experimental characterization and modeling of high
TRANSCRIPT
EXPERIMENTAL CHARACTERIZATION AND MODELING OF HIGH STRENGTH MARTENSITIC STEELS BASED ON A NEW
DISTORTIONAL HARDENING MODEL
BY
ELIZABETH K BARTLETT
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2018
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© 2018 Elizabeth K Bartlett
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To my daughter, Gwyneth Webb, who helps me to have fun, laugh, and love through the most difficult times in my life.
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ACKNOWLEDGEMENTS
First, I would like to thank all the members of my supervisory committee
for their support. I would especially like to thank my committee chair, Prof. Oana
Cazacu, for her support and constructive criticisms. It was an honor and a
privilege to study her research. I would also like to thank Dr. Benoit Revil-
Baudard for his expertise in numerical methods.
I am so grateful for the support of the 96th Test Wing, my sponsoring
facility in the Science, Mathematics, and Research for Transformation (SMART)
scholarship program. Specifically, I would like to thank my commanders and
supervisor, Mr. Ron Lutz, as well as, Mrs. Linda Busch and Dr. Betta Jerome.
I also greatly appreciate the support of the Air Force Research Laboratory
(AFRL) in the experimental characterization of Eglin steel: Dr. Geremy Kleiser
and Dr. Philip Flater for their training in quasi-static testing, Dr. Brad Martin and
Dr. Xu Nie for their expertise in dynamic split Hopkinson pressure bar (SHPB)
testing, Dr. Rachel Abrahams and Dr. Sean Gibbons for their instruction in
microscopy and material science, and Mr. Richard Harris for his expertise in
material characterization at the Advanced Weapons Effects Facility (AWEF).
I extend my gratitude to my friends and family for listening to me ramble
on and on about subsequent yield surfaces and various hardening models.
Finally, I would like to thank Gwyneth Webb, my 12-year old daughter, for her
encouragement and patience over long nights and weekends in room 171 of the
University of Florida (UF) Research and Engineering Education Facility (REEF).
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TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS ..................................................................................... 4
LIST OF TABLES .................................................................................................... 7
LIST OF FIGURES .................................................................................................. 8
LIST OF ABBREVIATIONS .................................................................................... 13
ABSTRACT ......................................................................................................... 16
CHAPTER
1 INTRODUCTION ............................................................................................. 18
1.1 Background of Ultra High Strength Martensitic Steel..................... 18 1.2 Survey of the Experimental Studies on Martensitic Steels ............ 23 1.3 Eglin Steel, ES-1 ........................................................................... 32 1.4 Elastic-Plastic Modeling ................................................................. 38 1.5 Goals of Current Research ............................................................ 48
2 MICROSCOPY ................................................................................................ 50
2.1 Optical Microscopy of ES-1 ........................................................... 51 2.2 Material Characterization using SEM ............................................ 55
3 MECHANICAL CHARACTERIZATION OF ES-1 ......................................... 63
3.1 Hardness of Eglin Steel ................................................................. 63 3.2 Quasi-Static Mechanical Characterization ..................................... 65 3.3 Dynamic Experimental Characterization of Eglin Steel .................. 82 3.4 Cyclic Experimental Characterization of ES-1 ............................... 99 3.5 Summary of the Experimental Characterization of ES-1 ............. 104
4 ELASTIC-PLASTIC MODEL FOR EGLIN STEEL ..................................... 105
4.1 Development of the Yield Criterion .............................................. 105 4.2 Asymmetric Hardening ................................................................ 112 4.3 Implementing the Proposed Yield Function ................................. 137 4.4 Implications of the Proposed Yield Function ............................... 143
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5 FINITE ELEMENT ANALYSIS ..................................................................... 145
5.1 Review of Finite Element Analysis .............................................. 145 5.2 Implementation of the Elastic-Plastic Model in FEA .................... 149 5.3 Finite Element Analysis of Ultra High Strength Martensitic Steel 151 5.4 Elastic-Plastic Model Predictions ................................................ 170
6 CONCLUSIONS AND RECOMMENDATIONS .......................................... 178
REFERENCES ................................................................................................. 184
BIOGRAPHICAL SKETCH ............................................................................... 194
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LIST OF TABLES
Table Page
1-1 Monotonic and cyclic material properties. ................................................ 30
1-2 Chemical composition by percent weight of AF-1410, SAE 4340, and ES-1. .............................................................................................................. 33
1-3 Mechanical properties under uniaxial, quasi-static tensile loading of AF-1410, SAE 4340, and ES-1. ..................................................................... 34
2-1 Heat treatment schedule for ES-1. ........................................................... 50
2-2 ES-1 grinding and polishing schedule. ..................................................... 51
2-3 Chemical composition of ES-1 for EDS. ................................................... 56
3-1 Eglin steel material characterization test matrix. ...................................... 63
3-2 Yield stress and quasi-static compression test data. ................................ 70
3-3 Coefficients involved in Swift and Voce Hardening Laws for ES-1. .......... 70
3-4 Summary of quasi-static tension test data................................................ 72
3-5 Hardening law parameters tensile round specimens. ............................... 73
3-6 Tension-compression asymmetry ratio of ES-1 by plastic strain. ............. 76
3-7 Hardening law parameters flat tension specimens. .................................. 77
3-8 Johnson and Cook material constants for forged, cast, and cast and HIP’d ES-1. ........................................................................................................ 94
4-1 Cazacu and Barlat yield criterion parameters. ........................................ 140
4-2 Points for linear interpolation of the Cazacu and Barlat asymmetry parameter, c. .......................................................................................... 141
5-1 Finite element analysis simulation matrix. .............................................. 152
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LIST OF FIGURES Figure Page
1-1 Tensile yield stress vs. strain-to-failure of current steels and expected strengths of third-generation UHSS. ........................................................ 19
1-2 The unit cell of single-crystal microstructures of solid steel. ..................... 20
1-3 Illustration of plastic deformation mechanisms. ........................................ 22
1-4 The fractional strength differential parameter vs. the carbon content by percent weight for different SAE 4300 series. .......................................... 24
1-5 The effect of tempering temperature on the strength differential of quenched and tempered 4340 UHSS. ..................................................... 25
1-6 The effect of tempering on the flow stress of various steels. .................... 28
1-7 Absolute flow stress vs. plastic strain of SAE 4340 from hysteresis loops of cyclic testing. ........................................................................................ 31
1-8 STF vs. tensile yield stress for different steels including conventional steels and UHSS. ..................................................................................... 35
1-9 ES-1 symmetric plate impact experimental setup of Martin et al. ............. 37
1-10 Tresca yield surface in Haigh-Westergaard space with the hydrostatic axis and deviatoric plane. ......................................................................... 40
1-11 Von Mises yield surface in Haigh-Westergaard space with the longitudinal axis as the hydrostatic axis. ..................................................................... 41
1-12 Projection in the biaxial plane of the yield surfaces of Drucker, Mises, and Tresca yield criteria. ................................................................................. 42
1-13 Cazacu and Barlat yield surface in Haigh-Westergaard space. ............... 44
1-14 Projection in the biaxial plane of the Cazacu and Barlat yield surface corresponding to flow stress ratios of 3/4, 1, and 5/4. .............................. 45
2-1 Z-stack of several pores on the polished surface of cast eglin steel specimen using CDIC. ............................................................................. 52
2-2 Photomontage stitched from nine individual images of the polished surface of cast ES-1 specimen containing several pores. ..................................... 53
2-3 Polished surface of ES-1 materials at 200 times magnification in an optical microscope with brightfield illumination. ................................................... 54
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2-4 Nitol etched cast material surface at 100 times magnification in Keyence optical microscope under brightfield illumination. ..................................... 55
2-5 Orientation map of forged Eglin steel demonstrating prior austenite grain boundaries and martensitic lath structure.Figure 2-6. Orientation map of forged Eglin steel demonstrating prior austenite grain boundaries and martensitic lath structure. ......................................................................... 57
2-6 Orientation map of cast and HIPd Eglin steel demonstrating prior austenite grain boundaries and martensitic lath structure. ....................................... 58
2-7 Orientation map of cast Eglin steel demonstrating prior austenite grain boundaries and martensitic lath structure................................................. 59
2-8 Pole figures of (001), (101), and (111) for forged, cast and HIP’d, and cast Eglin steel. ................................................................................................ 60
3-1 Buehler Digital Hardness Tester MMT-3.Figure 3-1. Buehler Digital Hardness Tester MMT-3. ......................................................................... 64
3-2 AWEF quasi-static experimental characterization setup .......................... 65
3-3 Quasi -static cylindrical compression test specimens. ............................. 67
3-4 Quasi-static round bar compression test stress-strain results .................. 68
3-5 Comparison of the quasi-static compressive stress-strain response for the ES-1 materials. ......................................................................................... 69
3-6 Hardening of ES-1 under compression according to Swift and Voce ....... 71
3-7 Schematic of the quasi-static round tensile test specimen. ...................... 72
3-8 Stress strain curves for ES-1 quasi-static tension test results of forged, cast and HIP’d, and cast no HIP specimens. ........................................... 74
3-9 Fracture surface of the round specimens following tensile tests .............. 75
3-10 Fracture surface of the round specimens following tensile tests .............. 76
3-11 Schematic of the quasi-static (flat) pin-loaded tensile test specimen. ...... 78
3-12 Quasi-static tension characterization stress-strain response. .................. 79
3-13 Camera layout to capture 2-D measurements of the strain evolution on the face and the side of the flat specimens during quasi-static tensile testing. ................................................................................................................. 81
3-14 Width and thickness strain of flat specimens under uniaxial tension ........ 82
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3-15 Schematic SHPB with an illustration of the propagating strain waves. ..... 83
3-16 A free body diagram of a portion of the SHPB of length dx and cross sectional area A. ...................................................................................... 87
3-17 Raw data from the dynamic SHPB characterization ................................. 88
3-18 The SHPB test specimen subjected to forces at the incident and transmitted bar interfaces. ........................................................................ 89
3-19 UF REEF test equipment for dynamic SHPB characterizations ............... 90
3-20 Strain waves at the incident and transmitted strain gages ....................... 91
3-21 Stress strain response of forged, cast and HIP’d, and cast no HIP ES-1 during SHPB tests. ................................................................................... 92
3-22 Forged Eglin steel cylindrical SHPB test specimens ................................ 93
3-23 Comparison of quasi-static and dynamic stress-strain response ............. 95
3-24 Initial yield stress-strain-rate dependence of AF-1410, AISI 4340, and ES-1 observed by Last in comparison to the cast material. ........................... 96
3-25 Strain rate in forged, cast and HIP’d, and cast no HIP ES-1 during SHPB tests. ........................................................................................................ 97
3-26 Dynamic pulse shapers made of copper, steel, and Teflon ...................... 98
3-27 Strain rate of forged test specimens without a pulse shaper, with the first pulse shaper, and with the second pulse shaper. .................................... 99
3-28 Stress strain curves of forged specimens without a pulse shaper, with the first pulse shaper design and the second pulse shaper design. ............. 100
3-29 UF REEF materials lab for cyclic characterization of ES-1 .................... 101
3-30 Hysteresis loops of ES-1 under completely reversed displacement ....... 102
3-31 Uniaxial absolute flow stress of ES-1 at the limits of the elastic region under cyclic loading by plastic strain. ..................................................... 103
3-32 Monotonic and kinematic stress-strain curves of Eglin steel .................. 105
4-1 Illustration of properties of yield surfaces with an associated flow rule .. 109
4-2 Cazacu and Barlat initial and two subsequent yield surfaces using classical isotropic hardening of forged ES-1 in the deviatoric plane. ...... 113
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4-3 The first stress-strain cycle of forged ES-1 during cyclic testing at completely reversed displacement of 0.57, 0.62 and 0.67 mm and the uniaxial tensile and compressive flow stress. ......................................... 118
4-4 Flow stress at the limits of the elastic region by plastic strain in forged ES-1 under cyclic loading. ............................................................................ 119
4-5 The uniaxial back stress as a function of plastic strain for forged ES-1 under cyclic loading. ............................................................................... 120
4-6 Subsequent yield surfaces using classical isotropic, redefined isotropic, and kinematic hardening under uniaxial tensile loading. ........................ 121
4-7 Evolution of the material asymmetry parameter, c, for forged ES-1 using the limits of the elastic region under cyclic loading. ................................ 122
4-8 Uniaxial flow stress at the limits of the elastic region and yield surfaces in the deviatoric plane for several values of plastic strain in forged ES-1 under cyclic loading. ............................................................................... 123
4-9 Cazacu and Barlat subsequent yield surface in the deviatoric plane following uniaxial tension, compression, or pure shear strain. ............... 125
4-10 Decomposition of the yield surface with distortional hardening .............. 126
4-11 Cazacu and Barlat yield surface under both uniaxial tension and compression and the common component............................................. 128
4-12 Initial and subsequent Cazacu and Barlat yield surface under pure shear strain. ..................................................................................................... 129
4-13 Cazacu and Barlat initial and third invariant component of the subsequent yield surfaces using asymmetric isotropic hardening under uniaxial tensile loading in the deviatoric plane. ............................................................... 136
4-14 Cazacu and Barlat yield surface in the deviatoric plane under uniaxial tension, uniaxial compression, and pure shear using distortional hardening. .............................................................................................. 138
4-15 Cazacu and Barlat material parameter, c, by plastic strain for forged specimens, cast and HIP’d specimens and cast specimens. ................. 139
4-16 Cazacu and Barlat yield surfaces in the deviatoric plane ....................... 142
5-1 Stress distribution according to the model within forged ES-1 under quasi-static uniaxial tension. ............................................................................ 153
5-2 Stress and plastic strain for forged ES-1. ............................................... 154
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5-3 The longitudinal strain distribution from DIC and FEA forged specimens. ............................................................................................................... 155
5-4 The longitudinal strain distribution from DIC and FEA cast and HIP’d specimens. ............................................................................................. 155
5-5 The longitudinal strain distribution from DIC and FEA of cast specimens. ............................................................................................................... 156
5-6 Stress-strain curves of flat DIC and FEA specimens in tensile loading .. 157
5-7 The uniform, longitudinal stress distribution within the forged round finite element specimens under quasi-static compressive loading ................. 158
5-8 Stress-strain curves of round DIC and FEA compression specimens .... 159
5-9 The experimental and model load-displacement curves ........................ 162
5-10 Stress-strain curves for round DIC and FEA tension specimens ............ 163
5-11 The von Mises stress distribution in the forged FEA specimen .............. 164
5-12 Evolution of the axial stress vs. equivalent plastic strain for tension-compression cyclic loading. .................................................................... 166
5-13 Geometry of the thin-walled specimen used for free-end torsion loading with dimensions expressed in millimeters. ............................................. 168
5-14 Isocontour of the predicted axial displacement that develops during free-end torsion loading of a forged ES-1 material: ....................................... 169
5-15 Longitudinal elongation by shear strain for forged quasi-static specimens under quasi-static torsion. ...................................................................... 170
5-16 The finite element SHPB and an inset containing a close-up of an ES-1 specimen. ............................................................................................... 171
5-17 SHPB incident, transmitted, and reflected waves from forged specimen and FEA. ................................................................................................ 172
5-18 Equivalent plastic strain isocontour in the cylindrical Taylor impact specimen of forged ES-1. ....................................................................... 175
5-19 Linear relationship of deformed section length by deformed total length following experimental and simulated Taylor impact tests of specimens of ES-1. ...................................................................................................... 176
5-20 Stress strain response of ES-1 subject to dynamic strain rates in Taylor impact tests conducted by Torres et al. .................................................. 177
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LIST OF ABBREVIATIONS
AHSS Advanced high strength steel
AFRL Air Force Research Laboratory
AISI American Iron and Steel Institute
AUST SS Austenitic stainless steel
AWEF Advanced Weapons Effects Facility
BCC Body-centered cubic
BCT Body-centered tetragonal
C Celsius
CPB Cazacu, Plunkett, and Barlat
CDIC Circular polarized light differential interference
CP Complex-phase
DAS Dendritic arm spacing
DIC Digital image correlation
DP Dual-phase
EBSD Electron backscatter diffraction
EDS Energy dispersion spectrography
F Fahrenheit
FCC Face-centered cubic
FEA Finite element analysis
ft Foot
GPa Gigapascal
HIP Hot isostatic pressure
ksi Kilopounds per square inch
LVDT Linear variable differential transformer
14
m Meter
MHz Megahertz
mm Millimeter
MPa Megapascal
MRD Maximum multiple random distribution
ms Millisecond
MS Martensitic steel
OFHC Oxygen-free high thermal conductivity
Pa Pascal
psi Pounds per square inch
REEF Research and Engineering Education Facility
RGB Red, green, and blue
RVE Representative volume element
s Second
SEM Scanning electron microscope
SHPB Split Hopkinson pressure bar
STF Strain-to-failure
STN Strain-to-necking
STP Standard test procedure
TRIP Transformation-induced plasticity
TWIP Twinning-induced plasticity
UF University of Florida
UHSLA Ultra high strength low alloy
UHSS Ultra high strength steel
UMAT User material subroutine
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USAF United States Air Force
V Voltage
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Abstract of Dissertation Presented to the Graduate School Of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
EXPERIMENTAL CHARACTERIZATION AND MODELING OF HIGH
STRENGTH MARTENSITIC STEELS BASED ON A NEW DISTORTIONAL HARDENING MODEL
By
Elizabeth K Bartlett
May 2018
Chair: Oana Cazacu Major: Mechanical Engineering In this dissertation is presented an extensive experimental and theoretical
investigation into the mechanical behavior of ultra high strength martensitic
steels, with the overall goal of determining the effect of processing on the
mechanical properties.
For this purpose, experimental characterization of forged, cast, and cast
and hot-isostatically pressed (HIP) specimens was conducted for both quasi-
static and high-rate loadings. For quasi-static loadings, the influence of loading
history was quantified by performing monotonic compression, monotonic tension,
and cyclic tension tests. Moreover, for monotonic tensile tests the influence of the
specimen geometry on localization of the deformation and strain-to-failure was
investigated. While the forged material demonstrated the greatest initial yield and
flow stress for all loading conditions, the test results indicate that the ductility of
the cast material was significantly increased by the subsequent HIP. Irrespective
of processing, the martensitic steel studied displays a higher plastic flow stress in
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uniaxial compression than in uniaxial tension, this strength differential effect
evolving with accumulated plastic deformation.
Comparison between the quasi-static and dynamic flow stresses
indicated that the dynamic increase factor is approximately 1.1 for all three
materials.
On the basis of the experimental results, an elastic/plastic modeling
approach was adopted. To account for tension-compression asymmetry in
yielding the isotropic form of Cazacu and Barlat (2004) yield criterion was used.
As concerns hardening of martensitic steels, a new distortional hardening model
was developed. In contrast to the existing hardening models, the new model
proposed can account for the Bauschinger effects.
Finally, the theoretical model was implemented in a fully three-
dimensional, implicit finite element solver, and the model predictions were
validated through comparison with data that were not used for identification of the
model parameters.
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CHAPTER 1 INTRODUCTION
This dissertation is devoted to the experimental characterization and
modeling of the effect of the manufacturing processes on the mechanical
behavior of a high strength martensitic steel, Eglin steel or ES-1. In this chapter,
an overview of the state-of-the-art on the metallurgical studies, experimental
mechanical characterization, and modeling of high strength steels is presented.
The key open research issues are identified along with the gaps in knowledge
that are attempted to be filled by the research conducted as part of this
dissertation.
1.1 Background of High Strength Martensitic Steels
Steels, alloys of iron and carbon, are some of the most widely used
engineering materials due, in part, to its strength and ease of manufacture.
Generally, the production costs of steel remain low because of an abundance of
suppliers and well-established supply lines. Although steel is viewed as nearly
an obsolete material by some who are more interested in the attractive
technology of, say, composite materials, the modern steel industry is using
progressive research and providing new alloys with improved mechanical
properties. This continuing progress in steel alloys development is due in a large
part to ever-changing safety and environmental requirements affecting the
contemporary automotive industry. A large class of new steels with yield
strength greater than 1379 MPa—known as advanced high-strength steels
(AHSS) or ultra high-strength steels (UHSS) have been developed over the last
decades. It encompasses dual-phase (DP) steels, complex-phase (CP) steels,
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transformation-induced plasticity (TRIP) steels, twinning-induced plasticity
(TWIP) steels, and martensitic steels (MS).
