high-strain-rate property determination of high-strength …
TRANSCRIPT
The Pennsylvania State University
The Graduate School
College of Engineering
HIGH-STRAIN-RATE PROPERTY DETERMINATION OF HIGH-STRENGTH
STEEL USING FINITE ELEMENT ANALYSIS AND EXPERIMENTAL DATA
A Thesis in
Engineering Science and Mechanics
by
Jeremy M. Schreiber
© 2013 Jeremy M. Schreiber
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
May 2013
ii
The thesis of Jeremy M. Schreiber was reviewed and approved* by the following:
Ivica Smid
Associate Professor of Engineering Science and Mechanics
Thesis Advisor
Timothy J. Eden
Research Associate. Applied Research Laboratory
Thesis Advisor
Albert E. Segall
Professor of Engineering Science and Mechanics
Committee Member
Judith Todd
Professor of Engineering Science and Mechanics
Head of the Department of Engineering Science and Mechanics
*Signatures are on file in the Graduate School
iii
ABSTRACT
There is a great deal of interest in the behavior of metallic materials under high strain rate
loading. Finite Element Analysis (FEA) can be used to model these materials with a
reduction in the amount of experimentation needed for characterization. High strain rate
properties of materials are often difficult and expensive to obtain. There is a growing
interest in the high strain rate behavior of metallic materials. A finite element model of a
metallic ring under high strain rate loading was developed using the Johnson-Cook
constitutive material model in Abaqus CAE. High strain rate properties of AISI 4340 and
HF-1 steel were used for the analysis. The finite element model was coupled with the
split Hopkinson pressure bar technique, along with a novel experimental method of
characterization. The ring was modeled both axisymmetrically and in 3D to help ensure
accuracy in results. Failure was determined by defining a failure strain to start the process
of element deletion. Failure strain in the FEA was adjusted to induce failure in the ring.
It was found that element deletion would occur when the failure strain was below 1x10-
5.Results of both axisymmetric and 3D were found to be within 3% of each other with
respect to maximum von Mises stress, and failure modes were identical. The effects of
mesh type and defects are investigated.
iv
TABLE OF CONTENTS
LIST OF TABLES ........................................................................................................ vi
LIST OF FIGURES ...................................................................................................... vii
ACKNOWLEDGMENTS ...............................................................................................x
INTRODUCTION ...........................................................................................1 Chapter 1
BACKGROUND .............................................................................................4 Chapter 2
Finite Element Analysis .....................................................................................4 2.1
2.2 Strain Rate Testing ............................................................................................4
2.3 High Strain Rate Test Methods .............................................................................6
2.3.1 Split Hopkinson Pressure Bar ..................................................................................6
2.3.2 Explosive Materials ........................................................................................... 12
2.3.2.1 Research Department Explosive (RDX) ............................................................. 12
2.3.2.2 High Molecular Weight RDX (HMX) ................................................................ 12
2.3.2.3 Polymer Bonded Explosive (PBX 9501) ............................................................ 12
2.3.3 Other Methods of Achieving High Strain Rates ............................................ 13
2.3.3.1 Theta Sample ..................................................................................................... 13
2.3.3.2 Multi-Specimen Tensile Tester........................................................................... 15
2.3.3.3 Improvised Experimental Device (IED) ............................................................. 16
2.4 Material Constitutive Models ............................................................................... 19
2.5Materials ............................................................................................................... 20
2.5.2 AISI 4340 Steel .................................................................................................... 20
2.5.3 High Fragmentation (HF-1) Steel .......................................................................... 21
MATERIAL PROPERTIES AND CONSTITUITIVE RELATIONSHIPS .... 22 Chapter 3
3.1 Material Property Input ........................................................................................ 22
LINEAR ELASTIC MODELING.................................................................. 26 Chapter 4
STRAIN-RATE DEPENDENT PLASTIC DEFORMATION MODELING .. 30 Chapter 5
STRAIN RATE DEPENDENT AXISYMMETRIC MODELING ................. 33 Chapter 6
MESH REFINEMENT .................................................................................. 38 Chapter 7
7.1 Mesh Refinement ............................................................................................ 38
FAILURE PARAMETERS AND ELEMENT DELETION ........................... 41 Chapter 8
v
8.1 Failure Model .................................................................................................. 41
3D MODELING ............................................................................................ 42 Chapter 9
3D ECCENTRICITY MODEL .................................................................... 43 Chapter 10
10.1 Eccentricity Introduction ................................................................................. 43
SAMPLE PREPARATION FOR SPLIT HOPKINSON PRESSURE BAR Chapter 11
TESTING ...................................................................................................................... 50
11.1 Sample Design ................................................................................................ 50
11.1 AISI 4340 ........................................................................................................ 50
11.2 HF-1 Steel ....................................................................................................... 51
METALLOGRAPHIC ANALYSIS ............................................................. 53 Chapter 12
12.1 Sample Preparation .......................................................................................... 53
12.2 Etchants ........................................................................................................... 53
12.3 AISI 4340 Microstructure ................................................................................ 53
12.4 HF-1 Microstructure ........................................................................................ 55
SPLIT HOPKINSON PRESSURE BAR ANALYSIS .................................. 58 Chapter 13
13.1 Parameterization of Data ................................................................................. 58
13.2 Experimental AISI 4340 Properties ................................................................. 59
13.3 Experimental HF-1 Properties ......................................................................... 60
MODELING OF EXPERIMENTAL AISI 4340 .......................................... 62 Chapter 14
MODELING OF EXPERIMENTAL HF-1 .................................................. 64 Chapter 15
CONCLUSIONS: ........................................................................................ 66 Chapter 16
FUTURE WORK ........................................................................................ 69 Chapter 17
REFERENCES .............................................................................................................. 70
Appendix A: Model Inputs ............................................................................................ 73
Appendix B: Axisymmetric Failure Strain Variation in Johnson-Cook AISI 4340 ......... 77
Appendix C: Axisymmetric Failure Strain Variation in Experimental AISI 4340 ........... 83
Appendix D: Axisymmetric Failure Strain Variation in Experimental HF-1 ................... 89
Appendix E: Etchant Recipes......................................................................................... 95
vi
LIST OF TABLES
Table 2.1 Recommended Testing Methods for Various Strain Rates (9) ...........................5
Table 2.2 Typical Chemical Composition of AISI 4340 Steel (25) ................................. 20
Table 2.3 Typical chemical composition of HF-1 Steel (26) ........................................... 21
Table 3.1 Selected Material Properties for AISI 4340 Steel ............................................ 22
Table 3.2 Johnson-Cook Plastic Deformation Parameters .............................................. 23
Table 3.3 Johnson-Cook Failure Parameters .................................................................. 23
Table 4.1 Dimensions of the cylindrical ring in millimeters. .......................................... 26
Table 4.2 Maximum von Mises Stress found in each model ........................................... 29
Table 12.1 Summary of microstructural findings related to hardness data. ..................... 57
Table 13.1 Material property data for AISI 4340 calculated from Split Hopkinson
Pressure bar compared against Johnson-Cook published data for HRC30 AISI4340 steel.
...................................................................................................................................... 60
Table 13.2 Parameterized Johnson-Cook data for HF-1 steel. ........................................ 61
Table 15.1 Summary of Modeling Results for Johnson-Cook AISI 4340, Experimental
AISI 4340, and HF-1 ..................................................................................................... 65
vii
LIST OF FIGURES
Figure 2.1 Strain Rate categories of each test type and the assumptions that are made in
each experiment(9). .........................................................................................................6
Figure 2.2 Original Hopkinson bar apparatus for measuring pressures from explosive
detonation. (11) ...............................................................................................................7
Figure 2.3 General Arrangement of Split Hopkinson Pressure Bar as described by
Kolsky. (13) ....................................................................................................................8
Figure 2.4 Example of a typical modern day SHPB. This schematic diagram is similar to
the SHPB used in CITEL (33). ...................................................................................... 11
Figure 2.5 Drawing of theta sample from Conway's patent in 1974 (32)......................... 14
Figure 2.6 Theta sample after impact with tungsten block. Notice stress peak in bottom
right corner of sample. ................................................................................................... 14
Figure 2.7 Section view of the cupholder apparatus illustrating the plunger, lid, bottom
plate, outer support, and dog bone sample. ..................................................................... 16
Figure 2.8 3D solid model rendering of the blaster high strain rate device. ..................... 18
Figure 2.9 Modified ASTM E8-11 Test Article Design .................................................. 18
Figure 3.1 Stress-strain relationship for J-C AISI 4340 Steel at various strain rates. ....... 24
Figure 3.2 Stress-strain relationship of experimental data compared to Johnson-Cook
published AISI 4340 data at room temperature and a strain rate of 1x104. ...................... 25
Figure 4.1 SolidWorks drawing of cylindrical ring rendered in Abaqus CAE ................. 26
Figure 4.2 Baseline SolidWorks Simulation Xpress results showing an increase in stress
where the thin “notch” section of ring is located ............................................................ 28
Figure 4.3 Abaqus Linear Elastic results agree with SolidWorks Simulation Xpress
results. ........................................................................................................................... 28
Figure 5.1 Ring section before loading with tetrahedral mesh. ....................................... 31
Figure 5.2 Ring Section after loading. Notice the localized stress peaks in the corners of
the notched section ........................................................................................................ 31
Figure 5.3 Ring Section with top view after loading. Only one row of elements across the
notch may lead to the localized stress peaks in the notch. ............................................... 32
Figure 6.1: Computer Rendering of Axisymmetric Model.............................................. 34
Figure 6.2 Degree of failure estimation with respect to fracture strain ............................ 35
Figure 6.3 Axisymmetric model before loading shown with CAX3 axisymmetric triangle
elements. ....................................................................................................................... 36
Figure 6.4 Axisymmetric model after loading with a large damage initiation value (no
element deletion). .......................................................................................................... 36
Figure 7.7.1 Axisymmetric Model after mesh refinement ............................................... 38
