DEVELOPMENT OF CONSTITUTIVE MODELS FOR CYCLIC PLASTICITY 

AND CREEP BEHAVIOR OF SUPER ALLOYS AT HIGH TEMPERATURE 

Final Report 

Submitted by 

Walter E. Haisler 

Aerospac-e Engine<ering Department 

Texas A&M University 

College Station, Texas 

to 

NASA Lewis Research Center 

Cleveland, Ohio 

September 30, 1983 

N861439,6 

1111111111111111111111111 III 

Research Foundation 

Project 4246 

Grant No. NAG 3-31 

REPRODUCED BY: N(§, . u.s. Department of Commerce 

National Technical Information Service· 

Springfield, VirginIa 22161 

), 

\ 

TABLE OF CONTENTS 

ABSTRACT 

SUMMARY OF RESEARCH 2 

PUBLICATION LIST 15 

APPENDIX (Selected Papers) 16 

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ABSTRACT 

This report. presents the Final Report on NASA Grant NAG 3-31 

entitled "Development of Consti tuti ve Models for Cycl ic Plastic i ty and 

Creep Behavior of Super Alloys at High Temperature. An uncoupled 

constituti ve model for predicting the transient response of thermal and 

rate dependent, inelastic material behavior has been developed. The 

uncoupled model assumes that there is a temperature below which the 

total strain consists essentially of elastic and rate insensitive 

inelast ic strains only. Above this temperature, the rate dependent 

inelastic strain (creep) dominates. The rate insensitive inelastic 

strain component is modelled in an incremental form with a yield 

function, blow rule and hardening law. Revisions to the hardening rule 

permit the model to predict temperature-dependent kinematic-isotropic 

hardening behavior, cyclic saturation, asymmetric stress-strain response 

upon stress reversal, and variable Bauschinger effect. The rate 

dependent inelastic strain component is modelled using a rate equation 

in terms of back stress, drag stress and exponent n as functions of 

temperature and strain. A sequence of hysteresis loops and relaxation 

tests are utilized to define the rate dependent inelastic strain rate. 

Evaluat~on of the model has been performed by comparison with 

experiments involving various thermal and mechanical load histories on 

5086 aluminum alloy, 304 stainless steel and Hastelloy X. 

1 

SUMMARY OF RESEARCH 

Consti tuti ve models may generally be grouped into three 

categories: microphenomenological, nonlinear viscoelastic, and 

classical plasticity. Each group can be further separated into unified 

and uncoupled theories, where the two differ in their approach to the 

treatment of rate independent and rate dependent inelastic deformation. 

The unified approach separates the total strain rate as 

I . T € = ,;e + E + E (1 ) 

. where E represents the total strain rate and superscripts E, I~ and T 

represent the elastic, inelast ic and thermal components, respectively. 

Alternatively, the uncoupled theories partition the inelastic strain 

into plastic and creep components. This can be expressed as 

(2) 

where superscripts P and C represent the rate independent "plasticity" 

and rate dependent "creep" strain terms, respectively. Such uncoupling 

of the strain components provides for simpler theories to be developed 

(along with simple tests to define material constants) but does preclude 

any creep-plastiC i ty interaction except through ad hoc rules. More 

complete discussions and literature reviews are provided in the papers 

and reports contained in the Appendix. 

The constitutive model developed in this research is based on 

uncoupled strain components as written in equation (2). The model 

.assumes that there is a temperature below which the total strain 

consists essentially of elastic and rate insensitive inelastic (plastic) 

strains only. Above this temperature, the rate dependent inelastic 

strain (creep) dominates. The rate insensitive plastic strain is 

modelled by an incremental model with a yield function, flow rule and 

hardening law. Provisions are made for cyclic saturation and variable 

Bauschinger effect. The rate dependent inelastic component is modelled 

using a rate equation in terms of back stress, drag stress, and exponent 

n as functions of temperature and strain. The rate dependent and 

independent inelastic components are selected so that their sum is 

always equal to the total inelastic strain at all temperatures. 

Rate Insensitive Strain Component 

The rate insensitive inelastic (plastic) strain component is 

modelled using a modified and improved incremental theory applicable to 

cyclic thermomechanical material response. The model makes use of four 

major components. For simplicity, the following equations will be 

presented in uni~ial form; their extension to multiaxial form is 

presented by Allen . 

First, we assume a relation between stress and elastic strain 

(3 ) 

2 
or in incremental form 

~ PeT I (t p"t ct' rr 

.6()::. IE r..'HI.'t (~€-lJe -IJ[ -IJ£ / + JE E - f - £ - f' ) (4) 

where cr is the uniaxial stress, E is elastic modulus, and £ is the 

uniaxial strain. Superscripts P, C, and T denote plastic, creep, and 

thermal components respeceively while t denotes values at the start of 

the load step, and t+At denotes values at the end of a load step. 

The von Mises yield criterion is used herein in combination with a 

variable ratio isotropic-kinematic hardening law. The modified 

hardening rule used in this research permits the' model to predict 

temperature dependent kinematic-isotropic hardening behavior, cyclic 

saturation, asymmetric stress-strain response upon stress reversal, and 

variable Bauschinger effect. The essential features of the hardening 

rule are described in the thesis by Cronenworth included in the 

Appendix. 

While metals like aluminum exhibit a similar strain hardening 

behaviOr for reyield after the initial yield, some metals (stainless 

steels, for example) exhibit different forms of hardening behavior for 

each half cycle of loading even after several loading cycles, i.e., the 

stress-atrain response is asymmetr ic. To account for the asymmetry, 

both halves of a stress-strain cycle are used as input to the model and 

the hardening ratio ~ is allowed to be a function of the direction of 

loading. Typical experimental data and the linearized representation 

are shO'ifn in Fig. 1. For temperature T1 , curve ABC corresponds to the 

first half cycle ~r response while curve ,DE (or D'E') corresponds to the 

second half or reverse loading response. 

For a comb ined hardening law, the hardening ratio, ~ ,is the 

ratio of isotropic to kinematic'hardening such that ~=O constitutes a 

kinematic hardening law where the yield surface retains its initial 

size, shape and orientation thus simply translating in principal stress 

space. Isotropic hardening,p=l, indicates that during plastic flow the 

yield'surface expands uniformly about the origin and never translates. 

Using a values for the hardening ratio such as 0 ~ tS ~ 1 constitutes 

a combined hardening rule where the yield surface is allowed to both 

expand and translate. The hardening ratio may be variable if ,it is 

defined as a function'of plastic strain and temperature using 

err - ( :2. 0; - a; ) 

(2V; - ~o;) 

~ (f~ r) = (5) 

where the stress values are defined in Fig. 2 for each temperature. If ~ 

is a function of plastic strain as well as temperature, several cycles 

of stress-strain data are required to characterize this parameter. 

Typical curves for the hardening ratio are also shown in Fig. 2. As ~ 

approaches a limiting or asymptotic value with respect to' plastic 

strain, we note that the cyclic hysteresis loops will saturate. 

The growth law for the yield surface size K may be obtained from 

using 

3 . ' 
I 

E 

0' 

u

y1 

O'y2 

0' 

E'l 

Tl 

I'1 T2 

G 

...,... E 

Fig. 1 Uniaxial isotherma1 single cy~le reverse 

loading curves and their linear' representations 

4 

" . 
a 

a 

o 

o 

y 

o 

r 

T = constant 

e: 

stress 

o - reyield r 

~. T2>T1 

~·------T1 ------------------

-------------------------- -p E 

Fig. 2 Hardening ratio 

5 
K = 0; + ~ (0-- 0; ) (6 ) 

where ~ represents the_ yield stress and c;- is the applied stress. In 

equation (6), k= J«(£P.) T). For the case where loading and reverse 

loading response is different,'we employ the following procedure at the 

reverse loading pOint. Consider that base diagrams of K vs lP have 

been obtained as shown in Fig. 3. Recall that the "a" curves are the 

ones that result from reverse yielding while the "b" curves are from the 

virgin stress-strain curves. If a reverse yielding occurs at £P::" i P , 

we assume the "b" curves ate linked to the global diagram as showg at 

the bottom of Fig. 3, always linking-on the complete curve starting from 

zero plastic strain. When another reverse yielding takes place, either 

the "a" or "b" curves are linked to the global diagram at that 

particular value of plastiC strain that has been accumUlated. A similar 

procedure is carried out for the global uniaxial stress vs. plastic 

strain diagram. . 

The experimental requirements required to characterize the 

hardening law depend upon the metal. For the most general case, 

isothermal cyclic stress-strain tests are run till the hysteresis loops 

stabilize. Cyclic tests are required for several temperatures so that 

the tempe-rature dependence can be adequately characterized. Loading 

rates must be selected so that time-dependent effects are negligible. 

Rate Dependent Inelastic Strain Component 

In the present, the rate sensi ti ve strain component is obtained 

from hysteresis loops and stress relaxation tests at various 

temperatures. Fig. 4 shows a typical set of these tests and details how 

the ·rate dependent deformation is extracted from the saturated cyclic 

data. A reference temperature hysteresis loop is defined at a 

temperature below which the total strain consists essentially of elastic 

and rate insensitive plastic strains only. Above this temperature, the 

rate dependent inelastic strain (creep) dominates. Exper imental tests 

condUcted by Yuen (see Appendix) seem to indicate that such an 

uncoupling is feasible for Hastelloy X. The basic concept of decoupling 

strains according to Fig. 4 is to take that part of the inelastic strain 

which is strongly time dependent and call it t c ; the remaining part 

is assumed to be all rate insensitive and incorporated into i P . 

Micromechanical consideration suggest a model for the rate 

dependent inelastic strain of the form 

where fC represents the rate dependent inelastic strain rate, .J1.. is 

the back stress, K is the drag stress, and n is a temperature-dependent 

"constant." The back stress, drag stress, and exponent n are generally 

function 'of temperature and strain (or stress). However, for some 

6 
K 

K1·-O y1 

K2=a y2 

K 

-p 

t 

o 

____ a 

i T1 

b 

.--a 

I T2 

b 

T2 > 11 

Base Diagram 

______ T

2 

-----------------.-~p 

Glob:!l Diagram 

Fig. 3 Construction and evolution of K vs. ~ diagrams 

7 

, 

t 

i 1-' 

( 

i . 

i 

t 
o 

rate independent 0 

plastic strain 

elastic~/ strain " 

, 

· . 

~ ___ --/TI 

dependent 

strain 

e: 

Fig. 4 Experimental characterization of the rate 

dependence of the model 

8 
materials a good assumption is that the drag stress and n are constant 

at sufficiently high temperatures. The experimental determination of12, 

K, and n is based on stress relaxation tests at low temperature and 

stress drop/strain transient test at high temperature. Details of these 

experiments are presented in the thesis by Yuen (see Appendix). 

Evaluation of the Model 

The capabilities of the constitutive model developed during this 

research have been evaluated through several hypothetical numerical 

experiments as well as by comparison with experimental results. Only a 

few of these results are presented here. More complete comparisons and 

details of the experiments, as well "as experimental laboratory 

facilities developed during this research, are presente~ in the 

Appendix. 

In the first example, we illustrate the effect of a change in 

strain hardening upon stress reversal, which is typical of many alloys. 

Figure 5 illustrates the stress-strain curves for a theoretical single 

cycle reverse load test. The input can be taken from either first or 

saturated cycle data. Curve B represents the initial strain hardening 

behavior while curve A represents the high rate of strain hardening seen 

upon stre~s reversal. If a single cycle reverse load test is performed 

with this theoretical data, the results will be similar to those shown 

as output in Fig. 5. The revised model using both halves of the cyclic 

stress-strain response (curves A and B) is a much more realistic 

representation of actually observed experimental behavior for many 

metals than the combined hardening rule previously used by the current 

researchers. Even the revised mod.el representation using only curve B 

is an improvement over existing hardening rules which "square-off" the 

behavior upon reyield. 

In the next example, we consider" a hypothetical fully-reversed 

cyclic uniaxial strain history of +/- 0.25% strain with two different 

assumptions for the hardening law. In Fig. 6a, a combined hardening law 

wi th S or o. 5" was used while in Fig. 6b results are presented for a 

variable hardening ration given by tS =-0.$ - ~/. 7 f P • It is seen 

that different rates and amounts of cyclic saturation are easily 

predicted by the model. 

In the third example, we consider a uniaxial specimen subjected to 

a thermomechanical loading history as shown in Fig. 7. The stressstrain curves do not include any asymmetric hardening ( ;6 = 0.5', ), but 

are considered to be dependent on temperature as shown. Results from 

the constitutive model for modulus, strain hardening, and yield surface 

size as functions of plastic strain and temperature are shown in Fig. 8 

(t. in Fig. 8 corresponds to t. in Fig. 7). It is interesting to note 

th~ change" in yield surface si~e as a "function of plastic strain. As 

discussed previously, the curves are modified each time there is a 

stress reversal which causes yield or reyield. This physically 

represents the rounded shape of the stress-strain" curve after reverse 

yielding seen" in experiment in terms of the mathematical modelling, 

this corresponds to the linking of the original yield surface curve onto 

9 

:,,~ 
i I 

2<~OU 

20QOO 

11~00 

I~COC 

12500 

lOCOC ' 

7500 

5000 

2500 

O,OCO 

18000 

ISOOC 

1200C 

9000 

6000 

'000 

'6000 

-9000 

'12000 

-15COO 

-I BOOO 

C.002 

__ --- II 

B 

A - strain hardening 

upon stress reversal 

B - initial strain hardening 

0,006 O,OOB C.CIC 

SfAAIH • IN/IN 

Input (Isothermal) 

~_A __ ·_A-- __ __ Haisler 

model 

revised model using B 

only 

revised model using both -2ICOO A & B 

.2"COj\i ____ ..-___ .,.-___ -.-___ ...... _~ 0.0000 0,0015 O,OOJO 0,0050 

,rAAEN .. IN/IN 

Output 

Fig. 5 Change of strain hardening with stress reVel" <;,11 

10 
S 

r 

II 

f 

5 

, 

I 

r 

, 

I 

...... 

...... 

"""1 

20000 

IS01)O 

10000 

5000 

0 

-soeo 

-10000 

-15000 

-20000 

-0.002 -0.001 0.000 0.001 0.002 0.001 

-25000~~,~~~~~~~ __ ~~~~~~~~~~----~--T 

-0.001 

SIRliIN - INIIN 

Combined hardening with B constant 

21000 

111000 

15000 

l2000 

8000 

0000 

'000 

5 

r 

R 

~ 0 

S 

5 

I -5000 

r 

, 

I - -6000 

-9000 

-l2000 

-15000 

-18000 

-21000 

-2qOOO~~, __ ~~~~ __ ~ __ r-~ __ ~~~ __ ~TT~~ __ ~~-----T 

-0.001 -0.002 -O.OOl 0.000 0.001 0.002 O.OOl 

IfllA IN - INIIN 

Combined hardening with B a function 

of plastic strain 

Fig. 6 Modeling of cyclic saturation using combined hardening law 

,~C-.~--''''''''''~ 
, 

. 

IIOIJOG 

c.c:o;:~':' o,or:r.~ a.CCL~.l G,ro.::~" O.C':~~ O.con" 

'''"'11"1 

Load 

T-800 

'-"--" ------- ...... ~7V~~+---:I'--,··-i-+--. time 

Lc~-,--+-----+---+-~- time 

Temperature 

1000 

800 

Fig. 7 Input for Example 3 

.2ao~~ 

l751lC ts 

150lla 

12!CO 

I~COO 

7'500 

5000 

Z:ClC 

/. 

a 100']1} 

-I2sao 

-15000i...,. ___ --r ___ + ___ ~-__ ~--~-

-a.OUJ / -O,lllJl 0.01.:0 a.cci _ -., (l.CO.? 

'TRAIN - (NI IN r-' 

Fig. 8a Cyclic Thermomechanical loading 

r 

I 

25000 

225Q.1 

20000 

17'500 

E lsoea L 

o 

, 

U 

A li!'50Q , 

• c 

~ 

5 10000 I 

! 

, 

'naa 

500' 

;soo 

T=SOO 

T .. 1000 

a.OOl1 a.OOI 0.002 0.003 0.00<1 0.009 0._ 

Fig. 8b 

STRESS 

Cyclic thermomechanical loading 

MPa KSI 

300 

40 

250 

200 30 

150 

100 

10 

50 

0 0 

-50 

-10 

·100 

·20 

·'50 

·200 -JO 

-250 

Fig. 9 

12 

fA 'VA. Lead 

~3~'r _---- Temperalure / 

2 

-- Ex.peflmenr 

I 

I 

I 

I 

I 

I 

3 

I 

I 

I 

I 

I 

I 

I. 

I 

I 

I 

I 

I 

5 

% STRAIN 

Comparison of revised model to experiment 

for aluminum cyclic load test 

I 

! ~. 

'~: .. , 

! ~Ii 

.4 

, ~ 
the current global diagram. The squares in Fig. 8 pOint out where this 

process takes place each time. 

The numerical experiments discussed above serve to indicate the 

potential response possible with the constitutive model. The model has 

also been compared to experiments involving several different materials 

and thermomechanical loading histories. Three of the comparisons are 

presented here. The first involves a- uniaxial test of 5086 aluminum 

alloy between -room temperature and 275 0 F. Stress-strain curves were 

obtained for one complete cycle (with -asymmetric strain hardening 

accounted for) and, from experiment, the hardening ratio was found to be 

approxima tely 0.45 for the first loading cycle and approximately 0.0 

(kinematic) for' all reloading cycles. Figure 9 presents the cyclic 

loading and temperature history. We observe that the theoretical 

resul ts are in very good' agreement wi th experiment and the high rate of 

strain hardening during load reversal is modelled accurately. 

In the next test, we attempt to model the evolution of hysteresis 

loops for a mechanically cycled 304 stainless steel specimen at 1,000oF. 

We note from Fig. 10 that the loading-reloading response i$ quite 

asymmetr ic and' the difference al ternates throughout many cycles. In 

this ca;:;e, tension preceeds compression; however, the same effe'ct is 

observed if compression is applied first. From experiment, the 

hardening ratios were determined to be approximately 0.2 for the first 

quarter cycle and 0.35 for the second -quarter cycle. -Resul ts for the 

first and fourth cycles indicate reasonably good- correlation with 

experiment. 

In the last case, we consider the prediction of cyclic saturation 

of Hastelloy X at room temperature when cycled between +/- 1.1% strain 

under strain control. Figure 11 presents the experimental results while 

Fig. 12 shows a comjJarison betVleen. theory and experiment. In Fig. 12, 

experimental data is shown only for the first one-half -cycle and the 

last (saturated) cycle. The model results depict the gradual saturation 

response as well as the limiting response. A hardening ratio function 

given by ~=O.o7S-itJ.37StP was uS'ed in tension while that used 

for compression was ~:' 0./0' - O. S" i,P . Notice that as model 

predictions approach saturation, some fluctuation is seen on the 

compression side. This is thought to be due to increasing numerical 

error at this pOint as no equilibrium iterations are performed in the 

finite element analysis. Most of the difference in experiment and 

theory is due to difficUlty in modeling of the second quarter cycle, 

otherwise the model predictions are reasonably good.· 

13 
S 

f 

A 

E 

I 

I 

Il0000 

.U:Jtl~O 

i> 

f 

-O.llOS 

S 10000 T 

R 

E 

S 

S 

I 

P 

$ -U::I I 

CYCLE NO.1 

a.ooo C.llOS 

&T""IW fIN/IN) 

CYCLE NO.4 

.,..' ~ .. ' 

_.-_.-......... 

Exp 

Model 

o.ou; 

i 

i 

/ I 

,: 

;' 

1' •• i"~.1 

.. ' 

-~oa~~----~i----~I-----T-----ri----~i----~i 0.000 0.003 O.C06 0.009 0.012 0,o,5 0.018 

sr~tH rUUINJ 

Fig. 10 Comparison of model and experiment for 304 

stainless steel at 10aO°F using first cycle 

data as input. A hardening ratio of 0.2 

in tension and.O.35 in compression was used 

14 

Fig. 11 Experimental hysteresis loops for Hastel1oyX at room temperature 

Fig. 12 Cyclic saturation of Hastel1oy-X at room 

temperature 
PUBLICATIONS LIST 

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

Materials,"·Computers and Structures, Vol. 13, 1981, pp. 124-135. 

Cronenworth, J., "Development and Evaluation of an Uncoupled, 

Incremental Constitutive Model for Elastic-Plastic Behavior at Elevated 

Temperature," TheSiS, Texas A&M University, College Station, Texas, May 

1982. 

Bradley, W.L., "A New Uncoupled Viscoplastic Constitutive Model," 

presented at the Symposium on Nonlinear Constitutive Relations fOr High 

Temperature Applications, The University of Akron, Akron, Ohio, May 19-

20, 1982. 

Haisler, W., "Application of an Uncoupled Elastic-Plastic Creep 

Consti tuti ve Model to Metals at High Temperatures," presented at the 

Symposium on· Nonlinear Constitutive Relations for High Temperature 

Applications, The University of Akron, Akron, Ohio, May 19-20, 1982. 

Yuen, S., "Development of an Uncoupled Viscoplastic Constitutive Model 

for Cyc~ic Plasticity of Hastelloy X at High Temperature," TheSiS, Texas 

A&M UniverSity, College Station, Texas, May 1983. 

Haisler, W. E. and Cronenworth, J. , "An Uncoupled Viscoplastic 

Constitutiv~ MOdel for Metals at Elevated Temperature," presented at the 

24th AIAA Structures, Structural Dynamics and Materials Conference, Lake 

Tahoe, Nevada, May 2-4, 1983. 

15 

t I, 

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APPENDIX 

(Selected Papers and Reports) 

16 

- r 

j. 

t 

1 

I, 

t 

\ 

I 

t 

i 

, ' 
- '.. _ ...... ,. ~ 

Haisler, W.E. and Cronenworth, J. , IIAn Uncoupled Viscoplastic 

Consti tuti v-e Model for Metals at Elevated Temperature ,II presented at the 

24th AIAA Structures, Structural Dynamics and Materials Conference, Lake 

Tahoe, Nevada, May 2-4, 1983. 

17 
1 

CHAPTER I 

INTRODUCTION 

Modern computational methods for the stress analysis of structures 

are well established. For behavior that is linear and even geometrically nonlinea~, the finite element method has proven to be a very 

capable tool for the structural engineer. However, nonlinear material 

analysis is much more difficult and not yet as fully developed as proven 

by the large amount of research done in this area in recent years. Of 

particular interest to this research, the solution of thermal and rate 

dependent elastic-plastic material behavior is quite difficult. Applications of this technology are needed in components such as nuclear 

reactor pressure vessels and gas turbine blades. 

Historically, the study of plasticity of metals began in 1864 with 

the publishing by Tresca [1] of a preliminary account of experiments 

on punching and extrusion. This led him to state that a metal yielded 

plastically when the maximum shear stress attained a critical value. 

Application of Tresca's yield criterion was investigated by Saint-Venant 

[2] to determine stresses in cyclinders and tubes. He also recognized 

that there was not a one-to-one relation between stress and total plastic strain. Levy in 1871 proposed multiaxial relations between stress 

and plastic strain [2]. 

Advancement to a more satisfactory yield criterion was constructed 

on the basis of purely mathematical considerations by von Mises [3] 

This thesis follows the style and format of the Journal of Applied 

Mechanics. 
in 1913. Further work was done in the 1920's by Prandtl [4], Nadai, 

Lode, and von Karman [2). It was in the early 1930's that two important generalizations of the available theory were made. The first was 

made by Reuss in 1930 [5] who allowed for the elastic component of 

strain. The second was made by Schmidt (1932) and Odquist (1933) [2] 

who showed how to incorporate work hardening, into the framework of the 

Levy-Mises equations. Thus by 1932 an elastic-plastic theory had been 

constructed to model these properties of an isotropic metal at room 

temperature [2]. 

Work in this area in the 1940's and beyond was done by several 

prominent researchers like Hill [2], Drucker [6], Prager [7], and 

Zieg~er [8]. Recently, the basic trend has been to extend the rate 

independent theories to include rate effects by adding on a creep term. 

This leads to an uncoupled or partioned theory, and many are now in 

use, today in finite element structural analysis codes. Most recently 

with the recognition that the rate independent and rate dependent inelastic deformations are not autonomous phenomenon, a number of "unified" constitutive models have been developed and are still undergoing 

active development [9]. Present day efforts include the extension of 

classical plasticity to predict rate dependent behavior and the modifi- . :-'. . . 

"cation of linear viscoelasticity to model nonlinear material response 

2 

[10] . Many theories are' available but none have shown clear superiority 

. ",;'",o~'~~,:the~thers,~J::o~;iling mat~~i~l' b~havior"ove~ wide ranges of 

" temper~ture, an(.{i~ad'.'-' .,' " ", -. .' 

',~. ':''':'~: ::_, ~~ -~ ~.:~;. :~~,:::-~;' 

ofthi~ research is extend the classical incremen-

'.l,", 

t, ::'~1 

J 

n LJ 

'-~l ", 

"",], " , 

',.:", I 
constitutive relations necessary to model the nonlinear behavior of 

crystalline materials experiencing thermoelastic-plastic-creep. Among 

the requirements for a sound constitutive theory for the media mentioned above are that it incorporate the ability to model transient temperature response; cyclic behavior including cyclic saturation, the 

change in strain hardening upon stress reversal, and the Bauschin,ger 

effect; and rate dependence. 

In fulfillment of these objectives, this thesis will proceed in 

the following manner. First, a brief literature review of the available theories is presented along with a discussion of the details of 

the constitutive model used in this research. Next, the development 

of the model is accomplished by derivation of the uniaxial theory including experimental data requirements and computational considerations. 

Several theoretical examples are presented along with comparison to 

experiments on several different metals at elevated temperature. 

3 

CHAPTER II 

BACKGROUND 

Present Status of Constitutive Modelling 

.To characterize the structural response of any general three-dimensional body, one must satisfy mechanics (conservation of mass and 

momentum, and kinematics), thermodynamics, and constitution. Since 

mechanics and thermodynamics are well established for most continua, 

it becomes the physical characteristics or constitution of a body that 

determines whether one can successfully analyze its structural response. Furthermore, because of the widely varying material properties and environments that structural media are subjected to, constitutive modelling is usually restricted to specific types of media. An 

all encompassing model that works equally well for all materials is 

simply not feasible. One such category of material response, and the 

one that this research will deal specifi~ally with, is the elasticplastic-rate-dependent crystalline material at elevated temperature. 

The major types of constitutive laws available to model the rnaterial behavior discussed above can be grouped into three categories, 

microphenomenological, nonlinear viscoelastic, and classical plasticity. Also, there is an existing subdivision within the categories 

labeled unified and uncoupled theories, where the two differ in their 

approach to the treatment of rate independent and rate dependent inelastic deformation. The unified approach separates the total strain 

as 

T 

e: , (1) 

4 

i 

I' 

r 

t 

I 

I 

I 

I 

t 

l 

r 

r 

i I 
5 

where e:represents the total strain and superscripts E, I, and T represent the elastic, inelastic, and thermal components respectively. AIternatively, the uncoupled theories partition the inelastic strain into 

plastic and creep components. This can be expressed as 

T 

e: , (2) 

where superscripts P and C represent the rate independent "plasticity" 

and rate dependent "creep" strain terms respectively. Researchers like 

Walker [9] and Krieg [11] question the partitioning as it has no underlying physical basis and does not account for creep and plasticity interaction. 

"The uncoupling of inelastic behavior into rate independent and 

rate dependent components is unsatisfactory to the material scientist 

because it is not microphysically justified [11]. Although many recent 

unified theories have been proposed, they still are not yet proven to 

be more successful overall than the uncoupled theories. 

For example, Walker's theory [9] (considered one of the better 

unified models) reproduces cyclic stress-strain behavior at elevated 

temperatures very well. Unfortunately, the theoretical hysteresis loops 

at smaller strain rates or lower temperatures are too square in comparison with the actual experimental loops. Thus, Walker's theory does 

not characterize adequately the classical plasticity that is occurring 

in this load-temperature range. Walker also proposes an extension to 

include temperature dependence, but no attempt has been made to model 

transient temperature response [10]. His theory is restricted in its 

·~l 

.J 

J 

J 

- . . 

..... j 

.. _I 

! 

___ J . 

:-1' 

. I' 

J 
,-

, 

6 

use to an elevated temperature environment. The author does not want 

to leave the impression that Walker's theory is an inferior one as it 

is not. This discussion is simply to point out that the uncoupled 

theories are not necessarily inferior to the unified ones. 

As discussed below, present theories attempt to model these physical mechanisms in a variety of ways. 

Microphenomenological theories. The mechanisms for microphenomenological theories are discussed in detail by Allen [12]. These 

theories represent an element of material called a po1ycrysta1 as being composed of a large number of randomly oriented monocrysta1s. 

Statistical averages of the properties of each monocrysta1 and their 

intera'ctions determine the behavior of the po1ycrysta1 [13]. These 

crystalline materials form lattice structures that contain many imperfections called dislocations which vary in density and location depending upon the processing used in manufacturing the material. Plastic deformation occurs by slip on certain crystallographic planes and 

is explained in terms of dislocation theory of plastic deformation. 

For example, dislocation interaction explains how strain hardening can 

occur in any crystal [14]. 

One example of a microphysically based constitutive law is an 

elastic-viscoplastic theory based on two internal state variables by 

Bodner, et a1. [15]. The authors state the constitutive equations 

ability to represent the principal features of cyclic loading behavior 

including softening upon stress reversal, cyclic hardening or softening, cyclic saturation, cyclic relaxation, and cyclic creep. One 

limitation of the formulation though is that the computed stress-strain 
7 

curves are independent of the strain amplitude and therefore too "flat" 

or "square". 

Another example is in the paper by Miller [16] on modelling of 

cyclic plasticity with unified constitutive equations. He also recognizes the shortcoming of many theories in predicting hysteresis loops 

"J' " ~--, 

which are "overs quare" in comparison to observed experimental behavior. 1.". 

Improvement is accomplished by making the kinematic work-hardening ;l' , , 

coefficient depend on the back stress and the sign of 'the nonelastic 

strain term. Results compare favorably with that observed in 2024-T4 

aluminum alloy. Miller states his approach appears consistent with an 

existing physical explanation in terms of annihilation of previouslygenerated dislocation loops upon reversals in the direction of dislo- J

. ,',,' 

:: . . cation motion. 

A theory that is similar in format to Miller's is by Krieg, 

::-J 

j 

Swearengen, and Rohde [11]. The model uses two internal state variables 

to reflect current microstructure and is based upon models for dislocation process in pure metals. It is an extension of an equation of 

state theory originally attributed to Kocks [17] where the inelastic 

flow rule is taken to be a power function incorporating a kinematic 

and isotropic internal variable. They assume that all the net mechanical effect of the complex dislocation processes can be contained by 

one or two readily ~easurable macroscopic variables. The theory cannot accurately model cyclic hardening or softening behavior and the , '.t;.,:. . ,;. ,:-,", ~ ' .. _ '" - . - , '~ . 

strain hardening·behavi~r is necessa~i1y "square" in nature because of • -, , ~ •• .' - - \\;;,," '. • _: r' ,r ;,";., " 

"'the, powe~law assumption.. A~~~" applications,:of the model, over a ,wide 

is not advised. 

'. ""J' 

, 'i:~ 

; :"';"" 
8 

Nonlinear viscoelastic theories. Nonlinear viscoelastic or thermodynamically based theories are usually distinguished by their single 

integral or convoluted form. This type of constitutive model employs 

the first and second laws of thermodynamics along with physical constraints to complete the formulation [12]. A detailed review of sevI 

eral existing theories is presented in both [10] and [18]. 

One of the more promising theories is credited to Walker [9,18]. 

It is a unified integral viscoplastic theory developed by modifying the 

constitutive relation for a linear three parameter viscoelastic solid. 

The theory contains clearly defined material parameters, a rate dependent equilibrium stress, and a proposed multiaxial model. An important shortcoming of Walker's theory is its failure to model transient 

temperature conditions, but Allen and Milly [10] conclude that his 

theory is the best presently available for predicting cyclic response 

at elevated temperatures under isothermal conditions. 

Other nonlinear viscoelastic theories discussed by Walker [18] 

are by Cernochy and Krempl, Valanis, and Chaboche. 

Classical plasticity theories. The type of constitutive law considered in this research is derived from classical incremental plasticity. It is termed a macrophenomenological theory as it derives its 

state variables purely from experimental results without direct 

reference to the microstructure of the material. This type of constitutive law can be defined as one that attempts to describe the elasticplastic behavior of a material based on properties obtained from a 

single stress state and then 'use them to establish relationships between the general. stress and strain tensors [13]. Most incremental 
plasticity theories have four major components: (1) a stress-elastic 

strain relation, (2) a yield function describing the onset of plastic 

deformation, (3) a hardening rule which prescribes the strain-hardening 

of the material and the modification of the yield surface during plastic flow, and (4) a flow rule which defines the component of strain 

that is plastic or nonrecoverable [19]. 

Research in this area is voluminous. The inviscid plasticity is 

well established while the extension to include rate and thermal effects is not. References [10,12] and [13] summarize some of the important research efforts in their field of study. Of the classical 

plasticity theories reviewed in [10], the most promising ones belong 

to Zienkiewicz and Cormeau, and Allen and Haisler. The former is a 

rate dependent unified theory which allows for nonassociative plasticity and strain softening but does not model the Bauschinger effect 

or temperature dependence. The latter is an extensioriof classical 

plasticity to model both rate and thermal effects. It is a two state 

variable uncoupled theory with clearly defined material parameters and 

extension to multiaxial form. Still another example is a model proposed by Popov and Petersson [20,21J. Excellent agreement with experiment is shown in the isothermal, rate independent case. Snyder and 

Bathe [22] have proposed a modification to classical plasticity which 

does model both rate and thermal effects in the monotonic load case 

but is restricted to a kinematic hardening rule. Allen [12] suggests 

that the theory proposed by Yamada and Sakurai [23,24] may be the best 

for modelling the type of behavior described herein. Temperature dependence of material properties, a combined hardening rule, and an 

9 

J 

J 

J 

] 

--I 

-, 

.- ) 

"I 

.J 
, .. 

, 

11 

CHAPTER III 

DEVELOPMENT OF THE MODEL 

The author gives the following justifications for using the uncoupled approach proposed by Allen and Haisler. One can partition 

the total inelastic strain into components as long as their sum yields 

the total nonrecoverable deformation without significant adverse effects [12J. The theory is able to model behavior over a wide range 

of load and temperature up to at least one-half the melting point for 

several metals. It is also one of only a few to address transient 

temperature conditions. Also, with the revised hardening law presented herein, the rate independent inelastic deformation is modeled 

as well as or better than the unified theories reviewed. Reasonable 

experimental data requirements are another strong point of this theory. 

