1. The
value is mainly dependent on the gap size
between filaments.

�� Empiral factor ranging from 0 to 1. The
value is mainly dependent on the gap size
between filaments.

xi
Acronyms
ABS Acrylonitrile Butadiene Styrene
CLT Classical Laminate Theory
CNC Computer Numerically Controlled
FDM Fused Deposition Modeling
FEA Finite Element Analysis
PLA Polylactic acid
SLS Selective Laser Sintering
1
1. Introduction
Additive manufacturing processes allow users to both create oneoff prototypes without
requiring expensive tooling and create parts that are impossible or extremely challenging
to build using alternative methods. Components created with 3D printed technologies
such as fused deposition modeling (FDM) are constructed by drawing a molten plastic
filament along a predefined path in a single layer in the xy plane. The next layer is
printed on top of the preceding layer. This stack of printed layers is somewhat
analogous to a composite lamina contained within a laminate. Due to imperfect bonding
between the filaments both in the xy layers and between the yz layers, the material
strengths are reduced in both the xy plane and zplane as compared to a part
manufactured with injection molding or subtractive manufacturing such as milling.
This project seeks to analyze the reduced mechanical properties of a 3D printed material
by analyzing 3D printed samples created in FEA software as a laminate. Three test
samples will be modeled, a tensile test to ASTM D638 [1], a compressive test to ASTM
D695 [2], and a bending test to ASTM D790 [3]. These samples will first be modeled as
an isotropic model, as though it they were milled out of a sheet of ABS material, then
modeled as a FDM3D printed composite. These results will be compared to each other,
as well as to existing material property testing of FDM3D printed materials.
1.1 Additive Manufacturing
Additive manufacturing is a process where structured components are produced by
depositing a primary material and possible a secondary support structure to create an end
product. This allows shapes to be created that are challenging or impossible to
manufacture by any other means. This also allows for the relatively rapid creation of
low volume or “one off” structures, without the need for expensive tooling dies or jigs.
Printing with plastic materials such as acrylonitrile butadiene styrene (ABS) or
Polylactic acid (PLA) has become affordable for the home user with entry level pre built
or ready to build kits (RepRap or Makerbot are two popular examples). Kit prices start
at $500 [4].
2
Many types of 3D printing technologies exist including, stereo lithography, selective
laser sintering (SLS), etc. and each has its strength and weaknesses. This project focuses
on the fused deposition method (FDM) method of printing, which entry level FDM3D
printers such as the RepRap and Makerbot currently use.
The part generation process begins with the generation of a 3D CAD model, either
created by a user, or through scanning of a physical component. The model is input to a
program which first slices the model into a series of Zthickness planes, and then the
path of the extruder is created for each layer, as shown in Figure 1. The Zthickness is
determined by a combination of machine capabilities (the layer cannot be thinner than
the minimum Zheight step), print speed (thicker layers means faster prints), and print
quality (the thinner the layer means less deviation from the nominal part dimensions).
The resulting data is converted to machine code for the 3D printer to interpret.
Figure 1: 3D Object Slicing and Path Generation
FDM3D printing machines are typically a 3axis Computer Numerically Controlled
(CNC) enclosure in which parts are printed in layers in the XY direction, then the
extruder raises up a preset amount in the Zdirection and prints the next layer. The
material is deposited via the extruder, which is simply a heated nozzle that feeds the raw
filament to the extruder head at a predetermined rate (analogous to a hot glue gun in
3
operation). The material is heated past its melting point (270°F for ABS [5]) in the
extruder, then molten material is deposited onto the print bed. The print bed is the table
the extruder deposits the material onto. Support material may be added to the part in
order to support overhanging material before it is fully cured. This support material may
be removed during the cleaning process after the part is finished. The process is
depicted below in Figure 2.
Figure 2: FDM3D Printing Process
Ideally, commercially produced plastics (ABS or PLA) have isotropic properties, the
parts produced using these materials exhibit the same properties when manufactured
using subtractive manufacturing techniques, such as milling. Components printed with
3D printers tend to exhibit lower tensile, compressive, and bending strength than their
conventionally manufactured counterparts. Analytical methods presented in reference
[6] claim the strength of parts in the XZ and YZplanes is less than the strength in the
XYplanes.