Figure 7.7.2 Remesh of the Three Dimensional Model................................................... 38
Figure 7.3 Final mesh of the axisymmetric model meshed with 500 CAX3 3 node linear
axisymmetric triangle element. ...................................................................................... 39
viii
Figure 7.4 Final meshing of three dimensional model with ~6,500 8-node linear brick
elements. ....................................................................................................................... 40
Figure 7.5 Final meshing of three dimensional model with ~6,800 10 node linear
tetrahedral elements. ...................................................................................................... 40
Figure 9.1 Results of both brick and tetrahedral models after 45 microseconds. The
tetrahedral model predicts a larger maximum von Mises stress than the brick elements.. 42
Figure 10.1 Top down view of the inner diameter shift for eccentric models. ................. 43
Figure 10.2 Eccentric 3D model of ring section meshed with linear brick elements after
45µsec. Maximum von Mises stress is found to be 1.491x109 Pascal. ............................ 44
Figure 10.3 Eccentric 3D model of ring section meshed with quadratic tetrahedral
elements after 45µsec. Maximum von Mises stress is found to be 4.555x109 Pascal. ...... 45
Figure 10.4 Section view of more refined eccentric ring showing a 2 degree taper from
top to bottom. ................................................................................................................ 46
Figure 10.5 Fracture initiation in the thinnest section of the ring occurs at 15µs. This is
somewhat shorter than predicted in the less refined models. The notch section is
completely missing in the thinnest section at 30µs. ........................................................ 47
Figure 10.6 Fragmentation at 45µs. Breakup of the thick section occurs at the same time
predicted by the less refined 3D model. ......................................................................... 48
Figure 10.7 Fragmentation behavior at 60Us. The axial fragmentation of the larger
sections of the ring appear follow the Mott fragmentation theory (23). ........................... 48
Figure 11.1 Front view of metal samples. Left side is AISI 4340 and the right side is HF-
1 steel. ........................................................................................................................... 51
Figure 11.2 Side view of metal samples. Left side is AISI 4340 and the right side is HF-1
steel. .............................................................................................................................. 52
Figure 12.1 Non-etched photomicrographs of AISI 4340 showing well dispersed non-
metallic inclusions at different magnifications. .............................................................. 54
Figure 12.2 Optical photomicrographs of AISI 4340 steel after 7 seconds of etching with
2% Nital. The etchant revealed a martensitic microstructure. ......................................... 55
Figure 12.3 Optical photomicrographs of HF-1 steel prior to etching. It appears that the
non-metallic inclusions are smaller than the AISI 4340, but seem to be well distributed.56
Figure 12.4 Optical photomicrographs of HF-1 steel alloy at varying magnifications. It
appears that the overall grain size is much larger than in the AISI 4340, and shows a
mixture of fine and coarse pearlite. ................................................................................ 57
Figure 13.1 Split Hopkinson data printout for AISI 4340 steel showing a lower than
expected yield stress at 1x103 strain rate. ....................................................................... 59
Figure 13.2 Split Hopkinson data printout for HF-1 steel. .............................................. 61
Figure 14.1 Typical failure behavior of experimental AISI 4340 ring. It appears that
brittle fracture occurs at the notched section. ................................................................. 62
Figure 15.1 Initial failure of HF-1 steel sample. Notice only leading edge of notched
section has failed. .......................................................................................................... 64
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Figure B1 Axisymmetric Failure Strain of 1. Stress in Pa. (AISI 4340) .......................... 77
Figure B2 Axisymmetric Failure Strain of 0.1. Stress in Pa (AISI 4340) ........................ 77
Figure B3 Axisymmetric Failure Strain of 0.01. Stress in Pa (AISI 4340) ...................... 78
Figure B4 Axisymmetric Failure Strain of 0.001. Stress in Pa (AISI 4340) .................... 78
Figure B5 Axisymmetric Failure Strain of 0.0001. Stress in Pa (AISI 4340) .................. 79
Figure B6 Axisymmetric Failure Strain of 0.00001. Stress in Pa (AISI 4340) ................ 79
Figure B7 Axisymmetric Failure Strain of 1x10-6. Stress in Pa (AISI 4340) .................. 80
Figure B8 Axisymmetric Failure Strain of 1x10-7. Stress in Pa (AISI 4340) .................. 80
Figure B9 Axisymmetric Failure Strain of 1x10-8. Stress in Pa (AISI 4340) .................. 81
Figure B10 Axisymmetric Failure Strain of 1x10-9. Stress in Pa (AISI 4340) ................ 81
Figure B11 Axisymmetric Failure Strain of 1x10-10. Stress in Pa (AISI 4340) .............. 82
Figure B12 Axisymmetric Failure Strain of 1x10-15. Stress in Pa (AISI 4340) .............. 82
Figure C1 Axisymmetric Failure Strain of 1. Stress in Pa. (AISI 4340 Exp.) .................. 83
Figure C2 Axisymmetric Failure Strain of 0.1. Stress in Pa (AISI 4340 Exp.) ................ 83
Figure C3 Axisymmetric Failure Strain of 0.01. Stress in Pa (AISI 4340 Exp.) .............. 84
Figure C4 Axisymmetric Failure Strain of 0.001. Stress in Pa (AISI 4340 Exp.) ............ 84
Figure C5 Axisymmetric Failure Strain of 0.0001. Stress in Pa (AISI 4340 Exp.) .......... 85
Figure C6 Axisymmetric Failure Strain of 0.00001. Stress in Pa (AISI 4340 Exp.) ........ 85
Figure C7 Axisymmetric Failure Strain of 1x10-6. Stress in Pa (AISI 4340 Exp.) .......... 86
Figure C8 Axisymmetric Failure Strain of 1x10-7. Stress in Pa (AISI 4340 Exp.) .......... 86
Figure C9 Axisymmetric Failure Strain of 1x10-8. Stress in Pa (AISI 4340 Exp.) .......... 87
Figure C10 Axisymmetric Failure Strain of 1x10-9. Stress in Pa (AISI 4340 Exp.) ........ 87
Figure C11 Axisymmetric Failure Strain of 1x10-10. Stress in Pa (AISI 4340 Exp.) ...... 88
Figure C12 Axisymmetric Failure Strain of 1x10-15. Stress in Pa (AISI 4340 Exp.) ...... 88
Figure D1 Axisymmetric Failure Strain of 1. Stress in Pa. (HF-1 Exp.) .......................... 89
Figure D2 Axisymmetric Failure Strain of 0.1. Stress in Pa (HF-1 Exp.) ........................ 89
Figure D3 Axisymmetric Failure Strain of 0.01. Stress in Pa (HF-1 Exp.) ...................... 90
Figure D4 Axisymmetric Failure Strain of 0.001. Stress in Pa (HF-1 Exp.) .................... 90
Figure D5 Axisymmetric Failure Strain of 0.0001. Stress in Pa (HF-1 Exp.) .................. 91
Figure D6 Axisymmetric Failure Strain of 0.00001. Stress in Pa (HF-1 Exp.) ................ 91
Figure D7 Axisymmetric Failure Strain of 1x10-6. Stress in Pa (HF-1 Exp.) .................. 92
Figure D8 Axisymmetric Failure Strain of 1x10-7. Stress in Pa (HF-1 Exp.) .................. 92
Figure D9 Axisymmetric Failure Strain of 1x10-8. Stress in Pa (HF-1 Exp.) .................. 93
Figure D10 Axisymmetric Failure Strain of 1x10-9. Stress in Pa (HF-1 Exp.) ................ 93
Figure D11 Axisymmetric Failure Strain of 1x10-10. Stress in Pa (HF-1 Exp.) .............. 94
Figure D12 Axisymmetric Failure Strain of 1x10-15. Stress in Pa (HF-1 Exp.) .............. 94
x
ACKNOWLEDGMENTS
I would like to thank Ellis Dunklebarger for his assistance with machining and surface
grinding of my steel samples. I would also like to thank Dr. Dan Linzell, Lynsey Reese,
and Kendra Jones at the Civil Infrastructure Testing and Evaluation Laboratory (CITEL)
for their assistance with the Split Hopkinson pressure bar testing. I would also like to
thank Dr. Eden and Dr. Smid for advising and funding me throughout this work, and Dr.
Segall for the thoughtful corrections to my thesis.
1
INTRODUCTION Chapter 1
Understanding the failure mechanism of high strength steels under large strain
rates is of great importance in areas such as friction stir welding, armor design, and
ballistics. Material fragmentation at very high strain rates is difficult to predict and
requires extensive characterization.
In the past 60 years, many semi-empirical material models have been developed
by researchers to predict failure at increased strain rates (1,2,3,4,5). Unfortunately, these
models rely on high strain rate data that must be obtained using the Split Hopkinson
Pressure Bar (SHPB) or destructive testing (blast). The SHPB measures stress pulse
propagation through a metal bar to predict the stress-strain relationships of a material
(6,7).
Many assumptions must be made about the sample-bar coupling and the elastic
behavior of the bars (7), and there are issues with accuracy in tension testing above strain
rates of 103 s
-1(7). Characterization of test articles is time consuming and expensive.
Standard practice is to build and detonate experimental test articles, collect fragments,
and examine the fragment pattern and size distribution. In some cases, it is beneficial to
determine the failure initiation site to help in the design of the part. This type of analysis
proves to be very difficult, if not impossible with destructive test methods. (8).
Computer modeling is being implemented to help identify failure modes, and
reduce the number of experiments.
2
Computer modeling can reduce development costs and increase the speed at
which parts can be modified for further examination. Coupling high strain rate data from
SHB experimentation, a computer model was developed using current failure models to
predict failure. Three dimensional linear elastic modeling of a steel ring was first
considered using two different programs; Abaqus CAE and SolidWorks Simulation
Xpress. Results of this initial modeling were found to be approximately 25% different.
This was most likely due to the variability in the two different mesh sizes. High strain
rate properties were then input into the model using the Johnson-Cook constitutive
material model in Abaqus only with published data by Johnson and Cook.
This modeling led to further refinement and the addition of failure parameters and
the use of a fracture strain value to initiate failure. The fracture strain parameter is simply
the amount of plastic strain the ring will experience before failure is assumed to occur. To
ensure that the model was converging properly, an axisymmetric model was developed
simultaneously. After ensuring convergence between axisymmetric modeling and 3D
modeling, the time at failure, maximum von Mises stress, and fracture strain was
determined. Failure was determined by the onset of element deletion in the model.
Initial modeling of this fracture model showed evidence that the solver had issues
with predicting failure in thicker areas of the model. For this reason, a defect was
introduced into the 3D model. The defect was a shift in the inner diameter of the ring,
forming an eccentric ring. This ring predicted failure in the thin section of the model, as
expected. The time at failure for both the eccentric ring and the defect free ring were
identical. Both rings failed at 45µsec. Maximum von Mises stress was similar for both
3
rings as well. Once convergence of this model was confirmed, only the axisymmetric
model was used for analysis due to its very short run time.
SHPB testing was conducted on two alloys, AISI 4340 and HF-1 steel. These
properties were input into the axisymmetric model to determine the strain at fracture. It
was found that these alloys did not show any real dependence on the failure strain
parameter. Data acquired from the SHPB may not be correct and is suspect. Further
analysis of the high strain rate properties of these alloys is necessary.
4
BACKGROUND Chapter 2
Finite Element Analysis 2.1
The use of finite element modeling (FEM) has increased dramatically in the past
twenty years with the advent of higher performing computers, advances in modeling
software and improved constitutive models. This allows FEM on systems that once
seemed nearly impossible to solve. The degradation and failure of a material under
extremely high strain rates could only be estimated from dangerous experiments and
empirical predictions that are at times impractical and inaccurate (8). Today, there are
numerous fracture and failure models, often pre-loaded in the commercial FEM software.
One can simply pick and choose the model that best represents the material system.
These models can readily be modified for other material systems and can be translated to
many practical applications in industry.
2.2 Strain Rate Testing
Strain rate is defined as the rate of change of strain with respect to time. Materials
behave differently at high strain rates than at quasi-static strain rates. Different methods
are used to achieve desired strain rates. These methods, such as using conventional load
frames for quasi-steady state testing and SHPB for higher strain rates, are chosen to best
represent the strain rate and operating conditions of the material with the greatest amount
of reproducibility. Table 2.1 shows some examples of testing methods that are used at
various strain rates. Figure 2.2 shows which category each strain rate test falls under and
what assumptions must be made for that test.