Derivation of the Constitutive Equations 

The constitutive equations of the Allen and Haisler model are derived in a uniaxial incremental form relating the total stress incre-

~ent to the total strain increment. Development of the theory in this 

form is a logical approach as it lends itself to much simpler evaluation without introducing unnecessary complications of a multiaxial 

theory. Many components of the work required are much more easily done 

in. uniaxial form such as the computer code development and experimental model verification. 

As stated earlier, most classical plasticity theories have four 

major components. First, there is a relation between stress and 

elastic strain 

PRECEDING EAGE BLANK NOT:-FILt.fED 
12 n 

(3) 

or in incremental form 

(4) 

:'_.', -] .. 

~here a is the uniaxial stress, E is the elastic modulus, ,and E is the 

uni~xial strain. Superscripts P, C, and T denote plastic, creep, and l 

thermal components respectively ~hile t denotes values at the start of 

the load step, and t + ~t denotes values at the end of a load step. ]. 

Note that the elastic modulus'is measured at the end of the step because the time step is finite rather than infinitesimal as described ] 

by Allen [12]. A graphical decomposition of the total strain is shown J 

in Fig. 1. The term "zero time" denotes a loading input short enough 

to negate any time dependent deformation but long enough to disregard 

inertial effects (a few seconds for many metals) [12]. Conversely, 

the long time curve characterizes the rate dependent deformation. Recall that there is no physical basis for uncoupling the inelastic deformation, but it is valid as long as coupling effects are insignificant or the uncoupling is done properly. 

Secondly, a yield function describes the onset of plastic deformation. A possible functional form, supported by experiment, is given 

by 

(5) 

where a and K represent the center and radius of the yield surface respectively, !d~ is the history of the equivalent uniaxial plastic 

:-'1 

--I 
i _ 

I, •• 

,-

r- -

i 

! 

II 

i 

,--

, 

I 

o 

"zero time" 

E 

E 

1 Total strain decomposition 

13 

longtime 
14 

strain and T is the temperature. Note that microphysically a. is a state 

variable combining the effects of the back and drag stresses while 

-p dE is a second state variable representing the dislocation arrangement. This arrangement of dislocations and dislocation loops will be 

discussed later as to their effect on reyield and strain hardening behavior upon reyield. 

A hardening rule prescribes the strain hardening of the material 

and the modification of the yield surface during plastic flow and can 

be stated in a combined isotropic-kinematic form as 

da = d).J(o-a.) (6). 

where·d).J is a scalar. Lastly, if yielding does occur we need further information concerning the rate of deformation to complete the 

description of the material behavior. This information is provided by 

the flow rule which defines the component of strain that is plastic or 

nonrecoverable and can be written as 

dA~ 30 (7) 

where dA is a scalar determined from material data. Equation (7) 

is called an associated flow rule because it is the less general case 

containing the partial derivative of the yield function rather than a 

general plastic potential. It is also known. as a normality condition 

because it can be interpreted as requiring the normality of the plastic strain increment "vector" to the yield surface in n stress dimens ions [26]. 

:."! . 

m··· 

o 

~j 

J 

J 

: .. J .. .. 

·1 

·,,_1 

,I 
I 

1 _.~ 

r-- . I 

Differentiation of (5) is the first step in deriving the constitutive equations and will yield 

dF dF do - - da. dO dO (8) 

The above consistency condition requires that loading from a plastic 

state must lead to another plastic state [27]. Substitution of (7) 

into (4) 

do 

and then (9) into (8) yields 

dF - - da. dO 

Solving for dA from above 

dK 

+ 2Kar dT. 

dF t+6t C T t pt Ct Tt dF dA ={ -[E (dE-dE -dE) + dE(E -E -E -E )] - - da dO dO 

and substituting (11) into (4) yields 

C T t pt Ct Tt 

-dE - dE }+ dE(E -E: -E: -E: ). 

Rearranging the last equation to obtain 

(9) 

(10) 

(11) 

(12) 

15 

i 

i 

~ . 

I 

----.. --~-"------------~--------.------,........---,--------------- - -.- .. ---:----,------.-.-- --"-- --------- ~-- - --'~.-'-'----"-"'-'-""';:--__;_-"7"- ,----~ .~-,----. ._----:--- i 
16 ~ 

: ___ 1 

.] 

j : .. ~ 

'~l , 

] 

] 

-aF da -2K~ --p _2KaK dT 

+ Et+6t aF ao a 0:: 

do:: at 

ao aF Et+M aF 

ao ao 

(13) ;] 

Using the normality condition (7) one can write ] ;:~ 

(14) ..J 

Then equation (4) can be rewritten as ] 

---I 

J 

and equation (13) can a1se be rewritten as 

, I 

El do ~J '" 

J " 

where ,-.-.. ', 

,cJ 

p 

e = E da + 2K~~ dd~ + 2KaK dT + Et+6t ~ 2E. 

ao d;\ o~ A aT d;\ ao 30 (17) --I .... 

;:"J 
Rearrange above to obtain 

do = [Et+tlt 

Et+tlt ~ iI Et +tlt 

dO dO 

e 

Et+tlt .£K ~ dE 

C T ] (dE-d,E -dE ) 

dO dO t pt Ct Tt 

+ [dE - ----6---] (E -E -E -E ) 

Now make the assumption made by Hunsaker [28] that 

(18) 

(19) 

P which defines C as a scaling parameter requiring (do - CdE ) to be 

perpendicular to the yield surface outer normal. The verification of 

this assumption is discussed by Allen [29]. Rewrite equation (8) 

P 

dF do = ~ da. + 2K d~ dE + 2K dK dT dO dO dT dE 

(20) 

and equate with (19) to obtain 

_dF da.' dL J dK 

dO + 2K~ dE + 2KdT dT. 

dE 

(21) 

Also, substitute the normality condition (7) into (21) will give 

dF dF ~ do dF 2KdK J 

2KdK CdA --= da + dE + dT dO dO dO dO '""=P dT dE 

(22) 

or 

_.p 

C dF .£K = -1:. ~ do dF da + 2KdK ~+ 2KdK dT 

dO dO dA dO dO dA ~ dA , dT dA dE 

(23) 

Using the relations in (23), equation (17) can now be rewritten as 

17 

, 

.1 

'f 

I 

! 

~ 

:: ! 

I 

, 

! 

I -.----------------------------------- .... _ .. --.. --.. .---... . ----.- -- --.... -'-----.lr 
w-here 

e 

~ da da C = ---:----

dA dF .EX. 

da da 

Applying (7), the above becomes 

c do 

-p 

de: 

do 

dA ~ da 

(24) 

(25) 

(26) 

Physically, C is the slope of the uniaxial stress vs. equivalent 

uniaxial plastic strain diagram during an isothermal load increment. 

How-ever, during a nonisothermal load step 

-P dO = cde: + aT dT (27) 

or 

(28) 

The above statement is required because the uniaxial stress is a function of both the plastic strain history and temperature during nonisothermal loading. 

Finally, substitute (24) and (28) into equation (18) to obtain 

Et +6t ~ lE Et +6t 

do = [E t+6 t _ ___ .:...30.:...-3:...;a=--____ _ 

H' lE lE + Et +6t .£E. dF 

dO da dO dO 

C T ] (ds-de: -de: ) 

18 ',', :21;'~ : 'i. 

"m'" 

'.'1: 

:L' 

] 

] 

J 

,.] 

~-. 

J 

:',::1 

J 

] 

:'1 

',j 

'~;l 

, , ' J 

'I 

II 

, 

, 

,: 

[ 

~ 

I 

t 

I' 

l 
L. _~ 

i 

\ -: t._ 

, -

i-·-

!. 

l, __ '_ Et +tit E dF dE 

+ [dE - dO dO 

H' dF aF Et+tit aF aF --+ -- ao ao ao ao 

EtHt ~E~ 

+ 

ao dO aT ] dT. (29) 

H' .£! .£! + Et+Lit dF dF -- ao ao dO dO 

Now for the yield surface translation scalar, substitute (6) into 

(8) and solve for d~ as follows 

~ do aF [d~ (o-a.)] 2K aK + = 2K aK dT 

dO dO -p aT dE 

d\.l aF aF 2K aK -p 2K aK dT - (o-a) = - do - dE - dO dO a~ aT 

d~ = 

dF do 

dO 2K~ d~ 

dF (o-a.) dO 

2K aK dT 

aT 

(30) 

(31) 

(32) 

An outline for the uniaxial computer program utilizing the above 

equations is contained in the Appendix. 

Yield Function 

The von Mises yield criterion is used herein an can be written in 

terms of principal stresses o. as 

1. 

F(o.) 1. 

~{(O -0 )2 + (02-0

3

)2 + (0 -0 )2 } = K2 

1 2 3 1 (33) 

where K represents the current yield surface size. It has been shown 

to be in excellent agreement with experiment for many ductile metals, 

19 
20 

for example aluminum, cold-worked mild steel, medium carbon and alloy 

steels [2]. The von Mises yield criterion can be written in uniaxial 

form (with the use of a combined isotropic-kinematic hardening law) as 

.. 

:~l 

2 2 F(o-a) = (a-a) = K (34) 

:·1 

1.:.-:' ~J 

where a is the yield surface center. 

Note that the assumed yield function is dependent only on the 

second deviatoric stress invarient, i.e., independent of hydrostatic 

stress and with the assumption of initially isotropic materials. Also, "] 

the temperature dependence is handled through thermally dependent material properties and is isotropic in nature, but there is no rate de- ] 

pendence in this form of the yield function. 

Hardening Rule J 

Laws governing the influence of plastic deformation on the yield 

surface and strain hardening of a material are called hardening rules. 

A significant amount of research has been done in this area, and there 

are many different rules in use. 

Consider Fig. 2 as a comparison of several hardening rules for a 

given isothermal load history. Although isotropic hardening will 

successfully model loading histories in which stress reversals do not 

occur, it is not satisfactory to model the Bauschinger effect or cyclic 

phenomenon. Conversely, kinematic hardening will model the Bauschinger 

effect, but neither hardening rule predicts the increased strain hard- ;,-1 

-j 

ening upon reyield as their "square" hysteris loop predictions show. 

:" I 

.J 

Oak Ridge· (ORNL) [30] and combined hardening rules predict overall 
Stress 

------

--- -- ISOTROPIC HARDENING 

Stress 

------

--- ---

ORNL MODEL 

Stress 

-I------,I-------+-S(raio 

POPOV IPETERSSON MODEL 

------- experiment 

-- ...... 

Stress -----

21 

-+-f-----------I-------"I .... S tra in 

f 

L------- -- / 

./ 

KINEMATIC HARDENING 

Stress 

-hf-------If-------H--Strain 

L_- - - - - -- - Mroz MODEL 

/' 

/ 

"/ 

I 

,..- ,/ 

Stress 

--------

-+-'-----~f__---~-St rain 

----- --

COMBINED HARDENING 

____ theory 

Fig. 2 Plasticity models 
22 

response somewhat better. Hunsaker [13] suggests the mechanical sublayer model for loadings in which stress reversals may occur. Of all 

the plasticity models displayed in Fig. 2, the multisurface approach 

of Petersson and Popov [20,21] clearly gives the best theoretical representation of the experiment. 

Thus, if one is interested in the exact stress-strain behavior 

over the entire load history and not just an end result, a hardening 

law which accounts for increased str,:lin hardening upon reyield becomes 

a necessity. 

At temperatures below roughly one-half the melting point for many 

metals, a change of loading direction in stress space plays a very importapt role. In many hardening rules emphasis is placed on how to 

describe the evolution of the yield surface while little has been done 

as to how the plastic modulus is affected by stress reversals in cyclic 

loading [31]. Above this temperature the stress-strain behavior is 

controlled by rate dependent deformation and experimentally observed 

hysteresis loops are relatively "square". Thus the simulation of the 

strain hardening is not as critical. 

Many authors mention the "oversquareness" of predicted hyteresis 

loops. Among these are Miller [16], Dafalias and Popov [31], Walker 

[9], and Popov and Peters son [20,21]. It seems that this phenomenon 

is of some interest to the researchers in this field. Miller improves 

his unified time-dependent constitutive equations by making the 

kinematic work-hardening coefficient a function of the back stress and 

the sign of the nonelastic strain rate and shows favorable agreement 

with that observed in 2024-T4 aluminum alloy. Another approach by 

1 ,-.. J 

"\ 

, ! ••• J 

: 7 

! "j 

-, 

,\ 

" 

,-•• ,1 

,

-:-") 

'j 

,·1: ',::-1, 
23 

Dafalias and Popov which was extended by Petersson and Popov involves 

enclosing the initial yield surface within a larger bounding surface. 

Both surfaces are allowed to translate and deform in stress space, 

and the proximity of the two determines the plastic moduli. Two other 

hardening rules which provide for a high strain-hardening rate upon 

reyielding are the Mroz and mechanical sublayer models [13J. 

Since Miller's theory [16] is microphenomenologically based, he 

seeks to make his model consistent with .existing microphysical explanations. He states that dislocations moving in a given direction will 

form dislocation loops by interaction with precipitate particles, and 

when the direction of dislocation motion. is reversed a dislocation can 

annihilate a nearby loop which was previously left behind. A relatively rapid change of strain in the nature of the stress field encountered 

by the reversing dislocation would result and lead to a large value 

of the slope do/dE. As the strain continues to reverse, dislocations 

will no longer be encountering such oppositely-signed loops and do/dE 

will gradually decrease. Miller models this by associating the directionality of the dislocation debris with the back stress R and the 

direction of current dislocation motion with the sign of the difference 

a/E-R. A similar argument is given by Polakowski and Ripling [14J. 

Although the combined isotropic-kinematic hardening rule used by 

Allen and Haisler [25] cannot represent the high rate of strain hardening accompanying stress reversals which cause yielding, it does 

account for thermally dependent material properties and the Bauschinger 

effect. Thus it was felt that with some modifications, all cyclic 

characteristics could be modeled. To summarize, the two important 
24 

shortcomings of their hardening law in representing cyclic behavior 

are its failure to account for the high rate of strain hardening upon 

reyielding after a stress reversal and cyclic saturation. 

In terms of actual modelling of the two shortcomings discussed 

above, several revisions and improvements have been added to the model. 

First as discussed in regards to experimental data requirements, additional stress-strain input is required to characterize various strain 

hardening rates. Metals like aluminum exhibit a similar strain hardening behavior for reyield after the initial yield. Conversely, stainless steel can exhibit two different forms of hardening behavior even 

after several loading cycles. A program flag controls which input 

stress-strain curve to use to generate hardening parameters depending 

on the material being modeled and the yield (load) history. Secondly, 

the hardening ratio w.as allowed to be a function of plastic strain and 

in some cases the direction of loading. This allows modelling of cyclic saturation. 

More specifically, the hardening rule revisions can be explained 

in two different discussions. The first deals with strain hardening 

upon reyield and the second with cyclic saturation. 

Use of the combined hardening law in Allen and Haisler [25], 

predicts cyclic behavior like that shown in Fig. 3. The hardening 

ratio, 8 is the ratio of isotropic to kinematic hardening. Setting 

8=0 constitutes a kinematic hardening law where the yield surface retains it initial size, shape, and orientation thus simply translating 

in principal stress space. Isotropic hardening, 8=1, means that during 

plastic flow the yield surface expands uniformly about the origin and 

1 

~'J 

l ··.·i 

i -. ! 

I 

I ! . 

.i .1:> 

.. 

'I ., 

] 

J 

]/ "-. -'-'--~-.-- ,---- '-', --.. -. ---_ .. --"--- . - ".-, --------'- ~-~.~---,,--.-
o 

K 

"----

--

20' 

Y 

--:--

20 

o 

j 

------------

-----

isotropic 

(S=l) 

combined 

(6=C) 

kinematic 

(8=0) 

-p • e: 

Fig. 3 Definition of standard hardening rules 

25 
26 

never translates. Using a value for the hardening ratio such as 

o < e < 1 constitutes a combined hardening rule where the yield surface 

is allowed to both expand and trans~ate. 

Notice in Fig. 3 that the reyielded strain hardening behavior is 

"squared-off" as discussed above. The revised model has the ability 

to reproduce strain hardening behavior that is dependen~ on direction 

or history of yielding. This is observed in experiment as the high 

rate of strain hardening upon stress reversal in a cyclic load test. 

In other words, the material demonstrates two distinct forms of behavior as shown in Fig. 4. Initial strain hardening character is usually quite "sharp" in comparison with the rounded shape of the reverse 

loading curve. Popov and Petersson [21] found it essential in dealing 

with cyclic steel behavior to use two stress-strain curves as input. 

The first is from a monotonic test on virgin material and the second is 

from a hysteresis ioop which develops after several loading cycles. 

For a large number of common metals, a cyclic load history leads 

to a limiting periodic response in which the stress-strain curve for 

each consecutive cycle is the same. This is termed cyclic saturation 

and is illustrated in Fig. 5 [32]. Capability to portray this phenomenon is not contained in many common hardening laws. With the addition of a hardening ratio that is a function of plastic strain and 

in some cases direction of loading into the existing model, this 

phenomenon can be modeled much more successfully than a combined hardening rule with a constant hardening ratio. 

" . ',-, , 

n".··.·· ;:. 

'-

J 

J 

n.·,.,: 

J 

D

" 

~.'. '.' 

" 

~.:--.,. ]. 

" 

1 

J 
(J 

E 

Fig. 4 Typical experimental results showing difference 

in first and second quarter cycle response 

- --.-----~---.,-~ -.--". - ~ -".- .- -.-.' ----- -- -,-.----~ . - ---.. -- ---.. - .-.- .. -------' .. -.--".-----._---- '---.. -._-- "'- -----------" , 

27 

i 

l-

,_ '.""r •• _r .~._ •• __ ._ ._,,_. _________ .f 

"p 
a 

---++r-f----------€ 

Fig. 5 Mechanical response of typical ductile 

metal exhibiting cyclic saturation in cyclic 

loading 

28 l 

: ... 1 

] 

j 

-:'.' 

J ~~'. 

:J :",' 

. 

J 

;] 

J 

J 

J 

] 

~'.J 

] 

:-1 

."j 

'·'1 ,.' 

~->. 

:..J 

;, ··i 

., 

:.J 

""1 

J 

J '. 

", t 

" Ii' 

" b 

w· 

'i 

i· 

", 

" i 

I~ 

! 

't 

ti 

r Ii 

," I! , 

, n 

'i ij 
The reader is referred to the sections on experimental data requirements and theoretical examples for more details concerning the 

above discussion. 

Creep Strain Increment 

The creep strain is defined as the uncoupled rate dependent inelastic deformation in the present theory. Several methods are available to characterize this strain component. Traditional characterization has been based on the use of standard creep tests run at constant 

stress and temperature ... Creep strain (or creep strain rate) is written as a function of time, stress, and temperature through the use of 

power law functions, exponentials, hyperbolic functions, etc. obtained 

by appropriate curve fitting techniques. Additional ad hoc creep 

hardening rules have been devised to model rate dependent behavior 

during reverse and cyclic loading. Oak Ridge National Laboratory 

used such models for 304 stainless steel with some success [33J. In 

many cases it is more expedient to use tabulated creep strain data as 

opposed to curve fitted data; although this requires numerical interpolation between a set of isothermal, constant stress creep curves. 

In the present research, an alternate approach has been taken 

involving hysteresis loops and stress relaxation tests at various 

temperatures. Fig. 6 shows a typical set of these tests and details 

how· th~ rate dependent deformation is extracted" from them. 

~'. '. ,-

A"reference temperature hysteresis loop is defined at a tempera- '-,', - .-~. ~ .. -

ture below which the total strain consists essentially of elastic and 

rate inde~ende~t~last~cstrains only. Above this temperature, the 

29 
rate independent 

plastic strain 

elastic --{,' 

strain " 

, 

(J 

o 

dependent 

strain 

T ref 

Fig. 6 Experimental characterization of the rate 

dependence of the model 

30 

J 

] 

J 

] 

:- I 

.j 

,'.J 
rate dependent inelastic strain (creep) dominates. Experimentally, 

Bradley and Haisler [34,35] have shown for Hastelloy-X that such an 

uncoupling appears feasible. 

Micromechanical considerations suggest a model for the rate dependent inelastic strain of the form 

(35) 

·C where £ represents the rate dependent inelastic strain rate, r2 is the 

back stress, K is the drag stress, and n is a constant. The back 

o 

31 

stress, drag stress, and exponent n are generally functions of temperature and strain. However, for some materials a good assumption is 

that the drag stress and n are constant at sufficiently high temperatures. 

At the tips of the hysteresis loops, the stress and back stress 

have reached maximum values ° and r2 respectively. If we let 01 . max max 

and 02 denote the maximum stress values at two different strain rates 

. . £1 and £2' then the value of n can be computed as 

(36) 

where r2max is assumed to be constant between £1 and £2. Similarly, 

the constant K can be determined by rewriting (35) in the form 

o 

1 

K = (0 -0 )~ - In o 1 max 1 (37) 

It should be noted that a number of the unified models reviewed 

in Chapter II (for example, Walker's [9]) utilized similar functional 

.. ______ " _. ___ , ___ " ______ __ -" ___ ,, - - .--- :-----0-_. ___ --. ___ --;-__ ,. .. _.- - ---c--~___;_----r,--. ------=-" ,- -:-'-----<.-'-.-- .-~--- _.' ~. --- --- -- ".-- -

~[ 

i' , 

I 

I 
.... -" .. _--------- -.----~...:....~ ----~---.-

32 

forms and experimental characterizations of the inelastic strain component compared to that used in this research for rate dependent inelastic strain. Although the functional forms used to model the unified inelastic strain give excellent agreement at high temperature 

(where rate dependence dominates), they give overly square stressstrain response at low temperature (where rate independent plasticity 

dominates). This is due primarily to the failure of the present unified models to account properly for rate independent behavior. In 

the present research, this difficulty is hopefully overcome by modelling the rate dependent inelastic and rate independent inelastic strain 

more accurately with appropriate definition of a temperature range 

where each is applicable. 

Experimental Data Requirements 

One of the requirements for a good constitutive model is that is 

have reasonable experimental data requirements. Characterization of 

model parameters should follow easily from standard tests. 

All experimental data tests are performed at sufficient levels of 

the primary variables (strain and temperature) in order to bracket 

their magnitudes in the particular test of interest. Consideration 

is also given to allow accurate linear interpolation between temperatures. 

The first set of tests required is uniaxial isothermal single cycle reverse loading tests like those shown in Fig. 7. These tests 

are performed at fast enough strain rates so that the rate dependent 

component of deformation is negligible. Characterization of both 

-n t:-

:,. 

fl 

;.~1 

] 

] 

J 

] 

"-' 

. ..1 

-/! 

'-J 

I 

( 
I 

E 

cr 

cryl 

°y2 

A 

E'I 

II 

I'} I2 

G 

--------__________ ~~~ s 

Fig. 7 Uniaxial isothermal single cycle reverse 

loading curves and their linear representations 

input to the model 

33 
initial and subsequent strain hardening behavior is the reason for the 

reverse loading tests. Piecewise linear representations of the data 

obtained from experiment are input into the model with the number of 

linear segments used depending on the accuracy desired. 

For example as shown in Fig. 7 at T=Tl , the initial elasticplastic behavior is characterized by curve ABC. Similarly the subsequent elastic-plastic behavior is characterized by curve DE. The 

piecwise linear representations become curve ABC and D'E' where D'E' 

was obtained by shifting curve DE to the yield pLint B. A similar 

procedure is performed for all other temperatures as required. Note 

that if the difference in strain hardening behavior is not considered 

critical, (the difference between curves ABC and DE is insignificant) 

only curves ABC and AFG would be required input. 

34 

If the hardening ratio is constant with respect to plastic strain, 

it can also be determined from the above tests using the relation 

s - o - (20 -0 ) r y a 

(20 -20 ) 

o Y 

(38) 

where stress values are defined in Fig. 8 for each temperature. If 

6 is a function of plastic strain as well as temperature, several 

cycles of stress-strain data would be required to characterize this 

parameter. Typical input curves for the hardening ratio are also 

shown in Fig. 8. 

A piecewise linear description of the temperature dependence of 

the coefficient-of thermal expansion is also required if it varies 

significantly for the temperature range of interest. 

.~ e~ 

~::~~: ~ . 

'J 

n '~ :, , 

<-

.i;.; 

J 

J 

] 

J 

~I 

-.J. 

] 

J 'r 

J 

--I .-

:::J 
] . 

·''::.l 

,---: 

r 

I 

o 

o 

o 

o 

y 

o 

r 

s 

s 

T constant 

- reyield 

stress 

~----------- ------------------------- T 

~--------------------------------~.-- -p E 

Fig. 8 Hardening ratio 

35 
,"' --,-.-... ~- ----......:. <,----

36 

Experimental characterization of the rate dependent terms is the 

same as for Walker's model [9]. Steady state hysteresis loops at different strain rates under fully reversed strain controlled conditions 

as shown previously in Fig. 6 (p. 30) are required. These are all performed until saturated values are established. Also required are 

stress drop creep and relaX'ation tests from initial point on a steady 

state hysteresis loop. All constants are a function of temperature 

and require the above tests to be performed at all temperatures of interest unless interpolation is used. 

For comparative purposes, a brief description of the experimental 

data requirements for several other constitutive models is presented. 

'The first tests required by Krieg, et al. [11] are stress-drop 

tests during steady state creep. This means that a fraction of the 

applied stress is removed rapidly, and then this reduced stress is 

held constant until the accrued strain is sufficient to permit the 

strain rate immediately after the drop to be obtained by extrapolation. 

The stress is then increased to its original value to re-establish 

steady state creep. This procedure allows the response of a speciman 

to drop tests of various magnitudes to be obtained from a single creep 

stress. For full characterization, this test procedure must be repeated at several nominal creep stress levels and temperatures. 

Steady state and primary creep data must also be utilized. 

Miller [16] states that with a general purpose constitutive 

equation a fairly wide variety of data is required. Calculation of 

a complete set of material constants for an alloy on which hysteresis 

loop data are available can be a very lengthy task with a substantial 

J 

J 

] 

J 

J 

] 

J 

] 

] 

.'J 

··,--1 

'J , f 

-

~-l I ! 

:. I 

: ~J I ,_. I: -- I 

I 

, ' 
37 

amount of experimentation required for his model. 

The characterization of model parameters for the theory of Bodner, 

et al. [15] requires two monatomic stress-strain curves at different 

steady strain rates. For cyclic loading an additional constant is 

required, but determination of that constant is unclear in their presentation. 

Construction ofl Yield Surface Size and Stress vs. Equivalent Uniaxial 

Plastic Strain Diagrams 

Calculation of the yield surface size and equivalent uniaxial 

-P stress is accomplished by constructing K and 0 vs. E diagrams and 

inter.polating on these diagrams at known values of EP and T. From a 

piecewise linear representation of a "zero time" stress-strain curve, 

\ 

the equivalent uniaxial plastic strain is given by 

-i' 

E i (3.9 ) 

The yield surface size and equivalent uniaxial stress are given by 

K. 

1 

o. 

1 

o + 8(0 -0) 

y xi y 

o 

x. 

1 

where 0 represents the yield stress. 

y 

(40) 

(41) 

For the revised hardening law, two each of the above diagrams are 

required. The first diagrams for each case (K and 0) will be labeled 

base diagram and are simply the functions calculated from equations 

(39) - (41) for the input stress-strain curves. These change only if 

.. -, .-. -....-------.. -:-----~----~ -~----~- --- - .- - ~- ~ ----......... ~- - ~ --...... ---- -- -.--- --- -~--- "-- - -----.". -_. ----

i 

~ 

I' 

t 

r 

.' i 

I 

I 

-,---- ----------t 
the hardening ratio is a function of plastic strain. The second set 

or global diagrams evolve throughout the solution process. They are 

initialized to equal the base diagrams but are modified each time a 

load reversal which causes yielding occurs. The modification process 

consists of linking the base curves to the global curves at the values 

--P of E , K and cr corresponding to the yield stress of interest. Inputting two stress-strain curves at each temperature comes into playas 

--P their exists for example two K vs. E base diagrams at each input temperature. Factors such as type of material and load history determine 

which base curve is linked to the global diagram. All program interpolation for calculation of model parameters is done on the global 

diagrams. 

The above procedure is best explained by a figurative example. 

-p Using Fig. 7 (p. 33) as the input stress-strain curves, the K vs. E 

base diagram for a constant hardening ratio is shown at the top of 

Fig. 9. Recall that the "a" curves are the ones that result from reverse yielding while the "b" curves are from the virgin stress-strain 

curves. Note that the Allen and Haisler model uses only the "b" 

curves. 

-P A reverse yielding occurs at E -p = E causing the "b" curves 

o 

to be linked to the global diagram as shown at the bottom of Fig. 9. 

When another reverse yielding takes place, either the "a" or "b" curves 

-p 

are linked to the global diagram at that particular value of E. A 

-p similar procedure is carried out for the global cr vs. E diagram. 

Computational Considerations 

For completeness of the theoretical presentation, the gradients, 

transition step, thermal strain increment, and elastic strain 

.' ".'. ~ .. 
i 

i 

, 

\, . ~ 

'.,. 

--

L·· -

r 

K 

t l 

K =0 - .l,yl 

K 

Fig. 9 

a 

b 

Base Diagram 

1 r 2 

T 1 

-p 

E: 

o 

------------------------~~--p 

Global Diagram 

-p Construction and evolution of K vs. E: diagrams 

E: 

-p 

E:' 

39 
40 

increment are all discussed below with regards to computational considerations. They are also presented by Allen [36]. 

Gradients. Linear. interpolation is performed to compute values 

between input temperatures. The yield surface temperature gradient 

can be obtained by 

(42) 

where the subscripts Hand L denote the values of a particular variable 

for the temperatures above and below the current temperature T .. 

Similarly 

(43) 

where the stress values are obtained from the a vs. ~ diagram. The 

-p slope of the K vs. € diagram can be obtained by 

aK (aK ) {(aK ) _ (aK ) } (T - T) H 

= (44) (T - a?' a? a£P -p T

L

) 

H H ae: t H 

and the slope of the a vs. 

-P 

€ diagram by 

(TH - T) 

H' = (H') - {(H') - (H') } (TH - T ) . (45) H H L L 

Transition step. Special treatment must be given to the transition step from elastic to elastic-plastic behavior. The portion of 

the assumed elastic stress increment da which will cause yielding 

~ 

is ~da and the strain increment to bring the total strain to the 

J 

1 

] 

J 

J 

J 

J 

.-:"] 
ri 

!---' 

yield surface is ndE. Thus for the transition step, the stress and 

strain status are modified to become 

(46) 

E -+ E + ndE (47) 

where 

-B + B

2

-4AC 

1;; = 2A 

(48) 

A dada 

B = 2(0-a.)da (49) 

C (a-a.) (a-a.) (Kt+llt) 2 

t 

+ 1;;da _ Et+t.t Et 

a E 

n Et+~t dEE 

(50) 

The above equations are presented here for completeness of the theory. 

A full derivation of them is given by Allen [12]. 

Thermal and elastic strain increments. The thermal strain increment is given by [36] 

(51) 

t+~t t where a.T and a.T are the coefficients of thermal expansion at the 

beginning and end of a load step respectively, Tt 

1 

are the 

temperatures at the beginning and end of a load step respectively, 

and TR is the reference temperature for the unstrained state. 

41 

---- ",~ . -, - .----.---.. - - -. --.. - -.- - ----------.---------.. - - --.-~--. ----.--. -------'--'--'- -- ---.- ---- '--'---c------.-.---.. --- -.-. .. --.-.--.-: .... --. -- .... : _ ... -.... -.--, .... - .. - --. -. --.-'-':":'-1: 
The elastic strain increment is calculated by [36] 

1 

E t+lit t pt Ct Tt d~ = E {da-dE(s -s -s -s )} 

Extension to MultiaxialTheory 

(52) 

The approach used to derive the uniaxial constitutive equations 

makes it very simple to extend them to the multiaxial case. For exampIe, equation (29) could be simplified considerably by, for instance, 

dividing out terms like aF/ao. This is not done, however, to retain 

generality and ease of extension to the multiaxial case. To convert 

equation (29) to a three dimensional form simply let uniaxial values 

of stress and strain become the respective stress and strain tensors. 

. Recall the yield function is stated in terms of principal stresses in 

equation (33). The general elastic constitutive matrix is then substituted for the elastic modulus. 

Gradients are still determined from uniaxial input stress-strain 

data, in fact the only additional experimental data required is 

Poisson's ratio. The equivalent uniaxial plastic strain is 

d~ = 13 dE~. dE~. ~J ~J 

(53) 

in tensor notation. In engineering notation this equation can be 

written as 

d7' 

+ 6(d~P )2 + 6(dsP )2 ~ 6(dsP )2]}~ xy xz yz (54) 

A full derivation of the multiaxial theory is contained in 

reference [12]. 

42 

0l ,J 

"J'.' ,", 

, '.:~ 

l 

:'j 

:] 

1 

.. J 

"l ' . 

J 

",J 

~") 

J 

") 

) .... J 
CHAPTER IV 

EVALUATION OF THE MODEL 

Evaluation of the constitutive model detailed in this thesis is 

now performed. A section on theoretical model capabilities is presented followed by evaluation against experimenL 

Theoretical Model Capabilities 

Several examples of the capabilities of the model proposed by 

Allen and Haisler are given in references [36] for the uniaxial case 

and [12] for the multiaxial case. The purpose of the following 

examples is to demonstrate. the improvements in the revised model by 

using theoretical illustrations of experimentally observed behavior. 

Example 1 - Change of strain hardening with·stress reversal. The 

revised model has the ability to reproduce strain hardening behavior 

that is dependent on direction of yielding. Fig. 10 illustrates the 

input stress-strain curves for a theoretical single cycle reverse 

load test. The input can be taken from either first or saturated 

43 

cycle data. On the input diagram curve B represents the initial strain 

hardening behavior while curve A represents the high rate of strain 

hardening seen upon stress reversal. If a single cycle reverse load 

test is performed with this theoretical data, the results will be 

similar to those shown as output in Fig. 10. The revised model using 

curves A and B is a much more realistic representation of actually 

observed experimental behavior for many metals than the combined 

hardening rule of Allen and Haisler. Even the revised model 
'i 

"I 

! 

i,l 

'1' 

. ~ 

25000 

22500 

20000 

17500 

15000 

S 

T 

R 

f 

S 

5 12500 

I 

,. 

