Simplifications and improvements in unified constitutive equations for creep and plasticity—II. Behavior and capabilities of the model

0956-7151/90 $3.00 + 0.00 Copyright © 1990 Pergamon Press plc

Acta metall, mater. Vol. 38, No. 11, pp. 2117-2128, 1990 Printed in Great Britain. All rights reserved

SIMPLIFICATIONS A N D IMPROVEMENTS IN UNIFIED CONSTITUTIVE EQUATIONS FOR CREEP A N D PLASTICITY--II. BEHAVIOR A N D CAPABILITIES OF THE MODEL G. A. HENSHALLt and A. K. MILLER Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, U.S.A: (Received 13 September 1989; received for publication 27 April 1990) Abstract--The capabilities of the MATMOD-BSSOL unified constitutive equations, developed in Part I of this paper, are demonstrated through comparisons of computed results against a wide variety of mechanical test data for pure aluminum and A1-Mg alloys. Flow stress plateaus and Class I steady state creep in the alloys are accurately simulated. These results confirm the viability of modeling solute strengthening through the short range back stress variable. Independent predictions of primary creep, the temperature dependence of the strain rate sensitivity, creep-plasticity interaction, stress relaxation, and the effects of changes in the applied stress are presented. The predictions generally compare favorably with the data. Discussion of these results emphasizes that the internal behavior of the model is consistent with the physical processes believed to control nonelastic deformation, thus providing confidence in predicting mechanical responses to complex deformation histories. The model's breadth, accuracy, and the effort required to evaluate the material constants are also discussed. R6sum6---Les possibilit6s des 6quations constitutives unifi6es MATMOD-BSSOL, d6velopp6es dans la premi6re partie de cet article, sont d6monstr6es par la comparaison des r6sultats calcul6s ~t un grand hombre de donn6es d'essais m6caniques pour l'aluminium pur et les alliages A1-Mg. Les paliers de contrainte de fluage et le fluage en r6gime permanent de classe I sont simul6s avec pr6cision dans ces alliages. Ces r6sultats confirment la viabilit6 de la mod61isation de la r6sistance m6canique du solut6 par la variable:contrainte en retour fi courte distance. On pr6sente des pr6visions ind6pendantes du fluage primaire, de la variation de la sensibilit6 de la vitesse de d6formation vis fi vis de la temp6rature, de l'interaction fluage-plasticit6, de la relaxation de contrainte, et des effects de variations de la contrainte appliqu6e. Les pr6visions sont g6n6ralement en accord favorable avec les donn6es exp6rimentales. La discussion de ces r6sultats souligne que le comportement interne du mod61e est en accord avec les processus physiques qui sont susceptible de contr61er la d6formation non 61astique, ce qui conf6re de la fiabilit6 fi la pr6vision des r6ponses m6caniques pour des 6chantillons ayant subi des traitements de d6formation complexes. Le champ d'application due mod61e, sa pr6cision et I'effort requis pour 6valuer les constantes du mat6riau sont aussi discut6s. Zusammenfassung--Die M6glichkeiten der in Teil I dieser Arbeit entwickelten vereinheitlichten Grundgleichungen, MATMOD-BSSOL, werden demonstriert, indem berechnete Ergebnisse mit einer groBen Reihe von mechanischen Prfifergebnissen an reinem Aluminium und an AI-Mg-Legierungen verglichen werden. FlieBspannungsniveaus und station~ires Klasse-I-Kriechen lassen sich genau simulieren. Diese Ergebnisse best/itigen den Weg, die Mischkirstallh/irtung fiber eine Variable der kurzreichenden R/ickspannungen zu beschreiben. Unabh/ingige Voraussagen zum prim~ren Kriechen, die Temperaturabh/ingigkeit der Dehnungsratenempfindlichkeit, Kriechen-Plastizit~its-Wechselwirkung, Spannungsrelaxation und Einfl/isse durch A,nderungen in der/iuBeren Spannung werden vorgelegt. Die Voraussagen entsprechen den Messungen ausgezeichnet. Die Diskussion dieser Ergebnisse zeight, dab das interne Verhalten des Modelles mit den physikalischen Prozessen, die als verantwortlich fiir das nichtelastische Verhalten angesehen werden, vertr/iglich ist. Dieses Vertrauen ist notwendig, wenn das mechanische Verhalten bei komplexen Verformungsgeschichte vorausgesagt werden soil. Die Breite des Modells, die Genauigkeit und der ffir die Auswertung der Materialkonstanten notwendige Aufwand werden auch diskutiert.