5
Table 2.1 Recommended Testing Methods for Various Strain Rates (9)
Strain Rate (s-1) Testing Technique
Compression Tests
<0.1 Conventional Load Frames
0.1-100 Special Servohydraulic Frames
0.1-500 Cam Plastometer and Drop Test
200-104 Split Hopkinson Pressure Bar
103-105 Taylor Impact Test
Tension Tests
<0.1 Conventional Load Frames
0.1-100 Special Servohydraulic Frames
100-103 Split Hopkinson Pressure Bar (in tension)
104 Expanding Ring
>105 Flyer Plate
Shear and Multiaxial Tests
<0.1 Conventional Shear Tests
0.1-100 Special Servohydraulic Frames
10-103 Torsional Impact
100-104 Split Hopkinson Pressure Bar (in torsion)
103-104 Double-notch Shear and Punch
104-107 Pressure-shear Plate Impact
6
Figure 2.1 Strain Rate categories of each test type and the assumptions that are
made in each experiment(9).
2.3 High Strain Rate Test Methods
2.3.1 Split Hopkinson Pressure Bar
The Split Hopkinson Pressure Bar technique is used to acquire high strain rate
properties of materials through the measurement of stress waves created by a high speed
impact onto the material. This method has gone through many developmental iterations
over its lifespan. In 1872, John Hopkinson conducted stress wave experiments with iron
wires and determined that a wave propagates through a material under impulse loading
(10). These experiments were followed by Hopkinson’s son, Bertram Hopkinson. B.
Hopkinson developed a device to measure the pressure produced in the detonation of
7
high explosives or bullet impact (11). A drawing of B. Hopkinson’s original pressure bar
is shown in Figure 2.3.
Figure 2.2 Original Hopkinson bar apparatus for measuring pressures from
explosive detonation. (11)
This method was crude and had many drawbacks that needed to be addressed in
order to make accurate measurements. These problems were addressed by R.M Davies in
1948. It was found that most of the issues in the apparatus were caused by the inability of
the pressure bar to accurately measure pressure under rapid time changes of 1
microsecond or less (12). Inaccuracies were inherent to the design of the bar as well.
Pressure applied nearly instantaneously, affects the length, radius, and Poisson’s ratio of
the bar which affects the arrival time of the pressure wave, skewing results (12).
8
One year after Davies publication, H. Kolsky presented a modified version of the
Hopkinson Pressure Bar called the Kolsky bar, or as it is more commonly known, the
Split-Hopkinson Pressure Bar shown in Figure 2.4 (13).
Figure 2.3 General Arrangement of Split Hopkinson Pressure Bar as described by
Kolsky. (13)
Kolsky’s method addressed the issues that Davies had published. The operation of
the equipment is very simple. A detonator applied a transient pressure to the anvil. The
anvil transmitted this pressure through an extension that impacts the sample. This
extension is where the Split Hopkinson Pressure Bar gets its name from. The pressure
waves are captured using two special microphones, a cylindrical condenser microphone
and a parallel plate condenser microphone (13).
9
2.3.1.1 Theory
If the pressure applied to the cylindrical bar does not exceed the Young’s modulus
of the bar material, the propagation velocity of the compression pulse can be calculated
using Equation 1. Propagation velocity is v, E is Young’s Modulus, and ρ is density (13).
Equations 2-9are required to determine kinetic energy, radial velocity, and axial
displacement of the bar(19).
√
By assuming that amplitude of pressure is given by Equation 2 where P is
pressure amplitude and V is particle velocity, the displacement as a function of time is
given by Equation 3 (13). The parallel plate microphone measures the relationship for P
as a function of x and time, where x is defined as the distance traveled by the wave.
Radial displacement of the bar, γ, is given in Equation 4, where υ is Poisson’s
Ratio and r is the bar radius (13). The radial displacement is captured using a cylindrical
condenser microphone. In modern equipment, condenser microphones are replaced with
strain gages.
10
When the incidence pulse, PI, travels through the bar and sample, there are two
contributions that need to be accounted for. These contributions are from the reflected
pulse, PR, and the tension pulse, PT, that travels back through the bar after the incidence
pressure pulse reaches the end of the bar (13). This relationship is given in Equation 5.
Displacement of both the main transmission and extension bars with respect to
time can be calculated using Equation 6 and Equation 7. Equation 6 integrates the sum of
the incidence pulse with the reflected pulse from zero time to a certain time t and
multiplies with material constants to give main bar displacement. Equation 7 integrates
the transmission pulse from zero to time t and multiplies by material constants to give
extension bar displacement (13)
∫
∫
Strain can be calculated by dividing the difference in extension bar displacement
and main bar displacement by sample thickness, z (13). This is shown in Equation 8.
Kolsky found the results from the cylindrical microphone to be unreliable due to
the distortion of the bar during testing. Results from experimentation showed that the
pressure-time integral in Equation 9 was in agreement with the parallel plate microphone
11
(13). Kolsky concluded that the parallel plate condenser microphone was more accurate
and reliable than the cylindrical condenser microphone.
∫
2.3.1.2 Modern Day Example
The SHPBs that are used today are not that different than the bar used by Kolsky.
Instead of using condenser microphones to measure data, two strain gages are used. One
strain gage is placed on the incident bar and one is placed on the transmitter bar. This is
shown in Figure 2.4. By using strain gages instead of condenser microphones, data
analysis can be simplified by reducing all data analysis to a simple fast fourier transform
that is done in seconds by an attached PC once all dispersion corrections are taken into
account. Dispersion corrections are unique to each machine and are taken into account by
the software during the initial calibration of the unit.
Figure 2.4 Example of a typical modern day SHPB. This schematic diagram is similar to the
SHPB used in CITEL (33).
12
2.3.2 Explosive Materials
Many methods have been used to achieve very high strain rates. High strain rates
in pipes and other containers can be achieved by filling them with plastic explosives and
detonating the plastic explosives. This method gives the researcher many different
explosive velocities to choose from. Most plastic explosives are based on Research
Department Explosive (RDX). Higher energy content explosives are High Molecular
Weight RDX (HMX) and Polymer Bonded Explosive 9501 (PBX9501) (14,15).
2.3.2.1 Research Department Explosive (RDX)
RDX is a nitroamine explosive that was developed in the late 1890’s in an attempt
to create an explosive more powerful that trinitrotoluene (TNT). It has been used in
military and industrial applications since World War II due to its stability and its ability
to be mixed easily with other explosives. RDX has an explosive velocity of 8750m/sec
(15).
2.3.2.2 High Molecular Weight RDX (HMX)
HMX is similar to RDX, but has a higher explosive velocity (9100m/sec). In
comparison, TNT has an explosive velocity of 6900 m/sec (16). HMX is one of the most
powerful conventional explosives currently available. It is very difficult to manufacture,
and for this reason, it is typically reserved to high end military applications. HMX
becomes machinable when mixed with a polymer or polymer based explosive such as
RDX.
2.3.2.3 Polymer Bonded Explosive (PBX 9501)
PBX 9501 is a polymer bonded explosive containing 95% HMX. This variant of
the PBX series can be machined to precise dimensions by either pressing or standard
13
machining techniques. It was developed by Los Alamos laboratories in the 1960’s and
70’s, and is one of the most widely studied explosives due to its use in explosive lenses
for nuclear weapons (15).
2.3.3 Other Methods of Achieving High Strain Rates
At the start of this work, there was an interest in the development of a new
method for testing materials at high strain rates. The main criteria for the test apparatus
were the following:
Must be safe to use
The test must be repeatable
It must be cheaper to operate than current test methods
It must allow for testing in uniaxial tension
Three designs were developed with varying complexity and operation methods.
These designs are known as the Theta Specimen, the Multi-Specimen Tensile Tester, and
the Improvised Experimental Device.
2.3.3.1 Theta Sample
The theta sample was originally developed in the 1970’s by Professor Joseph
Conway in State College, PA. Figure 2.5 shows the theta sample. As a load is applied to
the top and bottom of the theta sample, a tensile force builds in the “neck” in the center of
the sample. It was thought that by dropping a weight on the sample moving at high
velocity, high strain rates could be achieved in the neck. Modeling of the theta sample
using finite elements revealed that the method may have some inherent problems. Sample
14
fabrication and loading are difficult to control and may lead to large scatter in the test
data. Figure 2.6 shows a stress peak in the bottom right corner of the model. Machining
may lead to stress concentrations in the part. The amount of machining work alone makes
this method prohibitively expensive, and is the main reason that the theta sample has not
been widely adopted for tensile testing (31).
Figure 2.5 Drawing of theta sample from Conway's patent in 1974 (32).
Figure 2.6 Theta sample after impact with tungsten block. Notice stress peak in
bottom right corner of sample.
15
2.3.3.2 Multi-Specimen Tensile Tester
The next method that was developed and built was the multi-specimen tensile
tester. This device consisted of five main parts; the plunger, the lid, the bottom plate, the
outer support, and the dog bone as shown in Figure 2.7.
This model tested three samples at one time. This was done to keep the bottom
plate aligned during loading. The tester was loaded by applying a force to the plunger at
the desired strain rate. This method has numerous drawbacks. Three samples must be
tested at once, which increases cost and complexity of the tester.
There is an inherent amount of error in testing three samples at a time. As the
samples are loaded, the load is assumed to be P/3. This can only be assumed until one of
the samples breaks. Once the first sample breaks, the load on the remaining samples
increases to P/2, and they make experience a shear force as the bottom plate twists in the
outer support.
For this reason, modeling and testing of this device was stopped and a new design
was considered. This new design is known as the Improvised Experimental Device (IED).
The next section discusses the IED.
16
Figure 2.7 Section view of the cupholder apparatus illustrating the plunger, lid,
bottom plate, outer support, and dog bone sample.
2.3.3.3 Improvised Experimental Device (IED)
The last design to be developed is the Improvised Experimental Device (IED).
This device creates high strain rates via chemical potential. A preliminary design of the
test fixture is shown in Figure 2.8. The design consists of a pressure chamber and an
action/manifold that holds a test article. A smokeless powder rifle cartridge is fired in the
rifle receiver section. Expanding gas from the cartridge travels down the manifold into
the pressure chamber. At the bottom of the pressure chamber is a circular disk which
holds one end of the test article. As the pressure increases, the disk is ejected from the
bottom of the pressure chamber. This action applies uniaxial tension on the test article.
Plunger
Lid
Outer Support
Dog
Bone
17
Powder charge and composition can be varied to obtain the desired strain rate. Micro-
strain gages are mounted to each test article to measure strain. Strain gages may fall off
or not react during the test, so redundancy in this design is a must. A pressure transducer
is mounted inside of the pressure chamber to monitor gas pressure during the experiment.
Data Acquisition Software (DAQ) is used to record strain, ultimate tensile stress, and
velocity. A view port can be added to the fixture to provide access for high speed video.
Data can then be utilized to parameterize high rate dynamic material constitutive models.