The evolution of UHSS can be generally considered as comprising three
distinct generations. The first-generation UHSS steels are well accepted and are
currently widely in use in myriad applications. Examples include DP, CP and
TRIP. Other steels that are currently in transition from development into
production include TWIP and austenitic stainless steel (AUST SS). These
second-generation UHSS have high strength, comparable to MS, and display
ductility similar to mild steel. Third-generation steels are currently in research
and development, the aim being to fill the gap in terms of properties which is
shown in the strain-to-failure (STF) vs. yield stress graph of Figure 1-1.
Figure 1-1. Tensile yield stress vs. strain-to-failure of current steels and expected strengths of third-generation UHSS.
Eglin steel is a third-generation martensitic steel that has been developed
for use under dynamic conditions where exceptional strength and toughness are
required (e.g. as a casing material that needs to survive the high-impact speeds
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during deep penetration events). Iron is the base metal material and can take on
the crystalline forms (allotropes in Figure 1-2): body-centered cubic (BCC) and
face-centered cubic (FCC), depending on its temperature. The BCC form of iron
is called ferrite (or -iron). Below 910oC, the BCC allotrope of pure iron is stable.
Above this temperature the face-centered cubic allotrope of iron, called austenite
(or -iron) is stable. Rapid cooling (quenching) of the austenite form of iron in oil
or water at such a high rate that carbon atoms do not have time to diffuse results
in the FCC austenite transforming into a body-centered tetragonal (BCT) form,
called martensite that is supersaturated with carbon. The shear deformations
taking place produce a large number of dislocations, which is a primary
strengthening mechanism of the resulting martensitic steels.
Figure 1-2. The unit cell of single-crystal microstructures of solid steel. A) FCC, B) BCC, and C) BCT.
Moreover, the type of crystal structure present dictates the stress-strain
response of the material and its ability to deform plastically (i.e. sustain
permanent deformation). Irreversible or plastic deformation occurs by
crystallographic slip or twinning, mechanisms illustrated schematically in
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Figure 1-3. During crystallographic slip, a lattice plane glides by an adjacent
plane along a specific direction. Slip occurs on densely packed planes, and
therefore the slip systems are different depending on the single-crystal structure.
An imperfection in the three-dimensional lattice is known as a dislocation. The
energy required for the activation of crystallographic slip is significantly reduced
in the vicinity of a dislocation. For example, in pure iron, the crystal structure has
relatively little resistance to the iron atoms slipping past one another, and so pure
iron is quite ductile, or soft and easily formed. In steel, small amounts of carbon,
other elements, and inclusions within the iron act as hardening agents that
prevent the movement of dislocations that are common in the crystal lattices of
iron atoms. Another mechanism for plastic deformation in which the lattice
structure rotates about a plane of symmetry is known as twinning (see
Figure 1-3). Cubic single-crystal structures generally require more energy to
produce twinning and so twinning is less prevalent until temperatures dip well
below ambient (<196oC) or very high strain rates (>106 /s) are produced (Blewitt
et al., 1957, Huang et al., 1996).
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Figure 1-3. Illustration of plastic deformation mechanisms. A) slip and B) twinning.
Additionally, manufacturing processes can affect the microstructure and
subsequent mechanical response of martensitic steels, and specifically ES-1.
Forging is a forming process that involves applying large compressive forces with
a power hammer, press, or die to work a piece to deform it into a predetermined
shape. Deformation of the metal is accomplished using hot, cold, or even warm
forging processes. Ultimately, the manufacturer will look at a number of criteria
before choosing which type of forging is best for a particular application. Forging
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reduces the retained porosity of the material, reduces segregation, and realigns
grains to increase the specimen strength.
Casting is a process in which a liquid metal is poured into a mold that
contains a hollow cavity of the intended shape. The metal and mold are then
cooled, and the metal part (the casting) is extracted. Casting including
a solidification process, the solidification phenomenon controls most of the
properties of the casting. Moreover, most of the casting defects occur during
solidification, such as gas porosity and solidification shrinkage. To reduce this
retained porosity, some castings are subjected to hot isostatic pressing (HIP). In
the HIP process, the casting is subjected to both elevated temperature and
isostatic gas pressure in a high-pressure containment vessel.
1.2 Survey of the Experimental Studies on Martensitic Steels
At present, most of the studies on martensitic steels report only the
mechanical response under tensile loadings (e.g. Little et al., 1978). However,
early researchers have investigated the mechanical response under both tensile
and compressive loadings, and concluded that martensitic steels have higher
flow stress in compression than in tension. A brief summary of the main findings
is presented in the following.
Leslie and Sober (1967) conducted experiments on a dozen different
materials including SAE 4300 steels under uniaxial tension and compression, at
temperatures between -195oC and 23oC, and strain rates between 10-5 and 10-1
s-1. It was found that untempered martensites are more resistant to plastic flow in
compression than in tension during initial yielding and the early stages of plastic
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flow. To quantify this tension-compression asymmetry in the plastic regime,
these authors introduced a parameter, called the fractional strength differential
parameter (FSD), which is defined as:
C T
C T
FSD
, (1-1)
where σC denotes the flow stress in uniaxial compression and σT is the flow
stress in uniaxial tension. It was observed that for SAE 4300 series 4310, 4320,
4330, and 4340 steels there is a linear relationship between the fractional
strength differential parameter and the carbon content by percent weight (see
also Figure 1-4).
Figure 1-4. The fractional strength differential parameter vs. the carbon content by percent weight for different SAE 4300 series.
Several explanations for the observed strength differential in martensitic
steels have been proposed. One of the explanations is that SD effects are due
25
to solute-dislocation interaction, namely that the resulting martensitic BCT
structure, specifically the ratio c/a of the cell (see Figure 1-2) leads to non-linear
local elastic strains. Using a conservative estimate of the carbon–dislocation
interaction energy, Hirth and Cohen (1970) demonstrated that the resulting SD
values are between 3 and 6%, a range similar to that observed experimentally.
Chait (1971) conducted uniaxial tension and compression tests of 4340
and H11 high-strength steels and 410 martensitic stainless steels at room
temperature. A constant value of the SD parameter (see Eq. (1-1)) was
observed over a large range of plastic strains for all steels except the stainless
steel. Furthermore, Chait (1971) reported that for SAE 4340 the SD parameter
C TSD decreases with increasing tempering temperatures as shown in
Figure 1-5.
Figure 1-5. The effect of tempering temperature on the strength differential of quenched and tempered 4340 UHSS.
26
Resistance to plastic strain increases with increasing strain rates. For
example, Nadai (1963) reported that for 0.35 carbon steel there is a two-fold
increase in flow stress as the strain rate was increased 10,000 times. More
comprehensive surveys on the strain-rate sensitivity of various materials,
including high-strength steels were conducted by Maiden and Green (1966), Last
et al. (1996), and Johnson and Cook (1983).
Specifically, Maiden and Green (1966) conducted an investigation of the
rate-sensitivity of several metallic and non-metallic materials using a medium
strain-rate machine and a split Hopkinson pressure bar (SHPB) (for a description
of this dynamic testing technique and apparatus, see Chapter 3) over a range of
strain rates 10-3 to 104 s-1. They found that aluminum alloys 7075-T6 and 6061-
T6 were rate insensitive within this range. The rate-dependent behavior of 6A1-
4V titanium was described using a power law type relation between the uniaxial
compressive flow stress normalized by the quasi-static compressive yield stress
and the strain rate, , i.e.
1 n
C
k
. (1-2)
In the above equation, k and n are constants. For 6A1-4V titanium, k=1
and n = 0.2.
Johnson and Cook (1983) presented a new equation for description of the
effects of temperature and strain rate on the flow stress of materials based on
data collected in quasi-static uniaxial tension tests and SHPB tensile tests at
various temperatures. The materials tested included oxygen-free high thermal
conductivity (OFHC) copper, Cartridge brass, Nickel 200, Armco tool, Tungsten
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alloy, and DU-.71Ti depleted uranium. This empirical relationship was validated
using Taylor impact tests in which strain rates are typically between 104 and 105
s-1. The isothermal form of Johnson-Cook equation is:
*1 lnn
J J P J PA B C (1-3)
In the above equation, AJ, BJ, CJ, and n are material parameters, and *
p is
the relative plastic strain rate defined as:
*
0
p
p
p
, (1-4)
Where p is the dynamic plastic strain rate while 0p is the quasi-static plastic
strain rate.
Last et al. (1996) characterized the high strain-rate response of Hy100,
Hy130, and AF-1410, all martensitic steels. Hy100 was rate sensitive regardless
of temperature. However, Last et al. (1996) found that the rate-sensitivity of
HY130 and AF-1410 increased with temperature as shown in Figure 1-6.
28
Figure 1-6. The effect of tempering on the flow stress of various steels.
Understanding and modeling the behavior of engineering materials, and in
particular steels is a key for estimating the service life and safety of structural
components.
The stress-strain response of most engineering metals under cyclic
loading is often significantly different from the monotonic response. In fact, the
stress-strain response observed during the first cycle of loading is often
significantly different from the second, and the second is different from the third,
until the material stabilizes. In the subsequent hysteresis loops of Ti-8A1-1Mo-
1V (Collins, 1981), the difference between the stress-strain response in the 100th
and the 1000th cycle is less pronounced than that of the first and third.
Eventually, the same flow stress is observed in subsequent cycles (i.e. the
material has stabilized) and a stable hysteresis loop is observed. Thus, the
29
stress evolves under subsequent reversals until it reaches stability as
demonstrated by the stable hysteresis loop.
For martensitic steel SAE 4340, Collins (1981) reported stable hysteresis
loops under completely reversed strain. Initially the material was subject to a
tensile strain beyond yielding until the desired strain amplitude was attained.
Then, the displacement was reversed, unloading and reloading in compression
within the elastic regime to the flow stress in compression, and then to the same
strain amplitude, but this time in compression. Next, the displacement is
reversed again, this time back to tension. The ability of a material to resist
repeated straining is characterized through a cyclic stress-strain curve, loci of the
maximum values of stress and strain from stable hysteresis loops at different
levels of strain amplitude.
For SAE 4340 the monotonic flow stress is greater than the cyclic stress,
meaning that SAE 4340 softens under cyclic loading by up to 30%.
Landgraf (1970) conducted a comprehensive experimental study in which
he compared the hardening behavior of various metals, including martensitic
steels, under monotonic and cyclic loadings. Moreover, for any given material he
identified a power law type hardening model and reported the values of the
respective coefficients. In Figure 1-1 are presented these results. Note that the
high-strength materials (including martensitic steels) typically soften under cyclic
loading. In addition, Landgraf (1970) also reported the monotonic and cyclic
stress-strain behavior.
30
Table 1-1. Monotonic and cyclic material properties. Material Condition Monotonic
Yield Strength (MPa)
Cyclic Yield
Strength (MPa)
Monotonic Strain
Hardening Exponent
Cyclic Strain
Hardening Exponent
Cyclic Behavior
OFHC annealed 21 138 0.400 0.150 Hardens
partial annealed 255 200 0.130 0.160 Stable
cold worked 345 234 0.100 0.120 Softens
2024 Al T4 303 448 0.200 0.110 Hardens
7075 AL T6 469 517 0.110 0.110 Hardens
Man-Ten steel as-received 379 345 0.150 0.160 Both
SAE 4340 350 BHN* 1172 758 0.066 0.140 Softens
Ti-8Al-1Mo-1V duplex annealed 1000 793 0.078 0.140 Both
Waspaloy Ref 11 545 703 0.110 0.170 Hardens
SAE 1045 595 BHN 1862 1724 0.071 0.140 Stable
500 BHN 1689 1276 0.047 0.012 Softens
450 BHN 1517 965 0.041 0.150 Softens
390 BHN 1276 758 0.044 0.170 Softens
SAE 4142 670 BHN** 1620 N/A 0.140 N/A Hardens
560 BHN 1689 1724 0.092 0.130 Stable
475 BHN 1724 1344 0.048 0.120 Softens
450 BHN 1586 1069 0.040 0.170 Softens
380 BHN 1379 827 0.051 0.180 Softens
*Quenched and tempered **As-quenched
The Bauschinger effect, illustrated in hysteresis loops, is so named after
Bauschinger (1886) who observed that “after the stress is reversed from tension
to compression both the elastic range and a yield point for the reversed direction
of straining have completely disappeared.” In other words, a material subject to
uniaxial tension will harden in tension, but upon load reversal, the flow stress in
compression is decreased. If the material is loaded in uniaxial compression, it
hardens in compression and the tensile flow stress is decreased. The
Bauschinger effect is also seen in specimens that have been pre-strained. As an
example, Paul (1968) presented the stress-strain response of copper in uniaxial
tension tests and subsequent compression tests at room temperature from tests
by Nadai (1963).
31
To study the Bauschinger effect, the limits of the elastic region under
uniaxial tension and compression were calculated based on the stable hysteresis
loops of SAE 4340 presented by Collins (1981) using 0.1% offset following load
reversal or the maximum absolute flow stress prior to load reversal. Figure 1-7
plots the absolute flow stress in uniaxial tension and uniaxial compression under
completely reversed strain for SAE 4340. Notice the Bauschinger effect in Figure
1-7 for SAE 4340.
Figure 1-7. Absolute flow stress vs. plastic strain of SAE 4340 from hysteresis loops of cyclic testing.
Bridgman (1952) collected extensive experimental data at high hydrostatic
pressure and concluded that the yield stress of metals is unaffected by
hydrostatic pressures up to 2.8 GPa. However, Bridgman demonstrated
immense increases in ductility under high hydrostatic pressure. For example, he
32
estimated true strain of 4.4 106 in the neck of steel tension specimens under
hydrostatic pressure of 9.8 GPa.
With an understanding of the microstructure and mechanical behavior of
martensitic steels, a brief history and survey of Eglin steel, specifically, follows in
Section 1.3.
1.3 Eglin Steel, ES-1
AF-1410 is a high-strength, high-toughness, super alloy steel for use in
aerospace applications. In an effort to find a lower-cost substitute for AF-140,
Ellwood National Forge Company developed a new class of high-strength, low-
alloy martensitic steel called Eglin steel (ES-1). ES-1 received the patent
US7537727 Eglin Steel—A Low Alloy High Strength Composition by Morris
Dilmore and James Ruhlman (2009).
Table 1-2 presents the chemical composition of a high-alloy steel AF-
1410, UHSLA standard SAE 4340, and that of ES-1 for comparison.
33
Table 1-2. Chemical composition by percent weight of AF-1410, SAE 4340, and ES-1.
Element AF-1410* (%)
SAE 4340** (%)
ES-1 (%)
Cobalt 13.5-14.5000 0.000 0.000 Nickel 9.5-10.5000 1.65-2.000 5.000 Chromium 1.8-2.2000 0.7-0.900 2.380 Molybdenum 0.9-1.1000 0.2-0.300 0.550 Carbon 0.13-0.1700 0.37-0.430 0.250 Manganese <0.1000 0.6-0.800 0.850 Silicon <0.1000 0.15-0.300 1.250 Aluminum 0.0000 0.000 0.000 Vanadium 0.0000 0.000 0.180 Tin <0.0020 0.000 0.000 Lead <0.0020 0.000 0.000 Zirconium <0.0020 0.000 0.000 Boron <0.0005 0.000 0.000 Sulphur <0.0050 <0.040 0.012 Phosphorus <0.0080 <0.035 0.015 Tungsten 0.0000 0.000 1.980 Copper <0.1000 0.000 0.500 Calcium 0.0000 0.000 0.020 Nitrogen <0.0015 1.65-2.000 0.140
Iron Bal. Bal. Bal.
*AF-1410 chemical composition from Little (1979). **SAE 4340 chemical composition from Lynch (2011).
Notice the total alloy content of AF-1410, SAE 4340, and ES-1 are as high
as 29, 6, and 13%, respectively. Also, note that ES-1 uses roughly half as
much nickel and cobalt as other superalloys (e.g. AF-1410), which makes it a
low-cost alternative. Although the percentage of nickel was decreased in ES-1
compared to AF-1410, substituting silicon to help with toughness and particles
of vanadium carbide and tungsten carbide for additional hardness and high-
temperature strength ensured that ES-1 has the required properties. Moreover,
ES-1 contains less carbon than SAE 4340, and thus the tetragonality of ES-1
(ratio of c/a of its BCT structure, see also Figure 1-2) is just 1.01.
In addition to smaller amounts of expensive alloys, Eglin steel’s low cost is
in part a result of its manufacturability and machinability. Ingots of ES-1 are
manufactured by several processes depending on the application requirements;
34
these processes include electric arc, ladle refined, vacuum treated, vacuum
induction melting, vacuum arc re-melting, and/or electro-slag re-melting.
However, the ingot manufacturing process does affect the mechanical properties.
Final Eglin steel products can be produced by forging, casting, extrusion, rolling
or other conventional methods in accordance with the patent.
Before publishing the patent, Eglin steel was tested in quasi-static and
dynamic tension at ambient and high temperatures. The Rockwell hardness and
Charpy V-notch energy were also measured. Table 1-3 presents the ultimate
tensile strength, initial yield stress, and strain-to-failure in uniaxial tension of ES-1
along with the same properties of other martensitic steels: AF-1410 and SAE
4340.
Table 1-3. Mechanical properties under uniaxial, quasi-static tensile loading of AF-1410, SAE 4340, and ES-1.
Ultimate Tensile Strength
(%)
Yield Strength (%)
Strain to Failure (%)
AF-1410 1620 1482 12.0 SAE 4340 1793 1496 10.0 ES-1 1701 1337 18.4
* AF-1410 mechanical properties from Little (1979) ** SAE 4340 mechanical properties from Lynch (2011).
Note that the largest percentage difference in both ultimate tensile and
yield strength between the three UHSS is approximately 10%. The reported
strain-to-failure of ES-1 is significantly higher than AF-1410 and SAE 4340. SAE
4340 has the highest initial yield strength and ultimate tensile strength.
The graph in Figure 1-8 shows the tensile yield stress vs. STF of several UHSS
including ES-1. Notice that Eglin steel is a high strength steel with twice the
ductility of other martensitic steels.
35
Figure 1-8. STF vs. tensile yield stress for different steels including conventional steels and UHSS.
Boyce and Dilmore (2009) used a servohydraulic motor outfitted with a
custom-built load cell to characterize the stress-strain response of flat specimens
of AerMet100, 4340M, HP9-4-20 and ES-1c in both the quasi-static and dynamic
regimes. They used strain gages in a custom built load cell on the upper and
lower threaded sections of the test specimens to measure the load in addition to
the standard quartz load cell that introduces oscillations in the force data.
Secondly, a rubber pulse shaper was used in the high strain-rate experiments (1
to 200 s-1) to reduce the elastic oscillations in the test. Using both of these
methods the oscillations were dampened enough to directly determine the 0.2%
offset from the stress-strain curve. All of the alloys exhibited slight strain-rate
sensitivity (magnification factors between 1.06 – 1.10 at 2% strain). The data
was used to obtain the coefficients of a power law hardening for strain-rate
sensitivity (see Eq. (1-2)), and a semi-logarithmic relationship, which is more
typical for ferrous materials. It is worth noting that among all the alloys on which
36
the authors reported data, Eglin steel was the only alloy that shows increase in
ductility with increasing the strain rate.
In addition to the experimental characterization completed by Dilmore and
Ruhlman (2009) and included in the patent, several other researchers conducted
dynamic characterization experiments on ES-1. Importantly, Torres et al (2009)
completed twenty Taylor impact tests with 164-caliber (41.66 mm) and 215-
caliber (54.61) cylindrical specimens to determine the stress-strain response of
Eglin steel at strain rates between 103 and 104 s-1 which are higher than those
that can be achieved with the SHPB apparatus, but lower strain rates that occur
in plate impact tests. To interpret the data, use was made of the one-
dimensional theory proposed by Jones et al. (1998). However, it was found that
a 1-D analysis is no longer accurate for strains larger than 10%. Nevertheless, it
was possible to estimate the dynamic flow stress at 10% deformation, which
appears to be approximately 2.5 GPa.
