Mechanics and Materials Center

TEXAS A&M UNIVERSITY

College Station, Texas

INTEGRATED RESEARCH IN CONSTITUTIVE MODELLING

AT ELEVATED TEMPERATURES

FINAL REPORT

V

W.E. Haisler

and

D.H. Allen

IN AS A-CB-17,7 2 37) INTEGRATED RESEARCH IN

CONSTITUTIVE. HODELLING AT ELE:VATEB

TEHPEBATUBES, SJ\BT 1 . Final Eeport (Texas

ASM Oniv.) 332 p CSCL 20K

G3/39

N86-30227

TH.BU

N86-30232

Unclas

42932

Part 1

MM 4998-86-13 June 1986
INTEGRATED RESEARCH IN CONSTITUTIVE MODELLING

AT ELEVATED TEMPERATURES

FINAL REPORT

W.E. Haisler

and

D.H. Allen

Aerospace Engineering Department

Texas A&M University

College Station, Texas 77843

submitted to

NASA Lewis Research Center

Cleveland, Ohio

Technical Monitor: R.L. Thompson

MM 4998-86-13 June 1986
TABLE OF CONTENTS

PAGE

1. INTRODUCTION 1

1.1 Summary ................................................ 1

1.2 Acknowledgement ....;....;................:......:...... 1

2. RESEARCH OUTLINE 1

2.1 Summary of Completed"Research .......................... 1

2.2 Numerical Integration Techniques ....................... 2

2.3 Thermodynamics and Internal State"Variables ............ 6

2.4 Experimental Lab Development .;.......... 11

2.5 Comparison of Models at Room Temperature ............... 16

2.6 Comparison of Models at Elevated Temperature.......:;:. 20

2.7 Integrated Software Development 26

2.8 Conclusion .......;.;;.......;;;;;. 32

2.9 References .....;......;..........;;....;...;;..;.;..... 33

3. PUBLICATIONS LIST 34

4. PROFESSIONAL PERSONNEL INFORMATION 34

4.1 Faculty Research Assignments ........................... 34

4.2 Student Research Assignments ........................... 35

5. INTERACTIONS 35

5.1 Papers Presented ....................................... 35

5.2 Related Travel and Consultation ........................ 35

6. APPENDIX - INTERIM TECHNICAL REPORTS 36

6.1 Numerical Considerations in the Development

and Implementation of Constitutive Models

6.2 Evaluation of the Numerical Stability and

Sensitivity to Material Parameter Variations

for Several Unified Constitutive Models

6.3 On the Use of Internal State Variables in

Thermoviscoplastic Constitutive Equations

6.4 A Comparison of Current Models for Nonlinear

Rate - Dependent Material Behavior of Crystalline Solids.

6.5 An Experimental Comparison of Current Viscoplastic

Models at Elevated Temperature
1. INTRODUCTION

1.1 Summary

This report details research performed under NASA Grant NAG3-491,

with startup date of November 15, 1983 and completion on December 31,

1985. The research centers on evaluation of thermoviscoplastic

constitutive models for metals at elevated temperatures. The primary

intent of the research was threefold in nature: 1) to improve existing

experimental facilities within the Mechanics and Materials Labs at Texas

A&M University in order to perform the complex experiments required

under the grant; 2) to compare existing models to experiment for the

materials to be discussed herein; and 3) to extend existing models where

necessary to better predict complex material response. The research

group consists of Drs. W.E. Haisler and D.H. Allen, assisted by five

graduate research assistants.

1.2 Acknowledgement

The authors express their thanks for the support provided for this

research by the NASA Lewis Research Center. The technical grant monitor

is Dr. R.L. Thompson.

2. RESEARCH OUTLINE

2.1 Summary of Completed Research

The important details of the research are outlined in this

section. The findings are divided into the following six sections: 1)

numerical integration techniques; 2) thermodynamics and internal state

variables; 3) experimental lab development; 1) comparison of models at

room temperature; 5) comparison of models at elevated temperature; and

-1-
o) integrated software development. These topics are summarized in the

following seven sections (2.2 through 2.7). Further details are given

in the technical reports in the appendix. A summary of important

findings is given in Section 2.8. _ J/&&f

N86-30228 X>,~

2.2 Numerical Integration Techniques _/ 7

The sensitivity of the unified constitutive theories proposed by

Bodner, Walker, Krieg, and Miller, to numerical integration techniques

and slight changes in material parameters was investigated. Evaluations

were based upon numerical simulations of Hastelloy-x at 1800°F, in

uniaxial form, by specifying input strain histories and comparing output

stress histories. The constitutive models selected for this study were

chosen primarily because of the availability of a prescribed methodology

for material parameter evaluation. In addition, they appear to be the

most qualitatively attractive theories available and thus warrant

further evaluation.

The investigation begins by presenting the various models in

uniaxial differential equation form, followed by a description of the

methods used for material parameter evaluation. The main thrust of the

research was to assess the degree of sensitivity of each model,- to

integration techniques and/or variations in the respective material

constants. An example is shown in Fig. 1 for Bodner's model.

For the numerical time integration study, four commonly used

algorithms were selected. These included Explicit Euler Forward

Difference, Implicit Trapezoidal, 4th Order Runge-Kutta, and Trapezoidal

Predictoi—Correction methods. In addition, four different strain rate

input histories were selected to insure that evaluations were not biased

-2-
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-3-
to one specific simulation. The strain histories considered were not

developed^ to model any specific physical phenomena. However, they are

representative of conditions under which a material could be strained

either in the laboratory or the field.

The integration investigation was divided into two tasks. The

first task included a comparison of the numerical stability of each

constitutive theory in order to assess the degree of nonlinearity of the

prescribed growth laws. The comparisons were based, upon predicted

stress-strain behavior for constant strain rate tests simulating

monotonic loading. Explicit Euler Forward integration was used to

obtain stress histories for various time steps. It was concluded that

Miller's model was the most sensitive to step size variation because of

the oscillatory response of the stress-strain curve for the larger At

values. It is postulated that the oscillatory nature of Miller's model

can be traced back to the hyperbolic sine function that is used to

characterize the inelastic strain rate and other growth laws.

In addition, it was found that the predicted stress response for

each model appears to be "self correcting" for large time increments

when the constitutive equations are integrated by the Euler method.

This "self-correcting" phenomenon appears to be an intrinsic property of

the prescribed growth laws and the coupled nature of the equations.

The second task of the integration investigation was to evaluate

the constitutive theories in terms of solution stability, accuracy, and

computational efficiency when numerically integrated using the various

algorithms considered. A test matrix consisting of integration method,

time step sizes, and strain input histories was established in order to

make qualitative and quantitative comparisons. For each strain history

-4-
considered, a baseline or "pseudo correct" solution was obtained

analytically," sTrice" no"experimental data were available.

The computational efficiency of each integration method was

obtained through selection of equivalent At's. Euler's method was

observed to be the most efficient, followed by the Implicit Trapezoidal

method, then the *4th order Runge-Kutta method, and finally the

Trapezoidal Predictor-Corrector method. In addition, each model was

evaluated for its computational efficiency. The results indicate that

Walker's model resulted in the fastest execution time, followed by

Krieg, Bodner, and Miller, respectively.

In terms of qualitative and quantitative comparisons the following

observations were made: 1) In general, for equal time step size and

equivalent computation integration times, the Mth order Runge-Kutta

method consistently overpredicts the state of stress while the

Trapezoidal method generally underestimates the stress history. The

Euler and Predictor-corrector methods appear to provide accurate results

at any point in the simulation; 2) the amount of stress-overshoot (for

the larger step sizes) appears to grow with increasing strain rate; and

3) no "self-correcting" tendency was observed when either the Mth order

Runge-Kutta or Trapezoidal method was used.

Another important issue this research addressed was what amount of

degradation in predicted response could be expected due to variations in

material constants. Two procedures were developed to study this. The

first procedure characterized the sensitivity of each constant within a

particular constitutive model. Each constant's sensitivity to variation

was determined by adjusting it by 5% and comparing predicted results.

The second procedure provided an "upper bound" on the effect that

—5—
experimental error would have on the determination of the material

parameters and ultimately, predicted response. The effect of

experimental uncertainty was studied by adjusting numerical test data by

±5? and then using the adjusted data to determine new material

constants.

From the results of the sensitivity study, the following

observations were made: 1) For the load history considered,

constitutive models can have up to a 12$ variation in predicted response

for as little as a 5% variation in a given material constant; and 2)

Miller's model was shown to be the least sensitive to material parameter

variation possibly due to its strong microphysical basis. On the other

hand, Walker's model was the most sensitive, showing a 30$ overprediction in stress during the relaxation period of the load history

considered. Bodner's and Krieg's models were found to be between these

two extremes. Further details on this phase of the research can be

found in Appendices 6.1 and 6.2.

It should be noted that the conclusions reached from this study

are based on the use of uniaxial simulations. It is expected that the

conclusions would hold for multiaxial situations. However, further

evaluation is needed in this area.

2.3 Thermodynamics and Internal State Variables

As a part of the research effort it was concluded that a casting

of the currently available constitutive models into a common framework

would help in identifying the relative merits and/or deficiencies of

each model. For this purpose it was decided to cast the models into the

framework of internal state variable (ISV) theory, in which a set of

-6-
variables which are not observable are utilized to account for

thermpdyjiarnicallyjlis.sipati ve processes such as Dislocation moverne^nt,_

grain boundary sliding, deformation twining, phase changes, etc. This

framework has been used to determine qualitative differences in the

models studied under the current grant. Furthermore, in this process it

was found that certain thermodynamic issues could be clarified.

The specific ISV framework uses the approach initially proposed by

Coleman and Gurtin [1], in which the observable state variables are

supplemented with a set of internal state variables:

k k

a.. = ay^n,. t), K = 1,2, ...n , (1)

where a.. are a set of n internal state variables, in this case

designated as second order tensor functions of space x and time t.

The method of Coleman and Noll [2] may be utilized to obtain

thermodynamic constraints on the class above of materials including a

broad range of thermoviscoplastic metals. The results are:

3h

°

 = p '

where o, is the stress tensor, e is the infinitesimal strain tensor,

K JL K _L

p is the mass density, and h is the specific Helmnoltz free energy,

given by

HE u-Ts = h(emn, T, a*..) , (3)

-7-
where u is the internal energy, T is the temperature, and s is the

entropy. The Helmholtz free energy is then expanded in a Taylor series

in strain, temperature, and the ISV's and substituted into (2) to obtain

the stress-strain relation:

Cijkl (ekl ~ ell ~ akl

T where C... is the elastic modulus tensor, and e is the thermal strain

tensor. The above equations serve as a framework of comparison for all

of the models considered in this research. These are adjoined with ISV

growth laws of the general form:

ij

 (emn'

 T« "

The ISV's represent locally averaged measures of dislocation arrangement

(back stress) dislocation density (drag stress), etc. ISV growth laws

(5) represent the principal differences between the models studied in

the current research effort. These equations have been itemized in

Table 1 for ten different current models. Qualitative differences in

the models are also discussed in the table.

The main content of this part of the research has been to review

and clarify the continuum and thermodynamics based internal state

variable model for application to thermoviscoplastic metals. In this

process the following points have been made:

1) the definition of an internal state variable utilized in this

model has been clarified;
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2) internal state variables in metals represent local averages of

dislocation arrangement, dislocation density, and intergranular damage,

3) in the context of the ISV definition given here, inelastic

strain may also be interpreted as an internal state variable;

U) the path dependent Helmholtz free energy may be expanded in a

second order expansion in elastic strain and temperature in order to

obtain a stress-strain equation of state;

5) rate dependence enters the constitutive equations implicitly

via the inelastic strain, as demonstrated by the nonlinear standard

solid analog; and

6) a three-dimensional generalization of the standard solid may

be used as a means of comparison of the general form of several

currently proposed models.

Further details of this development are contained in Appendix 6.3-

2.4 Experimental Lab Development

This section describes the test equipment used to carry out the

tests required for this work. The first sub-section describes the test

set-up used for the creep and strain transient tests based on a dead

weight creep frame. Topics to be covered include the load frame, data

acquisition, and temperature'* measurement and control. The next subsection describes the test set-up used for the monotonic strain rate,

fully reversed cyclic tests, and transient tests during cyclic loading

as well as creep and strain transient tests based on a computer

controlled test set-up.

-11-
Creep Frame Test Set-up

Load Frame. The load frame utilized was a creep frame produced at

Texas A&M University. The initial configuration included a constant

load cam and a constant stress cam as described by Garofalo, et al. [3].

The lift for the pan was provided by an ATS (Applied Test Systems) 2081

cyclic load module. Several modifications were made to this set-up. The

constant stress cam was .removed and a second constant ' load cam

installed. This increased the maximum weight limit from 400 Ibs. to 800

Ibs. This also increased the stability of the loadpan during load-up

and provided a seven to one load magnification. The load pan was

increased in size and supports were added to further aid stability

during load-up. The cyclic load table was removed and replaced with a

Century-Fox model CF-59 5 ton capacity hydraulic jack. This provided

more lift capacity and gave more room for machine deflection during

load-up. The final configuration of the machine is shown in Fig. 2.

Further improvements included a more advanced load pan which would

provide automated removal and replacement of portions of the load.

Jones, et al. [4] have described such an apparatus. This modification

is needed to improve the results of the strain transient tests. A

second improvement was a more advanced method of load-up. For this

purpose, a pneumatic jack was included.

Data Acquisition. The load was measured with a Strain sert TLN20-

256K Tension Link driven by a Hewlett Packard 8805A Carrier Preamp. The

strain was measured with an ATS model 4112 LVDT (Linear Variable

Displacement Transducer) and the extensometer was driven by an ATS model

6974 signal conditioner. Two Hewlett Packard 8803A Low Level Preamps

-12-
ORIGINAL '"PAGE IS

OF POOR QUALITY

Fig. 2. Creep Frame Final Configuration

Fig. 3. MTS 880 llOKip Testing System

-13-
and two low pass filters were necessary to achieve the necessary

resolution. Load and strain measurements were recorded on a Hewlett

Packard series 7700 strip chart recorder.

This data acquisition system was very prone to drift, noise, and

nonlinearity. All strain amplification, filtration, and data

acquisition equipment were eventually replaced with a Kiethly model 197

logging digital multimeter. This set-up proved much more stable.

However, it lacked the ability to take in a large number of data points.

Much improvement is needed in strain data acquisition to produce a

system which is stable, can take in large numbers of points, and has

sufficient resolution. A common set of transducers and a common data

acquisition system should be utilized for both load frames. This would

negate errors arising from using two different systems.

Temperature Measurement and Control. An ATS 2961 Clamshell oven

and an LFE series 230 temperature controller were used for temperature

control. Temperature measurement was handled by two 20 gauge K-type

thermocouples. These were placed in contact with the sample in the

middle of the gauge section on diametrically opposite sides. One

thermocouple was used as input to the temperature controller. The other

thermocouple was input to a Fluke 2176A digital thermometer for readout.

Improvements in this system would include the ability to monitor

the temperature at several points simultaneously along the gauge

section. The temperature should also be input to a data acquisition

system. The method of thermocouple attachment should also be upgraded.

The optimum method of thermocouple attachment would be individual

welding of the leads to the sample surface. This is described in the

ASTM Thermocouple Handbook [5]. Such a procedure was attempted with 20

-14-
gauge thermocouple wire and a Duracom thermocouple welder. The samples

tended to fail prematurely at the,,thermocouple welds. Therefore, this

method of thermocouple attachment was abandoned. Another possible

method using smaller 28 gauge thermocouples was used with the straincontrolled set-up and will be discussed in that section. A common

temperature measurement and control system for both load frames would be

another improvement. This would eliminate relative errors between the

two systems.

Computer Controlled Test Set-up

Load Frame. The load frame utilized in this set-up was an MTS

(Materials Test System) model 880 electro hydraulic testing machine.

MTS 652.01, water-cooled, hydraulic grips allowed fully: reversed cyclic

tests to be carried out at high temperature. The frame was controlled

by a Digital Micro PDP-11. Computer programs were written to run

monotonic tension tests, cyclic tests, cyclic tests with hold times,

creep tests, and creep stress drop tests. The load frame is shown in

Fig. 3.

Data Acquisition. The Micro PDP-11 also handled data acquisition

functions. An MTS 661.21A-02 10 kip load cell was the load transducer.

An MTS 632.1J1B-02 axial extensometer was the strain transducer. This

device had quartz extension rods which contacted the sample at two 120°

punch holes. All data were stored on 5.25 inch floppy diskettes and

could be retrieved in hard copy or graphical form.

Temperature Measurement and Control. An MTS 652, three-zone,

clamshell furnace and three Research Incorporated 63911 Process

Temperature and Power Controllers were used for temperature control.

Temperature Measurement was handled by six 28 gauge K-type

-15-
thermocouples. These were placed in contact with the sample. Three

thermocouples were fed into a Fluke 2176A Digital Thermometer for

readout. These were placed with one each at the top, middle, and bottom

of the gauge section. The other three thermocouples were fed into the

temperature controllers. These were placed in the center of the furnace

zone each was to sense. One thermocouple was placed in the center of

the gauge section and one on each grip.

The thermocouples were supported at the grips by fiberglass

thread. They were fixed to the sample by self-supporting means. The

thermocouples at the top and bottom of the gauge section were wound

around the sample. The thermocouples used in the center of the gauge

section were brought into the oven from different directions and tied to

each other. These thermocouples were then wound around the sample. The

thermocouples used in the center of the gauge section were brought into

the oven from different directions and tied to each other. These

thermocouples were then wound around the sample for contact. Welding

the thermocouples to the sample would have produced harder contacts with

more reliable temperature measurement. However, as mentioned earlier,

premature failure occurred at the welds.

Further details about the experimental development are given in

Appendix 6.5.

2.5 Comparison of Models at Room Temperature

The purpose of the room temperature testing was to compare three

theories based, on considerations of microstructural behavior. These

theories are those of Krieg, et al., Bodner, et al., and Miller, et al.

(references [18], [107], and [115], respectively, in Appendix 6.U).

-16-
Each theory is proposed in uniaxial form to simplify experimental and

analytical analysis. Since each theory is proposed for a different

material, a set of experiments is performed with a candidate material in

order to create a common experimental data base with which the

mathematical representation of each model can be characterized. These

theories are subsequently analyzed qualitatively and quantitatively over

a broad range of material behavior to determine their suitability for

predicting inelastic deformation.

The experimental data base requirements for each theory are as

follows. The theory of Bodner, et al., requires several constant strain

rate tensile tests and several creep tests. The theory of Miller,

et.al., requires several creep tests, several constant strain rate

tensile tests, and some information about the material such as the

melting point and the activation energy. The theory of Krieg, et al.,

requires a constant strain rate tensile test, a creep curve complete

with primary and secondary regions, and several stress-drop tests in

which the stress is reduced rapidly during steady-state creep in order

to examine the resulting transient strain response. In each case the

actual number of tests, unless specified, is dependent on the amount of

data deemed necessary for accurate evaluation of the material constants.

The candidate material chosen is the aluminum alloy 5086 in the form of

uniaxial bars, and the material tests are performed at room temperature.

In determining the material constants for each theory from these

tests, the procedure proposed by the authors is followed. It is found

that some of these procedures may be difficult to implement. In

particular, difficulties are encountered in interpreting the stress-drop

test, in which the quantity of interest is a zero strain rate or merely

-17-
the absence of a resolvable strain rate after unloading. It can be seen

that the results of this test are somewhat dependent-on-the resolution

of the experimental equipment. In addition, the values of several

constants in the theories are set arbitrarily, possibly leading to

inaccuracies in the ensuing predictions. Many procedures require

extensive data reduction and graphing, making an interactive data

acquisition and computer system an indispensable tool. Finally,

correction of the value of any one constant can be a difficult task, as

some of the constants of each theory may be interdependent.

Each constitutive theory is presented by highly nonlinear,

numerically "stiff" differential equations. Since the objective of this

research is not to determine numerically efficient integration

techniques for these equations, a stable first order forward integration

scheme was used.

Numerical simulations were performed to compare the predictive

capability of each theory to experimental results for constant strain

rate tensile tests, constant strain rate cyclic tests, and complex

strain rate history tests. As an example, Fig. 4 compares theory to

experiment for the first cycle of a constant strain rate cyclic test.

The experiment is modelled fairly well, but it can be seen that the

elastic-inelastic transitions are too sharp, possibly indicating a need

for more complex hardening laws. In addition, the Bauschinger effect

may not be modelled well by Bodner's theory and Miller's theory due to

the lack of a representation for the back stress. The discrepancy in

the width and height of the hysteresis loop predicted by Krieg's theory

may be due to the ambiguous measurements of the back stress in the

stress-drop tests, leading to possible inaccuracies in the material

constants. Results and conclusions of comparisons of other tests may be

found in detail in Appendix 6.5 of this report.

-18-
EJ KRIE& STRAIN RATE - V-1E-4. CYCLE I

© MILLER. STRAIN RATE - V-lE-4. CYCLE 1

A BODNER. STRAIN RATE ^^/-lE-4. CYCLE L

+ EXPt STRAIN RATE = V-1E-4. CYCLE I

-a 07 -0.05 -0.03 -aoi aoi

Strain (in/in) *10

0.05 0.07

Fig. 4. Comparison of Models to Experiment for IN 718 at 1100 F

-19-
2.6 Comparison of Models at Elevated Temperature

The theories used in this work include Bodner's anisotropic model [61;

Krieg, Swearengen, and Rohde's model [71; Schmidt and Miller's model [81; and

Walker's exponential model [61. These models have been chosen because they

are widely publicized, examples of constant calculation methods have been

presented, and the attempt is being made to extend them to non-isothermal

modelling.

Bodner's anisotropic model utilizes an inelastic strain rate equation

extended from classical plasticity theory. The inelastic strain is assumed to

be exponential in stress. A directional parameter has been included as an

incrementally isotropic variable. The growth laws are microphenomenologically

based and model hardening, dynamic recovery, and static thermal recovery.

Bodner's model is given below:

D — 0 exp [- - (-)2n] sgno

Z = 2l+ ZA = Z1 + B sgno l

*T T Z ~

 Z? r1

Z

J. — / "7 "7 -

1

 \ I I l\ "7 t *- \ •*• = m,(Z,- Z ) W - A, L,( j-)

l

v

 1 p 1 lv 2^

jf\\ it A -f t \ L. I \ £. _ — -^M

B = M,(Z, sgno-Z") W - A, Z, (-Ly-1)

 fc

 sgn Z" (6)

Z J l l Zj

Krieg, Swearengen, and Rhode's model is based on microphenomenological

considerations. The inelastic strain rate equation is based on a power law

formulation. The growth laws model hardening and static thermal recovery.

This model is given below:

(o-B)

2 A (B-B )2

B = A1 e1- A2(B-B0)^(e

 J ° -1) sgn(B)

-20-
D = A4|ir| - A5(D-DQ)

n (7)

Schmidt and Miller's model is a microphenomenologically based model which

uses a hyperbolic sine inelastic strain rate equation. The growth laws for

back stress and drag stress model hardening and static thermal recovery.

Parameters have been added to this model which account for strengthening due

to solutes. Schmidt and Miller's model is given below:

.1 ' I'8 1.5 n

sol

T ' i i n B = H, I - H,B sinh (AilB)) sgn(B)

D = H2 I^KC^ |B| - ^ D3) -H2 C2 B' sinh

Fh (Jin 2

sol ~ ' sol max

 CAHV~ B

Walker's model has an exponentially based inelastic strain rate

equation. The back stress growth law models hardening dynamic recovery, and

static thermal recovery. The dynamic recovery term has been modified to

handle negative strain rate sensitivity. The drag stress variable allows for

cyclic hardening or softening. Walker's model is presented below:

.1 _ exp (o-B) -1. e =

Lsgn(o-B)

B = n2 er-B [n3+ n4 exp(-n5|log(—) | ] R

Ro

D = Dj+ D2 exp (-n7R)

R= |e!| (9)

-21-
2.6.1 Material Response

The material used in this work is Inconel 718 at 1100° F. All samples

have been sent through a common heat treatment prior to being tested. This

material system has several distinctive characteristics. The material

responds with negative strain rate sensitivity. Discontinuous yielding or the

Portevin-Le Chatelier effect is observed. The material cyclically worksoftens and has a fatigue life of 5 to 30 cycles when cycled at strain limits

over ± 1 percent strain. The yield point of the material is in the region of

95 to 110 ksi.

2.6.2 Experimental Considerations

The material samples were configured to meet ASTM standard E606-77T. The

tests were carried out on a computer-controlled, MTS 880, 110 kip, electrohydraulic testing machine. A Digital Micro PDP-11 computer controlled the

tests and acquired the data. An MTS 652 furnace system with three-zone

clamshell furnace and hydraulic grips was utilized. An MTS 632-41-02 hightemperature axial extensometer was utilized for displacement measurement.

Temperature was monitored with K-type thermocouples and a Fluke 2176A digital

thermometer. Welding of the thermocouples to the samples initiated failure

and a self-supporting, surface contact method was used for thermocouple

attachment.

The test program consisted of the following tests:

(a) 2 monotonic tension tests to 1.556 strain

([1] 3.15 xlO"3sec"1I2] and 7.25xlO~6sec~1)

(b) 5 fully reversed cyclic tests to ± .8% strain

(strain rates between 1x10 sec" and 7x10 sec" )

-22-
(c) 5 constant load creep tests

(applied stresses between 120 ksi & 140 ksi)

(d) Back stress measuring tests during cyclic loading and during secondary

creep

(e) a complex history test

2.6.3 Determination of Parameters

Beek [4] has shown that the parameter calculation process presented with

each model can produce constants which may produce undesirable model

response. A method for determining material parameters which lessened this

problem and provided some commonality in the calculations was used. A set of

initial assumptions was made based on the expected response of the material.

These initial assumptions included the following:

(1) Back stress was responsible for hardening in monotonic tension;

(2) Drag stress was responsible for cyclic softening;

(3) Thermal recovery could be neglected for rapid tests

(4) Drag stress thermal recovery could be calculated from slow cyclic tests;

and

(5) Back stress thermal recovery could be calculated from creep tests.

Hand calculations based on these assumptions provided initial estimates

of the material parameters. Computer-aided iterations were then used to tune

the response to match specific data points. The parameters which resulted

from the iteration process indicated that the following initial assumptions

would have been more appropriate:

(1) the inelastic strain rate equation constant could be set to provide the

proper scaling and strain rate sensitivity;

-23-
(2) back stress hardening produced monotonic hardening;

(3) solution strengthening parameters produced negative strain rate

sensitivity;

(4) drag stress hardening could be used to set cyclic work softening

characteristics; and

(5) thermal recovery effects were small and masked by solute strengthening.

A comparison of the response of the solute strengthening corrected models

to experiment is shown in Fig. 5. These results provide several conclusions

concerning various parts of the models. The theories of Walker and Bodner

with exponentially based inelastic strain rate equations handle the negative

strain rate sensitivity most effectively. The theories of Krieg, et al.,

Miller, and Bodner produce oversquare stress-strain curves.

-24-
ORIGINAL

OF POOR

-Illl •• C*»\f« !•**• • l«l*l

~MM* *M«k •!»• >M» ie**tl1IW IQI*t|l •Illoc*

•Illl M COVill l^wt •1110**

liO.O

171.0

100.0

f>

u

u

•^

OT

-.013 -.o:s -.oo» s.o .co» .oc«

Strain (in/in)

.C'J .CI S

-SO.i

-100.

-ISO.

•.0<2 '.COB -.004 0.0 .004 .OCB

Strain (in/in)

.812 .31t

-loi n

-ItlLlftl

1*0.0

I JVC

100.0

»0.0

? »»•«

n o.o

n

u

-5C.O

-J5.0

-ICO.:

-i:5.

-.31? -.C:B -.cc4 co .CD* .oce .ci:

Strai n (in/in )

.ocs -.03* o.o .po« .;os

Strain (in/in)

Figure 5. Comparison of Models to a Complex History Test

25
The addition of dynamic recovery terms to the directional growth law

appears to help the strain rate sensitivity problem. Bodner's corrected model

shows less response to strain rate jumps. This is due to hardening based on

plastic work rate. Walker's correction for solute strengthening comes closest

to reproducing the strain rate sensitivity. The correction applied to

Bodner's model produces unstable response and a correction similar to Walker's

should be considered. The correction factors negated the effects of thermal

recovery during hold times for Bodner's and Krieg, et al's. models. The drag

stress growth law of Walker provided the closet fit to data over several

cycles at the higher strain rates. Lower strain rates showed Bodner's model

handling cyclic response most effectively. Walker's model suffers from the

lack of drag stress thermal recovery at the lowest strain rate.

These results show that the combination of exponential strain rate

equations with dynamic recovery work best for this material system. Isotropic

thermal recovery and exponential growth of the isotropic internal state

variable are useful for modelling cyclic response.

Further details of this portion of the research are given in Appendix

6.5.

2.7 Integrated Software Development

Because of the vast number of constitutive theories available, only four

candidate models were selected for this research. These include the work of

Walker[6,9], Krieg, Swearengen, and Rohde[7], Bodner[6], and Schmidt and

Mi Her[10]. These models were chosen primarily because they appear to be the

most qualitatively attractive theories available and thus warrant further

evaluation. For this paper, only the procedure developed for Walker's model

will be presented.

-26-
The model proposed by Walker is a viscoplastic theory which uses an

exponential type inelastic strain rate relation. The growth law modeling back

stress is of the hardening/recovery format and accounts for both dynamic and

static thermal recovery. The drag stress term models isotropic hardening,

thus taking into account the cyclic hardening or softening characteristics of

a material. The uniaxial differential form of Walker's exponential model may

be written as:

exp(-2jj9-) - 1

e = 1 sgn(0 - B) . (10)

B = n^1 - B(ln3 + n, exp(-njlog(^-)] R + n6} , (11)

D = D, - D2 exp(-n7R) ,

 Ro (12)

R - |e!| . (13)

where a is the applied stress, e is the inelastic strain, B is the back

stress, and D is the drag stress. A superposed dot above the variables

denotes differentiation with respect to time. In addition, e , n2, n^, n4,

n5« n6» n7* ^2 anc* ^2 are material parameters. Therefore, this model requires

nine constants to be evaluated, along with selecting Young's Modulus E and the

strain aging parameter RQ .

The tests required to determine the constants for Walker's model using

the following procedure include: 1) A series of constant strain rate steady

state hysteresis loops under fully reversed strain controlled conditions; 2)

cyclic hold tests performed on the unloading branch of the cyclic tests; and

3) long term monotonic tension tests. The monotonic tension tests may not be

necessary if acceptable values of the limiting stress o^. can be obtained

from the first half cycle of the cyclic tests.

-27-
The determination of the constants begins by plotting o-j. versus

In (e ) . A nonlinear representation signifies that strain aging and/or

thermal recovery effects are present and thus need to be modeled. If o,. is

not obtained experimentally, it can be estimated in a manner similar to that

proposed by Lindholm, et al.[6].

Under conditions of uniaxial tension loading, when e is assumed to be a

constant and equal to the applied strain rate, eq. (10) may be written as:

o = D In (se1) + B . (14)

Using the evolution equation defining the back stress (eq. (11)) and on the

physical basis that D remains constant during monotonic loading,

then do/de (or 0) may be written as:

R I

o = n2 - e[n3 + n4exp(-njlog (-^-)|) + n6/e ] . (15)

Thus, equations (14) and (15), can be combined yielding

0 = -No + [n2 + NDln(ee )] , (16)

where

N = n3 + n, exp(-n5|log(-^)|) + n^e1 . (17)

R o

Therefore, equation (16) indicates that a plot of 0 versus o will be linear,

having a slope of N and an x-intercept of o^ .

The constant n5 is computed by determining where the effect of strain

aging is considered negligible and may be written as:

-28-
n5 = -[ln(t)/|log (-7-))] . (18)

The constants R0 and Rt represent the strain rate at which the strain aging

correction is a maximum and minimum, respectively, and T denotes the residual

correction at rate Rj . It should be noted that T also affects the rate of

decay of the strain aging correction and selection of too small of a value

will result in a very localized correction.

The next step in this procedure is to compute the dynamic and static

thermal recovery constants n^, n^, and ng using equation (17). Equation (17)

can be rewritten a number of times, corresponding to the different monotonic

tests as:

N. = n3 + nl)fi + n6/ej , (19)

where

f. = exp(-njlog (-7-)]) . (20)

R o

Thus, the three parameters, n3, n4, and ng, may be obtained simultaneously

using a multiple linear regression scheme.

The constant ^ is computed on the basis that B saturates to B^m at

large inelastic strains. Hence, B=0 and equation (11) reduces to

n2

n3

If it is assumed that the ratio o /B will remain the same for the

limiting condition at sufficiently large inelastic strains, then

°exp _ °lim

exp lim

-29-
Substituting eq. (21) into (22) and solving for n2 results in

n - 2

exp

where a,. values are obtained from long term monotonic tension tests

or e-plots and o and B values come from cyclic hold tests.

The initial value drag stress DQ and the inelastic strain rate sealer B

are determined by rewriting equation (14) using the limiting values of o and

B as:

-

 Blim = W') + Do1n<6> - (24)

Since Blim is given by equation (21), it can be substituted into (24)

resulting in

= D0ln(e1) + D0ln(e) , (25)

where

"2

6~R ~ °iim TT~ • (26)

Equation (25) indicates that a plot of 6 B versus ln(e ) should be linear,

having a slope of 0Q and an intercept of D ln(e) from which B can be found

directly.

Up to this point, the only tests that were needed, in order to compute

the material parameters, have been monotonic tension and cyclic hold tests.

To obtain the isotropic hardening and recovery constants D^, $2»

 anc* nl»

saturated cyclic hysteresis data are required. By estimating the cumulative

-30-
inelastic strain from applied stress, E, and strain amplitude and assuming

that D saturates to Dj, then r\-j can be approximated as:

"7 = -ln(0/Ravg . (27)

where Rav_ is the average of R for a number of tests and T is an approximation

of zero.

On the physical basis that B saturates much more rapidly than D, equation

(14) can be written as:

(28)

where

Blim = n2/N ' (29)

The values of o-,. and N in equations (28) and (29) are estimated from the

0-plot after cyclic saturation has occurred. If several tests are used, Q±

would be computed as the arithmetic mean.