1. INTRODUCTION An improved version of the M A T M O D unified constitutive equations has been developed in Part I of this paper [1]. The new model, M A T M O D - B S S O L , combines most of the capabilities of previous versions

tPresent address: Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A.

in a way that minimizes the complexity of the equations and the number of material constants. These improvements are largely the result of new methods used to model solute strengthening. In particular, yield strength plateaus and Class I steady state creep are simulated through the effects of solute concentration on the evolution of short range back stresses. The purpose of this paper is to demonstrate the capabilities of M A T M O D - B S S O L in simulating and

2117

2118

HENSHALL

and MILLER:

UNIFIED

CONSTITUTIVE

EQUATIONS--II

Table 1. Summary of the equipment and experiments needed to determine the MATMOD-BSSOL material constants Test type and approximate number required

Apparatus Creep machine

Tensile machine

Measure

.

i

Constants

Stress-change at various temperatures (5-10) Temperature change (8-15) Constant stress creep (10-15)

(cs as a function of a~ (pure solvent metal or alloy) i, (pure solvent metal) i,, as a function of tr~ (alloy)

B, d

A4, C2, Cs, D~, D 2, m2, n, ql

Constant engineering strain rate test (8-15)

~ as a function of a at various temperatures (alloy)

Am, A3, C4, H I , H 2, H 3,

Q*, Tt

H 4 , m l , G2, fl

Reverse torsion machine

Constant engineering strain rate test (5-15)

~ as a function of o at various temperatures and strain rates-could also use diffusion data (alloy)

Cyclic test at a constant engineering strain rate (2)

Cyclic hysteresis loop (alloy)

F~o~. . . . Zm~x Q~ol

As, C4, Gz, //3, //4, qs

Notation: subscript "~" denotes constant structure; subscript "~," denotes steady state.

calculations presented here were made using material

predicting a variety of mechanical test data. All of the

tions are the same as those in the one-dimensional equations developed in Part I of this paper.

constants for alloys in the AI-Mg

These

system. This system

was chosen because solute effects are pronounced in these materials, thus providing the opportunity to test the new methods

used to represent these phenomena.

constants

were determined

by fitting mech-

a n i c a l t e s t d a t a , a s d e s c r i b e d e l s e w h e r e [2]. A s u m mary of the experimental data needed to determine the MATMOD-BSSOL constants for a particular

I n a d d i t i o n , a l a r g e a m o u n t o f d a t a is a v a i l a b l e f o r this system, allowing the model to be tested over a wide range of behavior.

m a t e r i a l is g i v e n i n T a b l e 1. A k e y p o i n t is t h a t none of the material constants are temperature or

The calculations presented in this paper were made using the three-dimensional formulation of

of

MATMOD-BSSOL oped by Tanaka

[2]. T h e N O N S S

method

devel-

a n d M i l l e r [3, 4] w a s u s e d t o n u m e r i -

cally integrate these equations. The values material constants in the three-dimensional

of the equa-

strain rate dependent; constants

was

for each material a single set

used

for

all

the

calculations.

Thus, although there appear to be more constants in MATMOD-BSSOL than for other advanced constitutive equations, when several temperatures are of i n t e r e s t t h e t o t a l n u m b e r o f c o n s t a n t s is o f t e n l e s s t h a n f o r o t h e r m o d e l s [5]. T a b l e 2 g i v e s t h e v a l u e s o f

Table 2. The MATMOD-BSSOL material constants for pure aluminum and AI-Mg alloys Constant

