Fully Elastic versus Elastic-Plastic Material Deformation In Finite
Element Analysis
by
Nicholas Szwaja
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
In Partial Fulfillment of the
Requirements for the degree of
MASTER OF Mechanical Engineering
Approved:
_________________________________________
Professor Ernesto Guitierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Troy, New York
August, 2012
For Graduation December 2012
© Copyright 2012
by
Nicholas Szwaja
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1.0 INTRODUCTION ....................................................................................................... 1
1.1PURPOSE .............................................................................................................. 1
1.2 BACKGROUND .................................................................................................. 1
1.3 FULLY ELASTIC ................................................................................................ 4
1.4 ELASTIC-PLASTIC ............................................................................................ 6
1.5 FINITE ELEMENT ANALYSIS (FEA) ABAQUS ............................................ 7
2.0 METHODOLOGY AND APPROACH ...................................................................... 8
2.1 METHODOLGOY ................................................................................................ 8
2.2 APPROACH .......................................................................................................... 8
2.2.1 MODELING ....................................................................................................... 9
2.2.1.1 CREATING TEST SPECIMEN IN FEA ........................................................ 9
2.2.1.2 MATERIAL PROPERTIES .......................................................................... 12
2.2.1.3 MESH ............................................................................................................ 14
2.2.1.4 BOUNDARY CONDITIONS AND LOADING .......................................... 15
3.0 RESULTS AND DISCUSSION, ............................................................................... 18
3.1 FEA RESULTS ................................................................................................... 18
3.1.1 FULLY ELASTIC FEA RESULTS ................................................................. 18
3.1.2 ELASTIC-PLASTIC FEA RESULTS ............................................................. 25
3.1.3 ELASTIC-PLASTIC REVERSE LOADING .................................................. 29
3.1.4 ACTUAL TEST RESULTS ............................................................................. 33
3.1.5 ELASTIC-PLASTIC CAVITY ........................................................................ 34
3.1.6 ELASTIC-PLASTIC PORES INVESTIGATION ........................................... 40
3.2 DISCUSSION ...................................................................................................... 43
3.2.1 FULLY ELASTIC FEA DISCUSSION ........................................................... 44
3.2.2 ELASTIC-PLASTIC FEA DISCUSSION ....................................................... 45
iii
3.2.3 ELASTIC-PLASTIC REVERSE LOADING FEA RESULTS ....................... 47
4.0 CONCLUSION.......................................................................................................... 49
5.0 REFERENCES .......................................................................................................... 51
iv
LIST OF TABLES
Table 1: MECHANICAL PROPERTIES ........................................................................ 12
v
LIST OF FIGURES
FIGURE 1: Tensile Pull Test Results of High Strength Steel ........................................... 3
FIGURE 2: Metallic Bonds at the Atomic Level .............................................................. 5
FIGURE 3: Test Specimen 1 dimensions for standard tensile pull test .......................... 10
FIGURE 4: Sketch of Test Specimen .............................................................................. 11
FIGURE 5: Created part revolved around center line ..................................................... 12
FIGURE 6: Fully elastic material properties inserted into ABAQUS ............................. 13
FIGURE 7: Yield Stress - Plastic Strain .......................................................................... 14
FIGURE 8: Mesh ............................................................................................................. 15
FIGURE 9: Fixed boundary conditions ........................................................................... 16
FIGURE 10: Applied force of 15,000 lbs at Reference Point ......................................... 17
FIGURE 11: Von Mises Stress ........................................................................................ 18
FIGURE 12: Stress shown high in the smaller area ........................................................ 21
FIGURE 13: Forces in all three directions ...................................................................... 22
FIGURE 14: Deflection of 0.00836 inches ..................................................................... 23
FIGURE 15: Stress-Strain curve of the fully elastic deformation ................................... 25
FIGURE 16: Elastic-Plastic Tensile Test Maximum Stress ............................................ 26
FIGURE 17: Maximum deflection of 0.05072 inches..................................................... 27
FIGURE 18: Elastic-Plastic Stress-Strain Curve ............................................................. 28
FIGURE 19: Reverse Loading under the Tensile Load ................................................... 29
FIGURE 20: Reverse Loading Elongation for Tensile Load .......................................... 30
FIGURE 21: Reverse Loading Compressive Load ......................................................... 31
FIGURE 22: Reverse Loading Compressive Elongation ................................................ 32
FIGURE 23: Elastic-Plastic Stress-Strain Curve 1 .......................................................... 33
FIGURE 24: Test Specimen with Cavity ........................................................................ 35
FIGURE 25: Von Mises Stress with Cavity .................................................................... 36
FIGURE 26: Max Deflection........................................................................................... 37
FIGURE 27: Cavity stress-strain curve ........................................................................... 38
FIGURE 28: Reverse Loading: Cavity Test Specimen ................................................... 39
FIGURE 29: Test Specimen with Pores .......................................................................... 40
FIGURE 30: Mesh with Pores ......................................................................................... 41
vi
FIGURE 31: Stress with Pores ........................................................................................ 42
FIGURE 32: Cross Section Cut of Stress ........................................................................ 42
FIGURE 33: High Stress Center of Pores ....................................................................... 43
FIGURE 34: Test Specimen Necking.............................................................................. 45
FIGURE 35: Reverse Loading Stress-Strain Curve ........................................................ 48
vii
ACKNOWLEDGMENT
I would like to thank Rensselaer Polytechnic Institute for the resources, professionalism,
and providing me with the opportunity to obtain a Masters in Mechanical Engineering. I
would like to give a special thanks to Dr. Ernesto Guiterrez-Miravete for all his help,
mentoring, and patience he provided throughout the completion of this project. Also, I
would like to give thanks to my family and friends whom continue to provide unconditional support throughout my educational program at Rensselaer Polytechnic Institute.
viii
ABSTRACT
Finite Element Analysis (FEA) is an important engineering tool used to assist in approximating and verifying how a component will react under various external and internal
loading conditions. The purpose of this master of engineering project is to investigate
and analyze fully elastic and elastic-plastic deformation of a High Strength Steel (HSS)
tensile test specimen in FEA ABAQUS under various conditions. In general, material
selection is a key component to engineering analysis. If the material selection is performed incorrectly and does not have the strength, ductility, or physical features to
withstand the load then failure will become a realistic result. There are many conditions
where the stress applied could extend beyond the yield point of the material but FEA
does not necessarily provide visual indication that the part has yielded. The initial
results showed that the fully elastic material property has a linear stress-strain relationship regardless of the load applied while the elastic-plastic has a linear relationship up to
yield point and then becomes nonlinear beyond yield point. This FEA investigation will
also include elastic-plastic analysis on reverse loading, cavity geometry, and random
pores within the HSS tensile test specimen. This will be accomplished by using different
modeling techniques to investigate how FEA ABAQUS analyzes elastic-plastic material
deformation under various loading conditions and material conditions.