Martin et al. (2012) conducted five symmetric plate impact tests, with the
impact velocities being between 400 and 1000 m/s using ES-1 impactors and
targets. Prior to conducting the tests, Martin et al. (2012) used ultrasonic
techniques to determine the average longitudinal and shear wave speeds of the
material. It was found that ES-1 has average longitudinal and shear wave speeds
of 5.88 and 3.20 km/s and bulk wave speed of 4.58 km/s (Poisson ratio 0.29).
The impactors were launched from a smooth bore powder gun and a velocity
interferometer (VISAR) was used to measure the velocity of particles on the back
side of the targets as shown in Figure 1-9.
37
Figure 1-9. ES-1 symmetric plate impact experimental setup of Martin et al.
The Hugoniot elastic limit σHEL is defined as:
0
2
L EHEL
C u
(1-5)
where ρ0 is the initial density, CL is the longitudinal wave speed, and uE is the
particle velocity where the elastic to plastic transition occurs. For ES-1, σ HEL is
approximately 2.35 GPa.
The spall strength (dynamic tensile strength) is defined as:
0
2
L pbC uS
(1-6)
where Δupb denotes the pullback signal.
Estimating the pullback signal as being Δupb~0.279 km/s, the spall
strength of ES-1 calculated with the above formula is: S~6.34 GPa.
Finally, in experiments conducted by Weiderhold, Lambert, and Hopson
(2010), the detonation of hollow explosive-laden cylinders of Eglin steel was used
to analyze fragmentation and develop a statistical failure model.
In summary, prior to the research done as part of this dissertation the
stress-strain response has been experimentally characterized only under uniaxial
loading. In reality, materials are subjected to complex three-dimensional loads.
38
As materials cannot be tested in every possible loading configuration, a model is
required to predict behavior under complex, three-dimensional loads. Ultra high-
strength martensitic steels exhibit a well-defined yield stress and plastic
deformation. Therefore, the most appropriate framework for modeling their
behavior is that of the elastic-plastic theory. The general form of the governing
equations is briefly presented in the following, along with the state-of-the art in
description of yield criteria and hardening laws for metallic materials.
1.4 Elastic-Plastic Modeling
1.4.1 Yield Criteria
Elastic-plastic models best describe polycrystalline materials that have a
well-defined yield stress that defines the onset of plastic deformation. Yield
criteria define the boundary between the elastic and plastic regimes, and can be
visualized in six-dimensional stress space by surfaces. In the interior of the yield
surface the material behavior is elastic, usually characterized by Hooke’s linear
relationship. On the yield surface, the material behavior is plastic, nonlinear, and
depends on internal variables such as loading history. The yield surface grows,
translates, and changes shape based on the hardening behavior of the material.
Therefore, the yield function (F) is commonly expressed as the difference
between the effective stress ( ), a function of the Cauchy stress and a measure
of the equivalent plastic strain, and the hardening (Y), which is a function of only
the equivalent plastic strain:
, , 0p p pF Y σ σ (1-7)
39
The earliest yield criterion for isotropic pressure-insensitive metals was
proposed by Henri Tresca in 1865. It states that yielding occurs when the
maximum shear stress reaches a critical value, f . This criterion can be
expressed in terms of the principal stresses as:
1 2 2 3 1 3max( , , ) 2 f . (1-8)
The Tresca yield criterion is independent of the hydrostatic pressure (or
mean stress) and therefore can also be written in terms of the deviatoric stress
and the surface can be presented in the three-dimensional principal stress space
(or Haigh-Westergaard space) as an infinite, hexagonal prism with the
longitudinal axis (Figure 1-10) coinciding with the hydrostatic axis, or as a regular
hexagon in the deviatoric or plane, (plane having the normal the hydrostatic
axis).
1 2 3, ,
40
Figure 1-10. Tresca yield surface in Haigh-Westergaard space with the hydrostatic axis and deviatoric plane.
Richard Edler von Mises proposed in 1913 another yield criterion for
isotropic materials which states that yielding occurs when J2, the second invaraint
of the stress deviators reaches a critical value. It follows that the criterion writes:
2 2 2 2
2 1 2 3
1/ 3
2TJ s s s (1-9)
where s1, s2, s3, are the principal values of the stress deviator s and T is the
yield stress in uniaxial tension. In the Haigh-Westergaard space, the von mises
yield surface is a right circular cylinder of generator the hydrostatic axis (Figure
1-11).
41
Figure 1-11. Von Mises yield surface in Haigh-Westergaard space with the longitudinal axis as the hydrostatic axis.
In 1949, Daniel C. Drucker proposed a yield criterion for isotropic
materials that involves both J2 and the third-invariant 3 1 2 3J s s s of the deviatoric
stress. It is expressed as:
3 2 6
2 3 fJ cJ (1-10)
where c is a material parameter.
The projections in the biaxial plane ( 3 0 ) of the yield surfaces of the
Drucker (1949) corresponding to c=2.25, Mises, and Tresca yield criteria in the
plane of the third principal stress are presented in Figure 1-12.
42
Figure 1-12. Projection in the biaxial plane of the yield surfaces of Drucker, Mises, and Tresca yield criteria.
Note that according to either criterion the yield stress in tension is equal to
the yield stress in compression.
In 1954, Hershey proposed an yield function later used by Hosford (1972)
expressed in terms of principal stresses.
The criterion involves a unique parameter n. It reduces to Tresca's
criterion for n=infinity and to von Mises for n=2.
1
1 2 2 3 1 3
1 1 1
2 2 2
nn n n
(1-11)
In 2004, Cazacu and Barlat formulated an isotropic yield criterion
represented by an odd function that depends on both invariants J2 and J3. In this
manner, it was possible for the first time, to account for tension-compression
asymmetry in yielding of pressure-insensitive metals. Its expression is:
43
13 322 33
3 3 2T
J J
c
. (1-12)
The coefficient, c, can be determined based on the flow stress in tension
( ) and compression ( ) as (see Cazacu and Barlat, 2004):
3 3
3 3
3 3
2
T C
T C
c
. (1-13)
The Cazacu and Barlat (2004) isotropic yield function reduces to von
Mises for . For , the Cazacu and Barlat (2004) yield criterion is a
triangular prism with a longitudinal axis coinciding with the hydrostatic axis
(Figure 1-13).
T C
0c 0c
44
Figure 1-13. Cazacu and Barlat yield surface in Haigh-Westergaard space.
The projections in the biaxial plane ( 3 0 ) of the Cazacu and Barlat
(2004) yield surface is presented in Figure 1-14 for various ratios of flow stress
( /T C =3/4, 1, and 5/4) which correspond to c=1.056, 0.000, and 0.082,
respectively.
45
Figure 1-14. Projection in the biaxial plane of the Cazacu and Barlat yield surface corresponding to flow stress ratios of 3/4, 1, and 5/4.
Porosity can produce pressure dependence on yielding. Gurson deduced
the yield criterion for a von Mises material containing randomly distributed pores.
The pores were either of spherical or cylindrical geometry. In 2009, Cazacu and
Stewart (2009) developed a yield criterion for isotropic materials that display
tension-compression asymmetry based on Cazacu, Plunkett, and Barlat (2006),
also known as CBP-06, rather than von Mises effective stress:
2
2
2
2
32 cosh 1 0
2
32 cosh 1 0
2
e mm
T T
e mm
T C
f f
f f
(1-14)
where
is the mean macroscopic stress, m
46
is the effective macroscopic stress,
is the absolute value of the matrix material yield stress under uniaxial
compression,
is the matrix material yield stress in uniaxial tension, and
f is the void volume fraction.
This is not an exhaustive list of yield criteria, and many other criteria exist
to model materials with different characteristics. For example, several models for
nonmetallic materials include the first invariant of the stress to account for
pressure sensitivity. Furthermore, many materials display orthotropic behavior
and are better described by orthotropic criteria such as that of Hill (1948) and
Cazacu, Barlat and Plunkett (2006).
In this research, a new elastic-plastic model based on hardening as a
function of the second and third invariants of plastic strain using Cazacu and
Barlat (2004) asymmetric yield criteria is developed with a completely new
approach to the definition of asymmetry. Therefore, a background of existing
hardening models is presented in the subsequent section.
1.4.2 Background of Hardening Models
The term hardening often evokes the uniaxial stress-strain response of
materials under monotonic, uniaxial tension. Under this loading, a simple power
law approach can be used to describe the hardening or the increase in flow
stress following initial yielding of the material. In three dimensions, hardening is
characterized by subsequent yield surfaces that satisfy the yield function (Eq
(1-15)). Paul (1968) describes the three-dimensional hardening as isotropic if the
e
C
T
47
subsequent yield surfaces are larger than the previous yield surface without a
change in shape or translation in stress space and kinematic if the subsequent
yield surfaces translate in Haigh-Westergaard space. In 1957, Hodge proposed
combined hardening with an isotropic component (Y) and a kinematic portion
resulting from a fictitious back stress ( ):
, , ,p p pF Y σ σ (1-15)
1.4.2.1 Isotropic hardening
The isotropic hardening, defined as the expansion of the yield surface of
metallic materials under monotonic loading is generally considered to be a
function of equivalent plastic strain and can be described utilizing the hardening
curve of the material observed in uniaxial tension. The equivalent plastic strain
p associated to a given yield criterion is defined by the work equivalence
principle (Hill, 1987). For example, if a material is governed by the von Mises
yield criterion, we have
2
3
p p pε : ε
. (1-16)
Typical rate-independent isotropic hardening laws are Swift (1952) law
and Voce (1948) law.
snp
s pY k (1-17)
V pCp
V VY A B e
(1-18)
In the above equations, ks and ns and AV, BV, and CV are the material
parameters.
48
1.4.2.2 Kinematic hardening
To account for the Bauschinger effect, kinematic hardening under cyclic
loading is usually modeled by a translation of the yield surface resulting from the
application of a fictitious back stress tensor (). It should be noted that a
nonzero back stress destroys the symmetry based on the principal deviatoric
stresses. Several kinematic hardening models have been proposed including
Melan (1938) based on the plastic strain tensor, Prager (1955) based on the
plastic strain-rate tensor, and Lubliner (1990) with a term dependent on the
equivalent plastic strain rate presented below with material constants c and a:
p
c
c
c a
p
p
p
ε
ε
ε ρ
(1-19)
Dafalias and Popov (1975) and Krieg (1975) defined two-surface models
in which the yield surface is limited by an outer bounding surface in stress space.
1.5 Goals of Current Research
The overall goal of the dissertation research is to determine, for the first
time, the effect of three different manufacturing processes on the mechanical
response of Eglin steel: Chapter 2 presents the microstructure and porosity of the
material under study in its initial condition. Chapter 3 presents an in-depth
experimental characterization of the material response under various loading
paths and strain rates. Chapter 4 includes a review of finite element analysis, a
development of the theoretical model, identification of material parameters, and
integration of the theoretical model into a fully three dimensional finite element
analysis. Chapter 5 explains the model validation using independent
49
experimental data and finite-element analyses as well as predictions of the
material response under more complex loadings and higher strain rates. Finally,
Chapter 6 concludes this dissertation and research with a brief summary and
recommendations for continuing this exciting research.
50
CHAPTER 2 MICROSCOPY
Each manufacturing processing procedure implies different consequences
for the material: forging is a costly process but reduces porosity, shrinkages and
cavities, while casting is a cheaper process, but voids and cavities are created
during the solidification of the material. Most of the defects induced by
solidification could be removed with an HIP operation. Therefore, the initial
condition of each material including chemical composition, porosity, prior
austenite grain size, and dendrite arm spacing (DAS) was characterized using
microscopy.
The cast Eglin steel was received at AFRL in 2-in thick plates. The HIP
process involved heating the cast specimens to 2125oF at 15 ksi. All the of cast
specimens were then subject to homogenization to decrease segregation,
austenization to break down iron carbides, quenched to form martensite, and
tempered to relieve residual stresses. The forged specimens were subject to a
milder heat treatment process due to the superior homogeneity of forged
materials as compared to cast materials. The heat treatment schedules for all
three manufacturing processes are summarized in Table 2-1.
Table 2-1. Heat treatment schedule for ES-1. Forged Cast and HIP’d Cast
Hydrogen Bake 315oC/12 hr
HIP @ 103 MPa - 1163oC/4hr -
Homogenization 1163oC/4 hr - 1163
oC/4 hr
Sub-critical Anneal 677oC/4 hr (Air)
Austenization 1010oC/1.5 hr (Vacuum)
Quench Fast-Quench (Oil) Temper 204
oC/4 hr (Recirculating air)
51
Next, four different ASTM standard test specimens were machined from
the ES-1 plates: 25.4-mm flat tension, 25.4-mm round tension, and round
compression with diameters of 5.08 mm and 7.62 mm. An ES-1 specimen of
each manufacturing process was sectioned, mounted in polyfast resin using a
Struers ProntoPress-22, ground to plane, and polished on a Struers autopolisher
in accordance with ASM Metallurgy and Metallography Handbook as
summarized in Table 2-2. Finally, the specimens were polished in a Struers
vibratory polisher with OP-U Silica Carbide suspension at 40% for 2 hours.
Table 2-2. ES-1 grinding and polishing schedule. Process Grit Speed
(rpm) Force
(N) Time (min)
Lubricant Surface
Grind 220 300 20 5 Water N/A Grind 320 300 20 5 Water N/A Grind 500 300 20 5 Water N/A Grind 1200 300 20 5 Water N/A Polish 9 μm
3 μm 150 150
20 20
4 4
Blue Blue
MD-Largo, MD-Dur Polish
2.1 Optical Microscopy ES-1
The best way to characterize porosity in ES-1 according to Foley (2016) is
optical microscopy of polished specimens using brightfield illumination.
Therefore, each specimen of ES-1 was examined under a Zeiss Axio Observer
Z1 optical microscope to evaluate the porosity. The cast specimens contained
several large pores resulting from partially fused dendrites that may have
resulted from inadequate feeding as volume shrinkage occurred during
solidification. The relatively large pores in the cast material were examined using
Circular polarized light differential interference contrast (CDIC) to emphasize the
depth of the pores. Several examples are displayed in Figure 2-1 and the entire
surface is shown in Figure 2-2.
52
Figure 2-1. Z-stack of several pores on the polished surface of cast eglin steel specimen using CDIC.
53
Figure 2-2. Photomontage stitched from nine individual images of the polished surface of cast ES-1 specimen containing several pores.
An image processing program, Image J, was used to count (Np) and
measure the average area (Ap) of pores above 1.175 μm2 in a 2.5 mm2 area of
the materials at 200 times magnification as shown in Figure 2-3. Assuming a
spherical representative volume element (RVE), the two dimensional measured
area volume fraction of the polished surface was used to calculate the RVE inner
54
(a) and outer (b) radii and the void volume fraction ( ). As expected the
void volume fraction of the forged material is insignificant while that of the cast
and cast and HIPd materials is 0.007 and 0.001, respectively.
Figure 2-3. Polished surface of ES-1 materials at 200 times magnification in an optical microscope with brightfield illumination. A) forged, B) cast and HIPd, and C) cast.
To measure the dendritic arm spacing (DAS), a sample of the cast
material was etched with Nitol and placed in a Keyence optical microscope using
brightfield illumination. Due to the tempering, the dendrites are more difficult to
identify, however, the DAS was approximated as 173 μm using the two dendrites
highlighted in Figure 2-4. The dendrites of the cast and HIPd and forged
specimens were severely deformed in the high pressure operations and thus the
DAS was not measured.
3
3v
af
b
55
Figure 2-4. Nitol etched cast material surface at 100 times magnification in Keyence optical microscope under brightfield illumination.
2.2 Material Characterization using SEM
The microscopy specimens were then mounted in an FEI Quanta 200F
scanning electron microscope (SEM) and subject to a vacuum for energy
dispersion spectrography (EDS) to determine the approximate chemical
composition of ES-1 developed by each manufacturing process shown in Table
2-3.
56
Table 2-3. Chemical composition of ES-1 for EDS.
Elements Eglin Steel* (%)
Forged Cast Cast and HIP’d
Nickel <5.000 1.00 0.98 1.07 Chromium 1.5-3.250 2.57 2.56 2.70 Molybdenum <0.550 0.60 0.67 0.69 Carbon 0.16-0.350 2.87 1.83 1.52 Manganese <0.850 1.06 0.50 0.58 Silicon <1.250 1.20 1.11 1.22 Vanadium 0.05-0.300 0.14 0.08 0.08 Sulphur <0.012 0.00 0.00 0.00 Phosphorus <0.015 0.00 0.00 0.00 Tungsten 1.17-3.250 1.00 0.87 0.75 Copper <0.500 0.00 0.00 0.00 Calcium <0.020 0.00 0.00 0.00 Nitrogen <0.140 0.00 0.00 0.00 Aluminum <0.050 0.00 0.00 0.00 Iron Bal. 89.55 91.41 91.39
*Nominal ES-1 composition from Boyce and Dilmore (2009).
The EDS indicates the molybdenum is slightly high and the tungsten low.
Errors in the light elements like Carbon are expected using EDS with various
elements. Then, the electron backscatter diffraction (EBSD) was conducted
within the SEM equipped with EDAX Hikari cameras to establish average grain
size, grain orientation, and texture.
The orientation maps in Figures 2-6 through 2-8 show the orientation of
BCC single-crystal grains with respect to the surface normal (z) using a red,
green, and blue (RGB) color scheme coupled to Miller indices as indicated in
Figure 2-5.
57
Figure 2-5. RGB color scheme coupled with Miller indices (001), (101), and (111) in cylindrical coordinates of the specimen cross section.
58
Figure 2-6. Orientation map of forged Eglin steel demonstrating prior austenite grain boundaries and martensitic lath structure.
59
Figure 2-7. Orientation map of cast and HIPd Eglin steel demonstrating prior austenite grain boundaries and martensitic lath structure.
60
Figure 2-8. Orientation map of cast Eglin steel demonstrating prior austenite grain boundaries and martensitic lath structure.
The prior austenite grain sizes (G) 5.0 was determined using the linear
intercept method and the circular intercept method outlined in ASTM E112-13
Standard Test Methods for Determining Average Grain Size for all three
materials.
61
Additionally, EBSD was used to create pole figures and analyze the
texture of each material. Texture is the preferential alignment of grains within a
material. Texture commonly develops from material processing, but can be
mitigated by tempering. The pole figures, one figure from a plane from each
family in the cubic crystalline structure (001), (101), and (111), are presented in
Figure 2-9. The BCC orientation of all three crystal planes for all three material
processes appears to be random, confirming the samples are isotropic. The
maximum multiple random distribution (MRD) of 4.917 in the forged specimen
indicates that the (111) plane is approximately 5 times more likely to be oriented
at approximately 75o polar angle and 200o azimuth.
After the initial condition of the materials was determined, an extensive
suite of experiments was conducted to characterize the mechanical behavior of
the materials. The experimental characterization is presented in Chapter 3.
62
Forged
Cast and HIP’d
Cast
Figure 2-9. Pole figures of (001), (101), and (111) for forged, cast and HIP’d, and cast Eglin steel.
63
CHAPTER 3 MECHANICAL CHARACTERIZATION OF ES-1
In this chapter are presented the investigation conducted into the
mechanical behavior of ES-1 for each condition (i.e. forged, cast, cast and
HIP’d). A suite of experiments was conducted on all three ES-1 materials to
characterize the room-temperature mechanical response for both quasi-static
and dynamic strain rates, under a variety of test conditions (e.g. quasi-static and
monotonic loadings and specimen geometries). The test matrix is given in
Table 3-1.
