high-strain-rate property determination of high-strength steel using finite element analysis and...

8/9/2019 HIGH-STRAIN-RATE PROPERTY DETERMINATION OF HIGH-STRENGTH STEEL USING FINITE ELEMENT ANALYSIS AND EXPERIMENTAL DATA A Thesis in…

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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 Requirementsfor the Degree of

Master of Science

May 2013


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The thesis of Jeremy M. Schreiber was reviewed and approved* by the following:

Ivica SmidAssociate Professor of Engineering Science and Mechanics

Thesis Advisor

Timothy J. EdenResearch Associate. Applied Research Laboratory

Thesis Advisor

Albert E. SegallProfessor of Engineering Science and Mechanics

Committee Member

Judith Todd

Professor of Engineering Science and MechanicsHead of the Department of Engineering Science and Mechanics

*Signatures are on file in the Graduate School


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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.


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TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................ vi

LIST OF FIGURES ...................................................................................................... vii

ACKNOWLEDGMENTS ...............................................................................................x

INTRODUCTION ...........................................................................................1Chapter 1

BACKGROUND .............................................................................................4Chapter 2

Finite Element Analysis .....................................................................................42.1

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 .... 22Chapter 3

3.1 Material Property Input ........................................................................................ 22

LINEAR ELASTIC MODELING.................................................................. 26Chapter 4

STRAIN-RATE DEPENDENT PLASTIC DEFORMATION MODELING .. 30Chapter 5

STRAIN RATE DEPENDENT AXISYMMETRIC MODELING ................. 33Chapter 6

MESH REFINEMENT .................................................................................. 38Chapter 7

7.1 Mesh Refinement ............................................................................................ 38

FAILURE PARAMETERS AND ELEMENT DELETION ........................... 41Chapter 8


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8.1 Failure Model .................................................................................................. 41

3D MODELING ............................................................................................ 42Chapter 9

3D ECCENTRICITY MODEL .................................................................... 43Chapter 10

10.1 Eccentricity Introduction ................................................................................. 43

SAMPLE PREPARATION FOR SPLIT HOPKINSON PRESSURE BARChapter 11

TESTING ...................................................................................................................... 50

11.1 Sample Design ................................................................................................ 50

11.1 AISI 4340 ........................................................................................................ 50

11.2 HF-1 Steel ....................................................................................................... 51

METALLOGRAPHIC ANALYSIS ............................................................. 53Chapter 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 .................................. 58Chapter 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 .......................................... 62Chapter 14

MODELING OF EXPERIMENTAL HF-1 .................................................. 64Chapter 15

CONCLUSIONS: ........................................................................................ 66Chapter 16

FUTURE WORK ........................................................................................ 69Chapter 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


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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. ..................................................................... 16Figure 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. ........................................................................................................................... 28Figure 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


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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 iscompletely 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

http://arl.psu.edu/shares/Material%20Processing%20Div/Technology%20Areas/Student%20-%20Jeremy%20Schreiber/Master's%20Thesis/THESIS%20CORRECTIONS%202013.docx%23_Toc352773179http://arl.psu.edu/shares/Material%20Processing%20Div/Technology%20Areas/Student%20-%20Jeremy%20Schreiber/Master's%20Thesis/THESIS%20CORRECTIONS%202013.docx%23_Toc352773179


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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.


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INTRODUCTIONChapter 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.


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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


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BACKGROUND Chapter 2

Finite Element Analysis2.1

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.


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Table 2.1 Recommended Testing Methods for Various Strain Rates (9)

Strain Rate (s-1

) Testing Technique

Compression Tests


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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).


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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).


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2.3.1.1 Theory

If the pressure ap plied 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.


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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


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(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).


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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


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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 i ts 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


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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.


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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.


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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


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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


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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


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2.5.3 H igh F ragmentation (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


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MATERIAL PROPERTIES AND CONSTITUITIVEChapter 3

RELATIONSHIPS

3.1 Material Property InputThe 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 x10 7.80 0.285 22 745x10 470x10

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.


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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


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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

P r e d

i c t e d M i s e s F l o w S t r e s s ( P a )

True Tensile Strain (ε) mm/mm

1x104

1x10-4

1x10-21

10

1x102

Selected

Strain

Rates

1x10-5


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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 1x10 4.

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

v o n M i s e s F l o w S t r e s s ( M P

a )

Strain (mm/mm)

J-C AISI 4340

EXP 4340

HF-1


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LINEAR ELASTIC MODELINGChapter 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


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initial stages of modeling, 1x10-6GPa 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.


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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.


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STRAIN-RATE DEPENDENT PLASTIC DEFORMATIONChapter 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.


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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


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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.


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STRAIN RATE DEPENDENT AXISYMMETRICChapter 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.


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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 1x10-5mm/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.


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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

D e g r e e o f F a i l u r e E s t i m a t i o n

Fracture Strain (mm/mm)


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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).


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The maximum von Mises stress in the axisymmetric model was 4.8x108 N/m2.

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.


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MESH REFINEMENTChapter 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


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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.


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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.


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FAILURE PARAMETERS AND ELEMENT DELETIONChapter 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.


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3D MODELINGChapter 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 wasfound. 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.


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3D ECCENTRICITY MODELChapter 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 diameterb. Center of inner diameter


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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 1.491x10

9

Pascal.


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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.


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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


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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 completelymissing in the thinnest section at 30µs.

t=10µs

t=15µs t=30µs

t=0µs


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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).


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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.


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SAMPLE PREPARATION FOR SPLIT HOPKINSONChapter 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


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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


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METALLOGRAPHIC ANALYSISChapter 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.


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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


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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


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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 welldistributed.

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


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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


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SPLIT HOPKINSON PRESSURE BAR ANALYSISChapter 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).

̇

̇


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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


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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.


ExperimentalAISI 4340

103 3500 0.216 0.53 1.39

J-C PublishedAISI 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.


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Figure 13.2 Split Hopkinson data printout for HF-1 steel.

Table 13.2 Parameterized Johnson-Cook data for HF-1 steel.


ExperimentalHF-1

482.6 2757 0.026 0.816 0.548


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MODELING OF EXPERIMENTAL AISI 4340Chapter 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.


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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.


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MODELING OF EXPERIMENTAL HF-1Chapter 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


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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 __

ExperimentalAISI4340

36 7.35 1422

Experimental HF-1 28 2.25 2299


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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-stif

high-strain-rate property determination of high-strength steel using finite element analysis and...

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