The reverse
loading condition resulted in tensile and compressive stresses that were equal and
opposite even in the plastic range. Work hardening should have increased the strength
of the test specimen so the compressive stress should have been smaller than the tensile
loading. The plastic data from the tensile test was not cyclic and only applied tensile
force so FEA ABAQUS did not know how to interpret the plastic range for compression.
FEA ABAQUS simply applied a negative force which resulted in an equal and opposite
stress. The cavity model had higher stress than the tensile and reverse loading conditions with the elastic-plastic data. FEA showed significant visual deformation within the
cavity of the specimen. The high stress and deformation were due to the reduction in
cross sectional area because stress is a function of force over the area. The pores modeled within the test specimen had minimal effect on the overall stress, strain or deflection
compared to the theoretical HSS test specimen that had no imperfections. The pores had
high stress because the pores were a source of high stress concentrations but the stress
ix
and strains not within the pores did not change.
The fully elastic and elastic-plastic
material selections each play a vital role in FEA ABAQUS while both can contain their
sources of errors. Regardless of the amount of stress applied, the stress-strain curve
acted linear for the fully elastic properties even beyond yield point. The elastic-plastic
material selection applies actual test data beyond the yield point and applies it to the part
being analyzed. The results showed nonlinear stress but when stresses outside the
uploaded values were applied FEA ABAQUS analysis resorted back to a linear stressstrain relationship.
x
1. Introduction
1.1 Purpose
The objective of this master of engineering final project is to investigate and analyze
Finite Element Analysis (FEA) ABAQUS when it is used to evaluate fully elastic and
elastic-plastic deformation on a High Strength Steel (HSS) tensile test specimen. A
tensile test specimen will be modeled in FEA ABAQUS as if it was being loaded in a
tensile test apparatus to validate the material properties. Multiple analyses will be
investigated including:
1. Fully elastic material properties under high tensile load
2. Elastic-plastic material properties under high tensile load
3. Reverse loading with elastic-plastic material properties
4 Cavity simulations with elastic-plastic material properties
5 Reverse loading of the elastic-plastic cavity model
6 Pores analysis using elastic-plastic material properties
The objective of this analysis is to obtain a better understanding how FEA ABAQUS
analyzes fully elastic and elastic-plastic material properties under various loading and
material conditions. By placing different conditions on a simple geometry (Cavity and
Pores analysis), it is easier to obtain a better understanding how FEA ABAQUS analyzes
the different models and how changing conditions affects the overall strength of the
component. The objective of the cavity is to gain more knowledge on how the HSS test
specimen will react to a different internal geometry. The objective of the pores analysis
it to verify the overall strength of the HSS test specimen when imperfections are present
within the material.
1.2 Background
One of the major topics within Finite Element Analysis (FEA) is how to interpreter
results and verify whether the results accurately depict the loading condition. Inaccuracies within the FEA results can stem from various sources including modeling, input of
1
material properties (fully elastic, elastic-plastic, thermal, fluid, etc.), mesh density,
unrealistic stress concentrations, loading conditions, environment, etc.
Engineers
understand FEA is an approximation tool and the analysis has to be validated by hand
calculations, test results, or inspection to confirm the validity of the results. Hand
calculations are sufficient for calculating theoretical solutions for a piece part component
under a single or simple loading condition. But as the design increases in complexity, as
various loads are applied, and as the environment complexity increases, the engineer
requires assurance that the FEA results are correct. Multiple loading and environment
conditions make hand calculations complex, where multiple assumptions have to be
made to simplify the solution which could lead to inaccurate results. The more assumptions made will lead to questionable results.
Material selection and understanding how material properties interact within FEA
ABAQUS is a major factor in confirming the results. This project will analyze the use
of fully elastic material properties in FEA ABAQUS and elastic-plastic material properties to investigate how FEA ABAQUS analyzes fully elastic and elastic-plastic
deformation under a high loading condition of High Strength Steel (HSS). The predicted
FEA results will look like Figure 1 that shows a stress-strain curve for HSS from actual
test data. Hand calculations are used to verify the linear geometry to obtain a better
understanding how hand calculations compare to FEA results. Figure 1 depicts the fully
elastic range and the transition into the plastic range for HSS. Another method to
validate the FEA results is by hand calculations. Hand calculations are simpler in the
elastic range because of the linear relationships that can be developed between stress,
strains, and deflections. Hand calculations can be used in the plastic range but nonlinear
analysis is not simple and typically requires numerous assumptions.
2
100000
80000
Tru
e 60000
Str
es
s
(ps
i)
Rate_1_Spline-N ew Plastic Strain (shifted by E=33,496,800)
Rate_2_Spline-N ew Plastic Strain (shifted by E=33,008,889)
40000
Rate_3_Spline-1 Plastic S train (shifted by E=34,233,333)
Rate_4_Spline-1 Plastic S train (shifted by E=34,291,200)
Rate_5_Spline-1 Plastic S train (shifted by E=35,581,000)
20000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Plastic Strain (in./in.)
Figure 1: Engineering Stress vs Engineering Strain
Figure 1 shows various engineering stress vs engineering strain curves because the test
specimen was pulled at different strain rates. Various strain rates has its advantages for
work hardening of materials. The curve means a lot of different things too many people
depending on the information they are trying to extract. The material’s elastic behavior
in tension (Young’s modulus, yield stress) will determine the stiffness and load bearing
capability while the plastic behavior (Ultimate strength, breaking strain) will give a first
3
0.4
indication about brittleness and notch sensitivity [2]. In compression (not shown in
Figure 1), the shape of the curve, especially near the yield stress is important with
respect to buckling resistance and other stability related behaviors [2].
For stability
calculations, more information is needed about the precise shape of the curve near and
past the yield point to determine the plasticity corrections [2]. In FEA, modulus and
plastic strains are required to be manually loaded into the system to ensure the model is
precise and an accurate result can be obtained. The values typically come from graphs
to accurately represent a curve.
1.3 Fully Elastic
Fully elastic deformation is defined as reversible alteration of the form or dimensions of
a solid body under stress or strain [3]. From Figure 1, the fully elastic range is the linear
portion of the curve up to 51,000 psi. In the elastic range, when the HSS tensile specimen is put under a load (tensile or compressive), the part will be stressed but will
maintain its ability to return to its original shape. At the atomic level, there is not
enough strain energy to break the metallic bonds [4]. As shown in Figure 2, under pure
axial load, the shear in the bonding has to be great enough to break the bonds between
the atoms or the atoms will return to their original form [4]. The stability of the bonds
will ensure the material will not plastically deform but once the internal energy is
enough to break the bonds there will be permanent deformation [4]. For a theoretical
analysis, a perfect test specimen has no defects or dislocations. Metals will have a
significant number of dislocations that will reduce the strength of the test specimen that
will decrease the amount of load that can be applied as shown in Figure 2. The molecular bonds will shift and form new bonds where there are defects within the material. A
material that has dislocations demonstrates that the bonds will slip and the specimen will
permanently deform at lower stress [4].