Table 3-1. Eglin steel material characterization test matrix. Strain Rate Manufacturing
Process Loading
Configuration Specimen Shape/
Pulse Shaper Number of Specimens
Quasi-Static
Forged Tension Round 4 Pin Loaded 2
Compression Round 4 Cast and HIPd Tension Round 4
Pin Loaded 2 Compression Round 4
Cast Tension Round 4 Pin Loaded 3
Compression Round 4 High-Strain Rate
Forged Compression None 3 Pulse Shaper 2
Kinematic Forged Cyclic Round 1 Cast Cyclic Round 1 Cast and HIPd Cyclic Round 1
3.1 Hardness of Eglin Steel
Following the microstructure characterization of the initial condition of the
ES-1 materials (see Chapter 2), a Buehler Digital Hardness Tester MMT-3 with a
pyramidal indenter in Figure 3-1 was used to measure the Vickers hardness of
each sample at the AFRL Dynamic Properties Laboratory, Advanced Weapons
Effects Facility (AWEF).
64
Figure 3-1. Buehler Digital Hardness Tester MMT-3.
The material resistance to plastic deformation, HV, was determined with
the following equation (Smith and Sandland, 1922)::
2
FHV
d , (3-1)
where F is the applied force and d is the average length of the diagonal of the
material indent.
Each sample was tested three times across the cross section of the
specimen at distances greater than 2.5 times the diagonal of the indentation from
the outer surface and previous indentations. The forged, cast, and cast and
HIP’d specimens had an average Vickers hardness of 529.2, 499.5, 501.7
kgf/mm2 (or 5189.68, 4920.00, and 4898.42 MPa), respectively. The Vickers
65
hardness measurements converted to Rockwell Hardness are approximately
51.1, 49.1, and 49.1.
3.2 Quasi-Static Mechanical Characterization
An Instron Model 1332 mechanical test frame and an Instron Model 3156-
115 load cell at the AWEF presented in Figure 3-2 were used for both quasi-
static tension and compression testing.
Figure 3-2. AWEF quasi-static experimental characterization setup A) Instron test frame with load cell, B) pin-connected tension specimen prepared for loading and C) compression specimen prepared for loading.
Prior to each quasi-static test, the test specimen was cleaned with
acetone to remove oils and debris and painted in a high contrast, stochastic
pattern using Rustoleum black and white flat enamel spray paints to create
unique facets on the surface of the entire gage length of the test specimens.
A
B
C
66
Two Point Gray high-speed cameras with 5 megapixel sensors and 75mm fixed
focal length Fuginon lenses, stationary fiber optic illuminators, and a digital delay
generator were used to record synchronized images every four seconds.
Analysis of the recorded images using digital image correlation (DIC) commercial
software, ARAMIS v6.2, reveals the location of each facet in three dimensional
space and the local, full-field strain on the surface of the gage length of the test
specimen throughout the test. The tests were conducted under crosshead
displacement control at a rate of �̇� = 0.0127 𝑚𝑚/𝑠 and the load (F) was
recorded every 4 seconds using a 14bit Win600 Digital Oscilloscope.
Assuming plastic incompressibility, the true strain and true stress are
0
0 0
lnl
l
F l
A l
(3-2)
where l is the current length, l0 is the initial length, and A0 is the initial cross-
sectional area.,
3.2.1 Characterization of ES-1 under Uniaxial Compression
At the AWEF, compression tests were conducted in accordance with
ASTM E9 using standard cylindrical specimens of dimensions shown in
Figure 3-3 with lubricant on each face to minimize radial forces at an average
strain rate of 0.000333 s-1. Each test was repeated four times.
67
Figure 3-3. Quasi -static cylindrical compression test specimens.
Note the consistency in the stress-strain response displayed in Figure 3-4
for all three ES-1 materials.
7.62 mm
7.62 mm
68
Figure 3-4. Quasi-static round bar compression test stress-strain results A) forged specimens, B) cast and HIP’d specimens, and C) cast
specimens.
A
B
C
69
As expected based on the microstructural analysis, the stress-strain
behavior of the cast and cast and HIP’d materials in uniaxial compression are
similar, and the forged material is the strongest (where the stress-stain curves for
the three ES-1 materials are superposed in Figure 3-5). For example, at 13%
strain, the average compressive stress for the forged material is 2063 MPa,
compared to 1937 MPa and 1961 MPa for the cast and cast and HIPd materials,
respectively.
Figure 3-5. Comparison of the quasi-static compressive stress-strain response for the ES-1 materials.
70
For each ES-1 material, in accordance with ASTM E9, the yield point, the
beginning of nonlinear stress-strain behavior was determined consistently using
0.2% strain offset. The yield stress σy of each material are given in Table 3-2.
Table 3-2. Yield stress and quasi-static compression test data. Manufacturing Process Specimen σy (MPa)
Forged X29978 1616 X29979 1366 X29980 1503 X29981 1496 Cast and HIP’d X29876 1448 X29877 1340 X29878 1189 X29879 1336 Cast X29928 1395 X29929 1403 X29930 1345 X29931 1340
From the stress-strain curves, both Swift (see Eq. (1-17)) and Voce
hardening laws (see Eq. (1-18)) were identified (see also Figure 3-6). The
numerical values of the parameters involved in each law are given in Table 3-3.
Table 3-3. Coefficients involved in Swift and Voce Hardening Laws for ES-1. Manufacturing Process k
(MPa) n
Av (MPa)
Bv
(MPa) Cv
Forged 2436 0.0708 2044 504.5 63.33 Cast and HIPd 2465 0.0921 1934 556.4 66.20 Cast 2349 0.0777 1934 548.5 67.48
Note that hardening under compression of each ES-1 material is best
characterized using Swift law.
71
Figure 3-6. Hardening of ES-1 under compression according to Swift and Voce A) forged ES-1, B) cast and HIP’d, and C) cast ES-1.
72
3.2.2 Quasi-static Characterization of ES-1 in Uniaxial Tension
At the AWEF, quasi-static, uniaxial tension tests were conducted in
accordance with ASTM E8 standard using standard dog bone specimens
illustrated in Figure 3-7 at an average strain rate of 0.000365 s-1. For each
ES-1 material, the yield stress σy,, the stress at which necking is observed σu,
and the strain-to-necking (STN) are reported in Table 3-4.
Figure 3-7. Schematic of the quasi-static round tensile test specimen.
Table 3-4. Summary of quasi-static tension test data. Manufacturing Process Specimen σy
(MPa) σu
(MPa)
STN
Forged X29938 1601 1835 0.050 X29939 1640 1829 0.052 X29940 1456 1841 0.050 X29941 1465 1838 0.054 Y08613 1535 1782 0.041 Y08614 1506 1758 0.043 Cast and HIP’d X29836 1483 1733 0.047 X29837 1455 1710 0.044 X29838 1411 1736 0.049 X29839 1431 1726 0.049 ES1_C_H1 1425 1646 0.036 ES1_C_H2 1481 1681 0.052 Cast X29888 1428 1711 0.062 X29889 1456 1716 0.074 X29890 1447 1741 0.093 X29891 1450 1722 0.035 ES1_C_1 1400 1644 0.030 ES1_C_2 1390 1652 0.021 ES1_C_3 1407 1649 0.027
As for uniaxial compression loadings, the forged ES-1 exhibits higher
yield stress than the other ES-1 materials, the increase factor being of 1.1, the
uniaxial tensile response of the cast and cast and HIP ES-1 materials being
similar (see also Figure 3-8). The Swift and Voce hardening laws were also
6.35 mm
25.4 mm
31.75 mm
6.35 mm
73
identified based on the uniaxial tensile results. The numerical values of the
respective coefficients are reported in Table 3-5 and capture fairly well the
stress-strain curve until the onset of necking for all three manufacturing
procedures (Figure 3-8). Furthermore, the initial tensile yield stress of the forged,
cast, and cast and HIP’d ES-1 under uniaxial tensile loading corresponds to the
stress documented by Boyce and Dilmore (2009), Dilmore and Ruhlman (2009),
Van Aken et al. (2014), and Lynch (2011) at similar quasi-static strain rates. The
STF of the materials produced by the three manufacturing processes: forged,
cast and HIP’d, and cast were 5.15 %, 4.75 %, and 7.77%, respectively, similar
to that reported by other authors including van Aken et al. and Lynch. However,
the STF reported by Boyce and Dilmore was much higher than that of any of the
manufacturing processes examined in this study possibly due to additional heat
treatment processing.
Table 3-5. Hardening law parameters tensile round specimens. Manufacturing Process k
(MPa) n
Av (MPa)
Bv
(MPa) Cv
Forged 2272 0.0621 1836 307.1 110.5 Cast and HIPd 2171 0.0655 1722 303.5 139.1 Cast 2106 0.0578 1717 268.0 137.3
74
Figure 3-8. Stress strain curves for ES-1 quasi-static tension test results of forged, cast and HIP’d, and cast no HIP specimens.
The fracture surfaces of the round tensile test specimens are generally the
typical cup and cone displayed in Figure 3-9 similar to the ES-1 fracture surfaces
observed by Boyce and Dilmore (2009). The cross section of the fractured
specimens remained circular, further evidence of the isotropy of ES-1
irrespective of manufacturing process.
75
Figure 3-9. Fracture surface of the round specimens following tensile tests A) forged, B) cast and HIPd, and C) cast.
Several recent studies have demonstrated that ductility measurements are
inversely proportional to maximum cluster pore volume and pore fraction at
fracture (Susan et al., 2015, Foley et al., 2016). Consistent with these findings,
note the large porous regions in the cast ES-1 fracture surface and the lack
thereof in the cast and HIP’d ES-1 material.
The purpose of completing quasi-static tension and compression tests
was to establish the symmetry properties of ES-1. Like other BCT martensitic
steels discussed in Section 1.2, Eglin steel demonstrates higher flow stress in
uniaxial compression as compared to tension. Furthermore, the tension-
compression asymmetry evolves with the equivalent plastic strain. At initial
yielding, the tension-compression asymmetry is not pronounced but, the
asymmetry becomes significant at the ultimate stress. It is important to note that
the tension-compression asymmetry is reported only in the range of equivalent
plastic strain where the deformation in the flat tensile specimen is homogeneous
(i.e. before necking). The tension-compression asymmetry ratio, t/c for several
values of equivalent plastic strain is reported in Table 3-6. The cast specimens
76
did not reach a plastic strain of 4% therefore the tension-compression asymmetry
ratio was not calculated or reported.
Table 3-6. Tension-compression asymmetry ratio of ES-1 by plastic strain. Manufacturing Process/Plastic Strain 0.2% 2% 4%
Forged 1.0169 0.9249 0.8924 Cast and HIPd 1.0936 0.9375 0.8859 Cast 1.0205 0.9015 N/A
The asymmetry ratio of ES-1 at 4% strain is similar to the strength ratio of
SAE 4340 0.90T
C
and other martensitic steels reported by Leslie and Sober
(1967).
3.2.3 Influence of Specimen Geometry on the Mechanical Response in Uniaxial Tension
In order to study the influence of the geometry of the specimen on the
mechanical response in uniaxial tension, tension tests were also carried out
using quarter scale standard flat pin-loaded specimens shown in Figure 3-10.
Figure 3-10. Schematic of the quasi-static (flat) pin-loaded tensile test specimen.
Figure 3-11 allows a comparison between the stress-strain response of
cylindrical specimens and flat pin-loaded specimens for each type of processing.
It is worth noting that, as predicted, the stress-strain response prior to strain
localization was similar for both specimen geometries: round and flat. Though,
the STF of the flat specimens was about half that of the round specimens. It is
hypothesized that the decrease in STF is at least partially a result of the subscale
50.8 mm
12.7 mm
3.175 mm
0.156 mm
3.175 mm
14.29 mm
12.7 mm 12.7 mm
77
specimens containing relatively larger flaws. The hardening law parameters are
presented in Table 3-7.
Table 3-7. Hardening law parameters flat tension specimens. Manufacturing Process k
(MPa) n
Av (MPa)
Bv
(MPa) Cv
Forged 2157 0.06 1774 260.2 113.6 Cast and HIPd 1983 0.0491 1664 205.8 125.9 Cast 2210 0.0707 1653 423.8 290.1
78
Figure 3-11. Quasi-static tension characterization stress-strain response A) forged, B) cast and HIP’d, and C) cast no HIP specimens.
79
Although the linear variable differential transformer (LVDT) can provide
strain measurements, DIC measurement was used to monitor the evolution of the
local strain. To gain understanding of the plastic deformation of the ES-1, two
cameras were disposed to monitor, independently, the evolution of the strain in
the gage length. The first camera shown in Figure 3-12 associated with 2-
dimensional DIC measurements captured the evolution of strain occurring in the
face of the specimen, i.e. the width strain (width) and axial strain (axial), while the
second camera monitored the evolution of strain occurring in the side of the
specimen, i.e. the thickness strain (thickness) and axial strain (axial). Both cameras
were synchronized using a delay generator to capture images at the same time
and hence the same axial displacement.
Figure 3-12. Camera layout to capture 2-D measurements of the strain evolution on the face and the side of the flat specimens during quasi-static tensile testing.
Lankford coefficients, also known as r-values, are a measure of the
thinning resistance of a sheet or plate during forming and are given as the ratio of
width to thickness strain rate width
thickness
r
.
Taking advantage of the experimental setup that allows the independent
monitoring of the evolution of the width strain and the thickness strain presented
80
in Figure 3-13, there is no need to adopt the usual hypothesis of plastic
incompressibility. Using a linear regression, the Lankford coefficient of one was
obtained using direct measurements of the width and thickness strain rates
meaning that the material is indeed isotropic.
With an understanding of the quasi-static behavior of ES-1, dynamic
SHPB compression tests were conducted to determine the rate-sensitivity as
discussed in the ensuing section.
81
Figure 3-13. Width and thickness strain of flat specimens under uniaxial tension A) forged, B) cast and HIP’d, and C) cast no HIP specimens.
82
3.3 Dynamic Experimental Characterization of Eglin Steel
The Split Hopkinson Pressure Bar (SHPB) or Kolsky bar is a test
apparatus for material characterization at high strain rates on the order of 103 s-1
since it was developed by Kolsky (1949). A diagram of a Kolsky bar included in
Figure 3-14 shows the striker bar, incident bar, and transmitted bar aligned in
series.
Figure 3-14. Schematic SHPB with an illustration of the propagating strain waves.
In the operation of a SHPB, the striker bar is put into motion to impact the
incident bar creating a compressive strain wave composed of a spectrum of
frequencies which backward propagates the length of the striker bar and forward
propagates the incident bar. Within the striker bar, the compressive wave is
reflected at the free end creating a tensile wave forward propagating back though
the length of the striker bar (Ls). Upon the arrival of the tensile wave at the
impact face of the striker bar, the striker bar is pulled away from the incident bar
concluding the compressive wave in the incident bar with a final wavelength of
2Ls/C0 where C0 is the longitudinal wave speed in the striker bar. When the
compressive wave impinges upon the specimen, the strain wave is both reflected
and transmitted at both interfaces creating a tensile reflected pulse that backward
propagates the incident bar and a diminished compressive wave that continues
83
forward propagation through the transmitted bar. Strain gages are adhered to
the surface of both the incident and transmitted bars to collect engineering strain
at a distance xi and xt, respectively, from the specimen throughout the dynamic
compression test. The placement of the strain gages is critical to eliminate
edged effects and avoid undesirable interaction of the reflected waves at the
strain gage locations. One dimensional wave theory is generally used in the
analysis of a SHPB to determine the average, uniaxial stress and strain within
the cylindrical test specimen throughout the test. Using the free body diagram of
a section of length dx of the incident or transmitted bar under stress shown in
Figure 3-15, the equation of motion becomes:
2
2
uA A Adx
x t
(3-3)
Where
ρ is the density of the incident and transmitted bars,
u is the longitudinal displacement of particles within the bars, and
A is the cross sectional area of the incident and transmitted bars.
Figure 3-15. A free body diagram of a portion of the SHPB of length dx and cross sectional area A.
Simplifying and assuming (1) both the incident and transmitted bars are
not plastically deformed (i.e. Hooke’s law is applicable) and (2) the bars are long
enough to assume uniaxial strain, the one-dimensional wave equation introduced
by Jean le Rond d’Alembert is:
84
2 2
2 2
u E u
t x
(3-4)
where E is the Young’s modulus of the bars.
By introducing the longitudinal wave speed, defined as 0
EC
the
general solution to the differential equation is:
0 0u f x C t g x C t (3-5)
for displacement forward (f) and backward (g) propagating waves. For the
forward propagating incident and transmitted waves and the backward
propagating reflected wave, the displacement at the interfaces of the specimen
(u1 and u2) can be calculated:
1 0
0
2 0
0
t
i r
t
t
u C dt
u C dt
(3-6)
Where
εi is the strain measured as the incident wave initially passes the strain gage on the incident bar,
εt is the strain measured in the transmitted bar, and
εr is the strain measured on the incident bar after the wave has reflected at the specimen interface.
Notice the data are time-shifted, such that the strain can be integrated
from time, t=0. Remember that the strain gages are positioned xi and xt away
from the specimen interface creating a voltage time graph like the one shown in
Figure 3-16. The physical interpretation is that the incident strain is measured
85
first when the incident wave reaches the first strain gage before it ever reaches
the first specimen interface. It is assumed that the incident strain wave does not
change before it reaches the first specimen interface. Therefore, it is forward
propagated to the first interface with the specimen simply by adding the time it
takes for the wave to propagate the distance from the incident strain gage to the
first specimen interface, incident time 0
ii
xt
C
. After the incident stain is
measured and the wave propagates to the first specimen interface, the amplitude
of the incident wave is then transmitted and reflected at both interfaces of the
specimen. The portion that is transmitted through the specimen, propagates
down the transmitted bar until it reaches the transmitted strain gage. Therefore,
the transmitted time 0
tt
xt
C
is subtracted from the time associated with the
transmitted wave. Finally, the portion of the wave that was reflected at the
interfaces with the specimen is then measured when it returns to the incident
strain gage. Therefore, incident time, is subtracted from the time associated with
the reflected wave measurement. Finally, the new time, which should coincide
with the time that each wave was interacting with the specimen interfaces is reset
at 𝑡 = 0. This model ignores any changes in the wave as it propagates between
the incident strain gage and the first specimen interface and between the second
interface and the transmitted strain gage and neglects the time the wave travels
within the relatively short test specimen and is only an accurate assumption if the
SHPB has been designed to minimize dispersion, distortion of the wave as it
propagates the SHPB system due to phase speeds of the frequency components
86
of the composite strain wave, to a negligible amount. Distortion is minimized
when the wave length is much larger than the SHPB diameter as explained by
Davies (1948).
87
Figure 3-16. Raw data from the dynamic SHPB characterization A) forged, B) cast and HIP’d, and C) cast specimens.
88
The average strain (εs) in the specimen of length (L) is calculated based
on the displacement of the two interfaces.
02 1
0
t
s i r t
Cu udt
L L
(3-7)
Assuming a planar impact at both specimen interfaces, the stress at each
interface can be calculated using Hooke’s Law and the total strain at each
interface 1 2;i r t . The specimen free body diagram is shown in Figure
3-17.
1
2
i r
s
t
s
E A
A
E A
A
(3-8)
Where As is the cross sectional area of the specimen.
The average stress within the specimen can be estimated based on the
stress at each of the specimen interfaces.
1 2
2 2
i r t
s
s
E A
A
(3-9)
Figure 3-17. The SHPB test specimen subjected to forces at the incident and transmitted bar interfaces.
The materials lab at the University of Florida (UF) Research and
Engineering Education Facility (REEF) contains a SHPB made of three 0.75-in
diameter Vascomax-300 steel bars aligned in series: striker bar, incident bar, and
89
transmitted bar, on a rigid optical table shown in Figure 3-19. Vascomax-300 has
a Young’s modulus of 190 GPa and a density of 8,000 kg/m3.
Figure 3-18. UF REEF test equipment for dynamic SHPB characterizations A) HDO 8058 500 MHz high-definition oscilloscope, B) amplifier, and
C) SHPB.