5 

I 10000 

7500 

SOOO 

2500 

A 

B 

A - strain hardening 

upon stress reversal 

B - initial strain hardening 

0.002 

O~~:~~~~ __ ;r~~~--__ r-~~ ____ ~~~~~~r-~~~~~ 

0.000 o.oo~ 0.006 0.008 0.010 

~3TRAI~ - INIIN 

Input (Isothermal) 

18000 

15000 

12000 

9000 

6000 

5000 

S 

T 0 R 

f 

S 

S -3000 

,. -6000 S 

I 

-SODO 

-12000 

-15000 

-18000 

-21000 

-2~000 

0.0000 

----------~ Haisler 

-------" 

0.0015 0.0030 

model 

revised model using B 

only 

revised model using both 

A & B 

O. 00~5 0.0060 

STRAIN - INIIII 

Output 

., 

" Fig. 10 Change· of strain hardening with stress reversal 

" 

~: ' i~_c--" i_ -~":~~ j f~~J L~~ .: .. -; "'-~-..-...' L_; ___ ~ I 

L----J 

.j 

'c=~~ __ ~~~_~~~~--_"~"_-'-""'_ 

~ 

.) L.J L:J ~ • ',1 

I~ ,'.-,.,' .... ) , !'t:p.q L<':r;':J 

~ 

~ 

!>-.;:::!] , Z:C:~3,," 

',';-
45 

representation using only curve B is an improvement over existing f': 

hardening rules which "square-off" the behavior upon reyield. 

Example 2 - Cyclic saturation. A hypothetical fully reversed. 

cyclic strain history of ± 0.25% strain was considered using four 

different values for the hardening ratio. The values used are shown 

in Fig. 11 following the discus~ion on hardening rules in Chapter III. 

An allowance for cyclic saturation is made by letting 8 be a function 

of plastic strain. This corresponds physically to allowing the yield 

surface to first translate and expand and then gradually saturate to 

a constant size as observed in experiment. Input stress-strain data 

is the same as the first one-quarter cycle input. 

The results for isotropic and kinematic hardening are shown in 

Fig. 12. Kinematic hardening will model the limiting periodic response, but there is no cyclic hardening of the material in achieving 

the saturated hysteresis loop. Conversely, isotropic hardening predicts that the material will eventually cycle to a limiting purely 

elastic response as the yield surface expands without bound. This 

same behavior is also predicted by a combined hardening law using a , 

, .. 

constant hardening ratio. It takes somewhat longer to cycle to the 

elastic response because of the component of kinematic hardening 

present. This is shown in Fig. 13. 

i _ Successful modelling of cyclic saturation is accomplished by 

lettinp, the hardening rat~o inthe combined hardening law be a function 

of plastic strain. For this example, the linear relationship shown in 

Fig. 11 was assumed with results as shown in Fig. 13. Both the cyclic 

hardening and limiting periodic responses are depicted. 
K 

o y 

B 

Isotropic 

(S=l. 0) 

Combined 

(S=0.5) 

Kinematic 

(S=O) 

~----------------------~------~-~ 

1.0+---________________________ __ Isotropic 

Combined 

Combined <S=0.5 - 4l.667~) ..:. .' ,~'.' 

O. L----------.....::::.f:-----------Kinematic 

-P:(S=O) 

, e: 

in Example 2 

.46 

J 

~l 
:1 

1 

.... ~"..-.,-~-~-,~, -~ ~ 

15000 25000 

12500 20WlO 

10000 

I ') 

15000 

7500 

10000 

5000 

5000 

5 2500 T 

II 

E 

S 

S a 

5 

r 

II 

f 

5 

, a 

I 

r 

1 -~oo 

r 

, 

1 -5000 

-5000 

-10000 

-1500 

-15000 

-10000 

-20000 

-12500 

-I5000~~,~ ____ -r~ ____ T-__ ~-4~~ __ -r __ ~~T-____ -T 

-O.OOl O.OOl 

-25000-l~,-:-:--~ __ ...,...._~ __ ,.., ___ ......jf..-.-__ ~'""T __ ~_"T"'~~~ 

-0.003 O.OOl -0. 002 -0.001 0.000 0.001 0.002 -0.002 -0.001 0.000 0.001 0.002 

STRAJIt - INIIN STRAIIt - I,,/IN 

Kinematic Hardening Isotropic Hardening 

Fig. 12 

laws 

Modeling of cyclic saturation using kinematic and isotropic hardening 

....,.,..,....-_. - ~~T--~ 

;r""'-- -,-: 

.p.. 

" 
i 

~ , 

I 

L2 ~~ 

i 

S 

f 

H 

f 

"""I 

2co~a 

ISC~O 

ICO'la 

SOOI} 

~ 01 / I I ~/y/f t ( ( J } .TII 7 ) ) 

I 

~ , 

I -soeo 

-10000 

-15000 

-20000 

-25000~~r~ __ ~-r ______ ~ ______ ~ ____ ~~ ______ T-____ --T 

0.00] -0. 001 -0.002 -0.001 0.000 0.001 0.002 

STRRIN - INII" 

Combined hardening with 8 constant 

5 

r 

R 

~ 

5 

5 

I 

~ 

5 

I 

21000 

18000 

15000-

12000 

8000 

6000 

5000 

0 

-5000 

-GOOD 

-9000 

"2000 

-15000 

-16000 

-21000 

-2ijOOO~~r ______ -r __ ~ __ -r~ ____ ~ ______ ~ ______ T-__ ~ ___ 

-0.00] -0.002 -0.001 0.000 0.001 0.002 O.OOl 

,1/111IH - IN/IN 

Combined hardening with B a function 

of plastic strain 

Fig. 13 Modeling of cyclic saturation using combined 

hardening law 

~'-:r~~_.,<~ E,~t-; .' ~~ t1 ~ 

1.,- '-"1 I': L..J I __ .~·I L.J L:.:J L:.:J 1",':''''1 L::J ~ r:' ~::' 'I :':';>':q H~~:}'J 

.~_~ _~ ~~ . .-~ ____ "",.,....,~~_."""."-" ... ...,..,-, .• --r'~. ' 

t~~, ',:::1 

.p. 

00 

f';-·';':"~I 
49 

Example 3 - Cyclic thermomechanical loading. This example is 

similar to one in reference [25] but is still applicable here to show 

the increased capabilities of the revised model in predicting the 

strain hardening behavior and also in reiterating its nonisotherrnal 

aspects. Depicted in Fig. 14 are the input stress-strain curves and 

therrnomechanical load history for a hypothetical test. Note that 

these curves do not include the capacity for variable hardening in 

order to maintain clarity of the example. The hardening ratio is 

a constant one-half. 

Results from the constitutive model are shown in Fig. 15 where 

t. in this figure corresponds to t. in Fig. 14. Modulus, strain har1 1 

dening, and yield surface size changes with temperature are all shown. 

It is also interesting to look at the resulting yield surface 

size vs. plastic strain diagram for this example also shown in Fig. 15. 

As discussed previously in construction of these diagrams, the curves 

are modified every time there is a stress reversal which causes yield. 

This physically represents the rounded shape of the stress-strain curve 

after reverse yielding seen in experiment. In terms of modelling, this 

--P corresponds to "linking" the original K vs. E: curves onto the current 

global diagram also explained in the section on construction of these 

diagrams. The squares in Fig. 15 point out where this process takes 

place each time. The curves at each temperature are modified for 

linear interpolation purposes. 
25000 

225;;;; T=800 

20000 

17500 

15:)00 T=lOOO 

12500 

I.OUOO 

7500 

5000 

2500 

O~-------r-------r-------r ______ -Y ______ ~ ___ 

O. Gooeo o.OCO?s 0.00150 o. C022S 0.00300 0.00375 

5TRR[·1 [NI [N 

Load 

time 

Temperature 

1000 

800 ~---+----~--~----.~----+---~----~--~r-~---- time 

t3 t4 

Fig. 14 Input for Example 3 

n 

50-1 

J

. 

.--

J 

J 

] 

J 

] 

: .. :1 

····1 ~ -. \' 

···1 

'.,,' 
, 

'! 

,I 

.1 

:! 

" 

'1 

i 

! 

I 

'-1 

! 

I 

i 

-'j 

'I 

':1 ,1 

~ l 

'I 

",: 

.: 

i 

::'1 

1 

20'C!~11 

17S".G 

15G~O 

125GO 

laooo 

7500 

5 

I 5000 R 

E 

5 

5 2500 

I -

0 

-2500 

-5000 

-7500 

-10000 

-12500 

-15000 

-0. OOl -0.001 O.OCO 0.001 

STRfllN - INI [lj 

O.OC2 

Fig. 15 

.....,...., 

0.003 

ts 

250(11) T=SOO 

22~Oil / 200~O 

17SQr) 

T=lOOO 

'r 

~ 15000]/ 

L 

11 

5 

U 

R 12500 f 

R 

C 

E 

5 1000G I 

Z 

t 

7500 

501J0 

2500 

O~~~~ __ ~~~~~ __ ~~~~~~~ 

0.001) G.OCI 0.OQ2 O.IlOJ 0.004 a.nos 0.005 

PLASTIC STRAIN 

Cyclic thermomechanical loading 

i 

\J1 

I-' 

, " 

52 

Comparison with Experiment 

As discussed earlier, characteristics of metals such as the high 

rate of strain hardening upon yield reversal are much more pronounced 

at high strain rates and temperatures below one-half the melting point. 

Modelling of this phenomenon is less critical at more elevated temperatures where plastic deformation is more fully rate controlled. 

More importantly, there is an intermediate temperature range where 

the coupling of plasticity and rate effects is the most critical test 

of the constitutive model. To evaluate the models capabilities in all 

of these environments, specific areas of testing are: (1) elevated 

temperature tests that evaluate the rate independent capabilities of J 

the.model, (2) cyclic load tests that evaluate the modelling of the ] 

Bauschinger effect, the change in strain hardening upon stress reversal, 

yield surface evolution, and cyclic saturation, (3) transient tempera- J 

tures tests that evaluate the modelling of thermally dependent material 

properties, (4) high temperature tests that evaluate the rate dependent I· ] 

capabilities of the model, and (5) thermoe1astic-p1astic-creep tests J 

that evaluate the capabilities in the intermediate temperature range. 

The high temperature materials testing laboratory in the mechanical engineering department of Texas A&M University was used to run all 

J tests. A Mechanical Testing Systems (MTS) machine along with an induction heating coils and generator, optical and thermocouple type 

temperature controllers, high temperature tension-compression grips, 

a high temperature diametral strain extensometer, and constant stress 

creep frames ,fully outfitted for high temperature testing are the main 

components available in this lab. 

J 

--I 

-) 

-.~ - -,_--r-____ ._. ---.--,- -.-- -.'----,.,.'-- ---~.-.---.-- -- - -- -_ - R - - ---.------ -- --- -- ----- - ------ -,---~-- -- ---- ---,------------,-------- --- --- ---------- ---- -------,-- - - . --------- -- - ,-~--,-- ----- .-- u 
53 

Cyclic thermomechanical load test on 5086 aluminum alloy. The 

1.'- first experimental evaluation is for a thermomechanical load history 

applied to a 5086 aluminum alloy. This test is very similar to ExampIe 3. Fig. 16 shows the results of comparing the Allen and Haisler 

model using several different constant hardening ratios to experiment. 

One can observe both of the shortcomings of the model discussed earlier. 

The theoretical strain hardening behavior upon reyield is much too 

flat or square and the cyclic hardening predictions are inadequate. 

By inputting a second stress strain curve to characterize the more 

rounded strain hardening behavior and making the hardening ratio a 

function of plastic strain as in the revised model, this experiment 

can be modeled very well. The actual function of plastic strain used 

for 6 is that 6=0.45 (constant) for the first quarter cycle and since 

the isotropic component of the yield surface saturated very quickly 

in the observed experimental results, the second quarter cycle was 

treated as being kinematic in nature. These results are shown in 

Fig. 17. 

Cyclic loading of 304 stainless steel at lOOOoF. Several other 

i, 

model features are demonstrated by this test. As seen in Fig. 18, 

the compressive strain hardening behavior in the first cycle is more 

rounded in shape than the tensile behavior. It is believed that the 

same behavior would be seen on the first and second quarter cycles 

if compression preceded tension. Unlike the previous test on aluminum, 

this difference seems to alternate throughout all four cycles presented in Fig. 18. Thus one must alternate between first quarter and 

second quarter cycle input stress-strain data to model this test well. I 

i 

J 

i· ._.< _____ , __ c-' --~-- [ 
. STRESS 

MPa Ksi 

300 

40 

250 

30 

200 

150 

20 

100 

'0 

50 

Y' 

13>"r- RT 

VA Load 

L:-

: 'Temperature 

./ 

// 

/ 

I 

I 

I 

I 

r 

I 

, 

.,..,/" I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

a o~ ____ --~------~~--~~~--~--~--~L-~ 4 5 

1 % STRAIN 2 3 

-50 -,0 

-100 

-20 

-150 

-200 \·-30 

-250 

Hardening RatiOS 

• f3:::'.0 (isotropiC) 

o fl' 0.9 

o f3.= 0.7 

A f3' 0.5 

__ Experiment 

Fig. 16 Co~a,ison of Allen and Raisl

e

, model 

to experiment for aluminum cycliC load test 

;C7\ .:.'\ .,:' 

-.1 

-.· .. 1 .;\ 
. ., 

, STRESS ., 

.. 

: __ "..J MPa Ksi 

300 

40 

250 

200 30 

150 

20 

100 

10 

50 

i .. 

0 0 

-50 

-10 

-100 

-20 

-150 

-200 -30 

-250 

r-" "'V7'~ Load 

1350r Temperature / R.T 

2 3 ------4 

-- Experiment 

----.. Revised Model 

Fig. 17 Comparison of revised model to 

experiment for aluminum cyclic load test 

. I . 

I 

. I 

I 

I 

I 

I 

5 

% STRAIN 

55 
I')CO:j 

-I~JO 

CYCLE ~!O. 1 

Exp 

Model 

CYCLE "10. 2 

3COOI'J 

'CUDO 

10000 

-IDaoa 

-,0000 

-30000 

-uooo 

..- .~ ... , 

/ " 

56 

I 

i 

: 

,. 

; 

-~~:rJ:J.",. ____ --, ____ --, ____ --, ___ _ 

-500011,'-___ .,., __ ...,.., __ ...,.. __ ...,.., __ ...,.., __ ...,..,_ 

-loaoc p 

5 

I 

0.000 c;.~:s 

srr.RIN IIN/ENI 

CY':LE NO.3 

.... ~ .. ~ .. 

... 

/ / ..... .... , ..... /' 

/~~:_/ 

,-----,----

.. 

./ 

... / 

;' 

I 

! 

I 

i 

i 

! 

j 

i 

I 

-0.005 -0.002 o.nOt o.oo~ 0.1'"07 0.010 0.Ol3 

STRAEN [[~!/E~I 

CYCLE !'lO. 4 

50000 

~coo;:; 

3CC'JC 

,oeao, 

lODOD 

P 

5 -lOOCU I 

-20000 

-30000-

-5(m0"l,-___ ....,.-__ ....,.-_ ~--~--,-~ 

o,aoo 0,003 0,006 U, C09 0.012 O. o~s O.OlB 

STRI:I[N IIP4/(NJ 

n :1~? n. nru n. fJl'I\I [1,007 ,'.1'110 

Fig. 18 Comparison of model and experiment 

stainless steel at 1000°F using first cycle 

input. A hardening ratio of 0.2 in tension 

in compression was used 

for 304 

data as 

and 0.35 

]" 

J 

J 

J 

J 

J 

] 

J 

~'I " 

,~ -I 

~J 

J 

OJ I 

I: 
57 

One possible explanation for this alternating strain hardening behavior 

could be that the initial yield sets up an asymmetric microstructure. 

This phenomenon could also be explained by reasoning that there is 

less resistance to grain boundary sliding in tension as compared to 

compression. It is clear that a similar test needs to be performed 

with the initial yield in compression and then alternating tension and 

compression. 

Next, the ability to vary the hardening ratio ,between alternating 

quarter cycles is demonstrated. In addition, Fig. 19 shows the results 

of using fourth cycle data as input. Both Fig. 18 and Fig. 19 show 

good correlation of the model with experiment. An interesting observatidn is that the elastic modulus during unloading decreases slightly in the experimental results. This is not taken into account by 

the constitutive model and is the reason for most of the discrepancy 

between experiment and model predictions. 

Cyclic saturation of hastelloy-X. In Chapter 3, there was a discussion concerning cyclic saturation. Recall that cyclic saturation 

is the limiting periodic response in which the stress-strain curve for 

each consecutive cycle is the same as shown in Fig. 5 (p. 28). Although most materials show a gradual hardening during this saturation 

process, some materials may cyclically soften. Microphysically this 

corresponds to the movement of dislocations from a random orientation 

into a cell structure which stabilizes their movement. This results 

in the limiting response of the material. At more elevated temperatures, the saturation effect naturally becomes faster as the mobility 

of the dislocation increases with increasing temperature. Hastelloy-X 

-.-----.--.-~ ~~-----"--L-'---~-'--' ,-. :-.--
58 

CYCLE NO.1 crCL.E NO.2 

20000 

lCOQO 

! oi.;I. ____ -J.L--_________ -+_ £ 

5 

~ 

'll .......... ,"" .. ...- , ~. ! .... ~.... Exp ------

Model 

·~~~OO"" ____ -'-____ 'r-___ .....".--__ _ 

C. ;'11:16 

$fr~!M rtN/Ha 

CYCLE NO.3 

~. 010 

~~;:~.:I'r-_____ ---'----.-----,r----" 

O.om 0.007 0, ~to 0.013 

STRAtN !W/tNt 

.. tOOC~ 

-2'lO00 

-~OO~ ___ ~ __ r-__ ~_' __ ~ ____ ~' ___ ~'_ 

-a.oO'S ·0,C02 0.001 0,00" ~,ry07 0.010 0.01l 

STRRt,. Il~/UlJ 

CYCLE NO.4 

.. 20000 

-30000 

-~oooo 

-&;000,"1,-__ ...,... __ ..-__ .-_--,. __ ,....,..., __ -. 

O.COO O,0~3 o.ua, 0.009 0.012 O,O,S 0.0:8 

srARn. IH-U[tII 

Fig. 19 

steel at 

ratio of 

Comparison of model and experiment for 304 stainless 

lOOO°F using fourth cycle data as input. A hardening 

0.15 in tension and 0.28 in compression was used 

J 

J 

J 

J 

J 

] 

] 

] 

:"J " 
; . , 

r-

, 

displays cyclic saturation as shown in Figs. 20-25. These curves are 

actual experimental results of load vs. diameter change from the MTS 

machine. The abrupt drops in stress at various points are not due to 

/ 

machine problems but are rather due to dynamic strain aging of the 

material. 

The center diagram in Fig. 20 shows the cyclic saturation discussed above. Uniaxial specimens were cycled under strain control 

(diametral strain) between equal tension and compression strain ranges 

until saturation occurred. In order to observe strain rate effects, 

most tests were run at three rates corresponding to test times of 10, 

30, and 300 seconds per quarter cycle. During the test the diameter 

change rate was constant so that actual axial strain rate was variable 

during the cycle; however, the above test times correspond to average 

3 -4 -5 axial strain rates of 1.1 X 10- , 3.67 X 10 ,and 3.67 X 10 

in/in/sec. These rates were chosen because they are typical of strain 

rates seen under normal operating conditions of hot gas turbines. Notice that at room temperature, the rate dependent inelastic strain 

is negligible as the hysteresis loops do not change with different 

diameter change rates. Fig. 21 and Fig. 22 show the results for the 

same strain history at 500°F and 900°F, respectively. To define the 

rate dependent inelastic strain, a reference hysteresis loop is 

utlized as was outlined in Chapter 3. Comparing Figs. 22 and 23, 

it is seen that rate dependence of the saturated hysteresis loops is 

insignificant below 900°F for the rates considered. Consequently, 

the reference temperature is chosen as,900°F. 

59 
i ~ L---

12000 

8000 

4000 

o 

-4000 

-8000 

-12000 

Load, Ibs.· 

Diameter, in. 

::I::::I::::~,::!: ::: ___ t:n~::::L::::::I:: :,+ o. ~~3· j;4-----H~!T ' 

30 secs/~ cycle 100 secs/~ cycle 10 secs/!;' cycle 

Fig. 20 Experimental hysteresis loops for hastelloy-X at room temperature 

L __ _ ,~-~ ~_--.J ~n_.J -'_-:1 - ,1 '----_. ____ cJ ~~--~ 2_-..;:1 

'" o 

~-.::~ 
i -----,-.~ -1 

. , · .• 1 

12000 

Load, 1bs. 

8000 

4000 ----- ---+---------- ._+- ., . I I'! .11 

.:-

o 

, 

\, i 

-4000 -

-8000 

-120001 Diameter, in. 

I~-~----~l- --- 1---

0.3956 

I 

10 secs/!r. cycle 30 secs/!,. cycle 300 secs/!r. cycle 

Fig. 21 Experimental hysteresis loops for hastelloy-X at 500°F 

[~l --- ~-! 

0\ 

...... 

~ .. , 
i~ __ , 

12000 

8000 

4000 -

o -

-4000 

-8000 

-12000 -

, I 

~...: ~ 

"""'~4~~I:."l:'"_LI::::!'~~·~~'~~~Ir;:''''''':-'·='''''"·'---'''''··---''·-7~-· 

Load, lbs. 

" 

0.3984 0.3984 

10 secs/~ cycle 30 secs/~ cycle 

-- 0.001 

Diameter, in. 

0.3984 

, 

300 secs/~ cycle 

Fig. 22 Experimental hysteresis loops for hastelloy-X at 900°F 

'-------' ~'~ L.J L-J L-.--J L-J ~ ~ L...J r'>"~l I":"j ;~"-:,)] ~i::J 

0', 

'''' 

f"', ':'1 
,~ s:;7J :::-_~:~l 

:' .: '-'~..,' 

-,' 

7:.':-~\ .1 -'~-l 

BOOO - Load, lbs. 

6000 

·4000 

2000 

o -

~',' : 

'72000 . '" 

'74000 -

. -6000 -

-BOOO -1- --;~ 0.4229 

\t 

1 

I 

--~~-

---'-, 

, .' J \ ~~--:-.l 

--I: \ 

_ ...Jirr~'" -;-:..a..:~ 

,t.';"~ •• # -_.-

=--t-~~ __ ----r' 

_ •. _.,...- . \ 1 

. , " ~ \ 

Diameter, in. 

0. 4229 

I ·0.4229 -

\ 300 secs/!.r. cycle 

30 secs/!.t. cycle 

Fig. 23 EXPerimental hysteresis loops tor hastelloy-X at 120

0

°£ 

10 secs/~ cycle 

r:-:--\ 

'" w 
6000 - Load, lbs .. I 

i 

~--

4000 

2000 

o 

-2000 

-.4000 

-6000 

, " "---'---':" 

_~_~-:=--....... ' i , ... u~_"''''-_-''''='"'''''·~?''-====- 7""':..-,,--

'~'-'--' 

I . "'1-

I ~. f 0.001 

I 

t- I .' 

--+- --

___ L Diameter, in. 

I . =-j='! I 0.4214 '. "I--! i ---.- .... -.----.. -~ ..: 0.4214 -.---.-----t- ... I .:: .... .... I i 

0.4214 

10 secs/~ cycle 30 secs/~ cycle 300 secs/~ cycle 

Fig. 24 Experimental hysteresis loops for hastelloy-X at l400°F 

l' -: .-"-~' 

'; 

~ t.............: ~ ~ L-J L..J L.:..J ~ L..-J L.:J ;: : .-) 1',~ .' ".1 t··'·:1 

(j\ 

~ 

~~ 
, 

66 [J .,':; 

An interesting phenomenon most clearly evident in Fig. 21 is the ~ 

:',3 

~-'/~. 

:;. . .".~ 

discontinuities or serrations seen in the curves. This is known as dy- ~ .... 

~..:::.: 

namic strain aging and is associated with interactions between moving 

dislocations and solute atoms [37] .. Dynamic strain aging is both tem- J ',,' 

s 

perature and strain rate dependent. 

At l200°F, the rate dependent inelastic strain is no longer 

g :~' " 

,~ 

negligible. Fig. 23 shows the somewhat faster saturation and more J ':. 

~-~ 

"square" shape of the hysteresis loops. These effects are even more 

pronounced at l400°F and l600°F as depicted in Fig. 24 and Fig. 25. -J 

~. 

At these elevated temperatures the rate dependence dominates and can 

be modelled very well with an equation like equation (35). The J 

hys~eresis loops are quite square in nature, and a high rate of strain J 

hardening upon stress reversal is not seen. A critical test of a 

constitutive theory would be in modelling the behavior in Fig. 23 ] 

where both rate independent plasticity and rate dependent creep are 

important. ] 

Theoretical modelling of cyclic saturation of hastelloy-X was ,C'] 

performed for the room temperature case. The results are shown in 

Fig. 26. Experimental stress-strain data is shown only for the first ] 

one-half cycle and the last (saturated) cycle. The model results ::.j 

depict the gradual saturation response as well as the limiting re- "..J 

sponse. A hartlening ratio of S = -0.375~ + 0.075 was used in tension cJ 

and S = -0.5~ + 0.10 was used in compression. Notice that when the 

model predictions approach saturation, some fluctuation is seen on the 

compression side. This is thought to be due to increasing numerical 

error at this point as no equilibrium interations are performed. Most 

j">RECEDING PAGE Bt:At\'K NOT, r:II:.MEO 
S 

T 

R 

E 

S 

S 

I 

p 

S 

I 

I uuGuO 4 

'I 

j 

90000 

BOUOO 

70000 

60000 

50000 

l!,0000 

30000 

20000 

10000 

0 

-10000 

-20000 

-30000 

-ijOOOO 

-50000 

-60000 

-70000 

-80000 

-90000 

100000 

-0.012 

I 

I 

----

I 

I 

I 

'~saturated I 

first cycle ~ 

-------

-0.006 0.000 

STRAIN - IN/IN 

---

experiment 

___ theory 

0.006 0.012 

Fig. 26 Cyclic saturation of hastelloy-X at room t~mperatur~ 

67 
68 

of the difference in experiment and theory is due to difficulty in 

modeling of the second quarter cycle, otherwise the model predictions 

are quite good. 

--. 

" ! ' " 

; 

j 

J ~ ~\ 

] ':c:, 

J ' " 

J 

J 

'] 

J 

l 

_,J 

] 

J 

"1 

,_J 

J 

J 

I 

: _.I 

;-:---; 

, 

r···· 

! 

i .. 

69 

CHAPTER V 

CONCLUSIONS 

An uncoupled, incremental constitutive model for elastic-plastic 

behavior of metals at elevated temperatures has been presented. Revisions to the combined kinematic-isotropic hardening rule allow for 

much improved modelling of cyclic phenomenon. Also, an alternative approach to characterizing the rate dependent inelastic deformation has 

been proposed. 

Successful evaluation of the constitutive model against experiment .has been accomplished for a limited range of tests. The 

B~uschinger effect, change in strain hardening upon stress reversal, 

yield surface evolution, and cyclic saturation are all modeled well 

for the isothermal, rate independent case at elevated temperature. 

Unfortunately, no comparison of the theory to experiment have been 

performed at this time for the load and temperature ranges where rate 

dependence is a significant factor. The need tb utilize an improved 

model like that presented herein depends on the application. For 

example, if the application is to life or fatigue predictions, then 

the detailed modelling of saturation and strain hardening observed at 

low temperatures may not be of utmost importance. The saturated 

hySteresis loop may be adequate to define long term fatigue response. 

Conversely, for low cycle applications, many researchers in this field 

are concerned with exact hysteresis loop predictions and the evolution 

to saturation. In this case, the improved model has important 

I 

i 

I 

i 

t 

I 

! 

t 

------_._\, l 
70 

,applications. Finally, in regards to the rate dependence of the model, 

experimental results seem to indicate that the uncoupling presented in 

Chapter 3 gives acceptable predictions, but further tests are needed 

to verify the model. 

'~:' i,'.' 

~£ 

," 'Q' - ' ,~ , 

] 

J 

J 

J 

] 

J 

';1 

-.I 

, ~'I 

J 

'----1 

.' 
REFERENCES 

1. Tresca, H., "Notes on Yield of Solid Bodies Under Strong Pressure," 

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

2. Hill, R., The Mathematical Theory of Plasticity, Oxford University 

Press, London, 1950. 

71 

3. von Mises, R., "Mechanik der Festen Koerper in Plastisch Deformablen 

Zustant," Goettinger Nachr., Math-Phys. Kl., 1913, pp. 582-92. 

4. Prandtl, L., "Spannungsverteilurg in Plastischen Koerper," Proceedings of the First International Congress on Applied Mechanics, 

Delft, Technische Boekhandel en Druckerij, Jr. Waltman, Jr., 1925, 

pp. 43-45. 

5. Reuss, E., "Bereuchsichtigung der Elastischen Formaenderungen in 

der Plastizitaetstherie," Zeitschrift fuer Angewandte Mathematic 

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

6. Drucker, D. C., "A Definition of Stable Inelastic Mat~rial," 

Journal of Applied Mechanics, Vol. XVII, 1959, pp. 55-65. 

7. Prager, W., "The Theory of Plasticity: A Survey of Recent Achievements," Proceedings of the Institution of Mechanical Engineers, 

London, Vol. 169, 1955, pp. 41-57. 

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

of Applied Mechanics, Vol. XVII, 1959, pp. 55-65. 

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

'Temperat~re with a Functional Theory of Viscoplasticity," presented 

at ASME Pressure Vessels Conference, San Francisco, August 12, 

1980. 

10. Allen, David H., and Milly, Teresa M., "A Comparison of Constitutive 

Models for Nonlinear Rnte Dependent Material Behavior of Solids," 

Virginia Polytechnic Institute and State University Report No. VPIE-80.16, Blacksburg, Virginia,. Sept. 1981. 

11. Krieg, R. D., Swearengen, J. C., and Rohde, R. W., "A PhysicallyBased Internal VAriable Model for Rate-Dependent Plasticity," 

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

12. Allen, David .H., "A Model for Predicting Response of Nonlinear 

Materials Subjected to Thermomechanical Loading," Dissertation, 

Texas A&M University, Aug. 1980. 

13. Hunsaker, B. Jr., "An Evaluation of Four Hardening Rules of the 

Incremental Theory of Plasticity, " Thesis, Texas A&M University 

Dec. 1973. 

____ -. . _ .. _ .. ' ________ . _______ . _______________ . ______ ._ .. ____ .. ______ ', __ ~. ____ -=-. ____ ~ ____ ~ ___ ~._. ,_r_" _______________ , _.., __ . ___ .': ____ . ___ • ____ - ___ . _____ ~ __ --~-- ~,-"- _. __ r~--~·--· ~--,--_. -,--...... - "'7'- -,' 

i 

l 

! 

I 
72 J 

14. Polakowski, N. H., and Ripling, E. J., Strength and Structure of 

Engineering Materials, Prentice-Hall, Inc., Englewood Cliffs, 

N.J., 1966. 

15. Bodner, S. R., Partom, I., and Partom, Y., "Uniaxial Cyclic Loading of Elastic-Viscoplastic Materials," Journal of Applied 

Mechanics, Vol. 46, December 1979, pp. 805-810. 

16. Miller, A. K., "Modelling of Cyclic Plasticity with Unified Constitutive Equations: Improvements in Simulating Normal and 

Anomalous Bauschinger Effects," Journal of Engineering Materials 

and Technology, Vol. 102, April 1980, pp. 215-222. 

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

Journal of Engineering Materials and Technology, Vol. 98, 

January 1976, pp. 76-85. 

18. Walker, K. P., "Research and Development Program for Nonlinear 

Structural Modelling with Advanced Time - Temperature Dependent 

Constitutive Relationships," First Quarterly Technical Narrative, 

Pratt & Whitney Aircraft Group Report No. PWA-5700-6, January 1980. 

19 .. Allen, David H., "Course Notes for MM641 - Plasticity Theory," 

Texas A&M University, unpublished, Summer 1980. 

20. Petersson, Hans, and Popov, Egor P., "Constitutive Relations 

for Generalized Loadings," Journal of the Engineering Mechanics 

Division, ASCE, Vol. 103, No. EM4, Aug. 1977, pp. 611-27. 

21. Popov, Egor P., and Petersson, Hans, "Cyclic Metal Plasticity: 

Experiment and Theory," Journal of the Engineering Mechanics 

Division, ASCE, Vol. 104, Dec. 1978, pp. 1371-87. 

22. Snyder, M. D., and Bathe, K. J., "Formulation and Numerical Solution of Thermo-Elastic-Plastic and Creed Problems," National 

Technical Information Service, No. 82448-3, June 1977. 

23. Yamada, Y., "Constitutive Modelling of Inelastic Behavior of 

Numerical Solution of Nonlinear Problems by the Finite Element 

Method," Computers and Structures, Vol. 8, 1978, pp. 533-43. 

24. Yamada, Y., and Sakurai, T., "Basic Formulation and a Computer 

Program for Large Deformation Analysis," Pressure Vessel Technology, Part I, ASME, 1977, pp. 341-52. 

25. Allen, David H., and Haisler, Walter W., "A Theory for Analysis 

of Thermoplastic Materials." Computers and Structures, Vol. 13, 

1981, pp. 124-35. 

26. Zienkiewicz, O. C., The Finite Element Method, McGraw-Hill, 

London, 1977. 

l ""I' 

c:'1 

] 

J 

J 

] 
27. Fung, Y. C., Foundations of Solid Mechanics, Prentice-Hall, Inc., 

Englewood Cliffs, N.J., 1965. 

73 

28. Hunsaker, B. Jr., "The Application of Combined Kinematic-Isotropic 

Hardening and the Mechanical Sub layer Model to Small Strain Inelastic Structural Analysis by the Finite Element Method," 

Dissertation, Texas A&M University, August 1976. 

29. Allen, David H., "A Note on the Combined Isotropic-Kinematic 

Work Hardening Rule," International Journal for Numerical Methods 

in Engineering, Vol. 15, 1980, pp. 1724-28. 

30. Pugh, C. W., Corum, J. M., Lin, K. C., and Greenstreet, W. L., 

"Currently Recommended Constitutive Equations for Inelastic 

Design Analysis of FFTF Components," ORNL TM-3602, Oak Ridge 

National Laboratory, Oak Ridge, Tenn., Sept. 1972. 