4
Figure 2: Metallic Bonds at the Atomic Level
Fully elastic deformation is typically governed by Hookes Law [ 5] that states:
σ = Eε
Where:
σ = Stress
E = Modulus of Elasticity
ε = Strain
Hooke’s law is a linear relationship that relates stress to strain by using the modulus of
elasticity of the material [5]. Fully elastic analysis only pertains to stress and strain up to
the yield point. Past yield point, the HSS should start to plastically deform.
In FEA, to analyze items within the fully elastic range only the Modulus of Elasticity
and material density are required to perform a structural analysis. The analysis is
assumed to be accurate up to yield point but will continue to act linearly beyond the
yield point. The linear relationship past the yield point will result in an inaccurate result
for high stresses and high strains.
5
1.4 Elastic-Plastic
Plastic deformation is defined as permanent change in shape or size of a solid body
without fracture resulting from the application of sustained stress beyond the elastic limit
[6].
Prior to looking at the physical defects of elastic-plastic analysis, it is important to
review the plastic stress-strain relationship at the atomic / crystal level. Plasticity in
metals is usually a consequence of dislocations within the structure [6]. Plasticity will
occur beyond the yield point of the material. For HSS the yield point is 54,000 psi
which is found in MIL-S-22698, Rev C, and condition AH-36. For many ductile metals,
including HSS, tensile loading applied to a test specimen will cause it to behave in an
elastic manner [7]. Once the load exceeds the yield strength threshold, there is more of a
rapid increase in stress than in the elastic range, and when the load is removed some
amount of the extensions still exists which results in permanent deformation [7].
Plastic deformation in practice typically means the component was under designed and
was not capable of handling the environment for which it was meant to operate in. This
will require additional rework or in a worse case scenario a complete redesign to ensure
that the component does not plastically deform and still has enough strength to support
the operational loading conditions. A change in material to increase the strength would
be the simplest solution to this problem but the engineer must take into consideration
availability of the metal, cost, and the other physical properties such as ductility, brittleness, ultimate strengths, etc.
There is cost increase associated with both redesign
options.
Plastic deformation can be beneficial as when used to produce work strain hardening by
metal working, also known as strain hardening or cold working [8]. Work hardening
increases the number of dislocations [8] forming new, stronger crystal bonds. Energy is
almost always applied fast enough and in large enough magnitude to not only move
existing dislocations, but also produce a great number of new dislocations by jarring or
working the material sufficiently enough [8].
6
Elastic-plastic analysis uses the Modulus of Elasticity from the elastic material properties but FEA ABAQUS requires yield stress and plastic strains of the material in the
plastic range to be manually loaded into the material properties. If the plasticity data has
not been entered into FEA ABAQUS, the stress/strain relationship will continue to be
linear. This will not provide an accurate result of stress in the plastic range. The concern
with mathematically analyzing plastic deformation is that the existing theories are
constructed largely on the basis of mathematical considerations and involve a number of
arbitrary assumptions of uncertain validity [7]. More experimental data needs to be
obtained to validate the numerical solutions. FEA is a method of using the experimental
data from plastic deformation (yield stress and plastic strain) and analyzing the deformation in the plastic range [7]. As a result, this reduces the number of assumptions and
leads to a more accurate result. Significant work has been done to provide a more
analytical method for calculating stress in the plastic range [7] [9] but a significant
number of assumptions would have to be made that will lead to an inaccurate mode of
comparison. Each of the stress-strain curves will be provided within their respective
sections. The curves will look like Figure 1, page 3 and will start to obtain nonlinearity
after the yield point of 51,000 psi.
1.5 Finite Element Analysis (FEA) ABAQUS
FEA ABAQUS is an approximation tool and is only as accurate as the operator allows it
to be. The operator has to input all of the material technical data that can be found in
various material science books, military and commercial specifications, and various
published works. If the material data is not manual uploaded, the program will not know
it exists, i.e. density, thermal conductivity, plasticity. Defects and dislocations have to
be modeled into the program or FEA ABAQUS will assume a theoretical perfect metal
with no defects.
This project will use FEA to run an analysis mimicking a tensile test on High Strength
Steel (HSS). The tensile test will be place under various loading conditions to investigate how FEA analyzes the results. Test Specimen 1 from ASME E8, Figure 3 will be
7
modeled in FEA ABAQUS and pulled using fully elastic material properties and then
using elastic-plastic material properties from HSS. The results will demonstrate how
FEA ABAQUS analyzed elastic and plastic deformation at a high load.
This analysis will also include how FEA analyzes reverse loading, cavity model, and
imperfections in the form of pores using the elastic- plastic material properties. The
elastic-plastic data is only for a tensile loading condition and does not include other
variations as this FEA investigation will produce.
2. Methodology and Approach
2.1 Methodology
The method for investigating the various materials and loading conditions is to model a
HSS test specimen in FEA ABAQUS. Once modeled the test specimen will be put
under various material conditions and loading. The first portion of the investigation will
involve a HSS tensile test sample (Figure 3), modeling it in ABAQUS, and applying a
large tensile load on the sample. The first analysis will only focus on fully elastic
material properties up to the yield point and then investigate how FEA performs with
large loads beyond the elastic range. The second portion of this investigation will be to
use the elastic-plastic material properties to perform the same tensile test under various
conditions including reverse loading, cavity simulation, and pores analysis. Once the
analysis is complete, a comprehensive comparison will be performed to derive a conclusion on the fully elastic versus elastic-plastic deformation in FEA.
2.2 Approach
This project will provide step by step direction from modeling the test specimen to
analyzing the results to developing the stress-strain curves in FEA ABAQUS. The fully
elastic and elastic-plastic material properties are loaded into FEA ABAQUS and the
HSS test specimen is loaded in tension, compression, modeled with pores and a cavity.
The end results of this project will investigate how FEA ABAQUS reacts to high loading
conditions for fully elastic and elastic-plastic deformation, especially beyond yield point.
8
2.2.1 Modeling:
2.2.1.1 Created Test Specimen in FEA
Perhaps one of the most important tests performed on a material is the tensile test. In a
tensile test, one end of a rod is clamped in a loading frame while the other end is subjected to a controlled displacement [10]. A transducer is typically connected in series
with the specimen to provide an electronic reading of the load corresponding to the
displacement [10]. The analysis included in this report will model a cylindrical test
specimen from ASTM E8 as shown in Figure 3 and Figure 4.