Reference [7] encountered problems with the “dogbone’ portion of the tensile test
sample shearing due to gaps forming within the printed part, as well as the slicing
software creating offset contours. Offset contours slice the curves in a sample into a
series of smaller stepped curve to approximate a larger curve, similar to an analog to
digital conversion. They used test samples for tensile testing to ASTM D3039 instead to
solve the problem. The same P400 ABS material was used to create injection molded
samples to compare against the 3D printed samples in this case. The 3D printed parts
4
were printed 12 layers thick, with the orientation of each layer varying as follows: [0/0]
(axial) for all 12 layers, alternating [45/45], alternating [0/90], and [90/90] (transverse)
for all 12 layers.
The injection molded sample had the highest tensile strength (3,771 psi), and the second
highest was the [0/0] (axial) orientation. Next, the [45/45] orientation was slightly
higher than the [0/90] orientation and the weakest was the [90/90] (transverse)
orientation. Decreasing the air gap to 0.003inches resulted in a higher strength in all
samples, most notably in the [45/45] samples and the [0/90] samples. This experiment
also tested three compressive samples (cylinders) consisting of an injection molded
sample, and two 3D printed samples with [45/45] alternating layers, one built in the
axial direction with the layers stacked in series with the compressive load, and the other
in the transverse direction, with the layers stacked in parallel to the compressive load.
The 3D printed samples were weaker than the injection molded sample. The transverse
specimen had 15% lower compressive strength than the axial specimen. The
compressive strength testing concluded the maximum compression strength is
approximately double the tensile strength.
1.2 Classic Laminate Theory
Since FDM3D printed materials are deposited in layers they may lend themselves to
composite theory. A composite is composed of a stack of plies, of which each lamina
consists of a fiber enclosed by a matrix material [8]. A new coordinate system is
defined, a 12Z system, with the fibers oriented longitudinally in the 1 direction, the 2
direction normal to the 1axis, but orthogonal, and the Zdirection remains the same as
the XYZ coordinate system, with the lamina stacked in the Zdirection. These
coordinate systems are depicted in Figure 3. Two models exist, the specially orthotropic
case where the global XYZ coordinate system aligns with the principle material axis
(12Z) and the generally orthotropic case, when the principle material axis are not
coincident with the global material axis. Specially orthotropic only occur when the
laminates are stacked in the [0/90] orientation [8].
5
Figure 3: Coordinate Axis
Classical lamination theory (CLT) is used for analysis of composite materials. This
theory is valid for thin laminates with a small displacement in the transverse direction.
This theory shares the same classic plate theory assumptions of the Kirchhoff hypothesis
as well as perfect bonding between layers. Perfect bonding assumes there are no flaws
or gaps between layers, lamina cannot slip relative to each other, and the laminate acts as
a single lamina with combined properties of the layers.
Consider a laminated plate subjected to normal forces on the edges, Nx and Ny and shear
force Nxy as well as bending moments Mx, My, and Mxy, as depicted in Figure 4,. There
are three sets of strains occurring in each laminate as shown below in Eq [1] to [3]
��� = ���º + � ∗ ��� [1]
��� = ���º + � ∗ ��� [2]
��� = ���º + � ∗ ��� [3]
6
Figure 4: Forces Acting on a Lamina [9]
Where ���º is the inplane normal strain, ���
º is the inplane shear strain. ���º corresponds
to the membrane strains due to normal and shear forces, k corresponds to the bending
curvature, and z is the thickness. For CLT, z is assumed to be at the middle of the
laminate, with anything above the middle laminate having a +z value, and anything
below having a negative value. The constitutive equations for each ply of the laminate is
shown in Eq [4].
�
�������
�
�
= [��]� �
���������
� [4]
Where [��]is the stiffness matrix, which is composed of the compliance matrix [S], the
global coordinate stresses on the left hand side of the equation and the global strains on
the right hand side. By integrating each stress over its respective lamina, setting it equal
to the normal and bending forces and moments, then substituting the constitutive
equations the compliance matrix is created in Eq [ 5]
7
⎩⎪⎪⎨
⎪⎪⎧���������
��
���⎭⎪⎪⎬
⎪⎪⎫
= �� �� �
�
⎩⎪⎪⎨
⎪⎪⎧��º
��º
���º
�������⎭
⎪⎪⎬
⎪⎪⎫
[ 5]
[A] is the laminate extensional stiffness, [B] is the laminate crosscoupling stiffness, and
[D] is the laminate bending thickness. If the there is no external coupling, then [B] is
zero. To calculate the stresses and strains within each laminate, the stresses in the global
xy coordinate system must be transformed to the local 12 coordinate system using
transformation matrix [T] in Eq [6].