Pure AI

A1-1.15% Mg ~

AI-3.1% Mg~

AI-5.8% Mg ~

B d Tt Q* E0 EI E2

1.0 x 104 2.0 461 K 35,500 cal/mol 1.16 x 104 --4.392 --1.55x 10-3

1.0 × 10 4 2.0 461 K 35,500 cal/mol 1.16 x 104 -4.392 - I . 5 5 x 10-3

1.0 × l 0 4 2.0 461 K 35,500 cal/mol 1.16 x 104 -4.392 --1.55×10 3

1.0 x 104 2.0 461 K 35,500 cal/mol 1.16 x 104 --4.392 - 1 . 5 5 x 10 3

HI A1 m~ /'/2

5.0 5.00 x 1024 8.0 3.33 x 10-4

5.0 3.08 x 102t 8.0 2.00 x 10-5

5.0 5.52 x 1018 8.0 2.00 x 10-5

5.0 5.00 x 1016 8.0 2.00 x 10 -5

H3 A3 //4 C4

1.20 × i0 3 1.50×10 -s 4.00 x 10 -4 1.50 x 10 -6

7.00 x 10 3 5.00x10 9 9.56 x 10-4 1.50 x 10 -6

8.70 x 10-3 5.00x10 9 1.97 × 10 - 3 1.50 × 10-6

1.42 x 10-2 5.00x10 9 3.33 x 10 -3 1.50 x 10 -6

n Dt D2 qt

5.0 6.00 x 1022 5.00 x 103 3.1

5.0 4.06 x 10t5 2.00 x 108 3.1

5.0 1.26 x 1015 2.00 x 108 3.1

5.0 8.00 x 1014 2.00 x 10s 3.1

m2

1.9

1.9

1.9

C: A4 Cs

1.25 x 104 5.50 x 103 7.53 x 107

6.00 x 103 1.22 x 104 7.53 x 107

1.9

3.30 x l0 s 1.35 x 104 7.53 x 107

2.00 x 103 1.80 x 104 7.53 × 10 7

Gz q3

3.00 x 10-s 0.5

2.00 x 10-4 0.5

2.40

0.5

2.87 x 10 -4 0.5

Fsol.m,~ Zmx fl Q~l

-----

1.33 9.83 x 10I° 25.0 30,000 cal/mol

3.75 9.83 x 10m° 25.0 30,000 cal/mol

7.00 9.83 x 10j° 25.0 30,000 cal/mol

X 10 - 4

"Constants affecting cyclic deformation are only approximate due to the lack of cyclic data.

HENSHALL and MILLER:

UNIFIED CONSTITUTIVE EQUATIONS--II

the material constants used to perform the calculations for pure aluminum and three AI-Mg binary alloys. The constants for pure aluminum and the 5.8% Mg alloy were obtained from best fits to the data. For the 1.15% and 3.1% Mg alloys many of the material constants were estimated by interpolation from those of pure aluminum and the 5.8% Mg alloy using procedures described elsewhere [2]t. These procedures greatly reduce the effort required to evaluate the material constants. It is noteworthy that the MATMOD-BSSOL material constants have also been determined for mild steel and two HSLA steels by Adebanjo [6], and the constants for a single-crystalline precipitation-strengthened nickelbase superalloy (CMSX-3) have been determined by Miller et al. [7]. The calculations presented in this paper have been divided into two major categories: simulations and independent predictions. The former refers to cases in which the data being simulated have been used to determine some of the material constants. These results demonstrate the accuracy with which MATMOD-BSSOL simulates the phenomena that were specifically used to derive the equations, and focuses on the solute-related phenomena of flow stress plateaus and Class I steady state creep. Independent predictions are calculations in which the data have not been used to evaluate any of the material constants. These results demonstrate the extent to which the physics of nonelastic deformation have been properly represented in the equations. The predictive capabilities, therefore, are used to gauge the success of MATMOD-BSSOL in modeling complex deformation histories.

2. MATMOD-BSSOL SIMULATIONS Many of the improvements and simplifications present in the MATMOD-BSSOL equations concern modeling of the effects of solutes on the stresstemperature (flow stress at constant nonelastic strain vs temperature) and steady state creep curves. The MATMOD-BSSOL simulations of these phenomena are presented and discussed in this section. Figure 1 compares the MATMOD-BSSOL simulations of the stress-temperature curve at several nonelastic strains with the data of Henshall and Miller [8] for A1-5.8% Mg. MATMOD-BSSOL accurately simulates the complex stress-temperature curves over the entire range o f temperature and strain. The largest error in the simulated flow stress in these simulations is only about 15%, and is probably caused by difficulties in precisely evaluating all of the material constants, not by fundamental errors in the equations. The tBecause of a lack of data, the constants for the alloys were not optimized for cyclic deformation. Thus, predictions of cyclic behavior using the constants in Table 2 are not expected to be accurate for the A1-Mg alloys. AM 38/11--G

60

f ....