The final parameter to determine is D2« When R=0 equation (12) reduces

to

D0 = D, - D2 , (30)

from which D£ may be computed directly, completing the constant calculation

procedure.

The computer algorithm used to compute the material parameter proceeds in

the following manner:

1) Values of o,. and N from monotonic tension tests are computed using

-31-
equation (16) and a least squares procedure.

2) After selecting R0, R, and t, n5 is evaluated using equation (18).

3) The parameters n^, n^, ng are determined from equations (19) and (20)

using a multiple linear regression scheme.

4) The back stress hardening coefficient n2 is computed directly by

equation (23).

5) The constants DQ and 6 are evaluated using equations (25) and (26)

via a least squares procedure.

6) After computing the cumulative inelastic strain from the experimental

data, n7 is evaluated directly from equation (27).

7) DI and D2 are computed using equation (28) and (30).

A method for obtaining the material parameters for Walker's model has been

presented which is a synthesis of both physical and numerical

approximations. The associated computer algorithm allows the user to use

either a totally automated procedure or engineering intuition at selected

points when computing constants. Numerical simulations of Inconel 718 at

1100°F are to be presented.

2.8 Conclusion

Several general conclusions can be made from the current research. These

are as follows:

1) Euler's method is the most efficient for integrating the models of

those methods considered herein;

2) Walker's model is the most economical to integrate, while Miller's

model is least economical and also tends to be unstable;

3) ISV theory can be used as general framework for comparing all of the

models in the current literataure;

4) a laboratory has been developed at Texas A&M which is capable of

performing all of the complicated elevated temperature experiments necessary

to perform the research detailed herein;

5) for aluminum at room temperature, none of the models considered

herein appears to be very accurate in predicting uniaxial cyclic response;

6) at 1100°F Walker's model appears to be most accurate for predicting

the uniaxial cyclic response of IN718.

-32-
2.9 References

1. Coleman, B.D., and Gurtin, M.E., "Thermodynamics with Internal State

Variables," Journal of Chemical Physics, Vol. 47, pp. 597-613, 1967.

2. Coleraan, B.D., and Noll, W., "The Thermodynamics of Elastic Materials

with Heat Conduction and Viscosity," Archive for Rational Mechanics and

Analysis, Vol. 13, p. 167, 1963.

3. Garofalo, F., Richmond, 0. and Damis, W.F., "Design of Apparatus for

Constant-Stress or Constant-Load Creep Tests," Journal of Basic

Engineering, 84, 285 (1982).

4. Jones, W.B., Rhode, R.W. and Swearengen, J.C., "Deformation Modelling

and the Strain Transient Dip Test," Mechanical Testing for Deformation

Model Development, STP 785, ASTM, 102-118 (1982).

5. ASTM Thermocouple Handbook, STP 470A, ASTM, (1974).

6. Lindholm, U.S., Chan, K.S., Bodner, S.R., Weber, R.M., Walker, K.P.,

and Cassenti, B.N., "Constitutive Modeling for IsotropicMaterials,"CR174980, NASA (HOST), (1985).

7. Krieg, R.D., Swearengen, J.C. and Rhode, R.W., "A Physically-based

internal Variable Model for Rate Dependent Plasticity," Proc. ASME/CSSME

PVP Conference, 15-27, (1978).

8. Schmidt, C.G., A Unified Phenomenological Model for Solute Hardening,

Strain Hardening, and Their Interactions in Type 316 Stainless Steel, Ph.D.

Dissertation, Stanford University, Department of Materials Science and

Engineering (1979).

9. Walker, K.P. and Wilson, D.A., "Creep Crack Growth Predictions in INCO

718 Using a Continuum Damage Model," 2nd Symposium on Nonlinear

Constitutive Relations for High-Temperature Applications, Cleveland, OH,

1984.

10. Schmidt, C.G., A Unified Phenomenological Model for Solute Hardening,

Strain Hardening, and Their Interactions in Type 316 Stainless Steel, Ph.D.

Dissertation, Stanford University, Dept. of Materials Science and

Engineering, 1979.

-33-
3. PUBLICATIONS LIST

3.1 Papers Published

1. Haisler, W.E., and Irabrie, P.K., "Numerical Considerations in the

Development and Implementation of Constitutive Models," Proceedings Second

Symposium on Nonlinear Constitutive Relations for High Temperature

Applications, pp. 169-186, June, 198*1 (Appendix 6.1).

2. Allen, D.H., and Beek, J.M., "On the Use of Internal State Variables in

Thermoviscoplastic Constitutive Equations," Proceedings Second Symposium on

Nonlinear Constitutive Relations for High Temperature Applications, pp. 83~

102, June, 1981 (Appendix 6.3).3.2. Papers Submitted for Publication

3.2 Papers Accepted for Publication

1. G.H. James, P.K. Imbrie, P.S. Hill, D.H., Allen, and W.E. Haisler, "An

Experimental Comparison of Several Current Viscoplastic Constitutive Models

at Elevated Temperature," Proceedings Third Symposium on Nonlinear

Constitutive Relations for High Temperature Applications, June, 1986

(Appendix 6.5 in part).

2. P.S. Hill, G.H. James, P.K. Imbrie, W.E. Haisler, and D.H. Allen, "An

Automated Procedure for Material Parameter Evaluation and Uncertainty

Analysis for Viscoplastic Constitutive Models," Proceedings Their Symposium

on Nonlinear Constitutive Relations for High Temperature Applications,

June, 1986.

3.3 Papers to be Submitted for Publication

1. Beek, J.M., and Allen, D.H., "A Comparison of Current Models for

Nonlinear Rate-Dependent Material Behavior of Crystalline Solids," to be

submitted to Journal Engineering Materials Technology (Appendix 6.4).

2. Imbrie, P.K., Haisler, W.E., and Allen, D.H., "Evaluation of the

Numerical Stability and Sensitivity to Material Parameter Variations for

Several Unified Constitutive Models," to be submitted to Journal

Engineering Materials Technology (Appendix 6.2).

1). PROFESSIONAL PERSONNEL INFORMATION

^4.1 Faculty Research Assignments

1. W.E. Haisler - Numerical code development, computer software/ hardware

development, overall program coordination.

2. D.H. Allen - experimental equipment development, experimental testing

program, constitutive model development.

-34-
4.2 Student Research Assignments

1. J.M. Beek - room temperature testing, model comparisons at room

temperature.

2. P.K. Imbrie - numerical code development, computer software/

hardware development.

3. G.H. James - experimental equipment development, elevated tempera-ture

testing.

4. P.S. Hill - evaluation of material constants, computer software

development.

5. C. Wong - elevated temperature testing.

5. INTERACTIONS

5.1 Papers Presented

The following papers have been presented:

1. W.E. Haisler and P.K. Imbrie, "Numerical Considerations in the

Development and Implementation of Constitutive Models," Second Symposium on

Nonlinear Constitutive Relations for High Temperature Applications, June,

1984.

2. D.H. Allen and J.M. Beek, "On the Use of Internal State Variables in

Thermoviscoplastic Constitutive Equations," Second Symposium on Nonlinear

Constitutive Relations for High Temperature Applications, June, 1984.

The following papers are to be presented:

1. G.H. James, P.K. Imbrie, P.S. Hill, D.H. Allen, and W.E. Haisler, "An

Experimental Comparison of Several Current Viscoplastic Constitutive Models

at Elevated Temperature," Third Symposium on Nonlinear Constitutive

Relations for High Temperature Applications, June, 1986.

2. P.S. Hill, G.H. James, P.K. Imbrie, W.E. Haisler, and D.H. Allen, "An

Automated Procedure for Material Parameter Evaluation and Uncertainty

Analysis for Viscoplastic Constitutive Models," Third Symposium on

Nonlinear Constitutive Relations for High Temperature Applications, June,

1986.

5.2 Related Travel and Consultation

1 . Drs. Haisler and Allen are consultants with the General Electric

Aircraft Engine Business Group on NASA LeRC contract nos. NAS3-23698 and

NAS3-24538.

2. Dr. Allen and several of his students have visited Drs. U.S. Lindholm,

S. Bodner, and K. Chan on three occasions during the course of the contract

at Southwest Research Institute, San Antonio.

-35-
APPENDIX

-36-
APPENDIX 6.1
N86- 30229 ii*

NUMERICAL CONSIDERATIONS IN THE- DEVELOPMENT n

AND IMPLEMENTATION OF CONSTITUTIVE MODELS

W.E. Haisler and P.K. Imbrie /erzT X ///

Aerospace Engineering Department /VoO "5 /5» TV

Texas A&M University

College Station, Texas 77843

Several unified constitutive models were tested in uniaxial form by

specifying input strain histories and comparing output stress histories.

The purpose of the tests was to evaluate several time integration methods

with regard to accuracy, stability, and computational economy. The sensitivity of the models to slight changes in input constants was also in-

"vestigated. Results are presented for IN100 at 1350°F and Hastelloy-X at

1800°F.

INTRODUCTION

The characterization of the constitutive behaviour of metals has its

roots in the early work of Tresca, Levy, vonMises, Hencky, Prandtl, Reuss,

k Prager, and Ziegler (Refs. 1-8). These early models are incremental in

1'* nature, assume that plasticity and creep can be separated, and they incorporate a yield function, flow rule, and hardening rule to define the plastic

strain increment. These original incremental theories have been expanded

and modified by many researchers so that they provide adequate, and often

very good predictions of rate-independent plastic flow (see for example

Refs. 9-10). However, they are sometimes criticized as having no formal

micromechanical basis upon which to make the assumption of an uncoupling 4,

of the inelastic strain into rate-independent (plastic) and rate-dependent

(creep) strain components. Nevertheless, the classical incremental theories

are widely used.

During the last ten years, a number of unified constiutive models have

been proposed which retain the inelastic strain as a unified quantity without aritifical separation into plasticity and creep components. These include the models developed by Bodner (Refs. 11-13), Stouffer (Refs. 14-15),

Krieg (Ref. 16), Miller (Ref. 17), Walker (Refs. 18-19), Valanis (Refs. 20-

21), Krempl (Ref. 22), Cernocky (Ref. 23-24), Hart (Ref. 25), Chaboche (Ref.

26), Robinson (Ref. 27), Kocks (Ref. 28), and Cescotto and Leckie (Ref. 29).

The applicability of these viscoplastic constitutive theories (mostly to

high temperature applications) has been investigated by several researchers.

Walker (Ref. 19) compared the predictive capability of several models (Walker,

Miller and Krieg) for Hastelloy-X at 18008F. More recently, Milly and Allen

169
(Ref. 30) provided a qualitative as well as quantitative comparison of the

models developed by Bodner, Krieg, Walker and Krempl for IN100. Both Refs.

19 and 30 conclude that these models generally provide adequate results

for elevated isothermal conditions, they provide poor and overly-square

results at low temperature, the material constants are often difficult to

obtain experimentally, the resulting rate equations are "stiff" and sensitive to numerical integration, and the models do not provide any satisfactory transient temperature capability. Beek, Allen, .and Milly (Ref. 31)

have shown that all the unified viscoplastic models mentioned above can be

cast.in a functionally similar form (in terms of internal state variables).

None of the published literature provides a thorough evaluation of current viscoplastic constitutive models with comparison to experimental response for complex input histories. Such an evaluation is difficult at present for many reasons, namely: 1) Material constants' for most models are

usually available only for a single material and often for a single temperature; 2) The experimental procedures given by model developers for determining material constants from experimental data are often sketchy at best;

3) Material constants for some models are often obtained by trial-and-error

and are not based on experiments; and 4) There is a lack of good experimental

data against which the models can be evaluated (that is, test data which is

significantly different from that used to generate the material constants).

The purpose of the present paper is to report some preliminary evaluations of several of the unified viscoplastic models (Bodner, Krieg, Miller,

and Walker). These four models are evaluated with regard to 1) their sensitivity to numerical integration and 2) their sensitivity to slight changes

in input material constants.

CONSTITUTIVE MODELS CONSIDERED

The constitutive theories which have been studied to date include Bodner's

(Refs. 11-15), Krieg's (Ref. 16), Miller's (Ref. 17), and Walker's (Refs.

18-19). These particular models were selected for this initial study because material constants for Hastelloy-X were available for three of the

models. Other models are currently being considered as material constants

become available. Each model is listed below in uniaxial form using a consistent notation as presented by Beek, Allen and Milly (Ref. 31). In Ref.

31, it is shown that all of the current viscoplasti; r.cdals considered r.s.;-

be written in uniaxial form as

o - E(e - Oj - £T) (1)

where <j is stress, E is Young's modulus, e is strain, c^ is the inelastic

strain (internal state variable), and e* is the thermal strain. Each viscoplastic theory postulates a particular growth law for the internal state

variable(s) and the inelastic strain is obtained by time integration of

the growth law for a,, i.e.

170
l» f (2)

where

•

 dal •

ai

 = dT " at

(£'

 T*

 a2*

In equations (2) and (3), c is time, T is temperature, 0.2 is the back stress

(related to the dislocation arrangement and produces kinematic hardening

or the Bauschinger effect), and a^ is the drag stress (which represents

the dislocation density and produces isotropic hardening).

Bodner's Theory

The growth law for the inelastic strain in Bodner's model may be written in uniaxial form as

where

(5)

(6)

The quantities E, DQ, n, m, Zj, A, Zj and r are material constants. As

noted before, the variable ct-j is similar to the drag stress used in

many models (a measure of isotropic hardening or dislocation density).

It is noted that the model contains no parameter representing the back

stress and cannot account for the Bauschinger effect in kinematic hardening materials. The material constants are tabulated for IN100 at 1350eF

(732°C) in Table 1 (taken from Ref. 14).

171
Krieg's Theory

The inelastic strain growth law for the model developed by Krieg and

coworkers may be written in terms of state variables representing back

stress and drag stress:

a - a.

a.

sgn(a - a2> 17)

a [exp(C5 c ) - 1] sgn (a2) (8)

*

 C7(0t3 - a3

(9)

The model contains ten constants (C,, C- . . . , C_, E, a, , and n).

These have been evaluated by Walker (Ref. 19) for Hastelloy-X at 1800°F

(982°C) and are tabulated in Table 2. It should be noted that equations

(7), (8) and (9) form a coupled set of ordinary differential equations.

Miller's Theory

The growth laws for Miller's model may be written in uniaxial form as

B81 sinh

O - cu

ot.

.1.5

sgn(C - (10)

1 [sinh (11)

1 sinhU^j (12)

Miller's theory contains nine constants which are tabulated for HastelloyX at 1800°F (982°C) in Table 3 (see Ref. 19).

172
Walker's Theory

Walker's nonlinear viscoplastic theory can be cast in the following

uniaxial form

'[a - oJ

al " I—aT~^ j sgn(a

 " V (13)

(14)

n

8 1*11 " n9 I"l'a3 " nlO

 (a3 " a3

o

where R is the cumulative inelastic strain

3d,

R =

o

C VVf «

dtf (16)

The general model requires sixteen constants (E, n, m, q, nj, n2> . . . ,

n ? a. and a~(t=0). In determining the constants for Hastelloy-X at 1800°F

1 V ^Q J

(982°C), Walker made several simplifying assumptions [including 03 = constant «• a3(t=0)] which reduces the number of parameters to those shown in Table 4

(see Ref. 19). Further, the constants reported in Ref. 19 were developed from tests

using strain rates in the range 10~3 to 10~6 sec"1 and strain ranges of ±0.6%.

NUMERICAL TIME INTEGRATION STUDY

The integration of the constitutive relationship given by equations

(1). (2) and (3) forms an integral and extremely important part in any numerical solution of a nonlinear field problem. It has been observed by

many researchers that the coupled system of ordinary differential equations

defining the state variables may be locally "stiff" and thus are sensitive

to the time step size and numerical algorithm. The accurate integration

of these stiff equations can be accomplished by various means: use of small

time steps, higher-order or multi-point integration schemes, subincrementation

173
procedures (Refs. 33-35), "smart" algorithms which attempt to select appropriate time steps in order to achieve accuracy and stability (Refs. 36,37),

algorithms tailored for individual constitutive theories (Refs. 32,37), or

combinations of these approaches. In general, the computation time required

for the accurate solution of materially nonlinear problems is directly related to the numerical integration scheme used.

Regarding the constitutive models reviewed herein, Walker (Ref. 32)

uses a stable, iterative implicit scheme which takes advantage of the functional 'form of the integrand in the development of the reccurence relation.

Miller originally used Gear's method (Ref. 36) to integrate the stiff equations in his theory but later concluded in Ref. 37 that an implicit backward difference method was more economical and preferable to either Gear's

method or the explicit Euler forward integration method. The type of numerical integration scheme used by Bodner and Krieg is not known.

'The selection of an appropriate time integration scheme to be used in

a computer code is very important but is often based on the answers to such

questions as: "What is available in the present code?", "What will work

most of the time?", "What can we use that most users will understand?",

"What is the cheapest and easiest to use?", and the like. The usual response given is "it depends on the problem being solved!"

In general, equation (3) may be integrated between time t and t + At

by writing

/

t+At

I •

(17)

or

t+At

+ At) - at) =• d dt (18)

I

where a^ is defined by the particular constitutive theory being used. The

present investigation considers four integration schemes: explicit Euler

forward integration, implicit trapezoidal method, trapezoidal predictorcorrector (iterative) method, and Runge-Kutta 4th order method. The approximations for each of these methods is given in Table 5.

Each of the integration schemes in Table 5 were used to obtain stresstime and stress-strain responses for the four constitutive models considered

herein when subjected to the uniaxial, alternating square-wave strain-rate

history shown in Fig. 1. Figure 1 shows the 35 second response obtained

174
by Krieg's theory for Hastelloy-X at 1800°F using a time seep of 0.1 seconds. For this time step, the Euler and trapezoidal predictor-corrector

methods provide essentially the same results and are virtually identical

to that obtained for all methods using a time step of 0.005 seconds. The

4th order Runge-Kutta method generally overestimates the peak response while

the trapezoidal method underestimates the response. Figure 2 presents results for three integration methods such that the total computation time

for a 35 second response solution is approximately the same. For equivalent computation times, the Euler method provides the most accurate results

although smaller time steps are required. Similar results are observed for

Miller's model.

Figures 3 and 4 illustrate that various constitutive models may behave

appreciably different using the same integration method (in this case the

Euler method). In Fig. 3, Miller's theory (for Hastelloy-X at 1800°F) gives

considerable oscillatory response for a time step of 0.005 seconds while

Walker's theory shown in Fig. 4 gives a much smoother response for the same

time step. Comparing Figs. 3 and 4, it is seen that a smaller time step

is required (with Euler integration) in Miller's theory than in Walker's

theory.

Figure 5 presents results for IN100 at 1350°F usinj Bodner's model.

Time steps were chosen for each integration scheme to obtain solutions which

required approximately equal computation times. These results, when compared to solutions with much smaller time steps, indicate that the Euler

method provides the most accurate results. Again, the time step used is

smaller than that for the other methods but the computation time is the

same (for integrating the constitutive equations).

SENSITIVITY STUDY FOR MATERIAL CONSTANTS

In the previous section, results were presented which showed how the

numerical integration method used to integrate the constitutive equations

could affect the accuracy and computation times of predicted results for

stress-time and stress-strain responses. In this section, we consider

another important parameter in the application of any constitutive theory.

Namely, "how does the accuracy to which material constants are determined

from experimental test data affect the predicted response?"

Figures 6 and 7 present results for Walker's model (Hastelloy-X at

1800°F subjected to an alternating square-wave strain-rate history as shown)

wherein specified input material constants have been adjusted by 5%. Figure 6 shows the effect of a -5% change (error) in the stress exponent n

(the most sensitive parameter). Figure 7 shows that a 4-5% error in all test

data required to compute material constants results in significant predicted

response errors, up to 30% over-prediction in the stress at a time of 35

seconds (during the relaxation period).

Figures 8 and 9 present similar results for Krieg's model (Hastelloy-X

at 1800°F) and Bodner's model (IN100 at 13508F), respectively. Both results

indicate that the most sensitive parameter is the stress exponent "n" and

175
that a 5% error in specifying n may produce significant errors in the predicted response. Miller's model appears to be much less sensitive to errors in input material parameters.

Figure 10 provides a comparison of the Miller, Krieg, and Walker models

for the Hastelloy-X test at 1800°F (using constants obtained by Walker for

all models). The Euler method was used with a time step of 0.0005 seconds

which provides a solution with no significant truncation error. The results

obtained here show approximately 10-15% differences in peak stress amplitudes between the three constitutive models. Since no experimental results

are available at this time, no conclusions can be drawn as to which model

more accurately represents observed test data. However, the results do

point out that significant differences (greater than 15%) can be obtained

for stress peaks and stress relation values through the use of different

constitutive models.

CONCLUSIONS AND FUTURE WORK

The results of this study are not complete since only a portion of the

available constitutive models and numerical integration schemes have been

considered. However, some tentative conclusions can be reached. First,

it appears clear from the present investigation, and the work of others,

that simple integration schemes (like the Euler forward defference method)

are often preferable to more complex schemes from the standpoint of accuracy, computation time, and ease of implementation. Although not reported

herein, our work in progress indicates that Euler's method used with a simple

subincrementation strategy provides the most accurate and economical solution for most constitutive models.

The sensitivity study on material constants indicates that most viscoplastic constitutive models are significantly sensitive to one or more material constants derived from laboratory tests. It has been shown that a

5% "error" in laboratory measurements may lead to errors of 25%, or greater,

in predicted stress responses. Although most model developers have finetuned their models and input material constants for specific material/temperature/strain-rate combinations, it is not clear that end-users will be

able to do so when called upon to develop material constants for a new situation. The problem can be negated to some extent by defining more explicit

tescing procedures for obtaining material constants and by guidelines defining which constants are most sensitive to experimental error.

Our current and future work concerns the application of several integration schemes to the other constitutive theories, investigation of subincremental strategies, and consideration of "smart" integration methods

which detect local "stiffness" and adjust time steps but without significant computational expense. The material parameter sensitivity study will

be continued by considering other constitutive theories, and more importantly, by comparison with laboratory tests which involve complex therraomechanical loadings including transient temperature inputs.

176
ACKNOWLEDGEMENT

The authors gratefully acknowledge the financial support for this research by NASA Lewis Research Center under Grant no. NAG3-491.

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Comptes Rendus del' Academic des Sciences. Vol. 59, p. 754, 1864.

2. Levy, M., "Memoire sur les equations generales des mouvemencs interieurs des corps solides ductiles au dela des limites ou 1'elasticite

pourrait les ramener a leur premier eata.," C.R. Acad. Sci. (Paris),

Vol. 70, pp. 1323-1325, 1870.

3. von Mises, R., "Mechanik der Festen Koerper im plastisch deformablen

Zustant," Goettinger Nachr., Math.-Phys., Kl, pp. 582-592, 1913.

4. Hencky, H., "Zur Theorie Plastischer Deformationen und die Hierduch

la Material Hervorgerufenen Nach-Spannungen," Z. ang. Math. Mech.,

Vol. 4, pp. 323-334, 1924.

5. Prandtl, L., "Spannungsverteilurg in Plastischen Koerpen," Proceedings of the First International Congress on Applied Mechanics, Delft,

Technische Boekhandel en Druckerin, Jr. Waltman, Jr., pp. 43-45, 1925.

6. Reuss, E., "Bereuchkisichtigung der Elastischen Formaenderungen in

der Plastizitaetstheorie," Zeitschrift fuer Angewandte Mathematic und

Mechanik. Vol. 10, pp. 266-274, 1930.

7. Prager, W., "The Theory of Plasticity: A Survey of Recent Achievements,"

Proceedings of the Institution of Mechanical Engineers, London, Vol.

169, pp. 41-57, 1955.

8. Ziegler, H., "A Modification of Prager's Hardening Rule," Quarterly

of Applied Mathematics, Vol. XVIII, pp. 55-65, 1959.

9. Allen, D.H. and Haisler, W.E., "A Theory for Analysis of Thermoplastic

Materials," Computers and Structures, Vol. 13, pp. 125-135, 1981.

10. Haisler, W.E. and Cronenworth, J., "An Uncoupled Viscoplastic Constitutive Model for Metlas at Elevated Temperature," Proceedings of the

24th AIAA Structures, Structural Dynamics and Materials Conference,

Lake Tahoe, Nevada, May 2-4, 1983.

11. Bodner, S.R. and Partorn, Y., "Constitutive Equations for ElasticViscoplastic Strain-Hardening Materials," Journal of Applied Mechanics,

Vol. 42, No. 2, pp. 385-389, 1975.

12. Merzer, A. and Bodner, S.R., "Analytical Formulation of a Rate and

Temperature Dependent Stress-Strain Relation," Journal of Engineering

Materials and Technology. Vol. 101, pp. 254-257, 1979.

13. Bodner, S.R., "Representation of Time Dependent Mechanical Behaviour

of Rene 95 by Constitutive Equations," Air Force Materials Laboratory,

AFML-TR-4116, 1979.

14. Stouffer, D.C., "A Constitutive Representation for IN100," AFWAL-TR81-4039, Air Force Materials Laboratory, June 1981.

15. Stouffer, D.C. and Bodner, S.R., "A Relationship Between Theory and

Experiment for a State Variable Constitutive Equation," ASTM Special

Technical Publication 785. 1982 (also AFWAL-TR-80-4194, 1981).

177
16. Krieg, R.D., Swearengen, J.C., and Rohde, R.W., "A Physically-Based

Internal Variable Model for Rate-Dependent Plasticity," Proceedings

ASME/CSME/PVP Conference, pp. 15-27, 1978.

17. Miller, A.K., "An Inelastic Constitutive Model for Monotonic, Cyclic,

and Creep Deformation: Part I—Equations Development and Analytical

Procedures" and "Pare II—Application to Type 304 Stainless Steel,"

ASME Journal of Engineering Materials and Technology, pp. 97-113,

April 1976.

18. Walker, K.P., "Representation of Hastelloy-X Behavior at Elevated

Temperature with a Functional Theory of Viscoplasticity," Presented

at the ASME Pressure Vessels Conference, San Francisco, 1980, also

ASME Journal of Engineering Materials and Technology, 1981.

19. Walker, K.P., "Research and Development Program for Nonlinear Structural Modeling with Advanced Time-Temperature Dependent Constitutive

Relationships," PWA-5700-50, United Technologies Research Center,

(also NASA CR-165533), 1981.

20. Valanis, K.C., "A Theory of Viscoplasticity Without a Yield Surface

- Part I. General Theory," and "Part II. Application to Mechanical

Behavior of Metlas," Archives of Mechanics, Vol. 23, pp. 517-533,

1971.

21. Valanis, K.D., "On the Foundations of the Endochronic Theory of Viscoplasticity," Archives of Mechanics, Vol. 27, pp. 857-868.

22. Krempl, E., "On the Interaction of Rate and History Dependence in Structural Materials," Acta Mechanica, Vol. 22, pp. 53-90, 1975.

23. Cernocky, E.P. and Krempl, E., "A Nonlinear Uniaxial Integral Constitutive Equation Incorporating Rate Effects, Creep and Relaxation,"

International Journal of Nonlinear Mechanics, Vol. 14, pp. 183-205,

1979.

24. Cernocky, E.P. and Krempl, E., "A Theory of Thermoviscoplasticity

Based on Infinitesimal Total Strain," International Journal for Solids

and Structures, Vol. 16, pp. 723-741, 1980.

25. Hart, E.W., "Constitutive Relations for the Nonelastic Deformation

of Metals," ASME Journal of Engineering Materials and Technology,

Vol. 98-H, p. 193, 1976.

26. Cailletaud, G. and Chaboche, J.L., "Macroscopic Description of the

Microstructural Changes Induced by Varying Temperature: Example of

INI00 Cyclic Behaviour," Third International Conference on Mechanical

Behaviour of Materials, Vol. 2, pp. 22-32, ICM3, Cambridge, England,

August 1979.

27. Robinson, E.N., "A Unified Creep-Plasticity Model for Structural Metals

at High Temperatures," ORNL-TM75969, October 1978.

28. Kocks, U.F., "Laws for Work-Hardening and Low-Temperature Creep,"

Journal of Engineering Materials and Technology, Vol. 98-H, pp. 76-

85, 1976.

29. Cescotto, S. and Leckie, F., "Determination of Unified Constitutive

Equations for Metals at High Temperature," Proceedings of the International Conference on Constitutive Laws for Engineering Materials,

pp. 105-111, 1983.

30. Milly, T.M. and Allen, D.H., "A Comparative Study of Nonlinear RateDependent Mechanical Constitutive Theories for Crystalline Solids at

Elevated Temperature," VPI-E-82-5, Virginia Polytechnic Institute and

State University, March 1982.

178
31. Beek, J.M., Allen, D.H., and Milly, T.M., "A Qualitative Comparison

of Current Models for Nonlinear Rate-Dependent Material Behaviour of

Crystalline Solids," MM 42461-83-14, Mechanics and Materials Research

Center, Texas A&M University, November 1983.

32. Walker, K.P., "Research and Development Program for Nonlinear Structural Modeling with Advanced Time-Temperature Dependent Constitutive

Relations," First Quarterly Technical Narrative, NAS3-22055, January

1980.

33. Bushnell, D., "A Subincremental Strategy for Solving Proglems Involving Large Deflections, Plasticity and Creep," in Constitutive Equations

in Viscoplasticity; Computational and Engineering Aspects, AMD Vol.

20 (ASME), p. 171, 1976.

34. Krieg, R.D. and Key, S.W., "Implementation of a Time Independent Plasticity Theory into Structural Computer Programs," in Constitutive

Equations in Viscoplasticity: Computational and Engineering Aspects,

AMD Vol. 20 (ASME), p. 125, 1976.

35. Zaphir, Z. and Bodner, S.R., "Implementation of Elastic-Viscoplastic

Constitutive Equations into NDNSAP with Applications to Fracture Mechanics," presented at the MIT Conference on ADINA, 1979.

36. Gear, C.W., "The Automatic Integration of Ordinary Differential Equations," Communications of ACM, Vol. 14, p. 176, 1971.

37. Miller, A.K. and Shih, C.F., "An Improved Method for Numerical Integration of Constitutive Equations of the Work Hardening-Recovery Type,"

ASME Journal of Engineering Materials and Technology, Vol. 99-H, p.

275, 1977.

Table 1. Material Constants Used in Bodner's Model

for INIOO at 1350°F (732°C)

Bodners notation Beek and Allen's notation Numerical Value

E E

n n

Z

l

 zl

m m

D

o DO

A A

r r

Z

t

 ZI

£ (t=0) Ct,(t=0)

Z0 a3(t-0)

21.3xl06 psi

0.7

l.lOSxlO6 psi

•3 •• ^ 2.57xl03

 psi

 x

10* sec"1

l.9xlO~3 sec"1

2.66

0.6xlOb psi

0.0

0.915xl06 psi

179
Table 2. Material Constants Used in Krieg's Model

for Hastelloy-X at 1800°F (982°C)

Walker's notation

for Krieg's constants

n

AI

A2

A3

A*

As

E

Ko

n

c(t-O)

Q(t=0)

K(t-O)

Table 3.

Miller's notation

n

B6'

Hi

AI

H2

C2

A2

E

e(t-O)

R(t=0)

Do

Seek and Allen's notation

Ci

C2

C.,

Cs

c?

E

a3o

n

ai(t»0)

a2(t-0)

a3(t-0)

Numerical Value

1.0

4.49

L.OxlO6 psi

6.21xlO~6 psi"1 sec"1

4. 027x1 0~7 psi~2

100 psi sec1/" ,

4.365 psi1"

11 sec1 /Q'2

13.2xl06 psi

59,292 psi sec1/a

4.49

0.0

0.0

59,292 psi

Material Constants Used in Miller's Model

for Hastelloy-X at 18008F (982°

Beek and Allen's notation

n

B8'

HI

H2

C2

A2

E

Q.I (t=0)

a2(t=0)

as (t=0)

C)

Numerical Value

2.363

2.616xlO"5 sec"1

IxlO6 psi

1.4053xlO~3 psi""1

100 psi secl7n

5,000 psi

4.355xlO~12 psi"3

13.2xl06 psi

0.0

0.0

8,642 psi

180
Table 4. Material Constants Used in Walker's Model

for Hastelloy-X at 1800°F (982°C)

Walker's notation Seek and Allen's notation Numerical Value

o

n ' a2

ni ni

n

a n2

n9 *

n? n?

n n

m o

E E

c(t=0) cn(t=0)

JJ(t-O) a2(t-0)

K(t=0) a-j(t-O)

ne,n9 ,nio,q

-1,200 psi

0 psi (not used)

IxlO6 psi

312.5

2.73xlO"3 psi1 sec"1

4.49

1.16

13.2xl06 psi

0.0

0.0

59,292 psi

0 (not used)

n R

Table 5. Numerical Integration Approximation for Aa. = / a. dt

Method Approximation

Euler Forward Difference

Trapezoidal Rule

Trapezoidal Predictor-Corrector

Runge-Kucta 4th Order

At a

At • y[a

Same as trapezoidal except iterate

2K2

 + 2K3

At

K2 =• At a1(t+At/2,a1(t)+K1/2)

K.J =« At a1(t+At/2,aL(t)+K.2/2)

K - At

181
CO

_UJ

C/3

20.0 rI 2.0 -

OF POOR QUALfTY

&t • 3.1003 00 SEC KBIEG lOOtl IEULESI

at- O.iOOO 00 SCC UK I EC nOOU 11 HOPE JO IOm.1

At« o-iooo oo stc MIIEO MOCL ip»eoicTou-com»ecto»i

61 • 0.1000 00 3EC ««IEC HOOEL I4(H ORDER RUHCC-KUlTO I

24.0 28.0 32.0 36.0 40.0

-20.0»-

Fig. 1 Comparison of integration methods for Krieg's theory (Hastelloy-X

at 1800°F)

• at- :.!:::-:: ir: •

I:ij^"i. :

at • C.1C33 30 iiC ""1G "001^ '

a: . u.:!S: 33 5CC «»!!0 "OOtl :« 3«0t» <U«C!-<i,:Il>i

A: • 1.5-00-;; 5-: ^I

41 • J.IOOQ r: SEC *!

it . o.;:so co i£: »i

II* "ODE. .£'JL£I:

i.£* "3CCI. 'EUI.E»I

•«.3r

.: / r

1 8. Of-

# Strain tin in) -

3.0

-a.:

\ -~r r r— -"•=

.:.:;:l "-;

I ' i Strai n (in-in )

/

";:

I

.^f^y x- -- - -f- — — ^^f

-^Tx • ^- i - .... r- t -i^ •— — • •

Fig. 2 Comparison of integration methods Fig. 3 Stability and accuracy of^

for Kreig's thoery with equal com- Euler's method for Miller's

putation time allowed for each method theory (Hastelloy-X at 1800°F)

(Hastelloy-X at 1800°F)

- 182
*: • i.::oc 30 tic

• it • :.:rso ;s ss:

*j:r. :!'.xt«:

-"Oof. : iw.su

x.» ;.2:cc-ci sec

A: • o-«ooo-;i st:

«. 3.5030-31 :::

v:

Fig. 4 Stability and accuracy of Fig. 5 Comparison of integration methods

Euler's method for Walker's for Bodner's theory with equal

theory (Hastelloy-X at 1800°F) computation time allowed for each

method (1N100 at 1350°F)

V)

OT

20. Oi12.0f41- 0.5000-02 StC UH|.K(lt ItOOCl ICULtK) COMECT SOLUTION

at - o.sooo-oz stc ««.«« nooei itutMt »«toe or « sojusreo »» -sx

« - o.sooo-oz stc WUJCM noon nw.«« »«.ue Of » wjjusteo BT -sx

-12.Ot-

-20.0

28.0 32.0 36.0

Time (sec)

40.0

Fig. 6 Sensitivity of Walker's theory to -5% change in input constants

(Hastelloy-X at 1800°F)

183
OF POOR QUALITY

•7}

41 • 1-~i~.~0-3J i-C -SL'E" ••'CSEt 'EliLE.'l CORREC T 5CI.J":3N

At - O.SOOO-OZ UC MBLHER nQOtL 1E.ULERI ot.1. IE5t QRIB OOJUStEO 8t ./«'/,

28.0 32.0 36.0

Time (sec)

*0.0

-20.01-

fig. 7 Sensitivity of Walker's theory to 5% change in experimental test

data used to generate constants (Hastelloy-X at 1800T)

4t- 0.5000-02 SEC "HIEu .lOOEl lEULEKi CO»»ECT SOLU'iON

4t - 0.5000-02 SEC KRIEG "OOEL lEUlEm VOLUE OF N BOJUStEO 81 -SZ

At - 0.5000-02 SEC URIEC "OOEl lEULEHI vflLUE OF K BOJUS'EO 91 -57.