�
��������
�
= [�] �
�������
�
�
[6]
With these equations, all the stresses and strains can be found within a laminate.
1.3 Modifications to CLT
CLT assumes perfect bonding between layers, Reference [10] modeled the bonding
between P400 ABS filament for the purposes of FDM3D printing and concluded the
extruded filaments cannot be maintained at high enough temperatures long enough for
complete bonding to occur in the FDM3D printing process. Consequently, the
mechanical properties of the bonding zone between filaments are not the same as the
ABS filament in the printed part.
As mentioned before, the process of creating a 3D printed part starts with a CAD model
that represents the final desired part, this model is imported to a program which slices
the model into layers of a finite thickness. The layer size is controlled by the user,
within the limitations of the printer. The path for the extruder head is converted into
machine code, which the 3D printer can interpret. These paths account for the thickness
of the extruded material, a property of the extruder head and the gap between parallel
8
paths. This value is typically set to zero, but it can be modified to a positive (increasing
gap) or negative (interfering gap) value.
Testing in reference [5] on FDM3D printed ASTM D638 samples discovered a larger
(positive) gap value between filaments resulted in unpredictable results, due to the
additional voids incorporated into the samples. Reference [7] found decreasing the gap
to 0.003inches resulted in higher strengths in test samples compared to the default zero
value.
Reference [11] explores treating an ABS FDM3D printed material as a composite, but
argues composite theory requires perfect bonding between layers, which assumes the
plastic filaments are perfectly bonded, and any voids left between the filaments are
analogous to voids in the matrix material. The authors conclude the FDM3D printed
materials can never reach perfect bonding and propose introducing three new variables:
��,the area void density, ��, the linear void ratio, �,and �� empiral factors ranging from
0 to 1. The area void density uses the rule of mixtures to calculate this value. The value
is mainly dependent on the gap size between filaments. The linear void ratio is the void
ratio along the 2direction. The proposed modification is shown in Eq[ 7]Eq[ 9] below.
��� = (1 − ��) ∗ ��� Eq[ 7]
��� = � ∗ (1 − ��) ∗ ��� Eq[ 8]
��� = �� ∗ (1 − ��) ∗ � Eq[ 9]
Where E11 is the elastic modulus in the 1direction, E22 is the elastic modulus in the 2
direction, Epl is the elastic modulus of the plastic filament, G12 is the shear modulus of
the laminate, and G is the shear modulus of the plastic filament.
In [11], Three sets of test specimens were created with all layers in the [0/0]
(longitudinal), [45/45], and [0/90] (transverse) direction. Performing a uniaxial test
according the ASTM D3039/D yields the experimental values of E11, E22, G12, and the
9
Poisson’ss ratio ν12. The test also consisted of taking slices of the samples and
measuring the voids. Samples included gap sizes of 0.001968, 0.001, 0.0, and +0.100
inches. A depiction of the gaps in the filament is shown in Figure 5, with a zero gap on
top, a positive gap in the middle, and a negative (interference) gap on the bottom. The
test compared the theoretical calculations (assuming pure geometric shapes) and
experimental measurements using computer software to measure the voids. The
��theoretical value was found to be between 4.6% to 12.5% of the experimental value.
The �� theoretical value was found to be 6.6% different than the largest interference
size, then 35.7% for the smaller interference size, and 71% for the zero gap. Any gap
larger than zero results in a �� value of 1. The � value was found to be 0.96 for the
largest interference size, 0.82 for the smaller interference size, and zero for both the zero
gap and positive gap.
Figure 5: Gap Representation
The unidirectional laminate tests yielded a difference between theoretical and
experimental values for E11 and E22 between 0% and 16.48%. In addition to the uni
axial tests, additional tests in the [0/90], [15/75], [30/60], and [45/45] were performed.
Of these results, [0/90] laminates had the highest elastic modulus (254,137 psi)
experimental and 249,276 psi theoretical), and a [45/45] had the lowest elastic modulus
(249,276 psi experimental and 184,589 psi theoretical). The difference between
theoretical and experimental values varied between 3.1% to 7.1%.
10
1.4 Experimental Testing of FDM3D Printed Materials
Reference [12] investigated the tensile strengths of FDM3D printed materials in
accordance with ASTM D638. The same model file was printed on various FDM3D
printers with 100% fill, with the set variables being the difference in layer thickness and
the orientation of the print material. The printers used for this testing is shown in Table
1. The results for ABS shown in Table 2 shows the thinnest (2mm) layer height
specimens has the greatest tensile strength while the thicker layer height (4mm).