From these data, fracture energy can be determined. The fracture energy and constitutive
model become inputs for FE analyses. The test article shown in Figure 2.9 is a modified
design of ASTM E8-11, Subscale Dogbone. The test article can be machined at any
qualified machine shop. The circular disk will have to be replaced after several tests. This
design has not been built yet, but will be further developed in future work.
18
Figure 2.8 3D solid model rendering of the blaster high strain rate device.
Figure 2.9 Modified ASTM E8-11 Test Article Design
Rifle
Receiver
Gas
Manifold
Test Article
Circular
Disk
Pressure
Chamber Front View Section
View
19
2.4 Material Constitutive Models
In the past two decades, there has been much research done on the expansion of
ductile materials under high strain rate loading (1,2,3,4,5,17). These authors’ models are
very complex and are typically compared to experimental data to confirm accuracy.
Perhaps the most well-known constitutive material model is the Johnson-Cook
plasticity model (18, 19,20). Gordon Johnson and William Cook developed an empirical
dynamic model for ductile materials under various strain rates. They found that fracture
mainly depends on hydrostatic pressure rather than strain rate and temperature (18,19,20).
This is essential since materials fail differently when the strain rate is varied. However,
the Johnson-Cook model has its drawbacks. Campagne et.al. states that the Johnson-Cook
model gives plastic deformation information, not damage evolution and failure. There are
multiple options for a damage evolution model. A simple model can simply denote
failure as exceeding the shear strength of the material. More complicated models
typically determine failure with greater accuracy. Banerjee et.al. uses a Lagrangian
material point method simulation to model ductile failure of sealed steel containers
containing PBX9501 plastic explosive (17). The Lagrangian method was initially
developed by Sulsky et.al. and is a particle method that uses the state variables of a
material. These models become quite complex when additional effects are considered.
Some of these effects are introducing a plastic explosive quick burnoff model and
accounting for gas-solid interactions that may occur during the first few microseconds of
an explosion (17). Other researchers have used microstructural based assumptions in their
models (2,3,4). The current assumptions for failure in these types of models are that as
the strain increases, adiabatic shear bands form in the material or internal voids start to
20
form and coalesce, leading to catastrophic failure of the part (21,22,23,24). Failure
initiation is determined as an increase in porosity, and failure will occur when a
predetermined maximum porosity is reached.
2.5Materials
2.5.1 Introduction
Two different alloys were selected to be modeled, AISI 4340 and HF-1 steel.
These are both high strength steel alloys, and they are both used in munitions
applications.
2.5.2 AISI 4340 Steel
AISI 4340 is a high strength steel that is alloyed with chromium, molybdenum,
and nickel. Chemical composition is given in Table 2.1(25). This steel was chosen for
this work due to the large amount of information available under different heat treatment
cycles and strain rates.
Table 2.2 Typical Chemical Composition of AISI 4340 Steel (25)
Chemical Composition AISI 4340 (%)
C Ni Cr Mo Mn Si
0.37-0.43 1.65-2.00 0.7-0.9 0.2-0.3 0.6-0.8 0.15-0.30
21
2.5.3 High Fragmentation (HF-1) Steel
HF-1 steel is a high strength, high carbon alloy that is alloyed with manganese. It
is typically used in ordnance applications. Chemical composition is given in Table 2.2.
Table 2.3 Typical chemical composition of HF-1 Steel (26)
Chemical Composition HF-1 (%)
C Mn Si Cu Ni Cr Mo Al
1.00-1.15 1.70-2.10 0.70-1.00 0.35 max 0.25 max 0.20 max 0.06 max 0.020 max
22
MATERIAL PROPERTIES AND CONSTITUITIVE Chapter 3
RELATIONSHIPS
3.1 Material Property Input
The steel ring was initially modeled using only linear, elastic properties. During
these initial stages of modeling, only Young’s Modulus and Poisson’s Ratio were
required in the simulations. Material properties at quasi-static strain rates for selected
materials are shown in Table 3.1. More material properties were needed once plasticity
and failure at high strain rates were introduced.
Table 3.1 Selected Material Properties for AISI 4340 Steel
Elastic
Modulus (Pa)
Density
(g/cc)
Poisson’s
Ratio
%
Elongation
Ultimate
Tensile
Strength
(Pa)
Yield
Strength
(Pa)
AISI 4340 205 x109 7.80 0.285 22 745x10
6 470x10
6
3.2 Johnson-Cook Constitutive Material Model
The Johnson-Cook Constitutive Material Model is a purely empirical model that
is used to represent the strength behavior of materials subjected to large strain rates, such
as when a material is exposed to intense impulsive loading during the detonation of
explosives. This model is commonly used in finite element simulations of fracture and
failure of materials at high strain rates.
23
The model defines the yield stress, σy, of the material as:
[ ][
]
Where
Constants A, B, and C are experimentally determined material constants, where A
is the basic yield stress at low strains, B is the strain-hardening effects, and C is the strain
rate effects. Table 3.2 and Table 3.3 show Johnson-Cook parameters for AISI 4340 given
by Johnson and Cook (19,20).
Table 3.2 Johnson-Cook Plastic Deformation Parameters
Material A (MPa) B (MPa) C n m
AISI 4340 792 510 0.014 0.26 1.03
Table 3.3 Johnson-Cook Failure Parameters
D1 D2 D3 D4 D5
Melting
Temp
(K)
Transition
Temp (K)
Reference
Strain Rate
(mm/sec)
AISI 4340 0.05 3.44 -2.12 0.002 0.61 1793 255 7500
Plotting the Johnson-Cook plastic deformation parameters gives important stress-
strain relationships at varying strain rates. The Johnson-Cook model predicts an increase
in the yield strength as the strain rate increases. This is a major concern when developing
things such as munitions, or armor penetration rounds. Strain rates in these applications
can easily exceed 104s
-1. In these situations, quasi-static tensile data is insufficient in
24
designing these systems since the actual yield strength of the alloy will be much higher
than expected. These data are plotted in Figure 3.1.
Figure 3.1 Stress-strain relationship for J-C AISI 4340 Steel at various strain rates.
5.00E+08
6.00E+08
7.00E+08
8.00E+08
9.00E+08
1.00E+09
1.10E+09
1.20E+09
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Pre
dic
ted
Mis
es
Flo
w S
tre
ss (P
a)
True Tensile Strain (ε) mm/mm
1x104
1x10-4
1x10-2 1
10
1x102
Selected Strain Rates
1x10-5
25
Figure 3.2 Stress-strain relationship of experimental data compared to Johnson-Cook
published AISI 4340 data at room temperature and a strain rate of 1x104.
Figure 3.2 shows the relationship between experimental data from SHPB test in
Chapter 13 and published AISI 4340 data from Johnson and Cook. The experimental
AISI4340 data has a much steeper stress-strain curve than the Johnson-Cook data.
0
200
400
600
800
1000
1200
1400
0 0.02 0.04 0.06 0.08 0.1
von
Mis
es
Flo
w S
tre
ss (
MP
a)
Strain (mm/mm)
J-C AISI 4340
EXP 4340
HF-1
26
LINEAR ELASTIC MODELING Chapter 4
4.1 Initial Linear Elastic Modeling
A cylindrical ring was first drawn in SolidWorks and Abaqus CAE to simulate a
section of a larger part. Dimensions are given in Table 4.1. The computer aided design
(CAD) drawing of the ring is shown in Figure 4.1.
Table 4.1 Dimensions of the cylindrical ring in millimeters.
Outer Diameter Height Wall Thickness Notch Depth Notch Width
81mm 60mm 7mm 3.5mm 6.1mm
Figure 4.1 SolidWorks drawing of cylindrical ring rendered in Abaqus CAE
The first analysis was done on a ring of AISI 4340 steel. The material properties
of AISI 4340 steel were given in Table 3.1. A linear elastic analysis was performed using
Abaqus CAE and SolidWorks Simulation Xpress. The top and bottom of the ring were
fixed in the X, Y and Z directions. Any rotation was also fixed. The estimated radial
pressure exerted on the section from the PBX 9501 explosive is 1.2 GPa (27,28). In these
27
initial stages of modeling, 1x10-6
GPa was used for the internal pressure to keep the model
in the elastic regime.
A tetrahedral mesh was used in the Abaqus CAE model. The mesh is a 4-node
linear tetrahedron with Abaqus designation C3D4. SolidWorks Simulation Xpress did not
specify a name for its mesh. Models were created in both Abaqus CAE and SolidWorks
Simulation Xpress in order to compare the results in the elastic regime, and to ensure
convergence was occurring in the Abaqus model. Both the Abaqus model and the
SolidWorks model used the same part dimensions and material properties. SolidWorks
was used only as a baseline method for modeling. Using linear elastic modeling, both
Abaqus and SolidWorks Simulation Xpress predicted comparable values for maximum
von Mises stresses. Peak stresses were located at the notch area, as expected. Figure 4.2
and Figure 4.3 show the three dimensional rendering of the stress fields found using
Abaqus and SolidWorks Simulation Xpress. Table 4.2 lists the maximum von Mises
stress found in each analysis.
28
Figure 4.2 Baseline SolidWorks Simulation Xpress results showing an increase in
stress where the thin “notch” section of ring is located
Figure 4.3 Abaqus Linear Elastic results agree with SolidWorks Simulation Xpress
results.
29
Table 4.2 Maximum von Mises Stress found in each model
Finite Element Software von Mises stress (GPa)
SolidWorks Simulation Xpress 1.326x10-5
Abaqus CAE 1.035x10-5
There is a 25% difference in the estimated maximum von Mises stress between
the two parts. This discrepancy in maximum stress is to be expected due to the possible
differences in mesh density. SolidWorks Simulation Xpress uses a slider bar to adjust the
density of the parts mesh. This method does not allow for fine tuning of the mesh. Results
of the initial elastic modeling were determined to be close enough to permit further
evaluation of the model.
30
STRAIN-RATE DEPENDENT PLASTIC DEFORMATION Chapter 5
MODELING
Abaqus CAE was used in the subsequent modeling, as more advanced models
cannot be created in SolidWorks Simulation Xpress. A dynamic, explicit model of the
ring section was created in Abaqus CAE. Instead of constraining each end of the ring
completely, a single node was fixed in the X direction, to prevent “runaway” of the
model. Runaway is when the model is not constrained properly, leading to the model
moving in unexpected directions like a projectile. The rest of the model was free to
expand or move.
A time step of 3x10-5
seconds was used in this model since the pressures are so
high that failure would be expected to occur almost instantaneously. The estimated
pressure of 1.2GPa was used in this simulation since the model allows for plastic
deformation (27,28). A tetrahedral mesh with quadratic elements was used to increase
accuracy when excessive distortion is expected. This model is shown meshed before
loading in Figure 5.1. Figures 5.2.and 5.3 show different orientations of the ring after the
load is applied. Peak stresses are found at the thin notch section as expected. Localized
stress peaks are found along the edge of the notched section. This is most likely due to
the combination of the rough mesh and the stacking of the tetrahedral elements. Having
only one element spanning the notch may also lead to inaccuracies in the model during
analysis. These effects are easily removed by re-meshing with a finer mesh, or
partitioning the part and meshing using more accurate brick elements.
31
Figure 5.1 Ring section before loading with tetrahedral mesh.