31. Dafalias, Y. F., and Popov, E. P., "Plastic Internal Variables 

Formalism of Cyclic Plasticity," Journal of Applied Mechanics, 

Vol. 43, Dec. 1976, pp. 645-51. 

32. 'Caulk, D. A., and Naghdi, P. M., "On the Hardening Response in 

Small Deformation of Metals," Journal of Applied Mechanics, 

VoL 45, December 1978, pp. 755-64. 

33. Swindeman, R. W., and Pugh, C. E., "Creep Studies on Type 304 

Stainless Steel (Heat 8043813) Under Constant 'and Varying Loads," 

ORNL-TM-4427, June 1974, Oak Ridge National Laboratory, Oak Ridge, 

Tenn. 

34. Hais1er, W., "Application of an Uncoupled Elastic-Plastic Creep 

Constitutive Model to Metals at High Temperatures," to be presented at the Symposium on Nonlinear Constitutive Relations for 

High Temperature Applications, The University of Akron, Akron, 

Ohio, May 19 and 20, 1982. 

35. Bradley, W. L., "A New Uncoupled, Viscop1astic Constitutive 

Model," to be presented at the Symposium on Nonlinear Constitutive 

Relations for High Temperature Applications, The University of 

Akron, Akron, Ohio, May 19 and 20, 1982. 

36. Allen, David H., "Computational Aspects of the Nonisothermal 

Classical Plasticity Theory," Virginia Polytechnic Institute and 

State University Report No. VPI-E-80.29, Blacksburg, Virginia, 

Oct, 1980. 

37. Reed-Hill, Robert E., Physical Metallurgy Principles, D. van 

Nostrand Company, New York, 1964. 

; ; y 

74 

APPENDIX 

COMPUTER PROGRAM OUTLINE 

" ,. 

The following outline describes a basic flowchart of the uniaxial ; 

constitutive equations for a given total strain increment. Subincrementation (forward integration) with no equilibrium iterations is the 

solution technique used. Note that in the setup of any problem, the 

yield surface size and equivalent uniaxial stress vs. plastic strain 

diagrams are initialized using the first set of input stress-strain 

curves. 

A. Set up subincrementation on strain, temperature, and time increments. 

B. Compute elastic modulus and the ch"ange in the elastic modulus due 

to temperature increment. Linearlly interpolate between input 

values if necessary. 

C. Compute thermal strain increment. Linear interpolation may be 

necessary to obtain thermal expansion coefficient. [equation (51)] 

D. Compute creep strain increment based on stress at beginning of 

step and temperature at end of step. [CHAPTER III - Creep 

strain increment] 

E. Compute trial stress increment assuming elastic behavior. Add 

this to the str~ss at beginning of step to obtain total stress. 

F. Check for yielding 

1. Compute yield function. [equation (34)] 

2. Compute yield surface size for current value of equivalent 

- _ --_ --- •. _- --~~ ~ --i'"'-~-- ~----~--.'-~ ________ ' ___ -.~ --.~-..... ____ ,. ___ ~.-_ ~---.- ----'" __ '_'~'_'_,._ --;-~ --_. ---. ___ r -; ~ __ ' _____ r _~ •• 
75 

uniaxial plastic strain at temperatures at start and end of 

load step. 

3. Compare yield function value with current yield surface size. 

a. If elastic, go to step N'. 

b. If yielded and last step was elastic-plastic go to step J. 

c. If yielded and last step was elastic, go to step G. 

G. Update stress and strain to yielded portion for transition step. 

[equations (46) and (47)] 

1. Subtract assumed elastic stress increment from total since this 

step is elastic-plastic. 

2. Compute sand n factors. [equations (48) and (50)] 

3. Update total strain, elastic strain, and creep strain to yield 

surface. 

H. Compute creep strain increment based on yield stress and tempera-. 

ture at end of step. 

I. Modify global K and a vs. ~ diagrams. [CHAPTER III - Construction of K and 0 vs. ~ diagrams1 

1. If the hardening ratio is a function of plastic strain, recompute the base K and cr vs. €p diagrams. [equations (39) -

(41)] 

2. If this is the initial yield or yield has occurred without 

a stress reversal, no modification is necessary. Go to 

step J. 

3. If reverse yielding has occurred, modify the global diagrams. 

The particular material of interest determines whether the 

first or second base curves are used. 

1 

'l 

• •• _J 

···l 

I ... ! 

] 

-'1 

.J 

-I 

I '-_ J 

---1 

I 

I •• j 

I 

I 

i 

~ i 

, 

l 

.. 

~' 

i: 

I , 

i 

r 

1 

I 

f ---(I 
l .... 

; ,.' 

J. Experiencing elastic-plastic behavior. 

1. --P Compute aF/ao and the gradients H', ao/aT, aK/aE , aK/aT 

for the current value of plastic strain and temperature. 

[equations (42) - (45)) 

K. Compute stress increment. [equation (29)J 

In transition step from elastic to elastic-plastic behavior, only 

the leading term of equation (16) is used. 

L. Compute strains and update totals. 

1. Elastic strain. [equation (51)] 

2. Plastic strain increment is the total minus the elastic, 

thermal, and creep increments. 

3. Equivalent uniaxial plastic strain is the absolute value of 

the plastic strain increment for the uniaxial case. 

M. Update yield surface center. In transition step the yield surface 

translation scalar is calculated assuming isothermal behavior for 

temperature at end of step. 

1. Compute translation scalar. [equation (32)] 

2. Compute change in yield surface center. [equation (6)J 

3. Update yield surface center. 

N. Update values to end of subincrement. 

1. Stress. 

2. Total strain. 

3. Creep strain. 

4. Thermal strain. 

N' . Update values to end of load step for elastic case. 

1. Stress. 

76 

------.-------- --.-- ---_._----------- -._---_. -- ---.. -----._--_._---- --- .. -.- _.-._ .. _------... --_ .. _----- "'- .. ~ 
Yuen, S., "Development of an Uncoupled Viscoplastic Constitutive Model 

for Cyclic Plasticity of Hastelloy X at High Temperature," Thesis, Texas 

A&M University, College Station, Texas, May 1983. 
77 

2. Total strain. 

3. Elastic strain. 

4. Creep strain. 

5. Thermal strain. 

o. Repeat above procedure for each subincrement. 

i • I 

I 

~, : 

l 

I 

_.J 

·,1 

I 

j 

.. .1 

! 

.. , 

I 

! o..:_J 

i 

~' .. 

. . ,'," 

----" 

DEVELOPHENT OF AN UNCOUPLED, VISCOPLASTIC 

CONSTITUTIVE MODEL FOR CYCLIC PLASTICITY 

OF HASTELLOY-X AT HIGH TE}~ERATURE 

A Thesis 

by 

SHIK HUNG YU&'1 

Submitted to the Graduate School of 

Texas A&M University 

in partial fulfillment of the requirement for the degree of 

MASTER OF SCIENCe 

May 1983 

Major Subject: Mechanical Engineering 

--, ------- ---'-- -._- .---~-~~--.:~---.~----'~--.~-.'------ '~--'----- ---. :... .. -

.' ." : \ ! 

-,: 

,-I 

. " 

, 

DEVELOPMENT OF AN UNCOUPLED, VISCOPLASTIC 

CONSTITUTIVE HODEL FOR CYCLIC PLASTICITY 

OF HASTELLOY-X AT HIGH TEMPERATURE 

A Thesis 

by 

SHIK HUNG YUEN 

Approved as to style and content by: 

"lkdbz4~ Walter L. Bradley 

(Chairman of Committee) 

Walter E. Haisler 

(Member) 

Nay 1983 

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ABSTRACT 

Development of an Uncoupled, Viscoplastic 

Constitutive Model for Cyclic Plasticity 

of Hastelloy-X at High Temperature. (Nay 1983) 

Shik Hung Yuen, B.S., Texas A&M University 

Chairman of Advisory Committee: Dr. Walter L. Bradley 

An uncoupled viscoplastic model is presented based on the assumption that inelastic strain can be partitioned into rate insensitive 

and rate sensitive components. Unlike earlier uncoupled theories. 

this model recognizes that some of the inelastic strain which occurs 

during a load change is rate sensitive (or thermally activated). The 

rate insensitive. inelastic strain is defined in terms of a strain 

hardening function which is determined empirically from saturated 

hysteresis loops at room temperature. The rate sensitive inelastic 

strain is characterized using a standard viscoplastic model. The 

physical basis for the model and the experiments required to evaluate 

the material constants are also presented. Evaluation of the 

model is performed by comparing predicted hysteresis loops and stress 

relaxation behavior with experimental observation. The question of 

inelastic strain-rate continuity is addressed in terms of the 

uncoupled viscoplastic model as well as models developed by other 

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researchers . Finally, suggestions are made for further investigation. 

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ACKNOWLEDGEHENTS 

The financial support of this work by NASA Lewis Research Cent~r 

is gratefully acknowledged. I also gratefully acknowledge Dr. Walter 

Bradley for his support and guidance throughout this research effort. 

I give special" thanks to Dr. Walter Haisler for his help in this work. 

I would also like to thank Dr. Richard Schapery for kindly consenting 

to join my committee. Host of all. I appreciate the endless support 

and encouragement from my parents throughout my college career. 

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TABLE OF CONTENTS 

ABSTRACT .••• 

ACKNOWLEDGEMENTS • 

TABLE OF CONTENTS 

LIST OF TABLES • 

LIST OF FIGURES 

CHAPTER I. INTRODUCTION 

CHAPTER II. REVIEW OF EXISTING THEORIES • 

Present Status 

Available Constitutive Theories 

Micromechanistic theories 

Nonlinear viscoelastic theories 

Classical plasticity theories 

Uncoupled vs. Unified Theories 

Objectives 

CHAPTER III. CONSTITUTIVE MODEL DEVELOPMENT 

Physical Basis 

Description of Constitutive Model 

Experimental Data Requirements 

Numerical Scheme for Theoretical Prediction 

CHAPTER IV. EXPERIMENTAL PROCEDURE 

Material Description 

Specimen Preparation 

Experimental Set Up 

Description of Experiments 

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vi 

Page 

iii 

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vi 

viii 

ix 

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4 

4 

5 

5 

6 

8 

10 

11 

13 

13 

17 

23 

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31 f 

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31 I 

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31 I 

34 

39 
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Fully reversed, strain cycling experiments 

(hysteresis loops) • 

Stress relaxation tests 

Strain drop, stress transient tests 

Strain-rate cycling tests 

CHAPTER V. EXPERUiENTAL RESULTS • ANALYSIS AND DISCUSSION 

Data Reduction 

Hysteresis Loop Results 

Stress Relaxation Behavior 

Material Constants and Their Physical Significance 

CHAPTER VI. EVALUATION OF THE MODEL .•• 

Justifying the Uncoupled Inelastic Strain Model 

Comparison of Experiments with Theoretical Predictions 

Saturated hysteresis loops • 

Stress relaxation behavior 

CHAPTER VII. CONCLUSIONS 

CHAPTER VIII. RECOMMENDATIO.N FOR FUTURE WORK 

REFERENCES • 

APPENDIX A. 

APPENDIX B. 

APPENDIX C. 

APPENDIX D. 

VITA ••.. 

COMPUTER PROGRAM OUTLINE FOR 

HYSTERESIS LOOP SIMULATION 

COMPUTER PROGRAM OUTLINE FOR 

STRESS RELAXATION PREDICTION 

DERIVATION OF EQUATIONS USED FOR DATA REDUCTION 

VALUES FOR RATE INSENSITIVE, INELASTIC 

STRAIN FUNCTION • • 

vii 

page 

39 

40 

42 

44 

46 

46 

. . 47 

64 

76 

92 

92 

101 

101 

110 

121 

123 

125 

130 

132. 

133 

136 

142 
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LIST OF TABLES 

TABLE 

I Chemical Composition of Hastelloy-X • 

II Values for Back Stress (n) (for cr = cr ). max 

III 

IV 

--

n 

n. Drag Stress (K) and K •• 

Value.s for Young's Modulus and Poisson's 

Ratio at Different Temperatures : 

Inelastic Strain-rate Continuity at 

Various Temperatures .. • • • • 

'--->--~--'--.------'---'---'--"---'-------.- ,---~._-. 

viii 

page 

32 

77 

78 

97 

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LIST OF FIGURES 

Single nonlinear Maxwell element .• 

Schematic of stress-strain behavior at three 

temperatures ••.•• 

Experimental characterization of the three 

strain components for the uncoupled 

viscoplastic model 

Specimen geometry 

Experimental set up 

Schematic of experimental set up • 

Schematic for the strain drop, 

stress transient test 

Schematic for the strain-rate cycling test 

Saturated hysteresis loops for Hastelloy-X at room 

temperaturewith strain amplitudes of 0.05%, 

0.1%, 0.2%. 0.3%. 0.4%. 0.5%. 0.6% 0.8% and 1.0% 

Saturated hysteresis loops for 

Hastelloy-X at room temperature 

Saturated hysteresis loops for Hastelloy-X at 533K 

Saturated hysteresis loops for Hastelloy-X at 775K. 

(a) Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain . 

Saturated hysteresis loops for Hastelloy-X at 811K. 

(a) Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain . 

Saturated hysteresis loops for Hastelloy-X at 866K. 

(a) Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain . . 

Saturated hysteresis loops for Hastelloy-X at 922K. 

(a) Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain . 

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16 Saturated hysteresis loops for Hastelloy-X at 978K. 

(a) Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain · · 59 

17 Saturated hysteresis loops for Hastelloy-X at lO33K. 

(a) Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain · · 60 

18 Saturated hysteresis loops for Hastelloy-X at 1089K. 

Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain · · 61 

19 Saturated hysteresis loops for Hastelloy-X at 1144K. 

(a) Stress vs. total strain. (b) Stress vs. rate 

dependent (or rate sensitive) inelastic strain 62 

20 Stress relaxation tests for Hastelloy-X at room 

temperature executed at various points on 

saturated hysteresis loops · · · · · · . . · · · · 65 

21 Stress relxation tests for Hastelloy-X at 755K 

executed at various points on saturated 

hysteresis loops · · · · · · · · · · · · · · 66 

22 Stress relaxation tests for Hastelloy-X at 8llK 

executed at various points on saturated 

hysteresis loops · · · · · · .. · · · · · · · · 67 

23 Stress relaxation t~~ts-for Bastelloy-X at 866K 

executed at various points on saturated 

hysteresis loops · · · · · · · · · · · · 68 

\ 

24 Stress relaxation tests for Hastelloy-X at 922K , 

I executed at various points on saturated I 

hysteresis loops 69 I 

· · · · · · · · · · · · · · · j 

I 

25 Stress relaxation tests for Hastelloy-X at 978K i 

executed at various points on saturated 

hysteresis loops · · · · · · · · · · · · 70 

26 Stress relaxation tests for Hastelloy-X at 1033K 

executed at various points on .saturated 

'. hysteresis loops · · · .. · ... ..' .. " · · · · · · · · 71 , ~ " 

... ~ . .; ;_.' 

.. 27 Stress relaxation tests for Hastelloy-X at 1089K , 

.. . ~ i executed at various points on saturated ," hysteresis loops · · · · · · . . .' -- . ., 72 " 

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35 

36 

37 

38 

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42 

43 

Stress relaxation tests for Hastelloy-X at 1144K 

executed at various points on saturated 

hysteresis loops . • • . . • • . • 

Effective stress vs. rate sensitive, inelastic 

strain-rate at 1144K for a total strain-rate 

of lXlO-3s-1 • • . . • 

Stress and back stress as a function of 

strain for Hastelloy-X at 755K · · · · 

Stress and back stress as a function of 

strain for Hastelloy-X at 8IlK · · · · 

Stress and back stress as a function of 

strain for Hastelloy-X at 866K · · 

Stress and back stress as a function of 

strain for Hastelloy-X at 922K · · 

Stress and back stress as a function of 

strain for Hastelloy-X at 978K · · · · · 

Stress and back stress as a function of 

strain for Hastelloy-X at 1033K · · 

Stress and back stress as a function of 

strain for Hastelloy-X at 1089K · · · 

Stress and back stress as a function of 

strain for Hastelloy-X at 1144K · · 

Experimental and simulated hysteresis 

loops for Hastelloy-X at 755K · · · · 

Experimental and simulated hysteresis 

loops for Hastelloy-X at 81lK · · 

Experimental and simulated hysteresis 

loops for Hastelloy-X at 866K · · · 

Experimental and simulated hysteresis 

loops for Hastelloy-X at 922K · · 

Experimental and simulated hysteresis 

loops for Hastelloy-X at 978K · 

Experimental and simulated hysteresis 

loops for Hastelloy-X at 1033K · . · · 

· 

· 

· 

· 

· 

· 

· 

· 

· 

· 

· 

xi 

page 

73 

75 

· · · · 80 

· · · 81 

· · · · 82 

· · · · 83 

· · · · 84 

· · · · 85 

· · · 86 

87 

· · 102 

· · · · 103 

· · · · 104 

· 105 

· 106 

· · 107 

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-- " 44 Experimental and simulated hysteresis 

loops for Hastelloy-X at 10B9K . . . . . · · · · lOB 

45 Experimental and simulated hysteresis 

loops for Hastelloy-X at 1144K • . . . . . · · · · 109 

46 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at 755K · · · · III 

47 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at BllK · · · · lIZ 

48 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at 866K · · · · 113 

49 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at 922K · · · · 114 

50 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at 97BK · · · · as 

51 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at 1033K · · · · 116 

52 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at lO89K · · · · 117 

53 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at 1144K · · · · 118 

54 Experimental hysteresis loops for Hastelloy-X 

at 92ZK. Points A and B indicate where 

stress relaxation starts and ends. Point C 

indicates the flow stress upon reloading . . · · · · 119 

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1 

CHAPTER I 

INTRODUCTION 

---With the-tncreasing demand- for1I!aterials to function at elevated 

temperatures, an· understanding of material behavior under such conditions is essential. Structural components in the hot section of a 

turbine engine experience a complex thermal-mechanical history, often 

leading to premature failure. Similar problems are encountered in 

nuclear reactors. Components in critical areas are subjected to cyclic 

loading conditions and possible damage due to creep-fatigue interaction. 

To obtain design parameters under actual service conditions is inpractical in most cases due to the cost and time involved. Therefore, 

viscoplastic models must be developed to simulate material behavior 

under such conditions. 

Historically. the first experimental study on "viscoplasticity" 

was conducted by Vicat [1] in 1834. He found that at room temperature. 

iron wires loaded with a stress equal to one half the breaking stress 

exhibited a gradual increase in length. This is the first indication 

of the existence of rate dependent inelastic deformation in metals. 

which is now referred to as creep. 

Despite the fact that the first study of rate dependent inelastic 

deformation was performed more than a century ago, early plasticity 

theories were mainly concer~ed with rate independent inelastic 

This thesis follows the style and format of the Journal of 

Aoplied Mechanics . 

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deformation. (For a historical review of plasticity theories one ,is ! 

referred to books by Hill [2] and Martin [3]). 

In recent years, the need to apply materials at elevated temperatures necessitates the inclusi~n of rate effects in plasticity studies. 

Considerable research efforts have been undertaken to extend classical 

plasticity theories to model both rate and temperature dependent 

phenomena [4-9]. These modifications are accomplished by introducing 

an uncoupled rate dependent inelastic strain component as well as 

temperature dependence of material properties. Constitutive models of 

this nature are often referred to as uncoupled theories. 

More recently, there has been increasing attention toward the 

development of viscoplastic models which are consistent with microphysical observation. Research efforts in this area are reflected by 

the proposal of a number of unified theories [10-22], many of which 

are still undergoing active development. The major feature of these 

theories is -that all inelastic strain is considered to be rate dependent. Although physically appealing, many unified models fail to 

accurately predict cyclic behavior at lower temperatures. 

This thesis will proceed in the following manner. First, a 

review of the existing theories will be presented along with a comparison of the uncoupled and unified theories. This will be followed 

by a presentation of the uncoupled viscoplastic model and the physical ':- ". 

basis for this new model. A description of the various experiments used 

to evaluate the required material constants will be provided. Experimental results will be compared with theoretical predictions and the 

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question of inelastic strain-rate continuity will be discussed. 

Finally, suggestions for future work will be outlined . 

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CHAPTER II 

REVIEW OF EXISTING THEORIES 

Present Status 

As discussed in the previous chapter, the value of a constitutive 

model lies in its capability to predict complex thermomechanical 

histories. To satisfy such requirement, a model has to account for 

various thermally induced elastic and inelastic strains for cyclic 

loading as well as creep and stress relaxation responses at constant 

loads or strains. Furthermore, time and temperature dependent microstructural changes such as precipitation, grain growth, recrystallization, segregation, etc. have to be considered. Given the complexity of 

the problem, a general constitutive law which can fully describe all cf 

the above phenomena is simply not feasible. In this research effort, 

we will specifically deal with cyclic plasticity and stress relaxation 

behavior of Hastelloy-X at various strain-rate and temperature ranges. 

Three common approaches utilized in constitutive modelling 

include: (1) micromechanistic theories; (2) nonlinear viscoelastic 

theories; (3) classical incremental plasticity theories. The first 

approach has generally been favored by material scientists who attempt 

to characterize material behavior through microstructural considerations. The other two approaches are widely used in the mechanics 

community. Constitutive models are usually developed on the basis 

of continuum mechanics. Existing constitutive theories can also be 

classified as being "uncoupled" or "unified" depending on the way the 

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inelastic strain is modelled. 

In the following section, the three different approaches in constitutive modelling will be discussed along with a comparison of 

uncoupled and unified theories. 

Available Constitutive Theories 

~licrotilechanistic theories. The thrust of the micromechanistic 

·theories is to create a model which relates macroscopic parameters 

such as stress, strain-rate and temperature to underlying micromechanisms. It is generally accepted that inelastic deformation is a 

result of dislocation movements and interactions within a crystalline 

material. Constitutive laws are therefore based on various models of 

dislocation kinetics. For example, strain hardening is associated 

with the interactions of the long range stressfi~ldsof dislocations 

[23]. On the other hand, recovery can be explained in terms of thermally activated dislocation climb and cross-slip [24]. 

An example of a micromechanistic model is provided by Kreig, et 

ale [111. The constitutive model takes the form of a power law 

function. The current microstructure is reflected by a kinematic 

state variable and an isotropic internal variable, which allows the 

model to predict combined isotropic/kinematic response. Evolutionary 

laws are developed for the internal variables to account for strain 

hardening and recovery. Since recovery is assumed to be governed by 

dislocation climb, the constitutive model is limited to relatively 

high homologous temperatures. As pointed out by the authors, a short 

coming of the model is that predictions of stress-st:rain behavior are 
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too "square" compared with experimental observation. 

Another example is found in a unified model developed by Hart 

[12]. An important feature of the theory is that the inelastic 

s-frain is 'se'par-ated into recoverable (anelastic) and unrecoverable 

(plastic) components. The anelastic strain component is considered 

to be a state variable. The 'plastic strain, on the other hand, is 

governed by a number of internal variables. One state variable is 

called the "hardness" and represents isotropic strain hardening. 

Thermal recovery at high temperature is accounted for by modifying 

the "hardness" term with another function. A second state variable 

is used to represent the direction as well as magnitude of prior 

stress history and accounts for kinematic hardening. The model is 

based on the assumption of an isotropic material and since the 

internal variables are not functions of temperature, it can only be 

used under isotropic conditions. 

Other examples of micromechanistic theories are given by Bodner 

and 'Part om [13]. and Miller [14, 15]. 

Nonlinear viscoelastic theories. Nonlinear viscoelastic 

theories are generally constructed within a thermodynamic framework 

[25. 26] ,based on the First and Second Laws of Thermodynamics. These 

theories have been reported in both single integral and multiple 

integral form as well as differential equation form. In addition to 

thermodynamics, physical. constraints must be applied to obtain a 

meaningful constitutive model. 

One example is a viscoplastic model credited to Walker [161. It 

is developed in an integral form by modifying the constitutive 

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relation for a standard three parameter viscoelastic solid. The 

model employs two state variables, the equilibrium stress and the 

drag stress. The equilibrium stress represents kinematic hardening 

and is governed by a "'growth law-which contains both static and thermal 

recovery terms. The drag str,ess, on the other hand, represents 

isotropic hardening. In the original formulation, the drag stress was 

assumed to be constant and cyclic hardening and softening behavior was 

described in terms of changes in the back stress term. The drag 

stress term has recently been modified (17) to include a dynamic 

recovery term so that both internal variables can be used to describe 

cyclic hardening response. The evolutionary law for the back stress 

contains a term which allows the back stress to change with temperature during nonisothermal elastic excursions. The shortcomings of 

the model include its failure to accurately predict stress-strain 

behavior at relatively high strain-rates and the "over squareness" of 

its predictions in the case of cyclic response at small strain-rates. 

Another example is found in a model proposed originally by 

Schapery [27] and later·amplified by Valanis [18). It is an endochronic 

theory and does not require a yield surface. The curren-t- state of stress is 

expressed as a function of the strain history defined with respect to 

a "time scale" which is itself a material property. The model has 

been shown to be capable of describing a number of elastic-plastic 

phenomena (19). The endochronic theory has recently been modified by 

introducing the concept of "internal barriers" [20). It is proposed 

that an internal variable will not become activated until an 

activation barrier is overcome. Each internal variable is 

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"autonomous" in the sense that it is governed by its own time scale - -

--' 

which is defined by the internal variable itself. This version of 

the endochronic theory is termed an auto chronic theory. The theory 

is used mainly in a rate independent form under isothermal conditions 

and has not been shown to be adequate in modelling material behavior 

under cyclic load conditions. Efforts have been made by Wu and Chen 

[28] to include strain-rate effects in the model. 

Other examples are presented by Cernocky -and Krempl [21], Lui and 

Krempl [29], and Schapery [27]. A more detailed discussion of nonlinear viscoelastic theories is provided by Allen [30]. 

Classical plasticity theories. Classical plasticity theories are 

based on the concept of a yield surface. Various plastic strain 

histories can be described in terms of the expansion, translation and 

rotation of the yield surface. These theories are macrophenomenological in the sense that state variables are derived from experimental 

data without reference to the micromechanisms involved. 

Incremental plasticity theories are generally characterized by 

the following features: (1) a yield function distinguishing elastic 

and plastic deformation, (2) a relationship between the stress increment tensor and the strain increment tensor, (3) a description of 

the rate independent plastic strain increment and (4) a hardening 

rule which describes the evolution of the yield surface during plastic flow. 

. . : 

There has been an enormous amount of research in this category . 

A majority of the work is related to the development of various 

hardening rules to describe different loading histories [31]. Rate 
·"' . 9 

,> • ' 

independent plasticity theories are well established and recent efforts 

involve the extension of these theories to include rate and thermal 

.. ' /'" . effects. Recent research efforts in this area have been reviewed by 

Allen (32, 33]. Of the classical plasticity theories discussed in 

reference (32]. a more promising one belongs to Yamada and Sakurai 

(34. 35]. It includes temperature dependence of material properties, 

an uncoupled rate dependent inelastic strain component. and a combined hardening rule to account for the Bauchinger effect. However. 

the mathematical consistency of the model is questioned [32]. Another 

example is given by Snyder and Bathe [5]. The model is an extension 

of classical plasticity and has been shown to be quite accurate in 

modelling monotonic loadings. Since the model is a single state 

variable theory, it is inadequate to model combined isotropickinematic hardening under cyclic loading conditions. One other 

example is a model proposed by Sharifi and Yates [7], which accounts 

for creep. but d~es not include temperature dependence of material 

properties. 

One of the most comprehensive models is presented by Allen.and 

Haisler (6]. Their theory is an uncoupled. incremental. nonisothermal constitutive model based on the classical theory of plasticity. 

The theory is capable of modelling rate and thermal effects, the 

Bauchinger effect and can be extended to model multiaxial loading 

conditions. Recent addition of a revised hardening rule [36J 

results in improved capability of the theory to model cyclic 

behavior under various thermomechanical histories. 

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Uncoupled vs. Unified Theories 

As discussed in the opening chapter, two distinct approaches 

exist in constitutive modelling with respect to the treatment of the 

inelastic strain component. This results in the development of the so 

called "uncoupled" and "unified" theories. The unified approach is 

relatively new but has gained considerable support in the past few 

.yaars. .,It - is ,t,heraf.ore :worthwhile at. this point to discuss some of 

the advantages as well as drawbacks of the two approaches. 

The uncoupled approach separates the total inelastic strain into a 

rate independent component and a rate dependent component. An example 

'. of an uncoupled theory is one proposed by Strauss [37, 38) to model 

"creep" and "plasticity", effects in viscoplastic' materials. The model 

is termed a bilinear functional theory since the functional is linear 

in the rate dependent part as well as the rate independent part of 

the model. Other examples can be found in the preceding section under 

the category of classical plasticity theories. The uncoupled theories 

are widely used today in finite element structural analysis. In a 

recent review, Krempl [39) questioned the concept of "time independence" as a valid approach to describe material behavior at low homologous temperature since rate sensitivity is observable even at room 

temperature. Researchers like Walker [16], Miller [15] and Hart [12J 

criticized the uncoupled approach ,on the basis that creep and plas- , 

~ , 

ticity are controlled by the same physical processes. The uncoupled 

approach is further criticized for not taking into consideration the 

interaction between creep and plasticity. It is pointed out by 
,.', 

. . , 

,'':" . 

" .. " ," . ,~- -. . ..: ' .. ", ~ .' ~ 

'.' 

'., ".:"! 

.... ' -

. . . _.' _.~~. ___ ~L __ ~. __ --'- __ • __ , • __ ,. , •• 

11 

Krempl [40] that uncoupled models obtain their time dependent part 

from constant load creep tests and therefore are inadequate in 

modelling variable loading conditions. 

.,. __ Recognizing the above shot.:tc.omings. a number of unified models 

[10-22] have recently been proposed which consider all inelastic 

strain as rate dependent. Although physically satisfying, the unified approach is not without drawbacks. In modelling cyclic response, 

for example. predictions made by the unified models are often too 

"square" compared with experimental observation [11.15, 17]. In 

many cases, the evolution of the state variables is difficult to 

mQdel or experimentally characterize. Furthermore, unified models 

produce "stiff" .differential equations which often result in computational diffic.ulties. 

To summarize, neither the uncoupled nor the unified approach has 

shown clear advantage over the other. Since many unified models are 

still in a developing stage, it is possible that some of the problems 

discussed above will be overcome eventually. For the near future, 

however, it appears to be more advantageous to rely on an uncoupled 

theory whose form is consistent with known micromechanistic models. 

Objectives 

The objective of this research is to present an uncoupled 

viscoplastic model to characterize cyclic behavior of Hastelloy-X 

under different strain-rate and temperature conditions. The rate 

insensitive inelastic strain component is defined using hysteresis 

loops at room temperature. The rate sensitive inelastic strain 

! .~. __ ., --- .-- ,~--~. ~---:·----;--,·:--~-:-·;';'-':':7:,-':'-\--: -'"7 -.-- -- -..:.-.~~~;: ~:.~ -' 

.. ~,: .. :'" ~ ~·.·-_---:'-~"---:-:·:·-··--'-;~:~7:<~~··~ 

,,_,.' ' .. ,.;-. -,r-'-

'. :'_, ' ; ::"'. -.~:, \~~.::>::~ ::~.;.J' ':".' - -.,'. 

. ~ 

.. [t 

. c " 
.:'-:, - " 

, "',1 

",,-,'y .; ...... __ " ~_~'_,:C,_,.,._'_,:,,- 0,_ •• ,'_ • ..:.-.-.:.' __ ._;:":::._. ,.~""':"'....::...1.,_~':; 

component will be characterized using a standard viscoplastic model. 

Special emphasis is placed on the modelling of saturated hysteresis 

loops and stress relaxation behavior at various strain-rates and 

temperature ranges. Evaluation of tbemodel is performed by comparison with experimental results. 

"~--:":,",,"c'7'r"",--;--.,,--. .. ~-~--~---- '--";"-'.- "-.----- ,- --..,-.~.--

"', "-

, .--".-.::.-.. ~-:~~.":""::-:. ,-

,~~~.-:--, . 

12 

, 

i 

"J 

:'.,,; , 

.. ' 

.,. 

..... , 

',,".~ : 

. ~ ," ,; ... w ~~' 

. ,' .. ' 

0'" ,... 

:-' -

-r../'" 

13 

CHAPTER III 

CONSTITUTIVE MODEL DEVELOP~ffiNT 

·-mthis chapter, the physical. basis arid the experimental data 

requirements for the uncoupled viscoplastic model will be described 

in detail. As pointed out earlier, plasticity and creep are generally considered indistinguishable since both involve thermally ass isted overcoming of various dislocation barriers. In the discussion 

that follows, justification for the uncoupled approach will be given 

in terms of the nature of dislocation barriers and the role of thermal 

assistance in circumventing the obstacles under various temperature 

and strain-rate conditions. It is the author's belief that if 

uncoupling is performed properly, meaningful results can be obtained. 

Physical Basis 

Inelastic deformation of crystallise materials occurs by 

dislocation movement and various forms of dislocation-obstacle interactions. 

In the low temperature regions (0-0.2 T ), the yield strength m 

and flow stress are found to vary significantly with temperature. 

This is particularly true for body center cubic materials since the 

number of slip systems available decreases dramatically as the temperature falls. In this temperature range, plastic deformation is 

dominated by dislocation glide. The fairly small activation energy 

for thermally assisted dislocation movement is usually associated 

with the overcoming of lattice friction (i.e. Peierls stress) and 

-'-,- •• __ •• --. -,~.....,-~." • ...,....--"~.--. ----~------ -- ,---- --~-•• - '"'' ~.:- .-•• -:----.-~_:_-~ ... .--.-____ --.. -'. '_~:::~-.'--- .-_____ •• ___ ~~~. _._~"", _,n _ __ , __ .'" _. __ • ______ • __ , __ ._._._.~ _ ': __ ._ •• __ ~._. ". :'_ ~. 

-', ,- ...... : -- .. ·.:-;'-s-'·~~:~· --:;' .. 
,,' . ' .. '. 

1. _:': 

. . '.,~ '. -

,": . 

... . ' 

14 

the cutting of dislocation intersections [41] . 