Specimens were also tested with alternating layer filament orientation such as [0/90] and
[45/45] orientation. These tests were performed at a test rate of 5mm/mm, which is
higher than other comparable tests.
Table 1: Printers used for FDM3D Printing [12]
Number Type Filament Printer 1 MOST RepRap Natural ABS, Clear PLA Printer 2 LulzbotPrusa Mendel RepRap Natural ABS, Purple PLA, White PLA Printer 3 Prusa Mendel RepRap Black PLA Printer 4 Original Mendel RepRap Natural PLA
Table 2: Tensile Lab Test for 3D printed ABS Specimens [12]
Specimens Tested
Specimens Considered
Average Tensile
Strength (psi)
Average Strain at Tensile
Strength (mm/mm)
Average Elastic
Modulus (psi)
0.4 mm Layer height
30 24 4,090 0.0197 271,945
0.3 mm Layer height
40 39 4,003 0.0231 251,785
0.2 mm Layer height
40 35 4,307 0.0201 266,724
[0/90] Orientation
60 52 4,017 0.0192 270,785
[45/−45] Orientation
50 46 4,278 0.0233 252,220
Total 110 98 4,134 0.0212 262,083
11
2. Theory/Methodology
2.1 Approach
The approach used in this work consisted of first modeling a sheet of commonly
available ABS material, cut to the tensile test sample dimensions. Finite element models
of various samples were created to simulate tensile, compressive, and bending tests
assuming isotropic properties. These models were regarded as a baseline for
comparison. Next, finite element models were created to simulate the same tests on
FDM3D printed ABS composites and the results were compared to the baseline.
2.2 Finite Element Modeling of Isotropic Materials
2.2.1 Material Properties
First a material was selected. Many grades of ABS plastic exist, a general use grade,
most similar to 3D printer filament with all the appropriate material properties available
was critical to a successful FEA model. The material selected is: Styron Magnum™
8325 ABS, Sheet Coextrusion Grade by Dow. This plastic is described as:
MAGNUM® ABS resins are thermoplastic materials which provide an excellent
balance of processability, impact resistance and heat resistance as imparted by
the various polymer compositions. MAGNUM ABS resin are available in a wide
range of melt flow rates, impact strength and heat resistance for both high and
low gloss applications manufactured by injection molding, sheet or profile
extrusion and thermoforming. MAGNUM® 8325 ABS resin is a low gloss sheet
coextrusion grade. [13]
The isotropic material properties are shown in Table 3.
12
Table 3: Isotropic Material Properties [13]
Description Value Units Tensile strength, ultimate 5000 psi Tensile strength, yield 5400 psi Elongation at break 100% Elongation at yield 2.60% Tensile modulus 265000 psi Gage length 2 Inches
Next three material property tests (tensile, compressive, and bending) were selected, in
an effort to best categorize the mechanical properties. The three tests selected are
commercial ASTM tests for plastics, they are:
ASTM D638: Standard Test Method for Tensile Properties of Plastics [1]
ASTM D695: Standard Test Method for Compressive Properties of Rigid
Plastics [2]
ASTM D790: Flexural Properties of Unreinforced and Reinforced Plastics and
Electrical Insulating Materials for commercial materials [3]
2.2.2 Tensile Test
The material for the tensile test is assumed to be ¼inch nominal thickness sheet ABS.
Using sheets of ¼inch thickness lends itself to the avoidance of machining the thickness
of the material. This lends itself to using type I specimens, with the dimensions
specified in [1]. The model assumes the portion of the sample held by the gage does
slip, hence the ends are truncated to omit that portion. The shape of the tensile test
specimens is shown in Figure 6.
Figure 6: Tensile Test Specimen
13
Abaqus elements C3D20R are employed, they are described as: A 20node quadratic
brick, reduced integration. This mesh contains 3,115 elements. The mesh is sufficiently
fine to identify stress concentrations in the material and avoids excessive distortion in
the elements within the mesh geometry. The material properties are converted into true
stress and true strain from the nominal stresses and strains in order to satisfy the material
input requirements for Abaqus. The model is displaced an initial value of 1.5 inches in
the axial direction, and the other end has a fixed condition. The model is run as a half
symmetry model to decrease computation time; the mesh is shown in Figure 7.