9
Figure 3: Test Specimen 1 dimensions for standard tensile pull test
The dimensions from Figure 3, test specimen 1 from ASME E8 were then modeled into
FEA ABAQUS. Because this is a cylindrical specimen, the revolve function was used in
FEA. Half of the longitudinal cross section was modeled as shown in Figure 4 and
revolved around the y-axis (centerline).
10
Figure 3: Sketch of Test Specimen
With the sketch complete, the part is revolved around the centerline to develop the final
solid shape that will be analyzed as shown in Figure 5. Radii have been applied to all
sharp corners because sharp corners on any component are a source of high stress
concentrations. To alleviate this, radii are applied to the edges and this will provide a
more accurate result.
11
Figure 4: Created part revolved around center line
2.2.1.2 Material Properties
The material properties for High Strength Steel (HSS) came from MIL-S-22698, Revision C, and condition AH-36. Table 1 provides the mechanical properties found in MILS-22698:
Tensile Strength
71,000 – 90,000 psi
Yield Strength (min)
51,000 psi
Density
0.283 lb/in3
Poisson’s ratio
0.3
Young’s modulus
29,500,000 psi
Table 1: Mechanical Properties MIL-S-22698
To perform the fully elastic analysis one material had to be created. This material has
three physical properties that need to be applied to the test specimen which are Young’s
Modulus, Density, and Poisson’s Ratio.
12
Figure 5: Fully elastic material properties inserted into ABAQUS
To perform the elastic-plastic analysis, the elastic mechanical properties are still required
because this will provide the stress-strain relationship up to the yield point but additional
information is required to ensure the plastic range is accurately developed. To develop
the plastic range in FEA, the yield stress and plastic strain need to be manually loaded
into the material properties. Figure 7 shows a small portion of the yield stress and
plastic strain that was manually loaded. To perform this analysis 283 yield stresses and
plastic strains points were manually loaded into the plastic material properties module.
This is sufficient because of the small change between each stress-strain point.
13
Figure 6: Yield Stress - Plastic Strain
Note that the fully elastic material properties are also present in the material properties to
ensure that stress applied up to the yield point is elastic. The way ABAQUS analyzes an
elastic-plastic problem is by using the fully elastic material properties until the yield
point is achieved and then it starts to apply the plastic material properties.
FEA
ABAQUS will abort its analysis if there is too large of a gap in stress between the yield
point and the start of the plastic stress values.
2.2.1.3 Mesh
The mesh of the HSS test specimen is an important element to modeling the test specimen because a poorly meshed component could show stresses that are not realistic. If
the elements are too large then the stresses between each element can be magnified but if
14
the mesh is too fine then the part might take up too much memory space within the
computer and will take too long to run. A really fine mesh will not produce a more
accurate result than if the mesh density was correct. A mesh density study was performed by changing the seed size of each element, running the analysis and viewing the
results to see if the stresses changed with the change in seed size. This was done until
there was no significant change in stress. The final element size for meshing the test
specimen is 0.15 element seed size.
Figure 7: Mesh
2.2.1.4 Boundary Conditions and Loading
To best simulate the tensile test apparatus, one end of the test specimen is fixed in the
test fixture and does not move. The other end of the test specimen is slowly pulled in
tension to simulate the stress-strain curve of the HSS metal. For FEA to simulate this
test, one end of the test specimen is fixed from displacing or rotating in the x, y, and z
direction. This simulates the test specimen being threaded into the test apparatus where
it would not be able to deflect, rotate, or translate. The boundary conditions are applied
15
in Figure 9 that show the end of the test specimen is fixed from displacement and
rotation.
Figure 8: Fixed boundary conditions
A tensile load is then applied to the opposite end of the test specimen. To apply the
load, a reference was created at the top surface of the test specimen. A reference point
provides a point on the surface of the test specimen to apply the load. The load was then
applied to the reference point and pulled a positive 15,000 lbf in the y-direction. The
load is shown in Figure 10. The load was selected at 15,000 lbf because the calculate
force to yield the test specimen was 10,014 lbf. To ensure test specimen goes well
beyond the yield stress, a load of 15,000 lbf was applied. This same force can be applied
to the fully elastic and elastic-plastic model to determine how ABAQUS analyzes these
results. The minimum force is calculated below:
16
Figure 9: Applied force of 15,000 lbs at Reference Point 1
17
3. Results, Discussions, Conclusion
3.1 FEA Results
The FEA results include Von Mises stress, displacement, and stress-strain curves from
the FEA ABAQUS program. The results will include analysis for the fully elastic tensile
loading condition, elastic-plastic material properties under the various loading conditions.
3.1.1 Fully Elastic FEA Results
Figure 11 is the Von Mises stress as calculated in FEA ABAQUS. As shown in Figure
11, there is high stress in the center of the test specimen while the highest stress is due to
a notch which is due to the change in section areas. Stress is a function of force over the
area so as the area is increased the stress will decrease. The maximum stress is located
towards the 0.375 fillet and is not located at the center of the test specimen.
Figure 10: Von Mises Stress
Hand calculations were performed to validate the maximum stress:
18
The calculated Von Mises stress is slightly less than FEA ABAQUS Von Mises stress.
The stress at the center at the test specimen according to the hand calculation is 54,019
psi. When comparing the calculated Von Mises stress to the ABAQUS Von Mises stress
at the center there is an error of 2.7 percent. This is fairly accurate and the hand calculation validates the stress at the center of the test specimen.
But ABAQUS is showing a maximum stress of 67,503 psi. This is caused by a notch
effect from the changing of areas. Whenever there is a change in area along a continu19
ous part, there are areas of stress concentrations. The stress concentration factor, kt is
calculated from Reference [11]. Any discontinuity in a machine part (i.e. fillets) alters
the stress distribution in the neighborhood of the discontinuity so that the elementary
stress equations no longer describe the state of stress in that part of these locations [11].
The next set of equations from Reference [11] calculates out the stress concentration
factor, kt, then applies that to the Von Mises calculated value.
With a stress concentration factor of 1.25 applied to the calculated Von Mises stress the
high stress in this part is 67,524 psi while the FEA ABAQUS Von Mises stress was
67,507 psi. This is a percent error of approximately 0.03 percent. If the kt value went to
more decimal places and was more exact, the percent error would be slightly smaller.
20
Figure 11: Stress shown high in the smaller area
Figure 12 shows stresses under the pure axial force. A majority of the stress in the axis
of the applied load are residual stresses that can be shown in the x and z directions.
Figure 13 shows the forces in each direction.