50

_

I ....

I ....

A1-5.8;~ Mq

e

qX,x

,--4

40 -

X :30

_

I ....

- 1.28 x 10 .4 s "1

~ o

~

2119

MATMOD-BSSOL

o

" ~

LLI

DATA: "~

o 2OZ eN

rv-.~"x, "%.\

x loz e" ° 2z c,

- "~

20

.

o .02t( 6 N

10 0

O

200

400 TEMPERATURE

600

800

(K)

Fig. 1. The modulus-compensated flow stress as a function of temperature for A1-5.8% Mg deformed over a wide range of strains at an engineering strain rate of 1.28 × 10-4 s-l. The MATMOD-BSSOL simulations of the data of Henshall and Miller [8] are given by the solid lines. ability of MATMOD-BSSOL to simulate the stresstemperature behavior of pure aluminum for nonelastic strains ranging from 0.02 to 20% with a similar degree of accuracy has been demonstrated elsewhere [2]. The accuracy of the simulations in Fig. 1 is a result of modeling solute strengthening in a physically meaningful manner. Beginning with the 0.02% nonelastic strain curve, the plateau in the simulation is produced by the very rapid evolution and mechanically activated recovery of the short range back stress variable, RA, as shown in Fig. 2. (R B is negligible for strains this small.) The rapid evolution of RA, which leads to the large values shown in Fig. 2 at 0.02% nonelastic strain, represents the pinning and athermal break-away of dislocations from groups of immobile solute atoms [1, 2, 8]. It is interesting to note that a plateau is also observed in the stresstemperature curve for pure aluminum at strains below 0.2% [8], implying that short range back stresses also dominate the flow stress of pure metals at very small strains. This plateau is accurately simulated by MATMOD-BSSOL because R Aevolves rapidly relative to the isotropic hardening variables [2]. 30 . . . .

I ....

AI-5oSZ Hg ,¢ C3 "4 X

o

i .... MATHOO-BSSOL

- 1.28 X lO "4 s': 20

I ....

BEHAVIOR: o~-D--

~ ~

x

*

I0

--

20Z C N 10Z G" 2Z C" "

o.

O~

,.

O .... 0

I

200

400 TEMPERATURE (K)

600

800

Fig. 2. The total back stress, R A + RB, corresponding to the MATMOD-BSSOL simulations shown in Fig. 1.

2120

HENSHALL and MILLER: UNIFIED CONSTITUTIVE EQUATIONS--II

In addition to a plateau, the data for 0.02% strain show a small peak in the curve near 400 K . t This peak is caused by the interaction between mobile solutes and moving dislocations [8], and is accurately simulated through the FsoI variable, which represents this interaction [1]. At temperatures above 450 K the stress decreases rapidly with increasing temperature. This behavior is accurately simulated by the model because, as shown in Fig. 2, RA thermally recovers in this regime. The data at strains larger than 0.02% are also accurately simulated by MATMOD-BSSOL. The low temperature athermal behavior of the 5.8% Mg alloy at intermediate strains is produced in MATMOD-BSSOL by the large values of R A and R B. Comparing Fig. 2 with the simulations in Fig. 1 shows that the total back stress is a large fraction of a l E for each strain, especially for small strains, and is relatively temperature independent at low temperatures. The interactive solute drag peaks that are present in the data of Fig. 1 at intermediate and large strains are accurately simulated in M A T M O D BSSOL through/'so I and its interaction with Fp in the kinetic equation [see equation (4) of Part I]. The data show that the temperature at which the solute drag peak occurs, Tmax, decreases as the strain increases. The MATMOD-BSSOL simulations also exhibit this behavior because the back stresses begin to recover at ever lower temperatures as the strain increases, as shown in Fig. 2. Thus, at low temperatures and low strains the solute drag peak is superposed upon the athermal plateau, but at higher temperatures and higher strains the increase in stress due to interactive solute drag is partly cancelled by thermal recovery of the back stresses. Thus, MATMOD-BSSOL predicts (not just simulates) the decrease in TmaXthat occurs as the strain increases. This capability, which was not present in earlier versions of MATMOD, is evidence that the physics of solute strengthening have been properly represented in MATMOD-BSSOL. The methods described above for modeling solute strengthening were proposed by the authors in an earlier publication [8]. The simulations in Fig. 1 confirm the viability of these methods, specifically in simulating plateaus in the stress-temperature curve using the back stress variables, and simulating peaks in the stress-temperature curve at all strains with a single interactive solute strengthening variable. The generality of this approach is supported by MATMOD-BSSOL simulations of the stresstemperature curves of mild steel over a wide range of strains [6]. The accuracy of these simulations is better than was possible using a previous version of M A T M O D [9]. tThis peak can be seen more easily in Fig. 7 of the paper by Henshall and Miller [8]. :~The strain rates have been temperature compensated using the Arrhenius-like O' parameter defined in Part I of this paper.