Figure 7: Tensile Test Mesh
2.2.3 Compression Test
The compression test uses material samples of cylinders of 0.5 inch diameter and 1 inch
length as shown in Figure 8, per reference [2], for an unreinforced material of sufficient
size to make a specimen. The compression test uses the same ABS material as the
tensile test. In a laboratory, the test is run until the sample fails (typically ruptures),
executing this failure phenomenon in FEA increases the complexity of the model.
Hence this FEA Model is run until the onset of yielding. Hand calculations of the stress
in a cylinder subject to uniaxial loads below in Eq [ 10][ 12] estimate the yields load at
1060.288 lbf.
�� = 5400��� [ 10]
�������� = � ∗ ������� = 0.19635��� [ 11]
���������� = �������� ∗ ��
= 1060.288���
[ 12]
14
The model is executed as an axisymmetric model, since a cylindrical shape lends itself
well to axial symmetry and this drastically decreases the calculation time of the model
compared to a ¼symmetry, ½symmetry, or full model. The model is run until yield
begins to occur through an iterative process to find a load that demonstrates areas have
gone elasticplastic while still achieving convergence. Since a total force cannot be
applied in this type of model, an equivalent pressure is applied to the top surface of the
cylinder section. For radius of 0.25inches this is equivalent to 5400 psi. The elements
used are: CAX8RH: An 8node biquadratic axisymmetric quadrilateral, hybrid, linear
pressure, reduced integration. This mesh contains 2,500 elements. The mesh is
sufficiently fine to identify yielding in the outer edges of the material and avoids
excessive distortion in the elements within the mesh geometry. The bottom edge of the
cylinder section is fixed in the Ydirection. The 3D representation and associated mesh
are shown in Figure 8.
Figure 8: Compression Test Specimen and Mesh
2.2.4 Three Point Bend Test
The flexural bend test from reference [3] is a three point bend test applied to a simply
supported beam. This test is limited to specimens that fail on their outer surface within
5% of the strain limit. The nose or cylinder applying a load to the rectangular specimen
shall have a radius of 0.197 inches. The rectangular specimen must be 16 times the
15
depth of the sample, meaning a ¼inch sheet results is a 4inch wide sample. The mesh
for the model is shown in Figure 9.
Figure 9: Three Point Bending Test Mesh
The test is modeled with the two fixed pins with a radius of 0.197inches and a depth of
1inch supporting the bar specimen with a third displaced into the bar. The stresses in
the pins are expected to be less than yield, so they are modeled as elastic pins with
C3D8R: 8node linear brick, reduced integration, hourglass control elements. The pins
are assumed to be steel (AISI 1015) with an elastic modulus of 29,700ksi, an ultimate
yield strength of 55,800 psi, an ultimate tensile strength of 47,100 psi and a poisons ratio
of 0.29 [14]. The third pin is constrained to only slide in the vertical direction a
prescribed a distance sufficient to cause yielding on the test specimen. The distance
sufficient to cause multiple elements to become plastic without excessive displacement
is 0.8inches.
The plastic test specimen is modeled as a 6inch long x 0.25inch thick x 1inch deep bar
using C3D20R: 20node quadratic brick, reduced integration type elements. This mesh
contains 15,360 elements. The mesh is sufficiently fine to identify stress concentrations
in the material and avoids excessive distortion in the elements within the mesh
geometry. The bar is modeled in contact with the three pins, and the two fixed pins are
used in lieu of pin constraints on the bar, to allow the bar to slide and rotate around the
pins. The bar is modeled longer than the supported span requirement (4inches) at 6
inches to again accurately capture the sliding of the bar along the anchor pins.
16
2.3 Finite Element Model of 3D Printed Materials
2.3.1 Assumptions
The same geometries above were modeled in Abaqus as composites. The following
assumptions were used in the analysis:
The Abaqus FEA software assumes perfect bonding between layers, which
ABS FDM3D printed parts do not have, therefore the modifications to the
material properties presented in 1.3 are used.
Bonding strength between filaments in the 2direction (transverse) is
assumed to be the same in the 2Z direction. In other words E22=E23=E13,
G12=G22=G13, V12=V23=V13.
Continuum elements are used due to the possibility of a nonlinear analysis.
Parts are printed with a 100% fill, resulting in a 0 gap to 0.003 gap between
filaments.
Plastic properties are not input to this analysis, instead the composite failure
criterion are evaluated to predict failure.