The high stresses in Figure 12 are ex-
pected to be in the area with the smallest diameter or where there are stress
concentrations. When the test specimen is subject to higher loads, the cross sectional
area should be reduced in diameter and necking should start to take place which occurs
when the stress exceeds yield stress and reaches the plastic limit prior to reaching the
ultimate stress [13].
21
Figure 12: Forces in all three directions
Figure 13 shows the forces in all three directions. Considering this is only a uni-axial
loading condition, there are small residual forces in the z-axis and x-axis that do not
contribute to calculating the Von Mises stress in FEA ABAQUS. The residual stress is
due to the nodal bonds between each element. The link transmits load in all three
direction and applies a small load in areas where force may not be acting in that direction. This is a source of error between the hand calculations and ABAQUS.
FEA ABAQUS calculates the maximum deflection in the y-direction to be 0.00836
inches as shown in Figure 14. The highest deflection is at the end of the test specimen
that is pulled with the tensile force. The portion with no deflection has all of the boundary conditions applied that keeps the specimen from rotating or deforming at the base.
22
Figure 13: Deflection of 0.00836 inches
23
The hand calculation validates the ABAQUS result. ABAQUS has a deflection of
0.00836 inches while the hand calculation is showing 0.00916 inches. This is different
because ABAQUS uses the Von Mises stress that takes into consideration all three
directions while the hand calculation only assumed stress in the y-direction. Stress in all
three directions will result in a change in cross sectional area and does not affect the ydirection deflection by much. The difference in elongation in the y-direction is 9.5%.
Figure 15 shows the stress-strain curve of the test specimen using the fully elastic
material properties. The stress-strain curve is calculated by ABAQUS and is from the
input data used to run the model, force applied and material properties. The stress-strain
curve is linear, goes to a maximum stress slightly under 80,000 psi, and has a strain as
high as 0.002. The yield point of this material is 51,000 psi and from this graph the
linear relationship continues through the yield point. Only putting the Modulus of
Elasticity, Density and Poisson’s Ratio in the material properties provides a linear
relationship even when the material should plastically be deforming. Discussion section
3.2.1 will provide further rationale on why this takes place in FEA and what this graphically does to the outcome of the FEA results.
24
Figure 14: Stress-Strain curve of the fully elastic deformation
3.1.2 Elastic-Plastic FEA Results
The maximum stress in the FEA model is located in the center of the test specimen at
59,692 psi. For this analysis, the use of fully elastic material properties (Young’s
Modulus, Density, and Poisson’s ratio) are used until the yield point is reached. The
plastic material properties are then applied by ABAQUS beyond the yield point of
51,000 psi. Once the stress has reached the yield point, the plastic material properties
are applied to the test specimen. This provides a maximum FEA ABAQUS stress of the
test specimen of 59,692 psi.
25
Figure 15: Elastic-Plastic Tensile Test Maximum Stress
For the fully elastic analysis, the maximum stress was at the stress concentration where
there was a change in cross sectional area while the elastic-plastic analysis shows the
maximum stress to be in the center of the HSS test specimen. The stress between the
fully elastic and the elastic-plastic analysis is directly related to the material properties
loaded into each model. The fully elastic material property does not allow for nonlinear
interactions beyond the yield point and would continue to show invalid results. The
elastic-plastic analysis manually loaded the plastic strain and stress to obtain realistic
results beyond the yield point.
Maximum deflection in FEA ABAQUS is 0.007853 inches as shown in Figure 17. With
the plastic data, the model can react nonlinearly beyond the yield point which allows for
work hardening and will reduce elongation. The stress-strain curve in section 3.1.2.3
will no longer increase as more force is applied. At this point the ultimate stress has
been surpassed, material can no longer withstand a high load, and the test specimen will
continue to deflect at a high rate until the rupture point is reached. Without defining a
rupture point in FEA ABAQUS, the test specimen will continue to deflect and stretch to
an infinitesimal small diameter.
26
Figure 16: Maximum deflection of 0.05072 inches
Compared to the fully elastic that had a deflection of 0.00863 inches, the elastic-plastic
deflection is lower. As force is applied in the plastic range and the strain is increased,
the deflection will be nonlinear. Work hardening will reduce the deflection and that is
why the deflection is much lower than in the fully elastic model. As the part is continuously placed in tension, nonlinear deflection will increase at a different rate as shown in
Figure 17.
The stress-strain curve for the elastic-plastic case is shown in Figure 18. The elastic
range of the curve is linear until it reaches the yield point of 51,000 psi. At this point the
strain is approximately 0.005. Beyond the 51,000 psi stress, the plastic material properties that were manually loaded govern the shape of the curve. From inspection, the
curve on Figure 18 has a slight curve to it but only reads up to a strain of 0.10. This
happens because the maximum plastic strain that was manually loaded was up to 0.10.
27
Figure 17: Elastic-Plastic Stress-Strain Curve
The fully elastic curve, Figure 15 shows a linear relationship past the yield stress with
strains that are not comparable to the elastic-plastic curve. An engineer reviewing the
results and checking the model must determine which analysis is more reliable if plastic
deformation is acceptable. A factor of safety calculation would assist in determining if
the results are reasonable based on a ratio between the maximum calculated stress and
the yield stress. By taking the yield stress and dividing it by the maximum stress obtain
from ABAQUS, the engineer can validate his design. Other methods will be discussed
in Section 3.2.2.
The stress-strain curve in Figure 15 is similar to the values loaded in that material
selection for plastic deflection but not exact. The amount of force applied during the lab
tests is gradual and increases over time while the FEA ABAQUS results has the large
load applied throughout the whole step. This would account for the different curves.
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3.1.3 Elastic-Plastic Reverse Loading
The purpose of this analysis is to investigate how FEA ABAQUS analyzes reverse
loading of the HSS test specimen. To perform this analysis, two steps were created in
ABAQUS. The first step is the tensile condition where a 15,000 lbf load is positively
placed on the HSS test specimen until it plastically deformed. While the test specimen is
plastically deformed, step two applied a negative 15,000 lbf load to the same point to
compress the HSS test specimen.
Figure 18: Reverse Loading under the Tensile Load
Figure 19 shows the first step where the tensile load is applied. The center of the smaller
cross sectional area sees a stress of approximately 54,000 psi which is consistent with
the elastic-plastic FEA analysis. The highest stress shown reads approximately 59,692
psi which is consistent with the elastic-plastic model.
29
Figure 20 shows the change in length due to the tensile load. The deflection in the ydirection is 0.008403 inches which is also consistent with the elastic FEA analysis.
Figure 19: Reverse Loading Elongation for Tensile Load
Figure 20 shows the deformed and non deformed shape of the HSS test specimen. There
is significant change in elongation in the y-direction where as in the x and z-direction
there is minimal change to the cross sectional area.