Figure 3 presents the MATMOD-BSSOL simulation of the steady state creep data for pure aluminum. The simulations are reasonably consistent with the data over 21 orders of magnitude in the temperature-compensated strain rate.:~ The accuracy of the simulation demonstrates that the exponential form of the kinetic equation (and the resulting/~p and Px equations) provides the capability to represent steady state behavior over wide ranges of strain rate and temperature. The accuracy of the M A T M O D BSSOL simulation is slightly less than the fit of Luthy et al. [10] using a hyperbolic sine equation, and is approximately as accurate as the MATMOD-4V simulation (which also has a sinh form) presented by Lowe [11]. The MATMOD-BSSOL simulation of the steady state data for the 3% Mg alloy is presented in Fig. 4(a). The data are accurately simulated, including the transition between Class I (slope of about 3) and Class II (slope of about 5) behavior. Corresponding to the simulation shown in Fig. 4(a), Fig. 4(b) presents the behavior of the components of the stress provided by each of the MATMOD-BSSOL structure variables. ISO-p and ISO-2 refer to the components of the flow stress due to the isotropic strain hardening variables Fp and F~, and are defined V~

~

\l/d

+,_1

<,>

Figure 4(b) shows that RA, ~ dominates the flow stress throughout the Class I regime and the lower portion of the PLB regime. As discussed in Part I, these large values of RA simulate the domination of solute 1030

........

I

........

I

' o'

1024

t~ 1018

,(D

'

u

Pure AI O DB'~ OF Sorvl ,~ Grant

/o (300 - 800 KI ~ x OB~ OF"Forrler~ & Stonq (573 KI O1 • 08'~a oF GlbeJinq ~/ (673 K) o/ - o De¢o oF" Lushy et ~l° o/ (104 - 533 K] ~/ -

--

--

1012

106 10

-

,~,. . . . . . . I

........

10"4

I

,

,.

,10"3

~sslE Fig. 3. Steady state creep data and the corresponding MATMOD-BSSOL simulation for pure aluminum. The data are of: Luthy et al. [10], Ferriera et al. [30], Servi and Grant [31], and Gibeling [32].

HENSHALL and MILLER: UNIFIED CONSTITUTIVE EQUATIONS--II 1012 I0 II

a)

(b)

! .... I _

AI-3

........

I/

/

Hq

DATA:

1010

2121

- : ;:~,o~:a&eL~nqdon

'

1014

--'

....... I

........ I

....... I

....... I - -

HATHOD-BSSOL BEHAVIOR

~; /"0

1012

~-- i0 g

to I010

-

+ RA,ss x RB,ss o 1S0-o

_

x

ISO-~

to

• (i)

108 108

I0 7 106

108

105

, ,,I

. . . . . . . .

10"4

I

,

,

10"3 Crss/E

_

........

I0-7

I

10-6

......