3.3.1 Dynamic Stress Strain Response of ES-1 under Uniaxial Compression
The incident, transmitted, and reflected strain measurements collected at
500 MHz during the SHPB tests of forged, cast and HIP’d, and cast specimens
with a length and diameter of 5.08 mm are included in Figure 3-19 A, B, and C,
respectively, to provide insight into the duration and dispersion of the strain
waves.
90
Figure 3-19. Strain waves at the incident and transmitted strain gages A) forged, B) cast and HIP’d, and (C) cast specimen.
91
The stress-strain response of ES-1 under dynamic, compressive loading is
shown in Figure 3-20. The forged material demonstrates slightly higher flow
stress than the cast and HIP’d and cast no HIP materials and thus lower plastic
strain as the same kinetic energy (i.e. mass and impact velocity of the striker bar)
was introduced into the system for each test.
Figure 3-20. Stress strain response of forged, cast and HIP’d, and cast no HIP ES-1 during SHPB tests.
The deformed length of the post-test specimens was used to validate the
one-dimensional analysis. A photograph of the forged ES-1 cylindrical test
specimen initially and with 16% plastic strain following a SHPB dynamic
compression test included in Figure 3-21 proves the one-dimensional analysis
was accurate in the SHPB at the UF REEF. Additionally, to correct the wave
form for dispersion (Gong et al., 1990), a Fourier transform was used to
92
transform the data into the frequency domain, a variation of phase velocity using
the Pochhammer-Chree (Pochhammer, 1876 and Chree, 1889) solution was
introduced, and finally a reverse Fourier transform back to the time domain was
completed to confirm the UF REEF SHPB system permits very little dispersion.
Figure 3-21. Forged Eglin steel cylindrical SHPB test specimens A) before and B) after dynamic compression test.
Figure 3-22, the stress-strain response of the forged, cast and HIP’d, and
cast no HIP materials, permits comparison of the quasi-static and dynamic
behavior of Eglin steel.
93
Figure 3-22. Comparison of quasi-static and dynamic stress-strain response A) forged, B) cast and HIP’d, and C) cast no HIP ES-1.
94
Because of the difficulty in determining flow stress for small percentages
of plastic strain (Sharpe and Hoge, 1972, and Yadav et al., 1995) and to
eliminate the inertial effects in the dynamic stress-strain curves, the quasi-static
and dynamic flow stresses were compared for strains greater than 4% to
determine the forged dynamic flow stress is on average 1.10 times the quasi-
static stress for the same values of strain; the cast and HIP’d is 1.14; and the
cast no HIP is 1.12. These dynamic increase factors are nearly equal to the
dynamic increase factor of 1.18 reported by Johnson and Cook (1983) following
testing of SAE 4340. So, the dynamic increase factor can be used to solve the
dynamic portion of the isotropic Johnson Cook presented previously in Section
1.2.
Where the material constants, presented in Table 3-8, are similar to those
presented by Dilmore (2009) after characterizing the behavior of ES-1 at various
strain rates using a servohydraulic system with custom built load cells.
Table 3-8. Johnson and Cook material constants for forged, cast, and cast and HIP’d ES-1.
Manufacturing Process A (MPa) B (MPa) n C
Forged 1430 1142 0.2525 0.0124 Cast and HIP’d 1414 1184 0.2344 0.0065 Cast 1445 1149 0.2406 0.0069
Referring back to the literature, the initial yield stress of AF-1410, AISI
4340 and ES-1 observed by Last (1996) is presented in Figure 3-23 along with
the cast ES-1 experimental results. The initial yield stress of ES-1 reported by
Last corresponds well with the cast ES-1.
95
Figure 3-23. Initial yield stress-strain-rate dependence of AF-1410, AISI 4340, and ES-1 observed by Last in comparison to the cast material.
As shown in Figure 3-24, during the SHPB test, the strain rate increased
rapidly to a maximum of over 1,500 s-1 and declined to about 500 s-1 with an
average strain rate of approximately 700 s-1. To minimize the inertial effects,
forged ES-1 specimens were tested in a SHPB with a pulse shaper adhered to
the leading end of the incident bar as discussed in the following Section, 3.3.2.
96
Figure 3-24. Strain rate in forged, cast and HIP’d, and cast no HIP ES-1 during SHPB tests.
3.3.2 Dynamic Characterization of ES-1 at Constant Strain Rates
Without a pulse shaper, the strain rate in the test specimens increased
rapidly when the incident wave impinged the specimen and decayed
exponentially throughout the duration of the incident wave. Therefore, pulse
shapers, pieces of material placed on the leading end of the incident bar using
lubricant, were used to absorb the initial impact energy and induce an
approximately constant strain rate in the test specimen. The design of the pulse
shaper is critical to obtaining nearly constant strain rates and dynamic
equilibrium. Frew et al. (2005) present analytical models and experimental data
that demonstrate a broad range of incident pulses and can be obtained by
varying the design of the pulse shaper. For the Eglin steel specimens, two
different pulse shapers were designed to test at strain rates of approximately 500
s-1 and 2,000 s-1. The 500 s-1 pulse shaper was made from an 1/8-in thick,
97
annular piece of 4130 steel with four 0.012-in thick circular pieces of annealed
copper equally spaced around the center as shown in Figure 3-25A. The 2,000
s-1 pulse shaper was made from a 1/8-in thick, annular ring of 4130 steel, a
0.012-in thick annular ring of annealed copper and a circle of 0.01-in thick Teflon
as shown in Figure 3-25B.
Figure 3-25. Dynamic pulse shapers made of copper, steel, and Teflon A) 500 /s and (B) 2000 /s.
In order to obtain similar plastic deformation despite the energy absorbed
in the pulse shaper, the SHPB test was designed with an increased pressure of
55 psi and a 2-ft striker bar. Three SHPB tests were conducted on the forged
specimens using the pulse shapers: two at 500 s-1 and one at 2000 s-1. Notice
98
the more constant strain-rate behavior of the 500 s-1 pulse shaper presented in
Figure 3-26.
Figure 3-26. Strain rate of forged test specimens without a pulse shaper, with the first pulse shaper, and with the second pulse shaper.
The stress-strain response of forged ES-1 under constant strain rates of
500 s-1 and 2000 s-1 obtained using the pulse shapers are presented alongside
the forged ES-1 tested in the SHPB without a pulse shaper in Figure 3-27. The
flow stress of the forged material tested at the highest strain rate 2000 s-1 is
proportionately greater than at 500 s-1 and 700 s-1.
99
Figure 3-27. Stress strain curves of forged specimens without a pulse shaper, with the first pulse shaper design and the second pulse shaper design.
3.4 Cyclic Experimental Characterization of ES-1
Additional 25.4-mm long, round specimens were tested under cyclic
loading with an MTS 25-ton capacity test frame and load cell in the materials lab
at the UF REEF. The load frame, high-speed camera, and associated equipment
are shown in Figure 3-28.
100
Figure 3-28. UF REEF materials lab for cyclic characterization of ES-1 A) test frame and load cell, B) high-speed camera, and C) installed
specimen.
Prior to testing, the specimens were cleaned with acetone and marked
with a vertical line for alignment of the high-speed camera and targets evenly
spaced at 1 mm to track local displacement with the high-speed cameras. The
specimens were subjected to uniaxial tension followed by uniaxial compression
under completely reversed crosshead displacement control at a frequency of 0.1
Hz. The multi-test approach described in STP-465 was initialized with 0.2%
strain amplitude for 50 cycles to reach stability. Next, the strain amplitude was
101
then increased by 0.2% for each subsequent iteration. Images of the gage
length of the test specimen illuminated by a stationary LED array were recorded
using an Allied high-speed camera and 25 mm lens. The nested hysteresis
loops at various values of maximum plastic strain of the forged, cast, and cast
and HIP’d ES-1 materials are included in Figure 3-29.
Figure 3-29. Hysteresis loops of ES-1 under completely reversed displacement A) forged, B) cast and HIP’d, and C) cast.
The absolute uniaxial flow stresses in tension and compression at the
extremes of the elastic range under tensile plastic strain and compressive plastic
strain of forged ES-1 calculated using the maximum values or 0.1% offset as
required are presented in Figure 3-30. A large variability in flow stress was
observed depending on the method used to determine the flow stress (i.e.
proportionality limit or percent offsets between 0.01 and 0.2%). Nevertheless,
note the Bauschinger effect, softening in compression under tensile strain and
vice versa.
102
Figure 3-30. Uniaxial absolute flow stress of ES-1 at the limits of the elastic region under cyclic loading by plastic strain.
The loci of the maximum values of stress and strain from all the stable
hysteresis loops together form the kinematic stress-strain response of forged ES-
1. The kinematic stress-strain response is shown in Figure 3-31 alongside the
quasi-static response for comparison. Referring back to the literature, many
materials including martensitic steels undergo kinematic softening under cyclic
loading as demonstrated by a comparison of the quasi-static and kinematic
stress-strain curves presented by Landgraf (1970) included in Chapter 1 of this
dissertation. However, the kinematic softening of forged, cast and cast and
HIP’d ES-1 is small for the range of true strain, contrasting the extreme softening
of SAE 4340 at initial yielding.
103
Figure 3-31. Monotonic and kinematic stress-strain curves of Eglin steel A) forged, B) cast and HIP’d, and C) cast no HIP.
104
3.5 Summary of the Experimental Characterization of ES-1
The experimental characterization of Eglin steel revealed the forged
material flow stress was higher than the cast and cast and HIPd materials by a
factor of 1.1 irrespective of the sign of the loading and strain rate. The flow
stress of each material under quasi-static conditions was accurately described
using Swift hardening law. Additionally, the material regardless of manufacturing
route exhibited isotropy with comparable width and thickness strain rates within
the flat tension test specimens (r=1). The SHPB tests divulged a slight strain-
rate dependence well characterized by the Johnson Cook (1983) constitutive
model. Two characteristics were discovered during the cyclic testing: (1) Eglin
steel softens under repeated load reversals, and (2) the material exhibits a
pronounced Bauschinger effect. These material characteristics and associated
material parameters were used to develop the theoretical elastic-plastic model
presented in Chapter 4 of this dissertation.
105
CHAPTER 4 ELASTIC-PLASTIC MODEL FOR EGLIN STEEL
The main features revealed by the experimental data are used to develop
an appropriate theoretical model valid for general 3D loadings. As UHSS and
ES-1 steels have a well-defined yield point as seen in the stress-strain response
under uniaxial loading (see Chapter 3). To describe both ineslastic behavior and
the rate effects revealed by the SHPB tests and data, viscoplasticity theory was
used to model ES-1. In such theory, an appropriate yield criterion and hardening
law need to be defined.
, , 0p p pF Y σ σ (4-1)
As already mentioned, in the six dimensional stress space, the yield
criterion defines a surface such that for any state interior to it, the material
behavior is elastic. After initial yielding, the self-similar expansion of the
subsequent yield surfaces is usually considered to be governed by the equivalent
plastic strain. In this dissertation, isotropic hardening includes not just expansion
of the yield surface, but also distortion of the yield surface that preserves the
isotropic nature of the deformation. Rate effects can be introduced by either
considering an overstress approach or by considering that hardening is rate-
dependent. This latter approach is used.
4.1 Development of the Yield Criterion for ES-1
Several yield criteria and associated effective stresses have been
proposed in the literature (see Section 1.4), of which, most used are von Mises
(1913) and Drucker (1949), respectively.
106
2
13 2 62 3
3J
J cJ
(4-2)
where 𝐽2 and 𝐽3 are the second and third invariants of the deviatoric stress, s.
The deviatoric stress is the stress tensor less the hydrostatic stress tensor. The
second and third invariants of the deviatoric stress are the coefficients of the first
and zeroth order terms of the characteristic equation of the deviatoric stress
eigenvalue problem expressible in terms of the principal stress (1 2 3, , ),
principal deviatoric stress (1 2 3, ,s s s ), the Cauchy stress components
11 22 33 12 23 13, , , , , , or the deviatoric stress tensor components
11 22 33 12 23 13, , , , ,s s s s s s . The invariants are so named because their expression is
the same irrespective of the coordinate system. Let us recall that, in terms of
principal stresses 1 2 3, , :
2 2 2
2 1 2 2 3 3 1
1
6J
. (4-3)
While for general loadings, the expression is:
2 2 2 2 2 2
2 11 22 33 11 22 22 33 11 33 12 23 13
1 1
3 3J (4-4)
Likewise, the third invariant of the deviatoric stress is:
3 3 3 2 2 2
3 1 2 3 1 2 3 2 1 3 3 1 2 1 2 3
2 1 4
27 9 9J (4-5)
3 3 3 2 2 2
3 11 22 33 11 22 33 22 11 33 33 11 22 11 22 33
2 22
23 131211 22 33 11 22 33 11 22 33 12 23 13
2 1 4
27 9 9
2 2 2 23 3 3
J
(4-6)
107
Because von Mises (1913) and Drucker (1949) are functions of the
deviatoric stress invariants, martensitic steel material properties of pressure
insensitivity and isotropy are easily captured with these yield criteria. However,
these yield criteria are even functions of stress and therefore cannot capture the
experimentally observed tension-compression asymmetry. In the experimental
characterization, ES-1 exhibits higher yield stress in compression than tension
for a wide range of plastic strain. Therefore, a yield criterion that accounts for the
tension-compression asymmetry such as Cazacu and Barlat (2004) yield criterion
was used to describe the elastic-plastic behavior of the ES-1 steel alloys.
Similarly to Drucker (1949), this yield criterion is expressed as a function of the
invariants of the stress deviator, but contrary to the Drucker (1949) (see Eq.
(4-2)) which is an even function, the Cazacu and Barlat (2004) model is an odd
function of stresses. The effective stress associated with this yield function is:
13 322 3 YJ cJ
(4-7)
where c is an asymmetry parameter and Y is the shear yield stress. In Cazacu
and Barlat (2004) it was shown that c can be expressed using the flow stresses
in tension (T) and compression (C) as:
3 3
3 3
3 3
2
T C
T C
c
(4-8)
The main feature of the Cazacu and Barlat (2004) criterion is that it is an
odd function, which allows describing the tension-compression asymmetry using
the third invariant of the deviatoric stress. In the principal stress space, the
108
Cazacu and Barlat (2004) yield surface is an infinite cylinder with generator, the
hydrostatic axis. In the deviatoric plane, normal to the hydrostatic axis, the cross
section of the yield surface is a rounded “triangle” for a nonzero asymmetry
parameter 0c or a circle for 0c .
In this research a different method for determining this parameter and the
evolution of yield surfaces is proposed. Before discussing these aspects, the flow
rule and application of the criterion to description of the torsional response is
presented.
4.1.1 Associated Flow Rule
Drucker (1951) used the non-negative increment of plastic work to prove
normality (see Figure 4-1A).
0Pdσ :dε (4-9)
At the limiting case when the increment of stress is tangent to the yield
surface, the increment of plastic strain must be normal to the yield surface (i.e.
the increment of plastic work is zero).
The material being isotropic, it is sufficient to describe the plastic flow in
the coordinate system associated with the principal values of stress. The flow
rule for Cazacu and Barlat (2004) is:
32
2 3
P
i
i i
JJF F
J J
(4-10)
1
3222
32
232 3 322 3
3
2
3
i i
i i
JJJ c
JJF F
J J
J cJ
(4-11)
109
Let us recall that in 1951 Drucker proved that the work from a cycle of
loading and unloading between an arbitrary elastic stress and stress resulting in
an increment of plastic strain is positive only if the entire yield surface is on one
side of the hypertangent plane at the point of plastic stress.
0 Pσ σ* dε . (4-12)
Figure 4-1. Illustration of properties of yield surfaces with an associated flow rule A) normality and B) convexity.
The condition of positive work is satisfied at every point for a convex
instantaneous yield surface (see Figure 4-1B). For the entire yield surface to
remain convex the Hessian must be positive semi-definite for any fixed value of
hardening. The Hessian is positive semi-definite if the three eigenvalues at all
points of the yield surface are all non-negative.
222 2
32
2 2
2 3
0i j i j
JJF FH
J J
(4-13)
The local maximums of the convex yield surface in the deviatoric plane
occur along both the tensile and compressive meridians (projections of the
positive and negative principal stress half-axes, respectively, in the deviatoric
110
plane) where the determinant of the Hessian, and therefore an eigenvalue, are
both zero. Consequently, the Hessian is positive semi-definite if both the sum
and the product of the remaining, conjugate eigenvalues are positive (i.e. the first
and second invariants of the Hessian are both positive).
For the Cazacu and Barlat (2004) yield criterion, it is important that
convexity of the yield function be reinforced to find the acceptable range of the
asymmetry parameter.
1 1
32 2 22 22 2
23 322 3
1 1
3 32 22 22 2
53 322 3
3 3
4 2
3
3 32
2 2
9
i j i j i j
i i j j
JJ J JJ J c
H
J cJ
J JJ JJ c J c
J cJ
. (4-14)
It can be easily seen that:
2ij
ij
Js
(4-15)
2
2 1
3ik kl ij kl
ij kl
J
, (4-16)
and that:
32
2
3ik kj ij
ij
Js s J
(4-17)
2
3 2
3ik jl jl ik ij kl kl ij
ij kl
Js s s s
. (4-18)
111
In Cazacu and Barlat (2004), it was shown that the convexity requirement
results in an acceptable range of the asymmetry parameter 3 3 3 3
,4 4
c
.
4.1.2 Unusual Behavior in Torsion Revealed by Cazacu and Barlat (2004) Criterion
The Swift effect, put into evidence by Swift (1947), is the axial deformation
of a thin-walled cylinder during monotonic free-end torsion. This phenomenon
was previously attributed to anisotropy and/or its evolution. In 2013, Cazacu et
al. (2013) gave a different explanation. It was put into evidence that there is a
correlation between a material asymmetry and Swift effects. A theoretical
analysis was done using the CBP-06 yield criterion which accounts for tension-
compression asymmetry using the parameters, k and a.
1 1 2 2 3 3
a a a
s ks s ks s ks F (4-19)
Revil et al. (2014) conducted a finite element analysis using an elastic-
plastic model based on the anisotropic form of CBP-06 and described the
behavior of thin-walled cylinders of OFHC copper, high-purity α-titanium, and
AZ31-Mg alloy under torsion. It was confirmed that axial elongation occurs in
materials with a tension to compression flow stress ratio greater than one, and
contraction of materials with a ratio less than one.
In this work, analysis of the same problem is done using the Cazacu and
Barlat (2004) yield criterion. Let the applied loading be given by:
0 0
0 0
0 0 0
σ . (4-20)
112
Hence its deviator is:
0 0
0 0
0 0 0
s . (4-21)
Under this state of stress, the second invariant of the deviatoric stress is
equal to the shear stress and the third invariant is zero.
2 2 2 2
2 1 2 3
3 1 2 3
1
2
0
J s s s
J s s s
(4-22)
Substituting these values in the associated flow rule, we obtain:
1
22
2 11 3 3 33 2
2 3
30 0
2 33
32 0 02 3
3 3 223 3 2
0 03
c
J cd c
d d
ccc J cJ
Pdε
32 ∂J∂J∂F ∂σ ∂σ
∂σ (4-23)
Note that according to Eq. (4-23), if the asymmetry parameter, c, is zero,
then pure shear deformation occurs (i.e. no axial deformation). If the asymmetry
parameter is negative the cylinder will elongate, where if c is greater than zero, it
will contract. A numerical solution using finite element analysis will also be
presented in the next chapter.
4.2 Asymmetric Hardening
As already mentioned, based on the classical definition, isotropic
hardening is modeled, typically, by a function of equivalent plastic strain
parameterized using the quasi-static uniaxial tensile characterization. The
subsequent yield surface, symmetric about the origin, therefore enlarges to pass
through the subsequent, uniaxial tensile flow stress. Therefore, a similar
113
increment of hardening is added to each point on the yield surface. The initial
yield surface and two subsequent yield surfaces using classical isotropic
hardening for forged ES-1 are presented in Figure 4-2.
Figure 4-2. Cazacu and Barlat initial and two subsequent yield surfaces using classical isotropic hardening of forged ES-1 in the deviatoric plane.