Between 0.2 - 0.4 T

m, the thermal energy available is more than 

adequate to overcome such barriers, allowing dislocation glide to 

occur equally easily at various temperatures and strain-rates. While 

short range barriers such as forest dislocations and Peierls stress 

are easily overcome with thermal assistance at these temperatures, the 

thermal energy is relatively small and generally ineffective in overcoming long range barriers such as grain boundaries, cell walls and 

large incoherent second phase particles. The high strain hardening 

rate in this temperature range is generally believed to be the result 

bf dislocation pileups against long range obstacles. Recovery in 

this temperature range is controlled mainly by dislocation cross-slip 

[24], which depends on the stacking fault energy of the material, 

temperature, as well as stress. The activation barriers for dislocation cross-slip is relatively large compared to the thermal energy 

available, making this process quite sluggish. In this case, the 

flow stress depends more on the long range internal stress and the 

resultant strain hardening it produces than the strain-rate. In 

summary, between 0.2 and 0.4 T , thermally activated processes are 

m 

either so rapid (e.g. overcoming Peierls stress) or so slow (e.g. 

dislocation cross-slip) that very little rate sensitivity is observed. 

Deformation behavior in this temperature range can be very well 

described as rate insensitive and over a finite range of strain-rates 

will be essentially rate independent. It will be shown later that 

Hastelloy-X specimens tested under fully reversed strain control~ed 

conditions over a temperature range of 298 K to 533 K (0.2 - 0.35 T ). 

m 

.. _ .. '-:.-:- '--"--77~ -:--.. ---.~ -_.- .- -- ~- ~- "- --"-~---

- -' - .', ~ . 

. ~. -.' ._ :";i~- -> 
.- ',-

, " 

. '. ~ 

','.: 

, -5 -1 -3 1 and over a 30 X strain-rate range (3.3 X 10 s to 1 X 10 s-) 

have essentially identical hysteresis loops. At 755 K (0.49 T ), the 

m 

15 

hysteresis loops change slightly, though rate dependence over, the same 

strain-rate range is still not significant. 

As the temperature is raised above 0.5 T , rate effects on m 

inelastic deformation becomes apparent. Numerous dislocation models 

have been proposed to explain thermally activated deformation in this 

temperature range. One example is provided by Barrett and Nix [42] 

based on the assumption that diffusion controlled motion of jogged 

screw dislocations is the rate limiting deformation process at higher 

t~mperatures. Climb and annihilation of edge dislocations have also 

been suggested as dominant creep mechanisms [43]. Another example is 

given by Nabarro [44] and Herring [45] who independently proposed two 

similar models describing high temperature creep in terms of stress 

driven atomic diffusion from one ~rain boundary to another. Coble 

[46] suggested an alternative model involving atomic diffusion along 

grain boundaries. Additional deformation is ,generally provided by 

grain boundary sliding. Despite the large number of theories proposed, they all recognize the importance of thermal assistance to deformation at high temperatures. It will be shown later that the stressstrain behavior of Hastelloy-X is a sensitive function of temperature 

and strain-rate over this temperature range. 

In the following section, the rate insensitive, inelastic strain 

will be described in terms of dislocation pileups against long range 

barriers. The rate sensitive, inelastic strain will be associated 

with the additional increments of plastic deformation made possible 

-".-------'.--~.------__ ,. -_".' ___ ---___ •• _- "c ___ ,_,,~_. ____ • __ ~_, __ •• _". '_, _.,_~,_ 'r_._ ,. _______ _ .' . ; 

,0.; .s,··· "." :'; .... :. '"i ~'_ -': '. 

. ~':J ,.' ',",',. '. 

-r--.'< 
~~.'" 

'<~u~ 

-',' .. '" 

. .- ". ' ~, 

,,' ", 

16 

by the thermally assisted overcoming of these long range barriers to 

deformation. Dislocation cross-slip and climb are generally regarded 

as the dominant recovery processes at high temperatures. Nonetheless, 

thermally activated deformation resulting from other mechanisms will 

be implicitly included as part of the rate sensitive strain component . 

As discussed above, the rate sensitivity is seen principally in the 

circumventing of various barriers by cross-slip or climb rather than 

in the subsequent' glide to the next barrier. Nevertheless, all of 

the inelastic strain that results from the combined cross-slip (or 

climb) and subsequent glide will be included in the rate sensitive 

inelastic strain. 

The inelastic strain has been implicitly divided into a component which results in strain hardening and one which is associated 

with thermal recovery or softening. Even during the portion of the 

hysteresis loop where strain hardening is occurring, the inelastic 

strain may contain rate insensitive (strain hardening) as well as 

rate sensitive (thermal recovery) components. It should be noted that 

net strain hardening continues until the back stress reaches a level 

where recovery and strain hardening are balanced. This is analogous 

to the familiar description of high temperature creep as resulting 

from a bal-ance between the processes of work hardening and recovery. 

It will be shown later that a constant back stress does imply steady 

state creep. Ideally, the stress is dependent on the rate insensitive, 'inelastic strain and the rate sensitive, inelastic strain-rate. 

A transient dependence of stress on the rate sensitive, inelastic 

,strain (as well as strain-rate) is sometimes observed and is 

• I . 

": 

. " ,~ " 

""';:,," 

, ~" 

''-~ , 
17 

equivalent to primary creep. Such transients are believed to be assoeiated with the initial "shakedown" where dislocations are gradually 

being rearranged into more stable cell structures. 

In the next section the constitutive model will be defined in 

mathematical terms and the experimental requirements to characterize 

the various material constants will be described. 

Description of Constitutive Hodel 

It is general practice in constitutive modelling to separate the 

total strain into elastic, inelastic and thermal components, i.e. 

E = E + E. + E h t e 1. t (1) 

where Et , Ee' Ei and Eth denote the total strain, elastic strain. 

inelastic strain and thermal strain respectively. Uncoupled theories 

further partition the inelastic strain into rate independent and rate 

dependent components, which can be expressed as 

(2) 

where Eii and Eir represent the rate independent (or rate insensitive) 

inelastic strain and the rate dependent (or rate sensitive) inelastic 

strain respectively. Under isothermal conditions, equation (2) can be 

expressed as 

(3) 

where E , S , E .. , E. represent the total strain. elastic strain, rate 

t e 1.1. lor 

insensitive inelastic strain and rate sensitive inelastic strain 

-'.' 
- '. ~ .' ',-.',' 

- .,'~' ' 

" ',',," 

:"i ',", 

". '- .;': 

"". : _,' r 

,"-

.-: . ." ' 

~ --::' 

, ",1 

,,-.,-

" 

, ,. ,,', j 

, .-' " , '.," 

_- _____ "c".._"~·....:."_,_,_, 

18 

respectively. Equation (3) can be rewritten in rate form as follows: 

where S , t , Sio, to denote the total strain-rate, elastic straint e J. J.r 

rate, rate insensitive inelastic strain-rate and rate sensitive 

inelastic strain-rate respectively. It is assumed that at any point 

along the deformation path, the elastic strain-rate component can be 

described by the Rooke's Law as follows: 

t = olE 

e 

where a is the stress-rate and E is the modulus of elasticity of the 

specimen. As will be discussed later, all the experiments in this 

research program were conducted on a servohydraulic testing system 

(MIS) in the strain control mode. The rate insensitive, inelastic 

strain is modelled with an empirically determined strain hardening 

function as follows: 

E .. = f(a, cr ) cr 

J.J. max 

where f(a, cr

max) denotes the hardening function which depends on the 

current state of stress as well as the maximum stress before load 

reversal. It is given by 

dEii 

f(cr, a ) = dcr max 

The function is measured from saturated hysteresis loops in a 

", .... c~ _,' ,_ 

(4) 

(5) 

( 6) 

(7) 
-'. ,-',_. 

. ". ~ -. 

temperature range where rate sensitivity is not significant, i.e., 

essentially all inelastic strain is rate independent (or rate insensitive). It should be noted that the hysteresis loops even at these 

lower te~eratures are slightly asymmetric so the sign of the cr 

max 

value before the stress reversal as well as its magnitude must be 

19 

specified to define the particular f value" for a given value of stress 

in a load reversal. The rate sensitive, inelastic strain-rate is 

modelled by a power law function which can be expressed as 

(8) 

where n, nand K represent the stress exponent, back stress and dtag 

stress respectively. A constitutive law of the form stated above is 

commonly used in unified theories to model inelastic deformation 

behavior of crystalline materials [11, 14, 16]. It should be noted 

that unified theories assume all inelastic strain is rate sensitive 

and the constitutive law is used to model the "total" inelastic 

strain-rate rather than the rate sensitive inelastic strain-rate as 

described here. The stress exponent n is a mathematical constant 

which is generally a function of temperature. The back stress and 

drag stress represent two evolutionary parameters which are microstructure dependent. Equation (8) represents an "evolutionary 

I 

material law". It is assumed that the current state of a material 

can be described by a number of "hidden" microstructural parameters 

known as internal state variables. Ideally, adequate specification 

of the internal state variables allows prediction of material 

,. -;.-----... -: ~"'----.. ----~ __ ... T" ___ .. _~:. 

'~ , ' •• I •• ': •• '-' 

--- -' -------:"'~--~--.--~-. .,- .--- _._- .. -~---.-------,----~ .. 

" ' , 

·,--1 " . Ii, . ~ 
" 

.: . . _. _. __ ,. .. _ •• __ ::' ... - c.. ...... : ........ ~.:~~_.,_w .,: __ ~.,-,::",,_. __ ~_.:-_~ ,~',.-

,'.", ' 

- ~~ ..• --~,::,~~-=-..--.-,:"".- ... -",,~;.:,' 

20 

response without explicit knowledge of previous thermal-mechanical 

history. In this case, two internal state variables (i.e., ~ and K) 

are utilized to model microstructural changes associated with cyclic 

deformation. In classical plasticity theory, the drag stress reflects 

isotropic strain hardening and is equivalent to the expansion of the 

yield stirface. The back stress, on the other hand, correlates with 

kinematic hardening and is equivalent to the translation of the yield 

surface. Microscopically, the back stress is associated with dislocation pileups in pure metals and/or multiple loop formation around 

incoherent second phase particles in alloys. The drag stress represents short range barriers to deformation, in contrast to back 

stress, which represents iong range resistance to dislocation glide. 

An alternate though not contradictory interpretation of the drag stress 

and back stress is that they represent the respective effects of dislocation density and dislocation distribution. Equations (5), (6) and 

(8) can be recombined to yield the following relation: 

(9) 

The uncoupled model can be represented by a single ~~ell element as 

shown in Fig. 1. The spring represents the elastic strain which has 

an indirect temperature dependence through the Young's Modulus. The 

slider block represents the rate insensitive inelastic strain which 

is a function of the applied stress but is independent of time (or 

temperature). The rate independent nature of the slides block can 

be further demonstrated by rewriting equation (6) in the following 

.---- -... -~--.---~ .. -~-~---:------.. ,~~--.-~-----~---- -

-I 

-----.. t I 

1; 

:1 --- Ii 
-..:",. , 

e: 

E: =a/E 

e 

t'i=f(a,t: ) l. t 

€, =f(a,Et,T,t) lr 

~·ig. 1 Single nonlinear Maxvell element 

-~ ~ ,~.-. -.-.. -.--~-- ---.----~, ~'- ... --. .. - .. . ' .':'~-.----.---

21 
,":.,, ' 

'.":.< '( 

",- , 

,,''; .. , 

, ", 

: I"~ 

'".' 

, ,.", 

,.';', 

....... ' 

, -~~' 

. : . .' " 

.-./_' •• , _'_.,'--..~.~.,.......:.... __ ",,:,~,.~, __ c __ ... ~ ..... _., •• ' .:. ~ •• :-::.: '.";" __ -'-.- ....... ,:, ..... ~ __ ,_~_ ••• : 

22 

form: 

[:~iiL (10) 

ref 

where (~~i' J = [f(a, a )J-1 and represents the strain hardening max ~ T 

ref 

rate at the reference temperature (T f)1 and (ddO J denotes the 

re E: •• T ~~ 

strain hardening rate for the rate insensitive inelastic strain component at a higher temperature. Equation (10) merely states that the 

rate insensitive inelastic strain corresponding to a given stress (or 

strain) remains the same regardless of the temperature (or strainrate). At low temperatures, the "viscosity" of dashpot prevents any 

significant contribution from the rate sensitive inelastic component 

to the total deformation and the inelastic strain is essentially all 

rate independent. At higher temperatures and/or lower strain-rates, 

the dashpot becomes less viscous so that rate,dependent deformation 

or thermal recovery occurs which results in "softening" of the 

material. 

In summary, it is proposed that the total strain can be separated into an elastic component as well as an inelastic component which 

consists of rate sensitive and rate insensitive terms. The rate insensitive inelastic component results in strain hardening whereas the 

rate sensitive inelastic component occurs as a consequence of thermal 

I, : IT in our case is taken to be room temperature. , ref , , " 

',;, .",', .':'. ~. ", 
- . ~ '-

. - . 

i...:........!.-"O·:-;.-~.:. 

_ ... '; 

. , 

"'. " 

"~ :: ' , 

... _>1 

~"-~,' -~~---,,--.j- -~-"':'::. --.----.--- .. 

23 

recovery_ It is important to recognize that regardless of the deformation history, the sum of the three components at any stage must 

equal the total strain. 

In the next section, the experimental data requirements for the 

uncoupled model will be described. 

Experimental Data Requirements 

The value of a constitutive model lies mainly in its capability 

\ 

to predict complex thermo-mechanical histories. However, the effectiveness of a model can be severely hampered if it requires a large 

number of material parameters which have to be extracted from complex 

and lengthy experiments. A useful constitutive model should require 

a minimum number of material constants which can be determined by 

relatively simple testing procedures. 

The material parameters necessary to characterize the constitutive model previously discussed includes f(O', O'max)' )1(0', T, N), 

K(O', T, N), neT) and E(T) where Nand T refer to the number of cycles 

and temperature respectively. For the initial phase of this program 

only saturated hysteresis loop behavior will be evaluated, eliminating 

for the time being N as a variable. It is further assumed that for a 

saturated hysteresis loop, K will have a constant value around the 

loop whereas )1 will be allowed to vary as a function of stress. The 

rational for the assumption is that the drag stress is physically 

associated with the dislocation cell structure, which will achieve a 

stable configuration upon saturation. The back stress, on the other 

hand, is associated with metastable dislocation arrangements such as 

---.----.. -.--.-----. ..... ----,----.:~...: ::.----~-~--- --~,~..."..---.--....--~----:--.•. -.,,--

;,: .- ': .:; " .'" . :". ~', . ":',"::: . 
24 

pileups, multiple loops around particles, etc. Once a stable cell 

structure is formed (i.e •• at saturation). it is reasonable to assume 

that it does not change appreciably during strain cycling. 

The experiments required to evaluate the material constants are: 

(1) fully reversed, strain cycling experiments (i.e., hysteresis loops) 

at various temperatures and strain-rates; (2) stress relaxation tests 

at various points on the hysteresis loops; (3) strain drop, stress 

transient tests; (4) strain-rate cycling tests. 

The strain hardening function f(o, a ) is defined by saturated 

max 

hysteresis loops taken at a sufficiently low temperature (0.2 - 0.3 T ) 

m 

where rate sensitivity is not significant. Hysteresis loops at higher 

temperatures show a reduction in flow stress due to dynamic recovery 

(Fig. 2). The additional inelastic strain reSUlting from the recovery 

process can be obtained by simple subtraction as shown schematically 

in Fig. 3. 

At relatively low temperatures, the back stress can be determined 

by stress relaxation tests performed at various pOints around the 

hysteresis loop. It is assumed that thermal recovery is negligible and 

the back stress is taken to be the asymtotic value approached at the 

end of the stress relaxation test. In our experiments. each test was 

carried out to the point where the stress-rate (i.e., the slope of 

I 

the stress relaxation curve) approached zero. Thermal reco~ery becomes 

significant at higher temperatures and the assumption of constant back 

stress is no longer valid. In this case, the strain drop, stress 

transient test is used to evaluate the back stress. This approach is 

based on the observation of a "stress transient" which occurs during 

'.~, .'-
... :,', 

.. '" 

" [ 

"'-'':..-' 

..... ,-':':.-;'. 

'" --~~~.-........ -.. -. 

Fig. 2 Schematic of stress-strain behavior at three 

temperatures 

Fig. 3 Experimental characterization of the three 

strain components for the uncoupled viscoplastic 

model 

25 
. f', 

- ~. ' .. _. : _ ... _"" _"--'<--1-~._._~ __ ,. __ ,_."'~',_ 

'~ . ., .: 

26 

load reversal when the strain is held constant momentarily. The back 

stress is given by the point. where no transient is observed. 

Other techniques have been developed for measuring internal 

stress and a few examples will be given here for comparative purposes. 

One example is an "incremental unloading technique" developed by 

MacEwen. et al. [47]. The test is performed by incrementally unloading the stress and observing the subsequent relaxatiori behavior. 

Positive stress relaxation will occur initially and unloading will be 

continued to the point where no stress relaxation is observed. The 

stress where this occurs is defined as the internal stress. Two 

alternate approaches have been summarized by Ahlquist and Nix [48]. 

The first one is the "stress transient dip test" which is based on 

the same concept as the previous technique. During a constant stress 

creep test, the applied stress is reduced at the point of interest 

and the stress transient is observed while the strain is held constanto The internal stress is.gi-¥en-by the point where no transient 

is observed. The second method is referred to as the "strain 

transient dip test". In this case, the applied stress is reduced and 

held constant while the strain transient is monitored. The internal 

stress is indicated by the point where zero strain transient is 

observed. One las~ example is provided by Walker [17]. which is 

based on the same concept as the strain transient dip test. In his 

study of cyclic behavior of Hastelloy-X specimens, the back-stress 

" .- ~ .~ 

" . was determined by holding the stress constant at various points along 

the unloading branch of the hysteresis loop and observe the creep 

rate. The back stress is defined as the point where the creep rate 

--- -.- ~ _.- "--'- ,." --"--=-~'-"--:"'--'-"--'''-'--'-:--:-~,--- ~" 

.. :--.' ',: ; 

-_ .. _--_ ... ~ -..... ~-.~ ,'.--,.-" 

i" 

f 

t 

i 
, - '~ -. '. 

" .,.,. _____ "'.'~''''''''''''_':''_~ __ '_'.-'k'' ___ .~, 

goes to zero. As p'ointed out by Walker. the major disadvantage of 

this approach is that the creep strain observed in the region is very 

small, making an accurate assessment of the back stress difficult. 

He recommended the "strain drop, stress transient test" as an alternative since it is more sensitive and will allow a better estimation 

of the internal stress. 

At lower temperatures, the stress exponent n can be obtained by 

plotting In E. vs. ln (0 - n) from the stress relxation tests. At 

11' 

higher temperatures, n is obtained from strain-rate cycling tests 

performed at the plateaus of the saturated hysteresis loops. 

Finally. the drag stress K can be determined by substituting 

the known values of the back stress and the stress exponent into 

equation (8). 

The purpose of the above discussion is to provide a general 

description of the experimental parameters required for the uncoupled 

viscoplastic constitutive model and the necessary tests for their 

27 

evaluation. A more detailed discussion of the experimental procedures 

will be presented in the follOWing chapter. 

Numerical Scheme for Theoretical Prediction 

Before a constitutive model can be used to predict complex thermomechanical histories, it must be able to reproduce. experimental 

results. To first see if the uncoupled model is self consistent in 

being able to predict the original hysteresis loops, a numerical 

scheme has been devised. 

The uncoupled constitutive model is represented by equation (9) 

~, '" . 
, ; '. . ~ . 

.... .,"- .. " , .... 

,', . 

.. " 

," "' 

;' ". 

:,',.j 

.' ~' 

.;' ! 

.... -.. : 

'.'.', 

.: _ .... ::.~ ~;.:; .~~,:,:;<._~_..:.~ ____ ,~~,_ .;.., .. .::c:_'.:. 

28 

which can be rearranged in the following form: 

cr '" _E..;:,t_--:-s_g_n_(_o_-_~_)-!-.I o---='~=----~...J..I_n 1 (11 ) 

-E + f(o, 0 ) max 

or in incremental form as" 

1

0 ~In = ~ Et - sgn(o - ~) --i-- ~t (12) 

1 -E + f(o, 0 ) max 

The evaluation of ~ Et is given by Et~t where the total strain-rate 

(from experimental measurements) is given by 

-2 dO 

Do dt €: = --..,....w.----- t 1 _ ! do 

E dE

t 

(1-2v) 

(13) 

where Do" :~ and v represent the initial diameter, the diametral 

strain-rate and the Poisson's ratio respectively. It should be noted 

that the total strain-rate does not remain constant during the course 

of deformation but instead will change with the strain hardening rate . 

The point on the hysteresis loop where loading begins will be 

chosen as a "reference state" from which the simulation process will 

start. The purpose of the incremental scheme is to predict changes in 

material response from the preceding known state over a time interval 

~t. At the reference state, the required material parameters will be 

specified together with the value for the initial strain hardening 

rate from which the initial strain-rate is determined. Next, the 

"strain increment over the designated time interval will be computed. 
'~. 

-'.' , ~~'. 

-",.,<"-.,, . 

--"' 

;;: 

,-:., '~' 

. -".-

~ -, .. -'~ '';:::- . ,,-"" .-

29 

corresponding stress increment from the preceding reference state 

then be calculated using equation (12). With the stress and 

strain increments now known, the new strain hardening rate can be 

computedwhic-h allows the determination of a new strain-rate value 

from equation (13). This will in turn enable the value of the second 

strain increment to be calculated. Values of the internal state 

variables are updated accordingly to reflect the current "microstructural state" of the material. With the new values of internal 

variables and strain increment determined, the corresponding stress 

increment can again be computed. The procedure is then repeated for 

the next time increment. A program outline for this incremental 

procedure is presented in Appendix A. 

A second numerical scheme has been designed to predict stress 

relaxation behavior. The "reference state" is taken to be the point 

on the hysteresis loop where stress relaxation commences. Once the 

initial conditions (i.e., initial values of the stress and internal 

variables) are specified, the initial rate sensitive inelastic strainrate can be computed using equation (8), as shown below: 

(8) 

The stress relaxation tests in this research program were run under 

constant diameter conditions. The corresponding relation between 

the rate sensitive inelastic strain-rate and the stress-rate is given 

by 

-2v Eir = -E- cr (14) 

:_'-,,-,:" 

-"- " 

-ok 

.;L;\ 
.:.~ .:, 

,.', i 

.':,} i 

! 

. . . ., , 

", j. 

" . .. :-,:-:<':r..... __ 

30 

Knowledge of the rate sensitive inelastic strain-rate will enable the 

stress-rate to be determined from equation, (14). This will in turn 

allow the stress reduction over a time interval ~t to be evaluated. 

The ·semeft£ocedtire is-repeated by substitl1ting the new stress value 

into equation (8) and assuming the back stress and drag stress remain 

constant. As will be discussed later, this assumption is not valid 

at higher temperatures and will inevitably lead to erroneous prediction. A program outline for this numerical procedure can be found in 

Appendix B. 

, '.~ ... 
'.';', ,- ';'. i 

- .:.,' 

. -~- '-,: ,: 

" 

_', ,,'1. 

" : , .... __ ~~'_'4. _____ ' ____ ' •• _L. ~"-_.~: 

31 

CHAPTER IV 

EXPERIMENTAL PROCEDURE 

Material Description 

The material selected for this research program is Hastelloy 

alloy X. It has been a successful high temperature structural 

material which is widely used in applications such as turbL,e blades 

in jet engines. Considerable experimental effort has been made 

recently to evaluate the high temperature properties of the alloy 

for critical applications in nuclear reactors [49] and nuclear powered 

rocket engines [50]. 

Rastelloy-X is an austenitic nickel-base superalloy that possesses 

good strength and oxidation resistence up to 1470 K. The alloy is 

solid-solution strengthened. basically by the elements Cr. Mo and W. 

The chemical composition of Hastelloy-X is given in Table I. The 

alloy used in this study was supplied in the form of 19 mm (0.75 in.) 

diameter bar stock in the solution heat treated condition. The 

standard heat treatment is performed,at a temperature of 1450 K 

followed by rapid cooling. 

Specimen Preparation 

Specimens were machined from 19 mm (0.75 in.) diameter bar stock 

to the configuration shown in Fig. 4. Each specimen was annealed at 

1200 K for 30 minutes followed by air cooling prior to testing. The 

annealing process serves two purposes: (1) it will produce a more 

" .,-- ,~, 

, 

i 

I 

, r 

; 

- I 

, , ." ,~--.,~. ---~ 

-. 'f: 
.. '~',. 

" 

.. .' ~ . 

...... 

'.'; 

I 

.1 

: -'. -~ 

Table 

;~ '. -

-'- .~::- ./. , 

32 

1. Chemical Composition of Hastelloy-X , '-.; "'., 

Alloying Element Weight % Alloy 

Cobalt 0.50 - 2.50 

Chromium 20.50 - 23.00 

Molybdenum 8.00 - 10.00 

Tungsten 0.20 - 1.00 --

Iron 17.00 - 20.00 

Carbon 0.05 - 0.15 

Silicon 1.00* 

Manganese 1.00* 

Boron 0.010* 

Phosphorus 0.040* 

Sulfur 0.030* 

Nickel Balance 

*MaJ~imum 
I,: 

,; 

':;, ( 

; .. ~:I 'j 

"< ' ~ 

,I' 

f; 

~ ~ . 

. .' 

" 

, ·~i 

~ ,-' 

, 

,: 

; 

"\ 

I 

I: 

1._' 

I' 

" 

~ 

""""-

" t 

-

'.I III 

11. r.r.nT,,---J 

'. 

lbO.UU108.00 I 66.0041 

..... 1'56

.

00 

~ -"I - -, 

l l' --.....j 

L----IO.20DIA 

12.70R 

Fig. 4 Specimen geometry 

)00 

~ 

-.,. 

ODIA 

'I \ '1'11)-

L--

ALL DIMENSIONS ~0.01 

DIMENSIONS IN MILLIMETERS 

SCALE: FULL 

I,' 

w 

w 

i 

r· 

" 

,: 

';/' 
,c, 

~. ---.. -~-- .. -.-,"~,,'-,,~ . 

34 

"isotropic" microstructure, since this is an assumption made in the 

theoretical analysis; (2) it will provide a fully oxidized surface 

condition for the specimen. As will be discussed below, a stable 

oxidized surface condition is essential for accurate temperature 

measurement. The average hardness of the specimens is 54 RA 

(Rockwell hardness number) before annealing and 52 RA after annealing. Measurements were made at the gage section of the specimens. 

Experimental Set Up 

The experimental equipment utilized in this research program 

consists of four major components: (1) a 100 kip MTS Series 810 

materials test system; (2) a 2.5 ktol Lapel induction heating generator; 

(3) an LFE Series 230 temperature controller; (4) an Ircon Modline 

Series 6000 radiation thermometer (optical pyrometer). Picture of 

the experimental set up is given in Fig. 5 and a schematic is depicted 

in Fig .. 6. 

The MTS testing system is servo hydraulically controlled and has 

considerable flexibility. It is equipped with a function generator 

capable of simulating complex mechanical histories under one of three 

control modes (load, strain or stroke). To perform high temperature 

experiments the MTS"is equipped with special water cooled grips and a 

diametral extensometer utilizing quartz rods which can be used for 

high temperature strain measurement up to 1255 K. 

The Lapel induction heater is designed to deliver to an inductively coupled load a maximum of 2.5 k\.J of high frequency energy in 

the range of 250 kHz to 800 kHz. Induction heating offers the 

~. -..... -
·- ,,'-' 

. , " 

"" -'" 

Fig. 5 

OOBGRNAl' PAGE ~s 

Of POOR QU~UiV 

Experimental set up 

-------- - "--- -.. --.. ~--~-~.----

-: ,~~:-':~"C7-~-,-,~c;~---'~'~>;::,;~_: ::~.~,-~~~~~_?,::~:-,:;::.:::;;-~,o~-~-'-~:;7 - -' . ' '- 1:-.' _, .'\' _-.- -?;.~ ',-- "~" -.',.': 

35 
\' 

", 

;,; ~ 

" i 

I 

I 

,i ,,1 

",I 

'"I 

! 

,; 

" 

:1 

" 

" 

\ 

;, 

". 

" 

MTS MATERIALS TEST SYSTEM 

Readout 

Recorder 

Voltmeter 

Oscilloscope 

I 

Feedback 

Selector 

Feedback 

y... Servo 

Controller 

Function 

Generator 

Transducer 

Conditioner 

Transducer 

Conditioner 

Error " 

Transducer 

Conditioner 

,". 

POWER CONT~OL SYSTEM 

Crosshead 

~ ".~ Load Load Cell 

Output 

Induction Specimen Sensing Head 

Strain _0=-==== 

o 

____ - - --: ~' o " . 

ExtensometerJ 

Valve 

Driver 

--

f 

Control 

Signal 

Stroke 

Output 

r 

I 

L-.. 

} 

Actuator 

LVDT 

Transducer 

Power '---

Controller 

Output 

Proportional 

Controller 

Proportional 

Controller 

Output 

Inductiop 

Heater 

Fig. 6 Schematic of experimental set up w 

()\ 

~ '.; 

j, 

I"· 

f" ,: -

I 

!. 

. 

\ 

! 

;-. 
\ r ~--<: 

, ... , .. , 

37 

advantage of a rapid energy input into the specimen and is widely used 

for thermal or low cycle fatigue testing. 

For temperature control below 700 K an LFE controller is used. It 

is a proportional controller which utilizes a Type J (iron-contantan) 

thermocouple for temperature measurement. Power output of the induction heater is adjusted by the controller in proportion to the difference between the measured and the set point temperatures. For tempera-

: '" ture control above 700 K, an Ircon optical pyrometer is employed. It 

is a "noncontact" type controller which measures the temperature of 

an object by sensing the infrared radiation emitted from its surface. 

Temp.erature control is based on the same principle as the LFE controller. An output current is developed in the controller proportional 

to the deviation of the specimen temperature from the set point temperature, which governs the power output generated by the power controller inside the induction heater. 

Good facilities are a necessary but not sufficient condition to 

produce satisfactory results. Additional steps have to be taken to 

minimize experimental error which might occur as a result of the limitations imposed by the testing equipment itself. In the case of high 

temperature materials testing under cyclic loading conditions, the most 

important factors to be considered include: (1) measurement and 

control of temperature; (2) axiality of testing machine. 

In strain controlled low cycle fatigue testing, it is considered 

that the temperature should be controlled to within ~ 2 K for reprodu-

":'1 cible results. In this research program, temperature variation at ii 

1144 K was no more than ~ 1.1 K and better temperature control was I 

I· 

i 

I 

""" __ . ____ ;-__ ""-:::.~.~-.-c-~-"~:- _ ... -.-___ ._~ __ ---.~.~:;:. ~~.~ .. ~;~~-:;-:~~~ -'~.-"----~,~ --;~-~~;. :~~~" ~-,;=~~~~: .>.:~~-~--;~~~ 7'';.- ___ --:~.--.-:~- .~ ~-~ - -"- --~-_~ ..• "-. - -~ ::-_.-- ,---

'.':- "-'. - .-- -

·-·-.~:~·.:;I Ii 
,', 

_ ; ...... " ...... ' ,-:....,..'..,:.:.: ___ . ......:.....~ ........ _~~.C .. ::' ... '." " 

.-. \ 

38 

::' " 

.' ~, 

" ,: 

attained at lower temperatures. The accuracy of temperature measure-

,,",'" , .. ; ment by optical pyrometry depends on the emissivity of the material 

which is a function of both surface condition and temperature. For 

example, a nickel specimen with a smooth polished surface has an ." '0 

emittance value of 0.15 which increases dramatically to 0.90 when it 

becomes oxidized. Roughening of the specimen surface due to oxidation 

has a significant impact on the resulting emissivity va~ue and thus 

. .. ~ 

poses a severe temperature control problem. To overcome this difficulty,all specimens were annealed at 1200 K for 30 minutes prior to 

testing to produce a stable oxidized surface condition. The emissi-

~ity has also been found to vary with temperature, which necessicates 

readjustment to the emittance setting on the optical controller every 

time the set point temperature is changed. This disadvantage is 

overcome by calibrating, the optical pyrometer against a high precision, 

factory calibrated Type K (chromel-alumel) thermocouple which has an 

absolute accuracy of z 3 K. Calibration was performed in a conventional furnace using a small Hastelloy-X sample cut from the gage 

. i section of one of the original specimens . Additional thermocouples 

were attached to the sample to verify the absence of significant 

thermal gradient. The thermocouples were protected with Inconel 

overbraid and ceramic fiber insulation against oxidation and possible 

contamination by the furnace atmosphere. 

Induction heating offers the advantage of fast thermal response 

and relatively simple experimental set up. The thermal gradient at 

the middle of the specimen was minimized by concentrating the coil 

turns near the ends of the expense of heating 

"'.- ,,": 
0"':.· 

, ,,' ~'. 

'-:', "" "eO: ", 

,.:,'.<::'.' 0"'" 

.-, . 

", : 

39 

efficiency. 

For fully reversed low-cycle fatigue testing, it is recommended 

by ASTI1-E606 that bending strain should not exceed 5% of the minimum 

axial strain range. A trial specimen equipped with three equally 

spaced strain gauges around the circumference was used to check the 

axiality of the MTS. Special alignment procedure were employed to 

reduce an initial variation in axial strain measurements by the strain 

gauges from 30% to 5% maximum. 

Description of Experiments. 

In this section, the various experiments necessary to determine 

the material constants are described. The required tests are: 

(1) fully reversed, strain cycling experiments (hysteresis loops); 

(2) stress relaxation tests; (3) strain drop, stress transient tests; 

(4) strain-rate cycling tests. 

Fully reversed, strain cycling experiments (hysteresis loops). 

In this research program, saturated hysteresis loops are used to 

characterize the behavior of Hastelloy-X under cyclic loading conditions. The experiments cover temperature range from 294 K to 

1144 K 

to 1 X 

-5 -1 (0.2 - 0.75 T ) and a strain-rate range from 3.3 X 10 s 

m 

10-3s-1. It is important that the range of temperatures and 

strain-rates selected for the experimental study can adequately 

represent the conditions encountered in actual service. Generally 

speaking, turbine operations typically take place between 700 K and 

950 K and the strain-rate range covered in this study is consistent 

with those considered by other researchers [17, 50]. A tciangular 

- --.• ,", _'.~_O_'_~"_~_'_"_"""~~' -.---- '''~"--~'''.7"'- -r·,-,_·'--:_, -,0. ~~~ •••• ~"-'-

, 'T "",'~~ .. ~. 