20.0.-

73

23.0 32.0 36.0 -10.0

Fig. 8 Sensitivity of Krieg's theory to -5% change in input constants

(Hastelloy-X at 180CTF)

184
At* 0.1000-02 SEC SOONER 100CI 'CUIERI CORRECT 50LUHOH

At • 0.«000-OZ SEC 800NJR nOOEl IEUIERI VOLUE Or M HOJU5IEO 8T -SZ

at • 0.4000-0? SEC SOONER nOOEl ItUlEHI VBLUE Of ZI.ROJUSIEO ST -SX '

200.0

I 20.0

40.0

0

-40.0

-I 20.0

-200.0

4.0 / 6.0 .0 / 10.0 1 2.0 14.0 16.0 18.0

Time (sec)

i "'PI n n . .

.: .,.0,1 U LJ -

Fig. 9 Sensitivity of Bodner's theory to -57, change in input constants

(IN100 at 1350°F)

At • 0.5000-03 SEC "ll.L£* "OOEL

At • O.SOOO-OJ SEC KRIEC "OOEL IEULERI

At • 0.5000-03 SEC MRLKCR nOOEL (EULCRI

0)

U

_1

73

20.OfI 2.0 -

28.0 32.0 36.0

Time (sec)

-12.0 -

-20.0»-

Fig. 10 Comparison of stress-time predictions for Miller, Krieg and

Walker theories (Hastelloy-X at 1800°F)

40.0

185

N86-30230

EVALUATION OF THE NUMERICAL STABILITY AND

SENSITIVITY OF MATERIAL PARAMETER VARIATIONS

FOR SEVERAL UNIFIED CONSTITUTIVE MODELS

P.K. Imbrie

W.E. Haisler

D.H. Allen

Aerospace Engineering Department

Texas A&M University

College Station, Texas 77843

NASA Lewis Research Center

Cleveland, OH 44135

\

Grant No. NAG3-491

1-4998-85-6 May 1985
ABSTRACT

The sensitivity of the unified constitutive theories, proposed by

Bodner, Walker, Krieg, and Miller, to numerical integration techniques

and slight changes in material parameters was investigated. Evaluations

were based upon numerical simulations of Hastelloy-X at 1800° F in uniaxial

form by specifying input strain histories and comparing output stress

histories. The integration sensitivity study involved the use of both

single and multi-step integration schemes. The various algorithms were

compared with regard to accuracy, stability, and computational economy.

The material parameter sensitivity was studied by varying the material

constants by a specified amount and comparing predicted responses.

Numerical comparisons show that, of the numerical integration methods

studied, Eulers method is the most accurate, stable, and efficient procedure. The input sensitivity studies indicated that some constitutive

models are more sensitive to experimental errors than others and that a

5% error in certain material constrants may lead to 15-30% changes in

predicted response.
m

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support of this research by

NASA Lewis Research Center under Grant no. NAG3-491.
IV

TABLE OF CONTENTS

Page

ABSTRACT 11

ACKNOWLEDGEMENTS iii

TABLE OF CONTENTS iv

LIST OF TABLES vi

LIST OF FIGURES vii

CHAPTER I. INTRODUCTION 1

CHAPTER II. BACKGROUND 4

Constitutive Modeling 4

Integration Methods 6

The Origin of Stiffness 9

Material Testing 10

Objectives . 11

CHAPTER III. CONSTITUTIVE THEORIES CONSIDERED . . 13

Bodner, et al's. Theory 13

Determination of Material Parameters . 14

Walker's Theory 19

Determination of Material Parameters 20

Krieg, Swearengen, and Rhode's Theory 25

Determination of Material Parameters 27

Miller's Theory 29

Determination of Material Parameters 31
VI

LIST OF TABLES

Table Page

1 Material Constants Used In Bodner's Model

For Hastelloy-X at 1800° F . . 18

2 Material Constants Used in Walker's Model

For Hastelloy-X at 1800°F 26

3 Material Constants Used In Krieg's Model

For Hastelloy-X at 1800°F 30

4 Material Constants Used In Miller's Model

For Hastelloy-X at 1800°F 35
VII

LIST OF FIGURES

Figure Page

1 Plots Used to Obtain Material Parameters

For Bodner's Model 16

2 Back Stress and True Stress-Strain Curve

Used in Walker's Thedry 23

3 Plot Used to Determine The Material Constant

A For Miller's Model 33

4 Input Strain Rate Histories 41

5 Monotonic Loading of Hastelloy-X at 1800°F

Predicted by Bodner's Model (e = 0.001 sec.'1) .... 43

6 Monotonic Loading of Hastelloy-X at 1800°F

Predicted by Walker's Model (i = 0.001 sec-1) .... 44

7 Monotonic Loading of Hastelloy-X at 1800°F

Predicted by Krieg's Model (t = 0.001 sec'1) .... 45

8 Monotonic Loading of Hastelloy-X at 1P-00°F

Predicted by Miller's Model U =0.001 sec"1) .... 46

9 Inelastic Strain Predicted by Walker's Model

For Hasten oy-X at 1800°F 47

10 Back Stress Predicted by Walker's Model For

Hastelloy-X at 1800°F 48

11 Comparison of Integration Methods for Miller's

Model For Equivalent Time Steps >

(Hastelloy-X at 1800°F) 50

12 Stability and Accuracy of Euler Integration for

Miller's Model Using Various Step Sizes

(Hastelloy-X at 1800°F) 53

13 Stability and Accuracy of Euler Integration For

Walker's Model Using Various Step Sizes

(Hastelloy-X at 1800°F) 54

14 Stress-time Predictions of Miller's Model Using

Euler Integration (Hastelloy-X at 1800°F) 55
vni

Figure Page

15 Stability and Accuracy of Euler Integration

For Krieg's Model Using Various Step Sizes

(Hastelloy-X at 1800°F) . . . . 56

16 Stress-time Predictions of Walker's Model

Using Euler Integration

(Hastelloy-X at 1800°F) 57

17 Stress-Strain Predictions of Walker's Model

Using Euler Integration

(Hastelloy-X at 1800°F) 58

18 Comparisons of Integration Methods For Bodner's

Model With Equivalent Computation Time Allowed

For Each Method (Hastelloy-X at 1800°F) 59

19 Sensitivity of Bodner's Model to a -5% Change In

Input Constants (Hastelloy-X at 1800°F) 63

20 Sensitivity of Bodner's Model to a ±5% Change In

Experimental Test Data Used to Generate Input

Constants (Hastelloy-X at 1800°F) 64

21 Sensitivity of Walker's Model to a -5% Change In

Input Constants (Hastelloy-X at 1800°F) 66

22 Sensitivity of Walker's Model to a ±5% Change In

Experimental Test Data Used to Generate Input

Constants (Hastelloy-X at 1800°F) 68

23 Sensitivity of Krieg's Model to a -5% Change In

Input Constants (Hastelloy-X at 1800°F) 69

24 Sensitivity of Krieg's Model to a ±5% Change In

Experimental Test Data Used to Generate Input

Constants (Hastelloy-X at 1800°F) 70

25 Sensitivity of Miller's Model to a -5% Change In

Input Constants (Hastelloy-X at 1800°F) 71

26 Sensitivity of Miller's Model to a ±5% Change In

Experimental Test Data Used to Generate Input

Constants (Hastelloy-X at 1800°F) 73
CHAPTER I

INTRODUCTION

For many years there has been a substantial amount of research

performed in the area of modeling the constitutive behavior of inelastic deformation. Recently, new unified formulations have been

proposed which are attracting much attention because these theories

treat the inelastic strain as a unified quantity, incapable of being

separated into time dependent and time independent parts. However,

these new models have two potential problems which can effect their

predictive capabilities. These include sensitivity to the numerical

integration of the differential equations describing the inelastic

strain component, as well as a difficult and sometimes ill-defined

method of determining the material parameters.

Recently, Beek, Allen and Milly [1] have shown that a number of

proposed thermoviscoplastic constitutive models can be cast into a

common internal state variable framework. The first state variable

is the inelastic strain (e^) whose rate of growth is a function of both

current conditions (stress a and temperature T) and one or more internal

state variables. These unified formulations form a system of nonlinear

ordinary differential equations in the independent variable time (t).

Many researchers [2-16] have observed these equations to be well

behaved in some portions of the strain response and mathematically
stiff in other regions. Therefore, in order to obtain a stable and

accurate prediction of inelastic behavior, one must carefully consider

the method of integration. There are a variety of numerical

algorithms available, from the simple Euler integration scheme

wherein one must judiciously select an appropriate time step size to

insure solution stability and accuracy, to the more sophisticated

method of Gear [7-9] where the step size is automatically selected

after checking for the existence of stiffness.

The growth laws which are used to describe the stress-strain

response of a particular material rely on a number of material

parameters. The material parameters (or constants) are determined,

either implicitly or explicitly, from a series of uniaxial tests

which are prescribed by the model developer. Typically some

combination of the following tests is required: 1) constant strain

rate tensile tests; 2) constant strain rate tensile tests with

intermittent hold times; 3) constant strain rate cyclic tests;

4) relaxations tests; 5) creep tests; or 6) stress drop tests.

The application of test data in determining material constants varies

from model to model. In most cases, because of the lack of

standardization, it would appear that the end user will be burdened

with a trial-and-error method for determining specific material

parameters. This burden might be eased if.a quantitative assessment

of the model sensitivity to slight changes in material constants was

provided. However, there is no such information available in the

literature, nor is the problem even addressed.
The purpose of this research will be to evaluate numerical

simulations of several unified constitutive models, in uniaxial form,

by specifying input strain histories and comparing output stress

histories. Integration methods and slight changes to material

parameters will be the prime variables. This will allow for a

qualitative analysis of solution stability, accuracy, and sensitivity.

In fulfillment of these objectives, this thesis will proceed

in the following manner. First, a review of the pertinent literature

is presented in addition to a brief overview of the origin of

stiffness. Next, the constitutive theories considered herein are

discussed and the various methods used to determine their material

parameters are outlined. Finally, the numerical integration and

parameter sensitivity study is presented, wherein the use of

various integration algorithms selected for this study are

investigated and the effects of varying material constants,

simulating "experimental error", are analyzed. Evaluations are

based upon numerical simulations of Hastelloy-X at 1800° F, in

uniaxial form, when subjected to several strain input histories.
CHAPTER II

BACKGROUND

Constitutive Modeling

Constitutive modeling of inelastic deformation is not a new

concept, it has been in practice since 1864 starting with the

early work of Tresca [17]. Levy [18] and von Mises [19] added to

the study of plasticity by developing what have come to be known as

the Levy-Mises equations. The development of classical plasticity

continued with the work of Hencky [20], Prandtl [21], Prager [22],

and Ziegler [23] to name just a few. These early models were

incremental in nature, assumed that plasticity and creep could be

separated, and they incorporated a yield function, flow rule, and

hardening rule to define the plastic strain increment.

In addition to classical plasticity based-theories, other

fundamental bases were proposed to describe inelastic deformation.

These include the microphenomenologically-based theories and thermodynamically-based nonlinear viscoelastic theories. The aforementioned

models can be further categorized as unified or uncoupled theories,

where the two differ in their treatment of rate-independent and

rate-dependent strain components. The uncoupled theories partition

the inelastic strain into plastic and creep components; whereas,

the unified constitutive models retain the inelastic strain as a

unified quantity. However, it should be noted that the partitioning

method used in the uncoupled models is questioned because it has no

formal micromechanical basis and does not account for creep and

plasticity interaction.
Over the past decade a number of unified constitutive theories

have been developed. These new theories include the work of Bodner,

et al. [24-26], Cernocky and Krempl [27,28], Cescotto [29], Chaboche

and Cailletaud [30], Hart [31], Krempl [32], Krieg, et al. [33],

Kocks [34], Miller [35], Robinson [36], Stouffer [37,38],

Valanis [37,40], and Walker [41,42].

As stated earlier Beek, Allen, and Milly [1] and Krieg [16]

have shown that all thermoviscoplastic constitutive models can be

cast into a common internal state variable framework. For the

uniaxial case, a general form of these unified models may be

written as:

a = E(c - e1 - 6T) (1)

e

1

 = f(e,T,B,D,...,am) (2)

B = hB e1 - rB(T,B) (3)

6 = hD e1 - rD(T,D) , (4)

where a is the applied stress, E is Young's modulus, e is the strain,

e is the inelastic strain or first state variable (with both

plasticity and creep included), e is the thermal strain, B is the

back or rest stress corresponding to the dislocation arrangement

which produces kinematic hardening or the Bauschinger effect, and

D is the drag stress corresponding to the dislocation density which

produces isotropic hardening. In addition, f is the inelastic strain

rate function, T is the temperature, o are additional state variables

which are presently undefined,and hn, h~, rB, and rQ are the hardening
and recovery functions for the back or drag stress (designated by the

subscript B or D), respectively. A superposed dot above the

variables denotes differentiation with respect to time.

The abstract model presented above (equations (l)-(4)) contains

all three of the commonly used internal state variables; however,

specific theories may omit one or more of the growth laws which

characterize either the back stress, drag stress, or other variables.

In all cases, the inelastic strain is obtained via time integration

of the specified internal state variable growth law for e , i.e.,

t

- I e!(t')dt' , (5)

where e is defined in functional form by (2), t is the current time

of interest, and t1 is a dummy variable of integration.

Integration Methods

A number of integration strategies have been evaluated

[2,3,5,6,10-14] for integrating constitutive equations which have

"locally" stiff regimes. These algorithms have been implemented

into both large multi-axial finite element programs and small

uniaxial constitutive codes. For the purpose of this discussion,

only those publications dealing with integration on the constitutive

level will be reviewed.

Numerical schemes for the solution of the inelastic response

behavior were explored by William [2]. He used a generic constitutive

formulation in order to examine the algorithmic properties of various

numerical solution methods. These included the explicit forward
7

Euler method, the forward gradient method, the predictor-corrector

method, and the Newton-Raphson method. A qualitative comparison of

the four incremental solution methods was presented for pure creep

and pure relaxation behavior; however, no specific conclusions were

stated with regard to the "best" scheme.

An explicit trapezoidal method for handling deformation in

stiff regimes was proposed by Shin, Delorenzi, and Miller [3].

Results using the explicit trapezoidal scheme demonstrated both

improved stability and computational efficiency over either explicit

forward Euler integration or the trapezoidal method with a NewtonRaphson corrector,, It was shown that the predictor-corrector

method eventually converged to the same solution predicted by the

explicit trapezoidal scheme, while the Euler predicted response was

unstable.

Another promising numerical technique, called the a or NONSS

(Noniterative, Self-correcting Solution) method, has been developed

by Tanaka and Miller and is discussed in Reference [4]. The implicit

quantities are removed by a Taylor series expansion of the prescribed

growth laws through the use of an integration operator a. The

integration operator, a, has the range of (0,1). This alogrithm

also incorporates automatic time-step control and an error corrector.

Another scheme similar in style to the aforementioned is the 0-method

(described in Reference [5]) which incorporates the explicit forward

Euler method (0=0), the implicit trapezoidal scheme (0=^), or the

implicit Euler strategy (0=1). Numerical experiments indicated that

the forward Euler integration scheme (0=0) with an automatic stepping
8

and error control was computationally more efficient.

For integrating constitutive equations of the work hardeningrecovery type, Miller and Shin [6] have developed a method wherein

improved accuracy or larger time steps or some combination of both

can be obtained. This is accomplished through a Taylor series

expansion of the recovery function resulting in a more accurate

calculation of the internal variable(s) during each time step.

A comparison between this special algorithm and Euler integration

from Zircaloy-2 simulations verify the accuracy and stability of

this method.

Gear's method [7-9] has also been an effective tool for

integrating stiff differential equations. It is a multi-step

predictor corrector scheme whose order is automatically chosen.

Either an Adams' method or methods suitable for stiff equations

can be selected. Three alogrithms are available, STIFF-0, STIFF-1,

and STIFF-2. Chang and Chang [5] believe that Gear's package is

well suited to solving one-dimensional constitutive relations, but

is much too cumbersome for use on the structural level. Based on

this work, it may be tentatively concluded that Gear's method is not

suitable for implementation into finite element packages.

A very comprehensive study on numerical integration of stiff

constitutive models was reported by Kumar, Mor.iaria, and

Mukherjee [10]. Hart's equations for predicting inelastic behavior

were used to compare various numerical approximation strategies.

Several input histories of stress and strain were considered and

the accuracy and computational efficiency of the results were
compared. The study concluded that a simple Euler type one-step

method with automatic time-step control worked well; however, it

emphasizes that a one-step strategy is not necessarily the best.

A combination of the two-step Adams' method (for outside the

viscoplastic limit) and Euler's method (for inside the viscoplastic

limit), was found to be the most efficient in terms of computational

speed and accuracy.

The Origin of Stiffness

The purpose of this section is to familiarize the reader with

the concept of stiff equations through a physical and mathematical

definition. By understanding this concept, researchers may then

devise methods to test for stiffness or more simply concede to its

existence and develop appropriate solution algorithms.

A system of differential equations is said to be stiff if the

physical processes being modeled contain time constants with

different scales. In the case of a uniaxial test specimen, there

is grain boundary or dislocation movement in one time scale;

whereas the total stress-strain response is measured using a

different time constant. The stiffness of equations (2) thru (4)

arises from the following phenomenaon. The inelastic strain rate,

e , may be a strong function of a, B, and D in (2); that is small

changes in o, B, and D cause large changes in e , which then

influences o, B, and D through (1), (3), and (4). Ultimately, the

absolute stability of equations (2) thru (4) requires that the step

size used during numerical integration be not much greater than the
smallest time constant present in the physical system; however, for

computational reasons this is not always possible. Therefore,

integration methods must be used that are "stiffly stable".

The mathematical definition as described in [44] is as follows:

"The nonlinear system Y1 = F(X, Y) is said to be stiff in an

interval I if for xe I the eigenvalues A. of the Jacobian J

satisfy the following conditions."

1) Re \k < 0, k = l,...,m

2> \\

Where the A. , k=l,...,m are the eigenvalues of F.

The Ratio

(6)

is called the stiffness ratio

Basically this means that if the real eigenvalues are negative or

if the spread of real eigenvalues is large, then the equations are

stiff. Gear's method [7-9] uses this definition to test for the

existence of stiffness. However, this can be very expensive

expecially when large systems of equations are being analyzed.

Material Testing

Each constitutive model has a number of material parameters which

are determined through a set of complex tests; therefore, the end

user must be informed about the effect that small variations, i.e.

typical experimental error, have on the particular models' predictive
11

capabilities. In general, the testing of materials is a difficult

process in which the propensity for error is great. This experimental

error is known to occur in all stages of testing and will be discussed

herein; however, the reasons for its occurrence are beyond the scope

of this research.

Experimental error may develop in all phases of testing; however,

it occurs most frequently in the following areas. The nonrepeatability

of strain measurements for uniaxial strain rate, cyclic or creep

tests makes interpretation of results difficult even when new high

technology axial and diametral extensometers are used. When testing

requires elevated temperatures, error may result while measuring and

maintaining the specimen's temperature. Finally, experimental error

occurs in its most rudimentary form in the statistical variation of

the test sample's measurements. While these are only an example of

where experimental error occurs, they are representative of the

problems experimentalists must cope with and why absolute values of

material constants are difficult if not impossible to obtain.

Objectives

The objective of this research is to study the sensitivity of

the unified constitutive theories proposed by 1) Bodner, et al.,

2) Krieg, Swearengen and Rohde, 3) Miller, and 4) Walker to

numerical integration techniques and slight changes in material

parameters. Evaluations are based upon numerical simulations of

Haste!loy-X at 1800° F in uniaxial form by specifying input strain

histories and comparing output stress histories. The integration

sensitivity study involves the use of both single and multi step
12

integration schemes. The material parameter sensitivity is studied

by varying material constants by a specified amount and comparing

predicted responses.
13

CHAPTER III

CONSTITUTIVE THEORIES CONSIDERED

The constitutive models selected for this study include the work

of Bodner (et al.) [24-26], Krieg (Swearengen and Rohde) [33],

Miller [35], and Walker [41-42]. These models were chosen primarily

because of the availability of material parameters for the same

material (Hastelloy-X), the exception being Bodner's model (constants

for Bodner's model were obtained by simulating the required uniaxial

tests using Walker's model). A secondary reason for selecting these

models is that they appear to be the most qualitatively attractive

theories available [42, 43] and thus warrant further evaluation.

In the following sections, each model is presented in one-dimensional

form and the methods used to compute the material constants are

discussed.

Bodner, et al .'s Theory

The model proposed by Bodner, et al . is a microphenomenologically based theory for characterizing inelastic behavior. This

theory was developed to predict the response of a material which work

hardens isotropically. The model contains two internal state variables;

the inelastic strain and the drag stress. Bodner's model contains

no parameter which represents the back stress and thus cannot

account for the Bauschinger effect in kinematic hardening materials.

The uniaxial-differential form of Bodner's model may be written as

.

-- . 0

2 n D

 sgn(o) (7)

/3
14

D-D,

D = m(D1 - D)Wp - AD1 I -^- ] , (8)

where

Wp = o e1 . (9)

The material parameters to be determined are E, e , n, m, A, r,

DQ, D.J , and Dp, where D is the initial value of Du The constant e

represents a limiting value of the inelastic strain rate and is usually

4 - 1

assumed to be 10 sec unless the strain rates are very high,

Determination of Material Parameters., The material parameters

for Bodner's model are determined through a series of constant strain

rate tensile tests and constant stress creep tests. Control

parameters for both sets of experiments should encompass values

which are compatible with the numerical simulation to be performed.

The first step in the evaluation of the material constants, as

described by Stouffer [37] and Milly and Allen [43], is to determine

n from the constant strain rate tensile data. By observation, when

the stress reaches a saturated value during a constant strain rate

test equation (7) is satisfied if and only if D is a constant*

Therefore, by neglecting recovery via rapid loading of the test

specimen, D can be assumed to be in its fully work hardened state

and have a maximum value D,. Thus (7) may be rewritten, to account

for the steady state flow condition, as

r '1

In [-ln(-^d| e —)] = -(2n)lno + [2n lnD,+ln(£-4] (10).

o 1 Zn

If the experimental data base is good, the left hand side of equation
15

(10) must be a linear function of Ino- Thus, a plot of these two

variables (as depicted in Fig0 l(a)) yields a straight line where

the slope and intercept are defined by the quantities -2n and

[2nln D.J + ln((n+l)/2n)], respectively, from which n andD,, may be

determined.

To determine the parameters m and D , constant strain rate

tensile test data is used. If work hardening is assumed negligible,

(8) may be rewritten as

dD = m^-DjdW , (11)

which can be integrated to give ,

ln(DrD) = -mWp + InfDj-D^ , (12)

where W is the inelastic work defined by (9) and D is the initial

value of hardness D» To obtain D for measured values of stress and

inelastic strain rate, equation (7) is inverted and may be written as

D = a

l/2n

2n

(13)

Therefore, by using values of o and e from the nonlinear portion of

the tensile response data the quantity D may be determined by

equation (13)0 Since equation (12) is a linear representation, of

ln(D,-D) and W , a plot of these variables will be a straight line

(see Fig. l(b)) having a slope of m. Thus, knowing m, D may be

determined through simple substitution and rearrangement of equation

(12).
Denoted Experimental Data Points

16

•-i O

• u u

c

r—

I

O

In (o)

I

of—

O

w

a) Evaluation of Strain Rate

Exponent n.

b) Evaluation of Hardening

Parameter ITU

o

ar—

e• *

c.

D-D,

In

c) Evaluation of Hardening

Recovery Parameter r.

Fig. 1. Plots Used to Obtain Material Parameters

For Bodner's Model.
17

The last set of constants, A, r, and Dp are computed using data

obtained from constant stress creep tests. During secondary creep

the creep rate is approximately constant; therefore, the value of D

•

must be constant and D = 0. Thus, the hardening rate equation (8)

for secondary creep becomes

m(DrD)Wp = AD, (~) . (14)

linearization of (14) results in

/D-D

In [m(DrD)Wp] = r In^-g-^j + InfAD,) , (15)

where stable values of D are computed using equation (13) and Dp is

•

the minimum calculated value of D. By plotting ln[m(D,-D)W ] as

D-D2 ' P

a function of ln( -n- - ), similar to Fig, l(c), the variables A and r ul

may be determined. From equation (15), it may be seen that r

represents the slope of the straight line plot and ln(AD,)is the

intercept. Hence, A may be computed once the value of the

ln[m(D,-D)W ] intercept is known.

Since no experimental data for Hastelloy-X at 1800° F was

available, the material parameters for Bodner's model were computed

using Walker's theory (as described in the next section) to predict

the deformation response. Numerical simulations included both

constant strain rate and constant stress (creep) tests. The material

constants computed for Bodner's model are tabulated in Table 1.
18

Table 1. Material Constants Used In

Bodner's Model For Hastelloy-X

at 1800° F

Bodner's Constants Numerical Value

13.2 x 106 psi

1.0 x 104 sec"1

n 0.103

m - 0.178 psi

r 0.294

A 0.162 x 10"1 sec"1

D] 0.329 x 107 psi

D2 0.305 x 107 psi

e

r(t=0) 0.0

D(t--O) = D 0.317-x 107 psi
19

Walker's Theory

The model proposed by Walker is a viscoelastic theory developed

by modifying the constitutive relation for a three parameter viscoelastic solid. This model contains three internal state variables;

the inelastic strain, the back stress, and the drag stress. The

back stress introduces nonlinear kinematic hardening into the model

to account for the Bauschinger effect and the drag stress introduces

isotropic hardening into the model to account for cyclic hardening

or softening of the material. Creep, relaxation, and strain rate

effects are modeled by a power law function for the inelastic strain

rate. The uniaxial-differential form of Walker's model may be

written as

'I - ° ~ B / D\ f\(i\ e -

 J r—J (o - a) (to)

B = (n] + n2) e - (B - B - n] e) G

D = nft [e 1! - nQ le1! D - nin (D - Dn)

q

 , 10

where the value of G is defined to be

G

i -I, a

l

e

 I 3R

/ M

(n3 + n4R)

 MlTnTR

and R is the cumulative inelastic stran

.t

R E f ^ J

 i3t

dt'

(17)

(18)

n^B-Bf"1 (19)

(20)

Walker made several simplifying assumptions to equations (17) and (18),
20

including D = constant for the Hastelloy-x material. Therefore,

history dependence in this model is exhibited entirely by a

simplified growth law for the back stress.

B = (n1 + n2) e1 - (B-B-n^Mrige1 + ny iB-Bj"1'

1) (21)

Thus, the material parameters to be determined are E, n, m, n-j,

o

n2, n7, ng, B, and D.

Determination of Material Parameters. The material parameters

for Walker's model are determined through a series of constant strain

rate steady state hysteresis loops under fully reversed strain

controlled conditions at a constant temperature.

The first step in the evaluation of the material constants, as

o

described by Walker [41], is to determine the value of B. The

asymptotic steady state value of B may be obtained under conditions of

uniaxial tension loading at a known constant strain rate because

the back stress will saturate to a constant value as e -> » if n-j = 0.

• I

11m B = B + . "2 E

 0 m , (22)

£*-> » ngR + n7|B-B|m '

The back stress becomes rate-independent and will saturate to a

maximum value for rapid monotonic loading in tension and compression

. T

when R = E . Thus, the static thermal recovery term involving ny

in (22) may be neglected and maximum values of the back stress, in

tension and compression, may be obtained by

Bmax = °B + Vn9 (23)
21

and

Bn,ax = B -

For large strains, stress is assumed to approach an asymptotic

such that e = e; therefore, equation (16) may be rewritten as

o = B + D e1/n . (25)

The maximum values of stress in rapid tension and compression loading

are then obtained as

»

 + y-vi + D

 = <26>

and

'max

 = ° - Vn9 - D ' <27)

Hence, B may be obtained using the relation

• \ <°max

 + » > ' <28'

To determine the constants n and D equation (25) may be

rewritten in terms of steady state values of o and B in tenson as

t Rt \n

£ = f max ^ 'max j _ (2g)

If a, and o? denote the corresponding maximum tensile stresses at the

two maximum strain rates, c, and iL

 wnere

 B attains its maximum

saturated value, then equation (29) gives

(30)
22

and

The value of IJ „ used in equations (29)-(31) is determined max

from stress drop tests. If the material is cycled under steady

state strain controlled conditions around a closed hysteresis

loop as depicted in Fig. 2, positive creep will occur at point C

if the stress is held constant and alternately if the stress

is held constant at point I, negative creep will occur. Since the

saturation stress in tension, B , is desired, a point F on the max

hysteresis loop between points C and I must be found such that the

creep rate ceases instantaneously when the stress is held at a

constant value. Walker indicates that it is difficult to estimate

the value of B using the above procedure because the creep fflaX

rates are usually small on the unloading branch. He suggests that

a better estimate may be obtained by a relaxation test, noting

the point at which initial relaxation changes from positive (stress

decreases) to negative (stress increases), since relaxation

progresses more rapidly than creep.

Values for the material parameters n~ and ng may be determined

from the initial monotonic stress-strain curve. For rapid loading

rates n, = 0 and G = nq R, therefore the back stress may be expressed

as

B = °B + (Bmax * B) (1 - e" 9U " °/E)) ' (32)
23

*-Creep

-B = constant

• GJAHI - True Stress Curve, o/e

• KDEF - Back Stress Curve, B/e

Fig. 2. Back Stress and True Stress-Strain Curve

Used in Walker's Theory.
24

For tensile loading equation (16) may be written as

e o _ • 1 do • /, 1 do\ /o-B\n ,0 _v

 - E -

 e

 - FdTe = Y (1 - E d7} = (~rj (33)

and combined with equation (32) to give

n = U - o/E)- ln- r - - (34) 9

where y is the applied strain rate and do/de denotes the instantaneous

slope of the uniaxial monotonic stress-strain curve at the point

where the stress is o and the strain is e. An estimate of the

constant n« can then be made by rewriting equation (23) as

t o

(B1 - B)

n -- - -' (35)

2

 n

9

The final two constants to be determined are m and n~

Equation (21), with n, = 0, yields the expression for the back

stress rate for tensile loading as

o • o

B = n2 e1 - (B - B)(ng e1 + n^B-fif'1) . (36)

At the peaks of the hysteresis loops B saturates to a constant

value, JJ - 0 and o = 0 for tensile loading, so that equation (36)

gives

m n2 e4 = (B4 - B)ng i^ + n?(B4 - B) (37)

or

m

ny e, - (B, - B)nq e + n (B, - B)'" . (38)

c. 3 0 _7 D / D
25

If o- and Or denote the maximum tensile stresses at the two

smallest strain rates e. and e,-, then BA and B,-, which represent H O * f O

the saturated back stresses for these two strain rates, may be

determined by equation (25). By combining equations (37) and (38)

m is obtained in the form

o

"o £,1 - "Q e/i IB- - B) ,„ , 9 4 4

(39,

Knowing m, equation (37) may be solved for n^. Note that n^ is

assumed to be zero in these applications.

The material constants computed for Walker's model, simulating

Hastelloy-X, were evaluated by Walker [41] and are tabulated in

Table 2.