2.3.2 Material Properties
The properties were converted into true stress and strain in order to be input to the FEA
software. The material properties used were ABS filament properties shown in Table 4.
The material properties were input as lamina type material properties, which requires
additional material attributes as compared to an elasticplastic material, such as failure
stress or strain of the fibers and matrix. The experimental testing of 3D printed samples
yields the following properties for a small gap interference of 0.001inches [11] is
presented in Table 4. The tensile failure stress is assumed to be the average tensile
strength from testing in [12] or 4,133 psi. The compressive strength is approximately
double the tensile strength per [7], resulting in 8,266 psi. The shear failure stress is
conservatively estimated at 0.577 times the tensile strength using the von Mises criteria
or 2,384 psi. These failure criteria are the properties used for the fail stress material
inputs in the Abaqus FEA software.
17
Table 4: Filament Material Properties [11]
Description Value Units
E11 294,557 psi
E22 181,442 psi
ν12 0.39 
G12 59,465 psi
� 0.82 
�� 0.0434 
�� 0.3419 
Epl 323,434 psi
G 120,816 psi
ν 0.34 
2.3.3 Tensile Test
The tensile sample has the same geometry and loads as applied in section 2.2.2. The
sample is maintained at the 0.25inch total thickness with the numbers of layers varying
with the thickness. Since it is possible to include the shear effects in the model, a
continuum shell model is used, as opposed to a conventional shell. The plies thicknesses
and orientations to be tested are shown in Table 5. The plies are stacked parallel to the
load (axial) direction. This mesh contains 3,115 elements; the mesh geometry is the
same as the isotropic tensile sample test geometry. The elements used are SC8R: 8
node quadrilateral inplane generalpurpose continuum shell, reduced integration with
hourglass control, finite membrane strains.
Table 5: Tensile Test Configuration
Number of Plies Ply Thickness
(inches)
Total Thickness
(inches)
Orientation
16 0.015625 0.25 [0/0]
16 0.015625 0.25 [0/90]
22 0.0119 0.25 [0/90]
32 0.0078125 0.25 [0/90]
16 0.015625 0.25 [45/45]
22 0.0119 0.25 [45/45]
18
32 0.0078125 0.25 [45/45]
A representation of the plystack up for the 16 ply model in the [45/45] orientation is
shown in Figure 10. The mesh is shown in Figure 11.
Figure 10: Ply Stack Up
Figure 11: Composite Tensile Test Mesh
2.3.4 Compression Test
The composite compression analysis is created as a ¼symmetry model, to decrease
computing time and due to Abaqus not allowing composites to be axisymmetric models.
The 3D model is analyzed as a continuum shell since theses shells allow for linear and
nonlinear behavior and capture the through thickness response of composite laminate
structures. The same pressure of 5,260 psi for a total equivalent force of 1,032 lbf is
applied to the top surface of the cylinder. The bottom surface of the cylinder is fixed in
the Zdirection, and the X and Y faces have their respective symmetry boundary
conditions applied. The layers are stacked perpendicular to the loading direction (this
can be visualized as a stack of discs). This mesh contains 63,400 elements. The mesh is
sufficiently fine to identify stress concentrations in the material and avoids excessive
distortion in the elements within the mesh geometry. The elements used are SC8R: An
19
8node quadrilateral inplane generalpurpose continuum shell, reduced integration with
hourglass control, finite membrane strains. The various composite configurations to be
applied to the model are shown in Table 6.
Figure 12: Composite Compression 1/4Symmetry Mesh
Table 6: Composite Compression Laminate Configuration
Number of Plies Ply Thickness
(inches)
Total Thickness
(inches)
Orientation
64 0.015625 1 [0/0]
64 0.015625 1 [0/90]
84 0.0119 1 [0/90]
128 0.0078125 1 [0/90]
64 0.015625 1 [45/45]
84 0.0119 1 [45/45]
128 0.0078125 1 [45/45]
2.3.5 Three Point Bending Test
The geometry and loads are the same as stated in section 2.2.4. The composite bend
analysis uses the same material orientations as stated in Table 5, since the bar is the same
thickness as the tensile specimen (0.25inches). The plies are stacked perpendicular to
the load and pins. This mesh contains 12,960 elements. The mesh is the same geometry
20
as the isotropic three point bending test model. The elements in the model are SC8R, the
same used in section 2.3.3.