In this research a new description of isotropic hardening is proposed.
Here, it is hypothesized that isotropic hardening is a result of both a change in
size and shape of subsequent yield surfaces that does not destroy the isotropy.
In this research, the deformation of the subsequent yield surface is described
using the asymmetry parameter as a function of the second and third invariants
of plastic strain; and the enlargement is described using the hardening under
pure shear strain. With the evolution of asymmetry of the yield surface
dependent on the second and third invariants of plastic strain, the subsequent
114
yield surfaces vary based on the strain path. For example, the subsequent yield
surface following uniaxial tension may be different from the subsequent yield
surface following uniaxial compression.
For quasi-static monotonic loading, hardening is generally considered to
be a function of plastic strain though some models do include other variables like
fading strain history and other internal variables. However, for a plastically
incompressible, isotropic material, the hardening can be described using both the
second and third invariants of the total plastic strain, just as the yield surface in
Haigh-Westergaard space is defined by the second and third invariants of the
deviatoric stress (i.e. the hardening is a function of the second order plastic strain
tensor that can be completely described using both non-zero invariants). The
strain invariants, similar to the deviatoric stress invariants, are the coefficients of
the first and zeroth order terms of the eigenvalue problem in strain expressible as
a function of the components of the strain tensor 11 22 33 12 23 13, , , , , or the
principal strains 1 2 3, , . As a result of plastic incompressibility, the plastic
strain is deviatoric, that is the first invariant of plastic strain is identically zero (i.e.
1 1 2 3 0J ).
2 2 2 2 2 2 2 2 2
2 1 2 3 11 22 33 12 23 13
1 1
2 2J (4-24)
3 1 2 3
3 3 3 2 2 2
11 22 33 12 11 22 23 22 33 13 11 3 12 23 13
12
3
J
(4-25)
In the above equations, both expressions in terms of strain and principal
values are given. Upon further examination, the second invariant of plastic strain
115
is non-negative while the third invariant of plastic strain has the negative sign
function of the intermediate plastic strain, appropriate for the sign-sensitive
Bauschinger effect. Additionally, the second and third invariants of plastic strain
are suitable for describing asymmetric isotropic hardening because they are
constant in the elastic regime and continuous in the elastic-plastic.
4.2.1 New Description of Hardening
With a broader definition of isotropic hardening to include distortion of the
yield surface that does not annihilate the material isotropy, it is necessary to
examine the evolution of the yield surface under different load and strain paths.
Note that the evolution of tension-compression asymmetry can be directly
observed under reversed loading or inferred from the increment of plastic strain
in torsion testing of thin-walled cylinders using the associated flow rule. In
particular, the evolution of the asymmetry of ES-1 was evaluated using the cyclic
test data presented in Section 3.4.
For such cyclic loading, first ES-1 material was subject to tensile strain
and presumably uniaxial tensile stress, i.e.:
0 0
0 0 0
0 0 0
T
σ (4-26)
20 0
3
10 0
3
10 0
3
T
s . (4-27)
Under this state of stress, the second and third invariants are:
116
22 2 2
2 1 2 3
3
3 1 2 3
1
2 3
2
27
T
T
J s s s
J s s s
. (4-28)
Whereas the strain is:
1 0 0
10 0
2
10 0
2
T
ε (4-29)
22 2 2
2 1 2 3
3
3 1 2 3
31
2 4
4
T
T
J
J
. (4-30)
Initial yielding was observed as the specimen started to deform plastically.
At maximum displacement (0.57 mm), the flow stress was recorded and the
motion was reversed relieving the uniaxial tensile stress and then creating a state
of uniaxial compressive stress in the elongated specimen (assuming no bending
or buckling within the specimen gage length). In this stage of the test,
0 0
0 0 0
0 0 0
C
σ (4-31)
and its deviator is:
20 0
3
10 0
3
10 0
3
C
s (4-32)
117
while the second and third invariants are:
22 2 2
2 1 2 3
3
3 1 2 3
1
2 3
2
27
C
C
J s s s
J s s s
. (4-33)
Still under compressive stress, the specimen was returned to its initial
configuration 2 3 0J J and subject to compressive strain. Again, yielding
was observed as the specimen began to plastically deform, this time in
compression. The process was repeated 50 times in accordance with standard
test procedures.
1 0 0
10 0
2
10 0
2
C
ε (4-34)
22 2 2
2 1 2 3
3
3 1 2 3
31
2 4
4
C
C
J
J
(4-35)
To minimize the kinematic softening of the cyclic response, the first cycle
at each value of plastic strain of the cyclic test data of forged ES-1 in Figure 4-3
was used to find the limits of the elastic region after two load reversals between
uniaxial tension and compression.
118
Figure 4-3. The first stress-strain cycle of forged ES-1 during cyclic testing at completely reversed displacement of 0.57, 0.62 and 0.67 mm and the uniaxial tensile and compressive flow stress.
The offset, between zero (i.e. the proportionality limit) and 0.2% (the
ASTM standard), used to determine yielding following the load reversal greatly
affected the calculation of the flow stress and thus the evolution of asymmetry.
Still, the maximum absolute stress and 0.1% offset were used to determine the
limits of the elastic region in Figure 4-4.
σT(ε
T
P)
σT(ε
C
P)
σC(ε
C
P)
σC(ε
T
P)
119
Figure 4-4. Flow stress at the limits of the elastic region by plastic strain in forged ES-1 under cyclic loading.
As shown in Figure 4-4, the limits of the elastic region are increasingly
asymmetric with accumulation of plastic strain, a phenomenon known as the
Bauschinger effect. In kinematic hardening models, the asymmetry is commonly
modeled as a translation of the yield surface due to a fictitious back stress. The
back stress according to the most common kinematic hardening models
presented in Chapter 1 are functions of the increment of plastic strain tensor.
Therefore, under uniaxial tension, the back stress is also uniaxial. The uniaxial
back stress at various levels of plastic strain is shown in Figure 4-5.
1 1
3 3
1 1
3 3
3 3 2 3 3 2
3 3 2 3 3 2
T Cc c
c c
(4-36)
120
Figure 4-5. The uniaxial back stress as a function of plastic strain for forged ES-1 under cyclic loading.
An example of the translation of the yield surface using a kinematic
hardening law is presented alongside the classical isotropic hardening surface
and the asymmetric hardening surface for comparison in Figure 4-6. This shows
that there are two major disadvantages to these kinematic hardening laws. First,
the back stress is non-zero even after unloading (i.e. after all stresses have been
removed from the material). Secondly, a large disparity between the monotonic
and kinematic yield surfaces especially pronounced along the third compressive
meridian is evident upon initial loading (Figure 4-6). Additionally, there are also
limitations to the kinematic hardening models. The translation destroys the
isotropy, which is inappropriate for materials that remain isotropic even after such
loadings.
121
Figure 4-6. Subsequent yield surfaces using classical isotropic, redefined isotropic, and kinematic hardening under uniaxial tensile loading.
Another possible explanation for increasingly asymmetric flow stresses at
the limits of the elastic region under cyclic loading is that the evolution of
asymmetry depends on the strain tensor or some component thereof not just the
equivalent plastic strain. This possibility is attractive because a separate
kinematic hardening may no longer be required to model load reversals, the yield
surface remains symmetric about zero stress for isotropic materials, and the new
hardening law allows description of tension-compression asymmetry. To further
explore this idea, the asymmetry parameter was calculated assuming that the
yield surface did not translate. The evolution of the asymmetry parameter, c,
122
which was calculated at each level of plastic strain using Eq. (4-8), is included in
Figure 4-7.
Figure 4-7. Evolution of the material asymmetry parameter, c, for forged ES-1 using the limits of the elastic region under cyclic loading.
An innovative evolution law was proposed to describe the data. Namely,
the evolution of c was modeled assuming that it depends on the third invariant of
plastic strain. It was approximated:
30
1
tanhJ
c cc
. (4-37)
where the parameters of ES-1 are 0 1.7c and 1 0.002c . Note the asymmetry
parameter approaches c0 as the third invariant approaches infinity, and negative
c0 as it approaches negative infinity. The uniaxial flow stresses and superposed
corresponding yield surface evolution in the deviatoric plane for several values of
plastic strain are included in Figure 4-8. Note that under tensile (compressive)
plastic strain the yield surface hardens near the tensile (compressive) meridians
and softens near the compressive (tensile) meridians. Also notice that only
123
dependence of c on the third invariant of plastic strain was considered, but the
effects of the second invariant of plastic strain could also be of interest.
Additionally, it is worth mentioning that the elastic-plastic model with asymmetric
isotropic hardening results in stabilizing hysteresis loops under cyclic loadings.
This is in contrast with the classical isotropic hardening which cannot reproduce
such effects.
Figure 4-8. Uniaxial flow stress at the limits of the elastic region and yield surfaces in the deviatoric plane for several values of plastic strain in forged ES-1 under cyclic loading.
It is worth noting the difficulties associated with correct estimate of the
evolution of yield surfaces based on cyclic tests. For example, Paul (1968)
discusses three acceptable methods of determining the yield stress: Lode
extrapolation method, plastic strain offset, and the proportionality limit technique.
The method used for identifying the yield stress may create large differences in
the values of the asymmetry parameter, c, and, therefore, on the evolution of the
yield surface. Hence, it is proposed to obtain the evolution of c using pure shear
124
deformation data as described in the next section. However, the approach used
so far may be more realistic in modeling cyclic data through kinematic hardening.
4.2.2 Distortional Hardening with Asymmetry
The idea for the different method for determining the asymmetry
parameter sprang from the necessity to model the Bauschinger effect frequently
associated with kinematic hardening. However, the concept can be used to
better model monotonic loading as well. Since the asymmetry parameter is
dependent on the second and third invariants of plastic strain, the subsequent
yield surfaces will differ under monotonic uniaxial tension, monotonic uniaxial
compression, and pure shear strain as shown in Figure 4-9. These three
different experimental characterizations can be used to parameterize the yield
function with asymmetric hardening.
125
Figure 4-9. Cazacu and Barlat subsequent yield surface in the deviatoric plane following uniaxial tension, compression, or pure shear strain.
Let us recall that under monotonic pure shear strain, when the third
invariant of plastic strain is equal to zero, ultra high-strength steels strain harden
according to a power-type law as a function of shear strain (which, in this case, is
equal to the second invariant of plastic strain). Thus in general, the evolution of
the uniaxial yield stress and that of the uniaxial compressive yield stress can be
described:
2 3 0 2 2 3 3
2 3 0 2 2 3 3
,
,
T T T T
C C C C
J J J J
J J J J
, (4-38)
126
Where 2 3,T J J and 2 3,C J J are the tensile and compressive flow stresses
following uniaxial tensile and compressive stress, 0T and
0C are the initial
tensile and compressive yield stresses, 2 2T J and 2 2C J are the
components of tensile and compressive hardening resulting from the second the
invariant of plastic strain, and 3 3T J and 3 3C J are the components of
tensile and compressive hardening resulting from the third invariant of plastic
strain, respectively.
Similarly, for the remainder of the yield surface, the total hardening is a
combination of hardening as a function of the second and third invariants of
plastic strain. The decomposed elastic regions resulting from monotonic uniaxial
tensile loading are shown in Figure 4-10 alongside the subsequent yield surface.
The initial and subsequent yield surfaces are depicted together to illustrate the
material softening for regions of the yield surface near the compressive
meridians. The components are discussed in the following sections.
Figure 4-10. Decomposition of the yield surface with distortional hardening A) initial yield surface, B) second invariant component of the
subsequent yield surface, C) third invariant component of the subsequent yield surface, and D) the total subsequent yield surface.
3T
sT 0
3 2'J J
sT 2
sT 3
sT
Initial Yield Surface J2
ε Component of the
Yield Surface
J3
ε Component of the
Yield Surface
Subsequent Yield Surface
2 2'J J 3 2'J J
sz(MPa) s
z(MPa) s
z(MPa) s
z(MPa)
sy(MPa) s
y(MPa) s
y(MPa) s
y(MPa)
sC 2
2 2'J J
127
4.2.2.1 2J -component of the hardening law
As seen in Figure 4-10, the second invariant of total plastic strain is
associated primarily with a growth of the subsequent yield surface in Haigh-
Westergaard space and can be fully understood by a single fixed-end torsion test
or another test under pure shear strain for example using a butterfly specimen.
Based on the symmetry of the second invariant of total plastic strain, the 2J -
component of the subsequent yield surface is the same under uniaxial tensile
and compressive loading as shown in Figure 4-11. The uniaxial tensile and
compressive hardening, however, need not be equal (i.e. the model allows for
evolution of asymmetry under pure shear strain). For example, UHSS with
material tendency toward greater strength in compression than tension based on
the BCT microstructure, also results in a slight change in shape of the
subsequent yield surface (i.e. the increment of hardening associated with the
second invariant of plastic strain is somewhat greater along the compressive
meridians than the tensile meridians). This evolution of asymmetry with
accumulated plastic strain is a material property, c2, that can also be determined
from a single pure shear strain test using the associated flow rule. The
parameterization of the material asymmetry parameter, c2, and then the
hardening, Y, are discussed in the next subsection.
128
Figure 4-11. Cazacu and Barlat yield surface under both uniaxial tension and compression and the common component.
4.2.2.1.1 Material asymmetry parameter
It is assumed that pure shear deformation 3 0J can be produced by a
combination of shear and axial loading. There are six points on the yield surface
such that the strain increment is parallel to a shear axes and perpendicular to a
meridian as shown in Figure 4-12.
129
Figure 4-12. Initial and subsequent Cazacu and Barlat yield surface under pure shear strain.
Using the point on the yield surface in the 3-plane where 3
0
, the
strain, invariants of strain, state of stress, deviatoric stress, and the deviatoric
stress invariants are presented below.
0 0 0 0
0 0 0 0
0 0 0 0 0 0
ε (4-39)
2 1 2 3
3 1 2 3
1
2
0
J
J
(4-40)
0
0 0
0 0 0
σ (4-41)
J2
ε Component of the
Yield Surface
J '2
et
P( ) 3 2'J J
sT 2
sz(MPa)
sy(MPa)
sC 2
f1 f
2
f3
τ12
τ31
τ23
130
22
22
0 06 4
0 06 4
0 03
s (4-42)
22 2 2 2
2 1 2 3
3 2
3 1 2 3
1'
2 3
2'
27 3
J s s s
J s s s
(4-43)
The (‘) indicates the second and third invariants of deviatoric stress under
pure shear deformation.
Under pure shear deformation in the 3-plane under the described loading,
the evolution of asymmetry as a function of shear strain is calculated based on
the associated flow rule for the instantaneous yield surface where the through
plane strain is zero.
131
1 2 2 2 22 22
2
231 322 3
1 2 2 2 22 22
2
232 322 3
1 2 2
22
233 322 3
3'
2 6 4 18 3 4 3
3 ' '
3'
2 6 4 18 3 4 3
3 ' '
2'
2 9 3
3 ' '
J c
J cJ
J c
J cJ
J c
J cJ
0
(4-44)
Solving the third partial derivative for the material asymmetry parameter,
c2:
22
2 2 2
93
2 6c
. (4-45)
Using the Cazacu and Barlat (2004) yield criterion, uniaxial tensile and
compressive flow stresses following pure shear deformation as a function of the
second invariant of plastic strain are on the same subsequent yield surface with
the shear flow stress. Thus, the hardening of the uniaxial tensile and
compressive flow stresses following pure shear strain are:
132
13 3
1 2 3 223 23
222 2 3
2 2 0 01 1
3 32 2
13 3
1 2 3 223 23
222 2 3
2 2 0 01 1
3 32 2
23
3 ' ' 3 27 3
3 3 2 3 3 2
23
3 ' ' 3 27 3
3 3 2 3 3 2
T T T
C C C
cJ c J
J
c c
cJ c J
J
c c
(4-46)
4.2.2.1.2 Hardening under pure shear strain
The hardening law that relates the yield surface in the six dimensional
stress and strain spaces describes the six points of pure shear deformation (i.e.
3 0J ) common to all subsequent yield surfaces of the material.
1 13 33 32 22 3 2 3' ' 0J cJ J cJ
(4-47)
Note that the asymmetry parameter is not the material asymmetry
parameter, c2, but the asymmetry parameter as a function of the second and
third invariants of plastic strain defined in the following section. Equation (4-47)
means that all possible subsequent yield surfaces for any given value of
equivalent plastic strain, intersect at the six points where 3 0J (Figure 4-9).
Distortional hardening under general three-dimensional loads will result in the
same hardening behavior of the six points of pure shear deformation, but the
effective stress and plastic strain increments will vary. It is more conventional to
express the hardening as the shear stress in pure shear strain as a function of
the equivalent plastic strain.
133
n
P
Y k
(4-48)
In order to express the hardening as a function of equivalent plastic strain,
the axial stress is expressed as a function of the material asymmetry parameter
and the shear stress using Eq. (4-45). The fourth order polynomial is:
2 4 2 2 2 2 4
2 2 24 27 48 81 144 0c c c (4-49)
By solving the quadratic equation, the following equation is obtained:
2 2
2 22 2
2
2
3 27 16 9 3 32 27
2 4 27
c c
c
(4-50)
It is convenient to use the axial-shear stress ratio, r, defined:
1
22 2
2 2
2
2
3 27 16 9 3 32 27
2 4 27
c cr
c
(4-51)
The effective stress under pure shear strain can then be expressed as a
function of the axial-shear stress ratio.
11 3 3
3 3 22 32
2 3
1 2 1' ' 1
3 27 3J cJ r c r r
(4-52)
Finally, substituting the effective stress in pure shear strain into Eq. (4-47)
and expressing the hardening as a shear strain power law, the yield function is:
13 322 3
13 32
2 3
0
1 2 11
3 27 3
nP
J cJ
k
r c r r
(4-53)
134
The equivalent plastic strain is usually defined by using the work
equivalency principle based on the assumption that an equal amount of work is
required to harden the material to the same subsequent yield surface regardless
of the stress and plastic strain tensors. Since this proposed distortional model
results in a different subsequent yield surface for each state of plastic strain,
consistency lies in the six points of shear hardening. Therefore, the equivalent
plastic strain for monotonic loading is based only on the second invariant of strain
similar to the von Mises equivalent plastic strain. Recall, however, that even this
von Mises expression of equivalent plastic strain is only valid under monotonic
loading. Difficulty arises during strain reversals when the magnitude of total
plastic strain and, necessarily, the second invariant of plastic strain are reduced
because the flow stress at the point of loading should never decrease. The
hardening being based on the idea that compressive strain results in
compressive strain hardening and tensile strain produces tensile strain
hardening.
Nonetheless, based on monotonic loading, the equivalent plastic strain is
dependent on the second invariant. For this research, the scalar plastic strain is
described for monotonic loading as the square root of the second invariant of
plastic strain, similar to von Mises equivalent plastic strain.
2
1
222
2
1
2 2
P
P
P P
J
JJ
J
ε (4-54)
135
It is necessary that the increment of equivalent plastic strain be non-
negative. Therefore, the product of the absolute values of the derivative of
equivalent plastic strain and the increment of plastic strain are used in Eq. (4-56).
0Pd (4-55)
22
P
P Pd dJ
ε
ε (4-56)
Now, with the six points of shear strain and the material asymmetry, the
3J -component of hardening and the Bauschinger effect will be discussed in the
subsequent section.
4.2.2.2 3J -component of the hardening law
The third invariant of plastic strain then is responsible for both the
Bauschinger effect under reversed loading and the increasing asymmetry in the
yield surface under combined tension torsion loading determined using the
associated flow rule as shown in Figure 4-13. Since there is not a state of strain
with a nonzero 3rd invariant of plastic strain and a zero second invariant of plastic
strain, the other 3J -dependent component can be identified by subtracting the
first component from the total elastic region with a known contribution of the
second and third invariant of plastic strain. For simplicity, uniaxial tension and
uniaxial compression tests can be used to parameterize the asymmetry
parameter for positive and negative 3rd invariants of plastic strain, respectively.