, -:~", .. ,. 

;""; 

-, ~'. '-. 

,', ' 
','- : 

'-. " --:.1 

40 

waveform is used as the input function for strain cycling as recornmended by ASTM-E606 since it provides a more uniform rate of straining. 

The waveform selected is important to the resultant stress-strain 

response especially at high temperatures, where inelastic deformation 

is extremely rate sensitive. All experiments in this research program 

were performed in the strain-control mode. In this control mode, a 

servohydraulic machine (MIS in this case) has essentially infinite 

stiffness, thus avoiding the complications introduced by the elastic 

modulus of the testing system. 

Experiments were conducted at ten different temperatures and 

three different strain-rates. Below 0.5 T , rate sensitivty is not 

m 

important and hysteresis loops were run at selected temperatures 

(294 K. 533 K and 755 K). Above 0.5 T • the material becomes more 

m 

rate sensitive and experiments were conducted at 55 K intervals 

from 811 K to 1144 K to determine the required material constants. 

At each temperature, strain-cycling was performed at three different 

strain-rates (1 X,10- 3s- 1, 3.3 X 10-4s-1, 3.3X 10-5s- 1) with a strain 

range of ± 1%. 

As discussed previously,room temperatures hysteresis loops were 

used to define the rate insensitive, inelastic strain component, which 

has a cr dependence. Therefore,hysteresis loops were obtained at 

max 

'different strain amplitudes (0.05%, 0.1%. 0.2%, 0.3%, 0.4%, 0.6%. 0.8% 

and 1%) which correspond to different cr values. 

max 

Stress relaxation tests. The stress relaxation test is a technique commonly used for back stress determination. It is usually 

performed during the crosshead 

'" . 

,"'. T " , 
, ..• ';c .. ' 

"r.:· 

" ~ 

..... : 

41 

and monitor the stress variation with time while holding the _strain 

constant. Initially, the total strain consists of an elastic component 

as well as an inelastic component. Additional rate dependent inelastic 

strain (i.e. creep strain) occurs at the expense of- the elastic strain 

component. Since the elastic strain is a function of the applied 

stress, a stress reduction is necessary to maintain the total strain 

constant. As indicated by equation (8), the creep rate is governed 

by the stress (or effective stress) which decreases steadily during tr.e 

course of stress relaxation. Ideally, c~eep is terminated as the 

effective stress drops to zero; i.e., when the applied stress equals 

the back stress. Thus, the back stress can simply be defined as the 

asymtotic value reached at the end of a stress relaxation test where 

the stress relaxation rate is approaching zero. The stress relaxa~ion 

test offers a convenient way of determining the internal stress 

provided that it remains constant during the test. This condition 

cannot be satisfied at high temperatures where thermal recovery becomes 

important and can cause a substantial reduction of the internal stress. 

An early attempt has been made to analyze experimental stress relaxation data for Hastelloy-X using an indirect approach suggested by 

Gupta and Li [51]. The method failed to produce meaningful results 

for the back stress and stress exponent at 978 K and above, indicating the significance of recovery during stress relaxation at these 

elevated temperatures. 

In this research program, stress relaxation tests were only 

used in the temperature range of 755 K to 922 K (0.49 - 0.60 T ) at 

m 

different points around the saturated hysteresis loops to determine 

-.-- - ~ .-.--;-- :-------- --~- .. ------.- ---- _.-,-,._----_. - -, .... - --.~':"--- ._-- --,-:-- --, ---- .- . -- ... -,.- ~-- ~- --------

-,' ..-

. ': -_.' , __ : _.s 

_.J~ ~~:­ ~::. ...... ~ ~,- . 
42 

various values for the back stress. Due to the experimental set up, 

diametral strain rather than axial strain is held constant during 

stress relaxation and the resulting relation between the creep rate 

and the stress-rate is given by equation (14), repeated as below: 

(14) 

The value of the stress exponent n can be obtained by plotting 

ln E. vs. In(a. - m. The least square curve fitting technique can be ~r 

used to give the best estimation for n.. With the values for the back 

stress and stress exponent determined, the drag stress can be computed 

by rewriting equation (8) as follows: 

-1 

K = la - nll€. In ~r 

Strain drop, stress transient tests. As discussed above, the 

(15) 

stress relaxation test is unsuitable for back stress determination at 

high temperature due to thermal recovery. The strain drop, stress 

transient test is employed as an alternative in the temperature range 

of 978 K to 1144 K (0.64 - 0.75 T). A schematic of this approach is 

m 

shown in Fig. 7, which is based on the same principle as the "incremental unloading technique" [47] and the "stress transient dip test" 

[48] described in the previous chapter. During strain cycling, the 

unloading strain-rate is temporarily interrupted and while the strain 

is held constant, the stress transient is monitored. Referring to 

Fig. 7 (p. 43), if strain cycling is stopped at point A, positive 

stress relaxation will occur, indicating that the applied stress is 

\ 
"- . 

,,-, 

--,~ . , - :.~ 

(J 

E: 

t 

(J 

-+---.;;......;~ 0 = Q 

t 

Fig. 7 Schematic for the strain drop, stress transient 

test 

(J 

Fig. 8 Schematic for the strain-rate cycling test 

_______ 4_~_ .;;;o-,"j~~- - - - , 

: ... ' 

43 
44 

larger than the back stress. On the other hand, if the strain is held 

constant at point C, negative stress relaxation will occur since the 

back stress is now greater than the applied stress. The internal 

stress is given by point B where no stress transient is observed. 

Therefore, by introducing a series of load interruption on successive 

cycles at various points on the tension/compression stress reversal 

portion of the hysteresis loop, the internal stress value for the 

peak stress on the hysteresis loop can then be determined. The load 

transient during the first quarter second immediately after the 

interruption is in part due to machine characteristics -and thus 

should be disregarded. The transient after this time is due principally to real material behavior. 

Strain-rate cycling tests. The strain-rate cycling test is 

designed to determine the stress exponent in the temperature range of 

978 K to 1144 K (0.64 - 0.75 T). The test is performed at the 

m 

"plateau" of the hysteresis loop where the stress has saturated to a 

constant value. At the point of interest, the strain-rate is abruptly 

changed and the new flow stress is noted before returning to the 

original strain-rate. A schematic of the test is depicted in Fie. 8. 

It is assumed that the drag stress remains relatively constant during 

the strain-rate change and the reduction in flow stress is mainly a 

result of a decrease in the back stress. The physical basis for this 

assumption is that the drag stress is associated with the formation 

of a stable cell structure which is not expected to change appreciably 

during the strain-rate change. It will be shown later that the strain 

drop, stress transient test does indicate a decrease in back stress as 

-.':,f- :.-
45 

. ~ . -

, ",:;' 

the strain-rate is reduced. The fact that the stress remains constant 

at the plateau region indicates a balance between strain hardening and 

dynamic recovery and is equivalent to steady state creep. This implies 

. ,:.': 

that the total strain-rate in this region equals the rate sensitive 

inelastic strain-rate as described by equation (8). Referring to Fig. 

8 (p. 43). if a and a denote the flow stress at two different strain1 2 

rates E and E , then the value of the stress exponent n can be com1 2 

puted as 

n = In(E /E )/In[(a - ~ )/(a - ~ )] 

1 2 1 1 2 2 

(16) 

assuming K remains constant (during the period when cr drops to cr ), 

1 2 

~ and ~ being determined previously by the strain drop, stress 

1 2 

transient test. With the value of n now known, the drag stress can 

be calculated from equation (15). 

. ~ . 

'. ' .. 

': - , ,f',' 

46 

CHAPTER V 

EXPERIMENTAL RESULTS, ANALYSIS AND DISCUSSION 

The results of the experimental work for this research project 

will be described here in detail. The equations required for data 

analysis will be presented first, followed by the discussion of 

hysteresis loop and stress relaxation results. The various material 

constants and their physical significance will also be considered. 

Data Reduction 

Since diametral extensometer is employed in the experimental 

work, data reduction is performed in terms of diameter change. 

Engineering stress and engineering strain are employed in data analysis. Since the strain range in our experiments is limited to ± 1%, 

it would make little difference if true stress and true strain are 

used instead. Diameter measurements can be converted into axial 

strain using the following relationship: 

E - a (1-2v) _ 2 dD 

t - E D (17) 

o 

where a is the applied stress, E and Vare the Young I s Modulus and 

the Poisson's ratio, dD and D represent the diameter increment and 

o 

the initial diameter respectively. The axial inelastic strain is 

calculated as the difference between the total strain and the elastic 

strain, or 

. ---~ ------ --- .,--,-.-.-~­ ....... -.. -_. 

(18) 
-, ) 

The total strain-rate is represented by equation (13) which is 

repeated as follows: 

-2 dD 

. Do dt 

E = ----~-~~-----­ t 1 _ .!. dO' 

E de:t 

(l-2v) 

The inelastic strain-rate is given by 

-2 dD 

E:i = D dt 

o 

1 (l-2v) dO' - -E- de:

t 

where ~~ is the programmed diametral strain-rate and dO/de:t is the 

-?train hardening rate ,vhich is expressed as 

where dD/do is the diameter change rate with respect to stress. A 

complete derivation of the above equations is given in Appendix C. 

47 

(13) 

(19) 

(20) 

Once the strain hardening function f (O"Omax) is defined, the 

rate insensitive, inelastic strain-rate can be separated from the 

total inelastic strain-rate using equation (6). 

Hysteresis Loop Results 

The rate insensitive, inelastic strain is defined by the strain 

hardening function which is a dependent. Hysteresis loops were 

max 

run at room temperature for various strain rang~s which correspond to 

different a values. The experimental results are shown in Fig. 9

1 

max 

lStrain-rates shown in figures represent average values. 

- ,- -"-'--'----,-_':'-'_'--
700. 

600. 

sea. 

400. 

S 30a. 

T 200. 

R 

E 100. 

S 

S 0. ~----~~~~~~~~r-~-+--+-----­

( 

M -100. 

P 

A -200. 

) 

-30a. 

-400. 

-500. 

-see. 

-700. -1.2 ~~----------+-~----~-------+-- -.8 -.4 0. .4 .8 1.2 

-1. -.6 -.2 .2 .6 1. 

STRAIN(Y.) 

Fig. 9 Saturated hysteresis loops for Hastelloy-X 

at roon temperature with strain amplitudes of 0.05%, 

O.1%,O.2%,O;3%,O.4%,O.5%,O.6%,0.8%and 1.0% 

48 
. ,--" 

. ", .. '- , 

:.--' 

, . 

'.,: .' .. ; ", " 

," .. 

. . :: :.~ : -: 

,~ ".' 

- ,,>, ~ 

, 

:1 

with values of the strain hardening function tabulated in Appendix D. 

The effect of the maximum stress on the stress-strain response is 

apparent by comparing the values of the strain hardening function 

at different strain amplitudes. It can be seen that for a given 

stress, the value of the strain hardening function decreases as a 

function of cr 

max 

In other words, the strain hardening rate at a 

given stress level decreases as cr increases. As discussed in 

max 

Chapter III. strain hardening is physically associated with the 

building up of long range internal stress. the magnitude of which 

depends on the total strain (or applied stress). Upon load reversal. 

dislocations initially stopped by dislocation barriers become 

mobilized, making inelastic deformation easier in the opposite 

direction. A higher a value corresponds to a higher back stress 

max 

which results in a lower strain hardening rate upon load reversal, 

since inelastic deformation can occur easily .. On the other hand. a 

small back stress (or cr ) will not offer much assistance upon load 

max 

49 

reversal. which accounts for the higher strain hardening rate observed. 

The effect of the back stress results in a lower yield strength in 

reverse loading. This phenomenon is referred to as the Bauchinger 

effect which results in "kinematic hardening" in crystalline materials. 

The strain-rate dependence of Hastelloy-X at room temperature 

is studied by strain-cycling tests at three different strain-rates as 

shown in Fig. 10. The hysteresis loops at the higher strain-rates 

are identical while a slight reduction in the peak stress of the 

hysteresis loop is observed at the lower strain-rate. Nevertheless, 

the difference is small and the material can be considered rate 
-". ,.,.-. ;,," 

.: ",' . 

700. 

600. 

500. 

400. 

5 300. 

T 200. 

R 

E 100. 

5 

5 Ia. 

( 

M -100. 

P 

1=1 -200. 

) 

-300. 

-400. 

-500. 

-600. 

-70B. 

1.0X10-3s -1 

3.3X10-4S-1 

-1.2 -.8 -.4 Ia. .4 .8 1.2 

-1. -.6 -.2 .2 .6 1. 

STRAIN(%) 

Fig. 10. Saturated hysteresis loops for Hastelloy-X 

at room temperature 

50 

i 

, 

-t 

l 

! 
",- .' 

"insensitive". Hysteresis loop results at 533 K (Fig. 11) are almost 

identical with those at room temperature, indicating the alloy is 

basically rate insensitive over these temperature and strain-rate 

ranges. At 755 K (0.49 T ), the hysteresis loops change slightly m 

(Fig. 12a), though the difference observed is still not significant. 

51 

The higher strain hardening rate at these low temperatures are characterized by the rounded corners of the hysteresis loops, which is a 

good indication that inelastic deformation is mostly rate insensitive. 

The absence of significant dynamic recovery at 755 K is evidenced by 

the small amount of rate sensitive inelastic strain observed at this 

temperature (Fig. 12b). The rate sensitive inelastic strain component 

is obtained by subtracting the rate insensitive inelastic strain 

(defined by room temperature hysteresis loops) from the total ine.1astic 

strain. Over this temperature range, the shape of the loading portion 

of the hysteresis loop is similar to that of the unloading portion. 

This is expected to be the case if the overall deformation process 

remains the same during loading and unloading. At these relatively 

low temperatures, deformation occurs almost exclusively in the grain 

matrix via dislocation glide and limited recovery through dislocation 

cross-slip. Upon load reversal the direction of dislocation motion 

changes, but the relative significance of the deformation mechanisms 

remains the same. As will be discussed later, directionality of the 

loading path can affect the reSUlting stress-strain behavior when 

grain boundary sliding becomes important. 

At 533 K and 755 K,Haste11oy-X has been found to exhibit 

. \ 

inverse strain-rate sensitivity; i.e., the strain hardening rate of 
:.: :";, 

. 

", " . _. ' ,.1 

'-

'700 . 

600. 

500. 

400. 

S :300. 

T 200. 

R 

E 100. 

S 

S B. 

( 

M -100. 

p 

A -200. 

) 

-300. 

-400. 

-500. 

-600. 

-5 3.3X10 s 

-1 

3.3XlO _L. -1 s 

1. OXlO- 3s-1 

-'700. ~~--~---~-+--~~~~--+-~--~--~-

-1.2 -.8 -.4 B. .4 .8 1.2 

-1. -.6 -.2 .2 ~6 1. 

STRAIN(%) 

Fig. 11 Saturated hysteresis loops for Hastelloy-X 

at 533K 

52 

I 

I 

r 

I 

I 

1 
j; 

',I s:.: 

'I: 

'i 

,1 i 

j 

'"j 

\' 

,"\ 

:1 , 

:! 

',I ~ •• 

, 

,[ 

<. 

;~ .~ .. ---

700. 1 

500. --j 

400. 

5 300. 

T 200. 

R 

E 100. 

'.J.,. 

I' 3.3X10-\-1 

3.3XlO-II ·1 , s 

1.0XlO-3 -1 s 

~ 0. /I /I 

C 

M -100. 

P 

A -200. 

) 

-300. 

-400. 

-500. 

-600. 

-700. I I I, 

700. 

600 

500. 

400. 

5 300. 

T 200. 

R 

E 100. 

5 

5 0. 

( 

M -100. 

P 

A -200. 

) 

-300. 

-400. 

-500. 

-600. 

-700. 

(/f 

-10-" t.::; 

'.,. ' 

., .... 

3.3XlO· \ .. 1 

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STRAINC%) RATE DEPENDENT H£LASTIC STRAINC%) 

(a) (b) 

Fig. 12 Saturated hysteresis loops for Hastelloy-X at 155K. (a) Stress vs. 

total strain. (b) Stress VS. rate dependent(or rate sensitive)inelastic 

strain 

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54 

the hysteresis loop increases at reduced strain-rate, contrary to 

normal behavior. This phenomenon is believed to be resulted from 

dynamic strain aging ..... hich is associated r..1ith the formation of 

solute atom atmospheres around dislocations. Dynamic strain aging 

involves solute diffusion to dislocation sites. This can only occur 

.... hen the diffusion rate of solute atoms is similar to the dislocation 

velocity. Since atomic diffusion is temperature dependent and dislocation velocity is stress-rate controlled, dynamic strain aging 

can take place only at a suitable combination of temperature and 

strain-rate. At 10 .... temperatures, atomic mobility is low due to 

limited thermal assistance, and dynamic strain aging does not occur 

in general. As the temperature is raised, sufficient thermal energy 

will enable solute atoms to catch up with moving dislocations and the 

drag stress exerted by the dislocation atmospheres cause the material 

to strain harden. At higher temperatures, the dislocation atmospheres 

become dispersed as a result of an increased entropy driving force and 

dynamic strain aging is again unimportant. Since dynamic strain aging 

is possible only when the solute diffusion rate is compatible with 

the dislocation velocity, it .... 111 occur at different strain-rate 

ranges for different temperatures. The higher the temperature, the 

higher the strain-rate range in .... hich dynamic strain aging will take 

place. 

Experiments conducted on Hastelloy-X specimens reveal that 

inverse strain-rate sensitivity takes place in the temperature range 

from 533 K to 755 K (0.35 - 0.49 Td for the strain-rates used in this 

study. Another phenomenon associated with dynamic strain aging is the 
:' 

~ :. ,;. , . .. ' . . . ~. 

- ..... 

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so called Portevin-LeChatelier effect. This is manifested by 

irregularities in the stress-strain curve which is often accompanied 

by sharp load drops. This effect is observed from 533 K up to 922 K 

(0.35 - 0.60 T ) for Hastelloy-X specimens but becomes negligible m 

55 

above 811 K (0.53 T). It ,has also been found that at a given tempera- m 

ture the Portevin-LeChatelier effect is more pronounced at higher 

strain-rates for the strain-rate range considered in this research. 

The effects of dynamic strain aging have also been reported by other 

investigators [17, 50J. 

As the temperature is raised above 0.5 T , rate effects become 

m 

apparent. At 811 K and 866 K, the size of the hysteresis loops become 

smaller due to dynamic recovery (Fig. 13a and 14a). However J the 

hysteresis loops remain similar in shape to those at lower temperatures. indicating that a substantial portion of the inelastic strain 

is still rate insensitive. As shown by Fig. 13b and 14b, the rate 

'sensitive inelastic strain represents about only half of the total 

strain. At 922 K, the more intense thermal activity is reflected by 

the relatively "square" appearance of the hysteresis loops (Fig. 15). 

At 978 K and above (Fig. 16 - 19), dynamic recovery represents a 

dominant deformation process and the inelastic strain is essentially 

all rate sensitive. 

It has been observed that hysteresis loops are asymmetric; i.e. 

the material appears to be stronger in compression than in tension. 

This phenomenon can be explained in two ways: (1) material deformation 

is history dependent, i.e. the material has a memory of past events; 

(2) at high temperatures where history dependence is not. significant, 

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RATE DEPENDENT INELASTIC STRAIN( %) 

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Fig. 13 Saturated hysteresis loops for Hastelloy-X at BIIK. (a) Stress vs. 

total strain. (b) Stress'vs. rate dependent(or rate sensitive)inelastic 

strain 

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STRAIN(%) RATE DEPENDENT It£LASTIC STRAIN( %) 

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Fig. I), Saturated hysteresis loops for Hastelloy-X at 866K. (a) Stress vs. 

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RATE DEPENDENT INELASTIC STRAIN(%) 

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Fig. 1') Saturated hysteresis loops for Hnstelloy-X at 922K. (a) Stress vs. 

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total strain. (b) Stress vs. rate dependent (or rate sensitive)inelastic 

strain 

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RATE DEPENDENT INELASTIC STRAIN(%) 

(b) 

Fig. 17 Saturated hysteresis loops for lIustelloy-X at l033K. (a) Stress VS, 

total strain. (b) Stress vs. rate dependent(or rate sensitive)inelastic 

strain 

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STRAIN 00 RATE DEPENDENT INELASTIC 5TRAIN(~) 

(a) (b) 

Fig. 18 Saturated hysteresis loops for Hastelloy-X at lo89K. (a) Stress vs. 

total strain. (b) Stress V~. rate dependent(or rate sensitive)inelastic 

strain 

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RATE DEPENDENT INELASTIC STRAIN(%) 

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Fig. 19 Saturated hysteresis loops for Hastelloy-X at 114!IK. (a) Stress VS. 

total strain. (b) Stress VS. rate depertdent(or rate sensitlve)jnelastic 

strain 

" .;.' 

1. 

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the apparent asymmetry is caused by grain boundary sliding. In this 

research program, all specimens were initially loaded in tension 

followed by compression. It is expected that if loading starts in 

,compression, history effects will be manifested in the form of a 

higher peak stress in tensio~ than in compression. Hysteresis loop 

tests at low temperatures reveal a considerable amount of asymmetry 

during the first cycle. Experimental results show that below 978 K 

(0.64 T ), initial asymmetry may reach a maximum of about 17%. As 

m 

63 

strain cycling continues, the difference becomes progressively smaller 

and upon saturation, asymmetry is reduced to a minimum, though the 

alloy still maintains a higher peak stress in compression. The higher 

strain hardening rate in compression is the result of an increase in 

dislocation density during initial loading and their mutual interference with each other upon load reversal. As strain cycling continues, a stabilized cell structure begins to form which is equally 

effective against dislocation motion in eithe,r direction. As a result, 

the hysteresis loop becomes more symmetric upon saturation. Experimental results indicate that below 978 K (0.64 T ), the degree of 

m 

asymmetry of saturated hysteresis loops does not exceed 2%. At higher 

temperatures,the initial asymmetry is about 5% and remains relatively 

constant until saturation~ Recall that cyclic hardening is insigni~ 

ficant at high temperatues and saturation can occur in as few as two 

cycles. It can be seen from the above discussion that saturated 

hysteresis loops are more symmetric at lower temperatures despite 

the more significant asymmetry observed in the first cycle. The 

greater degree of asymmetry for saturated hysteresis loops at higher 

" , 

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temperatures is ascribed to grain boundary sliding, which occurs more 

readily for tensile than for compressive loading. For example, at 

temperatures above 978 K a "plateau" region is generally observed at 

the top of the hysteresis loop while the material seldom attains a 

constant stress in compression. Since the amount of asymmetry is 

relatively small, even at these higher temperatures, grain boundary 

sliding is not expected to play a major role in the overall deformation process. However, triple point cracking may occur as a result 

which can drastically reduce the fatigue life of the material. 

Stress Relaxation Behavior 

Stress relaxation tests for Hastelloy-X at various temperatures 

and strain-rates are shown 'in Fig. 20 - 28. Stress relaxation tests 

are utilized for back stress determination from 755 K to 922 K (0.49 

- 0.60 T ) where thermal recovery is not significant. At higher m 

temperatures. stress relaxation response provides a good indication 

of the degree of rate sensitivity the material experiences. 

As discussed earlier, stress relaxation is a result of rate 

dependent inelastic deformation or creep that occurs at constant 

strain. The rate sensitive inelastic strain-rate is a function of 

both the effective stress and the strain-rate prior to the start of 

the stress relaxation test. Room temperature stress relaxation 

tests were conducted to evaluate the relative significance of rate 

sensitive inelastic strain at this temperature. As shown in Fig. 20 

(p. 65), the relaxation rate is small initially and becomes insignificant after a few minutes. The low relaxation rate is a result of 

64 

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;'.,' 

. ~ ~,,- ;- ~ " 

.",. 

, : 

:500. 

.cell. 

3110. 

, " ....... 

:J. 

..' ,,' 

,,',' 

:,1" .' .' '.,' ~',.J :-,.:..~ 

.cell. 

3011 • 

200. 

---:---- s 

T 200. S 

T 

4 

6 9 IB 

Tlr£(l1lrt. ) 

1.OXIO-JS-1 

12 16 

14 19 

R 

E 100. 

S 

S B. e 

11 

P -100. 

A 

) -ZOO. 

-3110. 

-4IlII. 

-5011. 

8 4 9 

i! 6 

Tlr£II1IN. ) 

].JXIO-~S-l 

Ii! 16 

18 14 

R 100. 

E 

S 

S 8. ~------------~--------------- ( 

11 

P -100. 

A 

I 

-200. 

-3011. ].]XlO~5S-1 

_.cell. ~I--__ --~~~ __________ --r-__ --__ 

8 .. 2 6 9 18 

lIr£ll1lrt.) 

Ii! 16 14 19 

Fig. 23 Stress relaxation tests for Hastelloy-X at 866K executed at various 

points on saturated hysteresis loops 

" 

(J\ 

en 

,., 

\' ' 

" " 

L 

f 

to 

t'." 
. \. ! 

:SW'I ~'I 4IJil. 

4IJil. 300. 

300. 

~ ZOO. 5 CW. 

~ 5 CW. 5 

T T T A A A 100. [ 100. [ 100. [ 

5, 5 5 

5 e. s e . s e. . ( ( c 

" P -100. P " -100. " = P -100. R ,.~ R ~ A ) -cw. , -cw. , 

-cw. -300. -300. 

.,-4011. 1.0XlO-3S-1 1.lXIO-~S-1 --. -300. 1.3XlO-~S-1 

-500. ~I--------~----~----~~~--~ -500. e --. ~I------~--------------------~ Z 

4 o IZ 16 e 4 B IZ Iii " 4 o Ii Ie 14 Z Ii 14 IB Z Ii Ie 

TI~'"IIi. , 

Ie 

TlrE'"IIi. , TlrE'"lti. ) 

Fig. 24 Stre~s relaxation teats for Hastelloy-X at 922K executed at various 

points on saturated hysteresis loops 

IZ 16 

14 IB 

.. ·;:·-... .:-~~;:.k:"· 

. ~; . 

'1,-

:<~< " 

i 

r [ 

I 

t'-

t, 

1, 

L 

, 

>, 

1-: 0> 

\0 
i 

.J 

'~-.. 

", ,_-1 

i 

i 

";,j 

,I 

" I 

.; I 

····1 I 

I 

, 

,! 

"', \ 

. , 

';.!-.( 

.. ~ 

\ i 

"J 

-}-i 

';-', 

4W.l _.1 300. 

Z5B. 300. 300. 

200. 

alia. an. 150. S S S T T T 100. 

R 100. R 100. R [ [ [ 50. 

S S S S II. S II. S II. 

e , , " " " -511. P -100. 

~ P -100. P A 

I 

A A -100. 

I I 

-200. -200. -150. 

-200. 

-300. l.OXIO-1S-1 -300. 1.lXlO- 4S-1 3.JXlO-~S-1 -Z5B. 

-_. ~1--~~--~--____ --+-____ ~ ____ ~--1 

-ue. ~I--------+--+--.-~------------~ -:na . II. Z. 4. 6. B. 1111. III. Z. 4. 6. B. 1111. II. Z. 4. 6. B. 1111. lZ. 

1. 3. 5. 7. 9. ll. 1. 3. 5. 7. 9. ll. 1. 3. 5. 7. 9. ll. 

Tlt£'"II'1.1 Tlt£'"IN. I Tlt£l"II'I.1 

Fig. 25 Stress relaxation tests for Hastelloy-X at 978K executed at various 

points on saturated hysteresis loops 

:' : *''''_=>;,," __ -... .... , _______ .. _,_ ... '_ ...... ___ -, ~~---="'-'~~C."F j l • • ~ ;;~ 

...... 

a 

f' 

! 

;'.0 

y 

\: 

I 

l 

t' 

l 

; 

,~ . 

I:: 

I 

[:' 

I 

L 
:.j 

. i 

I 

I 

',; 

! 

f 

'j 

l 

I 

Ii 

-" ~ ~ -~ 

S 

300. 

Z50. 

ZIIJIJ. 

IS1ll. 

T 100. 

ft 

E SIll. 

'.: 

S 

S I 8.' ~---------------------------- 11 -58. p 

R -tOO. 

) 

-ISO. 

-z,oo. 

-Z50. 1.0X1O,3/j-1 

-300. ~I ________ ~~ __ ~ __ ~-P __ .-__ --~~ 

II. 2. 4. 6. 8. til. 

1. 3. 5. 7. 9.' U. 

TI!'EII'IIN. ) 

ii!S0. 

ZIIJIJ. 

150. 

s 100 • T 

A 

[ SO. 

S 

s ( O.,r-----------~---------------

1'1 

P -se. A 

) -tOO. 

-150. 

-zw. 3.3XlO-1

'B-1 

-Z50. I~ ____ ~--______ ~ __ r-____ +--' ____ ~ 

8. 2. 4. 6. 8. 10. 

t. 3. 5. 7. 9. at. 

H!'EII1Itt. ) 

i .. 

iDil. 

150. 

lim. 

11 

T 

R 50. E 

S 

5 8. f 

1'1 

P -'50. A 

) 

-100. 

-150. I 1. ]J(ln-~S-I 

-200. ~I----------~--~-----.--------~~ II. 2. 4. 6. 8. 10. 

1. 3. 5. 'I. 9. II. 

T1!'EII1IIf. ) 

Fig. 26 Stress relaxation tests for Ilastelloy-X at l033K executed at various 

points on saturated hysteresis loops 

~"r __ '. ______ '~·"-"_·~ \. 

" .' 

-"'." 

2i 

~~ i1! 

.o'U C> 

;poe;,) 

rm =I ...... -<00 

"'J 

.... 

::\ 

) 

, 

I, 

I 

t 

r 
ZOO. 

150. 

100. 

5 

T 

A 50. [ 

5 

5 e. C 

" p -se. A 

) 

-100. 

-150. 

-zoo. II. 

.5 

" 

,'j 

'j 

':,. .. 

.'.' 

, ' 

~ .... '._ ::.' J 

I. 2. 3. 4. 

1.5 2.5 3.5 4.5 

TltE("IN. ) 

5 

T 

A 

[ 

S 

6 

, 

" p 

A 

) 

5. 

17':i. 

150. 

125. 

100. 

15. 

50. 

25. 

e. 

-25. 

-se. 

-15. 

-100. 

-125. 

-150. 

-115. e., I. 2. 

.5 1.5 

TltEC"IN. ) 

'-'-, ,,'-

128. 

100. 

BIl. 

611. 

5 

T 40. 

A 

[ 28. 

S 

s e. , 

" -28. p 

A -40. ) 

-60. 

-00. 

3. ]XIO-~S-l -100. 

-120. 

3. 4. e. I. 2. 3. 4. 5. 2.5 3.5 .5 1.5 2.5 3.5 4.5 

lItE("IN. ) 

Fig. 27 Stress'relaxation tests for lIastelloy-X at lo89K executed at various 

points on saturated hysteresis loops 

-j =.~""':~-__ ,____ ,_.~~ ______ ,.~~ __ ..-r_._ .... ~ •. _ -
;1 

I 

-l~-~~~(: 

O"4'~!flt~~J~~i~;\~~::",t,,:~:.: ",'., 

100. 

lIB. 

68. 

S 411. 

T ','f" R 

E zo. S 

S 

( 

" P 

A 

) 

c': -Q!). 

1'.OXlo-1a-1 -fIB. 3. )XlO-~S-l 

'-IZO. I , • -100. • - . • 

II. I. Z. 3. 4. 5. 6. ,1. Z. 3. 4. 5. .S I.S Z.5 3.5 4.S 5.5 .5 loS Z.S 3.5 4.5 5.5 

TlrECMIN. ) TlrEC"I". ) 

,,' 

-. - ~.- '~ 

:~ 3B 

S 

T Z8 

R 

E III 

: 

( B~I--------------~~------- M -Ill 

P 

A-ZO 

1 

-~ ).3XlO-5S-1 

-Q!)~I------~----------~--~ II. 1. Z. 3. 4. S. 6. .S 1.5 Z.S 3.5 4.5 5.5 

TlNEIMI".1 

Fig. 28, Stress relaxation tests for Hastelloy-X at l144K executed at various 

points on saturated hysteresis loops 

_._._~_~_-:---"'O. ~_.,~ "-'-'~--"'r-,--~""'-' - -..-'-

.... J 

W 

\' 

c· 

;,', 

~ " 

'I' 

j"\ 

t I: 

f" 

i 

i 

! ,'.: I, 

f " r 

\ : 

i 

l 

I I, , ~ .. '" 

!-,' 

t" 

I', 

I, 

tf': [! ~ 

,. 

I.. 

t, 
. -'-..- !. .. 

; .. 

- ,-..: . 

74 

the high back stress (or small effective stress) due to the absence 

of significant thermal effects. When orie of the tests was extended 

for another nine hours, the stress was found to decrease by an additional 5%. At 755 K, the relaxation rate is slightly. higher but not 

significant. The observation compares favorably with hysteresis 

loop data over this temperature range. The internal stress at this 

temperature has a magnitude equal to about 80% of the applied stress. 

Above 811 K the stress relaxation rate increases steadily as a result 

of a lower back stress due to more intense thermal activity. As the 

temperature rises above 978 K the significance of thermal recovery is 

indicated by a sharp reduction in stress during the first several 

minutes of stress relaxation. Experimental results show that in this 

temperature range, the stress can drop by 50% or more for the first 

30 s of stress relaxation. In this temperature range,_ a plot of 1n Eir 

vs. In (cr - Q) often reveals a gradual reduction of the stress exponent "n" as indicated by the slope of the plot (Fig. 29). This 

observation indicates a possible reduction of the back stress and/or 

drag stress during relaxation as a result of thermal recovery. The 

" decrease in the stress exponent may also signal a possible change of ", . -

~ . .,-, .' . 

- . -.~ 

. -: -.' .... , 

the mode of recovery during stress relaxation. The stress exponent 

at the beginning of the stress relaxation test usually has a value of 

about 4, a typical number for power law creep or dynamic recovery. 

Toward the end of the test, the stress exponent drops to a value of 

2 or less, which is similar to values obtained for diffusional creep 

or static recovery. In other words, the recovery process has changed 

from one dominated by dislocation climb and cross-slip to one which 

,,'\ " . -,.-: ". . ~.' ," 

,~:-~ ;y-~~;~:~F( .:,~;.\, 

:,~. ~~-;:,~.~-. ~-.j:.. :' .. ~'";. '. .:':. ! 
" '. - : .. -~ 

'r, "',1.- 10 1 

EFFECTIVE STRESS(MPA) 

Fig .. 29 Effective stress vs. rate sensitive, inelas!~c_1 

strain-rate at 1144K for a total strain-rate of lXlO s 

.... : .. 