Figure 5.2 Ring Section after loading. Notice the localized stress peaks in the corners
of the notched section
32
Figure 5.3 Ring Section with top view after loading. Only one row of elements across
the notch may lead to the localized stress peaks in the notch.
The maximum von Mises stress was 0.53GPa. This stress is nearly at the elastic
modulus of the material, and was achieved in nearly one millionth of second. There are
two possible material responses:
1. The material undergoes almost instantaneous plastic deformation.
2. The material instantly fails without undergoing any plastic deformation.
Proving which one of these possibilities happens is not possible using computer
modeling. The material response is proved by experimental methods which involves pit
tests of the material. The next step will be to determine if the model is converging. This is
done best by introducing a simplified model such as an axisymmetric model that uses
symmetry elements to reduce calculation times.
33
STRAIN RATE DEPENDENT AXISYMMETRIC Chapter 6
MODELING
6.1Axisymmetric Modeling
An axisymmetric model was developed to reduce the calculation time of the
dynamic explicit model and to check the convergence of the 3D model. The run time of a
three dimensional dynamic model is normally over twenty hours with a refined mesh. An
axisymmetric model can be run in less than one half hour. The axisymmetric model was
an integral part of the simulation to reduce computational time and allow for the rapid
investigation of different parameters.
Axisymmetric modeling was also used in this study to determine a value for a
displacement based failure criterion that will delete elements when they exceed a certain
value of plastic strain. This value was thought to be the fracture strain of the material.
Axisymmetric modeling also allowed for shorter calculation times to determine rough
values for the fracture strain. The axisymmetric model is shown in Figure 6.1.
34
Figure 6.1: Computer Rendering of Axisymmetric Model
In the subsequent modeling, a dynamic explicit time step of 60µs was chosen for
both the axisymmetric model and the 3D model. This time step was chosen to be
somewhat longer than where the onset of failure occurs, but not long enough to give the
model time to delete all of the elements. The loading was an instantaneous pressure of
1.2GPa which was used in subsequent modeling. The boundary conditions were identical
for both models. Both the top and bottom faces of the model were fixed in the Y direction
and rotation was fixed in the X and Z directions.
Fracture strain for the material was determined by varying the fracture strain
value from 1mm/mm to 1x10-15
mm/mm until element deletion occurred throughout the
model. A fracture strain of 1x10-5
mm/mm was determined for the ring from axisymmetric
modeling. Models with a lower value for fracture strain produced the same fragmentation
behavior as 1x10-5
. This is shown in Figure 6.2.
35
Figure 6.2 Degree of failure estimation with respect to fracture strain
This value of fracture strain was used in the 3D model and specifies the onset of
element deletion. A stress analysis was conducted on a highly refined axisymmetric
model to compare with three dimensional results. Element type was a 4 node bilinear
axisymmetric quadrilateral with reduced integration and hourglass control. Failure was
predicted to start at approximately 32 microseconds. The peak von Mises stress was
estimated to be 1.41GPa.
Similar element shapes were used in the axisymmetric model to ensure uniformity
of the model. A 3-node linear axisymmetric triangle with Abaqus designation of CAX3
was used in this model. This element meshes more uniformly than a quadrilateral.
Figures 6.3 and 6.4 show images of the before and after loading conditions of the
axisymmetric model.
1.00E-101.00E-081.00E-061.00E-041.00E-021.00E+00
Deg
ree
of
Failu
re E
stim
atio
n
Fracture Strain (mm/mm)
36
Figure 6.3 Axisymmetric model before loading shown with CAX3 axisymmetric
triangle elements.
Figure 6.4 Axisymmetric model after loading with a large damage initiation value
(no element deletion).
37
The maximum von Mises stress in the axisymmetric model was 4.8x108 N/m
2.
This is a 6% difference in the amount of stress found in the 3D model. This agreement in
data for the two models suggests that the data from both models are consistent. The
model deformed severely at the notch area. This large deformation was expected due to
no failure criterion being used in this model. Remeshing was needed in the subsequent
modeling to refine the mesh and help the finite element solver find convergence.
38
MESH REFINEMENT Chapter 7
7.1 Mesh Refinement
Mesh refinement was instituted in both the axisymmetric and three dimensional
models. An optimal mesh for this type of analysis is shown in Figures 7.1 and 7.2. This
amount of meshing gave the best chance for convergence in the model, but may cause
issues with computer performance.
Figure 7.7.1 Axisymmetric Model after mesh refinement
Figure 7.7.2 Remesh of the Three Dimensional Model
39
It is easy to see that the models are much smoother than the previous unrefined
models. There is an increase of over 3,000 elements in the axisymmetric model and over
60,000 more elements in the three dimensional model. Mesh refinement leads to greater
accuracy in the model, however, too many elements will lead to excessive computation
time. To analyze the 3D model alone required nearly 80GB of RAM to finish the
analysis. Conducting a sensitivity analysis of both the axisymmetric and 3D model, the
final mesh refinement was reduced to less than 500 elements in the axisymmetric model,
and less than 10,000 elements in the 3D model. The final mesh refined models are shown
in Figures 7.3,7.4,and 7.5.
Figure 7.3 Final mesh of the axisymmetric model meshed with 500 CAX3 3 node linear
axisymmetric triangle element.
40
Figure 7.4 Final meshing of three dimensional model with ~6,500 8-node linear brick
elements.
Figure 7.5 Final meshing of three dimensional model with ~6,800 10 node linear tetrahedral
elements.
41
FAILURE PARAMETERS AND ELEMENT DELETION Chapter 8
8.1 Failure Model
Once the models were refined, it was appropriate to introduce the Johnson-Cook
failure parameters. The Johnson-Cook failure parameters were input into the Abaqus
material property interface. The failure parameters for AISI 4340 given by Johnson and
Cook are shown in Table 3.2. The inputs for both the axisymmetric and 3D models are
included in Appendix A.
8.1.1 Element Deletion
Failure was introduced through initiation of element deletion in the model.
Element deletion occurred when an element reached a certain failure strain that is
determined using the axisymmetric model and confirmed using the 3D model. This
failure strain was varied through a number of different values ranging from 1 to 1x10-
15mm/mm strain to identify the onset of element deletion. To use element deletion in
Abaqus, the STATUS variable must be called during analysis. This variable identifies the
element as deleted or still active in the simulation. This variable identifies each element
that has exceeded the predetermined failure strain and removes it from the analysis.
Appendix B shows the results of these runs for AISI 4340.
42
3D MODELING Chapter 9
After determining the fracture strain value from axisymmetric modeling, the three
dimensional model can be evaluated. Two different meshes were used in modeling to
check for convergence. The left mesh in Figure 7.4 is an 8-node linear brick element
with reduced integration and hourglass control, while Figure 7.5 uses 10-node quadratic
tetrahedral elements.
Results of the 3D calculation were similar result to those of the axisymmetric
model, but it appeared that the perfect rotational symmetry in the brick mesh gave
misleading results. It was thought that the solver was having issues determining a failure
initiation site in the thick section of brick model. The notch expanded first and elements
were deleted, but none of the elements in the thick section were deleted like the
tetrahedral mesh. Also, the tetrahedrally meshed model predicted a much higher
maximum von Mises stress than the brick mesh. This may be due to stress concentrations
from the tetrahedral mesh construction. In both models, a strain rate, of 1x104 was
found. This correlates well with the theoretical estimate in Figure 3.1. Figure 9.1 shows
the results of the calculations after 45µs.
Figure 9.1 Results of both brick and tetrahedral models after 45 microseconds. The
tetrahedral model predicts a larger maximum von Mises stress than the brick elements.
43
3D ECCENTRICITY MODEL Chapter 10
10.1 Eccentricity Introduction
Eccentricity was introduced into both models to simulate a possible machining
defect to induce failure in a localized region. The inner diameter was shifted 1mm to
remove concentricity and remove any rotational symmetry. This defect is to simulate a
machinist not indicating a part into concentricity after remounting the part in the lathe.
This shift in the part is more severe than most machining defects would be. Figure 10.1
shows a schematic drawing of the shift.
Fragmentation in the eccentric models was more prevalent near the thinned
regions and continued throughout both the thick and thin sections. The tetrahedral mesh
still predicted a larger maximum von Mises stress than the brick model. This discrepancy
in the von Mises stress may occur for a couple of reasons. The first reason may be that
the mesh refinement in one of the models was not as fine as it needed to be, and the
model had not converged. The other reason might be due to the construction of the
tetrahedrally meshed model. In the edges of the notched section, the corners of the
tetrahedrons may have given a false, higher than normal value for the stress. Mesh
refinement of the notch region may clear most of these excessively large localized stress
a. b.
Figure 10.1 Top down view of the inner diameter shift for eccentric models.
a. Center of outer diameter b. Center of inner diameter
44
peaks. Further refinement was not considered for this model due to an exponential
increase in run time, which exceeded the maximum run time allowed for the
supercomputer. Results for von Mises stress in each 3D model are shown in Figure 10.2
and Figure10.3.
Figure 10.2 Eccentric 3D model of ring section meshed with linear brick elements
after 45µsec. Maximum von Mises stress is found to be 1.491x109 Pascal.
45
Figure 10.3 Eccentric 3D model of ring section meshed with quadratic tetrahedral
elements after 45µsec. Maximum von Mises stress is found to be 4.555x109 Pascal.
10.2 Effects of Further Mesh Refinement and Eccentricity on Fragmentation
To ensure that convergence had occurred in the 3D model, the mesh was refined
to a much higher level than in the previous models. The ring was meshed using 1.38
million 8 node linear brick elements. Greater refinement of the mesh adds a significant
amount of run time. To reduce the run time, the model was shortened in the straight wall
sections above and below the notch area and eccentricity was increased. The height was
decreased to 40mm. In addition to shifting the inner diameter 1mm, the inner diameter
was given a two degree taper from the top of the ring to the bottom. These changes are
shown in a section view of the ring in Figure 10.4.
46
Figure 10.4 Section view of more refined eccentric ring showing a 2 degree taper
from top to bottom.
Figure 10.5 shows the results of the tapered eccentric ring at various time intervals. It
appears that failure occurred in the thinnest section of the notch at 15 microseconds. This
is faster than in the other 3D models. The decrease in failure time may be due to either
the increased mesh refinement or the effects of the new defects in the model. Stresses in
this model are similar in magnitude compared to the axisymmetric model. This is most
likely due to better convergence from the increased mesh density. At 30 microseconds,
the notched section is completely disintegrated in the thin section. Complete breakup of
the thick section of the ring occurs at 45 microseconds, as predicted in the less refined
model. This result is shown in Figure 10.6. Further propagation of fragmentation is
shown in Figure 10.7. The fragmentation behavior at this time step appears to follow the
Mott Theory (23).
2°
40mm
47
Figure 10.5 Fracture initiation in the thinnest section of the ring occurs at 15µs. This is
somewhat shorter than predicted in the less refined models. The notch section is completely
missing in the thinnest section at 30µs.
t=10µs
t=15µs t=30µs
t=0µs
48
Figure 10.6 Fragmentation at 45µs. Breakup of the thick section occurs at the same time
predicted by the less refined 3D model. Notice the beginning of axial fragmentation in the
thick wall section as predicted by Mott (23).