Krieg, Swearengen, and Rohde's Theory

The model proposed by Krieg, Swearengen, and Rohde uses a

power law function to model the inelastic strain rate and is based

upon the microstructural aspects of deformation mechanics. This

model contains three internal state variables; the inelastic strain,

the back stress, and the drag stress. The growth laws for the

back stress and drag stress are formulated as the differences in

hardening and recovery rates. The back stress is a kinematic

hardening state variable which is used to model the Bauschinger

effect and the drag stress is an isotropic hardening state variable

which attempts to account for the dislocation density or mechanical
26

Table 2. Material Constants Used In Walker's

Model For Hastelloy-X at 1800°F

Walker's Constants

E

n

m

nl

"2

n7

ng

o

B

D

e

!(t=o)

B(t=0)

D(t=0)

Numerical Value

13.2 x 106 psi

4.49

1.16

0.0 psi

1.0 x 106 psi

2.73 x 10"3 psi1""1 sec"1

312.5

- 1,200 psi

59,292 psi

0.0 •

0.0

59,292 psi
27

strength of the material. The uniaxial-differential form of

Krieg's model may be written as

•I = |0 - B|n

 (o _ B) (4Q)

D

n

o

0 ° \ 2

B = A1 e1 - A2 (B - B)|B - B|(eA3

(B"B) -1) (41)

D =A4|eI| -A 5 ( D - DQ)

n . (42)

It should be noted that equations (16) thru (18) vary in form from

those originally proposed by Krieg [33], instead the notation

adopted by Walker [42] is presented. The material parameters to

o

be determined are E, n, A,, A,,, A~. A4> A^, B, and DQ.

Determination of Material Parameters. The material parameters

for Krieg's model are determined through a series of constant

strain rate cyclic tests and stress drop tests.

o

The constants n, DO, B, and A, in Krieg's model are analogous

o

to the material parameters n, D, B, and (n, + n?) respectively,

in Walker's Theory. Therefore, the interested reader is referred

back to Walker's model for a detailed explanation of their

determination.

Under steady state conditions where o = 0, equation (40) shows

that B = constant for constant strain rates. That is,

B1 = Ol - D e]711 (43)

or

B, = o, - D ;!/n , (44)
28

Where B, and B? are saturated values of the back stress for strain

rates e, and k~ respectively. Therefore, by equating equation (41)

to zero and noting that e = e if a = 0, gives (for e positive)

.iLl!,e A3<B -»>2 - l) . (45)

By applying known values of the back stress (B, and Bp) for the

given strain rates (e, and e?) to equation (45) to give two

equations with the unknowns, (A,/A2) and A3, the constant (A,/A2)

may be eliminated so that A~ is determined by

injl + Ill

2

62

U,

0

- B

- B ) (eVVB) v\l

-')l

/O D } L.

VD-i ~ D)

A3 = __V £-l_J »-JL i : U. § (46)

which can be solved using an interative technique.

The constant A? may be determined from steady state hysteresis

loops where B = 0. Under these conditions, equation (41) may be

written as

A e

A = \ (47)

L-

(B - B)2 (e¥B-

B

' -1}

The final two constants to be determined are A. and A,.. The

parameter A. controls the amount of isotropic hardening produced

by a given amount of strain. As such, it plays an important role

in three different transient situations: 1) When a tensile
29

test is simulated, A. controls the rate of work hardening;

2) When a creep test is simulated, A. controls the amount of

primary creep; and 3) When a fatigue test is simulated, A. controls

the rate of cyclic hardening (or softening). A best-fit value for

A, can be tried and the resulting simulations compared with the

test data for the same loading, to select the one which causes

the closest agreement. Once a value for A- has been selected, the

constant AS may be determined from the following relation

V

A, = *— , (48) 5 (VV

where D, is the steady state value of the drag stress at a constant

strain rate, L

The material constants computed for Krieg's model, simulating

Hastelloy-X at 1800° F, were evaluated by Walker [42] and are

tabulated in Table 3.

Miller's Theory

The model proposed by Miller [35] is a phenomenologically based

theory which uses a hyperbolic sine function for the inelastic

strain rate. This model contains three internal state variables;

the inelastic strain, the back stress, and the drag stress. The

theory is capable of simulating many of the structurally significant

deformation phenomena including cyclic hardening/softening,

Bauschinger effect, strain rate effects, and annealing, among others.

The growth laws for the back stress and drag stress are formulated

as the difference between hardening and recovery rates.
30

Table 3. Material Constants Used In Krieg's

Model For Hastelloy-X at 1800° F

Krieg's Constants Numerical Value

E 13.2 x 106 psi

n 4.49

A1 1.0 x 106 psi

A2 6.21 x 10"6 psi"1 sec"1

A3 4.027 x 10"7 psi"2

A4 100 psi sec1/n

Ac 4.365 psi1-" sec17"'2

o

B - 1,200 psi

DQ 59,292 psi sec1/n

e

!(t=0) 0.0

B(t=0) 0.0

D(t=0) 59,292 psi
31

The uniaxial-differential form of Miller's model may be written as

I f /„ R\"-5 1 n

i

l

 = BC0' J sinh ( ^.J I sgn(o-B) (49)

S = H^1 - H^o1 { sinh A] |B| }n sgn (B) (50)

D = H^e1] (C2 + |B| - A2D

3/A1) - H^^O1

 {sinh(A2D

3)}n . (51)

The material parameters to be determined are E, B-G1, n, H,, Hp,

A,, A?, Cp» and DQ. The only constants which depend on temperature

are the coefficients 6-0' and DQ.

Determination of Material Parameters. The material parameters

for Miller's model are determined through a series of constant strain

rate steady state hysteresis loops under fully reversed strain

controlled conditions at a constant temperature.

The starting point for the calculation of the constants is to

find the values of B-G1, n, and A that satisfy one following steady

state equations

e*s = B01 [sinh (A ass)]n , (52)

The constant A does not appear explicitly in the final equations, but

it enters into the calculation of A, and Ap.

During monotonic loading under steady state conditions the back

• •

stress and drag stress reach their saturated values, thus B and D are

zero. Therefore, by equating equations (50) and (51) to zero and

solving for the resulting steady state plastic strain rate gives

e*s = BCG' [sinh(A1Bss)]n , (53)
32

.1

£SS

e.1

ss

BC0./„sinh

°SS~BSS

1.5

I \ "SS

BC0' [sinh(A2Dss

3

)]n

BSS - A2°SS

, (54)

(55)

Thru separation of variables in equations (52) through (55), it may

be seen that

A1BSS

°ss •

 Bss

Dss

1.5

. (56)

Hence, from equation (56) explicit relations for the constants A,

and A« may be obtained:

A a.

Al

 =

JSS

JSS

and

A = ( A ^3

2 ( ^BSS^SS '

By linearizing equation (52) one obtains the relation

J<.) = n In [sinh(Ao<;<.)] + In (Bre') .

u o oo \s

(57)

(58)

(59)

The constant A is chosen so that a plot of In (ECC) versus

In [sinh(Aoss)] falls on a straight line. If the test data points

fall on a straight line for a particular value of A, then Fig. 3

shows that (CD/AB)/(EC/EB) = 1, or

In - In

In e, - In EC

ln[sinh(AoJ] - ln[sinh(Ao5)]

ln[sinh(Aa1)] - ln[sinh(Ao5)]

. (60)

Where a,, a., and a& are the steady state asymptotic stress
33

In [sinh(Ao)]

Fig. 3. Plot Used to Determine The Material

Constant A For Miller's Model.
34

obtained from hystresis loops executed at strain rates e,, e*,

and e5 respectively. Once a value for A has been chosen, then

from equation (59) the slope of ln(e ) versus ln[sinh(Aoss)] plot

is n and the intercept is ln(Bce').

From stress drop creep tests the value of B<-<. can be

established and the constants A, and Ap may be computed from

equations (57) and (58) respectively. In additon, once A, and A«

are known, then D~~ = D~ may be found from equation (56).

The constant H-j in equation (50) governs the hardening rate

of the back stress. This constant is analogous to the constants

n, + r\2 in Walker's model; therefore, the interested reader is

referred back to Walker's model for a detailed explanation of

their determination.

The final two constants yet to be determined are H^ and Cp.

The parameter hL controls the mount of isotropic hardening

produced by a given amount of strain. A best-fit value for FL

can be tried and the resulting simulations compared with the

test data for the same loading, to select the one which causes

the closest agreement. The constant C~ sets a "floor" under the

drag stress and its value is estimated by a best-fit procedure to

the actual test data.

The material constants computed for Miller's model, simulating

Hastelloy-X at 1800° F, were evaluated by Walker [42] and are

tabulated in Table 4.
35

Table 4. Material Constants Used In Miller's

Model For Hastelloy-X at 1800° F

Miller's Constants Numerical Value

E 13.2 x 106 psi

n 2.363

BC0' 2.616 x 10"5 sec"1

H1 1.0 x 106 psi

A1 1.4053 x 10"3 psi"1

H2 100 psi sec1/n

Cp 5,000.0 psi

A 4.355 x 10"12 psi"3

e(t=Q) 0.0

B(t=0) 0.0

D(t=0) 8,642 psi
36

CHAPTER IV

EVALUATION OF SELECTED THEORIES

Evaluation .of the unified constitutive theories considered

herein is based upon numerical simulations of Hastelloy-X at

1800° F in uniaxial form by specifying input strain histories

and comparing output stress histories. Since this thesis addresses

two separate problems, a section on the numerical time integration

study is presented first, followed by a section detailing the

material parameter sensitivity study.

Numerical Time Integration Study

Accurate integration of the constitutive relationship given

by equations (1) through (4) is extremely important to the numerical

solution of a nonlinear field problem. However, the integration

process |is somewhat hindered because the coupled system of ordinary

differential equations defining the state variables may be "locally"

stiff and thus is sensitive to the time step size and numerical

algorithm. The instability problems associated with integrating

stiff O.D.E.'s can be minimized through proper selection of time

step size, when using standard integration techniques, or through

the use of specially tailored algorithms which check for stiffness

and automatically take the appropriate action.

In general, equation (5) may be integrated between time t and

t + At by writing
t.

J

37

_t+At

e

1 dt (61)

c

l(t) t

or

t+At

A e1 = e1 (t+At) - e^t) = / e1 dt , (62)

where e is defined by the particular constitutive theory being

employed.

Integration Algorithms Considered. Four of the more commonly

used integration schemes were selected for use in this study. These

include: Explicit Euler Forward Difference Method, Implicit

Trapezoidal Method, Runge-Kutta 4th Order Method, and Trapezoidal

Predictor-Corrector (Iterative) Method.

The basic presumption of the Euler method is that the inelastic

strain rate function is assumed to be a constant over an interval of

interest. For all but the highly-nonlinear portion of the stress strain

curve, this assumption is quite reasonable at constant strain rates.

However, when the material response is nonlinear, the assumption may no

longer be valid unless a sufficiently small time interval is chosen. In

addition,the Euler method is sensitive to numerical round-off. The

major advantage of using the Euler technique is its ease of implementation. The approximation to the solution of equation (5) using Euler's

method may be written as

A e1 = At t1 (t) . (63)
38

The Implicit Trapezoidal Method is another commonly used

integration strategy from the family of single step schemes.

Overall solution accuracy is improved by averaging the evaluated

function over the desired interval assuming a small enough time

increment has been selected. The Implicit Trapezoidal Method

approximates the inelastic strain variable as

e

!

(t+At)] (64)

The Runge-Kutta Method, which is also a single step scheme,

provides a straight forward high order approximation for computing

Ae for various locations along the interval in question. The

self-starting characteristic and small error per step of this

algorithm has made it popular among researchers for numerically

integrating differential equations. It is pointed out that, like

most other numerical procedures for solving differential equations,

the methods of Runge-Kutta are sensitive to variations in the

selected interval. Too small a value leads to an excessive number

of computations; on the other hand, too large of an interval may

well lead to an answer that differs significantly from the true one.

The 4th order Runge-Kutta integration method approximates the

solution to equation (5) as

Ae1 = 1/6 [^ + 2K2 + 2K3 + K4] , (65)

where

K, = At t!(t, e
39

= At

K3 = At e!(t + JjAt, eJ(t) +

= At e!(t + At, Kg) .

The last integration algorithm selected for use in this study is

the multi-step Trapezoidal Predictor-Corrector method. The objective

of this strategy is to predict Ae as accurately as possible using an

iterative procedure. As a first guess, e (t+At) is computed by the

Euler method

Predictor: e^t+At) = e^t) + At c!(t, e^t)). (66)

This value is then substituted into the Trapezoidal scheme

Corrector: e^t+At). = e^t) + Q

(67)

which may also be written as

Ac! =

J

e

J(t+At)]. (68)

.th- subscript j in equations (67) and (68) refers to the j

iteration. If the values for Ae determined by (66) and (67) differ

too greatly, equation (67) is re-evaluated using the most recent

computed value of e (t+At)j. This iterative process is continued

until the convergence criterion,

Aej+l

1

 rtol

(69)
40

is satisfied. Selection of an appropriate r , is important

because it governs solution accuracy, number of iterations per

interval, and ultimate solution time. For this study r. , was

chosen to be 0.000001. While the error of this iterative procedure

is greatly reduced, it is still sensitive to numerical round-off.

Strain Histories Considered. All numerical simulations were

performed under strain control conditions using one of four input

histories. These included: An alternating square-wave strain rate;

a ramp up and ramp down strain rate; and two complex multi-step

strain rate histories. Restrictions which were arbitrarily imposed

on each loading sequence included: 1) the total response time

would be 35 seconds; 2) the total strain would not exceed 0.5%; 3)

all simulations would begin with a positive (or tensille) load; and

4) the numerical experiments would end with a 10 to 15 second

relaxation period. While these strain rate histories (see Fig. 4)

were not developed to model any specific physical phenomena, they

are representative of conditions under which a material could be

strained either in the laboratory or the field.

Qualitative and Quantitative Comparisons. The integration

investigation was divided into two tasks. The first task included a

comparison of the numerical stability of each constitutive theory

in order to access the degree of nonlinearity of the prescribed

growth laws. The comparisons were based upon predicted stress-strain

behavior for constant strain rate tests simulating monotonic loading.

Explicit Euler forward integration was used to obtain the stress
to 4J S- (O i a: ex3 CI -r-

§•£

(T3 4J

o: oo

CVJ

m

s- c\j oo•M

 i

•— o

CL4->

E ««

O T-

O 31

O) si

I/o)

•f—O:n) 4Jnsto

O) OtoJ C 4-

>

•i- n}

Cto C

i. T- O) (O

•(-> 1-

1— •(->

«r oo

CD

(U4J<o • 4Joo i X >> <u C i— O E </> O -iC_) 01

oo

Iooo
42

histories presented in Fig.'s 5 through 8 for various time steps.

From Fig. 8 it may be concluded that Miller's model is the most

sensitive to step size variation because of the oscillatory response

of the stress-strain curve for the larger At values.

The oscillatory nature of Miller's model can be traced back

to the hyperbolic sine function that is used to characterize the

inelastic strain rate and other growth laws. In contrast, the power

law function used by Bodner, Walker, and Krieg, to model a material's

response appears to be much more stable for a wide range of time

i ncrements.

Figures 5 through 8 also reveal that as the step size is

increased, all of the constitutive theories tend to overestimate

the stress history in the 0.1 to 0.2 percent strain region and

then automatically correct themselves as deformation continues.

The "self-correcting" phenomenon of predicted stress appears to be

an intrinsic property of the prescribed growth laws as is illustrated

in Fig's. 9 and 10. Figure 9 is a plot of the inelastic strain

computed by Walker's model for At's of 0.0005 and 0.1 seconds. If

the smaller time step is defined to predict the correct results, then

the inelastic strain computed when using the larger At is initially

underestimated. Because of the coupled nature of equations (1)

through (4), subsequent integration reverses this trend and results

in an over prediction of the inelastic strain. While not shown in

Fig. 9, the difference between the two curves assumes a more constant

value as the total strain is increased. That is, the difference

(or error) does not grow. Figure 10 shows the identical situation
43

CO

en

co

o>

CO

At- 0.5000-03 SEC BODNER MODEL IEULER) CORRECT SOLUTION

At- 0-5000-02 SEC BOONER MODEL IEULER)

At - 0.5000-01 SEC BOONER MODEL IEULER)

At - 0.1000 00 SEC BODNER MODEL IEULER)

At «= 0.125D 00 SEC BOONER MODEL IEULER)

16.0

14.0

12.0

10.0

8.0

6.0

4.0

2.0

0.0

I i i 1,1,1, 1

0.0 .001 .002 .003 .00-s- .005 .006

Strain (in/in)

Fig. 5. Mcmotonic Loading of Hastelloy-X at 1800° F

Predicted by Bodner's Model (e = 0.001 sec" 1).
ORIGINAL PAGE *S

OF POOR QUAttTY

44

At «= C.5000-03 SEC

At = 0.5000-02 SEC

At = 0.5000-01 SEC

At = 0-1000 00 SEC

At «= 0-1250 00 SEC

WPLKsfi M3DEL .' EUL-R !

WflLKER MODEL (EUUERi

HflLKER MODEL IEULER)

HRLKER MODEL (EULER)

HflLKER MODEL (EULER)

en

0

CO

16.0

14.0

12.0

10.0

6.0

6.0

4.0

2.0

0.0

I I I

0.0 .001 .00: .003 .004 .005 .006

Strain (in/in)

Fig. 6. Monotonic Loading of Hastelloy-X at ]800° F

Predicted by Walker's Model (e = 0.001 sec"1).
45

CO

(«

o>

s-.

j_>

to

16.Op14.0

12.0

10.0

8.0

6.0

4.0

2.0

At =

At =

At =

At =

At <*

C.5CCO-03 SEC

0.5000-02 SEC

0.5000-01 SEC

O.I ODD 00 SEC

0.125D OC SEC

KR1HG

RRIEG nOOEL tEULER)

KRiEC MODEL (EULERI

KRIEG MODEL IEULER:

KRIEO MODEL IEULER)

?£"" E3LU~lDi>

0.0

1 I

0.0 .001 .002 .003 .004 .005 -06

Strain (in/in)

Fig. 7. Monotonic Loading of Hastelloy-X at 1800° F

Predicted by Krieg's Model (e = 0.001 sec-').
ORIGINAL PAGE IS

OF POOR QUALrTY

46

V)

to

c

At « D.50CD-D:. S£C

At •= 0-500D-02 SEC

At = O-SOOD-01 SEC

At = 0-1000 00 SEC

At = 0-125D 00 SEC

f:: •_ •. E R MODEL ! EULER '

niLLER MODEL ILULER)

niLLER MODEL IEULER1

HILLER MODEL IEULER 1

niLLER MODEL IEULER)

16.0

14.0

12.0

10.0

8.0

6.0

4.0

2.0

0.0

i I

0.0 .001 .002 .003 .004 .005 .006

Strain (in/in )

Fig. 8. Monotonic Loading of Hastelloy-X at 1800° F

Predicted by Miller's Model U = 0.001 sec-1).
47

6.0

5.0

o

X

£ 4.0

u

c

o

<o

O)

c

3.0

2.0.

1.0

At

At

0.0005 sec

0.1000 sec

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Percent Total Strain

Fig. 9. Inelastic Stain Predicted by Walker's Model

For Hastelloy-X at 1800° F.
48

2.5

2.0

CM

I

o

to

Q.

CO

in

to

QJ

oo

jx:

o

ro

CO

1.5

1.0

0.5

0.0

-0.5

At

At

0.0005 sec

0.1000 sec.

0.02 0.04 0.06 0.08 0.10 0.12 0.14

Percent Total Strain

Fig. 10. Back Stress Predicted by Walker's Model

For Hastelloy-X at 1800° F.
49

arises in the prediction of the back stress. The over/under

estimation of computed stress continues until the equations

( (1) through (4) ) naturally dampen themselves out, as is shown

in Fig. 6. It should be noted that the "self-correcting" nature

of the constitutive models was only apparent when the Euler method

was used to numerically integrate the prescribed growth laws.

Figure 11 shows that when either the 4th order Runge-Kutta or

Trapezoidal method was used, the peak values of stress were simply

over- or under- predicted and there was no self-correcting tendency

observed.

The second task of the integration investigation was to

evaluate the constitutive theories considered herein in terms of

solution stability, accuracy, and computational efficiency when

numerically integrated using various algorithms. A test matrix

consisting of integration methods, time step sizes, and strain

input histories was established in order to make qualitative and

quantitative comparisons. However, before this phase of testing

could begin both a baseline or "pseudo correct" solution for each

input history was needed and the integration algorithms needed

to be functionally verified so that, in the limit, they would

predict the same response.

For each strain history considered, a baseline or "pseudo correct"

solution was obtained analytically, since no experimental data

were available. The Euler method was used to numerically integrate

the prescribed growth laws for each constitutive model until

successive changes in step size predicted an identical response.
O) XJ - Uo

.

iO.) O

i— O

i— o

•r- CO

t- 4->

O (XO CO I 0 o JC i—

4-> r—

01 (raCOU

(O co S- Q

.

O) OJ

O) 4->

+-> 10

c .

H-. Q) C 4J o c CO O) (O >

CL-^

E 3 O CT O UJ

r\;

(ISM) s
51

This process was repeated for the various input histories, thus

developing a family of baseline solutions.

Verification of the integration algorithms was accomplished

by using an a priori assumption that all numerical approximation

procedures would yield the same output stress histories if the step

size was small enough. Results of this investigation indicate

that the various integration schemes converge to the same solution

for the strain histories and constitutive theories considered

herein. In addition, the computational efficiency of each method

was obtained through selection of equivalent At's. Euler's method was

observed to be the most efficient (in terms of execution time on

an Amdahl 470V/8 computer),foil owed by the implicit Trapezoidal

method (a factor of 2 slower), then the 4th order Runge-Kutta method

(a factor of 4 slower), and finally the Trapezoidal PredictorCorrector method (a factor of 12 slower). A plausible explanation

for the relative slowness of the Predictor-Corrector method was the

selection of an inappropriate convergence criterion.

In the ensuing paragraphs qualitative and quantitative

comparisons are made for the various test matrix parameters

presented earlier. Since the amount of data generated during this

study was voluminous, only a representative sample of the results

will be presented.

Figure 11 demonstrates the response predicted by Miller's

model for a uniaxial, alternating square-wave strain rate input

history. The growth laws were integrated using four different

numerical approximations and a constant step size of 0.1 seconds.
52

The results indicate, for monotonic loading and unloading, that

integration by the 4th order Runge-Kutta method consistently overpredicts the state of stress by as much as 15%. Conversely, the

Trapezoidal method generally underestimates the stress history by

4%. The Euler and Predictor-Corrector methods appear to provide

accurate results at any point in the experiment. Similar observations were made for Walker's, Krieg's, and Bodner's model, when

subjected to the same strain rate history. These results indicate

that for equal time steps, the 4th order Runge-Kutta method

provides the least accurate results.

Figures 12 and 13 illustrate that various constitutive models

may differ appreciably when the same integration method is used

(in this case the Euler method). In Fig. 12 it can be seen that

Miller's theory gives a considerable oscillatory response for time

steps greater than 0.05 seconds, while Walker's theory shown in

Fig. 13 gives a much smoother response for the same time step.

Figures 12 through 15 also demonstrate that numerical stability

and stress-overshoot are strain rate dependent. Note that in

Fig's. 12 and 14 the oscillatory response of Miller's model does

not occur until a strain rate of 0.001 in/in/sec is applied.

In Fig's. 13 and 15 the amount of stress-overshoot (for the larger

step sizes) appears to grow with increasing strain rate.

Figures T6 and 17 present results predicted by Walker's model

for a ramp-up-ramp-down strain rate history. Unlike the constant

strain rate histories, this type of loading appears to affect the

predictive capabilities of the model. As is illustrated, the
53

OF POOR

A: =

CO

Vi

CO

A: =

At =

A: =

16.0

12.C

3.0

0-500D-0£ SEC MILLE R MODEL IEULER '

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Fig. 12. Stability and Accuracy of Euler Integration For

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Euler Integration (Hastelloy-X at 1800° F).
59

tensile response is well behaved and the peak response is estimated

fairly accurately for all step sizes. However, the compression

response is oscillatory and inconsistent for At's greater than 0.05

seconds.

Figure 18 shows a 35 second response obtained with Bodner's

theory using various integration algorithms. Time steps were

selected to obtain solutions which required approximately equivalent

computation times. As can be seen, the Euler method solution compares

favorably with the baseline prediction, whereas the 4th order RungeKutta and Trapezoidal methods tend to over- or under- predict the

peak response. Again, the time step used for the Euler method was

smaller than those used in the other methods but the total computation

time was the same. Similar observations were made for Walker's,

Krieg's, and Miller's models when evaluated under the same conditions.

Material Parameter Sensitivity Study

Another important issue this research addressed was what amount

of degradation in predicted response could be expected due to

variations in material constants. Since each model has a number of

material parameters which are determined through a set of complicated

tests, the end user must be made aware of the effect small variations,

i.e. typical experimental errors,have on the predictive capabilities

of the constitutive theory in question. In addition, the experimentalist will better understand the importance of setting up and

conducting accurate experiments from which the material constants are

obtained.
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61

Two procedures were developed to study the effect of material

parameter variation. The first procedure characterizes the

sensitivity of each constant within a particular'constitutive

model. Hence, the experimentalist will know what parameters need

to be determined most accurately. Each constant's sensitivity to

variation was determined by adjusting it by a prescribed amount and

comparing predicted results.

The second procedure provided an "upper bound" on the effect

that experimental error would have on the determination of the

material parameters and ultimately, predicted response. Therefore,

the experimentalist will know to what degree of accuracy the test

data need to be obtained in order to insure accurate results.

The effect of experimental uncertainty was studied by adjusting

test data by a prescribed amount (simulating some type of error)

and then using that data to determine the material constants.

The overall material parameter sensitivity study was performed

by subjecting Bodner's, Walker's , Krieg's, and Miller's models to

an alternating square-wave strain rate history. Euler's method was

used to numerically integrate the prescribed growth laws using a

time step of 0.005 seconds.

All nine of the material constants in Bodner's model were

individually adjusted by -5% in order to identify the parameters that,

i

when changed, would have the most pronounced effect on the predicted

response. Results of the sensitivity study indicate that the two

most sensitive constants were the stress exponent, n, and the

hardening recovery constant, D~, with the first of the two causing
62

the largest deviation from the baseline solution. The predicted

response, after each parameter was changed by -5%, is shown in Fig. 19.

The constant n, in Bodner's model, is determined from a series

of constant strain rate tensile tests, where the stress has reached

a saturated value. Through the use of equation (10) or by plotting

the data as depicted in Fig. 1 (a), it can be seen that the only

experimentally measured quantity that affects the value of n is the

stress. The parameter D~ is obtained primarily from constant stress

creep tests. Equation (13), which is used to compute D2, shows that

the measured creep rate and the peak stress values (which are used to

calculate the material constant n) can affect the determination of

this parameter. Hence, the aforementioned experimental quantities

must be measured very accurately in order to insure a minimal error

in the predicted response.

In addition to studying the sensitivity of individual constants

in Bodner's model, the effect of a ±5°/> change in the test data

(simulating experimental error) used to compute all of the material

parameters was investigated. From the procedure discussed above, the

stress exponent was shown to be the most sensitive constant.

Therefore, the values of stress, determined from constant strain

tests, were adjusted by ±5% in order to produce the largest change in n.

Then the creep rates, from constant stress creep tests, were

modified by ±5% in order to obtain the largest variance in D2- Hence,

with new values of stress, creep rate, n, and D^, the other seven

material parameters were re-evaluated and the resulting stress

history, as shown in Fig. 20, was predicted. However, Fig. 20
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65

indicated that as a result of the way the constants are determined

in Bodner's model, a ±5% change in the test data used to compute

the material parameters has little effect on the predicted response.

The sensitivity of the nine constants used in Walker's model

were evaluated by individually changing each parameter by -5% and

comparing the predicted stress history to a baseline solution.

Identified as the two most critical constants were the drag stress

(assumed to be a constant value in this model), D, and the stress

exponent, n, with the latter of the two being the most sensitive.

Shown in Fig. 21 is the predicted response when each parameter

(D and n) was adjusted by -5%.

To compute both the stress exponent and the constant valued

drag stress, the two maximum tensile stresses at the two highest

strain rates (from steady state hysteresis loops) are needed. In

addition to the above data, the saturated back stress in tension

(Bm, )» obtained from stress drop tests, is required. Values for

niaX

n and D may then be determined from equations (30) and (31),

respectively. In order to insure accurate output stress history

predictions, precise experimental measurements of a and Bmiv are max

necessary. It should be noted that Walker [42] indicated that it is

difficult to estimate the value of Bmav because the creep rates are

Hid X

usually small and it is very difficult to discern when the creep

rate ceases instantaneously.

The effect that experimental error would have on the determination

of the material parameters and ultimately, predicted response, was

investigated for Walker's model. By adjusting the experimentally
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67

measured values of stress from constant strain rate hysteresis loops

and saturated back stress by ±5% to create a worst case scenario,

each of the nine constants were sequentially recomputed. The effect

of experimental uncertainty on the output stress history for Walker's

theory is shown in Fig. 22. It can be seen that a ±5% error in all

the test data required to compute the material parameters results in

significant predicted response errors, with up to 30% over-prediction in

the stress during the relaxation period.

In a similar fashion, Fig. 23 illustrates the sensitivity of

the stress exponent, n, and initial drag stress, D, to a -5% change

in Krieg's model. Since these constants are analogous to the

parameters, n and D, used in Walker's model, it is simply reiterated

that the measured values of a and EJ must be determined very max

precisely in order to insure an accurate estimation of the predicted

response. However, from Fig. 24 it can be seen that even after the

constants are re-evaluated to reflect a ±5% change in the experimental test data, the output stress history is virtually unaffected.

Hence, Krieg's theory appears to be more tolerant to experimental

uncertainty than any of the models previously discussed.

The nine constants in Miller's model were individually varied

by -5% in order to characterize their sensitivity. However, unlike

the preceding models, selectively changing any one parameter in

this set of constitutive equations had little effect on the predicted

response. In any event, the most sensitive constants were found to

be n and A2- Figure 25 illustrates the effect that a -5% change in

each of these parameters had on the computed stress history.
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72

In order to minimiz e the error in the computation of n, the

steady state asymptotic stresses obtained from hysteresis loops

executed at several strain rates must be measured very accurately.

In addition, to determine the parameter A^, one must precisely

establish the value of the steady state back stress from stress

drop creep tests.

Whil e Miller's model appears to be fairly insensitive to small

variations in particular material constants, when the parameters

were collectively re-evaluated (to simulate experimental error in

the test data) an appreciable deviation from the baseline solution

was noticed. Shown in Fig. 26 is the output stress history resulting

from a ±5% change in the steady state stresses and back stresses

that were used to recompute the respective constants. As can be

seen, a significant over-prediction in peak stresses occurs.
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(ISM)
74

CHAPTER V

CONCLUSIONS

In conclusion, the sensitivity of the unified constitutive

theories proposed by Bodner, Walker, Krieg, and Miller, to numerical

integration techniques and slight changes in material parameters was

investigated. The integration sensitivity study involved the use of

the Explicit Euler Forward Difference method, the Implicit

Trapezoidal method, the Runge-Kutta 4th Order method, and the

Trapezoidal Predictor-Corrector method. From the results of this

study, the following observations were made:

1) It appears that a simple integration scheme, like the Euler

Forward Difference Method, is preferable to more complex

schemes from the standpoint of accuracy, computation time,

and ease of implementation. The results also indicate

that under the conditions tested, the 4th Order Runge-Kutta

method provided the least accurate results.

2) The constitutive models may differ appreciably, in terms

of numerical stability, when the same integration method and

strain input history is used. Miller's model was shown to

be the most sensitive to time step variation, which is

apparently due to the hyperbolic sine function that is used

to characterize the inelastic strain rate function and other

growth laws.

3) The predicted stress response appears to be "self correcting"

for large time increments when the constitutive equations

are integrated by the Euler method. However, when either
75

the 4th Order Runge-Kutta or Trapezoidal method was used, the

peak values of stress were simply over- or under- predicted

and no self-correcting tendency was observed.

The material parameter sensitivity study involved the

characterization of the sensitivity of each constant within a

particular constitutive theory and provides an "upper bound" on the

effect that experimental error would have on the determination of the

material parameters and ultimately, predicted response. From the

results of this sensitivity study, the following observations were

made:

1) Small changes (5%) in the material constants may produce

significant and unacceptable errors in the predicted stress,

particularly the peak stress amplitudes during cyclic

loading, and the amount of residual stress during stress

relaxation.

2) From.this numerical study, it would appear that more precise

predictions can be achieved if the experimentally obtained

data can be measured more accurately. The type of tests and

data that are required to determine the material parameters

for each model vary, however some generalizations can be

made. For example, most theories require either monotonic

or cyclic loading of the specimen in order to obtain peak

or saturated values of stress for several constant strain

rates. In addition, most models require a stress drop or

constant stress creep test in order to determine the steady

state or saturated back stress. While other specific
76

experiments may need to be performed, those mentioned above

are probably the most critical to obtaining accurate

analytical predictions.

It should be noted that the conclusions reached herein are based

on the use of uniaxial simulations. It is expected that the

conclusions would hold for multi axial situations; however, further

evaluation is needed in this area.
77

REFERENCES

1. Beek, J,M., Allen, D.H., and Milly, T.M., "A Qualitative

Comparison of Current Models for Nonlinear Rate-Dependent

Material Behavior of Crystalline Solids," MM 4246T-83-14,

Mechanics and Materials Research Center, Texas A&M University,

1983.

2. William, K.J., "Numerical Solution of Inelastic Rate Processes,"

Computers & Structures, Vol. 8, p. 511, 1978.

3o Shin, C.F0, Delorenzi, H.G., and Miller, A.K., "A Stable

Computational Scheme for Stiff Time-Dependent Constitutive

Equations," Transactions of the 4th International Conference

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9. Gear, C«,W«,, "Algorithm 407 -- DIFSUB for Solution of Ordinary

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11. Cormeau, I.e., "Numerical Stability In Quasi-Static Elasto/

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and Creep in Elastic Solids—A Unified Numerical Solution

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15. Hughes, J.R., and Taylor, R.L., "Unconditionally Stable

Algorithms for Quasi-Static Elasto/Visco-Plastic Finite

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16. Krieg, R.D., "Numerical Integration of Some New Unified

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17. Tresca, H., "Notes on Yield of Solid Bodies Under Strong

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ou 1' elasticite pourrait les ramener a leur premier eata.,"

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Hierduch im Material Hervorgerufenen Nach-Spannungen,"

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Waltman, Jr., p. 43, 1925.
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22. Prager, W., "The Theory of Plasticity: A Survey of Recent

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N86-30231

APPENDIX 6.3
ON THE USE OF INTERNAL STATE VARIABLES IN

THERMOVISCOPLASTIC CONSTITUTIVE EQUATIONS

D.H. Allen and J.M. Beek

Aerospace Engineering Department

Texas A&M University

ABSTRACT

The use of internal state variables in modeling of inelastic solids is

gaining widespread usage in current research. Therefore, it is useful to construct a well-defined framework for internal state variable models which is

based in continuum mechanics. The objective of this paper is to review and

clarify the general theory of internal state variables and to apply it to inelastic metals currently in use in high temperature environments. In this

process, certain constraints and clarifications will be made regarding internal

state variables.