. ,', -~. 

7S 

'''-,' 

i--, . I 

I 

I 

i.-

,i? 

i .'-~ 

" ~~ 

:.~. . -,t 

,:;.;-

-.J" ~." .. 

~:o~ 

. I ,~' 
76 

involves atomic diffusion across grain boundaries (or subgrain boun-

.' ' .. ,' .. " daries) . 

. '-', 

It is observed that the alloy experiences negative stress relaxa-

'''''-'. ' 

tion on the compressive side of the hysteresis loop. As shown in 

equation (14), negative creep occurs in compression which can only be 

compensated by a "positive" stress-rate. An alti:!rnative explanation 

can be given by the fact that the back stress is less negative than 

the compressive applied stress thus resulting in negative stress 

relaxation. 

It has been found that the inelastic strain-rate at the beginning 

of the stress relaxation test is often less than that measured just 

prior to the start of the test. The "discontinuity" is more significant at lower temperatures and gradually diminishes as the temperature 

increases. Whether this phenomenon is real material behavior or 

merely the result of inaccurate measurements is a subject of much 

controversy. In the next chapter, arguments offered by other investigators will be discussed and an attempt will be made to explain the 

phenomenon in terms of the uncoupled constitutive model. 

Material Constants and Their Physical Significance 

The various materal constants required for the characterization 

of the rate sensitive inelastic strain component are summarized in 

Table II. The elastic constants tabulated in Table III are taken from ~ '. : 

ref. [52J. The Poisson's ratio given in Table III (p. 78) is based on . ~" 

room temperature data and is assumed to be constant. In practice, it 

.; h~s a weak temperature dependence. For example, the Poisson's ratio 
, , 

-1 

. , 

, .... , 

Table II. Values for Back Stess (~) (for 0 ~ 0 ), max 

n 

n, Drag·Stress (K) and K . 

11 K 1 

Temp. (MPa) (HPasn) 

Strain-rate n Strain-rate (K) (10-"S_I) (10-" s -1) 

10 3.3 0.33 10 3.3 0.33 10 

755 468 468 474 5.50 390 51] 1045 1. 78 X 

811 252 265 318 4.96 1186 1260 1704 1. 77 x 

866 255 226 237 5.57 905 978 1274 2.94 x 

922 161 164 l63 4.31 1690 1829 2597 8.17 x 

978 141 136 115 5.57 800 800 800 1.48 X 

1033 118 111 82 4.75 672 672 672 2:69 x 

1089 76 66 43 4.70 488 488 488 4.32 X 

1144 41 38 ' 28 3.63 532 532 532 7.85 x 

155- 922 K: ~, n & K obtained from stress relaxation tests. 

978-1144 K: n obtained from strain-rate change tests. 

11 obtained from strain drop, stress transient tests. 

K assumed to be constant for all strain-rates. 

n 

K-I 

(MPasn) 

Strain-rate 

(10-"s-l) 

3.3 

1Ol't 8.05 X 10l't 

10 15 2.39 X 1015 

1016 4.53 X 1016 

1013 1.15 X 101

" 

1016 1.48 X 1016 

1013 2.69 X 1013 

1012 4.32 X 1012 

10 9 7.85 X 109 

".-,' 

. " '-~ ~ ',-,' 

-. - " " ' .. < 

-.. :,' ." 

0.33 

4.03 X 1016 

1.07 X 1016 

1.98 X ]0 17 

5.20 X 1Ol't 

1.48 X 1016 

2.69 X 1013 

4.32 X 1012 

7.85 X 10 9 

:; , 

," ., ,,". 

"-'.<-_:: 

, ' . -, ~ .' ,- .. 

' .. ~ .~- . 

i-· 

\

'i' 

'-.' 

l 

~ ; 

- l 

" 

i· 

l-' 

I 

i 

, 
. - . " 

i_ ~." 

.... :' .... :~:' .. - '.' ,. -' ". 

.- .. " '--"~' .. ' .. -" ... :_- -----~~:.:....-~~ .. -~ .. :' 

Table III. Values for Young's Nodulus and Poisson's Ratio at 

Different Temperatures. 

Temperature TIT * Young's 

(K) m Modulus 

(GPa) 

294 0.19 197 

533 0.35 182 

755 0.4-9 166 

811 0.53 162 

866 0.57 158 

922 0.60 154 

978 0.64 150 

1033 0.67 146 

1089 -,. 0.71 142 

1144 0.75 137 

v = 0.32 (assumed constant for all temperatures) 

*Melting range is 1533 - 1628 K 

-.. - _.- -~.,~ - ',--,-" ,---, -,.-'. -.. 

", ,.,' 

78 
" , 

79 

for Hastelloy-X has a value of 0.34 at 1255 K [53]. Since the change 

is rather small, it will have a negligible effect on the resulting 

strain calculations. tHth the strain hardening function given in 

, :. ' 

Appendix D and the material constants defined in Table II '(po 77), 

the total strain at each temperature and strain-rate can be uncoupled 

into different components. 

It is noted in Table II (p. 77) that the stress exponent varies 

from 3.63 to 5.57. The value for n does not seem to be a sensitive 

function of temperature although slightly higher values are generally 

obtained at lower temperatures. This is in contrast to unified 

models where the n values, at lower temperatures may be as high as 

60-100. Strain-rate cycling tests conducted at lower temperatures do 

result in higher n values if analysis is made assuming all inelastic 

strain as being rate sensitive (or rate dependent) as the unified 

theory does. 

Values of the back stress measured at the peak of the hysteresis 

loops are shown in Table II (p. 77). At lower temperatures, the 

internal stress values are also evaluated at zero strain. Based on 

experimental data. back stress at other points can then be calculated , 

using equation (8). The results are plotted in Fig. 30-37. At 755 K, 

the higher strain hardening rate observed can be seen as a result of 

a high internal stress due to limited thermal recovery (Fig. 30) 

(p. 80). The back stress at this temperature is relatively independent of strain-rate (Table II) (p. 77), which explains the rate 

insensitive nature of the hysteresis loops. Between 811 K and 922 K, 

the internal stress declines steadily as a function of temperature. 

'-; . 

I 

I 

I' 

I 

~ ------""- ¥"~ -.--- -, I~ 

< 

-f 
. ;j 

I 

I 

1 

1.', 

. ~ ~ l ~, 

" 

"-~. '"". '" I" 

1iiDII. ~tr~w!J 600. 700. 5011J. !luck 3lnHHl 5011J. 600. 

-. -. --- 5011J. 

300. 300. -. /,..t/ I s s S 300. T ZOO. T ZOO. t 200. R R R 

E 100. E Ifill. E 100. 5 I S S e. S iii. I iii. ( ( ( 

1'1 -100. 1'1 -Ifill. 1'1 -Ifill. 

P P P A -200. II -200. A -3Je. ) ) ) 

-300. 

-300. -300. 

--. --. --. -500. 

-500. 1.IJXlO-1S- 1 -500. l. IX 10 -\;-J -£00. 'L 1XIQ-')S-1 

-£00. -£00. 1 ~. -700. ~I---------------+---------------- -1.2 -.8 -.4 e. .4 .8 1.2 -1.2 -.8 -.4 iii. .4 .9 1.2 -1.2 -.8 -.4 iii. .4 .8 1.2 -I. -.6 -.2 .2 .6 I. -1. -.6 -.2 .2 .6 I. -I. -.6 -.2 .2 .6 I. 

StmltH%) StmINIlO STRAINI%I 

Fig. 30 Stress and back stress as a function of strain for Hastelloy,...X at 155K 

-.. ,-............----~---. -~ ~ .. ',.~ 

00 

o 

I. 

;, 

, 

j. ,', 

r 

i 

, 
. -;.-: 

. f?,treas ,'. ~ 

~a~~ stress 

-.8 -;4 , :-.6 a. -.2 .2 

STRfIIN(~) 

''j 

I~ ;, 

) .OXlO-3S-1 

.4 .8 1.2 '.6 1. 

5 

T 

R 

E 

S 

S 

~ . 

4011. 

300. 

ZOO. 

1110. 

a. 

( -1110. 

" p 

A -zoo. I 

-300. 

-4011. 

-590. 

-6110. -1.2 

I 

-.8 -.4 

-I. -.6 

/" " 

3.3XlO-~S,..1 

a. .4 .8 1.2 -.2 .2 .6 I. 

STRfIIN(%) 

5 

T 

R 

[ 

5 

S 

c 

" 

5110. 

400, 

300. 

ZOO. 

1110. 

a. 

P -1110. A 

) -zoo. 

-300. 

-400. 

--500. -1.2 

,', 

~' .>.-,~ 

. ~ L" ~. . .~. ',L~- ~ 

-.8 -.4 a. -1. -.6 -.2 .2 

STRfIIN(%) 

Fig. 31,' Stress and back stress as a function of strain for Hastel1oy-X at BIlK 

3.3XIO-5S-1 

.4 .8 .6 I. 1.2 

co 

...., 

!, 

~ .. " 

i.-

," 

I: 

\ ' 

( 

i 

~ 

i 

t I: 

" 

j, 
.~! 

',":: 

!'O: 

:<t, 'r~mrFv~!~~I!~!\f~r~:i-'~( ~ :. ::: f'.'," 

i ,I 

1 

,,! \ 

,j 

; , .. 

:1 

. 1 

, '.,;'\' , 

, 

j 

! 

I 

.l 

500. 

400. 

"", 

Ii ZOO, 

T 

A 

"".~ ~R.ck St'-c~5 

... "" ,," E 11!0. B 

S e. I I t l J e 

" p -11!0. 

A 

) -ZOO. --- -"". 

-.011!0. 1.0X10-1S-1 

~. -1.2 ~I--~----------~----+---------~ -.8 -.4 e. .4 .8 1.2 

-1. -.6 -.2 .2 .6 I. 

STRAIHOO 

".,C 

'. - .' ~'.;- . -:-~ --.: 

5eII. 

~ 5eII. 

---' 400. 400. 

380. 300. 

Ii zoo. -- Ii zoo, T 

-- T 

A ,,- A 

E 11!0. " E II!0. 

B .t Ii 

S e. s e. e ( 

" P -11!0. " P -too. A A ) -zoo. ) -zoo. 

-300. -380. 

-400. 1. lXlO -~S-l -400. 1.lXlO-5S- 1 

~. -1.2 ~I----------~--+----+--~----~ -.0 -.4 e. .4 .0 1.2 -5eII. -1.2 ~I----------+---~~------------~ -.0 -.4 e. .4 .0 1.2 

-I. -.6 -.2 .2 .6 I . -1. -.6 -.2 .2 .6 I. 

STRAIN 00 STRAIH(~) 

,j Fig. 32 Stress and back stress as a function of strain for Hastelloy-X at 866K 

" 

, " 

,-,." . 

(Xl 

N 

i/. 

[ 

f. 

r· 

" , f,: . 

f 

~.<>:-

L. 
I·~. _ 

.rt.r'f(,·~l;;£f"'~~'" 

"·1 ' I 

'jl 

;.! 

j 

-j 

,'I 

I 

- 1.--

-300. 

-.0100. 1. mclO - 35- 1 

-500. 

-1.2 -.8 -.4 B. .4 .8 4.2 -1. -.6 -.2 .2 .6 I. 

5TRAIN(%1 

-300. 

-400. 

-I, _I 1. '!XlO S 

-500. I I, -1.2 -.0 -.4 8. .4 .Il 1.2 

-to -.6 -.2 .2 .6 L 

5TRAINI%1 

-31:'0. 

-400. i. ]XI ()-~S-I 

-500. -1.2 -.0 -.4 B. .4 .0 1.2 -I. -.6 -.2 .2 .6 I. 

STRAIIiI%) 

Fig. 33 Stress and back stress as a function of strain for Hastelloy-X at 922K 

" 

j 

! 

:>=~_=_. ._ '_'_'_<_'_."l!'_,u,~., ___ ."", ~ -.--.-

00 

"'fI:lll 

"gffii 02 

if! 

.g~ 

>G> FM 

:!w 

co 

w 

,-::,r' 

. ;-

:.,l '~ 

, ~ 

i !'l '. < . C. -: .. " ! ~ 

< "':,' 

! . 

j' 

f~~ f; 

i 

\~ 

'I' 
lir~~~~tl~",;~0i~j,?",." . 

,;t; _iMffiliMlM,,kih,: if> 

. '.~. '.' ",.,' . 

::r. - ... 1>.','." , 

lti 

~i~~ i~if 

?Jff 

'i~i ,~ 

~~! l.t'1j 

·~i~1 it", 

Il~~;u~ ~ 

~~' I~~~~ 

I~·~t 

iV.~ 

j;iJir: 

,ii'! 