Figure 10.7 Fragmentation behavior at 60µs. The axial fragmentation of the larger sections
of the ring appear follow the Mott fragmentation theory (23).
49
Adding a defect like eccentricity to the model did predict the onset of failure at
the thinnest region. Eccentricity did not change the time at which the material started to
fail in the thick region, which was the goal of this work. Damage propagation after the
onset of failure in the eccentric model was much different than in the “non-damaged”
model. This is more important information for further material analysis, than for
determining the failure strain. By having such good agreement in the axisymmetric model
and the non-damaged linear brick model, the 3D modeling no longer needed to be
considered in further materials analysis. Axisymmetric modeling was satisfactory in
determining fracture strain in other material systems.
50
SAMPLE PREPARATION FOR SPLIT HOPKINSON Chapter 11
PRESSURE BAR TESTING
Split Hopkinson bar testing was conducted for two different alloys, AISI 4340
and HF-1 steel. These alloys were heat treated, machined, and surface ground to a certain
aspect ratio to ensure adequate coupling during the SHPB test. Sample dimensions are
critical for the SHPB, so they are machined to tolerances of
11.1 Sample Design
The material used in this experimental analysis was prepared so it will have
adequate coupling of the longitudinal and radial deformation for use in the SHPB test.
This is done by machining samples to a diameter and thickness given by Equation 10.
The aspect ratio of ls/ds is what ensures a good couple between the sample and the SHPB
transmission and extension bars, allowing stress waves to propagate properly through the
samples.
√
11.1 AISI 4340
A 12.7mm diameter by 1.82m long bar of AISI 4340 steel was purchased from
McMaster-Carr Co.. This bar was in the annealed condition with a Rockwell hardness of
C22. The bar was cut into 15cm long pieces and heat treated. Normalizing and tempering
were done in a Thermolyne FA1738-1 electric furnace. The parts were normalized at
800°C for 1.5 hours to transform the microstructure into austenite. After normalizing, the
51
parts were oil quenched. Tempering was done at 650 °C for 2.5 hours to reach a
Rockwell hardness of C34-36. An Accupro Rockwell hardness tester was used to
measure hardness.
After heat treatment, the bars were turned down to 9.525mm diameter to remove
the decarburization layer caused by the heat treatment. Disks were parted off at ~5.7mm.
These disks were surface ground to a final thickness of 4.445mm
11.2 HF-1 Steel
A rod of HF-1 steel was turned on a lathe to a diameter of 6.35mm and parted into
disks with a thickness of ~5.1mm. These disks were also surface ground to a final
thickness of 2.946mm. The hardness of these disks was found to be Rockwell C34-36. A
sample of both the AISI 4340 and the HF-1 are shown in Figure 11.1 and Figure 11.2.
Figure 11.1 Front view of metal samples. Left side is AISI 4340 and the right side is
HF-1 steel.
5mm
52
Figure 11.2 Side view of metal samples. Left side is AISI 4340 and the right side is
HF-1 steel.
5mm
53
METALLOGRAPHIC ANALYSIS Chapter 12
12.1 Sample Preparation
To prepare the samples for metallographic analysis, two of the metal disks were
cold mounted in epoxy. Both samples were polished to 0.06µm then imaged and etched.
After polishing, the samples were analyzed with bright field microscopy using a Nikon
Epiphot 200 inverted metallographic optical microscope. Magnification ranged from
100x to 1000x.
12.2 Etchants
Two different etchants were used in this analysis. Picral etchant was used for the
high carbon HF-1 alloy. This etchant is recommended for structures that contain ferrite
and carbide. This etchant does not reveal ferrite grain boundaries (29). Zephiran chloride
was added to the solution to increase the uniformity of the etch (29).
The AISI 4340 was etched with 2% Nital etchant. This is the most common
etchant for steel alloys. Alpha grain boundaries and constituents are revealed with this
etchant (29). The recipe for each etchant is found in Appendix E.
12.3 AISI 4340 Microstructure
The sample was analyzed before etching to identify the amount of non-metallic
inclusions dispersed throughout the material. Figure 12.1 shows the surface of the sample
at varying magnifications. It appears that the non-metallic inclusions are small and well
dispersed throughout the sample. There was no evidence of any agglomeration in the
sample.
54
Figure 12.1 Non-etched photomicrographs of AISI 4340 showing well dispersed non-
metallic inclusions at different magnifications.
The sample was then etched with 2% Nital for 7 seconds. Figure 12.2 shows the
results of etching at varying magnifications. Etching of the sample revealed a martensitic
microstructure throughout. This microstructure was expected due to the oil quenching
and tempering steps in the heat treatment cycle. A Rockwell hardness of C34-36 also
predicted a non-pearlite microstructure.
1000x 500x
200x 100x
55
Figure 12.2 Optical photomicrographs of AISI 4340 steel after 7 seconds of etching
with 2% Nital. The etchant revealed a martensitic microstructure.
12.4 HF-1 Microstructure
The HF-1 was also analyzed before etching to identify the distribution of non-
metallic inclusions throughout the sample. Figure 12.3 shows the surface of the sample at
varying magnifications. The non-metallic inclusions seem to be well distributed
throughout the sample, but appear to be smaller than the inclusions found in the AISI
4340 sample.
1000x 500x
200x 100x
56
Figure 12.3 Optical photomicrographs of HF-1 steel prior to etching. It appears that
the non-metallic inclusions are smaller than the AISI 4340, but seem to be well
distributed.
The sample was then etched with Picral etchant for 25 seconds. Figure 12.4 shows
the results of the etching at varying magnifications. The micrographs show a much larger
grain size than the AISI 4340, and a mixture of fine and coarse pearlite throughout. This
suggests that the material was air cooled. Rockwell hardness measurements of C34-36
would suggest a tempered martensitic microstructure as found in the AISI 4340, but the
high carbon content increase the amount of cementite on the grain boundaries, resulting
in increased hardness. The large amount of manganese also acts as a pearlite
strengthener. Table 12.1 summarizes the microstructure and hardness findings.
1000x 500x
200x 100x
57
Figure 12.4 Optical photomicrographs of HF-1 steel alloy at varying magnifications.
It appears that the overall grain size is much larger than in the AISI 4340, and
shows a mixture of fine and coarse pearlite.
Table 12.1 Summary of microstructural findings related to hardness data.
Material Microstructure Hardness Non-metallic inclusions
AISI 4340 Martensite HRC 34-36 Well Dispersed
HF-1 Fine and Coarse Pearlite HRC 34-36 Well Dispersed
1000x 500x
200x 100x
58
SPLIT HOPKINSON PRESSURE BAR ANALYSIS Chapter 13
13.1 Parameterization of Data
The SHPB was utilized to acquire high strain rate data for both the AISI 4340 and
HF-1steels. Data are presented in a true stress-true strain diagram The Johnson-Cook
constitutive parameters are found by curve fitting the data. The following equations are
used to find the parameters, A, B, n, C, and m.
By using the yield stress, A, and plotting trends in the stress-strain curve, parameters B
and the strain hardening exponent, n, can be found using Equation 11 (30). Equation 11 is
simply the Johnson-Cook equation with strain rate and temperature dependence assumed
to be negligible. B and n are found by plotting the logarithmic transformation of the
quasi-static stress-strain diagram between the yield stress, A, and the ultimate
compressive stress. The slope of this line is n and the intercept is equal to lnB (30).
(11)
Equation 12 gives the parameter C, and Equation 13 gives the parameter m (30).
59
Homologous temperature is defined as:
.
13.2 Experimental AISI 4340 Properties
Data collected from the SHPB experiment for AISI 4340 is shown in Figure 13.1.
The yield strength predicted from the experiment appears to be too low for this alloy.
This may be due to issues with the SHPB setup at the Civil Infrastructure Testing and
Evaluation Laboratory (CITEL).
Figure 13.1 Split Hopkinson data printout for AISI 4340 steel showing a lower than
expected yield stress at 1x103 strain rate.
Plotting and parameterizing this data gives the Johnson-Cook parameters needed
for the modeling of 4340. Results are shown in Table 13.1. The strain rate measured for
60
this test was 1422 (1/s). The strain rate is given in the data from the SHPB test. As shown
in Table 13.1, the experimental results do not agree with results found by Johnson and
Cook (19,20). This is attributed mainly to the SHPB results obtained from CITEL. It is
possible that the hardness of the experimental AISI 4340 may give much different results
than the published Johnson and Cook data, but the data should not be as skewed as what
was calculated.
Table 13.1 Material property data for AISI 4340 calculated from Split Hopkinson Pressure
bar compared against Johnson-Cook published data for HRC30 AISI4340 steel.
Material A (MPa) B (MPa) C n m
Experimental
AISI 4340
103 3500 0.216 0.53 1.39
J-C Published
AISI 4340
792 510 0.014 0.26 1.03
13.3 Experimental HF-1 Properties
Data collected from the SHPB experiment for HF-1 is shown in Figure 13.2. This
data predicts a reasonable value for yield strength of a high strength steel alloy. The
parameterization of this data is found in Table 13.2. The strain rate measured for this test
was 2299 (1/s). There is no publically available high strain rate data available for HF-1 to
compare against. Due to the data for the experimental AISI4340 being suspect, this data
should also be evaluated further using other equipment.
61
Figure 13.2 Split Hopkinson data printout for HF-1 steel.
Table 13.2 Parameterized Johnson-Cook data for HF-1 steel.
Material A (MPa) B (MPa) C n m
Experimental
HF-1
482.6 2757 0.026 0.816 0.548
62
MODELING OF EXPERIMENTAL AISI 4340 Chapter 14
Axisymmetric modeling was repeated using the new experimental AISI 4340
parameters. The new parameters had a significant effect on the failure behavior of the
steel ring. It appears that this steel does not show any noticeable dependence of the
variation of the fracture strain variable. The ring failed in the notch section at all fracture
strain values. An example of failure is shown in Figure 14.1. Appendix C shows model
results at different fracture strains. Different fracture strain values resulted in different
magnitudes of the maximum von Mises stress. Time at failure was 36 microseconds for
the experimental AISI 4340 steel properties. This is compared to 45 microseconds in the
published Johnson-Cook constiutive model data for AISI 4340.
Figure 14.1 Typical failure behavior of experimental AISI 4340 ring. It appears that
brittle fracture occurs at the notched section.
63
The use of experimental data results in failure at much lower strains than what is
found with the published Johnson-Cook parameters for AISI 4340 steel. Based on these
results, a brittle fracture would be expected due to the increased hardness and the
martensitic microstructure of this AISI 4340.
64
MODELING OF EXPERIMENTAL HF-1 Chapter 15
Modeling of the HF-1 steel was repeated in the same manner as the AISI 4340
steel. The parameters found in the SHPB test were input into the Johnson-Cook model.
An example of the results is shown in Figure 15.1.
Figure 15.1 Initial failure of HF-1 steel sample. Notice only leading edge of notched
section has failed.