It will be shown that the Helmholtz free energy can be utilized to construct constitutive equations which are appropriate for metallic superalloys.

Furthermore, internal state variables will be shown to represent locally averaged measures of dislocation arrangement, dislocation density, and intergranular gracture. Finally, the internal state variable model will be demonstrated to be a suitable framework for comparison of several currently proposed models for metals and can therefore be used to exhibit history dependence, nonlinearity, and rate as well as temperature sensitivity.

INTRODUCTION

The prediction of inelastic behavior of structural materials at elevated

temperature is a problem of great importance which has accordingly been given

a great deal of interest by the research community in recent years. These

materials exhibit substantial complexity in their thermomechanical constitution. In fact, so complex is their material response that it could be argued

that without useful a priori information, experimental characterization is futile. The purpose of this paper is to show how the thermodynamics with internal state variables can be utilized to emplace certain constraints on the

allowable form of thermomechanical constitutive equations, thus providing some

limited insight regarding experimental requirements.

Historically, there have been two distinct approaches to the modelling

of inelastic materials: 1) the functional theory [1], in which all dependent

variables are assumed to depend on the entire history of independent variables;

and 2) the internal state variable (ISV) approach [2], wherein history dependence is postulated to appear implicitly in a set of internal state variables.

Lubliner [3] has shown that in most circumstances ISV models can be considered

to be special cases of functional models. For experimental as well as analytic reasons numerous recently proposed models for the classes of materials
discussed herein have been proposed in ISV form. Therefore, in this paper

the ISV method will be reviewed as well as clarified and it will be shown that

this general framework is useful in modeling metals at elevated temperature.

The paper begins with a review of ISV theory, and this is supplemented

with a section describing the procedure for constructing macroscopically averaged internal state variables. These concepts are then applied to metals'

at elevated temperatures. Finally, applications to boundary value problem

solving techniques are discussed.

REVIEW OF THE INTERNAL STATE VARIABLE (ISV) APPROACH

The concept of internal state variables, sometimes called hidden variables, was apparently first utilized in thermodynamics by Onsager [4,5] and

numerous applications are recorded since the second world war [2,6-14]. Although not originally described for application to solids, the approach which

will be discussed herein is due to Coleman and Gurtin [2].

In the theory of internal state variables applied to solids the following state variables are required in order to fully characterize the state of

the body at all points x. and at all times t:*

1) the displacement field u. = u.(x, ,t) ; (1)

* 1 1 K.

2) the stress tensor a.. = a..(x. ,t) ; (2)

ij 13

 k

3) the body force per unit mass f. = f. (x, ,t) ; (3)

i i k

4) the internal energy per unit mass u = u(x, ,t) ; (4)

tc

5) the heat supply per unit mass r = r(x, ,t) ; (5)

6) the entropy per unit mass s = s(x, ,t) ; (6)

7) the absolute temperature T = T(x ,t) ; (7)

rC

8) the heat flux vector q. = q. (x, ,t) ; (8)

i i K

and

9) c = c (xm,t) , k = 1, 2 n ; (9)

•K

where c.^. are a set of n internal state variables which are necessary to

account for inelastic material behavior. Although they are listed here as

second order tensors, they may be tensors of other rank as well [15].

* For convenience, only infinitesimal deformations will be considered here,

although the general theory applies to finite deformations as well.
The method of Coleman and Noll [16] may be used to obtain the spatial

and time distribution of the body force f^ and heat supply r from the conser

vation of linear momentum and energy, respectively, assuming the displacements u-^ and the temperature T are specified independent variables. Subsequently, it is hypothesized that constitutive equations of state may be constructed for the state variables described in (1) through (8) in terms of

'

 amn(xk't})

u. and T and their spatial derivatives:

U(VC)

 - ^WV^'

 T(Vt)

,

 8m(

V

C)' a

mn(xk't)} :.

(11)

s(xk,t) = s(emn(xk,t), T(xk,t), gm(xk,t), aPn(xk,t)) ; (12) and

q.(xk,t) = q^Oc^t), T(xk,t), gjx^t), ajn(xk,t)) ; (13)

where g is the spacial temperature gradient T ,m and

e. . = %(u. . + u. .) . (14)

ij i.J J.1

The form of equations (11) through (13) implies that all constitutive equations are evaluated in the specified state (x^.t). For this reason 3jH ,

 u

»

 s> anc* 1± are termed observable state variables since they

can be determined from equations of state even though there is implicit

history dependence via the internal state variables o^ , which are defined

to be of the form:

«ij — ij( £mn' T' V*mn>

 ;

where time and spacial dependence have been dropped for notational convenience. If equations (15) are at all times integrable in time, then

the following form is equivalent to (15):

= /

J •

where t is the time of interest and t' is a dummy variable of integration. Therefore, it is apparent that a^j are not directly observable

at any time and must therefore be considered to be hidden or internal.

Although the above framework has been shown to be applicable to rate

dependent crystalline solids [17,18], it is often misconstrued that the absence of explicit, strain-rate dependence renders the model inappropriate

for use in viscoplasticity theories. It is alternatively hypothesized that

a. . = o. .(£ , e , T, g , a'c ) (17)

ij ij mn mn

 &m mn

is an appropriate form of thermomechanical constitutive equations (10). Although metals at elevated temperature certainly exhibit strain-rate dependence, there are several reasons why equations (17) are less desirable than

equations (10). First, equations (17) are not actually equations of state

since the inclusion of strain rate implies knowledge is required at some time

other than the current time t. Secondly, as demonstrated in discussions of
materials similar to (17) but without internal state variables [19], very

little useful information will come from thermodynamic constraints. Finally,

explicit strain rate dependence is actually redundant for the materials discussed herein, as will be shown later. Therefore, although this is certainly

a semantical issue, equations (10) through (13) and (15) are utilized as the

constitutive model in the balance of this paper.

It should also be pointed out that internal state variable growth laws

(15) could contain explicit strain-rate dependence:

a.. = nk.(£ , e , T, e , a£ ) , (18)

ij ij ran mn °m mn

as in the example of a rate independent elastic-plastic material , in which

equations of the above form are linear in strain rate:

a*. = fik. (e , T, e , <xA )e . (19)

ij ijpq mn °m mn pq

Such a form, although not excluded by the principle of equipresence [20],

is only necessary in the circumstance wherein specific rate independence is

required, as can be demonstrated by direct substitution of (19) into (16).

Furthermore, although the thermodynamic constraints will vary somewhat when

(19) are utilized [21,22], the results will be quite similar to those described

below.

. On the basis of the Coleman-Mizel procedure [23] it can be shown that

satisfaction of the first and second laws of thermodynamics for the class of

materials detailed above will lead to the following conclusions:

h 5 u - Ts = h(e , T, ak ) ; (20)

mn mn

where h is the specific Helmholtz free energy;

3h /-ION

s = - 3r - ; (22)

and

)

 •

 (23)

Equations (21) should not be interpreted as defining as hyperelastic material

since the Helmholtz free energy, described by (20), is dependent on the internal state and therefore path dependent.

Although not directly related to our problem, it is useful to note that

the path dependence of the Helmholtz free energy precludes the usefulness

of equations (21) in Rice's J-integral for fracture mechanics [24]. However,

in the case wherein the loading path is radial:

e. . = k. .1 ; o^. = k*.I ; I = V~£ . . e . . , (24)
v

where kjj and kjj are constant coefficients, then it is well known that equations (15) are directly integrable so that the free energy can be described

by

h(e , T, a1 ) = h(e , T, a1 (e )) =

mn pq mn pq mn

U , T)

mn

(25)

Thus, for the case of proportional loading only, the constitutive equations are derivable directly from a potential function and the J-integral

method is applicable.

THE LOCAL AVERAGING PROCESS

Constitutive equations (10) through (13) and (15) are theoretically

pointwise in nature; that is, they are applicable to fixed infinitesimal

material points. However, practically speaking, there is no way to construct

experiments on material points since at the microscopic level the continuum

assumption becomes invalid. Rather, it is considered acceptable to construct

constitutive equations by subjecting local specimens to surface deformations

(or tractions) which lead to spacially homogeneous stresses and strains so

that some local average of the pointwise observable state variables can be

determined directly from the effects on the boundaries of the specimens.

As shown in Fig. 1, the scale of the smallest dimension of a local specimen is generally constructed so as to be at least an order of magnitude larger

than the scale of the largest material inhomogeneity. This sizing helps preserve the continuum assumption while at the same time averaging out the effects

of point defects such as crystal lattice dislocations. Conversely, the scale

of the largest dimension of a typical specimen should be as small as possible

compared to the scale of the global boundary value problem of interest. This

constraint is necessary in order to preserve the notion that constitutive equations are indeed pointwise in nature, but v ^

it is pragmatic in that it is a simple

matter of economy.

The local rather than pointwise constitutive equations that result from experimentation are assumed to be of the

same form as pointwise equations (10)

through (13) and (15). For example, in

the uniaxial test described in Fig. 1

it is customary to define

'11

'11

global

domain of

interest

(26)

(27)

physical

inhomogeneitied

*

'

• ' , •

* > i .

•

local

specimen

X^ L

I

Figure 1
and

T = , (28)

where L is the local specimen gage length, A is the cross-sectional area in

the x9-x3 plane, and (a±, a2, a-j) is some arbitrary point on the surface of

the specimen. Utilizing these quantities, it is then hypothesized that

T

'

wherec.

-ja =i _ i

 rr

 r l £ a dx , , . dx

mn V J mn 1 2

 dx3

(29)

, (30)

and all quantities with bars represent the locally measured state variables.

Although equations (29) represent an often used way of relating pointwise equations to experimental results, the local averaging process is nevertheless fraught with shortcomings since definitions (26) through (28) all reresent nonunique relations between pointwise_state variables a±j , cjj , T, a^

and their locally defined counterparts 'ai j , £.j_j , T, and a^j . There are in

fact an infinite number_of distributions ^^(x^, x2, x-j) which will result

in identical values of ot^. However, assuming that the scale of inhomogeneities is small and that the distribution of o^ is random the specimen will

be statistically homogeneous and the relation between oTMn and o^n will be

reasonably one to one.

For example, suppose that during some monotonically increasing local

strain history £^ a particular internal state variable an such as a single

dislocation arrangement is governed on a pointwise basis by the almost discontinuous behavior shown in Fig. 2. Suppose further that the time t at

which the internal state begins to change

is determined by the pointwise stress

state. Then the number of dislocation

rearrangements occurring in the local

specimen as a function of time might be

distributed as shown in Fig. 3. If the

local specimen is large compared to the

scale of the dislocation, and there are

numerous dislocation rearrangements, as

is usually the case in testing of metals,

then the peak of the curve shown in Fig.

3 will be several orders of magnitude

greater than unity. It follows from

equations (30) that the locally averaged

value of the internal state variable

represented in Fig. 2 will be as qualitatively shown in Fig. 4.

t ,'d't

'11

-»t

1

"11'

Figure 2
APPLICATION TO METAL CONSTITUTION

In order to describe the class of metals discussed herein, the free en- p

ergy maybe expanded in terms of the elastic strain tensor c£^ and the temperature T in a second order Taylor series expansion as follows:

1_E« IJ- iT

£

ij =

 £ij * £ij - £ij ' '

T I where e^-j is the thermal strain tensor and e-n is the inelastic strain tensor,

considered to be an internal state variable [17,18,22,25], and

- W (T - V ' (32)

where the subscript R refers to quantities in the reference state, D. is

/32h\

 1J

the linear elastic modulus tensor, and Cv E -TI^=-2") is the specific heat at

constant elastic strain. Substitution of equation (32) into (21) will result

in

a, = D. . (e - s1 - £T ) . (33)

kl klmn mn mn mn

The above equations, together with internal state variable growth laws (15),

will be shown to be a suitable framework for comparison of all of the models

to be discussed herein.

Internal State Variables in Metals

It is now generally agreed in the literature that in single crystals there

are two locally averages internal state variables: the back stress (a )

ij

representing dislocation arrangement; and the drag stress (a?) representing

dislocation density; where the bars have been dropped for convenience and the

superscript has been converted to a subscript in order to avoid the confusion

which would arise if a state variable were raised to some power. For obvious

reasons the back stress is a second order tensor, whereas the drag stress is

a scalar. In specimens composed of multiple crystals it is generally agreed

that a third internal state variable loosely termed damage (a-j ) is necessary in order to account for intergranular mechanisms such as grain boundary

sliding and microvoid growth and coalescence that may occur at high temperature and/or large strain. Although damage is obviously a directionally related quantity and therefore tensorial in nature, it is difficult to distinguish phenomenologically between damage and drag stress since both are primarily stiffness reducing mechanisms.

Within the thermodynamic framework described earlier it is also possible

to define the inelastic strain tensor to be an internal state variable. However, this interpretation is not generally utilized within the materials
literature. It is hypothesized that the

rate of growth of the internal state variables does not depend on the inelastic

strain tensor so that

no. of dislocation

rearrangements in

a local 4

specimen

\j

 = • (34)

mn mn

Due to the form of equations (34) it is

said that since the inelastic strain tensor does not appear on the right hand

side it is not an internal state variable. However, within the framework defined herein, it is still possible to

construct an internal state variable

growth law of the form

Li -

;ij '= V£

mn'T'Val 'Va3

(35)

mn nin

-1

-Hwhich is precisely in agreement with definitions (15).

In order to qualitatively verify the

supposition that the inelastic strain tensor can be regarded to be an ISV, consider

the example of a uniaxial bar subjected to

applied displacements such that the end

tractions will be evenly distributed. It

is customary to deduce the inelastic strain

in an experiment of this type by utilizing

the output from a load cell to determine

the stress and then making use of equations

(33) to determine the elastic strain. This result and the total strain measured by an extensometer are then substituted into equations (31) to determine the inelastic strain. Nevertheless, this does not imply that the inelastic strain tensor in an observable state variable. This result can be arrived

at only in constitutive experiments such as uniaxial bar tests in which the

stress and strain tensors are spacially homogeneous. In heterogeneous boundary value problems, only two state variables may be input (temperature and

either stress or strain), and for this case equations (31) and (33) must be

supplemented with an ISV growth law of the form of equations (35) in order

to determine the inelastic strain tensor. Therefore, in the context of the

current thermodynamic framework the inelastic strain tensor may be interpreted

to be an ISV.

dt

Figure 4

A Framework for Current Metals Models

In order to establish that current models can be constructed from equations (33), consider the standard solid shown in Fig. 5. The governing differential equation for this analog is

n F . "M • TM

M

(36)
where by convention the stress is denoted

G and the strain is e. Equation (36) may

be written in the following equivalent

form

\

 [EM+EJ '

(37)

00 .

In accordance with the instantaneous linear elastic behavior of metals, it is assumed that E

E

M

+E°° = E = Young's modulus = constant, (38)

so that it is clear that equation (37) can

be integrated in time to give the following _

stress formulation j !

O(t ) I !

£(t ) = ——- + ei

L

(t ) . (39) f r

 & J- ; a

where £ is the inelastic strain, defined

by

/

I E ra_E £T , Figure 5

IT —lf^dt ' (40>

Equation (39) may be solved for the stress and substituted into equation (40)

so that it is clear that equation (40) is in accordance with ISV growth laws

(16). Further, it can be seen from the standard solid analog in Fig. 5 that

since o-E^E represents the stress in the Maxwell element, E1 is not observable, so that £•"- satisfies the two conditions required for it to be an internal

state variable.

Equation (39) may be written equivalently in the following strain formulation:

^ = E[£(tl) - £I(t1)] . , (41)

which is an equation of state compatible with constitutive equations (10) as

well as equations (33) . Since no other internal state variables are present

in this equation, and also, no additional internal state variables are present

in growth law (40) it is apparent that the standard solid analog with constant

coefficients E , n and E^ is a single internal state variable model.

It has been noted by several researchers that the standard solid is an

appropriate analog for thermoviscoplastic metals if the springs and dashpot

are nonlinearized [26,27]. In order to demonstrate this feature, consider a

multiaxial extension of equation (36):
c+Ka=G£+M £ , (42)

pq pqmn mn pqmn mn pqmn mn

where by convention the small strain tensor £.jj is used in conjunction with

the work conjugate stress tensor a^ . In order to model metals

and Mpqmn are required to be nonlinear in some as yet undetermined way. In

addition, in accordance with, constraint equation (38), it is required that

K?! M . - = D. ., . , (43)

ijmn mnkl ijkl

where Djjkl is the linear elastic modulus tensor. Equations (42) may be rewritten in a strain formulation equation of state form as follows:

where £r is the inelastic strain tensor, defined by

•L

 s

 /' ^ [a -G £ ]dt . (45)

pq pqmn mn

Substituting equations (43) and (44) into equations (45) will result in

t

e

1. = / {KT! [£ - E1 ] - M?1 G £ } dt , (46)

ij J ijmn mn mnj ijpq pqmn mn

—oo

so that equations (46) are in accordance with growth laws (.16). The number

of internal state variables contained in the model will depend on the degree

of nonlinearity proposed in the nonlinear tensors Kpqmn, Gpqmn, and Mpqmn,

and this will be discussed in the following section. However, before continuing, it should be pointed out that the constitutive equations developed in

this section assume that the elastic and inelastic strain tensors may be linearly decoupled. It has been shown that this assumption is invalid for finite

deformation [28]. However, even under finite deformation conditions the inelastic strain is decoupled from the elastic strain in such a way that the

inelastic strain tensor may be considered to be an internal state variable.

Current Models for Metals

The framework for metals models discussed in the previous section can

be used to describe numerous models currently under development [26,27,29-58].

For example, the microphysically based isothermal model proposed by Krieg,

et al. , [30] is of the form described by equations (33):

where
e

1. = e -<

1J O

pq pq

and £0 and m are material constants, and ajj is the deviatoric stress tensor

and a[ is the deviatoric component of the back stress tensor. Since equations (48) contain the stress tensor, substituting equations (33) into (48)

will result in equations consistent eith growth laws (15). In addition, Krieg,

et al., give the back stress and drag stress to be, respectively,

a

 -1

pq pq

and

2 * \ j *S ~

 r

R '

where Aa and A^ are hardening constants, and ra and rR are recovery functions

of temperature and internal state variables. It can be seen that since ISV

growth laws (49) and (50) are consistent with equations (15) , the model proposed by Krieg, et al., contains three internal state variables: the inelastic

strain tensor, the back stress tensor, and the drag stress tensor.

Furthermore, classical plasticity theories can be described by the general form

a

ij

 = Dijmn (£mn ' ^ > <5l>

where

%

 = * 3^7 • <")

•

A is a scalar valued function of state, and F is a scalar valued state function for inelastic behaviour often taken to be the yield function. If F is

described by the von Mises yield criterion [53], given by

F(a - a ) = Js(a - a ) (a - a ) = k2 , (53)

J ij

 1J L±j 1J l±j

where a^ is a tensor describing the yield surface center in stress space and

k is a constant representing the yield surface size, then equations (52) can

be written as
eij =

 x(0ij-al..) ' (54)

IJ

resulting in a kinematic hardening model with constant yield surface size.

Substitution of equations (51) into the above will yield a result consistent

with rate independent ISV growth laws (19).

Furthermore, if the yield surface translation is derived from the Ziegler

modification [60] of the Prager work hardening rule [61], it may be described

by

~

 ai '

ij ij

•

where y is a scalar valued function of state. By use of equations (51), equations (55) can also be shown to be consistent with' equations (19). Therefore,

a classical plasticity-based kinematic hardening model contains two internal

state variables: the inelastic strain tensor and the yield surface translation tensor representing the back stress.

In order to further illustrate the applicability of equations (33), (35)

and (15) to current models for metals, ten of these models have been cast in

uniaxial form in Table 1 , wherein it is shown that although the framework for

each model is identical (Valanis ' model is in simplified form), the ISV growth

laws vary widely both in number and form.

CONCLUSION

The main content of this paper has been to review and clarify the continuum

and thermodynamics based internal state variable model for application to thermoviscoplastic metals. In this process the following points have been made:

1) the definition of an internal state variable utilized in this model

has been clarified;

2) internal state variables in metals represent local averages of dislocation arrangement, dislocation density, and intergranular damage,

3) in the context of the ISV definition given here, inelastic strain may

also be interpreted as an internal state variable;

4).the path dependent Helmholtz free energy may be expanded in a second

order expansion in elastic strain and temperature in order to obtain a stressstrain equation of state;

5) rate dependence enters the constitutive equations implicitly via the

inelastic strain, as demonstrated by the nonlinear standard solid analog; and

6) a three-dimensional generalization of the standard solid may be used

as a means of comparison of the general form of several currently proposed

models.
Further ramifications of the ISV model discussed are also of importance,

although not detailed herein. For example, this model may be utilized to construct a coupled heat conduction equation which may be utilized to predict

heat, generation in thermoviscoplastic metals [62]. Furthermore, the concept

of internal state variables may be utilized to construct models for the mechanical constitution of composites with damage [63,15,65,66].

ACKNOWLEDGEMENT

The authors gratefully acknowledge support for this research which was

sponsored by NASA Lewis Research Center under grant no. NAG 3-491.

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9. Biot,.M.A.: Variational Principles in Irreversible Thermodynamics with

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10. Ziegler, H.: An Attempt to Generalize Onsager's Principle, and Its Significance for Rheological Problems. Z. angew. Math. Phys., Vol. 9,

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13. Schapery, R.A. : Application of Thermodynamics to Thermomechanical, Fracture and Birefringent Phenomena in Viscoelastic Media. Journal of Applied Physics, Vol. 35, p. 1941, 1964.

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Irreversible Thermodynamics. Proceedings 5th U.S. National Congress

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Elastic-Visco-Plastic Materials. Journal of Applied Physics, Vol. 41,

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18. Kratochvil, J., and Dillon, O.W. , Jr.: Thermodynamics of Elastic-Plastic

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23. Coleman, B.D., and Mizel, V.J.: Thermodynamics and Departures from Fourier's

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42, No. 2, pp. 305-389, 1975.

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34. Bodner, S.R., Partom, I., and Partom, Y.: Uniaxial Cyclic Loading of

: Elastic-Viscoplastic Materials. J. App. Mech., 1979.

35. Cescotto, S., and Leckie, F.: Determination of Unified Constitutive

Equations for Metals at High Temperature. Proc. Int. Conf. on Constitutive Laws for Eng. Mater., pp. 105-111, 1983.

36. Robinson, D.N.: A Unified Creep-Plasticity Model for Structural Metals

at High Temperatures. ORNL-TM-5969, October 1978.

37. Miller, A.K.: An Inelastic Constitutive Model for Monotonic, Cyclic, •

and Creep Deformation: Part I - Equations Development and Analytical

Procedures and Part II - Application to Type 304 Stainless Steel.

ASME J. Eng. Mater. Tech., Vol. 98-H, p. 97, 1976.

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39. Liu, M.C.M., and Krempl, E.: A Uniaxial Viscoplastic Model Based on

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391, 1979.

40. Cernocky, E.P., and Krempl, E.: A Theory of Viscoplasticity Based on

Infinitesimal Total Strain. Acta Mechanica, Vol. 36, pp. 263-289, 1980.

41. Cernocky, E.P., and Krempl, E.: A Nonlinear Uniaxial Integral Constitutive Equation Incorporating Rate Effects, Creep and Relaxation. Int.

J. Nonlinear Mechanics, Vol. 14, pp. 183-203, 1979.

42. Valanis, K.C.: A Theory of Viscoplasticity Without a Yield Surface Part

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43. Valanis, K.C.: A Theory of Viscoplasticity Without a Yield Surface Part

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44. Valanis, K.C.: On the Foundations of the Endochronic Theory of Viscoplasticity. Arch. Mech., Vol. 27, pp. 857-868, 1975.

45. Allen, D.H., and Haisler, W.E.: A Theory for Analysis of Thermoplastic

Materials. Computers and Structures, Vol. 13, pp. 124-135, 1981.

46. Allen, D.H.: Computational Aspects of the Nonisothermal Classical Plasticity Theory. Accepted for publication in Computers and Structures,

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47. Kocks, U.F.: Laws for Work-hardening and Low-Temperature Creep. Journal

of Engineering Materials and Technology, Vol. 98-H, pp. 76-85, 1976.

48. .Cailletaud, G., and Chaboche, J.L.: Macroscopic Description of the Microstructural Changes Induced by Varying Temperature: Example of INI00

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•'.'.'.''•/ iour of Materials, Vol. 2, pp. 23-32, ICM3, Cambridge, England, August

: .'•-. 1979.

49. Lee, D., .and Zaverl, F., Jr.: A Generalized Strain Rate Dependent Constitutive Equation for Anisotropic Metals. Acta Metallica Vol. 26, No.

11, p. 1771, 1978.

50. Laflen, J.H., and Stouffer, D.C.: An Analysis of High Temperature Metal

Creep: Part I - Experimental Definition of an Alloy and Part II - A

Constitutive Formulation and Verification. ASME J. Eng. Mat. & Tech.,

Vol. 100, p. 363, 1978.

51. Zienkiewicz, O.C., and Cormeau, I.C.: Visco-Plasticity — Plasticity

and Creep in Elastic Solids — A Unified Numerical Approach. Int.

Journal for Numerical Methods in Engineering, Vol. 8, pp. 821-845,

1974.

52. Perzyna, P.: Fundamental Problems in Viscoplasticity. Advan. Appl.

Mech...Vol. 9, pp. 243-377, 1966.
53. Perzyna, P., and Wojno, W.: Thermodynamics of Rate Sensitive Plastic

Materials. Arch. Mech. Stos. , Vol. 20, p. 5, 1968.

54. Perzyna, P.: The Constitutive Equations for Work-Hardening and RateSensitive Plastic Materials. Proc. Vibr. Probl., Vol. 4, pp. 281-290,

1963; Bull. Acad. Polon. Sci., Ser. Sci. Tech., Vol. 12, pp. 199-206,

55. : Perzyna, P.: On the Thennodynamic Foundations of Viscoplasticity. Symposium on the Mechanical Behavior of Materials Under Dynamic Loads.,

San Antonio, Texas, 1967.

56. Yamada, Y., and Sakurai, T.: Basic Formulation and a Computer Program

for Large Deformation Analysis. Pressure Vessel Technology, Part 1,

ASME, pp. 341-352, 1977.

57. Yamada, Y.: Constitutive Modelling of Inelastic Behaviour and Numerical

Solution of Nonlinear Problems by the Finite Element Method. Computers and Structures, Vol. 8, pp. 533-543, 1978.

58. Snyder, M.D., and Bathe, K.J.: Formulation and Numerical Solution of

Thermo-Elastic-Plastic and Creep Problems. National TechnicalInformation Service, No. 82448-3, June 1977.

59. von Mises, R.: Mechanik .der Festen Koerper im Plastich Deformablen

Zustand, Goettinger Nachr. , Math.-Phys. Kl., pp. 582-592, 1913.

60. Ziegler, H.: A Modification of Prager's Hardening Rule. Quarterly of

Applied Mathematics, Vol. XVIII, pp. 55-65, 1959.

61. Prager, W.: The Theory of Plasticity: A Survey of Recent Achivements.

Proceedings of the Institution of Mechanical Engineers, London, Vol.

169, pp. 41-57, 1955.

62. Allen, D.H.: A Prediction of Heat Generation in a"Thermbviscoplastic

" Uniaxial Bar. Texas A&M Mechanics and Materials Center, Report no.

.:.-', MM 4875-83-10, July, 1983 (to be published in International Journal

of Solids and Structures).

63. Schapery, R.A.: On Viscoelastic Deformation and Failure Behavior of Composite Materials with Distributed Flaws. Advances in Aerospace Structures and Materials, ASME AD-01, pp. 5-20, 1981.

64'. Talreja, R. : A Continuum Mechanics Characterization of Damage in Composite Materials. The Technical University of Denmark, Lyngby, to

be published.

65. Allen, D.H., Groves, S.E., and Schapery, R.A.: A Damage Model for Continuous Fiber Composites. In preparation.
TABLE 1. COMPARISON OF UNLAXIAL MODELS

Theory

Cemocky

and

Kreopl

Swear eng en.

and

Rohde

Bodner

et al.

Walker

Miller

-

StreM-Soaln

lULwtian

(Tl) o-Efc-c1^

1

]

(T3) o-Etc-c1]

(T7) o-ELt-c1]

(T10) o-Efc-c1]

(T14) o-Ete-rc1-.

1

]

*

Internal State Variable Qrcwch Law

(T2) i1-^-

'O O fll 5S

. i r|o-0l|-|c2 $2

/i»/ \ A ; !• _| 1 | ___ /_ \ «^ *%•» Vl*+y c •• C| i — — — i sgn^o~o» j yg ^^

•1 *o 1 * . ^n f~"

c o 2

(T5)1'

2

 i^Cji^o2!*

 5"l

-l]sgn(o1) g g

•J T 3^ £^

(T6) a->°c £ L|c 1- Cn[a*>-c»o 1° fTM ^'*1

o ^ ,„

•< t«

r nr "2o

(T8)1 i1- -

2

-D0e~l-"'EuL";zJ ,gn(o)

_ __ _r

(T9) i2.[Z1-a2]Wp.AZ1[!2

T-lJ

1 I P°-0llln

(Til)1 cM— ^- 8gn(o-a2)

(Ti2)2_.l-[nl^2]i1

 n I j .1 , «5R + ' ^

O

i°i 0

n

i

l £ iTf "3^4 "rfii^^D+'V-i-l,,! 1

(T13) 42-n8li |-n,|i |o2-nloC.2-2o]

<

'

IT F l°"»i 1 i ^1 n

(T15) 1 -Bfl' luinhf

 1 11>5

\**^/ K ^Ow IB .Liltl \ • f \I oK nimfi u ^ vt • • 1 | ^

L °2 J l

1,2 . . I , n

(T17)2 oj-HjI^ICC.+ lo.l- ^ oh-H,C7B»'[sinh(A,.h]n

* * •> 1. Aj A £ * X A

.i

CQEBBBltS

1. C-G(c.T) is obtained

from extrapolation of

relaxation data.

2. k is curve-fit to

R.

Rxe «• ' J

1. W -oc1

p

1. R is the cumulative

inelastic strain:

2. The growth law for a.,

eq.(T13), is not

presently used in the

model; a, i* assumed

to be a constant.

. Q

1, e'- e W for T>0.6 TO

Material

P

~ ""

R3 °

E.cl.c2.c3.

o, ,n

0

E,D0 .n.m.Z1 ,

Zj.A.r

:

E.n.'n „ .„ .

n4.n5.n6.n7.

0 0

!

E...n.^.

-fir^j lfin( ^5= ) * i] "2'

t2'A2'<1

e'- • L ' BJJ 1

far T<0.6 Tm

Tm is th« naltlng teqp.

k is the gaa constant.

^
Ceccocco

and

Leckie

Hare

Rob in con

Valanis

Allen

and

Haisler

(TIB) o-E[c-cI-cT)

(T22) o-E[» -c1-*1]

(T26) «-E[«-iX -cT]

(T29) o-Elt-t1-^]

(T31) o-E[e-tZ-*T]

1 I l°"0l

J o^ ogn o ai

(T20)2 &j- j b^k1 - rao1

(T21) &2-hf - tf

\ T r7TM/2r!°-° nM

iCT23> c -«*|_jj 1 sgn(o-oL)

L

 " J oo

(T.A)2 . . 3 .1 "Lcrl

 f* o 2

L * Q jf^ *^i

C ^1

1*1^ k./ 2 ~ ftf r T * * 9 f" ^ '"•

(T25) o,- cljl la IJT • • . »j ^

L

 2J 1 y °2 f •< tf

1 \ / •• n

1 V 7 11

.1 if i o''iln"L .

["'2'. I I Jn^B"""1

L^'MI

(T30) i^kj^Co.c)* + k2 f 2 (o,t)

(T32) cI-x[o-o1] + g[o-0l]

(T33) i^ito-oj^]

T2

(T34) 42 -f(tA )

dcccnainad function*.

1. Tha drag atraaa ia

takan co ba a oonatant,

v. hanca thara 1* no

o_ aa in othar •odala.

lhara iat hOMwar* a

chird Intamal ctaca

variable, tenad <^ .

2. T ia the abeoluta teop.

R ia the (aa constant.

I; GO i* the initial value

-4-

1. Repreacnta iiapllfiad

£001 of Vtlanla* Model.

fitting and interpolation of acreaa-otraln

and creep data

required CD ^*f a1n

material peraBater*

i, g, and i.

E

C.m.f.Q.k.

X.C

E.ii.K.n.l,

fj

1 sgn(x)

rl x>

fr{ 0 x -

l-l x «

2 c COD be aubeticuted directly into growth law for «nd a. to

obtain a form ccnalatoc with internal state variable growth laws (14).

Note: parentheses ( ) imply "function of, uhereaa brackets [ ] imply

multiplication.
N86-3023E1

APPENDIX 6.M
A COMPARISON OF CURRENT MODELS FOR NONLINEAR RATE-DEPENDENT

MATERIAL BEHAVIOR OF CRYSTALLINE SOLIDS

A Thesis

by

JOACHIM MICHAEL BEEK

Submitted to the Graduate College of

Texas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

May 1966

Major Subject: Aerospace Engineering
1 i. 1

ABSTRACT

A Comparison of Current Models for Nonlinear Rate-Dependent

Material Behavior of Crystalline Solids. (May 1986)

Joachim Michael Beek, B.S., Texas A&M University

Chair of Advisory Committee: Dr. D. H. Allen

This thesis reviews three theories for prediction of

inelastic deformation which are based on considerations of

the microstructural behavior of materials. These theories

are those of Krieg, et al., Bodner, et al., and Miller,

et al. The thesis opens with a review of the mechanics of

continua with internal state variables and a review of the

historical development of constitutive modelling. A

detailed discussion, including chronological development,

review of theory, and method of determination of material

parameters, of each model follows. An experimental data

base.is established from which the material parameters of

the constitutive equations are calculated. Finally,

computer simulations of various load histories are

performed and compared to experiment, and conclusions of

the ability of each theory to model inelastic deformation

are drawn.
ACKNOWLEDGMENTS

I wish to thank Dr. David H. Allen for his guidance,

patience, and support throughout this research effort.