30
The next step was the compressive step that reverse loaded the HSS test specimen while
the test specimen was plastically deformed.
Figure 20: Reverse Loading Compressive Load
Figure 22 shows the second step of the reverse loading. The second step applies the
15,000 lbf load in the negative y-direction, compressing the HSS test specimen. Just as
in the first step, the maximum compressive stress is 54,000 psi stress and the maximum
stress at the stress concentrations around the fillet.
31
To show the change in length, Figure 21 shows the change in elongation along with the
original shape to show the compressive force acting on the HSS test specimen.
Figure 21: Reverse Loading Compressive Elongation
FEA has the change in length to be 0.008403 inches. The stress and elongation is
equivalent to the tensile loading which means FEA ABAQUS analyzed the tensile and
compressive loading as being equal and opposite. Upon reverse loading, the stress
returned to zero while the strain did not return to zero as shown in Figure 35. This will
be further discussed in Section 3.2.3.
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3.1.4 Actual Test Results
Actual test results can be provided and obtained by various material testing companies
that provides various industries with physical material properties. The test data provided
in Figures 23 depict various curves for different strain rates. A slower strain rate provides many benefits especially when obtaining strain hardening data because more stress
can be applied increasing the ultimate strength and rupture point. Strain hardening is
also dependent on the ductility of the material and HSS is a moderate ductile material.
Figure 23: Elastic-Plastic Stress-Strain Curve 1
Figure 23 only shows a small portion of the total curve but these curves best represent
what was loaded into FEA ABAQUS to obtain an accurate stress and strain relationship
33
during periods of high stress. The linear portion (fully elastic) occurs during very low
strains for High Strength Steel (HSS). The nonlinear portion used in FEA ABAQUS for
the elastic-plastic analysis contained in this report is at Rate_1 that has a sharp transition
into the plastic range. The transition is sharp based on the Modulus of Elasticity because
of HSS ductility and its ability to deform under higher stresses.
The plastic range appears to have a slight curve but almost acts linear which occurs
because of the strain increments. The smaller the increment, the more linear the curve
looks because of the small steps between strains in the plastic range. This small range
was chosen for the FEA analysis to obtain a more accurate method for comparing the
values FEA values to the graphs. With just reviewing a small section of the graph, more
imperfections can be viewed. The fully elastic range is difficult to evaluate based on
Figure 23 due to the small strain for the elastic range. To evaluate the fully elastic range
hand calculations were performed but for the elastic-plastic analysis, for any discussion
on accuracy, the FEA results would have to be compared to Figure 23. Nonlinear
calculations require several assumptions that could create discrepancies between the two
methods of evaluations.
3.1.5 ELASTIC-PLASTIC CAVITY
Now that there is a general understanding on how the elastic-plastic analysis works in
FEA ABAQUS, the next topic investigates the same test specimen with a cavity inside
the middle of the geometry as shown in Figure 24. The cavity represents a casting where
the piece being created could have a large void or cavity in the middle. This would be
done for a majority of reasons especially to save weight.
Figure 24 shows a cut in the test specimen along the y-axis to show how the cavity was
machined into the part. A square section was removed from the part module and revolved around the y-axis. Once the HSS test specimen was revolve, the cut out created a
cylindrical void within the center of the piece. The result is as shown in Figure 26.
34
Figure 24: Test Specimen with cavity
The cut out to create the cavity is centered within the test specimen and reduces the
weight by approximately 25%.
The part was re-meshed with the elastic-plastic material properties applied and pulled in
tension to replicate a tensile test.
Figure 25 is the Von Mises stress as calculated by FEA ABAQUS. The maximum stress
is 70,792 psi which is significantly larger than the Von Mises stress with no cavity in the
test specimen (59,692 psi). The increase in stress is due to the reduction in surface area.
Stress is a function of force over area so as the area decreases the stress will increase.
35
Figure 25: Von Mises Stress with Cavity
The stress concentrations at the fillets are showing minimal stress and the high stresses
are more concentrated at the center of the test specimen. The reduction in cross sectional area at the center cavity created a large stress that is significantly larger than the fillet
area, which is approximately at 54,000 psi.
Figure 26 shows the deflection of the test specimen with the cavity. The deflection of
the test specimen is 0.2733 inches. Compared to the no cavity deflection at 0.05092
inches the deflection is significantly larger. Because elongation is a function of strain,
the larger strains at the center of the test specimen will induce a larger deflection.
36
Figure 26: Max Deflection
The stress-strain curve in Figure 27 is linear up to 51,000 psi (yield point) and then
becomes nonlinear. The stress goes up to 89,000 psi and the strain is approximately 0.13
which is a significant change from the solid test specimen that went up to 60,000 psi
with a strain of 0.10. The shape of the curve is different than the solid test specimen and
the cavity test specimen has a smoother nonlinear curve. Again, this is not exact to what
was loaded for the plastic stress-strain in the material selection. This is due to the
different ways the load is applied between the actual test and FEA. FEA ABAQUS did
not allow for a higher load over 15,000 lbf because the elements strains were too high.
The elements were being pulled past their plastic limits and to obtain the full curve the
strains had to be below the values loaded.
37
Figure 27: Cavity stress-strain curve
In the plastic region, the curve appears to start off linear but then transitions to an
arching curve. The elastic-plastic material data loaded had a fairly flat transition into the
plastic range which is demonstrated in Figure 27. After the test specimen was pulled a
little further, the curve started to develop into a parabolic shape. With only a small
portion of the complete stress-strain curve loaded into FEA ABAQUS, the transition
from elastic to plastic deformation was not a smooth curve.
The test specimen with the cavity was then subjected to the reverse loading condition to
investigate if the test specimen would react differently under compression. Figure 28
shows the complete cycle from tensile stress to compressive stress.
38
Figure 28: Reverse Loading: Cavity test specimen
The stress in the reverse loading cycle with the cavity was slightly smaller than the
tensile test at 70,490 psi. Figure 28 is an illustration on how the test specimen’s cross
sectional area changes dimensions, first a reduction and then an expansion. Again, the
highest stress is in the center where the cavity is located but the stress is equivalent in
tension and compression. No material values were loaded into the material selection for
compressive stress or strains so ABAQUS did its best effort to analyze the compressive
strengths that is to reverse load the force. This would make the stress-strain equal and
opposite based on the direction of the force.
39
3.1.6 Elastic-Plastic Pores Investigation
Materials are not perfect and imperfections affect the quality of the steel. HSS and
similar steels have pores, cracks, cavities, and other imperfections that affect the final
results of the stress-strain tests. This portion of the report will investigate the elasticplastic material property with pores modeled randomly throughout the specimen. Figure
29 shows the cross sectional area of a single plane cut of the test specimen.