The leading edge of the notched section has failed first. This occurs at around 28
microseconds. This may be indicative of a more ductile failure than the experimental
AISI 4340. The microstructural analysis of this alloy shows a pearlitic structure. Pearlite
is more ductile than martensite, so it should be a less brittle fracture. Table 15.1
summarizes the modeling results for the Johnson-Cook AISI 4340, experimental AISI
4340, and HF-1 steel. It appears that the HF-1 steel started to fail before the AISI 4340
65
steel, but had a maximum von Mises stress that was in between the Johnson-Cook AISI
4340 and the experimental AISI 4340.
Table 15.1 Summary of Modeling Results for Johnson-Cook AISI 4340,
Experimental AISI 4340, and HF-1
Model Time at Failure
(µs)
Mises Stress at
Failure (GPa)
Strain Rate from
SHPB test (s-1
)
Published J-C
Parameters AISI 4340
45 1.49 __
Experimental
AISI4340
36 7.35 1422
Experimental HF-1 28 2.25 2299
66
CONCLUSIONS: Chapter 16
Three dimensional and axisymmetric FEA models were created and used to
predict failure of a steel ring. These models were designed to predict the behavior of the
material under high strain rate loading using the Johnson-Cook failure model. HF-1 and
AISI 4340 were both considered. The main conclusion of this work is that computer
modeling can be used to predict failure in high strength steel.
It was found that Abaqus can model failure using the Johnson-Cook constitutive
model for high strength steel using multiple techniques. Comparing linear elastic
simulations between two different programs is possible. Abaqus CAE and SolidWorks
Simulation Xpress found similar results. This is a simple method to check the
convergence of a model.
Axisymmetric modeling can be used to verify the convergence of 3D models and
replace 3D modeling when no defects or non-symmetric changes are used. By using
axisymmetric models of 3D parts, multiple properties can be investigated and changed
much faster than a full 3D model. The axisymmetric Johnson-Cook AISI 4340 data
predicted a fracture strain of 1x10-5
for the model. This value was input into the 3D model
and predicted similar results. Fragmentation was estimated to start at 45µs with a strain
rate of 1x104s
-1 throughout the model.
Modeling in three dimensions proved to be more difficult than using symmetry.
Mesh size played an important role in the convergence of the results. Convergence is
heavily dependent on the amount of mesh refinement in the model. This is especially
evident in models with tetrahedral elements, where over-stiffening can occur, leading to
false stress peaks in the results. Linear brick elements, which are not as sensitive to over-
67
stiffening, predicted a maximum von Mises stress much closer to the results found in the
axisymmetric model.
Experimental data gathered using the SHPB may be suspect for the AISI 4340.
The yield stress for 4340 is much higher than what was predicted using SHPB. This issue
was evident when modeling the 4340 using the axisymmetric model. There was no
dependency on the fracture strain parameter in the model. Fracture occurred at every
value of fracture strain tested. It is possible that the strain at fracture is much lower than
what was predicted using these experimental properties, but it is more likely that the
SHPB results are flawed and should be repeated. The HF-1 Johnson-Cook parameters
appeared to be more realistic for a steel of that hardness, but the data should still be
suspect since it was measured using the same SHPB as the AISI 4340. Similarly, HF-1
did not show a dependence on the fracture strain during axisymmetric modeling, which
may be due to poor SHPB results.
Throughout this work, there have been some software issues that directly or
indirectly affected the results of the model. These problems needed to be addressed on
multiple occasions, and it is worthwhile to conclude this work by listing a few of them.
The first software issue found in Abaqus is the limited amount of high strain rate models
that are preloaded into the program. Users are limited to the Johnson-Cook model for
ductile materials rather than have the ability to directly use models such as the Zerilli-
Armstrong model to predict high strain rate behavior. Other models can be input into the
software via the use of a subroutine, but this method is very time consuming and requires
the user to debug lines of code and experimentally validate the results.
68
Another issue was found when refining the mesh. When changing from linear to
quadratic elements, it was found that quadratic brick elements are not available in a
dynamic, explicit calculation, but quadratic tetrahedrons were. It would be nice to have
the option to use quadratic brick elements in this type of model to help achieve
convergence with less elements and to increase accuracy.
During the analysis of the eccentric model, the tetrahedral elements predicted a
much larger stress than the brick elements did. The reason is mainly due to over-
stiffening of the tetrahedral element. Over-stiffening is usually evident in linear elements,
but in this case full integration was used, which may have caused the model to not
converge even with quadratic elements. Less tetrahedral elements need to be used in
curved models which decreases run time. Unfortunately, these run time gains are lost
when the mesh needs significant refinement to achieve convergence. Some of these
issues are easily avoided, but a number of them pose a problem for the user and should be
addressed.
69
FUTURE WORK Chapter 17
Additional work is needed for better understanding of the fracture mechanism of
the ring and model validation. Future work should include the following:
Greater mesh refinement and optimization of the model
Increased total model time to see fragmentation behavior after failure
initiation
Refine time step to determine a more accurate failure time
Introduce a time-dependent and non-uniformly oriented loading condition
Further develop modeling of defects in the ring
Conduct more SHPB experiments for increased material property accuracy
Conduct fragmentation experiments
70
REFERENCES
1. Campagne, L., Daridon, L. and Ahzi, S A Physically Based Model for Dynamic
Failure in Ductile Materials.. Strasbourg : Mechanics of Materials, 2004, Vol. 37.
2. Stepanov, G.V. and A.V., Shirokov. Modeling of Crack Propagation Kinetics. No. 4,
Kiev : Problemy Prochnosti, 2008. 0039-2316/4204-0426.
3. Rusinek, A. and Zaera, R. Finite Element Simulation of Steel Ring Fragmentation
Under Radial Expansion. s.l. : International Journal of Impact Engineering, 2006, Vol.
34.
4. Altenhof, W., et al Numerical Simulation of AM50A Magnesium Alloy Under Large
Deformation.. Windsor : International Journal of Impact Engineering, 2003, Vol. 30.
5. Mishra, A., et al. High-strain-rate Reponse of Ultra-fine-grained Copper. s.l. : Acta
Materialia, 2008, Vol. 56.
6. Rand, J. L. (1967). An analysis of the split hopkinson pressure bar. White Oak
Maryland: United States Naval Ordnance Laboratory.
7. Gama, B., Lopatnikov, S., & Gillespie, J. (2004). Hopkinson bar experimental
technique: A critical review. Applied Mechanics Reviews, 223-250.
8. Rushton, N., et al. Internal Explosive Loading of Steel Pipes. s.l. : Thin-Walled
Structures, 2008, Vol. 46.
9. S.Nemat-Nasser, Introduction to High Strain Rate Testing, Mechanical Testing and
Evaluation, Vol 8, ASM Handbook, ASM International, 2000, p 427–428
10. Hopkinson, J. Original papers by J. Hopkinson, vol. II. Cambridge University Press.
1901 p 316-324
11. Hopkinson, B. A Method of Measuring the Pressure Produced in the Detonation of
High Explosives of by Impact of Bullets. Philosophical Transactions of the Royal Society
London 1914. Pg.437-456
12. Davies R.M. A Critical Study of the Hopkinson Pressure Bar, Philosophical
Transactions of the Royal Society London 1948. 375-457.
13. An Investigation of the Mechanical Properties of Materials at very High Rates of
Loading. Kolsky. H. Philosophical Transactions of the Royal Society London 1949. pg.
676-700
14. Wikipedia. RDX. www.wikipedia.com. [Online] [Cited: December 1, 2011.]
15. 9501, PBX. www.wikipedia.com. [Online] December 1, 2011.
16. Trinitrotoluene. [Online] [Cited: ]
www.ch.ic.as.uk/vchemlib/mim/bristol/tnt/tnt_test.htm.
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17. Banerjee, Biswajit Material Point Method Simulations of Fragmenting Cylinders..
Newark : 17th ASCE Engineering Mechanics Conference, 2004.
18. Johnson-Cook Model. [Online] [Cited: 3 14, 2011.]
www.kxcad.net/ansys/ANSYS/AUTODYN.
19. Johnson, G.R., Cook, W.H A Constituitive Model and Data for Metals Subjected to
Large Strains, High Strain Rates and High Temperatures.. The Hague, The Netherlands :
Proc. 7th Int. Symp. on Ballistics, 1983.
20. Johnson G.R., Cook, W.H. Fracture Characteristics of Three Metals Subjected to
Various Strains, Strain Rates, Temperatures, and Pressures. s.l. : International Journal of
Engineering Fracture Mechanics, 1985, Vol. 21
21. Benzerga, A., & Leblond, J. (2010). Ductile fracture by Void Growth to Coalescence.
Advances in Applied Mechanics.
22. Gurson, A. (1977). Continuum Theory of Ductile Rupture by Void Nucleation and
Growth. Transactions of the ASME, 2-15.
23. Mott, N., & Linfoot, E. (2006). A Theory of Fragmentation. In D. Grady,
Fragmentation of Rings and Shells Shock Wave and High Pressure Phenomena (pp. 207-
225).
24. Mott, N. (1947). Fragmentation of Shell Cases. Proceedings of the Royal Society of
London, 300-308.
25.AISI4340Steel.Matweb.[Online]www.matweb.com/search/DataSheet.aspx?MatGUID.
26. Mulherin,J. et.al. Fracture Mechanics Study on 155mm M107 Projectile Made from
Isothermally Transformed HF-1 Steel. U.S. Army Armament Command. Frankford
Arsenal.1976
27. . Forbes Jerry W. Pressure Wave Measurements During Thermal Explosion of HMX-
based High Explosives., et.al. Livermore CA : s.n.
28. Yoh, J., et al. Test-based Thermal Explosion Model for HMX. s.l. : Proceedings of the
Combustion Institute, 2006, Vol. 31.
29. Van der Voort, G. (1999). Metallography, Principles and Practice. Materials Park:
ASM International.
30. Yafu, F., Qu-dong, W., & Jun-sheng, N. (2010). Experimental Measure of
Parameters: The Johnson-Cook Material Model of Extruded Mg-Gd-Y Series Alloy.
Journal of Applied Mechanics, 1-5
31. Davis, J. (2004). Tensile Testing. Materials Park, OH: ASM International.
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32. Conway Jr., J. C. (1974). Patent No. 3,842,664. United States of America
33. Lee, W., Chen, T., Lin, C., & Luo, W. (2011). Dynamic Mechanical Response of
Biomedical 316L Stainless Steel as a Function of Strain Rate and Temperature.
Bioinorganic Chemistry and Applications, 12.