I would also like to thank Dr. R. A. Schapery and

Dr. W. L. Bradley for kindly consenting to join my

committee and for their support. Gratitude is due to my

wife for her patience and to all others, too numerous to

mention here, who aided in this work.

This research was supported by NASA Lewis Research

Center under Grant no. NAG3-491.
TABLE OF CONTENTS

Page

INTRODUCTION 1

THE EQUATION OF STATE APPROACH 6

LITERATURE REVIEW AND GENERAL DISCUSSION OF MODELS 16

Classical Plasticity Models 16

Nonlinear Viscoelasticity. Models 21

Microstructural Models 25

KRIEG, SWEARENGEN, AND ROHDE'S MODEL 33

Chronological Development 33

General Theory 35

Evaluation of Material Constants 41

BODNER AND PARTOM'S MODEL 54

Chronological Development 54

General Theory 57

Evaluation of Material Constants 62

MILLER'S MODEL 72

Chronological Development 72

General Theory 75

Evaluation of Material Constants 81

DETERMINATION OF MATERIAL CONSTANTS 85

Material Constants for Krieg, et al • 90

Material Constants for Bodner, et al 95

Material Constants for Miller, et al 102

COMPARISON OF THEORY TO EXPERIMENT 107

Qualitative Review of Predictive Capabilities 107

Review of Integration Technique 109

Comparison of Numerical Predict ions to ' Experiment».11 0

SUMMARY AND CONCLUSIONS 1 41

REFERENCES 144
V 1

Table 5.

LIST OF TABLES

Page

Table 1. Composition and Material Properties

of Al 5086 87

Table 2. Review of Material Test Requirements...,

Table 3. Pertinent Results of Experimental Tests,

Table 4. Calculated Values of Z used in the

Determination of Z,

,88

,92

,99

Calculated Values of Z used in the

Determination of A and r 100

Table 6. Calculations for Various Values of A 103

Table 7. Summary of Material Constants for

all Models. 1 06
v ii

LIST OF FIGURES

Page

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

1

2

3

4

5

6

7

8

9

1 0

Phenomena associated with inelastic

Phenomena associated with inelastic

The relationship between local inhomogenei t ies

The three parameter standard solid

A comparison of the hyperbolic sine, power

law, and exponential forms of the inelastic

strain rate equation

Strain response for rapid stress reductions

during steady-state creep

A stress-drop test

Four hypotheses for creep response following

a stress-drop

A qualitative graph of ln(e ) v. In o~a|

for determining c. and c?

A qualitative graph of the primary-toM

5

13

22

29

.43

44

46

.49

secondary creep ratio used in determining

c and c^ 52

Fig. 11 A qualitative comparison of a saturated stress

curve and a non-saturated stress curve 63

Fig. 12 A qualitative graph of constant strain rate

tensile data for determining n and Z 65

Fig. 13 A qualitative graph of Y v. o for determining

the saturated stress level 67

Fig. 14 A qualitative graph of In(Z^-Z) v. WP

for determining m and Z 69

Fig. 15 The effect of A on log n(e /6') v.

log, _[sinh( A <>/£) ] data. .T. 83

1 U 5 S
vii i

Fig. 16 Experimental constant strain rate tensile test

response ....................................... 89

Fig. 17 Experimental creep test response ............... 91

Fig. 18 Graph of ln(eX) v. ln|o-o| for Krieg's

model ............ ; ............................. 93

Fig. 19 Graph of Y v. o for Bodner's model ............. 97

Fig. 20 Graph of Inl-ln l /D ) ] v. ln(o ) for

Bodner's model. ........ .......... 7 ............. 98

Fig. 21 .Graph of ln[m(Z -Z)WP] v. ln[(Z-Z )/Z ]

for Bodner's model ................... . ........ 101

Fig. 22 Graph of log (e1 /6') v. log [ sinh( Ao /E ) ]

for Miller 's mode! ...... : ____ \. ........ 77 ..... 104

Fig. 23 Experimental constant strain rate tensile

test: E = UdO)"1' sec" ....................... 111

Fig. 2 4 Constant strain rate tensile test. Comparison

of theory of Krieg, e^t al . , to experiment ..... 113

Fig. 25 Constant strain rate tensile test. Comparison

of theory of Bodner, et al . , to exper iment .... 1 1 5

Fig. 26 Constant strain rate tensile test. Comparison

of theory of Miller, et al . , to exper iment .... 1 1 6

Fig. 27 Constant strain rate tensile test. Comparison

of all theories to experiment ................. 117

Fig. 28 Experimental cyclic loading test:

cycles 1-1 0 ................................... 119

Fig. 29 Cyclic loading test cycle 1. Comparison

of theory of Krieg, et al . , to experiment ..... 120

Fig. 30 Cyclic loading test: cycles 1 and 10.

Prediction of theory of Krieg, et al .......... 121

Fig. 31 Cyclic loading test: cycle 10. Comparison

of theory of Krieg, et al . , ' to experiment ..... 123

Fig. 32 Cyclic loading test: cycle 1. Comparison

of theory of Bodner, et al . , to exper iment .... 1 24
ix

F.tg. 33 Cyclic loading 'test: cycles 1 and 10.

Prediction of theory of Bodner, et al 125

Fig. 34 Cyclic loading test: cycle 10. Comparison

of theory of Bodner, et al., to experiment.... 126

Fig. 35 Cyclic loading test: cycle 1. Comparison

of theory of Miller, et al., to experiment....128

Fig. 36 Cyclic loading test: cycles 1 and 10.

Prediction of theory of Miller, et al 129

Fig. 37 Cyclic loading test: cycle 10. Comparison

of theory of Miller, et al., to experiment.... 130

Fig. 38 Cyclic loading test: cycle 1. Comparison

of all theories to experiment 131

Fig. 39 Cyclic loading test: cycle 10. Comparison

of all theories to experiment 132

Fig. 40 Experimental complex loading history 1 3 ^

Fig. 41 Experimental complex loading response 135

Fig. 42 Complex loading test. Comparison of

theory of Krieg, et al., to experiment 136

Fig. 43 Complex loading test. Comparison of

theory of Bodner, et al., to experiment 137

Fig. 44 Complex loading test. Comparison of

theory of Miller, et al., to experiment 139

Fig. 45 Complex loading test. Comparison of

all theories to experiment 140
INTRODUCTION

The prediction of inelastic material behavior in metals

is a problem of great importance which has accordingly been

given a great deal of attention by the research community

in recent years. Recent technological advances require

materials to function in severe chemical, mechanical, and

thermal environments such as nuclear reactors and gas

turbines. Superalloys have been developed that are able to

sustain loads at extreme temperatures; however, due to this

environment, these materials exhibit substantial complexity

in their material constitution in that they are highly

nonlinear, rate-dependent, temperature-dependent, and

history-dependent.

Numerous theories have been proposed to predict the

thermomechanical behavior of inelastic solids at elevated

temperatures. These theories are usually based on one of

the following concepts: 1) rate-dependent extensions of

classical plasticity theory retaining the concept of a

yield surface, 2) thermodynamics and/or nonlinear

viscoelasticity theory, and 3) considerations of the

The style and format of this thesis follow the Journal

of Engineering Materials and Technology.
microstructural behavior of the material. It is important

that each of these theories be able to model diverse

phenomena associated with inelastic deformation including

anelasticity (completely recoverable time-dependent strain

at zero load), the Bauschinger effect, cyclic strain

hardening/softening, rate sensitivity, creep, relaxation,

and tensile and compressive loading and unloading. These

are illustrated in Figures 1 and 2.

The purpose of this research is to review three

theories based on considerations of microstructural

behavior. Since each theory is proposed for a material

and temperature of its authors' choosing, a set of

experiments is performed by .this author with a single

candidate material in order to create a common experimental

data base with which the mathematical representation of

each model can be characterized. The theories are

subsequently analyzed qualitatively and quantitatively over

a range of material behavior to determine their suitability

in predicting inelastic deformation.

This thesis open's with a short review of the mechanics

of continua with internal state variables. This is

followed by a literature review and, utilizing the

framework of internal state variables, a detailed study of

the models chosen for discussion. The experimental program

associated with the research for this thesis is presented
as an assessment of the experimental data base requirements

for each model, accompanied by the results of the

experiments. A uniaxial comparison of the predictive

capabilities of the models follows and, finally,

conclusions from the complete survey are presented.
Rate sensitivity Relaxation

eo -f

t3>t2>t1>to

Creep Anelast ic i ty

Fig. 1 Phenomena associated with inelastic deformation

(uniaxial loading)
Cyclic strain softening Cyclic strain hardening

a

yc

Bauschinger effect

Fig. 2 Phenomena associated with inelastic deformation

(cyclic loading)
THE EQUATION OF STATE APPROACH

The concept of internal state variables, sometimes

called- hidden variables, is gaining widespread usage in

current research on the modelling of inelastic solids.

Therefore, it is useful to construct a well-defined

framework for internal state variable models which is based

on continuum mechanics.

The concept of internal state variables was apparently

first utilized in Onsager's work in thermodynamics [1,2]

and has found numerous applications since the Second World

War [3-12].

In the theory of internal state variables applied to

solids, Coleman and Gurtin [3] defined the elastic field

problem to be one in which the following state variables

are required in order to characterize fully the state of a

body at all points x. and at all times t: *

J

1) the displacement field u. = u.(x.,t); (1)

* For convenience, only Infinitesimal deformations will

be considered here, although the general theory applies to

finite deformations as well.
2) the stress tensor o. . = o-.(x.,t); (2)

1 J 1 J K

3) the body force per unit mass f. = f (x.,t); (3)

the internal energy per unit mass

u = u(x ,t);

5) the heat supply per unit mass r = r(x.,t); (5)

6) the entropy per unit mass s = s(x.,t); (6)

7) the absolute temperature T = T(x.,t); (7)

J

and

8) the heat flux vector q = q (x.,t). (8)

Thermodynamic constraints will lead to the conclusion that,

for elastic bodies, entropy is generated only through heat

conduction [33- For inelastic bodies, however, this

conclusion cannot be drawn because additional entropy is

generated. The state cannot be characterized solely from

the observable state variables; rather, a set of internal

k

state variables a.. is also necessary, and these are

determined by observation of the entire past history of
states of the body

where the superscript k ranges from one to the number of

internal state variables necessary to characterize fully

the state of the body. Although listed here as second

order tensors, they may be tensors of other rank as well

[13].

The method of Coleman and Noll [14] may be used to

obtain the spatial and time distribution of the body force

f. and the heat supply r from the conservation of linear

momentum and the conservation of energy, respectively,

assuming the displacements u and the temperature T are

specified independent variables. It is hypothesized

subsequently that constitutive equations may be constructed

for the remaining state variables in terms of u. and T and

their spatial derivatives:
,t)

3(xk,t) =

s(emn(

V

t)'T(xk't)'Sm(xk't)'a

mn(V

t))'

where g is the spatial temperature gradient T, and

is the infinitesimal strain tensor. The form of equations

(10) through (13) implies that all constitutive equations

are evaluated in the specified state x and t. For this

reason, o.. , u, s, and q. are termed observable state

variables since they can be determined from equations of

state, even though there is an implied history dependence

through the internal state variables a . These are

mn

defined to be of the form:
10

"ij - "ij'W^rXn'' <1 5 )

where time and spatial dependence have been dropped for

notational convenience and the superscripts k and p range

from one to the number of internal state variables

necessary to characterize fully the state of the body. If

equations (15) are integrable in t at all times, then they

can be rewritten as

where t is the time of interest and t' is a dummy variable

of integration.

The above framework has been shown to be applicable to

crystalline solids [15,16] and further discussion of this

subject is provided in reference [17].

On the basis of the Coleman-Mizel procedure [18] it can

be shown that satisfaction of the first and second laws of

thermodynamics will lead to the following conclusions:

h = u - Ts = h<enn.T.«j;n>, . (17)

where h is the Helmholtz free energy;

o ;1 = p dh/6ekl; (18)
1 1

s = -dh/dT; (19)

and

j8j * 0(gj

)> (20)

where k is the thermal conductivity tensor. Although

further results are obtainable [3,15-17], they are not

pertinent to the current research. The importance of the

results above is that, in order to construct a complete

description of constitutive equations (10) through (13) and

(15), it is necessary only to prescribe the Helmholtz free

energy. In order to describe the class of materials

discussed herein, the free energy is expanded in terms of

j?

the elastic strain tensor e.. and the temperature T in a

second order Taylor series as follows:

(T

 - V/2T'

where the subscript R refers to quantities in the reference

state, D. is the linear elastic modulus tensor,

1 J K J.

c = -T(d2h/dT2) is the specific heat at constant elastic

volume, and

£

E. = e. . - e1. - J., (22)

'
1 2

T I where e. . is the thermal strain tensor and e. . is the

inelastic strain tensor, which can be considered to be an

internal state variable [15,16,18-21]. Substitution of

equation (21) into (18) will result in

Dklmn (emn " emn ' Sn*' (23)

The above equations, together with internal state variable

growth laws (15), will be shown to be a suitable framework

for comparison of all the models discussed herein.

Theoretically, constitutive equations (10) through (13)

and (15) are applicable to fixed infinitesimal material

points. In practical terms however, it is not possible to

construct experiments on material points since the

continuum assumption becomes invalid at the microscopic

level. Rather, it is considered acceptable to construct

constitutive equations by subjecting local specimens to

surface deformations (or tractions) which lead to spatially

homogeneous stresses and strains. In this manner, a local

average of the pointwise observable state variables can be

determined directly from the effects on the boundaries of

the specimen.

As shown in Figure 3, the scale of the smallest

dimension of a local specimen is generally assumed to be at

least one order of magnitude larger than the scale of the
13

t \ t t

Physical

inhpmogeneities

Fig . 3 The relationshi p betwee n local inhomogeneitie s and

t h e continuu m assumptio n
1 1

largest material inhomogeneity, while the scale of the

largest dimension of the local specimen is much smaller

than the scale of the domain of interest. The former

preserves the continuum assumption, while the latter

conserves the notion that constitutive equations are indeed

i

)

pointwise in nature. The local constitutive equations that

result from experimentation are assumed to be of the same

form as pointwise equations (10) through (13) and (15); for

example,

where

(25)

(26)

and

T = T(a1,a2,a3), (27)

where L is the length of the local specimen, A is the

cross-sectional area in the x -x_ plane, (a ,a_,a_ ) Is an

arbitrary point on the surface of the specimen, and
15

* ' dxdX d (28)

It Is then assumed that equation (2*0 is a suitable

replacement for equation (10). There are obvious

shortcomings in this assumption in that equation (24)

represents a solution to a boundary value problem. This

implies volume averaging of local balance laws, which

violates the local nature of constitutive equations. In

addition, there may be a question of uniqueness resulting

from the fact that more than one global state of a given

internal state variable will map into the globally averaged

state variable described by equations (28). However,

assuming that the scale of inhomogeneities is small and

that the distribution of a is random, the specimen will

mn

be statistically homogeneous and the relation between a

ran

and a reasonably one-to-one. The above procedure

remains the only reasonable method for constructing

constitutive equations from experimental data bases.
16

LITERATURE REVIEW AND GENERAL DISCUSSION OF MODELS

Man's curiosity about his environment and his constant

striving to harness the power and resources of nature with

increasing efficiency have been the driving force behind

the study of the laws of nature and the attempts to use

these laws to predict the solutions of many problems.

The theory of elasticity dates to the 17th century with

Robert Hooke's discovery that the tension in a spring is

directly proportional to its extension. This heralded the

beginning of the characterization of mechanical behavior of

materials using mathematical constitutive models.

Contributions to the theory of elasticity were also made by

Galileo, the Bernoullis, Navier, Cauchy, Euler, and

Saint-Venant, to name a few [22].

Classical Plasticity Models

Often only a small part of material deformation is

elastic and, in general, a material will not behave

elastically at large strains. Consequently, in 1864 Tresca

[23] proposed a criterion to predict the onset of inelastic

deformation in metals based on the maximum shear stress

reaching a critical value. This was followed in 1913 by von

Mises [24] with a yield criterion based on an equivalent
17

stress state reaching a critical value; this has been

interpreted as the elastic shear strain energy reaching a

critical value [25].

Independently of each other, Levy [26] and von Mises

[24] formed the basis of classical plasticity by proposing

a three-dimensional relationship between the strain

increment and the stress tensor which became known as the

Levy-Mises equations:

dA o' , (29)

where de. . is the total strain increment, dX is a scalar,

and o!. is the deviatoric stress tensor, given by

°u ' 3°kx

6ij (30)

where 6. . is the Kronecker delta. In equations (29), the

elastic strain component is assumed to be negligible; these

equations consequently describe a fictitious, rigid,

perfectly plastic material. Equations (29) were extended

to take into account the elastic strain component by

Prandtl [27] in 1924, who solved the plane stress case, and

by Reuss [28] in 1930, who generalized the equations to

three dimensions:
18

(31 )

P

where de, is the plastic strain component and dA is a

scalar history-dependent material property. The total

strain increment de.. is then assumed to be the sum of the

P

plastic strain component de,. and the elastic strain

g

component de ..

These developments enabled the classical plasticity

theory to model elastic and rate-independent inelastic

deformation fairly accurately for most metals with only a

few restrictions. These were that the material be

subjected to monotonically increasing loads and low

non-transient homologous temperatures; that is,

temperatures less than three tenths of the absolute melting

temperature. At- higher temperatures, however, rate

dependence of material behavior becomes significant,

limiting the use of the existing plasticity theory.

Rate dependence was initially treated as an extension

of classical plasticity theory by Bingham [29] in 1922 by

using the concept of a yield function dependent on the

excess of the stress intensity over the yield stress for

the case of simple shear. This was generalized by

Hohenemser and Prager [30] in 1932 for the

three-dimensional case. Further progress in plasticity

theories was slow and advanced vastly and diversely only
19

after completion of World War II. Prager [31] and Ziegler

[32] proposed work-hardening rules for rate-independent

plasticity to allow the yield surface to translate to model

kinematic hardening. Drucker's stability postulate [333

serves as a basis for the incremental theories of

plasticity, in which the plastic strain increment is

determined by the stress and the stress increment. Ratedependent plasticity models proposed in the postwar period

included those by Freudenthal [34], Malvern [35,36], who

constructed a one-dimensional equation relating the stress

increment to the strain rate, Lubliner [37], who modified

Malvern's equation to include a limiting stress-strain

curve, and Perzyna [38-11], who generalized Malvern's

equations to three dimensions and finite strains.

Many classical plasticity models have been extended to

include rate dependence by adding an uncoupled ratedependent strain component to the rate-independent strain

component. No observable difference exists between the

physical mechanisms causing these phenomena, however, and

it has been argued that the two terms should be combined

[*)2]. Naghdl and Murch [43] attempted to include rate

dependence by introducing interdependent, although not

unified, plastic and viscoelastlc strain components.

Classical plasticity theories are usually described by
20

Dijmn (emn ' e

mn ' £L

)

 •

 (32)

T

where D. . is the linear elastic modulus tensor, e is

the thermal strain, and e , the inelastic strain, is

ran

given by

°

F/0°ran' (33)

where X is a scalar valued function of state and F is a

scalar valued function for inelastic behavior often taken

to be the yield function. If F is described by the von

Mises yield criterion [2*1], given by

where a . is a tensor describing the yield surface center

•*• J

in stress space and k is a constant representing the yield

surface size, then equations (33) can be written as

(35)

resulting in a kinematic hardening model with constant

yield surface size. Furthermore, if the yield surface

translation is derived from the Ziegler modification [32]

of the Prager work-hardening rule [31]• it may be described
21

by

°U

 = 'v (°ij ' aij}'

 (36)

where \i is a scalar valued function of state.

By use of equations (32), equations (35) and (36) can

be shown to be consistent with growth laws (15). It can

then be seen that classical plasticity theories are

consistent with the thermodynamic framework developed in

the equation of state approach.

Recently proposed theories include those by Snyder and

Bathe [44], Yamada and Sakural [45], Allen and Haisler

[46,4?], which attempts to model transient temperature

effects, Zienkiewicz and Cormeau [48], and Robinson [49].

Bodner and his associates proposed a model [50] which uses

a flow law similar to equations (31), although the internal

state variable growth laws are based on microstructural

considerations. Robinson's model and Bodner's model are

still under active development.

Nonlinear Viscoelasticity Models

Nonlinear viscoelasticity models are usually based on

thermodynamics and/or mechanical analogs composed of

springs and dashpot combinations such as the three

parameter standard solid, shown in Figure 4. In 1954, Biot
22

'M

Fig. ^ The three parameter standard solid
23

[6,7] proposed a theory of linear thermoviscoelasticity

based on the principles of irreversible thermodynamics.

Schapery [12] later modified this to include nonlinear

behavior and introduced a reduced time scale to solve the

resulting highly nonlinear equations. Valanis [51-53]

proposed a viscoplasticity theory, termed the endochronic

theory, which includes a history-dependent reduced time

parameter. This was later shown to be a special case of

Schapery's model [51*]. Some other models based on

thermodynamics are those of Coleman and Noll [55,1 4], Green

and Rivlin [56], and Green and Naghdi [57]. These tend to

be quite complex in nature and difficult to solve.

Many of the recent models are based on variations of

the three parameter standard solid. The standard solid

shown in Figure ^ has been demonstrated to be an

appropriate representation for thermoviscoplastic metals if

the springs and dashpot are nonlinearized [58,59]. The

governing differential equation for this solid is

(nM/EM) (EM + Eje. (37)

This may be rewritte n in the equivalen t for m

e = o/(E M + EJ + E M /[n M ( E M + Ej] (o - E.e). (38 )
21J

In accordance with the instantaneous linear behavior of

metals, it is assumed that

E,, «• E = E = Elastic modulus => constant, (39)

M °°

and equation (38) becomes

e = + e1, (40)

where e is the inelastic strain, given by

e - (EM/nME) (0 - E^E). (41)

The nonlinearity of the springs and dashpot can be

introduced by considering a multiaxial extension of

equat ion (37 ) :

o + K o= G e+ M e

pq pqmn mn pqmn ran pqmn mn

In order to model metals, K , G , and M are

pqmn pqmn pqmn

nonlinear material functions determined from experimental

data. In addition, constraint (39) requires that

Ki]mnMmnkl = D 1JK1'
25

where D. . is the linear elastic modulus tensor

Equations (42) and (43) lead to

Dijmn (emn ' emn

where the inelastic strain e..is given by

•I J

Substitution of equations (44) into (45) will result in

equations consistent with growth laws (15). It can then be

seen that nonlinear viscoelasticity models are consistent

with the thermodynamic framework developed earlier in the

equation of state approach.

Some recent models which are based on nonlinear

modifications of the three parameter standard solid are

those of Krempl, et al. [59-66], and Walker [58,67,68].

Microstructural Models

Microstructural models have generated increased

interest in recent years because their formulation is based

on microphysical considerations rather than a statistically

averaged macroscopic view. The primary micromechanisms

which cause inelastic deformation are dislocation glide and

climb, deformation twinning, diffusion, grain boundary
26

sliding, and microvoid growth and coalescence. These

mechanisms are difficult to model individually and are

usually reduced to three more general mechanisms:

dislocation arrangement, dislocation density, and damage.

In the concept of microphenoraenology it is assumed that the

three mechanisms can be characterized by internal state

variables. One internal state variable represents the back

stress for modelling the dislocation arrangement. Another

internal state variable, termed the drag stress, models

dislocation density. In addition, it can be argued that

the inelastic strain represents the third internal state

variable [15,16,19]. Current research is in progress to

account for damage in materials using a fourth internal

state variable [69-71], although it is difficult to

distinguish between damage and the drag stress since both

are primarily stiffness-reducing parameters.

Early research in this field was performed by Coble

[72], Nabarro [73], and Herring [74] in their work on

diffusion-controlled creep. Other important contributions

were made by Sherby, et al. [75-77], Garofalo [78], Argon

[42], Weertman [79], Alden [80], Kocks [81], and Hart [82].

Mukherjee, et al. [83], studied the effectiveness of

using a power law of the following form to model

dislocation climb:
27

D A (o/E)n, (46)

where D is the self-diffus 1vity, E is Young's modulus, and

A and n are temperature-dependent material parameters. The

authors concluded, however, that a model in terms of a

single dislocation mechanism may not completely

characterize high-temperature creep.

Gibbs [84] proposed a two-internal-state-variable

theory in which the inelastic strain rate is characterized

by an exponential function of temperature and an effective

stress:

e

1

 = A* exp{-[Cl - ( o- oo) lb2]/KT) , (47)

c a

where A is a structure factor, Q is the dislocation core

c

diffusion energy, k is the activation energy, T is the

absolute temperature, b is the Burger's vector, and o and

di

I are internal state variables representing, respectively,

back stress and drag stress.

It is generally recognized [85] that at intermediate to

high stresses and at temperatures above 0.5T , where T is

m m

the absolute melting temperature, stress is related to the

inelastic strain rate by a power law of the form

A o", (48)
28

where A and n are constants. At very high stress levels,

the relationship is

e » B exp(mo), (19)

where B and m are constants. Power law and exponential

relationships represent limiting cases for the more general

empirical relationship

e =• C [sinh(ko)]p, (50)

where C, p, and k are constants. Equation (50)

approximates a power law when ko<0.8, but it approximates

an exponential curve when ko>1.2. See Figure 5 for a

comparison of the forms of equations (48), (19), and (50).

Microstructural models are usually similar in nature,

differing only in the proposed relationship between stress

and inelastic strain (that is, equations (18), (19), or

(50)) and in the proposed internal state variable growth

laws. The growth laws for the internal state variables

representing the back stress and the drag stress usually

follow the framework established by Bailey [86] and Orowan

[87] in which a hardening term, proceeding with accumulated

deformation, competes simultaneously with a softening or

recovery term, proceeding with time. For example, the
29

exp nx

sinh x

Fig. 5 A comparison of the hyperbolic sine, power law, and

exponential forms of the inelastic strain rate equation
30

isothermal model proposed by Krteg, et al. [88], is given

in its uniaxial form by

o = E (E - e1), (51 )

where the inelastic strain is given by

e = C [|o - cj/c ] sgn( o - a ) , (52 )

where the growth laws for the back stress a1 and the drag

stress a are given as the differences of hardening and

softening rates:

T O O

= c_e - cija1 [expCc^) - 1] sgn(a1) (53)

and

a 3 f* I C" I ^ f^ i n ^/v Q2

 C 6 l£ I °7 l°2 a20

where GI , c^, c , c^, c^, c&, c^, n, and ct2 are

temperature-dependent material constants, and sgn() is the

signum function.

Substituting equation (51) into (52) will result in

equations consistent with growth laws (15). In addition,

since equations (53) and (54) are consistent with growth
31

laws (15), It can be seen that raicrostructural models agree

with the thermodynamic framework developed earlier in the

equation of state approach. Cescotto and Leckie [89]

proposed a model which has the interesting feature that no

particular forms are assumed for the inelastic strain rate

function and the hardening and recovery functions; only

weak hypotheses are required and the functions are defined

exper imentally.

Other recently proposed models include those of Miller,

et al. [90], in which a hyperbolic sine function is used

for the stress-inelastic strain rate equation, Walker [58],

in which a power law is used for the inelastic strain rate

equation, and Bodner [50], in which an exponential function

is used. An interesting aspect of these last two models is

that, although the respective growth laws for the back

stress and the drag stress are formulated from

microstructural considerations, Walker's inelastic strain

rate equation is based on a nonlinear modification of a

three parameter viscoelastic solid, while Bodner's

inelastic strain rate equation is based on a flow law

similar to the Prandtl-Reuss equation of classical

plasticity (see the literature reviews on nonlinear

viscoelasticity models and classical plasticity models,

respectively) .

Since theories based on microstructural considerations
32

have been given a great deal of study In recent years, the

theories chosen for comparison are taken from this field

and are those of Krieg, et al. [88], Miller, et al. [90],

and,Bodner, et al. [50]. The motivation for the choice of

these three models is that they are still under

development; they have been cast in a common thermodynamic

framework; the authors have included in their theories

detailed accounts of methods of determination of the

material constants; these models have received considerable

attention [20,68,91-93]; and research is in progress to

extend these isothermal models to include transient

temperature response [68,91,9^].
33

KRIEG, SWEARENGEN, AND ROHDE'S MODEL

Chronological Development

In 1977, Krieg [95] cast several current unified

constitutive models into the common framework of a skeletal

model in which general similarities, such as the use of

competing hardening and recovery rates for internal state

variable growth laws, are noted. These models include

those of Bodner, et al., Robinson, et al. , Hart, Lagneborg,

Miller, Pasley and Wells, Ponter and Leckie, and an early

development of Krieg's model, which is described as a

composite of the aforementioned models. All hardening and

softening functions are taken to be constants in the

skeletal model in order to make observations about

difficulties encountered i i numerically integrating

mathematically stiff constitutive equations. Finally,

integration techniques are proposed to alleviate these

difficulties.

In a development of this model, Krieg, Swearengen, and

Rohde [88] proposed a power-law kinetic relation in

multiaxial form to model pure aluminum at room temperature.

Th:.s model has three internal state variables: the

inelastic strain, the back stress and the drag stress. The

growth laws of the back stress and the drag stress are

formulated in the usual hardening rate/recovery rate
format. The hardening functions have not been defined and

are assumed to be constant; recovery is considered to be a

thermally-activated process and dynamic recovery is not

included. Furthermore, the response of pure aluminum at

room temperature is very nearly kinematic and, as a

consequence, the drag stress (the isotropic hardening

variable) is taken to be constant. Krieg, et al. , present

a test method in which all material constants, except

Young's modulus, are calculated by determining values of

the back stress from a set of stress-drop tests. In this

test, a portion of the load on a specimen during secondary

creep is removed and the resulting strain transient is

recorded. The authors have obtained reasonable results for

pure aluminum at room temperature for various load

histories, although the predicted hysteresis loops are

"over-square" and the model's use is limited in high strain

rate ranges.

In 1982, Jones, et al. [96], noted that, while most

unified creep-plasticity models require an extensive number

of tests for characterization of material parameters, the

model proposed by Krieg, et al., requires only a set of

stress-drop tests. It is pointed out that, although simple

in concept, the stress-drop test is difficult to perform in

practice. The critical measurement in this test, a zero

strain rate after unloading or merely the absence of a
35

resolvable strain rate, results in a strong dependence of

the parameters on experimental resolution. The authors

present the design of a test apparatus to minimize these

effects. In addition, there appear to be differing

opinions in the literature on the response of the creep

rate immediately after unloading. Jones, et a1., conclude

that stress-drop testing should be performed with great

care.

In subsequent research in 1983. Jones and Rohde [97]

refer to a new technique to reduce stress-drop data which

alleviates the difficulties mentioned above. Rather than

attempting to measure a zero strain rate, this method [98]

uses .a comparison of the instantaneous strain change and

the maximum strain change after unloading. This method

appears to simplify the calculations considerably and

merits further study.

General Theory

In their unified creep-plasticity model, Krieg,

Swearengen, and Rohde [88] state that inelastic deformation

can be caused by: 1) thermally-activated, stress-assisted

penetration of short-range obstacles to dislocation motion,

2) generation and immobilization of dislocations, leading

to increases in dislocation density, stored elastic energy,

and the flow stress, and 3) stress-assisted thermal
36

rearrangement of dislocations to reduce long-range internal

stresses. Krieg, et al., note that strain hardening at low

to intermediate homologous temperatures (below 0.5T , where

T is the absolute melting temperature) is both isotropic

and kinematic. Isotropic hardening, manifested by an

increase in the height of the reversed strain hysteresis

loop, has its mlcrostructural origins in dislocation

obstacle interaction, while kinematic hardening, manifested

by the Bauschinger effect, has its origins in dislocation

pile-ups. Mathematical descriptions of this behavior

require two internal state variables.

Deformation kinetics resulting from thermally-activated

processes are usually described by Arrhenius or hyperbolic

sine functions; however, over a wide range of stresses, the

analytically simpler power law provides a good

approximation. This motivated Krieg, e't al. , to propose the

following flow rule in multlaxial form in deviatoric

stress-strain space:

e

1

 = e0 (U|/R)m < 5 /J5|> . (55)

. T , , where e is the inelastic strain rate vector, || is the

Euclidean vector norm, e and m are temperature-dependent

material constants, and R is the drag stress, an internal

state variable which accounts for isotropic hardening. The
37

effective stress £ is given by

5 = § - g, (56)

where a is the back stress, an internal state variable which

accounts for kinematic hardening; and s is the applied

stress. The increase in flow stress associated with an

increase in hardening is taken to be proportional to inelastic

strain. Recovery is associated with dislocation escape from

glide planes by climb or cross-slip. In the absence of

external loads, this escape reduces the internal stresses

through annealing; in the presence of external loads, this

escape permits additional inelastic deformation through creep

or dynamic recovery. The evolution of a and R is given by

the difference between hardening and recovery rates as

a = A e1 - r (a/|a|) (57)

and

A = Aje ) - rR, (58)

where A and An are the hardening functions, and r and r

a » a n

are the recovery functions.

Krieg, et al. , assume the hardening functions A and A.
to be constant. The recovery rates are assumed to be

proportional to dislocation velocity of escape and

inversely proportional to the escape distance [99], where

the velocity and distance depend on the magnitude of the

back and drag stresses. Since annealing rate depends on

hardening mechanisms, one would expect the kinetics of

recovery for o and for R to be different. A unique

feature of this model is that it incorporates different

physically-based recovery kinetics for each internal

var iable.

Dislocation processes associated with the drag stress

are formations of dislocation tangles, or networks.