Figure 29: Test Specimen with Pores
To create the pores, the test specimen had random size small spheres cut into within the
center of the test specimen. This created a complete circle and that acted like a small
void or imperfection within the test specimen. Two planes, y-z and y-x and the pores
were randomly placed within the test specimen at different diameters.
The mesh, as shown in Figure 30 is different than the previous test specimens. A
complete hex mesh around the whole test specimen did not work because of all the
pores. The pores created a disturbance in the hex mesh and distorted the element shape
so another method was chosen to mesh the center section. The hex mesh was used at
40
both ends and where the pores were modeled, a partition was implemented and a tetrahedral (tet) mesh was generated for the center portion. The test mesh is extremely dense
to ensure the stress or any other analysis can be accurately shown. Due to the density
increase of the tet mesh, the time to run the analysis increased significantly (10 minutes
compared to 1 minute for the hex mesh).
Figure 30: Mesh with Pores
Now that the part is meshed, the model is now ready to be analyzed. The stress, as
shown in Figure 31 is not much higher than 59,000 psi which is consistent with the solid
model with no pores, theoretically perfect.
41
Figure 31: Stress with pores
The high stress is located at the pores, which act as stress concentrations throughout the
whole test specimen. Figures 31 and 32 show a cross sectional cut of the test specimen.
The tet mesh is extremely dense and this is evident in Figure 34 around the pores.
Figure 32: Cross Section Cut of Stress
42
Figure 31 also illustrates how the tet and hex mesh meet up. They are connected by a
series of nodes but the transition between tet and hex is not smooth which could lead to
inaccuracies of the stress at those nodes. The high stress, as shown in Figure 33 is
located on the outer circumference of the pores. The pores circular shape allows for
high concentrations around 0 and 180 degree of the centerlines of the pores.
Figure 33: High Stress Center of Pores
The circular pores show distortion as they went from circular to oblong. The deflection
of the model is the same as the elastic-plastic with no pores at 0.00783 inches. If more
pores were created then this could have affected the elongation but there were no changes to elongation. The same is true for the stress-strain curve.
3.2 Discussion
This discussion section will further investigate the FEA ABAQUS analysis for fully
elastic and elastic-plastic material properties for a HSS tensile test specimen. Simple
calculations verified the fully elastic material properties while actual test data ensured
accurate results in the elastic-plastic range for tensile loading. The reverse loading
condition with the elastic plastic material properties is an investigative study to analyze
how FEA analyzes a component being extended into the plastic range then immediately
43
compressed. Additionally the cavity and pores models demonstrated that the cavity has
significant effects on the stress-strain of the test specimen while the pores model showed
no change in the overall stress-strain that would cause the part to fail.
3.2.1 Fully Elastic FEA Discussion
As shown in the FEA results of Figure 11, the stress-strain relationship is linear regardless of the amount of force is applied. FEA accuracy is based on the amount of fidelity
incorporated into the design. By only adding material properties that pertain to elastic
mechanical properties, the model will react linearly, even beyond the yield point. The
hand calculations show there is a 2.9% variance in the Von Mises stress when compared
to the FEA and a deflection of 9.5% different. Calculating the percentile difference in
strength and deflection past the yield point is not practical because Figure 23 shows the
test specimen going into the plastic range and nonlinearly deforming. Linear stress as a
function of strain is not applicable in the plastic region and should be discarded in an
engineering analysis.
When comparing the stresses in ABAQUS, two sections of the HSS test specimen were
reviewed. The center section of the test specimen because it had a smaller cross sectional area and the fillet radius because this is an area identified by FEA as having a high
concentration of stress. The smaller cross section of the test specimen shows the FEA
ABAQUS model and the hand calculations are consistent with one another. There is a
2.9% difference in strength with the hand calculations which is fairly consistent with
each other.
The stress concentration around the fillet is an expected location for a high stress but not
a predicted failure mode. Based on Figure 23, the failure mode seems to be within the
center of the test specimen and is not around the fillet. When performing the elasticplastic deformation the highest Von Mises stress was located at the center of the test
specimen.
Fully elastic material properties did not allow for realist failure results
beyond the yield point and high stresses were shown in the wrong locations. Figure 34,
provided by www.mse.mtu.edu.
44
Figure 34: Test Specimen Necking
One of the limitations of FEA ABQUS is with the fully elastic analysis, showed no
visual distortion of the model. As the test specimen is pulled beyond the yield point and
into the plastic range, there is a reduction in the cross sectional area as the strain is
increased [5]. The fully elastic analysis is only accurate up to the yield point and past
the yield point the results are not representative of what actually occurs. Engineering
redesign of the component should take place to ensure the part does not yield.
3.2.2 Elastic-Plastic FEA Discussion
Performing an elastic-plastic FEA ABAQUS model requires manual input the modulus
of elasticity and plastic strain values from actual test data. The results in the plastic
range do not guarantee 100% accuracy just because they are loaded from actual test
results. The rate at which the load is applied and the amount of force applied plays a
role in the final shape of the stress-strain curve in the elastic-plastic regions.
45
Non time dependent analysis does not allow the operator to control the strain rate. FEA
ABAQUS will apply the force based on the step but will do it at a set rate. To vary the
strain rate, a time dependent problem needs to be modeled which increases the difficulty
of the FEA analysis and was not part of the scope of this investigation. For optimal
results, the operator should select a strain rate that best suits the application and the
environment from which the component is operated. Material selection catalogues and
other material documents provide the metal in various conditions including hardening
properties, yield strengths, ultimate strengths, and ductility.
FEA ABAQUS results only show the HSS test specimen elongating and does not show
the HSS test specimen necking or rupturing. This occurs because of the element structure within FEA. The elements are structured in such a way that true necking will not
occur but will continue to show a decrease in cross sectional area based on the element
geometry. Figure 23 shows the results from a real pull test that failed. The area within
the center of the HSS test specimen physically changed in diameter and reduced in size
creating the “necking”.
Ultimately the test specimen failed at this location. FEA
ABAQUS does not show necking or the fracture point. The elastic-plastic result shows
the area does decrease in size but that decrease in size is constant throughout the whole
test specimen. Compared to the fully elastic results that show no change in cross sectional area, the elastic-plastic results are more accurate but do not accurately depict the
point of failure of the test specimen. Typically in a structural design, any stress over
yield is undesirable and will require a redesign but in some applications plastic deformation is desirable, example forgings. During a forging, the operator wants the piece of
metal to plastically deform to obtain a new shape but if too much force is applied the
shape could be distorted and unusable. Strain rates play a vital role in obtaining the final
shape of a forging and assist in developing the final ultimate strength of the forging.