73
Appendix A: Model Inputs
A1: Axisymmetric Input
*Heading
** Job name: neg_ten Model name: Model-1
** Generated by: Abaqus/CAE Student Edition 6.10-2
*Preprint, echo=NO, model=NO, history=NO, contact=NO
** PARTS
*Part, name=Part-1
*End Part
** ASSEMBLY
*Assembly, name=Assembly
*Instance, name=Part-1-1, part=Part-1
*Node
*Element, type=CAX3
*Nset, nset=_PickedSet2, internal, generate
1, 244, 1
*Elset, elset=_PickedSet2, internal, generate
1, 392, 1
** Section: Axisymmetric
*Solid Section, elset=_PickedSet2, material="Steel 4340"
*End Instance
*Nset, nset=_PickedSet89, internal, instance=Part-1-1
1, 2, 7, 8, 9, 10, 11, 12, 52, 53, 54, 55
*Elset, elset=_PickedSet89, internal, instance=Part-1-1
6, 7, 37, 50, 54, 101, 103, 173, 176, 261
*Elset, elset=__PickedSurf88_S2, internal, instance=Part-1-1
1, 73, 92, 177, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202
204, 206, 208, 210, 215, 217, 219, 221, 223, 226, 228, 230, 232, 234, 236, 238
240, 242, 244, 246, 248, 251
*Elset, elset=__PickedSurf88_S3, internal, instance=Part-1-1
50,
*Elset, elset=__PickedSurf88_S1, internal, instance=Part-1-1
54,
*Surface, type=ELEMENT, name=_PickedSurf88, internal
__PickedSurf88_S2, S2
__PickedSurf88_S3, S3
__PickedSurf88_S1, S1
*End Assembly
** MATERIALS
*Material, name="Steel 4340"
*Damage Initiation, criterion=JOHNSON COOK
0.05, 3.44, 2.12, 0.002, 0.61, 1793., 255., 1.
*Damage Evolution, type=DISPLACEMENT
1e-10,
*Density
74
7870.,
*Elastic
2.05e+11, 0.33
*Plastic, hardening=JOHNSON COOK
7.92e+08, 5.1e+08, 0.26, 1.03, 1793., 255.
** STEP: Step-1
*Step, name=Step-1
Explosion
*Dynamic, Explicit, scale factor=0.005
, 4.8e-05
*Bulk Viscosity
0.06, 1.2
** BOUNDARY CONDITIONS
** Name: BC-1 Type: Displacement/Rotation
*Boundary
_PickedSet89, 2, 2
** LOADS
** Name: Load-1 Type: Pressure
*Dsload
_PickedSurf88, P, 1.2e+09
** OUTPUT REQUESTS
*Restart, write, number interval=1, time marks=NO
** FIELD OUTPUT: F-Output-1
*Output, field, number interval=10
*Node Output
A, AT, CF, RF, RT, U, UT, V
VT
*Element Output, directions=YES
BF, DAMAGEFC, DAMAGEFT, DAMAGEMC, DAMAGEMT, DAMAGESHR,
DENSITY, DMICRT, EDCDEN, EDT, EFABRIC, ELEDEN, ELEN, EMSF, ENER, ER
ERPRATIO, GRAV, HP, LE, MISESMAX, NE, NFORC, P, PE, PEEQ, PEEQMAX,
PEEQT, PEQC, S, SBF, SDEG
SFABRIC, SHRRATIO, STAGP, STATUS, TRIAX, TRNOR, TRSHR, VP
*Contact Output
CFORCE, CSDMG, CSMAXSCRT, CSMAXUCRT, CSQUADSCRT, CSQUADUCRT,
CSTRESS, CTHICK, FSLIP, FSLIPR
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
75
A2: Three Dimensional Input File
*Heading
** Job name: jeremudsd Model name: Model-1
** Generated by: Abaqus/CAE 6.10-2
*Preprint, echo=NO, model=NO, history=NO, contact=NO
** PARTS
*Part, name=Ring
*End Part
** ASSEMBLY
*Assembly, name=Assembly
*Instance, name=Ring-1, part=Ring
*Node
0.0327737853, 0.0432417728, 0.0162973069
409, -0.0169027951, 0.0264521372, 0.0322135948
*Element, type=C3D4
1, 401, 194, 400, 403
2, 400, 403, 9, 401
3, 403, 404, 3,
1127, 218, 23, 216, 68
*Nset, nset=_PickedSet2, internal, generate
1, 409, 1
*Elset, elset=_PickedSet2, internal, generate
1, 1127, 1
** Section: Ring Section
*Solid Section, elset=_PickedSet2, controls=EC-1, material="Steel 4340"
*End Instance
*Nset, nset=_PickedSet5, internal, instance=Ring-1
*Elset, elset=_PickedSet5, internal, instance=Ring-1
*Elset, elset=__PickedSurf4_S1, internal, instance=Ring-1
*Elset, elset=__PickedSurf4_S3, internal, instance=Ring-1
*Elset, elset=__PickedSurf4_S2, internal, instance=Ring-1
*Elset, elset=__PickedSurf4_S4, internal, instance=Ring-1
*Surface, type=ELEMENT, name=_PickedSurf4, internal
*End Assembly
** ELEMENT CONTROLS
*Section Controls, name=EC-1, ELEMENT DELETION=YES
1., 1., 1.
** MATERIALS
**
*Material, name="Steel 4340"
*Damage Initiation, criterion=JOHNSON COOK
0.05, 3.44, 2.12, 0.002, 0.61, 1793., 255., 1.
*Damage Evolution, type=DISPLACEMENT
1e-09,
*Density
76
7870.,
*Elastic
2.05e+11, 0.33
*Plastic, hardening=JOHNSON COOK
7.92e+08, 5.1e+08, 0.26, 1.03, 1793., 255.
** STEP: Explosion
*Step, name=Explosion
*Dynamic, Explicit, scale factor=0.005
, 7.5e-05
*Bulk Viscosity
0.06, 1.2
** BOUNDARY CONDITIONS
** Name: BC-1 Type: Symmetry/Antisymmetry/Encastre
*Boundary
_PickedSet5, YSYMM
** LOADS
** Name: Load-1 Type: Pressure
*Dsload
_PickedSurf4, P, 1.2e+09
** OUTPUT REQUESTS
*Restart, write, number interval=1, time marks=NO
** FIELD OUTPUT: F-Output-1
*Output, field, number interval=10
*Node Output
A, AT, CF, RF, RT, U, UT, V
VT
*Element Output, directions=YES
BF, DAMAGEFC, DAMAGEFT, DAMAGEMC, DAMAGEMT,
DAMAGESHR, DENSITY, DMICRT, EDCDEN, EDT, EFABRIC, ELEDEN, ELEN,
EMSF, ENER, ER
ERPRATIO, GRAV, HP, LE, NE, NFORC, P, PE, PEEQ, PEEQMAX, PEEQT,
PEQC, S, SBF, SDEG, SFABRIC
SHRRATIO, STAGP, STATUS, TRIAX, TRNOR, TRSHR, VP
*Contact Output
CFORCE, CSDMG, CSMAXSCRT, CSMAXUCRT, CSQUADSCRT,
CSQUADUCRT, CSTRESS, CTHICK, FSLIP, FSLIPR
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
77
Appendix B: Axisymmetric Failure Strain Variation in Johnson-
Cook AISI 4340
Figure B1 Axisymmetric Failure Strain of 1. Stress in Pa. (AISI 4340)
Figure B2 Axisymmetric Failure Strain of 0.1. Stress in Pa (AISI 4340)
78
Figure B3 Axisymmetric Failure Strain of 0.01. Stress in Pa (AISI 4340)
Figure B4 Axisymmetric Failure Strain of 0.001. Stress in Pa (AISI 4340)
79
Figure B5 Axisymmetric Failure Strain of 0.0001. Stress in Pa (AISI 4340)
Figure B6 Axisymmetric Failure Strain of 0.00001. Stress in Pa (AISI 4340)
80
Figure B7 Axisymmetric Failure Strain of 1x10-6. Stress in Pa (AISI 4340)
Figure B8 Axisymmetric Failure Strain of 1x10-7. Stress in Pa (AISI 4340)
81
Figure B9 Axisymmetric Failure Strain of 1x10-8. Stress in Pa (AISI 4340)
Figure B10 Axisymmetric Failure Strain of 1x10-9. Stress in Pa (AISI 4340)
82
Figure B11 Axisymmetric Failure Strain of 1x10-10. Stress in Pa (AISI 4340)
Figure B12 Axisymmetric Failure Strain of 1x10-15. Stress in Pa (AISI 4340)
83
Appendix C: Axisymmetric Failure Strain Variation in Experimental AISI 4340
Figure C1 Axisymmetric Failure Strain of 1. Stress in Pa. (AISI 4340 Exp.)
Figure C2: Axisymmetric Failure Strain of 0.1. Stress in Pa (AISI 4340 Exp.)
84
Figure C3 Axisymmetric Failure Strain of 0.01. Stress in Pa (AISI 4340 Exp.)
Figure C4 Axisymmetric Failure Strain of 0.001. Stress in Pa (AISI 4340 Exp.)
85
Figure C5 Axisymmetric Failure Strain of 0.0001. Stress in Pa (AISI 4340 Exp.)
Figure C6 Axisymmetric Failure Strain of 0.00001. Stress in Pa (AISI 4340 Exp.)
86
Figure C7 Axisymmetric Failure Strain of 1x10-6. Stress in Pa (AISI 4340 Exp.)
Figure C8 Axisymmetric Failure Strain of 1x10-7. Stress in Pa (AISI 4340 Exp.)
87
Figure C9 Axisymmetric Failure Strain of 1x10-8. Stress in Pa (AISI 4340 Exp.)
Figure C10 Axisymmetric Failure Strain of 1x10-9. Stress in Pa (AISI 4340 Exp.)
88
Figure C11 Axisymmetric Failure Strain of 1x10-10. Stress in Pa (AISI 4340 Exp.)
Figure C12 Axisymmetric Failure Strain of 1x10-15. Stress in Pa (AISI 4340 Exp.)
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Appendix D: Axisymmetric Failure Strain Variation in Experimental HF-1
Figure D1: Axisymmetric Failure Strain of 1. Stress in Pa. (HF-1 Exp.)
Figure D2: Axisymmetric Failure Strain of 0.1. Stress in Pa (HF-1 Exp.)
90
Figure D3 Axisymmetric Failure Strain of 0.01. Stress in Pa (HF-1 Exp.)
Figure D4 Axisymmetric Failure Strain of 0.001. Stress in Pa (HF-1 Exp.)
91
Figure D5 Axisymmetric Failure Strain of 0.0001. Stress in Pa (HF-1 Exp.)
Figure D6 Axisymmetric Failure Strain of 0.00001. Stress in Pa (HF-1 Exp.)
92
Figure D7 Axisymmetric Failure Strain of 1x10-6. Stress in Pa (HF-1 Exp.)
Figure D8 Axisymmetric Failure Strain of 1x10-7. Stress in Pa (HF-1 Exp.)
93
Figure D9 Axisymmetric Failure Strain of 1x10-8. Stress in Pa (HF-1 Exp.)
Figure D10 Axisymmetric Failure Strain of 1x10-9. Stress in Pa (HF-1 Exp.)
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Figure D11 Axisymmetric Failure Strain of 1x10-10. Stress in Pa (HF-1 Exp.)
Figure D12 Axisymmetric Failure Strain of 1x10-15. Stress in Pa (HF-1 Exp.)
95
Appendix E: Etchant Recipes
Nital (29):
2mL Nitric Acid (HNO3)
98mL ethanol
Immerse for up to 60sec
Picral (29):
4g picric acid
100mL ethanol
0.5-1% zephiran chloride (for uniformity and improved etch rate)
Immerse sample