Temperature-dependent changes in the networks can be

described by the climb recovery model of Friedel [99]:

rR = -K1 (Rn/kT) exp(-U/kT), (59)

where K. and n are material constants, k is the gas

constant, T is the absolute temperature, and U is the

activation energy associated with the particular

micromechanism of softening.

Dislocation processes associated with the back stress

are pile-ups or cell-wall bowing. Although both cross-slip

and climb are recovery mechanisms, only screw dislocations

can cross-slip and, consequently, the model's authors feel
39

that recovery by climb is a more complete description. The

kinetics of this process are again given by Friedel [99]:

-K2 a2 [exp(K a2/kT) - 1], (60)

where K and K are material constants.

By use of equations (60), in multiaxial form, and (59),

equations (57) and (58) become

2 = AaS1 ~ |a|

 K

2 [exp(K3|a2|/KT) - 1] (61)

and

R = Aje1! - K, [(R - Rn)

n/T] exp(-uVKT), (62)

n ' ~ ' I u

where R« represents an isotropic annealed state.

For conditions of uniaxial stress and constant

temperature, equations (55), (61), and (62) reduce to

(63)

T O O

d = c e - Cj^a [exp(c a ) - 1] sgn(a),

and
1*0

R = c6|cI| - c? (R - RQ)

n

, (65)

where GI , c2> c^, c^, o^, c& , c^, n, and RQ are

temperature-dependent material constants, sgn () is the

signum function, and the effective stress £ is

5 = 0-0 . (66)

Krieg, et al. , simplify the model further by assuming a

kinematic work-hardening material so that the isotropic

hardening variable R will remain constant. Equation (65)

can then be omitted and the uniaxial isothermal form of the

model becomes

o = E (e - e1), (67)

wher e e is give n by

1 |c| C 2 s g nU), (68 )

and

T O O

a = c e - c^ci [exp( c a ) - 1] sgn(a) , (69 )

wher e c , c? , c_, c^, and c,_ are-temperatur e dependen t
material constants, and the effective stress (, is given by

equation (66).

Evaluation Of Material Constants

Krieg, et al., present an experimental test and data

reduction procedure which allows all constants, except the

elastic modulus E, to be determined from a set of

stress-drop tests, sometimes called strain transient dip

tests.

Before describing the procedure to evaluate the

material constants, it would be instructive to review the

somewhat controversial stress-drop test. High-temperature

creep can be characterized in terms of an internal stress

and an effective stress [100], where the internal (or back)

stress is the driving force for recovery, and the effective

stress (the difference between the applied and back stress)

is the driving force for dislocation glide. Consequently,

it is of interest to be able to determine the back stress.

Since the back stress is a result of the dynamic

balance between strain hardening and recovery, the back

stress begins to change when the applied stress is changed.

Therefore, the transient response examined should be of

short enough duration to ensure a structure which is still

reasonably representative of the steady-state structure;

i.e., a structure in which the value of the back stress has
not changed appreciably from Its steady-state value. A

technique described by Ahlquist and Nix [100,101] involves

the dynamic measurement of back stresses by rapidly

reducing the applied stress from the steady-state value o.

and measuring the strain rate immediately after the

reduction. For a small reduction in stress Ao, the strain

rate immediately after the reduction is positive as shown

in Figure 6a. For a large reduction in stress Ao, the

strain rate is negative as shown in Figure 6b. For an

intermediate reduction in stress Ao, the strain rate is

zero and the new stress level o? describes the mean

internal (or back) stress o.. After a short time, recovery

events reduce the back stress, yielding a positive strain

rate. As can be seen from Figure 7, it may be necessary to

load and unload a sample several times during a test before

finding the stress reduction that will give the back

stress.

This procedure for determining the back stress applies

to thermally-activated creep. However, there are different

hypotheses for creep deformation where the concept of a

back stress may become questionable. Consequently, these

hypotheses give differing predictions of the transient

response after a stress drop. Poirier [102] summarizes

these into four categories, as shown in Figure 8, where the

responses due to stress drops of different magnitudes have
O].

02

O .

1

Aa is small:

G 2 > 0 .

(a )

o.

02

Aa is large:

. t

( b )

Aa is intermediate:

'..• 2 -o...

(c)

Fig. 6 Strain respons e for rapid stress reduction s durin g

steady-stat e creep
-I H

C

l

 C2 '3 C4 cf

Fig . 7 A stress-drop test
been superposed.

The athermal recovery-controlled creep case is

characterized by: 1) the absence of an effective stress,

2) creep proceeding only if the back stress is reduced to

the applied stress level by diffusion-controlled recovery,

and 3) stress reductions, regardless of magnitude, always

followed by a period of zero creep rate (see Figure 8a).

The thermally-activated creep, jerky glide, case is

characterized by localized obstacles to dislocation

movement and, consequently, no negative creep rate after

the stress reduction. However, there may be a period of

zero creep or positive creep, depending on the magnitude of

the stress reduction (see Figure 8b).

The thermally-activated creep, viscous glide, case is

characterized by "smeared" rather than localized obstacles.

Consequently, there may be positive, zero, or negative

creep after the stress reduction (see Figure 8c).

Finally, the thermally-activated creep case with a

combination of jerky and viscous glide is characterized by

a combination of localized and smeared obstacles. The

response of this case is similar to that of the previous

two and depends on the magnitude of the stress reduction as

well as the applied stress level (see Figure 8d).

Poirier concludes that, while the concept of a back

stress cannot be rejected on experimental grounds,
( a )

0

o . •

02

Arr<n

err

Ao>o c,

eff

i

Aa<o

eff

Aa>o

eff

( b )

Fig . 8 Four hypothese s for cree p respons e followin g a

stress-drop
02

Ao<o

eff

Ao>o

eff

Ao<o

eff

Ao>o

eff

(c)

01

02

O .

02

Ao<a

eff

Ao>0

eff

Ao<a

eff

Ao>o

eff

(d)

Fig. 8 (continued)
18

measurements of the back stress resulting from stress-drop

tests should be performed with great care and are valid

only for thermally-activated flow.

Similarly, Jones, et al. [96], conclude that due to

ambiguity involved in attempting to determine a zero creep

rate or merely the absence of a resolvable creep rate after

unloading, the stress-drop test should be interpreted with

great care on the basis of a specific deformation

hypothesis. With this in mind, one can now proceed to

evaluate the material constants c1 , c?, c_, c^, and GJ...

c and c2

Equation (68), which governs inelastic strain, can be

rewritten as

Inte1 ) = In(c1) + c2 injc|, (70)

where |£j is the difference between the applied stress o

. i

and the experimentally determined back stress a, and e is

the creep rate immediately preceding the particular stress

reduction associated with that value of a. The left hand

term of (70) must be linear in ln|c| if it is an adequate

representation of experimental data. A plot of data in the

form of (70) is shown in Figure 9. Graphical means or a

least-squares analysis of the data will give th.e values of
© Stress-drop

data

— Best fit line

through data

Ino-a

- I Fig . 9 A qualitativ e grap h o.f ln( £ ) v. ln|a-a | for

determining c. and G
50

c. and

c and (c /GJJ)

During steady-state creep, the creep rate e is

constant and, as a consequence of equation (68), £ must

also be constant. Since the applied stress o is constant

in a creep test, this implies that the back stress o is

also constant. With a = 0, equation (69) reduces to:

(c3/c,j) = Ca2/(c1 |c|°2)] Cexp(c5a

2) - 1], (71)

where the fitted form c.|s|

 2 has been used for the term

e to reduce experimental scatter. The experimental pairs

(^...a..) and (Cpfdp), where the subscripts 1 and 2 refer to

the highest and lowest values of a, may be used in equation

(71) to give two nonlinear equations in two unknowns.

These are then solved by iterative means to give the values

of c.- and the ratio (c_/c,.).

5 3 4

c3 and c^

It now remains to determine either c or cu, since only

their ratio is known at this point. In order to accomplish

this, information from a point on the primary creep curve

(where a is not zero) must be used since the available

steady-state information has been exhausted. Substituting
51

equation (68) into (69), separating variables, and

integrating leads to

{cjo,

Cexp(c5o

2) - 1]}~^da, (72)

wher*e o. is the creep stress, a is the variable of

integration, a. and t are the back stress and time,

respectively, at which the primary creep rate e,. is still

twic/e the eventual steady-state value. The value for a. at

data point can be determined by inverting equation

(68):

(73)

Sinc?e all the quantities inside the integral now have known

values, equation (72) can be integrated numerically to give

a value for c t. Since t is known, c and, consequently,

ch can be calculated easily. Although not explained by

KrieB, et al., a primary-to-steady-state creep rate ratio

of two appears to represent a transition point between the

region of rapidly decreasing primary creep rate and the

region in which the creep rate slowly approaches its

steady-state value, as shown in Figure 10.
52

.1

e

.1

ss

10 -

8 •

6 '

4 •

0 Experimental data

— Trend of data

points

Fig . 10 A qualitativ e graph of the primary-to-secondar y

cree p ratio used in determinin g c_ and
53

In summary, the Krieg, Swearengen, and Rohde theory

requires a constant strain rate test (for the elastic

modulus E), several stress-drop tests in the steady-state

creep region (constants c , c?, c^, and the ratio c_/c^)

and a creep curve complete with primary and secondary

regions (constants c and c^). This concludes the

determination of the material constants for the Krieg,

Swearengen, and Rohde model.
DODNER AND PARTOM'S MODEL

Chronological Development

In 1975, Bodner and Partora [50] expanded an earlier

model, capable of modelling elastic-perfectly plastic

behavior only [103], to include strain-hardening by

introducing plastic work as the measure of the hardened

state. The formulation of the inelastic strain rate

equation is based on a generalization of the flow rule of

classical plasticity and is motivated partly by dislocation

dynamics. The inelastic strain rate is proposed as an

exponential function of stress and an internal state

variable representing hardness, where the hardness

parameter is an exponential function of plastic work. The

proposed formulation corresponds to isotropic hardening and

therefore would not characterize the Bauschinger effect.

Bodner and Partom found good agreement of their theory with

experimental results for commercially pure aluminum.

In noting that most constitutive theories consider

isotropic and directional hardening effects as completely

separable and thereby controlled by different internal

state variables, Bodner and his colleagues [104] extended

their isotropic theory in 1979 to model uni.axial cyclic

loading by proposing an alternative approach. This
55

approach uses separate values of the hardening parameter in

order to account for the directional character of hardening

during cyclic loading; one value corresponds to tensile

loading, the other corresponds to compressive loading.

Bodner, et al., found good agreement with uniaxial

experimental data for OFHC copper and commercially pure

titanium and aluminum at room temperature in modelling

phenomena such as cyclic strain hardening and softening.

The Bodner-Partom theory subsequently was generalized

to a framework for the multidimensional stress case by

using a general anisotropic form of the flow law [105].

In another development of the theory [106], an

additional term was introduced to the constitutive

equations in order to account for the thermal recovery of

hardening, a term essential for modelling secondary creep.

This development was used to model the superalloy Rene 95

at high temperatures.

A parameter to account for damage was introduced in

1980 [71]. Bodner examined the isotropic case first and

then generalized the damage parameter to the anisotropic

case in a manner similar to his earlier treatment of

anisotropic hardening [105].

A preliminary method to determine the material

constants of the uniaxial constitutive equations directly

from experimental data rather than from trial-and-error
56

curve-fittin g was presented in 1981 in a study of Rene 95

at high temperatures [69]. This was further developed into

a detailed systematic method and applied to IN100 at

elevated temperatures [107] for the case of isotropic

hardening with thermal recovery and no damage.

In an earlier paper [105], anisotropy was introduced in

a manner not automatically leading to plastic

incompressibility. Bodner and Stbuffer revised the theory

in 1983 [108] to enforce plastic incompressibility in order

to make the theory consistent with stability and

thermodynamic principles. Results from the revised theory

are essentially unchanged from the original work except

that plastic volume changes become zero.

An incrementally isotropic form of the flow law has

been proposed [109] in order to alleviate some of the

computational difficultie s encountered in using the full

anisotropic form. In this formulation, the scalar

hardening variable is taken to be the sum of the isotropic

component and a scalar effective value of the directional

component. Evolutionary equations are given for both

components, including thermal recovery terms, as well as

for isotropic damage. It is suggested that anisotropic

damage can be treated in a manner similar to anisotropic

hardening.

Recent work on the Bodner-Partorn theory includes
57

research into modelling transient temperatures [91,94], a

complete detailed review of the present constitutive theory

[110], and development of a systematic method for the

determination of material constants for the model in its

complete anisotropic form with damage [111].

General Theory

The unified creep-plasticity model of Bodner and his

colleagues [107] is motivated in part by work done in

dislocation dynamics in which the total deformation can be

separated into elastic and plastic components which are

generally non-zero at all stages of loading. A yield

criterion or loading and unloading conditions are therefore

not required.

The total strain rate is given by

where the elastic strain rate e.. is given by the time

derivativ e of Hooke's Law and the inelastic strain rate

. i

e. . is given by the flow rule of classical plasticity:

1.. - e1.. - A 0Jr (75)

.

where e.. and o!. are the deviatoric components of the

i J i J
58

inelastic strain rate and applied stress tensors,

respectively, and A is a scalar material function.

Squaring both sides of (75) gives

A

2

 = D P / J , (76)

P

where D_ is the second invariant of the inelastic strain

rate deviator

and J is the second invariant of the stress deviator

(78)

A fundamental assumption of the Bodner-Partorn theory is

that all inelastic deformations are governed by the kinetic

equat ion

f(J2,T,Zk), (79)

where T is the absolute temperature and Z are a set of

K

internal state variables. Extensive work in the field of

dislocation dynamics [11 2-1 1*1] has shown that dislocation

velocity and, therefore, inelastic strain rate can be
59

represented as a power function or an exponential function

of stress.

Bodner and Partom note that in order to represent

P

material behavior D? should vary inversely with the

measure of strain hardening, have a limiting value for

large J?, and be almost zero for low stresses. Whereas

several mathematical functions may fulfill these

requirements, the function chosen by Bodner and Partom is

D

P

2 = DQ exp[-(Z2/3J2)

n], (80)

where D is the limiting strain rate in shear, n is a

temperature dependent material constant, and Z is

interpreted as an internal state variable representing

hardness. It should be noted that, in subsequent

developments of the theory, an internal state variable

representing damage was introduced in the following manner

[71 ,109,110] :

exp{-[Z2(1 - u>)2/3J0]

n}, (81)

where to represents damage in the material.

Equations (75),(76), and (80) can then'be combined to

giv e an expression for the .inelastic strain rate:
60

*. - {D2 exp[-(Z2/3J2)

n]/J2}1/2 ojj. (82)

The work-hardened state, that is, the resistance to

plastic flow, is assumed to be represented by a single

P P variable Z which depends on the plastic work W , where dW

is ode . The plastic work rate was chosen by Bodner and

Partom as the measure of hardening over the more common

choice of the inelastic strain rate because Z is directly

related to the stored energy of cold work and this choice

leads to relative simplicity of the resulting equations

[110].

It is postulated that the evolution of Z is governed by

the current values of stress, hardness, and the absolute

temperature:

Z = F(J2,Z,T), (83)

where the particular form chosen by Bodner and his

colleagues is

Z = ra (Z1 - Z) WP - A Z1 [(Z - Z2)/Z1J

r, (84)

where m, Z , Z , A, and r are temperature-dependent

material constants and the constant Z is designated as the

initial value of Z. The constants Z and Z? correspond to
61

the maximum value of Z and the minimum recoverable value of

Z, respect ively.

The first term in equation (84) is the hardening term,

where the negative part can be interpreted as dynamic

recovery, and the second term is the thermal recovery term,

designed to be negligible during rapid loading histories.

This representation, equation (84), corresponds to

isotropic hardening and, as a result does not predict the

Bauschinger effect. The Bodner-Partorn theory was later

modified [1093 to include directional hardening by taking

the scalar hardening variable Z in the inelastic strain

rate equation (82) as the sum of the scalar isotropic

hardening variable Z and a scalar effective directional

4

hardening variable Z :

Z = Z1 + ZA, (85)

where Z is now given by (84) and the evolution equation

a

for Z has the general form of (84) but is tensorial in

character.

For the purposes of this research, the isotropic

hardening model wit h no damage [107], equations (82) and

(84), is considered.

For uniaxial stress and constant temperature, equation

(82) reduces to
62

DQ exp[-i(o/2)"2n] sgn(o) (86)

and equation (84) remains unchanged due to its scalar

nature:

Z = m (Z1 - Z) WP - A Z1 [(Z - Z2)/Z1]

r

. (84)

The material constants to be determined are Dfl, n, m, Z ,

A, Z , r, and Z , the initial value of Z.

D

0

Evaluation of Material Parameters

The quantity DO is the maximum value of strain rate in

shear and has its physical basis in the upper bound of the

dislocation velocity. The value of Dn can be set as

1O8 sec"1 for eI >103 , 1O6 sec"1 for 101 < el <103 , and

1 O14 sec"1 for £I <101 .

Z and n

The first step in the evaluation of material constants

is to determine the strain rate sensitivity parameter n and

the maximum value of the hardness, Z , from tensile data.

For material behavior as shown by curve (a) in Figure 11

where the stress saturates to a maximum value, in the

region where both the stress and strain rate are constant Z
63

(a) stress is

saturated

(b) stress is not

saturated

Fig . 11 A qualitativ e compariso n of a saturate d stress

curv e and a non-saturate d stress curv e
must also be constant in order to satisfy equation (86).

For short duration tests with no recovery, the material

must be in a fully work-hardened state to obtain the

maximum value of stress; that is, Z must have its maximum

value Z.. Rewriting equation (86) for this steady flow

condition gives

-2nln(o) + [2nln(Z ) + ln()]. (87)

The left hand term of (87) must be linear in In (o) if it

is an adequate representation of experimental data. A plot

of data in the form of equation (87) is shown in Figure 12.

Graphical means or a least-squares analysis of the data

will give the values for n and Z .

However, if the tensile curves are not quite saturated,

as shown by curve (b) in Figure 11, simply using the

largest value of stress attained may give incorrect values

for Z and n. In this case, equation (86) is solved for o

to give

o = K Z, (88)

where
Constant strain rate

tensile data

Best fit line

through data

In (a)

Fig. 12 A qualitative graph of constant strain rate

tensile data for determining n and Z
66

[2ln(7|D0/eI)]~1/2n. (89)

In the absence of hardening recovery, the hardening

evolution equation (84) becomes the first order linear

differential equation

dZ = m (Z1 - Z) dWP, ' (90)

which can be combined with the differential of equation

(88) to give

do = K1 m (Z1 - Z) dWP. (91 )

By defining a new parameter Y as

Y = do/dWP (92)

and substituting equation (88) for Z, equation (91) can be

rewritten as

m Z1 -mo . (93)

In order to be an adequate representation of experimental

data, Y must be linear in o, as shown in Figure 13- The

saturation stress is given by the stress level where Y = 0.
67

H Test 1

0 Test 2

Best fit line through

data points

saturation stress

(test 1)

saturation stress

(test 2)

Fig . 13 A qualitativ e grap h of Y v. o for determinin g the

saturate d stress leve l
68

Wit h this value of the saturation stress, one can now

return to the method outlined above to determine n and Z .

Z and m

The value of the hardening variable Z can be calculated

for corresponding values of stress and inelastic strain

rate by inverting equation (86):

Z = o C2ln(7|D0/eI)]1/2n. (91)

Equation (90), the hardening evolution equation with no

recovery, can be integrated to give

ln(Z1 - Z) - ln(Z1 - ZQ) - mWP, (95)

where Z0 is the initial value of Z. Several values of Z

can be calculated for each test by using equation (9^) at

different stress levels and their corresponding values of

e taken from the region in the o~e graph where recovery

p

is minimal. In this manner a plot of ln(Z - Z) against W

can be made for several constant strain rate tensile tests.

Since this plot must be linear in order to represent

material behavior adequately, as shown in Figure 1H, the

constants m and Z can be calculated easily either by

graphical means or by a least-squares analysis.
69

ln(Z1-Z0)

Q Test 1

© Test 2

Best fit line through

data points

-m

W'

Fig. 1 *J A qualitative graph of ln(Z -Z) v. W for

determining m and Z0
70

Alternatively, if a Y-o graph was used In the earlier

determination of n and Z , then the average of the slopes

of the linear curve-fits of the "Y-o data will be -m. Z is

then given by calculating values of Z in the immediate

post-yield (i.e., post-elastic limit) region of a

stress-strain curve.

Z

2

During secondary creep, since stress and inelastic

strain rate are constant, the value of Z must also be

constant in order to satisfy equation (86). These

stationary values of Z can be determined from equation (9*O

for several creep tests. The lowest calculated stationary

value of Z can be used for the value of Z , the minimum

value of Z, although Bodner states that Z =Z for many

applications since the initial condition (Z ) is the fully

recovered state (Z2) [110].

A and r

Since Z = 0 during secondary creep, the hardening

evolution equation becomes

m (Z1 - Z) WP = A Z1 [(Z - Z2)/Z1]

r

. (96)

.

The values of o, e , and Z ur.ed In the determination of Z
71

• P above may be used in a plot of ln[m (Z - Z) W ] against

ln[(Z - Z )/Z ]. The constants A and r can be calculated

from these data by assuming a linear representation,

although a bilinear representation may also be possible in

some cases [107],

In summary, the Bodner-Partorn theory requires several

constant strain rate tensile tests (constants E, n, Z , m,

and ZQ) and several creep tests (constants Z«, A, and r),

where the actual number of tests is dependent on the amount

of data deemed necessary for accurate evaluation of the

constants. This concludes the determination of material

constants for the Bodner-Partorn theory.
72

MILLER'S MODEL

Chronological Development

Miller's model [90], introduced in 1976, is based on a

combination of creep phenomenology and microstructural

considerations. In stressing the need to model

steady-state creep rates accurately, Miller proposes to use

the Garofalo hyperbolic sine relation for steady-state

creep as a basis for his constitutive equations.

Representing isotropic and kinematic hardening,

respectively, a drag stress variable and a back stress

variable are introduced to describe non-steady-state

inelastic responses. The growth laws for the back stress

and the drag stress follow the standard Bailey-Orowan

format of competing hardening and recovery rates. Miller's

growth laws feature, a constant kinematic work-hardening

coefficient but a variable isotropic work-hardening

coefficient. A procedure for evaluation of the material

constants, based partly on trial-and-error curve-fitting,

is also provided. Miller found reasonably good agreement

of his theory with experimental results for type 30^4

stainless steel at room temperature.

This model contains both kinematic and isotropic

hardening terms, an arrangement which leads, in Miller's

view , to a rather complex set of equations. Miller and
73

Sherby [115] simplified this in 1978 by eliminating

kinematic hardening and allowing only isotropic hardening

to be modelled. While this representation gave fairly

realistic simulations for pure aluminum, it became apparent

that an additional strengthening term was needed for

modelling of solute-strengthened materials. The solute

strengthening term, which is not an internal state

variable, was added to the drag stress in the inelastic

strain rate equation.

In 1980, Miller noted that several existing unified

constitutive theories, including his own, predict

"over-square" hysteresis loops with respect to

experimentally observed behavior [116]. To eliminate this

discrepancy in Miller's model, the work-hardening

i

coefficient in the back stress growth law, previously a

constant, is made an exponential function of the back

stress and the direction of the effective stress. The

predicted hysteresis loops using the improved equations

compare more favorably with experimental results.

Schmidt and Miller further improved the model by

introducing an additional solute strengthening term

[117,118]. This interactive term is multiplied by the drag

stress to accompany the already present non-interactive

solute strengthening term (see reference [115]) which is

added to this product. The non-interactive term models the
influence of solutes on yield strength while the

Interactive term models the influence of solutes on strain

hardening. These improvements allow the model to be used

over a broader range of loadings where solute effects are

significant.

Miller and his colleagues have devoted much effort to

research in microstructural topics which have led to the

particular formulation and features of the model: the

mechanisms of solutes and their effects on the yield

strength and strain hardening behavior of a wide variety of

alloys [119], the physical factors governing power law

creep as a basis for the form of the inelastic strain rate

equation [120], and experimental cyclic torsion data for

support of many assumptions and predictions of the model

with respect to steady-state back stresses and cyclic

strain hardening/softening [121].

Recent work includes extending the theory to multiaxial

form by Miller [122], and in a different manner by Kagawa

and Asada [123]. In addition, it has been reported [91]

that the latest form of Miller's theory contains four

internal state variables in order to model the long range

and short range components of the drag stress and the back

stress. This form of the model may be useful in modelling

transient temperature response [91].
75

General Theory

In attempting to model a broad spectrum of deformation

phenomena, Miller uses a combination of micromechanics and

creep phenomenology to derive the constitutive equations of

his model [90]. It is postulated that all inelastic

deformation can be characterized adequately by

e

1

 = f[(o - R)/D], (97)

where e is the inelastic strain rate, f is the inelastic

strain rate function, R is the "rest stress" (or back

stress), and D is the drag stress. To complete the model,

it remains to determine the best possible specific function

for equation (97) and the evolution equations for R and D.

In stressing the need to model steady-state creep rates

accurately, Miller uses the Garofalo hyperbolic sine

relation for steady-state creep as a basis for deriving the

inelastic strain rate function:

* = B' [sinh(Aoe J]n, (98)

S S So

where the subscripts "ss" refer to the steady-state

condition, B' is a temperature dependent parameter, and A

and n are temperature-independent constants.

Since equation (97) must reduce to (98) for the special
76

case of steady-state creep, there must exist a function f

which causes the argument (o - R )/D to reduce to o ,

ss sss s ss

i.e.

f

1

[

<°SS

When this is true, equation (99) can be substituted for o

S 3

in equation (98) to give E as a function not only of

S 3

o , but also of R and D . Since steady-state creep is

s s sss s

only a special case, the subscripts "ss" can be dropped to

give

e

1

 = B' (sinh{Af1[(0 - R)/D]})n. (100)

By examination of warm-working data for several metals, the

function f. is determined to be

L(o - R)/D]1 >5. (101

Substituting this into equation (100) will result in the

explicit equation governing the inelastic strain rate:

e

1

 = B1 (sinh{[(0 - R)/D]1'

5J)n. (102)

The entire temperature dependence of the model lies in
.77

the constant B1, which is related to the activation energy

for plastic flow. Below 0.6T , where T is the absolute

m m

melting temperature, the activation energy decreases

linearly with temperature and B' is given as

B' = B exp{[-Q/0.6kT ] [ln(0.6T /T) + 1]}, (103)

m m

where B is a material constant, k is the gas constant, Q is

the activation energy for plastic flow, and T is the

absolute temperature. Above 0.6T , the activation energy

is approximately constant and B' is given as

B' = B exp(-QXkT). (104)

In either case, the expression for B1 can be abbreviated as

B' - B 9' , (105)

where 8' is the desired temperature-dependent factor.

The evolution equations for R and D are given in the

standard work-hardening/recovery format as

R = H1 e - f2(R,T) (106)

and
78

D - H'e1 - f_(D,T), (107) '2

where H and H' are constants, and f and f_ are recovery

functions.

In order to determine the recovery functions f ' and f_,

Miller applies equations (106) and (107) to the special case

of steady-state creep (where R=D=0) and substitutes

equation (98) for e to obtain

S3

f

2

( R ss' T )

 --

 HlSs ' H1 B' tsinh(Ao3 S )] n

 (1080

and

f3(D33.T) = H^egs = H' B' [sinh(Aogs)]n. (109)

In a manner similar to the derivation of the inelastic

strain rate expression, equations (97) to (100), the

evolution equations become

- HI B 6' £sinh(A 1 R)] n (110 )

a n d

B 6 ? [sinh(A 2 D

3

) ] n, (111 )
79

where A. and A? are material constants.

For the purposes of this research, a simplified version

of this model will be used. Miller and Sherby [115]

simplified the model by eliminating all kinematic hardening

terms, allowing only isotropic hardening to be modelled,

and incorporating a solute strengthening term:

B 9' (sinh{[(o/E)/(D + F)1/2]

1

'

5

})n, (112)

where the modulus-compensated stress o/E has replaced the

stress o from the earlier equations for mathematical

convenience, F is the solute strengthening parameter, s o j,

and D and F are placed under the square-root sign to

SOX

produce parabolic hardening, since parabolic hardening is a

better approximation for many materials than the linear

hardening of the earlier equations. The evolution equation

for D becomes

D = Hje 1 ) - H B 6' [Sinn(A3D

1'5)]n. (113)

where H and A are material constants.

It should be noted that a slight modification has been

introduced by this author to equation (113). In order to

be able to simulate cyclic loading, the inelastic strain

rate term in equation (113) has been placed in absolute
80

value signs. This follows the practice .Introduced by

Miller, et al . , in later works [90,116-118].

The variable F is introduced into the equations in

order to characterize the effects of solute strengthening

on material behavior that are found in many solute

strengthened alloys. These effects include plateaus in the

yield strength versus temperature curve, local maxima and

minima in the strain rate sensitivity versus temperature

curve, and peaks in the apparent activation energy. By

judicious selection of the functional dependence of F

3 O X

upon the other variables, one can ensure that the resulting

equations simulate these effects.

At low temperatures, solute atoms are so immobile that

they cannot keep up with the moving dislocations and hence

cannot exert any particularly strong drag force. At high

temperatures, the solute atoms are so mobile that they can

move easily with the dislocations and are again unable to

exert a strong drag force. At intermediate temperatures,

however, the average velocities of solute atoms and

dislocations are approximately equal, and the two can

interact strongly, producing a maximum strengthening

effect. From these physical considerations, one can deduce

that F .. should have a low value at low temperatures,

should increase with temperature and pass through a

maximum, and should fall to a low value at high
81

temperatures.

Furthermore, experimental evidence indicates that

strain rate sensitivity occurs in a fashion similar to

temperature sensitivity.

The particular statistical function which satisfies the

mathematical requirements is

Fsol = Fsol,raax exp(-{[log10(Z) - log, Q ( Z ) ]/2} 2)

1(10)~7 exp(-{[log1Q(Z) - 30D/10}2),

i'I i

where Z is e1 /6', F , and Z are constants, and

 ' sol,max max

the second term on the right hand side of the equation is

included to ensure a non-zero value of F , for conditions

sol

when Z>»Z

max

Finally, the material constants to be determined are A,

B, n, H, F , , and Z . In addition, T and Q must be

sol,max max m

determined for the temperature-dependent factor 9'.

Evaluation of Material Constants

Q and Tm

The values of the absolute melting temperature T and

the activation energy for plastic flow for the material of

interest may be determined from various sources in the
82

literature. With these values and by use of equation (103)

or (101), the temperature-dependent factor e1 may be

calculated for a given absolute temperature T.

A

In order to derive the relation between the constants

A, B, and n and steady-state creep rates, one returns to

the observation that, for the special case of steady-state

creep, the inelastic strain rate equation (112) reduces to

the Garofalo equation (see equation (98)):

eL - B 8' [sinh(Ao /E)]n, (115)

9 3 -S 3

where e is the steady-state creep rate and o /E has

ss ss

replaced o •. Equation (115) may be rewritten as

3 S

s/e')

 = logio(B)

nlog 1ft[sinh(Ao /E)]. (116 )

I (J S3

If this equation is an adequate representation of

experimental data, then the constant A can be chosen such

that log.-(e^/e1 ) is linear in log,n[sinh(Ao /E) ], as

Iv/S S I U 33

shown in Figure 15.
83

Value of A is too small;

curve is concave

upward.

log10[sinh(A0ss/E)]

Value of A is too

large; curve is

concave downward.

log10[sinh(Aoss/E)]

Value of A is

correct; curve is

linear.

lo81()[sinh(Aass/E)]

Fig . 15 The effec t of A on log, r t( e 1 /e 1) v.

i 0 ss

l o g [sinh(A o /E) ] data

\ \J o 3
B and n

Since the value of A is now known, one can easily

determine the constants B and n from equation (116), either

by graphical means or by a least-squares analysis.

sol,max

 an max

The maximum value for F is assigned arbitrarily and

3 O -L

is based on an intuitive analysis of the problem of

interest. Z is set by Miller as Z - 1(10)10 for his max . max

choice of function for F ,.

sol

H

Finally, the value of the work-hardening coefficient H

is determined on a trial-and-error basis by a best fit of

the model through several experimental constant strain-rate

stress-strain curves.

In summary, Miller's theory requires some information

about the material of interest (for T and Q), several

m

creep tests (constants A, B, and n), and several constant

strain rate tensile tests (for e and H). This concludes

the evaluation of material constants for Miller's model.
85

DETERMINATION OF MATERIAL CONSTANTS

Each theory is evaluated and compared to experiment by

its authors for a particular material under particular

temperature conditions. In general, the choice of material

and conditions varies from theory to theory. One of the

objectives of the current research is to create a common

experimental data base for one material at one temperature

in order to characterize the mathematical representations

and to compare to experiment the numerical predictions of

each theory.

Many of the phenomena associated with inelastic

deformation such as rate sensitivity, the Bauschinger

effect, and creep become significant in crystalline

structures at elevated temperatures, that is, above three

tenths of the absolute melting temperature. At the start

of this research it was intended to perform testing on the

nickel-based superalloy IN718 at 1350°F. Testing was

restricted to room temperature conditions, however, since

extensive difficulties were encountered with the available

equipment with testing at high temperatures. This

restriction led to the choice of the aluminum alloy 5086 as

the candidate material. This alloy displays many of the

phenomena of inelastic deformation and is already above
86

three tenths of Its melting temperature at room

temperature. Its composition and some properties are

listed in Table 1.

A review of the material test requirements of each

theory is shown in Table 2. It should be noted that the

stress-drop tests are performed during steady-state creep

and, as such, can be combined with the creep tests. The

constant strain rate tensile tests were performed on an MTS

810 hydraulic material testing system; the creep tests and

the stress-drop tests were performed on a constant-load

creep frame which was designed and constructed at Texas A&M

University. Data acquisition in both cases was by

graphical means, although a digital system was also

available for the creep frame. It was decided to use a

constant-load creep frame rather than a constant-stress

creep frame due to difficulties associated with accurately

determining the applied stress when using the latter

system. As can be seen by the creep response below, the

effects of this choice on the results are negligible.