The FEA ABAQUS stress-strain curve is not 100% accurate to the actual test data
because the force applied in ABAQUS is not gradual but instant which is different from
the actual test results. Because the material values loaded into the ABAQUS analysis
are from actual test results, it is difficult to manipulate the results or the loading condi-
46
tions. Strain rates play a vital role in determining the final material properties and which
could affect the analysis. Selection of which test results would vary based on the
application and environment. If the applied load is applied at a quicker rate then a
higher strain late curve should be selected to run the analysis.
What if the operator selects the wrong stress-strain curves from Figure 23? As shown, a
faster strain rate does affect the yield point. Once stress is applied beyond the yield
point, the HSS tensile test specimen will continue to plastically deform until failure. If
the operator does not understand this then the part could be under designed or over
designed which could yield negative results.
3.2.3 Elastic-Plastic Reverse Loading FEA Results
The reverse loading condition utilized the elastic-plastic material properties and graphs
from Figure 23. Plastically deforming the test specimen in pure tension and then reverses loading the test specimen into compression was investigated to see how ABAQUS
applied plastic strain properties under high compressive stress.
The elastic-plastic data was for the tensile range and did not contain negative stress or
plastic strains values. FEA ABAQUS results show the stress and strain acted linearly
during compression and the compressive strengths/elongations were directly proportional to each other as shown in Figure 22.
47
Figure 35: Reverse Loading Stress-Strain Curve
Figure 35 shows the stress-strain curve for the elastic-plastic reverse loading model.
Step 1 is the tensile loading on the test specimen which is consistent with Figure 18.
Step 2 applies the compressive force and the stress-strain curve changes direction and
the negative force creates a decrease in stress. Again, just like in the elastic-plastic
tensile analysis there is not a smooth transition between the elastic and plastic range in
step 1 and in between step 1 and 2.
The stress and strain in step 2 does not reach negative stress but acts linearly until it the
stress reaches zero. After the part was plastically deformed in step 1, the assumption
was the part would remain nonlinear. From the plot, step 2 is linear and never shows
signs of nonlinear plastic deformation. Since the elastic-plastic stress-strain values were
provided for the tensile range, ABAQUS did not have values for plastic stress and strain
on compressive values. Just like the fully elastic model, if no plastic data is loaded into
the material selection, the stress will act linearly. This is what occurred in the reverse
loading condition.
48
The elastic-plastic test specimen with the cavity showed a large increase in stress and
strain throughout the cavity. The reduced cross sectional area caused an increase in
stress because as the area decreases in diameter, the stress increases. The expected
results would be the part would fail sooner because the cavity reduced the robustness of
the test specimen. The higher stress is well beyond the yield point and the factor of
safety is well below 1.0. Figure 30 is a visual representation on what happens to the part
as it is pulled in tension and compressed in compression. The stress-strain curve produced the same plot except the stress was much higher, by 20,000 psi within the cavity.
Although it was able to withstand higher stress, the part will rupture earlier and will not
be able to withstand the higher stresses.
The elastic-plastic model with the pores demonstrated that with imperfections within the
HSS test specimen does not greatly affect the stress when compared to the model with
no imperfections. The stresses were identical as the specimen with no pores and high
stresses existed within the pores themselves but these are isolated to small areas. The
strain was not affect by the pores because there was no change in elongation. Implementing the mesh was a key component to the design of the HSS test specimen. Hex
mesh did not work due to the pores so a combination of hex and tet mesh was utilized.
The tet mesh was significantly denser than a hex mesh and created more elements within
the center of the test specimen because they were not uniform in size. From inspection,
the stress distribution did not seem out of the ordinary between the hex and tet mesh
although the transition from hex to tet was not a smooth transition.
With no increase in stress for the overall test specimen it validates the tensile tests as a
repeatable test. Imperfections will exist but as long as there are no significant cavities,
the tensile test will provide accurate, repeatable data points.
4.0 Conclusion
FEA ABAQUS is an approximation tool that provides representative values to complicated problems. FEA is only as accurate as the fidelity implemented by the operator.
The full elastic model showed a slight variation in stress and elongation, 2.3% and 9.5%
49
respectfully when compared to hand calculations. The fully elastic stress-strain curve is
a linear relationship which is not accurate past yield point. The elastic-plastic material
properties showed that for pure tension, the plastic strains and stress governs the shape
of the stress-strain curve. Also, the Von Mises stress and elongation was reduced when
compared to the fully elastic analysis because of the nonlinear properties. The reverse
loading of the elastic-plastic material properties demonstrated nonlinear deformation
through the tensile loading condition but under the compressive load, the stress-strain
relationship was linear and never fell below zero stress. Two additional cases were
analyzed with the elastic-plastic material properties. The first analysis included a large
cavity within the center of the test specimen. The cavity showed extremely high stress
due to the reduction in area. The stress-strain curve was equivalent to the plot with no
cavity except for the increase in stress values. The second investigation included the
HSS test specimen with random pores imbedded within the body of the specimen that
represented imperfections. The stress, strain and elongation did not change for the part
and the only significant increase in stress was within the pores. The pores showed visual
elongation but the stress-strain curve was the same as plot with no pores. The few pores
did not affect the overall strength of the test specimen.
This investigation demonstrated FEA ABAQUS’s ability to solve problems using elastic
and elastic-plastic material properties. In general, the results showed similar results as
that predicted but certain cases in the plastic range, especially compression in the reverse
loading did not react as expected.
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5.0 References:
[1] Labs HSS Tensile Material Tests Results
[2] C Van Hengel, Fibre Metal Laminates: An Introduction, 2001 Kluwer Academic
Publisher
[3] Chao, Isaac A simple geometric model for elastic deformations, Caltech, TU Munchen Germany
[4] George E. Dieter. Mechanical Metallurgy, McGraw-Hill Inc, New York New York,
1986.
[5] Karlsruhe - Allemagne, Tensile, Charpy, Creep and Structural Tests, 2003, vol. 6911
[6] Yang, Qiang, Mechanical basis and engineering significance of deformation reinforcement theory, State Key Laboratory of Hydroscience and Hydraulic Engineering,
Tsinghua University, Bejing 100084, China)
[7] Batdorf, S B; Budiansky, Bernard, A mathematical Theory of Plasticity Based on the
Concept of Slip, 1980
[8] Gil Sevillan, Large Strain Work Hardening and Texture, Progress in Materials
Science
[9] Robert D. Cook. Concepts and Applications of Finite Element Analysis. University
of Wisconsin – Madison, 1974
[10] David Roylance, Stress-Strain Curves, Massachusetts Institute of Technology,
August 23, 2011.
[11] Joseph E. Shigley, Mechanical Engineering Design, Seventh Edition, McGraw-Hill
2004
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