21
3. Results and Discussion
3.1 Isotropic Material
3.1.1 Tensile Test Results
The resulting von Mises stresses from the isotropic FEA model is shown in Figure 13.
The maximum von Mises stress is 6,057 psi. The center of the bar goes fully plastic
when a displacement of 0.24375inches has been applied. The plastic strain is shown in
Figure 14.
Figure 13: Isotropic Tensile Test Model
Figure 14: Isotropic Tensile Test Plastic Strain
22
3.1.2 Compression Test Results
The pressure on the top of the cylinder equivalent required to produce yielding was
5260.52 psi. This corresponds to a total force of 1032.9 lbf, 2.5% less than the predicted
yield force. The maximum von Mises stress is 5,263 psi, which occurs at the edges of
the cylinder, as shown in Figure 15. The presence of plastic strains depicted in Figure
16 through the cylinder demonstrates yielding has begun to occur.
Figure 15: Isotropic Compression Test von Mises Stress
Figure 16: Isotopic Compression Test Plastic Strain
23
3.1.3 Three Point Bending Test Results
The resulting von Mises stresses in the 3D model from the isotropic bend test are shown
in Figure 17. The max von Mises stress in the specimen was 6,156 psi, located at the
center of the bar. The top image shows the straight on view, the middle image shows the
surface in contact with the pin (the pin is removed for clarity), and the bottom image
shows the bottom of the specimen. Figure 18 shows the areas with plastic strain.
The maximum von Mises stress in the steel pins was 1,011 psi, an order of magnitude
less than the yield strength (55,800 psi), therefore the assumption of the pins as linear
elastic was valid.
Figure 17: Isotropic Bend Test von Mises Stresses
24
Figure 18: Isotropic Bend Test Plastic Strain
3.2 3D Printed Material
3.2.1 Tensile Test Results
The maximum von Mises stress for the first sample in Table 5 was 98,720 psi. Yielding
begins to occur when the displacement is greater than 0.25inches. The maximum von
Mises stress and maximum principle stress for all the tensile samples is shown in Table
7. There is a stress concentration near the radius of the test sample and this is to be
expected for a test sample. There is also a stress concentration at the fixed end of the
sample, which could possibly be mitigated by modeling the grippers of the machine, but
since no slip is assumed and the stresses do not affect the area of concern, then this
concentration can be disregarded. The rest of the models called out in Table 5, resulted
in the same stresses for all samples. This is due to Abaqus computing the composite as
an equivalent thickness material by combining the layer properties into one equivalent
layer. Since the strains and their respective stresses are calculated by integrating the [��]
matrix over each ply thickness and there are no applied moments, only normal forces,
then the summation of the resulting values over the over the entire laminate results in the
same [��]matrix when the thickness of the entire laminate remains the same. For
example, there are an equal number of plies of each orientation in the 16 ply [0/0] model
(eight 0º plies and eight 90º plies) and the 22ply [0/90] (11 of each ply). In both cases,
the [��]�º and [��]��º matrix are each half the thickness of the laminate regardless of the
number of plies.
25
Plastic deformation properties cannot be input to laminate material configurations, only
the fail stress or strain, which defines when the fibers and matrix fail within a laminate.
For this uniaxial test, the number of plies did not change the result, only the orientation
of the material resulted in material property differences. The strongest material appears
to be the [45/45] oriented material, regardless of the number of plies. The samples
showed failure according to the TsaiWu, TsaiHill, and max stress criteria in the
smallest load step, which corresponds to 0.25inches of displacement in the X or 1
direction. The area which have failed according to the TsaiWu criteria are shown in
Figure 20.
Table 7: Composite Tensile Analysis Results
Number of Plies
Orientation Max von Mises Stress (psi)
Max Principle Stress (psi)
16 [0/0] 98,720 98,610
16 [0/90] 97,850 100,700
22 [0/90] 97,850 100,700
32 [0/90] 97,850 100,700
16 [45/45] 58,200 58,080
22 [45/45] 58,200 58,080
32 [45/45] 58,200 58,080
Figure 19: Tensile Test von Mises Stress (16ply [0/0])
26
Figure 20: Composite Tensile Test TsaiWu Failure (32ply [45/45])
3.2.2 Compression Test Results
The max von Mises stress for a compressive sample was 5,478 psi, in the 64ply [0/0]
laminate. The results are shown in Table 8. Like the results in section 3.2.1, the number
of plies had no effect on the computed von Mises stress of the material. The possible
contributors to this situation were discussed in section 3.2.1 the strongest orientation
was in the [0/90] orientation.