The results of the constant strain rate tensile tests

can be seen in Figure 16 for four applied strain rates

-7 -1 -ty -1 ranging from M10) sec to 4(10) sec . It can be

seen that the material exhibits some strain rate

sensitivity and considerable work-hardening.
87

Table 1

Composition and Material Properties of Al 5086

Material: Al 5086 H11 1

Composition: 0.15/6Cr, Q.UO% Mn, H.Q% Mg, remainder Al

Melting temperature T : 858K

Elastic modulus E: 10.313C10)3 ksi

Ultimate stress o : ^0 ksi

Activation energy for plastic flow Q (at room temp.):

27,500 cal/mole

Form of specimens: uniaxial bar with --inch diameter

gauge section and threaded ends.
88

Table 2

Review of Material Test Requirements

Type of test Krieg Bodner Miller Total

Constant strain rate

tensile test

Creep

Stress-drop

1

1

3-1

3-1

3-1

-

3-1

3-1

-

3-1

3-1

3-1
89

a EXP, STRAIN RATE » 4E-4

O EXPi STRAIN RATE - 4E-5

4 EXPi STRAIN RATE = 4E-6

+ EXP. STRAIN RATE = 4E-7

on

CD a

CO"?

asj-

-O

CO •

a

a 02 0.06 0.10 0.14 C. 18

Strain (in/in)

0.26 0.30

Fig. 16 Experimental constant strain rate tensi-le test

response
90

tne results of the constant-load creep tests can be

seen in Figure 17 for five applied stresses ranging from

32.725 ksi to 38.850 ksi. Pertinent results of the

constant strain rate tensile tests and the creep tests, as

well as the stress-drop tests, are summarized in Table 3.

Material Constants for Krieg, et al .

In determining the material constants for this and the

other theories under consideration, the procedures

discussed in a previous section are followed. The reader

may consult that section for greater detail.

The values of c. and c_ are found by applying a linear

curve-fit to a graph of ln(e ) v. ln|o - a j data, where

. T

e is the strain rate immediately preceding the

stress-drop, o is the applied stress, and a is the back

stress. This is shown in Figure 18; the slope of the .

linear fit is c and the vertical intercept is ln(c..). In

this manner, c is calculated to be 7.177 and c? is

Under steady-state creep, the growth law for the back

stress becomes (equation (71) repeated)

[exp(c5a

2) - 1]. (71)

Applying this equation to two experimental cases results in
91

2.8

0=38.850 ksi

0 2000 4000 6000 8000

Time (sec)

Fig. 17 Experimenta l Cree p Test Respons e

10000
92

Table 3

Pertinent Results of Experimental Tests

Constant strain rate tensile tests

Elastic modulus = 10.313(10)3 ksi

Elastic limit = 28.000 ksi

0.2% offset yield stress = 30.250 ksi for 4(10) sec"1

30.000 ksi 4(10)"5 sec"1

29.750 ksi 4(10)"6 sec"1

29.500 ksi 4(10)"7 sec"1

Creep tests

Steady-state creep rate -

1.205OO)"8 sec"1 for 32.725 ksi

4.167OO)"8 sec"1 35.433 ksi

3.125OO)"8 sec"1 36.437 ksi

4.435(10)"8 sec"1 36.705 ksi

4.546OO)"8 sec"1 38.850 ksi

Stress-drop tests

Strain rate before applied stress back stress

stress drop

0.1000(10)~7 sec"1 34.402 ksi 33.107 ksi

0.4243(10)~7 sec"1 35.433 ksi 33.804 ksi

2.658(10)~7 sec"1 36.705 ksi 34.659 ksi
93

(sec )

-15

-20

-25 '

Q Experimental

data

Best fit line

1 1 1 1 ( 1 1- 1

0 0.2 0.4 0.6 0.8 1.0

ln|a-a| (ksi)

Fig. 18 Grap h of IrKe 1) v. ln|<j-a | for Krieg's Mode l
two nonlinear equations in two unknowns which can be solved

for (c /CjJ and c by Iteration. Solving for the cases

with the highest and lowest values of a (see Table 3)

yields (c-/^) =• 3.255(10)25 ksi2sec and c = 0.03030

ksi"2.

To determine c_, a point is chosen on the primary creep

curve where the creep rate is still twice the eventual

steady-state rate. The effective stress c at this point is

calculated by inverting the inelastic strain rate equation

(68) (equation (73) repeated here):

)

1/Cz

. (73)

The value of the back stress is then given by a = o - £.

Using the creep test with o = 35.^33 ksi gives the

following information:

o1 = 35.^33 ksi,

ef = 8.333(10)~8 sec'1,

t1 = 450 sec.

This gives $ = 1.717 ksi and a = 33.716 ksi, where the

subscript 1 refers to the point on the primary creep curve.

The growth law for a, equation (69), can be integrated

numerically by separating variables (equation (72)

repeated):
95

[exp(c 5 a

2

) - 1]}~ 1d a . (72 )

Q

With a and o. known, c.,t is calculated as 2.126(10) ksi

ij

sec. From this, c = 5.185(10) ksl and

GJJ = 1.593(10)~21 ksi"1sec"1.

This concludes the material parameter determination for

the theory of Krieg, et al.

Material Constants for Bodner, et al.

For the present strain rate range of interest, DQ is

H -1

set as 1(10) sec

To determine n and Z , a saturation stress must be

determined analytically since the experimental

stress-strain curves are not saturated (see Figure 16).

This is accomplished by extrapolating to the point of zero

P

Y on the Y v. o graph, where Y=do/dW and is given by

differentiating a curve-fit of W = WP(o). It is found

P P that quadratics provide the best curve-fit for W = W (o):

W

P = 4.679 - 0.3679o + 0.7^62(10)"2o2 (117)

for e = 4(10) sec , and
97

Y

15 t

10 +

5 i

Data for

e=4E-7 sec

Data for

e=4E-5 sec

-1

-1

Best fit line

through data

4(10)~5sec"1

4(LO)"7sec~lx

\

\\

\\

\\

\\

H 1

28 30 32 34

H 1 1 1 -H 1 h

36

a (ksi)

38 40

Fig. 19 Graph of Tf v. o for Bodner's Model

WSCEDWG PAQE SUWK WOT
O Data points 

- Best through fit data line 

Fig. 20 

for ~ddne 
99

Values of c are calculated from e = e (1 - ^-r^O

and the resulting values of Z are shown in Table 4 below,

Table 4. Calculated Values of Z used

in the Determination of Z_

e (sec'1 ) o (ksi) e1 (sec"1 ) Z (ksi)

4(10)" 7 28.250 3-6474(10)"" 7 34.102

4(10)" 6 28.375 3-5152(10)" 6 34.090

4(10)~ 5 28.500 3.3536OO)" 5 34.060

4(10)" ^ . 28.688 3-2243(10)" ^ 34.081

An average value of 34.00 ksi is taken for Z .

The value of Z?, the minimum value of Z, is taken to be

the lowest value of Z for steady-state creep, although one

may simply set Z •= Z-. The latter is chosen, giving

Z2 - 33.50 ksi.

The values of the recovery coefficient A and the

recovery exponent r are determined by calculating a value

of Z for the steady-state region of each creep test by

using equation (94) (see above). Calculated values of Z

are shown in Table 5 below.
1 00

Table 5. Calculated Values of Z used

in the Determination of A and r

ogg egg Z ln[m (Z^Z)^] ln[ (Z-

(ksi) (sec"1) (ksi) •

32.735 1.205OO)'8 39.758 -2.091 -12.38

36.437 3.125(10)"8 44.192 -1.520 -12.37

36.705 4.435(10)~8 44.489 -1.491 -12.14

These values of Z are then used to construct a

ln[m (Z1 - Z) W ] v. ln[(Z - Z )/Z ] graph, as shown in

Figure 21. The slope of a linear curve-fit for this graph

is r; the vertical intercept is ln(AZ.). In this manner, A

is calculated to be 1 . 447 (10 )~7sec~1 and r is 0.2290.

This concludes the material constant determination for

the theory of Bodner, et al.
101

© Data points

'Best fit

line

-12.5

-12.0

-11.5

-1.2 -1.6

1 1 1 1 1

-2.0

ln[(Z-Z2 )/Z1 ]

.P-.

Fig. 21 Grap h of ln[m (Z^-Z ) W ] v. ln[(Z- Z )/Z ]

for Bodner's Mode l
102

Material Constants for Miller, et al.

The values of the activation energy for plastic flow Q

and the melting temperature T can be found in a materials

handbook. For Al 5086 at room temperature, Q is 27500

cal/raole and T is 858K. m

Since room temperature is less than 0.6T for Al 5086,

tn

equation (103)t repeated below, is used to calculate the

temperature-dependent factor e': -

8' = exp{[-Q/0.6kT 3 [ln(0.6T /T) + 1]}, (103)

IB ID

where T and T are absolute temperatures. This gives

8' as S.T.IdO)'1 9.

The value of A is chosen such that a graph of

log..(eo_/e') v. log n[sinh(AoOQ/E) ] for steady-state

IOS 3 I » S3

creep data is linear. This is shown below in Table 6 and

in graphical form Figure 22 for several values of A.
103

Table 6. Calculations for Various Values of A

Ss

 10SlOU

ss/9l) °S3

(sec" 1 ) (ksi) A = 50 A=-200 A-1000

1.205C10)" 8 10.1 i| 32.725 -0.7977 -0.1687 1.076

4.167(10)~ 8 10.68 35.443 .-0.7628 -0.1292 1.191

3.125(10)" 8 10.55 36.437 -0.7506 -0.1153 1.233

4.435OO)" 8 10.71 36.705 -0.7474 -0.1116 1.244

4.546C10)" 8 10.72 38.850 -0.7225 -0.0826 1.335

.,.•.•- ...

As can be seen in Figure 22, t- =; curveT'is rtear-i-y linear for

A = 50, with no significant improvement for A < 50.

With this value of A, the values of B and n can be

determined by constructing a linear curve-fit for the data

in Figure 22. The slope of the curve-fit is n; the

vertical intercept is log (B). In this manner, n is

calculated to be 7-701 and B is 2.410(10)16 sec"1.

The maximum value of F , is set arbitrarily as

sol

F , = 1(10) . Simulations of experimental tests have

sol,max

shown that the choice of this value does not influence the

results significantly.

The work-hardening coefficient H is set arbitrarily to

giv e the best fit of simulations of constant strain rate

tensile tests to experimental data. This value is set as

7.5(1 O)"1* ksi.
CO

-I

• U

 W

)

sOx

60

O

uO

O

O

i

II

o

-I 1-

oo

O

60

O

o

o

O

o

o

i

o

Fig. 22 Graph of 108,0(6^/8*) v. Iog1 Q[ sinh ( A ogg/E ) ]

for Miller's Model
105

Finally, the initial value of the drag stress, D.

is calculated by inverting the inelastic strain rate

equation (1 12) and using values of e and <? at yield. In

this manner, D. is calculated to be 3-386(10) .

X D X v

This concludes the material constant determinat5 on for

the theory of Miller, et al.

A sun ,ary of the material constants for each theorv is

given in Table 7.
1 06

Table 7. Summary of Material Constants for all Models

Krieg , etal i

Bodner, et al

Miller, et al

E = 10.313(1O) 3 ksi

c, = 7.17 7

.459(10 )

~

9

c = 5.1 SSdO) 1* ksi

0 = 1.593C10)" 2 1 ksi"1sec"1

c = 0.03030 ksi

-2

E - 10.313(10) ° ksi

n = 10.30

Z1 = 46.59 ksi

m - 1.558 ksi"1

ZQ = 3^.00 ksi

Z = 33.50 ksi

"

1

A = 1.4^7(10 ) sec

r = 0.2290

E = 10.31 3d °)3 ksi

6' = 8.74(1O)" 1 9

A = 50

n = 7.701

B = 2.4 1 0(1 O)1 6 sec"1

-7

sol,ma x

"

2 H - 7.5( 1 0) ks * i

D

ini t = 3.386(10 )

-5
107

COMPARISON OF THEORY TO EXPERIMENT

The third and final objective of this research is to

compare the numerical predictions of each theory to

experiment by integrating the constitutive equations

subject to specific load histories. Before proceeding to

the quantitative comparisons, however, there follows a

brief qualitative review of the predictive capabilities of

each theory.

Qualitative Review of Predictive Capabilities

All the theories under consideration are similar in

that they are unified theories; that is, the rate-dependent

creep and the rate-independent plasticity components have

been combined into one inelastic strain term. Each theory,

in the form under consideration, is isothermal in that it

is not able to model transient temperature response.

Furthermore, each model is able to exhibit strain rate

sensitivity through the exponent in the inelastic strain

rate equation.

The theory of Krieg, et al., contains two internal

state variables: the inelastic strain and the back stress.

The back stress represents kinematic hardening, which

allows the Bauschinger effect to be modelled. Krieg,
108

etL__a 1. , have assumed the hardening function in the back

stress growth law to be constant, which may cause

"over-square" hysteresis loops. This may be alleviated by

introducing a dependence of the hardening function on the

current value of the stress or back stress. Due to the

lack of a representation for isotropic hardening (the drag

stress), this theory may not be able to model cyclic strain

hardening. Finally, it can be seen that anelasticity

(strain recovery at zero load) can be predicted by this

theory by setting o = 0 in the inelastic strain rate

equation (68); relaxation is predicted by setting e =0;

and creep is predicted by setting o •= 0.

The theory of Bodner, et al. , also contains two

internal state variables: the inelastic strain and a terra

representing isotropic hardening. The isotropic hardening

growth law contains dynamic as well as thermal recovery.

The presence of isotropic hardening allows cyclic strain

hardening to be modelled, although the absence of kinematic

hardening may preclude the ability to model the Bauschinger

effect. Finally, it can be seen that anelasticity cannot

. i be modelled since e is zero when setting o = 0 in the

inelastic strain rate equation (86); creep and relaxation

can be predicted, however, by setting o = 0 and e = 0,

respect ively.

The theory of Miller, et al., contains two internal
1 09

state variables: the inelastic strain and the drag stress.

The drag stress represents isotropic hardening, which

allows cyclic strain hardening to be modelled. This model

is similar to that of Krieg, et al . , in that the hardening

function in the drag stress growth law is assumed to be

constant, which produces "over-square" hysteresis loops.

The model is similar to that of Bodner, et al . , in that the

lack of a back stress may preclude the ability to model the

Bauschinger effect as well as anelastic i ty . Finally, creep

and relaxation are modelled in a manner similar to that of

the other two theories.

Review of Integration Technique

Each theory is represented mathematically by a set of

differential equations. Since these differential equations

are numerically "stiff", causing higher-order integration

schemes to become unstable [121)], a first-order forward

integration scheme with small, variable time steps was

used. For example, Krieg's inelastic strain rate equation

((68) repeated here)

*

e = c | o - a | 2 sgn(o - a) (68)

becomes
1 10

c

I(t + At) = coCt ) - a(t)|°2At + e:(t) (119)

where t represents an Initial state and t + At represents

the incremented state. Similarly, the growth law for the

back stress (equation (69)) becomes

a(t + At) - [c [(eI(t + At) - eJ(t))/At] -

Clja

2(t)[exp(c5a2(t) ) - 1 ] sgn( a (t) )} At + a(t).(120)

Finally, the stress is updated by

o(t + At) = E {e - [(eX(t + At) - e1(t))/At]}At +

o(t). (121)

This concludes the review of the integration technique used

in this research.

Comparison of Numerical Predictions to Experiment

In this section, the ability of each model to predict

the response to a specific load history is analyzed. The

first experimental test to be modelled is a constant strain

rate tensile test with e = 4(10) sec , shown in

Figure 23.
111

B EXPi STRAIN RATE = 4E-S

/~\ m

cn^'

^^

(!) °.

£_

CO •

0.02 o.06 a 10 o.u a is 0.22

Strain (in/in ) *10''

a 26 0.30

Fig. 23 Experimental constant strain rate tensile test

e = i4(10)~5 sec"1
1 12

Krieg, et al. ; Constant Strain Rate Tensile Test

It can be seen from Figure 24 that this theory does not

compare favorably with experiment for this constant strain

rate tensile test. It is believed by this author that this

is due largely to the method of determining the material

constants. As discussed in an earlier section, the ability

to determine a zero creep rate or merely the absence of a

resolvable creep rate in a stress-drop test depends greatly

on the resolution of the data acquisition equipment and is

extremely difficult and time consuming at best. A

parameter variation study showed slight improvements in the

stress-strain curve in Figure 24 but led to computational

difficulties in simulating cyclic load histories and

complex load histories. It can be concluded from this that

improvements in the measurement of back stresses are

required. For example, the methods of Blum and Finkel

[98], in which the instantaneous strain change and the

maximum strain change after unloading are compared, and of

Walker [58], in which one applies hold times on the

unloading side of a saturated hysteresis loop and

extrapolates to the point of zero relaxation, merit further

study.

Bodner, et al.: Constant Strain Rate Tensile Test

The theory of Bodner, etal . , compares favorably to
1 13

Q KRIEC. STRAIN RATE - 4E-5

O EXft STRAIN RATE • 4E-5

0.02 0.08 a ID au a 18 0.22

Strain (in/in) *10~'

a.36 a 30

Fig. 2^4 Constant strain rate tensile test. Comparison of

theory of Krieg, et al., to experiment.
1 1

experiment, aa can be seen in Figure 25. Work-hardening is

predicted well although the elastic-inelastic behavior

transition is too sharp. It Is found that this is due to

the theory predicting values of Z that are too high at low

stresses, possibly requiring a modification of the

hardening term in the growth law for Z.

Miller, et al.; Constant Strain Rate Tensile Test

In determining the material constants for this theory,

the strain rate exponent n was calculated to be 7.7,

resulting in the response shown in Figure 26. A parameter

variation study indicated that a value of n «= 6 appeared to

fit the experimental data much better. This indicates a

problem inherent in many constitutive theories: the

determination of material constants is not rigorous and the

difficulties are compounded by the fact that many constants

are interdependent and, in fact, may be assigned arbitrary

values. In the interests of further numerical comparisons,

a value of n = 6 is chosen for this model. With this

choice, this theory predicts the yield point and

work-hardening fairly well.

Finally, a comparison of all the models for a constant

strain rate tensile test can be seen in Figure 27.
1 15

0 EXPi STRAIN RATE » 4E-5

O BQONERi STRAIN RATE • 4E-S

0.02 0.06 0.10 0,14 a 18 0.22

Strain (in/in) *10"

a 26 a.30

Fig. 25 Constant strain rate tensile test. Comparison of

theory of Bodner, et al., to experiment.
116

a MILLEfe STRAIN RATE - 4E-5. N-7.7

O MIU-ERi STRAIN RATE - 4E-5. M-S

« EXPi STRAIN RATE - 4E-5

0.02 a 08 a to au a is a 22

Strain (in/in) *10'(

a 26 a so

Fig. 26 Constant strain rate tensile test. Comparison of

theory of Miller, eta l . , to experiment.
1 17

B KRIE& STRAIN RATE = 4E-5

S EXPi STRAfN RATE = 4£-5

< HOLER. STRAIN RATE = 4E-5

+ BOONEft STRAIN RATE = 4E-5

aoa 0.08 0.10 ai4 0.18 0.22

Strain (in/in) *10~'

0. 28

Fig . 27 Constan t strai n rat e tensil e test. Compariso n of

all theorie s to experiment .
1 1 8

Many theories use creep tests or constant strain rate

tensile tests as information for the material parameter

determination; consequently, one would expect fairly

accurate simulations .of these types of load histories. A

more rigorous comparison, however is a cyclic load history,

• —'1- 1

as shown in Figure 28 with e = _+1 (10) sec for cycles

1-10. The experimental data exhibit a pronounced

Bauschinger effect and considerable cyclic strain

hardening, although it appears that the hysteresis loop

saturates fairly quickly.

Krieg, et a1.: Cyclic Loading Test

The cyclic response of the theory of Krieg, et al., is

shown in Figure 29 for cycle 1 . It can be seen that the

theory does indeed predict a Bauschinger effect, although

it is not as pronounced as the experimental data. The

model predicts a harder material than is the actual case,

resulting in the very thin and, as expected, "over-square"

hysteresis loop. This may be due to the ratio c /c^

(calculated from stress-drop data) being too large.

A comparison of the prediction for cycles 1 and 10 can

be seen in Figure 30, where, as expected, the model does

not display significant cyclic strain hardening. However,

since the material exhibits cyclic strain hardening, the

difference between the saturated experimental and
1 1 9

Q EXPi STRAIN RATE - V-lE-4. CYCLES 1-10

-a o? -aos -aoa -aoi aoi aro aos ao?

Strain (in/in) *10"

Fig. 28 Experimental cyclic loading test: cycles 1-10
1 20

EJ KRIEd STRAIN RATE - V-1E-4. CYCLE I

G EXPi STRAIN RATE = V-lE-4. CYCLE I

CO i

CO i

3

§

Xi-l

8

$-\

-ao? -aos -aca -aoi 0.01 0.03

Strain (in/in) *10''

0.05 a 07

Fig. 29 Cyclic loading test: Cycle 1. Comparison .of

theory of Krieg, etal . , to experiment.
Q KRIEGt STRAIN RATE •> V-lE-4. CYCLE 10

0 KRIEC, STRAIN RATE - -/-1E-4. CYCLE 1

121

-0.07 -0.05 -aoa -aoi aoi ara

Strain (in/in) *10~'

0.05 a o?

Fig. 30 Cyclic loading test: cycles 1 and 10. Prediction

of theory of Krieg, etal .
1 22

predicted loops diminishes somewhat, as shown in Figure 31•

Bodner, et al.; Cyclic Loading Test

Figure 32 shows that this theory models cycle 1 fairly

well, although the elastic-inelastic behavior transition is

too sharp (as noted in earlier tensile data also). The

absence of the Bauschinger effect suggests that a

directional hardening term is necessary. This can also be

seen when examining the Y v. o graph used earlier in

calculating some of the material constants. Although the

theory assumes a linear relationship between Y and o, the

experimental data suggest a bilinear relationship. This is

examined in greater detail by Bodner, et al., in recent

work [109-111]. The upper slope in this bilinear

relationship governs directional hardening and the lower

slope governs Isotropic hardening. It can be seen easily

that including such a directional hardening term could

provide the Bauschinger effect and also decrease the height

of the hysteresis loop.

The aforementioned observations can also be seen in

Figures 33 and 34, which compare prediction for cycle 1 to

prediction for cycle 10 and prediction to experiment for

cycle 1.0, respectively.
H KRIEGt STRAIN RATE • -/-1E-4. CYCLE 10

O EXPt STRAIN RATE •> »/-l£-4. CYCLE 10

123

CD i

-O

CO

§

-a 07 -0.05 -a 03 -aoi 0.01

Strain . (in/in)

i i I r

a 03

*1Q-'

0.05 a 07

Fig. 31 Cyclic loading test: cycle 10. Comparison of

theory of Krieg, et al.. to experiment.
121

EJ BOOfCfii STRAIN RATE = -Y-1E-4. CYOE 1

O EXP, STRAIN RATE = V-lE-4. CYCLE \

-do? -a 05 -a 03 -0.01 aoi 0.03

Strain (in/in) *1CT

a 05 0.07

Fig. 32 Cyclic loading test: cycle 1. Comparison of

theory of Bodner, etal. , to experiment.
Q 30DNER. STRAIN RATE - -Y-1E-4. CYCLE 10

© BOOMER. STRAIN RATE - -/-IE-4. CYCLE 1

-0.07 0.05 a o?

Strain Cin/in) *10"

Fig. 33 Cyclic loading test: cycles 1 and 10. Prediction

of theory of Sodner, et al.
0 BOQNERi STRAIN RATE - V-lE-4. CYCLE 10

O EXPi STRAIN RATE " V-1E-4. CYCLE 10

126

KlCO'

\~t

V) i 05!

L

CO

§

-0.07 -0.05 -0.03 -a 01 a 01 a 03

Strain (in/in) *10~'

0.05 a 07

Fig. 3^ Cyclic loading Lest: cycle 1.0. Comparison of

theory of Bodner, et al., to experiment.
127

Miller, et al. ; Cyclic Loading Test

The similarities between this model and that of Bodner,

et al. , can be seen in Figure 35 in that the theory does

not model the Bauschinger effect due to the absence of a

directional hardening term. It can also be seen that the

assumption of a constant hardening function for the drag

stress causes an "over-square" loop, although this is not

as pronounced here as with the other two models.

Comparing the predictions of cycle 10 and cycle 1 shows

extensive cyclic strain hardening (see Figure 36). This is

much more apparent in the theory-to-experiment comparison

for cycle 10, as shown in Figure 37, and suggests a need

for a revision of the hardening function.

Finally, a comparison of all models for cycles 1 and 10

is shown for completion in Figures 38 and 39, respectively.
128

B MILLER. STRAIN RATE " V-lE-4. CYCLE 1

O EXPi STRAIN RATE - V-lE-4. CYCLE 1

8

«H

3

i/H

-

CO"*

CO

«

Q)

4^

cn

g

°

§

§

F I

-ao? -aos -ao3 -aoi aot aoa

Strain (in/in) *10~'

\ I i I aos ao?

Fig. 35 Cyclic loading test: cycle 1. Comparison of

theory of Miller, etal. , to experiment.
1 29

B MILLERi STRAIN RATE • V-lE-4. CYCLE 10

© MILLER STRAIN RATE » V-1E-4. CYCLE I

CD"5

(D

CO -

3

in

-ao? -aas -aoa -aoi aoi ao3

Strain (in/in) *10~'

a os a o?

Fig. 36 Cyclic loading test: cycles 1 and 10. Prediction

of theory of Miller, et al .
130

B MILLER, STRAIN RATE - V-lE-4. CYCLE 10

O EXP. STRAIN RATE • */-lE-4. CYCLE 10.

-a 07 -a os -a 03 -aoi aoi a 03

Strain (in/in) *10"'

0.05 0.07

Fig. 37 Cyclic loading test: cycle 10. Comparison of

theory of Miller, eta l . , to experiment.
131

Q KRIEd STRAIN RATE • V-1E-4. CYCLE I

O MILLERi STRAIN RATE - V-lE-4. CYCLE 1

« BOmCR. STRAIN RATE = V-lE-4. CYCLE 1

*• EXPi STRAIN RATE • V-1E-4. CYCLE t

-ao? -aos -0.03 -aoi aoi ao3

Strain (in/in) *10~'

aos a 07

Fig. 38 Cyclic loading test: cycle 1. Comparison of all

theories to experiment.
132

B KRIEGt STRAtN RATE - -/-1E-4. CYCLE 10

O NrLLER. STRAIN RATE - V-1E-4. CYCLE 10

4 BOONERi STRAtN RATE = V-1E-4. CYCLE 10

+ EXPi STRAIN RATE • -V-1E-4. CYCLE 10.

-a 07 -aos -acre -aoi aoi ao3

Strain (in/in) *10~'

aos a 07

Fig. 39 Cyclic loading test: cycle 10. Comparison of all

theories to experiment.
133

The final set of comparisons involves the prediction of

the response to a complex loading history. Many models are

designed specifically for creep response or cyclic response

with somewhat limited capability for a general, complex

response. The loading history is shown in Figure 40; the

experimental response is shown in Figure 41.

Krieg, et al.; Complex Loading History

Figure 42 shows that this model over-predicts the

stress, indicating that the Inelastic strain is growing too

slowly. This implies that the back stress is growing too

quickly and that the hardening/recovery ratio is too large.

Further evidence of this is demonstrated by the

considerable amount of relaxation in compression. As

before, this discrepancy may be due to the method of

material constant determination.

Bodner, et al.: Complex Loading History

Figure 43 shows that this theory predicts the complex

loading response very well. The model appears to predict a

harder material with less relaxation than the experiment,

again suggesting that the hardness Z is too large. A large

Z causes a large e, leading to an over-prediction of

stress. Similarly, during relaxation (when e = 0), a

large Z causes a small o, leading to little relaxation.
0)

4-1

CO

)-l

4-

»J

CO

•H

J

CO o

XI 10

a) <r

•H I

i-H O

O. r-l

O. ^

< '

O U

en

o H

t

O

o

o

ro

H 1 1 1 1-

Fig. HO Experimental complex loading history
00-95 00 TW 00 -»2 00 "8 DO'S- 0012- OOTDV- 00-9S-

§

-s

- 2 0

- QJ

cn

o OJ

° E

-a'

-d

(ISM)

Fig . i|1 Experimenta l comple x loadin g response
D Q

UJ D

<

t— UJ

01

in <r

i in

1C u}

El O

00-9S OO'OV oo-»z 00 "8 00-8- 00 -V2- 00 "OT- 00 "9S-

(ISM)-

- OJ

-d

Fig. ^2 Complex loading test. Comparison of theory of

Krieg, et al., to experiment.
00-85 OOTJV QOtZ OQ-9-

(ISM)

m• ge

if

Q ©

00 -9SCO

8

o

Fig. ^3 Complex loading test. Comparison of theory of

Bodner, etal. , to experiment.
138

Miller, e t a1.: Complex Loading History

Figure 44 shows that this model over-predicts the stress

during loading and unloading. This may be due to the fact

that the hardening constant was set arbitrarily by fitting

theory to experiment for a given strain rate, whereas the

complex loading test involves strain rates upto two orders

of magnitude higher. The large amount of relaxation is

then brought on by the large value of stress.

Furthermore, It should be noted that, at approximately

40 seconds, the model experiences a negative stress rate

immediately after the applied total strain rate is reduced.

Again, this is due to the large value of stress causing the

inelastic strain rate momentarily to be larger than the

. T

total strain rate. This decreases o until e is less than

e, at which point the response returns to normal.

A solution to this could be an improved method for

determining H, as well as a more complex hardening law for

the drag stress.

Finally, a comparison of all theories to experiment for

the complex loading history is shown in Figure 45 for

complet ion. .
00-95 OOTW 00^3

•

OC LU

a e

oo^e 00*8- oo •95-

(ISH)

1-3

- OJ

(f)

-si

-d

Fig. HH Complex loading test. Comparison of theory of

Miller, et al., to experiment.
g a o |

• . §

is • S

 a

 " IS

liii

B G < +

00-9S

-r

I

i

o

- Q)

CO

g OJ

° E

-CD

(ISM)

Fig. H5 Complex loading test. Comparison of all theories

to experiment.
SUMMARY AND CONCLUSIONS

The objective of the research in this thesis has been

to review and implement three theories for the prediction

of inelastic deformation in crystalline structures. These

theories are based on considerations of the raicrostructural

behavior of the material and are the theories of Krleg,

e t a1., Bodner, et al., and Miller, et al.

The thesis opens with a review of the mechanics of

continua with internal state variables and a review of the

historical development of constitutive modelling. It was

shown that the concept of internal state variables is a

suitable framework for comparison of all the models

discussed herein and that the primary differences in these

models are only in the number of internal state variables

chosen and the growth laws used to obtain these variables.

A detailed discussion of each model chosen for

comparison follows, in which the chronological development

of the model is given and the theory on which the model is

based is reviewed. This section closes with a description

of the method of determination of material parameters and

an assessment of the material test requirements of each

model.

In order to perform a numerical comparison of the
1 H2

models, the aluminum alloy Al 5086 was chosen as the

candidate material. An experimental data base was

established, from which the material constants for each

model were calculated.

Finally, the uniaxial mathematical representations of

each model have been encoded and implemented in computer

programs in order .to simulate various load histories.

These simulations are compared to experimental data and

comments are made on their accuracy.

In conclusion, it has been demonstrated that the

theories discussed herein are suitable models for the

prediction of inelastic deformation in crystalline

solids. It is also apparent, however, that further

research is necessary in several areas in the development

of these theories.

For example, the theories lack, to varying degrees,

clear, concise, and rigorous methods to determine material

constants from simple experiments. The theory of Krieg,

et al., uses the somewhat controversial and difficult to

implement stress-drop test, although the calculation of

constants is simple. This dependence on stress-drop test

results may adversely affect the performance of this model.

The theory of Miller, et al., uses standard creep tests,

but values of several constants are set arbitrarily to

obtain an optimum curve-fit. Although this is also true to
a lesser degree for the theory of Bodner, et al. , it is

found that this theory provides the best balance between

ease of testing and determination of material constants for

the theories discussed herein. In addition, it is found

that all three theories exhibit substantial material

parameter sensitivity to small variations in input.

It can be seen in the previous section that the models

in general compare favorably to experiment. However, it is

clear that the full representations of the models,

incorporating both isotropic and directional hardening

terms, may be more appropriate for modelling phenomena such

as the Bauschinger effect and cyclic strain hardening than

the simplified versions examined herein. In addition,

these theories are proposed mainly for intermediate to high

temperature conditions and may have failed to model fully

the phenomena at the temperature used in this research

(0.3Tm).

Another area of improvement is in the use of internal

state variables. Many hardening functions are taken to be

constants, whereas functions of the current stress level or

back stress level may be warranted to prevent an

overprediction of stress and an "over-square" transition

from elastic to inelastic behavior.

Based on these assessments, it is felt that the theory

of Bodner, etal. , provides the best balance between ease
1 Jill

of implementation and accuracy of results for the material

and test conditions of interest for this research. The

theory of Krieg, et al., is affected adversely by the

complexity of the required material tests, while the theory

of Miller, et al., is affected adversely by the lack of

rigor in calculating constants.

As a result of this research, it Is the experience of

this researcher that data acquisition and control equipment

of the highest resolution and accuracy is an absolute

necessity to ensure credible experimental results,

particularly in specialized tests such as the stress-drop

test.

Finally, it is hoped that the research presented in

this thesis may answer questions relating to the modelling

of inelastic deformation in crystalline solids and lead to

further research in this field.
145

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VITA

Joachim Michael Erdmann Beek was born on 21 January

1961 in Augsburg, West Germany, the son of Karin Dorothea

Beek and Karl Friedo Beek. After graduating from London

Central High School, High Wycombe, Bucks., England, in

1978, he enrolled at Texas A&M University, where he

completed his B.S. in Aerospace Engineering in 1982. During

his undergraduate years he worked two summers at the

University of Stuttgart in Stuttgart, West Germany, and one

summer for an oil exploration company in East Africa.

His graduate work has centered on constitutive

modelling of inelastic deformation in crystalline solids.

Together with his wife Belinda, he may be contacted

through Lockheed Engineering and Management Services

Company, 2^00 Nasa Rd. 1, Houston, Texas, 77258.
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