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 Gutierrez-Miravete, 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 FDM-3D 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 FDM-3D 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: FDM-3D 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 1/4-Symmetry 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 (16-ply [0/0]) ................................................. 25

Figure 20: Composite Tensile Test Tsai-Wu Failure (32-ply [45/-45]) .......................... 26

Figure 21: Composite Compression Test von Mises Stress (64-ply [0/0]) ..................... 27

Figure 22: Composite Compression Test Tsai-Hill Criteria (128-ply [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]) Tsai-Hill Failure ................................... 31

vi

ACKNOWLEDGMENT

I would like to thank my project advisor, Ernesto Gutierrez-Miravete 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 FDM-3D printed parts. The analysis revealed the FDM-

3D printed parts indeed yield at a lower load than their comparable isotropic

counterparts. Treating a FDM-3D 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 uni-axial loaded sample, only changing the orientation of the

laminate results in changes. Thus, modeling a FDM-3D 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 X-direction Lbf

Ny Normal force in the Y-direction Lbf

Nxy Shear Force Lbf

Mx Moment about X-axis Inch-lbf

My Moment about Y-axis Inch-lbf

Mxy Torsional Moment Inch-lbf

��� Strain in X-direction Inch/inch

���º In-plane normal strain, X-direction Inch/inch

��� Bending curvature, X-direction inch/inch2

��� Strain in Y-direction Inch/inch

���º In-plane normal strain, Y-direction Inch/inch

��� Bending Curvature, Y-direction inch/inch2

��� Shear strain inch/inch

���º In-plane shear strain Inch/inch

��� In-plane torsion inch/inch2

z Layer thickness Inch

�� Normal Stress in X-direction Psi

�� Normal Stress in Y-direction Psi

��� Shear Stress Psi

[��]� Stiffness matrix for generally orthotropic

lamina

-

[A] Laminate extensional stiffness matrix -

[B] Laminate cross-coupling 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 1-2-

direction (shear direction)

Psi

��� E11 is the elastic modulus in the 1-direction Psi

�� Area void density -

��� Elastic modulus of the plastic filament Psi

��� Elastic modulus in the 2-direction Psi

�� Linear void ratio, along the direction of the

filament in the 2-direction

-

��� Shear modulus of the laminate in the 1-2

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 one-off 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 pre-defined path in a single layer in the x-y 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 x-y layers and between the y-z layers, the material

strengths are reduced in both the x-y plane and z-plane 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 FDM-3D printed composite. These results will be compared to each other,

as well as to existing material property testing of FDM-3D 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 FDM-3D

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 Z-thickness planes, and then the

path of the extruder is created for each layer, as shown in Figure 1. The Z-thickness is

determined by a combination of machine capabilities (the layer cannot be thinner than

the minimum Z-height 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

FDM-3D printing machines are typically a 3-axis Computer Numerically Controlled

(CNC) enclosure in which parts are printed in layers in the X-Y direction, then the

extruder raises up a preset amount in the Z-direction 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: FDM-3D 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 X-Z and Y-Z-planes is less than the strength in the

X-Y-planes.

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 -0.003-inches 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 FDM-3D 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 1-2-Z system, with the fibers oriented longitudinally in the 1 direction, the 2-

direction normal to the 1-axis, but orthogonal, and the Z-direction remains the same as

the X-Y-Z coordinate system, with the lamina stacked in the Z-direction. These

coordinate systems are depicted in Figure 3. Two models exist, the specially orthotropic

case where the global X-Y-Z coordinate system aligns with the principle material axis

(1-2-Z) 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 in-plane normal strain, ���

º is the in-plane 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 cross-coupling 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

x-y coordinate system must be transformed to the local 1-2 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 FDM-3D printing and concluded the

extruded filaments cannot be maintained at high enough temperatures long enough for

complete bonding to occur in the FDM-3D 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 FDM-3D 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 -0.003-inches resulted in higher strengths in test samples compared to the default zero

value.

Reference [11] explores treating an ABS FDM-3D 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 FDM-3D 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 2-direction. 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 1-direction, 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 uni-directional 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 FDM-3D Printed Materials

Reference [12] investigated the tensile strengths of FDM-3D printed materials in

accordance with ASTM D638. The same model file was printed on various FDM-3D

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 FDM-3D 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

FDM-3D 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 ¼-inch nominal thickness sheet ABS.

Using sheets of ¼-inch 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 20-node 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 axi-symmetric model, since a cylindrical shape lends itself

well to axial symmetry and this drastically decreases the calculation time of the model

compared to a ¼-symmetry, ½-symmetry, 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 elastic-plastic 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 0.25-inches this is equivalent to 5400 psi. The elements

used are: CAX8RH: An 8-node 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 Y-direction. 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 ¼-inch sheet results is a 4-inch 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 0.197-inches and a depth of

1-inch 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: 8-node 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 0.8-inches.