136
Figure 4-13. Cazacu and Barlat initial and third invariant component of the subsequent yield surfaces using asymmetric isotropic hardening under uniaxial tensile loading in the deviatoric plane.
Notice the 3J -component of the subsequent yield surface under uniaxial
tensile loading in Figure 4-13 shows hardening near the tensile meridians and
softening near the compressive meridians, the Bauschinger effect. Furthermore,
the surface remains isotropic, centered at the origin and maintaining three-fold
symmetry about the projections of the principal axes.
Next, to parameterize the evolution of asymmetry under axial strains, the
component of the uniaxial tensile flow stress associated with the third total plastic
strain invariant is determined by linear decomposition.
3 3 2 2 0
3 3 2 2 0
P
T T T T
P
C C C C
J J
J J
(4-57)
sy(MPa)
sz(MPa)
sT 3
137
Note that the tensile flow stress is characterized for both positive and
negative values of the third invariant of plastic strain using the uniaxial
compression test data. Now, the asymmetry parameter is determined as a
function of the uniaxial tensile flow stress which is a function of the second and
third invariants of total plastic strain.
33
2 2 32322
3 2 2 3
3
3 9 3 327 ' 3 3
27 ' 2 2 9 2
TT
T T
Jc
J
(4-58)
4.3 Implementation of the Distortional Hardening Model For illustrative purposes, ES-1 was first modeled using the Cazacu and
Barlat (2004) yield criterion with conventional isotropic hardening and then, for
comparison, using the von Mises yield criterion with the proposed distortional
hardening model.
It was assumed that ES-1 was a Cazacu and Barlat (2004) material at the
outset of this research. Therefore, using the flow stress of Eglin steel in tension
and compression, the material constant (c) was determined at several values of
plastic strain to characterize the progression of the tension-compression
asymmetry. The evolution of the material parameter, c, with the accumulated
plastic strain is shown in Figure 4-14.
138
Figure 4-14. Cazacu and Barlat material parameter, c, by plastic strain for forged specimens, cast and HIP’d specimens and cast specimens.
The yield surface was plotted in the deviatoric (y-z) plane using the
constant, c, and the normalized stress in uniaxial tension as shown in Figure
4-15.
139
Figure 4-15. Cazacu and Barlat yield surfaces in the deviatoric plane (A) forged, (B) cast and HIP’d, and (C) cast no HIP specimens.
140
If only dependence of P is considered, variation of c can be
approximated as:
ln P Pc A B C (4-59)
where the constants A, B, and C were determined using the experimental data
and are presented in Table 4-1.
Table 4-1. Cazacu and Barlat yield criterion parameters. Coefficients Mises Cazacu Constant Linear Cazacu Log Cazacu
A 0.0 0.000 0.000 -0.166 B 0.0 0.000 -13.170 0.000 C 0.0 -0.226 0.047 -0.966 A 0.0 0.000 0.000 -0.269 B 0.0 0.000 -21.220 0.000 C 0.0 -0.124 0.315 -1.318 A 0.0 0.000 0.000 -0.208 B 0.0 0.000 -13.60 0.000 C 0.0 -0.161 0.183 -1.216
In order to more accurately determine the value of c at much higher values
of local plastic strain, the linear interpolation procedure was used (see Table
4-2).
141
Table 4-2. Points for linear interpolation of the Cazacu and Barlat asymmetry parameter, c.
Equivalent Plastic Strain Forged Cast and HIP’d Cast no HIP
0.00 -0.2120 0.0000 -0.0682 0.02 -0.3119 -0.2079 -0.1335 0.04 -0.3387 -0.3152 -0.1511 0.06 -0.3548 -0.3792 -0.1617 0.08 -0.3663 -0.4250 -0.1693 0.10 -0.3753 -0.4605 -0.1752 0.12 -0.3827 -0.4895 -0.1800 0.14 -0.3889 -0.5140 -0.1842 0.16 -0.3944 -0.5352 -0.1877 0.18 -0.3992 -0.5538 -0.1909 0.20 -0.4034 -0.5705 -0.1934 0.22 -0.4073 -0.5855 -0.1963 0.24 -0.4109 -0.5992 -0.1986 0.26 -0.4141 -0.6118 -0.2008 0.28 -0.4171 -0.6234 -0.2028 0.30 -0.4199 -0.6342 -0.2046 0.32 -0.4226 -0.6443 -0.2064 0.34 -0.4240 -0.6538 -0.2080 0.36 -0.4274 -0.6627 -0.2095 0.38 -0.4296 -0.6711 -0.2110 0.40 -0.4317 -0.6791 -0.2124
This distortional hardening model in conjunction with a von Mises yield
criterion reveals very interesting trends. Recall that the initial yield surface for a
Mises material is circular in the deviatoric plane. Under pure shear deformation
as in a fixed-end torsion test, the proposed hardening model predicts that the
circle will grow in size, but retain the same circular shape as shown in Figure
4-16. However, under axial loading 3 0J , the yield surface would begin to
take the form of a rounded triangle. Under tension, the surface would have
maximum curvature at the intersection of the three tensile meridians and reach a
minimum curvature at the intersection with the compressive meridians.
Conversely, under compression, the surface would reach maximum curvature at
the intersection with the three compressive meridians and reach a minimum
curvature at the intersection with the tensile meridians.
142
Figure 4-16. Cazacu and Barlat yield surface in the deviatoric plane under uniaxial tension, uniaxial compression, and pure shear using distortional hardening.
Furthermore, for a von Mises material with such distortional hardening, the
evolution of asymmetry is independent of the second invariant of plastic strain
2 0c . Therefore the yield function reduces to the familiar form.
13 322 3 YJ cJ
(4-60)
where c is an odd function of the third invariant of plastic strain. For this model,
the asymmetry parameter was approximated again using the hyperbolic tangent
143
function from the cyclic characterization and power law hardening in pure shear
strain.
31.7 tanh0.002
Jc
(4-61)
4.4 Implications of the Proposed Distortional Hardening Model
While the idea for this new model for hardening came from the need to
better account for the Bauschinger effect, induced tension-compression
asymmetry, and how it affects the intrinsic asymmetry due to the structure of the
material, the new model has broader applicability. Let us point to its advantages.
First, as three different monotonic stress-strain curves are used to parameterize
the subsequent yield surfaces, the model is accurate under monotonic uniaxial
tension, uniaxial compression, and shear strain. Second, the model captures the
trend of induced tension-compression asymmetry under reversed loading often
seen in dynamic or cyclic applications or the effect of pre-strained components.
Third, the model predicts the direction of the plastic flow under general three-
dimensional loads.
Expressly under dynamic conditions, reversals of three-dimensional
loadings are possible as waves propagate through the material. With accuracy in
uniaxial tension, uniaxial compression, and pure shear plastic strain under
monotonic loading as well as capturing the trend of softening under reversed
loading, it may be possible to provide the most accurate predictions under three-
dimensional, dynamic conditions using this relatively simple yield function.
This model provides new insights and paths to be taken for further
understanding of asymmetric hardening. Much more research is required to
144
understand the effects of strain rate, temperature, and orthotropy on the evolving
yield surface. Additionally, more research is required to glean more predictive
capability from the model. For example, is the extent of asymmetry predictive of
aging, spring-back, strain localization, or failure?
Creep and high strain-rate tension tests could reveal the rate-sensitivity of
each component of the asymmetric hardening laws.
Finally, while this research concerns isotropic behavior, the applicability of
this new hardening model to anisotropic materials is a promising research path.
The theoretical yield function parameterized using experimental data was
then implemented in a numerical solution in a user material subroutine within
ABAQUS implicit. The finite element analysis is discussed in Chapter 5.
145
CHAPTER 5 FINITE ELEMENT ANALYSIS OF EGLIN STEEL
The capabilities of the model developed for Eglin steel to capture the
mechanical behavior for general loadings will be assessed. To solve boundary-
value problems for plastically deformable materials, generally the finite element
framework is used. Therefore, first a brief overview of this framework including
the strong and weak form of the balance of linear momentum and finite element
discretization is presented along with the user material subroutine (UMAT)
containing a time-integration algorithm used for implementing the model in
commercial FEA software (ABAQUS, 2009). Next, finite element simulations of
the mechanical response of the material for a variety of loading conditions for
both quasi-static and dynamic loadings are presented. Comparison between
simulation results and data attest to the adequacy of the formulation proposed.
Moreover, the model is used to gain insights into the behavior of ES-1 for
loadings that are not accessible to direct experimentations. Such numerical tests
are useful to engineers in the field who need to assess the behavior of the
material for combined loadings involving shear.
5.1 Finite Element Formulation
The implicit solver of the commercial software ABAQUS (see ABAQUS,
2009) was used in the finite element analysis. Let Φ ,tX define the motion of a
deformable material from the reference to the current configuration. This
software uses an updated Lagrangian formulation, that is Φ ,t+ t at the
increment n+1 corresponding to the instant t+ t is calculated based on Φ ,t at
146
the increment n at the instant t by imposing the velocity field. Thus, the
deformation gradient, Φ
XF
, is:
,t t
iij t
j
XF
X
(5-1)
and all the dependent variables i.e. the Cauchy stress σ , the velocity, v, and the
strain-rate tensor D are expressed as functions of the material coordinates X.
Let be the domain occupied by the plastically-deformable material with
the boundary at time t. For isothermal processes, the governing equations
include the balance of linear momentum and the constitutive equations relating
the stress to the rate of deformation or strain-rate, D. Boundary conditions, such
as tractions T on t and velocity v on v , with t v , are associated
with the governing equations. The strong form consists of the balance of the
linear momentum presented here with zero body forces and traction boundary
conditions:
div σ a on
T n σ T on (5-2)
Where is the current density, and a is the acceleration (the second derivative
with respect to time of the displacement vector, =a u ). The above equations
depend only on the velocity field v, because the stresses can be expressed in
terms of velocities by the constitutive model. The exact solution to this set of
differential equations is difficult, if not impossible, to calculate. Hence, to
approximate the solution by discretization of the domain into finite elements with
n nodes, the strong form is multiplied by a vector-valued weighting function ( v )
147
that is zero on the boundary v and then integrated over the volume. Note that
since stress is symmetric:
: :v σ σ D
where 1
2
Tv v D = , the weak form can be written as:
:a vd vd T vdA
. (5-3)
An admissible trial solution for the element velocity ev is differentiable
and satisfies the boundary conditions (i.e. e vv on
v ). The acceleration ea is
given by the time derivative of the trial function.
1
,n
e k k
k
t N t
v X X v , (5-4)
1
,n
e k k
k
t N t
a X X a , (5-5)
where kN ( )X denote the shape functions having the desired level of continuity
such that ev is admissible. Using the Galerkin method, the weighting function
has the same form as the trial solution with the additional restriction v = 0 on
v .
1
n
k k
k
v N v t
X X (5-6)
The rate of deformation is the symmetric part of the spatial velocity
gradient (i.e. derivatives are taken with respect to the spatial coordinate x and
not X):
148
ij Ii I, j Ij I,i
1
1D = v N v N
2
n
ek
.
Substituting the weighting and trial functions back into the weak form, the
equation of motion is assembled into a global system of equations,
: 0
e e e
I e e I e I eN d N d N dA
a σ T (5-7)
Or in matrix form:
ext intM [a]= f f (5-8)
In the above equation the matrix M is defined as
e
IJ I JM N N
, (5-9)
while f dA
et
ext
I I eN
T and f dΩ
et
int
k k eN
. For more information on finite
element analysis the reader is referred to the textbook by Belytschko et al.
(2013).
Moreover, it is important to note that in ABAQUS, all stresses and strains
are rotated by R, the proper-orthogonal rotation corresponding to the polar
decomposition of the gradient of deformation (i.e. F = RU = VR) with the right
stretch tensor (U) and left stretch tensor (V) being symmetric and positive-definite
tensors before the UMAT is called.
ABAQUS uses of the Green-Naghdi rate (Green and Naghdi, 1965) which
is objective (frame invariant) and has the remarkable property that it reduces to a
time derivative in the rotated coordinate system. Thus, in ABAQUS, the finite-
deformation formulation of any elastic-plastic model has the same form as the
149
small deformation formulation because D reduces to ε in the rotated frame. For
proofs on the unified treatment of finite-deformation and small-deformation
theories, the reader is referred to Hughes (1984).
Without a UMAT, an elastic-plastic model using the von Mises (1913) yield
criterion is used to solve the global system of equations in ABAQUS implicit. The
UMAT described in the following section was developed to implement the
viscoplastic model with the Cazacu and Barlat (2004) yield criterion and the
proposed distortional hardening model in the finite element analyses.
5.2 Implementation of the Elastic-Plastic Model in FEA Recall in this research, the yield function is expressed as the difference
between the effective stress (a function of the Cauchy stress and the plastic
strain) and the hardening (a function of the plastic strain).
, ,P P PF Y σ ε σ ε ε (5-10)
The total plastic strain increment is commonly expressed as a summation
of the plastic strain increment and the elastic strain increment using Hooke’s law.
1
:P e P eC
(5-11)
Equations (5-10) and (5-11) are combined and solved using the iterative
Newton-Raphson numerical root-finding algorithm in the UMAT.
1 1 1 1 1
1
1 1 1
, , 0
: 0
P P P
n n n n n n
e P
n n n
F Y
σ ε σ σ ε ε
ε C σ ε (5-12)
At each increment of time 0 through n, the above system of equations is
solved using an iterative root finding algorithm called the Newton-Raphson
method using a Taylor series expansion within the UMAT.
150
1 1 1 11 1
1 1 1 1
1 11 1
1 1 1 1 1
, ,, : : 0
: : 0
m Pm m Pm
n n n nm P m m Pm
n n n nP
e m Pm e m Pm
n n n n n
F FF
σ ε σ εσ ε σ ε
ε C σ ε C σ ε
(5-13)
For the first iteration of the Newton-Raphson, the initial trial solution (m=0)
is assumed to be completely elastic. Hence, the trial solution for the stress
tensor is described by Hooke’s law and the plastic strain increment is zero.
1 1:trial e
n n n σ σ C ε (5-14)
If the yield function was not within the specified tolerance (usually 1 Pa),
the trial solution was updated (m+1) by a variation, 1
1
m
n
σ and 1
1
Pm
n
ε .
1 1
1 1 1
1 1
1 1 1
Pm Pm Pm
n n n
m m m
n n n
σ σ σ (5-15)
Recall that the increment of plastic strain is normal to the yield surface.
21 1 1
1 1 1 121
:
m
pm m m m
n n n nn
F F
ε σ
σ σ
Solving the system of equations for the variation in the stress:
11 1
1 1 1 1 11 1
: :
m m
m m m m
n n n n npn n
F FF
eσ P C σ εε σ
(5-16)
1
p
F F
eP C
σ ε (5-17)
11 1
1 1 1 1:Pm m Pm
n n n n
eε ε C σ ε (5-18)
This technique is better known as return mapping. Once the yield function
is equal to zero within a specified criterion, 1 Pa, the process is repeated for the
next time increment. The stress and plastic strain are updated until the
151
numerical analysis converges upon a solution at each increment of time. The
UMAT was used in the finite element analyses discussed in Section 5.3.
5.3 Finite Element Analysis of Ultra High Strength Martensitic Steel
First, verification of the implementation of the model using a right
rectangular prism was completed. Next, all of the experimental loading scenarios
were simulated and compared with the experimental data. Let us recall that the
flat specimens were used to characterize the uniaxial tensile response for model
identification. Validation of the model was performed by comparing the
simulation results with data from tests on round specimens. In addition, the
capabilities of the model to predict the shear response of ES-1 were investigated.
Finally, the model with a rate-dependent hardening rule was used to predict the
behavior for dynamic conditions. A summary of all the finite element simulations
is included in Table 5-1.
152
Table 5-1. Finite element analysis simulation matrix.
Strain Rate Manufacturing Process
Loading Configuration
Specimen Shape
Quasi-Static Forged Tension Round
Pin Loaded
Prism
Compression Round
Torsion Thin Walled
Cast and HIPd Tension Round
Pin Loaded
Compression Round
Torsion Thin Walled
Cast Tension Round
Pin Loaded
Compression Round
Torsion Thin Walled Dynamic Forged SHPB Round
Taylor Impact Round Cast and HIPd SHPB Round
Taylor Impact Round
Cast SHPB Round
Taylor Impact Round
5.3.1 Verification of the FE Implementation
The finite element analysis of a right rectangular prism (1 mm x 1 mm x 2
mm, see Figure 5-1) composed of two linear hexahedral elements with reduced
integration (C3D8R) subject to uniaxial tension was used to verify the UMAT
development. Symmetry and displacement boundary conditions were
prescribed.
153
Figure 5-1. Stress distribution according to the model within forged ES-1 under
quasi-static uniaxial tension.
Next, the axial force require to obtain the prescribed displacement was
used to calculate the stress. It was then verified that this stress-strain response
matches the input hardening law (Swift law) as well as the stress-strain curve
obtained directly at the integration point of one element (see Figure 5-2) for
comparison of the three curves. Note the implementation is correct with slight
differences associated with elastic contributions.
154
Figure 5-2. Stress and plastic strain for forged ES-1.
5.3.2 Comparison between Model Predictions and Experimental Results in Uniaxial Tension and Compression
The proposed model was further applied to simulate the plastic response
of the flat tensile specimen (for the geometry of the specimens, see Chapter 3)
and compared with the experimental observations acquired using DIC. The finite
element mesh for the flat tensile specimens consisted of 2829 linear hexahedral
element with reduced integration (C3D8R).
For each processing condition (i.e. forged, cast, and cast and HIP’d), the
finite element predictions of the strain isocontours were compared to the DIC
local strain maps obtained experimentally for the same axial displacement, the
onset of necking.
155
Figure 5-3. The longitudinal strain distribution from DIC and FEA forged specimens.
Figure 5-4. The longitudinal strain distribution from DIC and FEA cast and HIP’d specimens.
156
Figure 5-5. The longitudinal strain distribution from DIC and FEA of cast specimens.
Note that there is an overall agreement between model and data for all
materials. Next, the predictions of the global behavior in uniaxial tension were
compared to the experimental data for all materials. Note the excellent
agreement in Figure 5-6.
157
Figure 5-6. Stress-strain curves of flat DIC and FEA specimens in tensile loading A) forged, B) cast and HIP’d, and C) cast.
158
Next, the finite element predictions were compared to data for each
material under uniaxial compression. The finite element mesh of 3375
hexahedral elements is presented in Figure 5-7 and the specimen dimensions
are included in Chapter 3. Note the excellent agreement obtained for all
materials for this loading in Figure 5-8.
Figure 5-7. The uniform, longitudinal stress distribution within the forged round finite element specimens under quasi-static compressive loading
A) cross section and B) side.
159
Figure 5-8. Stress-strain curves of round DIC and FEA compression specimens A) forged, B) cast and HIP’d, and C) cast.
160
5.3.3 Model Validation: Comparison with Uniaxial Tension in Axisymmetric Specimens
In order to further assess the predictive capabilities of the plasticity model,
the model predictions have been compared to the uniaxial tension experiments
performed on axisymmetric specimens. The geometry of the specimen is given
in Chapter 3. Due to symmetry of the problem only an eighth of the specimen
was modeled using 6400 hexahedral elements with reduced integration. The
mesh was refined in the mid cross-section region as shown in Figure 1-11 in
order to capture the localization of the deformation. Boundary conditions were
applied such as to maintain the symmetry of the problem, and an axial
displacement of 3mm was imposed at the extremity of the specimen.
For the (A) forged, (B) cast and HIP’d, and (C) cast specimen, the
predicted load-displacement curves are compared with the experimental ones
(see Figure 5-9). It is worth mentioning the agreement between the model
prediction and the experiments. Also, in Figure 5-9, it could be seen that
numerical predictions considering a Swift law as an isotropic hardening law are
more accurate than the one using a Voce hardening law. Furthermore, the
discrepancies observed between the predicted load-displacement curve for the
elastic domain are due to the fact that the experimental displacement has been
acquired using the crosshead, i.e. the measured displacement encompass the
compliance of the load frame.