Table 8: Composite Compression Analysis Results
Number of Plies
Orientation Maximum Von Mises Stress (psi)
Maximum Principle Stress (psi)
64 [0/0] 5,478 3,541
64 [0/90] 4,565 4,221
84 [0/90] 4,565 4,221
128 [0/90] 4,565 4,221
64 [45/45] 4,746 4,048
84 [45/45] 4,746 4,048
128 [45/45] 4,746 4,048
27
Figure 21: Composite Compression Test von Mises Stress (64ply [0/0])
Figure 22: Composite Compression Test TsaiHill Criteria (128ply [45/45])
3.2.3 Three Point Bending Test Results
The calculated max von Mises stresses are shown in Table 9. The von Mises stress for
the 16 ply [0/0] orientation bar is shown in Figure 23. The top third of the figure is the
YZ view, the middle view is a top down view with the pin removed for clarity, and the
bottom 3rd is the bottom view of the bar. In the 16 ply [0/90] orientation, the effects of
the laminate are shown in Figure 24, which is depicted in the same manner as Figure 23.
This is due to each ply being discretized as a continuum shell at the midpoint and the
stress being extrapolated to each node at the top and bottom surface.
28
Table 9: Composite Bend Analysis Results
Number of Plies
Orientation Max von Mises Stress (psi)
Max Principle Stress (psi)
16 [0/0] 21,180 20,810
16 [0/90] 20,020 20650
22 [0/90] 20,070 20710
32 [0/90] 20,110 20760
16 [45/45] 13,130 12870
22 [45/45] 13,120 12890
32 [45/45] 13,120 12890
Thus, for the [45/45] orientation the von Mises stress due to bending decreases slightly
when the number of plies increases, and is significantly lower (Figure 25). However the
von Mises stress is still more than double the one calculated for the isotropic samples.
Figure 23: Composite Bend Test von Mises Stress (16 ply [0/0])
29
Figure 24: Composite Bend Analysis (16 ply [0/90]) von Mises Stress
30
Figure 25: Composite Bend Analysis (16 ply [45/45]) von Mises Stress
Maximum stress failure criteria states that failure occurs when any principal material
axis stress component exceeds its corresponding strength. This failure criteria is
independent of shear stress and it does not account for interaction between stress
components. This method tends to agree well with experimental data when there is uni
axial stress in the principle material directions. Conversely, the agreement with
experimental biaxial data is poor. Failure occurs when this value is greater than 1 in
Abaqus.
31
The TsaiHill failure criterion is based on the von Mises failure criterion, it does not
reliably predict failure for cases of �� = �� or ��� = 0. For failure not to occur
according to the criterion, the value must be less than 1. TsaiWu failure criterion is a
tensor based failure crtierion that uses experimentally fitted data to evaluate failure.
This method does account for stress interactions and is good for ductile materials. The
value must be less than 1 to avoid failure. A plot of the TsaiHill failure for the 16 ply
[0/0] is shown in Figure 26, which shows the beam exhibiting failure in most areas
beyond the support pin. The point at which failure first occurs is calculated by linearly
interpolating between the first failure step and the preceding step. Since these FEA
models are executed as Abaqus/standard models as opposed to Abaqus/explicit, the
model is step the displacement of the pin linearly over the step. E.g. a time step of 0
corresponds to zero load, 0.5 corresponds to 0.5 times the load, and 1 corresponds to the
full load.
Figure 26: Composite Bend Test (16 ply [0/0]) TsaiHill Failure
32
4. Conclusion
Results indicate FDM3D printed materials modeled in FEA software using a lamina
configuration, tend to exhibit properties of the laminate as a whole rather than exhibiting
failure in a single lamina when performing uniaxial pull test or compression test. All
the test samples, tensile, compression, and bending had higher stresses than their
respective isotropic counterparts for a given load. In addition, composite parts yielded at
a lower load than their respective isotropic test samples. FEA users must consider the
failure criterion for FDM3D printed laminates are different from that for an isotropic
elasticplastic material. Since perfect bonding between FDM3D printed layers cannot
occur, the modifications to the material properties appear to be reasonable to mimic the
behavior of FDM3D printed parts and close to experimental results than using isotropic
samples.
33
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Test Method for Compressive Properties of Rigid Plastics," West Conshohoken,
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[3] American Society for Testing and Materials (ASTM), "ASTM D79010: Standard
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