The plastic test specimen is modeled as a 6-inch long x 0.25-inch thick x 1-inch deep bar

using C3D20R: 20-node 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 (4-inches) 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 FDM-3D printed parts do not have, therefore the modifications to the

material properties presented in 1.3 are used.

Bonding strength between filaments in the 2-direction (transverse) is

assumed to be the same in the 2-Z direction. In other words E22=E23=E13,

G12=G22=G13, V12=V23=V13.

Continuum elements are used due to the possibility of a non-linear 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 elastic-plastic 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 -0.001-inches [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 0.25-inch 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 in-plane general-purpose 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 ply-stack 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 ¼-symmetry model, to decrease

computing time and due to Abaqus not allowing composites to be axi-symmetric models.

The 3D model is analyzed as a continuum shell since theses shells allow for linear and

non-linear 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 Z-direction, 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

8-node quadrilateral in-plane general-purpose 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 1/4-Symmetry 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 (0.25-inches). 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 0.24375-inches 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 0.25-inches. 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 22-ply [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 Tsai-Wu, Tsai-Hill, and max stress criteria in the

smallest load step, which corresponds to 0.25-inches of displacement in the X or 1-

direction. The area which have failed according to the Tsai-Wu 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 (16-ply [0/0])

26

Figure 20: Composite Tensile Test Tsai-Wu Failure (32-ply [45/-45])

3.2.2 Compression Test Results

The max von Mises stress for a compressive sample was 5,478 psi, in the 64-ply [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 (64-ply [0/0])

Figure 22: Composite Compression Test Tsai-Hill Criteria (128-ply [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

Y-Z 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 mid-point 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 bi-axial data is poor. Failure occurs when this value is greater than 1 in

Abaqus.

31

The Tsai-Hill 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. Tsai-Wu 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 Tsai-Hill 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]) Tsai-Hill Failure

32

4. Conclusion

Results indicate FDM-3D 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 uni-axial 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 FDM-3D printed laminates are different from that for an isotropic

elastic-plastic material. Since perfect bonding between FDM-3D printed layers cannot

occur, the modifications to the material properties appear to be reasonable to mimic the

behavior of FDM-3D printed parts and close to experimental results than using isotropic

samples.

33

5. References

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Test Method for Tensile Properties of Plastics," West Conshohocken, 2010.

[2] American Society for Testing and Materials (ASTM), "ASTM D695-10: Standard

Test Method for Compressive Properties of Rigid Plastics," West Conshohoken,

2010.

[3] American Society for Testing and Materials (ASTM), "ASTM D790-10: Standard

Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and

Electrical Insulating Materials," West Conshohoken, 2010.

[4] RepRap, "RepRap Mendel," [Online]. Available: http://reprap.org/wiki/Mendel.

[Accessed Nov 2014].

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material properties of fused deposition modeling ABS," Rapid Prototyping Journal,

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[8] R. F. Gibson, Principles of Composite Material Mechanics 3rd Edition, Boca

Raton: CRC Press, 2012.

[9] Efunda, "Classical Lamination Theory - Equilibrium," Efunda, Inc, 2014. [Online].

Available:

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Polymer Filaments in the Fused Deposition Modeling Process," Journal of

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Manufacturing Processes, vol. 6, no. 2, pp. 170-178, 2004.

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Fabrication of FDM Prototypes with Locally Controlled Properties," Journal of

Manufacturing Processes, vol. 4, no. 2, pp. 129-141, 2002.

[12] B. M. Tymrak, M. Kreiger and J. M. Pearce, "Mechanical properties of components

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[13] Matweb, "Styron Magnum™ 8325 ABS, Sheet Coextrusion Grade," [Online].

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http://matweb.com/search/DataSheet.aspx?MatGUID=46e0d29236da411a8083885

b6d01b020&ckck=1. [Accessed 27 Sep 2014].

[14] Matweb, "AISI 1015 Steel, cold drawn," [Online]. Available:

http://www.matweb.com/search/DataSheet.aspx?MatGUID=6cd3ff8c19bb42bda1fa

848e6d12bbb9. [Accessed Nov 2014].

[15] Makerbot Industries, "Makerbot Replicator 2 Desktop 3D Printer," [Online].

Available: https://store.makerbot.com/replicator2. [Accessed 15 10 2014].

[16] Amazon, "RepRap Prusa Mendel Iteration 2 Complete 3D Printer Kit," [Online].

Available: http://www.amazon.com/RepRap-Mendel-Iteration-Complete-

Printer/dp/B00DMRR4BG. [Accessed Nov 2014].

[17] Ultimaker, "Ultimaker 2," [Online]. Available:

https://www.ultimaker.com/pages/our-printers/ultimaker-2. [Accessed Nov 2014].