To overcome this issue, the experimental displacement has also been
accurately measured using the DIC technique. A virtual extensometer has been
placed in the gauge length of the specimen to precisely record the axial strain. In
161
Figure 5-10 is shown the comparison between the predicted stress-strain curve
and the experimental one with the strain calculated using DIC measurements. It
is worth noticing the very good agreement between the model prediction and the
experiments for the three materials.
162
Figure 5-9. The experimental and model load-displacement curves A) forged, B) cast and HIP’d, and C) cast materials.
163
Figure 5-10. Stress-strain curves for round DIC and FEA tension specimens A) forged, B) cast and HIP’d, and C) cast.
164
As an example, Figure 5-11 shows the predicted isocontour of the von
Mises stress for the forged ES-1 material at different axial displacements
corresponding to the occurrence of plastic strain (axial plastic strain of 0.2%,
Figure 5-11 A), to the ultimate stress (axial plastic strain of 4%, Figure 5-11 B)
and on the onset of failure (axial plastic strain of 9%, Figure 5-11 C). It is worth
noting that the stress field is still homogenous for an axial plastic strain of 5%,
while at the onset of fracture (10% axial plastic strain), it is predicted that stress
localization and necking has occurred.
Figure 5-11. The von Mises stress distribution in the forged FEA specimen A) initial yielding, B) ultimate stress, and C) fracture.
165
The predictive capabilities of the model have been assessed through
comparison with experimental results under uniaxial compression and uniaxial
tension with flat and axisymmetric specimens. In the following section, numerical
simulations of more complex loadings will be performed.
5.3.4 Predictive Capabilities of the New Distortional Hardening Model: Finite Element Analysis using Cyclic Loadings In order to illustrate the capability of the new hardening model for cyclic
loading, cyclic tension-compression simulations have been performed. The
specimen geometry is axisymmetric of diameter of 12.7 mm and height of 6.35
mm. Only one quarter of the specimen has been meshed with 3375 hexahedral
elements. A cyclic compression-tension loading controlled in displacement has
been applied to the specimen. The prescribed loading consists of an initial
compression 0.1u mm of the specimen follow by four cycles of tension-
compression loading 0.1u mm .
The predicted mechanical behavior under cyclic loading of the forged ES-
1 material is plotted in Figure 5-12. The absolute axial stress is plotted as a
function of equivalent plastic strain. It is important to notice the capability of the
new hardening model to capture the reduction in axial stress upon each load
reversal. This is due to the fact that the new hardening model depends on the
third invariant of plastic strain, 3J (see Chapter 4). For uniaxial tension loading,
3 0J , while for uniaxial compression loading, 3 0J .
166
Figure 5-12. Evolution of the axial stress vs. equivalent plastic strain for tension-compression cyclic loading.
5.3.5 Predictive Capabilities: Finite Element Analysis of Free End Torsion
As discussed previously in Section 1.4, during monotonic free end torsion
loading of thin-walled cylindrical specimens, the occurrence of axial plastic strain
has been observed. It was demonstrated that the occurrence of these axial
plastic strains are due to the tension-compression asymmetry displayed by the
mechanical behavior of the material. In Chapter 4, it was deduced from the yield
criterion that the increment of plastic deformation (in the principal axis) under free
end torsion is:
167
1
3222
23 322 3
30 0
2 33
32 0 03 2 3
230 0
3
P
c
JJJ c
F d cd d d
cJ cJ
ε (5-19)
According to Equation (5-19), the increment of thickness strain that
develops during free end torsion is proportional to 2
3
c. By symmetry and plastic
incompressibility, the axial strain is, therefore, directly related to the asymmetry
parameter, c. If c is zero (no tension-compression asymmetry), then pure shear
deformation occurs (i.e. no axial deformation). If the material has higher yielding
strength in uniaxial tension than in uniaxial compression (c>0), the specimen
walls thicken (positive thickness strain) and the length shortens (negative axial
plastic strain). The reverse holds true for greater yield stress in uniaxial
compression (c<0). For the ES-1 materials, irrespective of the processing, the
mechanical material has greater flow stress in uniaxial compression than in
uniaxial tension. Thus, it is expected that the specimen should elongate under
free end torsion.
Numerical predictions for free end torsion loadings have been performed.
The geometry of the specimen is shown in Figure 5-13 and was meshed with
4522 hexahedral elements (see Figure 5-14). The nodes at the lower extremity
of the specimen (y=0) were pinned, i.e., no displacement was allowed, while the
upper nodes (y=58.4) were tied to a rigid tool to impose torsion while ensuring
that all the upper nodes experienced the same boundary conditions. An angular
168
rotation of 0.3 radian was applied to the rigid tool, but the axial displacement was
not constrained.
Figure 5-13. Geometry of the thin-walled specimen used for free-end torsion loading with dimensions expressed in millimeters.
169
Figure 5-14. Isocontour of the predicted axial displacement that develops during free-end torsion loading of a forged ES-1 material:
A) undeformed and B) with 0.05 radians of axial rotation and axial displacement.
Figure 5-14 shows the axial displacement field for a rotation of the rigid
tool of 0.05 radian. For this angular displacement, an axial displacement of
0.047 mm is predicted. Therefore the specimen lengthened when subject to pure
rotation, a phenomenon known as the Swift effect. In Figure 5-15 is shown the
evolution of the axial strain with respect to the shear strain. The axial strain and
shear strain are calculated using
0
ln 1axial
u
L
and
0
γr
L
where r is the current radius, L0 is the initial length of the specimen, u is the axial
displacement, and is the twist angle. Due to the fact that the yield stress in
uniaxial compression is larger than the one in uniaxial tension, it is predict that
the ES-1 materials will elongate.
170
Figure 5-15. Longitudinal elongation by shear strain for forged quasi-static specimens under quasi-static torsion.
5.4 Comparison between the Model and Dynamic Characterization Data
In this section, the model developed previously will be applied to high
strain-rate loading and impact testing. Using the dynamic implicit solver of
ABAQUS, the model predictions were compared with experimental data obtained
with a SHPB apparatus and Taylor impact tests.
5.4.1 Comparison between Model and Dynamic Characterization Data
To characterize the dynamic behavior of the ES-1 materials, SHPB
compressive experiments were performed and discussed in Section 3.3. In this
section, the numerical prediction will be compare with the experimental results.
But, in this section, we will not compare the strain-stress curve obtained at high-
strain rate, but the signals that were recorded by strain-gages located on the
incident and transmitted bars. Therefore, the SHPB system was modeled in its
entirety (i.e. the striker bar, the incident and transmitted bars as well as the
specimen) and wave propagations through the bars were studied. The geometry
of each component of the UF REEF SHPB system is presented in Chapter 3.
171
Due to symmetry conditions, only one quarter of the bars were meshed. The
total size of the mesh is 6942 elements. Close to the location of the strain-gages
on the incident bar and the transmitted bars, the mesh is refined to obtain a fine
wave propagation signal. Contacts are defined between each of the system
components, and the specimen is initially in contact with the incident and
transmitted bars (see Figure 5-16). The initial velocity of the striker bar is 14 m/s.
As the striker bar impacts the incident bar, an elastic wave is created and further
propagates through the SHPB system and the cylindrical specimen of ES-1. It is
worth noting that the bars are considered to only deform elastically, while the
specimen is described by the developed elastic plastic model.
Figure 5-16. The finite element SHPB and an inset containing a close-up of an ES-1 specimen.
The predicted strain waves, i.e. the incident, reflected and transmitted
waves, are compared with the experimental ones (see Figure 5-17). Note the
very good agreement between the experimental data and the predictions. The
172
evolution of the strain wave with time is correctly predicted by the model. The
measured strain waves from the finite element simulation and the experiment
were compared in Figure 5-17 to validate the rate dependent modeling of Eglin
steel.
Figure 5-17. SHPB incident, transmitted, and reflected waves from forged specimen and FEA.
5.4.2 Model validation: Simulation of Taylor Impact Tests
To further validate the prediction of the developed model for high strain-
rate, finite element simulations of Taylor impacts have been performed. The
Taylor impact test developed by Taylor (1948) consists of launching a solid
cylindrical specimen at an elevated velocity of the order of 100 to 300 m/s
against a stationary rigid anvil. When the specimen impacts the rigid stationary
anvil an elastic compressive wave is generated at the impact interface and
travels back and forth through the specimen. For a sufficiently high impact
velocity, for which the magnitude of the compressive wave reaches the yield
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stress of the material, the impact end undergoes plastic deformation. Therefore,
the plastic front starts propagating from the impact interface. However, only a
portion of the specimen deforms plastically.
Torres et al. (2009) completed ten Taylor impact tests of cylindrical ES-1
specimens. Torres et al. used the analyses of the deformed specimens
developed by Jones et al. (1998). The initial diameter of the specimens was d0
=4.17 mm while its initial length was L0 = 31.317 mm. The impact velocities
varied between 148 and 200 m/s. As already mentioned, when recovered, the
length of the deformed Taylor impact specimens, Lf, is less than than its initial
length, L0. Furthermore, the recovered specimen consists of a rear portion of
length (lf) with a constant cross sectional area (A0), i.e. that has only deformed
elastically, and a portion that has deformed plastically in which the cross-section
(A) gradually increases as shown in Figure 5-19. The analysis depends entirely
upon three measurements of the deformed specimens: the cross sectional area
at several locations along the length of the specimen, the length from the rear of
the projectile to the corresponding cross sectional area (lf), and the total
deformed length of the projectile (Lf). Using plastic incompressibility, the
engineering plastic strain was calculated based on the variation of the cross-
sectional area.
0 1A
eA
(5-20)
As shown in Figure 5-19, Torres et al. (2009) reported the location of the
plane (lf/L0) of the cross section of the recovered specimen that was subjected to
engineering plastic strain e={2, 3, 4, 5, ..12%}. For more details about the
174
analysis of the Taylor impact experiments, the reader is referred to Jones et al.
(1998).
FEA simulations using the developed model in conjunction with the implicit
solver of ABAQUS have been performed for the ES-1 material at an impact
velocity of 148 m/s. Due to the symmetry of the problem, only a quarter of the
specimen was meshed with 38880 hexahedral elements. The mesh is refined in
the impact zone to accurately capture the deformation of the impacted surface of
the specimen. Boundary conditions were applied such as to maintain the
symmetry of the problem. The simulations were performed in two steps,
associated with the specimen launch and impact, respectively. In the first step,
the impact velocity was applied to all of the nodes such as to represent the free
flight or launch of the specimen. In the second step, the impact of the specimen
with the rigid anvil was reproduced by imposing a null forward velocity only to the
nodes belonging to the impact surface while the other nodes were not
constrained anymore. In this manner, the test conditions were reproduced with
fidelity. Furthermore, it is assumed that the impact ends when the axial velocity
at the end of the specimen becomes null. Figure 5-18 presents the equivalent
plastic strain isocontour in the Taylor impact specimen of forged ES-1.
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Figure 5-18. Equivalent plastic strain isocontour in the cylindrical Taylor impact specimen of forged ES-1.
The FEA predictions are in good agreement with the experimental data.
For an impact velocity of 148 m/s, the predicted normalized final length of the
specimen is 0
0.954fL
L
, while the reported experimental data by Torres et al.
(2009) is 0
0.957fL
L
. To directly compare experimental data and FEA
predictions, the axial location of the cross-section subjected to a plastic strain of
2.54,3.28,10.35%e have been reported in the plane (lf/L0, Lf/L0) in Figure 5-19.
It is worth noting the good agreement between the model prediction and
experiments.
176
Figure 5-19. Linear relationship of deformed section length by deformed total length following experimental and simulated Taylor impact tests of specimens of ES-1.
Using the one dimensional wave analysis of Jones et al. (1998) of the
deformed specimen dimensions, Torres et al. presented the stress-strain
response shown in Figure 5-20 for strain rates between 103 and 104 s-1. It is also
worth comparing the experimental results with the numerical predictions.
177
Figure 5-20. Stress strain response of ES-1 subject to dynamic strain rates in Taylor impact tests conducted by Torres et al.
The excellent agreement between the simulated and experimental data for
the Taylor impact tests assesses the predictive capabilities of the developed
model of ultra high strength martensitic steels under dynamic loadings.
In this chapter, FEA predictions of the developed model have been
performed for monotonic tests (e.g. uniaxial tension and free-end torsion), cyclic
tension-compression loadings, and dynamic events (SHPB and Taylor impact
tests). A good overall agreement between the prediction and available
experimental data was observed.
Chapter 6 includes a summary of this dissertation and research as well as
recommendations for future research.
178
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS
In this dissertation an integrated approach using experimental, theoretical,
and numerical techniques was employed to further the understanding of ultra
high strength martensitic steels. The overall goal was to determine, for the first
time, the effect on the mechanical response of ES-1 steel of three different
manufacturing processes, namely forging, casting, and HIPing. Prior to the
experimental characterization, the ES-1 materials were subject to microscopic
analysis. From the SEM observations of the ES-1 material texture, it was
concluded that processing did not introduce anisotropic features. Irrespective of
the processing, the observed texture was random indicating that the materials
are isotropic.
To characterize the mechanical behavior, a suite of mechanical tests were
conducted. The quasi-static tests 310 performed were monotonic
compression, monotonic tension, and cyclic tension tests.
As concerns the monotonic tests, first axisymmetric specimens (circular
cross section) with a 25.4 mm gage length were tested in uniaxial tension. The
average yield strength for the forged, cast and HIP’d, and cast axisymmetric
specimens using 0.2% offset method were 1541 MPa, 1414 MPa, and 1445
MPa, respectively. The strain-to-failure was of 10%, 8%, and 13% for the forged,
cast, and cast and HIP’d materials, respectively. Irrespective of processing, the
posttest cross section was circular. It was thus confirmed by mechanical testing
that the ES-1 materials studied are isotropic.
179
Moreover, for monotonic tensile loading the influence of the specimen
geometry on localization of the deformation and strain to failure was investigated.
Both global and local strain measurements using DIC were done. For DIC, two
orthogonal cameras were used such as to allow measurements of both the width
and thickness strains. Irrespective of the processing history of the material,
these strains were equal thus confirming the isotropy of ES-1. The true stress-
true strain behavior prior to strain localization was found to be very close to that
of the round axisymmetric specimens, but the strain-to-failure of the flat
specimens was about half that of the axisymmetric specimens.
Quasi-static uniaxial compression tests were also conducted on cylindrical
specimens with a length and diameter of 7.62 mm. It was found that ES-1
exhibits similar yield values in tension and compression. As the plastic strain
accumulates, however, the plastic flow stress in compression exceeds that in
tension. At 4% axial strain, the ratio of flow stress in tension to compression
approaches 0.89.
To elucidate the evolution of the asymmetry of the yield surface, cyclic
tests on ES-1 axisymmetric specimens with a 25.4 mm gage length were
conducted using an MTS load frame at the UF REEF. Under tensile strain, the
tensile flow stress increased while the compressive flow stress was diminished.
Under compressive strain the opposite trend was observed. It was concluded
that the evolution of asymmetry of the yield surface is a function of the plastic
strain tensor.
180
As concerns the dynamic tests, cylindrical specimens with a length and
diameter of 5.08 mm were tested in a SHPB system with a 19 mm diameter at
the UF REEF. A gas gun was used to accelerate a striker bar to impact the
incident bar at approximately 13 m/s creating a strain wave that deformed the
specimen at an average strain rate of 700 s-1. Irrespective of the processing
history, the hardening behavior of ES-1 subjected to high-rate strain was similar
to that under quasi-static loadings, but the strength was higher by a dynamic
increase factor of 1.1.
To describe the experimentally observed mechanical behavior a new
model was developed in the framework of the theory of viscoplasticity. To
account for tension-compression asymmetry in yielding the isotropic form of
Cazacu and Barlat (2004) yield criterion was used in conjunction with a new
hardening model. The key novel aspect of this new hardening model is that it
accounts for the distortion of the yield locus for both monotonic and cyclic
loadings. The noteworthy aspect is that the model is dependent on both
invariants of the plastic strain, and it is isotropic in nature. This is a departure
from the current practice where in order to account for Bauschinger effects an
additional tensorial internal variable, namely the overstress, is introduced. The
prime advantage of the proposed hardening model is that it is accurate under
monotonic loading, but also describes the Bauschinger effect.
Because of the slightly increased accuracy under monotonic loading and
significantly improved characterization of hardening under changes in strain path,
it is believed that the new model will provide more accurate results under
181
complex loadings. For this purpose, the capabilities of the new hardening model
were illustrated for combined loadings such as tension-shear; compression-
shear. The tests necessary for the determination of the parameters involved in
the new hardening model and the identification procedure were outlined.
Specifically, the dependence of hardening on the second invariant of strain can
be isolated and thus fully identified from pure shear tests. Such a law induces a
growth of the yield surface in the deviatoric plane. The strain-hardening
dependence on the third invariant of plastic strain cannot be isolated.
Nevertheless, from uniaxial tensile and compressive test data when strain-
hardening depends on both invariants, assuming linear decomposition, it
becomes possible to identify the dependence on the second invariant of the
plastic strain. Since the third invariant of plastic strain is not associated with
growth of the entire yield surface, to accommodate the prescribed deformation in
two-stage tests involving uniaxial load reversal, in the octahedral plane, the yield
surface normal at the six states associated to pure shear strain ought to rotate.
Therefore, it is predicted that in order to accommodate Bauschinger effects the
yield surface is distorted. For example, under uniaxial tensile strain the initially
“triangular” yield surface will extend along the tensile meridians and will contract
along the compressive ones (i.e. the rounded corners of the rounded “triangle”
intersect the tensile meridians). Under uniaxial compressive strain, the opposite
distortion occurs. Thus, the yield surface evolves differently depending on the
strain history (tension followed by compression, or compression followed by
tension).
182
This new model was implemented in a fully three dimensional, implicit
finite element code and then validated using independent experimental data.
Specifically, for quasi-static loading validation was done for pure shear loadings
while for dynamic loadings the Taylor impact loadings were simulated.
For comparison purposes, the von Mises yield criterion in conjunction with
a distortional model involving only dependence on the third invariant of the plastic
strain, and the Cazacu and Barlat (2004) yield criterion with evolving hardening
depending only on the accumulated plastic strain was also applied to ES-1
materials. The same experimental data was used for identification of these
formulations. Both formulations were also implemented in the same FE solver
(ABAQUS) using a UMAT. FE simulations of flat ES-1 specimens under uniaxial
tensile loading were conducted and a side-by-side comparison of the strain maps
using DIC in the experimental characterization and both FE analyses was
presented. It was shown that the Cazacu and Barlat (2004) criterion coupled with
the full distortional hardening model provides the most accurate representation of
the test results.
In addition, verification of the FE implementation and of the capabilities of
the model was done by simulating free-end torsion of a thin-walled cylinder. Due
to the tension-compression asymmetry of the material, elongation under torsion
was predicted.
Most importantly, this dissertation research provides new insights and
understanding of plastic deformation under combined loadings. While the new
model developed was applied to ES-1 materials, it has a much broader range of
183
validity. Future research involving the use of this hardening model in conjunction
with anisotropic yield criteria and its applications to strongly textured materials
should be of great interest.
184
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BIOGRAPHICAL SKETCH In May of 2000, Elizabeth Kay Bartlett graduated with a Bachelor of
Science in mechanical engineering from Mississippi State University. As a
graduate research assistant to Dr. Susan Hudson, she continued her education
and completed a Master of Science in mechanical engineering in May of 2002.
While working as a weapons test engineer for the United States Air Force
(USAF), Elizabeth became interested in the fragmentation of warheads in arena
tests. In May 2005, her only daughter, Gwyneth Webb, was born at Fort Walton
Beach Medical Center. While working full-time, she began taking classes at the
UF Research and Engineering Education Facility in Shalimar, FL. In August of
2016, she was awarded the Science, Mathematics, and Research for
Transformation (SMART) scholarship to complete her Doctor of Philosophy in
mechanical engineering under the advisement of Prof. Oana Cazacu.