iM~ t'{;.~{;, 

~~~--i' 

~r,f; 

I 

"'"";:"t' . 

. , 

400. 

~00. 

'S 200. 

T 

R E 100. 

S 

S' 0. 

I 

~ -,J 00. 

A 

) -200. 

-~00. 

-400 . 

- 1 .5 

StreB~ 

Hack st'"CD3 

1.0XlO-3S-1 

- .6 .5 1.5 

1 • 0. I . 

STRAINI'l.) 

: ... _i. 

400. 

!i00. 

S 200. 

T 

R E 100. 

S 

S 0. 

( 

~ - 100. 

A ) -200. 

-!i00. 

-401Z1 "- 1 .5 

- 1 • 

3.3X10-hS-1 

-.5 .5 0~ 

STRAIN(XI 

1 .5 

1 • 

~01Z1. 

200. 

S 

T 

R l01Z1. 

E 

S 

S IZI. 

I 

M 

P - llZllZl. 

A 

) 

-21Z11Z1. 

-3121121. 

1 .5 

- t 

""J 

.-:" .. ',,':'':'" 

-,=- " 

, -,. - . . .' . : ~~- ~ "." - --' 

3, 3XlO-~s-1 

-.6 .5 

liL 

STRAINIXI 

1 .5 

l. 

Fig. 34 ~tress and back stress as a function of strain for Hastelloy-X at 918K 

.' '--'-'~ ,",! 

00 

'",," 

;.-

I 

:'. 

., 

, 

I·. 

I 

:".-

. ~' 

, t" 

f·, 

I 

[> 

[ 

: ~. 

f",-

!. 

I.' 

. r:: 
;; j~"! I' 

.>:~?~ . 

" 

( ...... 

\ 

, \ 

I 

, , 

I 

-'.:; 

.~1 

';ji 

..... j 

;':'{i :1'< 

:;. 

';i;'~~:l 

t' . ~ .I 

. ,'I; 

~ i. ~"<\ 

.' .. ~ 

5 

T 

A 

E 

5 

5 

( 

" p 

300. 

25e. 

200. 

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STRAIfWO 

Fig. 35 Stress and back stress as a function of strain for Hastelloy-X at l033K '1 

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STRRIIiI%) STRRItII%) 

Fig. 36 Stress and back stress as a function of strain for.Hastelloy-X at lo89K 

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STRAIN( lO 

.6 

.B 

Fig. 37 Stress and back stress as a function of strain for Hastelloy-X at 1144K 

.. ' , " 

.. '.: .... " 

l. 

00 

"-J 

, 

~', 
88 

. :.',:' The effect of increased thermal assistance is reflected by an increas-

.-. " • : ~.' _ r 

- - . ing effective stress as well as a decreasing strain hardening rate - -

(Fig. 31-33) (p.81-83). At each temperature, however, the internal 

':" .. ·:l 

stress remains relatively constant, indicating that rate effects are 

still not very significant. At 978 K and above, thermal effects are 

important and the back stress usually saturates to a constant value, 

accounting for the appearance of a "plateau" region at the top of the 

hysteresis loop. A constant back stress indicates a balance betwee~ 

strain hardening and dynamic recovery and is equivalent to steady 

state creep. It should be noted, however, that strain cycling and 

creep tests involve different deformation histories. Strain cycling 

tests are strain controlled, i.e., they are carried out at constant 

strain-rate while the stress is monitored. Creep tests, on the other 

hand, are conducted at constant stress while the resulting strainrate (or creep rate) is monitored. A recent study on a nickel-base 

superalloy shows that at steady state, a tensile test (conducted at 

constant strain-rate) and a creep test will produce equivalent results 

provided the strain-rates are equal [54]. This requires that the ten-

.. " " . sile test be performed at a strain-rate equivalent to the steady-state 

. '. . -, ",.', 

creep rate for a given stress at a given temperature. 

It can be seen that in this temperature range, slower strainrates result in lower back stresses. At reduced strain-rate, more 

time will be available for dynamic recovery such as dislocation climb 

and cross-slip to proceed. 

The values of the drag stress K are tabulated in Table II (p. 77). 

Despite some scattering of the data, the drag stress is found in 
T "." 

...... -~ -." ~' " .; ,':.'. <-'- ,"" .. ",'. 

","" 

89 

general to decrease with increasing temperature. This indicates an 

increasing mobile dislocation density, possibly resulting from an 

increased cell size which is both the source of mobile dislocations 

and a place where they may be entrapped. At lower temperatures, 

the drag stress increases with decreasing strain-rate, again indicating the expected lower mobile dislocation density at lower strainraces. At higher temperatures (978 - 1144 K), it is assumed that 

K remains constant over the strain-rate range considered. Strainrate cycling tests performed at these temperatures suggest that the 

reduction in flow stress at lower strain-rate is a result of changes 

in' the back stress rather than the drag stress. It was subsequently 

confirmed by strain drop, stress transient tests that the internal 

stress decreases with decreasing strain-rate in this temperature 

range. From a physical standpoint, the abundance of thermal energy 

available at these temperatures justifies the assumption that the 

mobile dislocation density does not vary appreciably over the strainrate range studied. The physical significance of the mobile dislocation density as well as the drag stress can be better understood 

in terms of the Orowan equation as shown below: 

t. =4lPmbv ~r 

where ~ is a geometrical constant, p is the mobile dislocation 

m 

density, b is the Burgers vector and; is the average dislocation 

(21) 

velocity. v can be_expressed as a function of the effective stress 

as follows: 

n v = S(a - It) 

, -~·~~~~~~~~i"~:--- ~ ':'~- - j" 

,,- .... "" 

(22) 
, , ':- , 

-',J' 

-"", 

': ~'.,.: 

'. '",'.' 

• _,.0"' 

- -

.,- ~ . 

-, ' 

90 

where 6 is the average dislocation velocity at unit effective stress, 

Q is the internal stress and n is the stress exponent. Substituting 

equation (22) in equation (21) yields the following relationship: 

n 

= ¢ Pm b 6(0 - n) (23) 

A comparison of equation (23) with equation (8) shows that the mobile 

dislocation density is inversely propotional to the drag stress, i.e., 

p a K-n (24) 

m 

Values for K

n 

at various temperatures and strain-rates are tabulated in 

T~ble II (p. 77). The results indicated that at lower temperatures, K

n 

is relatively insensitive to temperature changes but increases as the 

strain-rate decreases. In other words, the mobile dislocation density 

increases with strain-rate but not a sensitive function of temperature 

(in the low temperature range). At higher temperatures, however, K~ 

is found to decrease steadily with temperature while remaining constant at various straLl-rates (over the strain-rate range considered). 

This in turn indicates that mobile dislocation density increases with 

temperature but is insensitive to strain-rate change in the high temperature range. Experimental data indicates that the rate sensitive 

inelastic strain-rate increases both as a function of temperature and 

strain-rate. This observation can be explained by referring to 

equation (21), which implies that the rate sensitive inelastic strainrate is controlled by the mobile dislocation density as well as the 

average dislocation velocity. At lower temperatures, the back stress 

is rate insensitive and equation (22) indicates that the average 

.' ,~:, .: . 

,"'C-'_·_, ',"': ",.-
. '''''~:: -

91 

dislocation velocity remains relatively constant,and therefore, has 

limited effect on the rate sensitive inelastic strain-rate. Thus, 

the increase in the rate sensitive inelastic strain-rate (at low 

",/ temperatures) is mainly the result of higher mobile dislocation 

density at higher strain-rates. At high temperatures, however, the 

mobile dislocation density as well as the drag stress is rate 

insensitive and therefore has no significant impact on the rate 

. sensitive inelastic strain-rate. In this temperature range, both 

the back stress and the effective stress are found to be an increasing function of strain-rate (Figs. 34-37) (p. 84 - 87), which results 

in a higher dislocation velocity and therefore a higher rate sensitive inelastic strain-rate. 

To summarize, the rate sensitive inelastic st~ain-rate is controlled by the mobile dislocation density (or drag stress) at lower 

temperatures; whereas at higher temperatures, the average dislocation 

velocity becomes the controlling factor. It should be noted that 

despite an increase in the "magnitude" of the rate sensitive inelastic strain with increasing strain-rate, it represents a smaller 

"percentage II of the total inelastic strain. This is consistent with 

.,. the general observation that a material is mpre rate insensitive at 

higher strain-rates. 

',-; 
."._,:.'. ,',. 

, ',", 

'_:'.-

: . . ,~. 

" .. ,' :'.:' 

: ~ '~' '-'.: .,' ".: '- .,::"": 

, ,-. - " 

'.: 

".-

', .. : ,_I:";: •. 

. ,",,:, ... '; .. <~.>~:~'~ ,. ~~::-:- :'-!'.' ,~-,-

.:.n " - - ,,-\_"~ '. -- ,-...... ..:..-----~.--- .-<- ,'.---

92 

CHAPTER VI 

EVALUATION OF THE HODEL 

Justifyine the Uncoupled Inelastic Strain Hodel 

It is generally accepted that inelastic deformation in crystalline materials is a result of dislocation movement on various slip 

planes. However, the physical processes involved are still not fully 

understood despite considerable experimental effort. An area still 

open to discussion is the role of thermal activation in dislocation 

dynamics under various temperature and strain-rate conditions. The 

question of inelastic strain-rate continuity (or discontinuity) 

during a change in total strain-rate has received much attention 

recently since it provides some important insight into the effect of 

thermal assistance on material deformation. 

A common approach to the study of inelastic strain-rate continuity is by means of the stress relaxation test. It is ,performed by 

interrupting a constant strain-rate test, holding the strain constant 

while the stress is monitored as a function of time. Measurement of 

the inelastic strain-rate immediately before and after the start of 

stress relaxation should provide an answer as to whether inelastic 

strain-rate continuity exists. While attractive in principle, the 

test is difficult to conduct experimentally. In the following discussion, arguments provided by other investigators will be presented 

and an attempt will be made to describe the phenomenon in terms of 

the uncoupled constitutive model. 

. ~~'.: .' 

-. - ~' ,. ~-, 
':, ... ' 

.~' .~ 

.. "-;" 

-~ --, 

-,' , 

'. : ~ , 

: ..... 

... -

. >-, 

" . .-

~. ; 

'-~ , 

.j 

.. . 

,'- . ' ' .. ~"" - " 

93 

In a recent study, Alden [55] verified the existence of inelastic 

strain-rate discontinuities in aluminum single crystals over a temperature range of 77 K - 593 K. The existence of strain-rate discontinuity at the start of a stress relaxation test can also be deduced 

theoretically from the time-independent deformation theory (recoveryrearrangement theory) developed by the same author [56]. In the timeindeRendent deformation theory, inelastic deformation is considered 

rate independent. It is assumed that stress alone is sufficient to 

produce strain without the direct aid of thermal activation. Thermal 

effect (or recovery) influences the deformation behavior indirectly 

through the loss and rearrangement of obstacle dislocations. The 

expression for the inelastic strain-rate is given in the theory as 

E: = 

6 + r A (6 > 0) i e r 

y 

Ei 

r = A (6 2. 0) e r 

y 

where e is the strain hardening rate in the absence of recovery y 

(assumed to be at 0 K), r is a generalized recovery parameter 

(25a) 

(25b) 

associated with both loss and rearrangement of obstacle dislocations, 

A is a free area function characterizing local slip behavior. 

r 

During a tensile test 6 > 0 and the inelastic strain-rate is given 

by equation (25a). During stress relaxation, however, 6 < 0 and 

equation (25b) is utilized. It cari be seen from equation (25) that 

the inelastic strain-rate ratio (i.e., ratio of the inelastic strainrate i~mediately after and before the start of stress relaxation) is 

given by 

I !. 

, 

'. ,'." ---~-.~,:~,.~::,-~~,,-, . "-"'~.~-~~-~~-,~.- -'-~---';-.: .. ~:;.:.-::;=.~~;~-;":- -~. ~; ~ .: :- ---i. :e , .+,'. .:t 
., , " ' 

. "'," .:. '-./:.'.: ;-- \' 

-', , 

... -. 

". ~-

.- J'_; 

:.-.-.-,.-" 

... .:; .. 

-'," , 

.;., 

: ';~' . . 

2', 

-~ 

. ~~ 

~. 

.~,,:; . 

'·I~;", i~, 

I 

r. .' 

___ ~',~ .• ~ ~~-'_._'::_;"'.:.~ • .',~·._c~ ,-"."·, •• ,.',.C" 

94 

r 

R = a + r (26) 

where R is the inelastic strain~rate ratio and 0' denotes the stressrate prior to stress relaxation. R will be less than unity except 

for the case where 6 is equal to zero. Therefore, inelastic ,strainrate discontinuity will always occur prior to saturation. It was 

found that Ris small initially and increases with increasing stress. 

Most recently, a study by Holbrook, et al. [57] on aluminum, 

iron and 304 stainless steel confirmed the existence of inelastic 

strain-rate continuity. This observation is in agreement with the 

thermally-activated deformation theory (reaction-rate theory) supported by the authors. In this theory, it is postulated that inelastic 

deformation results from the thermally assisted overcoming of local 

obstacles. The "therrilal activation barrier" (Le. activation energy) 

increases with dislocation density and decreases with stress. The 

expression for the inelastic strain-rate is given by 

(27) 

where ~G is the activation energy for dislocation motion past 

obstacles, k is the Boltzmann's constant, T is the absolute temperature and t is dependent on stress, temperature and the current 

o 

microstructure. It can be seen from equation (27) that inelastic 

deformation is controlled primarily by the activation energy 6G and 

is therefore rate dependent . 

It is assumed that all inelastic deformation is rate dependent 

and th~ inelastic strain-rate can be represented by equation (8) as 

• : .' - ---,~--'-~-~---:-----: ----:':'.----;:---:-'-:------:---- -.--~-~~~----:--, . --,-. 7;:O7··~;--·-"~--:--- '--',---;- -.- - -_--rl:"\' ...... ;-----

'. :':":." '.:-' ~ '-. , ..... , 
,. "", 

, 

, . . , 

:. : 

__ J ____ :.._ .,'-=-. __ ", _._.. .. ',~ , .... _. _. ~ __ . ___ _ 

95 

follows: 

E: • 

~r 1

0' - 12ln sgn(O' - ~) ---K-- (8) 

None of the parameters on the right side of the equation are expected to change immediately before and after the start'of stress 

relaxation .. Although the total strain-rate experiences a sudden 

change from positive (or negative) to zero as the crosshead is stopped, the inela~tic strain-rate will be continuous. The authors state 

that inelastic strain-rate discontinutiy is merely the result of 

inaccurate measurements and continuity can be verified with improved 

time and load resolution provided by microprocessor digital data 

acquisition. Inelastic strain-rate continuities were confirmed for 

aluminum, iron and 304 stainless steel at low strain-rates. The 

apparent discontinuity at higher strain-rates was believed to be the 

result of experimental limitations posed by the testing system. A 

small number of experiments were also performed on polycrysta1line 

nickel and 304 L stainless steel utilizing a strip chart recorder 

for data collecting purposes. Experimental results for nickel 

indicated inelastic strain-rate discontinuity, which was also considered to be a resolution problem rather than real material behavior. 

In this research program, the uncoupled constitutive model can 

be expressed in the following form: 

E. = E., + E. = f(a, a ) (; + sgn(a - i'"]) 110' K- rll n 

~ ~~ ~r max 

(28) 

.... --- ---,--- ~-- " .----;-,--:.".-------, .. -.. '.-----:1 .-, -;--- --~ -.c .. --_--· "-------.---:---- ';"""-- - ..... - --, 

I 

I 

\ 

i 

t 

I 

... :-'--'~--' il 
",,': 

" <: -:: 

. ", : .. :' \ 

-:,.,' 

Since cr goes from positive to negative as the constant strain-rate is 

interrupted for stress relaxation and since f(cr, (j ) is essentially max 

96 

zero just after a load reversal, the rate insensitive inelastic strainrate experiences a change from a positive value to zero. Since the 

stress is continuous at this point, the rate sensitive inelastic 

strain-rate would be expected to be continuous. Thus, a large decrease 

in inelastic strain-rate as one interrupts the strain cycling indicates. 

that the inelastic strain-rate prior to stress relaxation is principally rate insensitive. If the inelastic strain-rate before and aiter 

the interruption is essentially the same, this indicates that the 

inelastic strain during strain cycling is mainly rate sensitive. The 

subject of strain-rate continuity has been studied for Hastelloy-~ at 

three different strain-rates for a temperature range from 755 K to 

1144 K. The results are summarized in Table IV. It can be seen that 

for a given strain-rate. the strain-rate ratio (given by Ei+/Ei-) is 

an increasing function of temperature. For a g~ven temperature, the 

strain-rate ratio is found to decrease with increasing strain-rate. 

At 755 K, the inelastic strain-rate after the start of stress relaxation is considerably lower than that prior to stress relaxation. 

Although adequat:e thermal assistance is available to overcome local 

obstacles. long range barriers such as grain boundaries and cell 

walls have to be overcome by dislocation climb or cross-slip, which 

are quite sluggish at this relatively low temperature. Thus, the 

inelastic strain is mostly rate insensitive which accounts for the 

rather significant discontinuity observed at this temperature. The 

rate sensitive inelastic strain-rate is increased considerably at 

, ...... ; ..... "-

~- -'~' 
.' :-~ " .. :.,...., l 

~ -,. , 

97 

Table IV. Inelastic Strain-rate Continuity at Various Temperatures. 

-.' .. 

Stress t. - t. + 

Ei Temp. +r - ~ ~ 

6: i 

(K) (MFa) (10-'+s -l) (10-'+s -l) (%) 

755 434 8.76 .21 2 

558 9.57 .10 1 

811 492 9.51 1. 22 13 

866 386 9.18 2.12 23 

455 9.57 1. 01 11 

922 405 9.34 2.34 25 

458 9.59 1. 95 20 

978 330 9.40 9.01 96 

1033 277 9.80 9.80 100 

1089 185 9.61 9.61 100 

1144 119 9.83 9.83 100 

E. = plastic ~ test 

strain rate before the beginning of stress relaxation 

E1 + 

= plastic 

test 

strain rate after the beginning of stress relaxation 

:":: ,;''':.-;''' 

- " ,'-' .. ".-

- -' ."'-.'",~' . 

-._.", 

",- . '.... ~": 

. '. -' -

-', ;,'; ,', '; 

"'" ,! 

"'.~'-~-;-'-".:-"~'-;~--... ~. -~, -, .~.~~-.:~;":.:::.-"-'-': ... _- ~~' ~',~-: ~'~'.:.~: .. '~-:-~-'- ._- ----~:-~--. .:..'-_' -~, . 

_J. 
" ""·c,,,': . 

.. '~~---'-.-~-.'~"~' . 

. :" 98 

',',;, ,',-': .. ':: 

',,':'.,',' Table IV. (Continued) 

, .' 

E. - s. + I Temp. Stress E. + f" E. - ~ ~ ~ ~ 

(K) (MPa) (10-1+5 -1) (10-45 -1) (%) 

755 450 2.95 .41 14 

568 3.19 .24 8 

811 394 2.98 .40 13 

471 3.16 .69 22 

866 376 3.10 .68. 22 

439 3.20 .54 17 

922 381 3.14 1. 21 39 

431 3.22 1. 38 43 

978 327 3.19 3.09 97 

1033 236 3.27 3.27 100 

1089 154 3.21 3.21 100 

1144 96 3.29 3.29 100 

f, ' 

:r; , 

-.'--~--'--- - -- '.-.. ,_ ... _- .-, 
. ,I"'" 

;'. 

'. ;." 

, ;\' 

, .. 

" 

• ',.' • c .' ~ 

, , 

'; 

Table IV. (Continued) 

C) 

Tem~. Stress 

(K) (MFa) 

755 592 

811 373 

866 354 

922 352 

978 I 246 

1033 163 

1089 100 

1144 59 

99 

t. - E. + E. +r - s . 1. 1. 1. 1. 

(10- 5s-l ) (10-55 -l) (%) 

3.19 .82 26 

2.97 .80 27 

3.05 1.84 60 

3.15 2.38 76 

3.20 3.19 100 

3.27 3.27 . 100 

3.22 3.22 100 

3.29 3.29 100 

___ ._~ ___ • __ •• _____ , ___ ••• _ ' __ ~_."' __ T _-- __ .......... __ • 
.. ,'.;, 

. ~.'., -.' 

:. ':. .. :-: ~:- ...:..-:.:...:.::...-- ':>~-.:~' 

100 

lower strain-rates although it still represents a small fraction of 

the total inelastic strain-rate. The rounded corners of the hysteresis loops and the high strain hardening rates at this temperature 

are further evidence of the absence of significant thermally assisted 

flow. The rate sensitive inelastic strain-rate increases steadily 

with temperature, indicating the growing influence of thermal effects. 

At higher strain-rates, rate effects do not become important until 

the temperature is raised above 922 K. At the lowest strain-rate, 

however, the rate sensitive inelastic strain-rate component becomes 

dominant at temperatures above 866 K. In all cases, the inelastic 

srrain-rate is essentially all rate sensitive at 978 K and above. 

The high atomic diffusion rate in this temperature range implies that 

dislocation climb is the dominant recovery process. Inelastic strainrat~ continuity is generally observed in this temperature range due to 

the lack of rate insensitive inelastic strain . 

. , It should be noted that the information provided in Table IV 

(p. 97) is subjected to modification for two reasons: (1) time ",', 

resolution achievable by our recording system is not so good as that 

attained by some other investigators; (2) the effect of machine tran-

'. ~. ':: ,-.' t;'';:; • sient has not been studied in detail. To correct the first short- ... -.: ::." '. ~~ .. -. -. ;- ' .. ~ 

coming, a high speed digital computer has been installed which can 

provide greatly improved time and load resolution. Before any test 

can be performed to confirm the existence of inelastic strain-rate 

continuity (or discontinutiy), the complications introduced by 

machine characteristics have to be fully understood. Since continuity has to be studied at the point where the crosshead is stopped 

. ~., . 

, ',.", " ".~ ~. I: 
, :,., 

c' 

"' ".,.,;k·d 

':. ~.::.. ,,', , -. ",." '.1.":<., 

,~. "', 

'.,' 

. , • ,~ .. _ .•• __ ...... ,';_. :.":..:-:.. .... ~ ~_:.:~'..:.......~.::..::;......:~.:.:~.:,~ ... ,:.o: ,,; :'.,: . __ .. " _<~:; ___ ._. '-....... ___ ~,_~. __ •. _. ".,., .' , . . " _ . ....:..."'~_._~k ...!-:-'-'~'~_:'~ ~~ 

101 

"instantaneously", machine transient which lasts for even a small fraction of a second can effectively nullify any attempt to obtain a 

meaningful result. Until further experiments are conducted, however, 

._-Q'Itt' d-iscus-si~will be based on the data tabulated in Table IV (p. 97). 

Comparison of Experiments with Theoretical Predictions 

The uncoupled constitutive model as described in Chapter III may be 

used to predict strain cycling, stress relaxation, creep or other 

phenomena if used with the proper material constants. To first see 

if the model is self consistent in being able to reproduce the experimental hysteresis loops, all of the input strain cycling curves wer~ 

simulated using a numerical scheme discussed earlier. Stress relaxation simulations are also presented and comparisons are made with 

observed relaxation behavior. 

Saturated hysteresis loops. Saturated hysteresis loops are simu1ated using equations (11) - (13) and the numerical scheme presented 

in Appendix A. The strain hardening function and other material 

constants are obtained from Appendix D, Table II (p. 77) and Table 

III (p. 78). The experimental and simulated hysteresis loops at 

various temperatures and strain-rates are presented in Figures 38 -

45. Generally speaking, the experimental hysteresis loops and the 

simulated curves are in good agreement over the temperature and 

, ~<-" 

strain-rate ranges studied. Rounded cor,ners are well simulated at 

the lower temperatures where inelastic strain is mostly rate insensitive. The unified theory with its high lin" values a!vlays gives 

square corners at lower temperatures. At high temperatures inelastic 
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for Hastelloy-X at l144K 

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110 

strain is mostly rate sensitive and results obtained by the uncoupled 

model is the same as those obtained by the unified models. At high. 

temperatures and low strain-rat~s, predicted hysteresis loops may 

appear.~,be .too square compared to experimental results (Fig. 45) 

(p. 109). This "squareness" at low strain-rates is common among 

unified models. Walker [17] suggested that the "square" hysteresis 

loops are a result of the fact that the back stress reaches the 

saturated values more rapidly at low strain-rates. 

Stress relaxation behavior. Theoretical stress relaxation 

behavior is simulated using equations (8) and (14) and the numerical 

scheme shown in Appendix B. Material constants utilized are tabulated in Tables II and III (p. 77 and 79). Values for the back stress 

"rI" and the drag stress "K" are assumed to remain constant during 

the course of stress relaxation. Theoretical predictions as well as 

experimental results are shown in Figs. 46-53. At lower tempera-' 

tures, thermal recovery is less significant and nand K are not 

expected to change appreciably during stress relaxation. Predicted 

relaxation behavior at the lower temperatures is reasonably close to 

experimental observation (Figs. 46-49) (p. III - 114), indicating 

the above assumptions are valid. At higher temperatures, however, 

actual asymtotic stress values are much lower than predicted values 

(Fig. 50 - 53) (p. 115 - 118): At these higher temperatures, 

recovery takes place during the course of stress relaxation which 

results in lower values for rI and K. If the specimen is reloaded 

at the end of the relaxation test, a reduction in flow stress is 

generally observed. The raw data shown in Fig. 54 indicate the 

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TlI"'E (MIN. ) 

Fig. 46 Experimental and theoretical stress 

relaxation behavior for Eastelloy-X at 755K 

111 

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2.5 7.5 12.5 17.5 22.5 27.5 32.5 

TIMECMIN. ) 

Fig. 47 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at BllK 

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TII'ECMIN. ) 

Fig. 48 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at 866K 

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Fig. 49 Experimental and theoretical stress 

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114 

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Fig. 50 Experimental and theoretical stress 

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Fig. 51 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at l033K 

116 

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TIMECMIN. ) 

Fig. 52 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at lo89~ 

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Fig. 53 Experimental and theoretical stress 

relaxation behavior for Hastelloy-X at l144K 

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Fig. 511 Experimental hysteresis loops for Ifo.st.elloy-X [Ii 922K. Point.s A and B 

indicate where stress relaxation starts and ends. Point C indicnics the floW' 

st.ress upon reloading 

...... 

.-

\D 
120 

points along the saturated hysteresis loops where stress relaxation 

tests have been conducted. In each case, the reloading path after 

stress relaxation always stays below the original loading path. At 

point A, for example, the flow stress has a value of 349 MFa, which 

is reduced to 122 MPa (point B) after 16 minutes of stress relaxation. Upon reloading, the flow stress reaches a value of 328 MFa 

(Point C), a reduction of 21 MFa from the initial value of the flow 

stress. This provides direct experimental evidence that thermal 

recovery has taken place during stress relaxation. From a microstructural standpoint, the reduction in flow stress is mainly associated with the coarsening of the cell structure. The initial microstructural state prior to relaxation can usually be re-established 

after several cycles. 

'~." ...... , ...... -- --;'-:::, ' .. '--~'~ - .---. 

121 

CHAPTER VII 

CONCLUSIONS 

An uncoupled viscoplastic. constitutiv~ model has been presented 

to describe elastic-plastic behavior of metals under a variety of 

temperature and strain-rate conditions. The uncoupling of the total 

inelastic strain into rate sensitive and rate insensitive components 

allows modelling of cyclic deformation behavior over a wide range of 

temperatures. The model is capable of describing strain cycling, 

stress relaxation and creep under isothermal conditions. 

This model recognizes that the inelastic strain produced by a 

load change consists of both rate sensitive and rate insensitive 

components. In this regard, it is different from the models described in the preceding chapter which either assume inelastic strain 

is all rate independent as in the time-independent deformation theory 

or all rate dependent as in the thermally-activated deformation 

theory. Although the uncoupled approach may seem somewhat "art ificial", some physical justifications are given in Chapter III. As 

discussed previously, the role of thermal activation at low temperatures is seen mainly in the overcoming of short range obstacles and 

therefore has no major effect on the overall deformation process. 

On the other hand, thermally activated processes such as dislocation 

cross-slip and dislocation climb dominate at high temperatures. A 

major difficulty encountered by the unified models is that at lower 

temperatures they often predict hysteresis loops which are too 

"square". The "squareness" is due to.the assumption that all 

-~ .~-

- - " ... :- .:-.... 
122 

inelastic strain is thermally activated. The rate sensitive inelastic 

strain component of the uncoupled model is described by a po~er la~ 

formulation whicQ is standard to many unified models. Therefore, the 

two approaches are nearly the same at high temperatures ~here inelastic strain is essentially all rate sen,sitive while at lo~er tecperatures, the uncoupled model provides more satisfactory predictions. 

At present, only saturated hysteresis loops have been considered. 

While saturated hysteresis loops may be useful for long term predictions, low cycle fatigue applications may require the modelling of 

cyclic response prior to saturation. Two internal variables - the back 

stress and the drag stress, are employed to describe the current 

microstructural state of the material. Growth laws have to be 

developed for the internal variables before cyclic hardening (or 

softening) can be modelled. Comparison between experimental and 

simulated hysteresis loops shows good agreement in general. More 

experiments are necessary, however, to verify the predictive capability of the model for a complex thermomechanical lo~ding history. 

Stress relaxation predictions at lower temperatures are reasonable 

Hhile high temperature predictions indicate significant thermal 

effects. Recovery terms have to be incorporated into the growth 

laws for the state variables if the model is to be useful for high 

temperature stress relaxation predictions. 

.-~.' 

; 

r 

\ 

\ ' 

..... ~J 
123 

CHAPTER VII I 

RECOHHENDATION FOR FUTURE WORK 

.-1'heuncoupled constitutive model described earlier has been shown 

to satisfactorily model saturated hysteresis loops under isothermal 

conditions over a wide range of temperatures. In this chapter, 

experiments necessary for further development of the model will be 

outlined. 

As mentioned in the previous chapter, modelling of cyclic hardenlng prior to saturation is necessary to predict short term response 

such as low cycle fatigue. This will require the development of 

growth laws for the two internal state variables utilized in the 

model. Since cyclic saturation is basically associated with the 

formation of a stabilized cell structure, it will be reasonable to 

assume as a first approximation that the drag stress is mainly responsible for cyclic hardening while the back stress reaches a saturated value after the first cycle. To verify the above assumptions, 

tests should be conducted to measure the back stress at suitable 

intervals prior to saturation. It has been shown in the previous 

chapter that discrepancies exist between experimental and predicted 

stress relaxation behavior at high temperatures, indicating the 

need to incorporate thermal recovery into the growth laws for ~ and 

K. The concribution of grain boundary effects to the overall deformation process has not been considered due to insufficient information at this time. It will be interesting to consider the introduction of an additional state variable to model grain boundary sliding 

-,.-..--._ ... "---" .-- .---- "~------.- .. __ ._- ~--- -~-"--- -- ---------~--'--"--.--:---~~-----.-------.----- -~--.--"";".--------:----::----....,--------.--- ;- --- . - --;";:~ .... --- .-- ,- ----.--_ .. .. - . ..... . 
124 

as suggested by some investigators. Constant stress creep experiments should also be conducted at various points along the hysteresis loop and compared to predictions made by the model. Finally. 

tests have to be performed under non isothermal loading conditions 

to study the capability of the model to predict more complex thermalmechanical histories • 

.' .,' 
'., 

1. 

REFERENCES 

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Soumis a Diverses Tensions." Annales des Ponts et Chaussees, 

ler semestre, 1834, pp. 40-44. 

2. Hill. R.R., The Mathematical Theory of Plasticity, Oxford 

University Press, London, 1950. 

3. Martin, J.B .• Plasticity: Fundamentals and General Results, 

The MIT Press, Cambridge, Massachusetts, and London, England, 

1975. 

4. Pugh, C.E •• Corum, J .M., Lin, K.C •• and Greenstreet, W.L., 

"Currently Recommended Constitutive Equations for Inelastic 

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National Lab., Sept. 1972. 

125 

5. Snyder, M.D., and Bathe, K.J., "Formulation and Numerical 

Solution of Thermo-Elastic-Plastic and Creep Problems," 

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1981, pp. 124-135. 

7. Sharifi, P .• and Yates, D.N., "Nonlinear Thermal-ElasticPlastic and Creep Analysis by the Finite Element Method,lI AIAA 

Journal, Vol. 12, 1974, pp. 1210-1215. 

8. Zudans, Z., "Elastic-Plastic-Creep Response of Structures 

Under Composite Time History of Loadings, "Third International 

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9. Cailletaud, G., and Chaboche, J.L., "Macroscopic Description 

of the Hicrostructural Changes Induced by Varying Temperature: 

Example of IN100 cyclic Behavior, "Third International Conference on Mechanical Behavior of Materials, Cambridge, Aug. 1979. 

10. Lee, D. and Zaverl, F., Jr., "A Generalized Strain Rate Dependent Constitutive Equation for Ani~otropic Metals, "Acta 

Metallurgica. Vol. 26. No. 11, 1978, pp. 1771-1780. 

11. Krieg. R.D., Swearengen. J.C •• and Rhode. R.W •• "A PhysicallyBased Internal Variable Model for Rate-Dependent Plasticity," 

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

, ' 

. ';.- .: 

: .. -. ~, 

, ' 

-, 

12. Hart, E.W., "Constitutive Relations for the Nonelastic Deformation of Metals," Journal of Engineering Materials and Technology, Vol. 98, 1976, pp. 193-202. 

13. Bodner, S.R., and Partom, Y., "Constitutive Equations for 

Elastic-Viscoplastic Strain-Hardening Materials," Journal of 

Applied Mechanics, Vol. 42, No. i, June 1975, pp. 385-389. 

14. 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 

126 

304 Stainless Steel," ASME Journal of Engineering Materials and 

Technology, Vol. 98, 1976, p. 97-113. 

15. Miller, A.K., "Modelling of Cyclic Plasticity with Unified 

Constitutive Equations: Improvements in Simulating Normal and 

Anomalous Bauschinger Effects," Journal of Engineering Materials 

and Technology, Vol. 102, April 1980, pp. 215-222. 

16. Walker, K.P., "Representation of Hastelloy-X Behavior at Elevated Temperature with a Functional Theory of Viscoplasticity," 

presented at AS~ffi Pressure Vessels Conference, San Francisco, 

Aug. 12, 1980. 

17. Walker, K.P., "Research and Development Program for Non-Linear 

Structural Modelling with Advanced, Time-Temperature Dependent 

Constitutive Relationships," Final Report, NASA Report NAS3-

22055, Nov. 25, 1981. 

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

Surface. Part I. General Theory," Archives of Mechanics, 

Vol. 23, 1971, pp. 517-533. 

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

Surface, Part II. Application to Hechanical Behavior of Metals," 

Archives of Mechanics, Vol. 23, 1971, pp. 535-551. 

20. Valanis,'K.C., "Some Recent Developments in the Endochronic 

Theory of Plasticity-The Concept of Internal Barriers," 

Constitutive Equations in Viscoplasticity: Phenomenological 

and Physical Aspects, AMD Vol. ~1, ASME, New York, 1976. 

21. Cernocky, E. P., and Krempl, E., "A Nonlinear Uniaxial Integral 

Constitutive Equa~ion Incorporating Rate Effects, Creep and 

Relaxation," International Journal of Nonlinear Hechanics, 

Vol. 14, 1979, pp. 183-203. 

22. Delph, T.J., "A Comparative Study of Two State-Variable Constitutive Theories," ASME Journal of Engineering Haterials and 

Technology, Vol. 102, 1980, pp. 327-336. 
':" .' . ~, -' ", ' 

"',, 

,'t" '" :, '" 

, ::,,:'" 

',',! 

• 1 

. - ~. <, ,', ~ I 

•• < ' •• ~ • '; , .',: 

',r, 

" 

, r 

.: 

127 

23. Reed-Hill. R.E •• Physcial Metallurgy Principles. D. Van Nostrand 

Company. New York. N.Y .• 1973. 

24. Sherby, O.D •• and Burke. P.M •• "Mechanical Behavior of Crystalline Solids at Elevated Temperatures." Progress in Materials 

Science. Vol. 13. 1968, pp. 325-390. 

25. Schapery. R.A., "A Theory of Non-Linear Thermoviscoelasticity 

Based on Irreversible Thermodynamics," Proceedings,5th U.S. 

National Congress of Applied Mechanics. ASME, 1966. 

26. Biot. M;A., "Theory of Stress-Strain Relations in Anisotropic 

Viscoelasticity and Relaxation Phenomena. "Journal of Applied 

'f'hysiC"S, Vol. 25. 1'954, pp. 1385-1391. 

27. Schapery, R.A .• "On a Thermodynamic Constitutive Theory and 

Its Application to Various Nonlinear Materials," Proceedings. 

IUTAJiSymposium, Springer-Verlag. Wien, New York. 1970. 

28. Wu, H.C •• and Chen, L •• "Endochronic Theory of Transient Creep 

and Creep Recovery," Report NASA-CR-158487. NASA-Langley 

Research Center. 1979. 

29. Lui. M.C.M., and Krempl, E., "Uniaxial Viscoplasticity Based 

on Total Strain and Overstress," Renssalaer Polytechnic 

Institute, No. RPI CS 78-4, July 1978. 

30. Allen, D.H., "A Survey of Current Temperature Dependent ElasticPlastic-Creep Constitutive Laws for Applicability to Finite 

Element Computer Codes." Presented at the 21st AIAA/ASCE/ASME/ 

AHS Structural Dynamics and Materials Conference. Seattle, 

Washington, May. 1980. 

31. Haisler. W.E •• "Numerical and Experimental Comparison of Plastic 

Work Hardening Rules." Transactions of the 4th International 

Conference of Structural Mechanics in Reactor Technology, San 

Francisco. California. Aug. 1977. 

32. Allen, D.H., "A Model for Predicting Response of Nonlinear 

Materials Subjected to Thermomechanical Loading," Dissertation, 

Texas A&H University, Aug. 1980 . 

33. Allen. D.H., "A Tt,!rmodynamic Framework for Comparison of 

Current Thermoviscoplastic Constitutive Models for Metals at 

Elevated Temperature." to be presented at the International 

Conference on Constitutive Laws for Engineering Materials: 

Theory and Application. University of Arizona. Tucson. Arizona. 

January 10-14. 1983. 
, ',.' 

, ,_ /~, - _ -'.1 . 

~, .,':' .'. 

, " ; . ~ . 

128 

34. Yamada, Y., and Sakurai. T., "Basic Formulation and a Computer 

Program for Large Deformation Analysis," Pressure Vessel 

Technology. Part I. ASME. 1977. pp. 344-352. 

35. Yamada, Y., "Constitutive Modelling of Inelastic Behavior of 

Numerical Solution of Nonlinear Problems by the Finite Element 

Method," Computers and Structures. Vol. 8, 1978. pp. 533-543. 

36. Cronenworth, J •• "Development and Evaluation of An uncoupled, 

Incremental Constitutive Model for Elastic. Plastic Behvaior 

at Elevated Temperatures." Thesis. Texas A&M University, May 

1982. 

37. Strauss, A.M •• "A Bilinear Functional Theory of Viscoplasticity," 

Proc. West Virginia Acad. Sci., Vol. 44, No.2. 1972. pp. 

183-190. 

38. Strauss, A.M •• and Stouffer, D.C., "Sundry Aspects of Viscoplasticity," Presented at the Winter Annual Meeting of ASME, 

New York, Dec. 5-10. 1976. 

39. Krempl, E., "Cyclic Creep - An interpretive Literature Survey,lI 

Welding Research Council Bulletin No. 195, 1974. 

40. Krempl, E •• "Viscop1asticity Based on Total Strain. The Modelling of Creep With Special Considerations of Initial Strain 

and Aging,lt Transactions of the ASME. Vol. 101, Oct. 1979, 

pp. 380-386. 

41. Michalak, J.T., "The Influence of Temperature on the Development of Long-Range Internal Stress During the Plastic Deformation of High-Purity Iron," Acta Metallurgica, Vol. 13. March 

1965, pp. 213-222. 

42. Barret, C.R., and Nix,W.D., "A Model for Stready State Creep 

Based on the Motion of Jogged Screw Dislocations," Acta 

Metallurgica, Vol. 13, Dec. 1965. pp. 1247-1258. --

43. Weertman, J., "Dislocation Climb Theory of Steady-State Creep," 

Transactions of the ASM, Vol. 61, 1968. pp. 681-694. 

44. Nabarro, F.R.N., "Deformation of Crystals by the Motion of 

Single Ion~,11 Report of a Conference on the Strength of Solids, 

The Physical Society. London. 1948, pp. 75. 

45. Herring, C •• "Diffusional Viscosity of a Polycrystalline 

Solid," Journal of Applied Physics. Vol. 21. 1950. pp. 437. 

46. Coble. R.L., "A Model for Boundary Diffusion Controlled Creep 

in Polycrystalline Materials," Journal of Applied Physics, 

Vol. 34, 1963. pp. 1679-1682. 
,"; . 

129 

47. MacEwen, S.R., Kupcis, a.A., and Ramaswami. B •• "An Investigation of an Incremental Unloading Technique for Estimating 

Internal Stresses," Scripta Metallurgica, Vol. 3. 1969, pp. 441-

448. 

48. Ahlquist, C.N., and Nix,W.D., liThe Measurement of Internal 

Stresses During Creep of Al and AI-Mg Alloy," Acta Metallurgica. 

Vol. 19, April 1971, pp. 373-385. 

49. Booker, M.K., "Evaluation of Creep and Rti!Iaxation Data for 

Hastelloy Alloy X Sheet," Oak Ridge National Laborato"ry, Report 

ORNL-5479, February 1979. 

50. Jaske, C.E •• Rice, R.C., Buchheit, R.D., Roach, D.B •• and 

Porfilio, T.L., "Low Cycle Fatigue of Type 347 Stainless Steel 

and Hastelloy Alloy X in Hydrogen Gas and in Air at Elevated 

Temperatures," Final Report, NASA Report NAS3-20078. May 1976. 

51. Gupta. I, and Li, J.C.M., "Stress Relaxation, Internal Stress, 

and Work Hardening in some BCC Metals and Alloys," Metallurgical 

Transactions, Vol. I, August 1970, pp. 2323-2330. 

52. High temperature, High Strength, Nickel Base Alloys, The 

International Nickel Company, Inc., New York. July, 1977. 

53. Walker, K.P., "Research and Development Program for Nonlinear 

Structural Modelling with Advanced Time-Temperature Dependent 

Constitutive Relationships," Third Quarterly Technical 

Narrative, NASA Report NAS3-22055, Aug. 1980. 

54. Jensen, R.R., Howson, T.E., and Tien, J.K •• "Very Slow Strain 

Rate Stress-Strain Behavior and Resisting Stress for Creep in a 

Nickel-Base Superalloy," Superalloys 1980, Proceedings of the 

Fourth International Symposium on Superalloys, &~S, 1980. 

55. Alden. T .H., "Load Relaxation in Aluminum: 1. Theory of Plastic 

Deformation. II. Plastic Equation of State," Metallurgical 

Transactions A, Vol: 8A, Nov. 1977, pp. 1675-1679. 

56. Alden. T.H •• "Theory of Plastic Flow in Strain-Hardened Metals," 

Philosophical Magazine. Vol. 25, 1972, pp. '785-811. 

57. Holbrook, J.H., Rohde, R.W., and Swearengen, J.C., "The Continuity of Plastic Strain Rate," Acta Metallurgica. Vol. 29.1981, 

pp. 1099-1106. 

130 

- -! 

APPENDIX A 

','. COMPUTER PROGRMI OUTLINE FOR HYSTERESIS LOOP SIMULATION 

The following outline describes the numerical scheme used to 

simulate saturated hysteresis loops. Simulation always starts at 

the point where loading begins (i.e •• the point corresponding to 

maximum compressive stress). 

A. Enter required material constants. These include Young's 

Modulus. Poisson's ratio. initial diameter. programmed diametra1 strain-rate (dD/dt), stress exponent and drag stress. 

B. Enter initial conditons. These include initial values for 

·dD/dP, strain hardening function, stress. total strain and back 

stress. 

C. Enter required time step. 

D. Calculate strain hardening rate [equation (20)]. 

E. Calculate total strain-rate corresponding to strain hardening 

rate in step D. [equation (13)]. 

F. Calculate strain increment based on strain-rate in step E and 

time step in step C. 

G. Calculate corresponding stress increment [equation (12)). 

/ 

H. Update stress and strain values. 

1. Add stress increment to stress. 

2. Add strain increment to strain. 

I. Check to see if total strain range is exceeded. If exceeded go 

';.-: 

to step M to start unloading. 
. '"-.I---·--··~··-~-·-·~· 

,-

. :I~ '.'._'. ". ' .:~~~".~ .:_"~,~ ... _. __ .,--•• _.,_._.~: ___ ~:. ,,_ - ___ . - - ._.~ 

'-" " 

131 ..... 

J. Update values for. back stress and strain ha:rdening function. 

~ .:. ' .. ', Linearly interpolate between input values if necessary. 

K. Calculate new value for dD/dP. 

L • Repeat steps D through K. . . 

M. Hodi£y initial conditions corresponding to the point where 

unloading starts. 

N. Repeat step L. If total strain range is exceeded in step I. stop . 

• 1 

..... : 

-- .7I-~,: .. ~ 
" , 

,'~-~:- ,-.'-~,~,~ . .:..- .:'':'',,- :.~..::--~.::..:..~~. ~. -:,~. --'~'.:..- .;: '" ~ '~"-" 

", 132 .. " ,: J:.,_.-: 

" 

I.':' 

APPENDIX B 

COMPUTER PROGRAM OUTLINE FOR STRESS RELAXATION PREDICTION 

:r:ha"following outline describes the numerical scheme. used to 

predict stress relaxation behavior. Small time step is used initially 

to better simulate the initial portion of the stress relaxation curve. 

For computational efficiency, the time step is increased progressively 

as the stress approaches the asymtotic ,value. 

A. Enter material constants. These include Young's Modulus, back 

stress, drag stress and stress exponent. 

B. Enter initial values for stress and time limit. 

C. Calculate inelastic strain-rate [equation (8)]. 

D. Calculate stress-rate corresponding to inelastic strain-rate 

given in step C.[equation (14)]. 

E. Calculate new stress value by subtracting stress decrement (or 

increment) from previous stress. 

G. Check to see if increase in time step is necessary. 

H. Check to see if time limit has been exceeded. If time limit 

exceeded, '" stop. 

I. Repeat steps C through H. 

",: .. ": 

. 

" 

:',1 

_ .......... - ~., .' -;:.- .. ..,-~ ~ -..------~-~ .. -~.-.-.--. 

-., . 

, ;,. 

: : -' ~ 

" 

133 

APPENDIX C 

DERIVATION OF EQUATIONS USED FOR DATA REDUCTION 

In . .this appendix the equations used for data reduction will be 

derived. Since diametral extensometer is employed in our tescing, 

diametral strain is measured and converted into axial strain. The 

equations are summarized in Chapter V. 

Total diametral strairl can be separat~d into elastic and inelastic components ~s follows: 

(Cl) 

The diametral elastic strain can be converted into axial strain as 

shown below: 

a (E) = V E - v (C2) eO - e--E" 

where v is the elastic Poisson's ratio. Similarly, the diametral 

inelastic strain can be expressed in terms of the axial strain as 

where v, is the inelastic Poisson's ratio. Equation (C3) can be 

l. 

rearranged as follows: 

E = -2I(E) - (E ) l = -2 []O + vi] 

i L' t D e ~ ~o ~ 

The total axial strain can be given in terms of its. components as 

E = e: + e:~ t e J-

(C4) 

(CS) 

Substituting equation (C4) into equation (CS) and rearranging yields 

_."." ___ -._~1~::-'" '-_--::"'~--'"-:-~-.-"--~-' ----,..,-- ~·.c·----~-·"~-·- .. -------- .--.-~.-~ .. --.- ."-.-------~----.--"~----.~- . 
- ) 

;.; 

..... w-- ~--~ r --~- . 

E = E (1 - 2v) _ 2 dD 

tED 

o 

which is the same as equation (17). The inelastic strain can be 

obtained by subtracting the elastic strain from the total strain, 

i. e. 

134 

(C6) 

which is also given by equation (18). The strain hardening rate can 

be expressed as follows: 

da da 1 --- = ----~~----- = ----~~----

__ da(1_2V) 2dD (1-2v) 2 dD 

E Do E - Do da 

eC8) 

Equation (C8) is identical with equation (20). Differentiation of 

equation (C6) gives 

1 da 2 dD 1 da dEt 2 dD 

Et = E dt (l-2v) - Do dt = E dE

t 

d't - Do dt 

Substituting equation (C8) into equation (C9) results in 

-2 dD 

. n;;- dt 

E

t 

= ----:--'"1-----------

1 - 1. da (l-2\» E dE

t 

(C9) 

(CIG) 

which is the same as equation (13). Th~ inelastic strain-rate can 

be obtained from equation (C7) as 

= -2 [~ da +.L dDJ = -2 [~da dE t + _1_ dD) Ei E dt Do dt E dE dt D dt 

t 0 

(Cll ) 

Substituting equations (C8) and (CIG) into equation (Cl1) yields 

. -

.. - --- - - --- ~-
135 

~ 1 da 

1 - E dE:

t 

1 _ (l-~V) (C12) 

which is given by equation (19). 

i 

L 

, '. 

j i 

! 

i 

. ' 

"," - ,,- ._ .... ", , .. 

136 

"" ,-, 

'".":" ."" . 

APPENDIX D 

VALUES FOR RATE INSENSITIVE, INELASTIC STRAIN Fu~CTION 

a = 149 MFa 

max 

Stress feat a

max ) 

(MFa) (MFa -1) 

Loading 

-149 0 

-106 0 

106 0 

149 0 

Unloading 

149 0 

106 0 

-106 0 

-149 0 

a = 286 MFa 

max 

Stress f (a, a ) max 

(MFa) (MFa -1) 

- . Loading 

-286 0 

0 0 

106 0.187E-6 

212 0.406E-6 

286 0.161E-5 

Unloading 

.~~' .. ' . 286 0 

0 0 

-106 0.275E-6 

... ,. " -212 0.580E-6 

-286 0.148E-5 

----,.=-- .::"-~- -"~-----'-'::'-' "" . 
. " 

" ..... , ''''' .. 

. >.,; ....... ,. 

.' ~ .;., 137 

a = 392 MFa , . max ','." ....... ,. 

, .' , . . . 

. " '. Stress f(a, a ) max 

. ".' (MPa) (HPa- 1 ) 

./ Loading 

-403 0 

0 0.319E-6 

106 o .16SE-S 

212 0.2S9E-S 

31B 0.43SE-S 

392 0.124E-4 

Unloading 

392 0 

0 O.232E-6 

-106 0.IB6E-S 

-212 0.270E-S 

-31B 0.417E-S 

-403 o .107E-4 

a = 435 HPa 

max 

Stress f(a, a ) max 

(MPa) (MFa -1) 

Loading 

-446 0 

0 o .'6S2E-6 

106 0.258E-S 

212 O.432E-5 

31B 0.620E-5 

392 o . 126E-4 

424 0.246E-4 

435 0.680E-4 

Unloading . ,r.: 435 0 

0 0.S51E-6 : - . ~,. I 

,.'/" .-,' -106 0.214E-5 

-212 0.441E-5 

,'--'.'.J -318 O.694E-5 

-392 o .lOBE-4 

",.-'. ,~ i . , ,. 

-424 o .lB4E-4 

l -446 0.522E-4 

I 

., _'-'-:- ~.:~_r.-:::..,.------':-::'-:::,':·.· - .~_ ';" .. ~- _,~ ~-~-.~----.-:'-'--;':.:-J---:--~'-'---.:-~~-" '--.,~ .. ---, , _. 
138 

a 467 MFa 

max 

Stress fea, a ) max 

(MPa) (MPa -l) 

Loading 

-477 . 0 

-106 0.14SE-6 

0 0.113E-5 

106 0.25SE-S 

212 0.559E-5 

31B O.BBOE-S 

424 o .lB7E-4 

461 0.455E-4 

467 0.134E-3 

Unloading 

467 a 

106 a .14SE-7 

a a .104E-S 

-106 0.294E-5 

-212 0.546E-S 

-31B 0.B90E-5 

-424 o .165E-4 

-459 0.329E-4 

-477 0.913E-4 

.. _____ . __ .~_ . ___ ~ . ___ . __ .~ __ ~.:: .. ~~_":-----.-.:-_,-..::--_.,_ .. _-::..:-. __ --_-a-, 

'~,! .. . :-..' .. ' 

- ",-. 
'-

139 " '.~ 

. ""," a ::: 520 MFa 

max 

, .. :. 

",.:.': 

Stress f(a, a ) max 

(HPa) (MFa -1) 

Loading 

-530 0 

-106 0.681E-6 

0 o . 164E-5 

106 0.330E-5 

212 0.732E-5 

318 o .114E-4, 

424 0.197E-4 

477 0.333E-4 

509 0.754E-4 

520 0.196E-3 

Unloading 

520 0 

106 o .127E-6 

0 0.135E-5 

-106 0.383E-5 

-212 0.723E-5 

-318 0.110E-4 

-424 o . 177E-4 

-477 0.310E-4 

-509 0.541E-4 

-530 0.157E-3 

,'j 

." 

,': . 
.'.' .. 

140 

-~:".- _. . . 

- .. 

. . .. 

-. G = 562 MPa 

max 

:' " 

'-- .', Stress .f (0 t 

G

max ) 

(MPa) (MFa -1) 

." 

Loading 

-567 0 

-106 .0.986E-6 

0 0.225E-5 

106 0.394E-5 

212 O.658E-5 

265 O.B62E-5 ... . . 31B 0.126E-4 

371 0.164E-4 

424 0.220E-4 

477 0.32BE-4 

509 0.461E-4 

530 0.600E-4 

552 0.1lIE-3 

562 0.291E-3 

Unloading 

562 0 

106 0.783E-6 

0 0.174E-5 

-106 0.388E-5 

-212 0.677E-5 

-318 0.120E-4 

-371 o . 164E-4 

-424 0.217E-4 

-477 0.314E-4 

-509 0.403E-4 

-530 0.565E-4 

."-".:.,: -546 0.796E-4 

-557 0.132E-3 

-567 0.388E-3 
. ..,,:.~ .... ~.,,~-

141 

a = 610 HPa 

max 

Stress f(a, a ) max 

(HPa) . (MPa:- 1 ) 

Loading 

-615 0 

-106 0.141E-5 

0 0.288E-5 

106 0.429E-5 

212 O.723E-5 

318 o .126E-4 

371 0.175E-4 

424 0.232E-4 

477 0.307E-4 

530 0.448E-4 

562 0.629E-4 

583 0.868E-4 

6,05 o .162E-3 

610 0.229E-3 

Unloading 

610 0 

106 0.681E-6 

0 0.188E-5 

-106 0.367E-5 

-212 0.630E-5 

-318 0.103E-4 

-371 0.159E-4 

-424 o .-194E-4 

-477 0.274E-4 

-530 0.416E-4 

-562 0.572E-4 

-583 0.894E-4 

-605 o .186E-3 

-615 0.372E-3 

:.. .. ' 
, :", 

-", ",~ ":: - ~ 

,~~- :.'~ - " 

,' ... ' 

"" .. 

VITA 

Shik, Hung Yuen was born March 7, 1958, in Kowloon, Hong Kong, 

the son of Yin W. Yuen and Lau W. Yuen. He attended elementary and 

high school in Kowloon. His high school diploma was obtained from 

the Maryknoll Fathers' -SchooL in l1ay 1976. He .. as enrolled in the 

Southeastern Oklahoma State University in the fall of 1976. 

142 

Shik entered Texas A&M University- ·in· September 1977 and majared 

in Mechanical Engineering. His B.S. degree was awarded in May of 

1980. He began.graduate study in July 1980 and will receive his M.S. ______ ~ . ~~ ---...... 

degree in May 1983. 

Shik's permanent address is 1317 Ching Yung House, Cheung Ching 

Estate, Tsing Yi Is., N.T., Hong Kong. 

,~ "" 

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