I

10-5

...... I

10-4

....... I

10-3

COMPONENTS OF THE STRESS

Fig. 4. (a) Steady state creep data and the corresponding MATMOD-BSSOL simulation for A1-3% Mg. The data are of: Horita and Langdon [33] and Oikawa et al. [34]. (b) The components of the flow stress for the simulation shown in Fig. 4(a). drag in the Class I regime, which is represented by its effects on the short range back stress. The use of R A to simulate Class I steady state creep, instead of the F~oI terms used in earlier versions of MATMOD, allows MATMOD-BSSOL to simulate a Class I stress exponent that remains at a constant value approximately equal to 3. The transition from Class I to Class II creep is also simulated through the behavior of RA, as discussed in Part I of this paper. Although not clearly apparent in Fig. 4(b), the slope of the RA.~ curve changes from 3 to 5 at ~ss/O' approximately equal to 106. Another feature of the MATMOD-BSSOL simulation that concerns the transition between Class I and Class II behavior is that t h e ISO-p and ISO-2 curves cross. Thus, in the PLB regime and in the majority of the Class I regime the isotropic strength is strongly affected by ISO-p. In the Class II regime, which is just beginning at the lowest ~ss/~9', the isotropic strength is dominated by ISO-2. This behavior is consistent with the change in substructure that has been associated with the transition between Class I and Class II behavior; individual dislocations and dislocation loops dominate the substructure in the Class I regime, while subgrains dominate the substructure in the Class II regime [1, 12, 13]. Finally, note that the data of Horita and Langdon in Fig. 4(a) exhibit Class II behavior at high stresses. As discussed in Part I of this paper, this behavior was not incorporated into MATMOD-BSSOL because it would unnecessarily complicate the equations. Figure 4(a) demonstrates that this omission does not significantly decrease the accuracy of the simulations. One minor problem with MATMOD-BSSOL is apparent in the RB., curve in Fig. 4(b). The slope of the Rs.ss curve increases at very low strain rates or high temperatures. This non-physical behavior is

a consequence of not having a separate thermal recovery term in the/~s equation. The errors in Rs .... however, cause only minor errors in the flow stress because RB.ss is small. Furthermore, these errors occur only at temperature-compensated strain rates that are close to those at which diffusional creep dominates behavior and MATMOD-BSSOL becomes invalid. The minor improvement in modeling accuracy that would result from including a separate thermal recovery term in the /~B equation does not justify increasing the complexity of the equations. The capability of the MATMOD-BSSOL equations to simulate the steady state behavior of pure metals and Class I alloys is summarized in Fig. 5. The effects of solute concentration on the steady state creep curve are accurately simulated over the entire range of temperature-compensated strain rate. In particular, as the solute concentration increases MATMOD-BSSOL correctly simulates the increase in the stress or strain rate range over which Class I creep occurs. The steady state simulations for the alloys are largely controlled by thermal recovery of RA, which dominates the flow stress in the Class I regime. One advantage of simulating Class I creep through RA, instead of through Fso~, is that the stress exponent in the Class I regime is then independent of solute concentration. As shown by Miller and Sherby [14], using Fso~ to produce Class I creep causes the stress exponent in the Class I regime to decrease as the solute content increases. 3. INDEPENDENT PREDICTIONS

3.1. Simple loading histories Steady state creep data were used to determine the thermal recovery constants in MATMOD-BSSOL, but primary creep data were not used to evaluate any of the constants. Calculations of the primary creep

2122

HENSHALL and MILLER:

UNIFIED CONSTITUTIVE EQUATIONS---II

1015 * P u r e AI

..

. ^l-3z.q _ o ^,-5~ .q

1012

-

• (J.)

0~

lO g

1o6

/ / / f' - I

'/

I

/je

--

m

CA =l

10-5

'/

/ ]

I I tlllll 10-4

....... , , ,,]7 I 10-3

O'sslE Fig. 5. Summary of the steady state creep data and simulations for pure aluminum and A1-Mg alloys. The data for pure aluminum and A1-3% Mg are from the sources listed m Figs 3 and 4. The data for AI-I% Mg are from Horita and Langdon [33], and Oikawa et al. [34]. The data for the AI-5% Mg are from these same sources, plus from Oliver [35] and Mills [36]. Some of the individual data points from each set of AI-Mg data have been omitted for clarity.

Inverted transient behavior, which is due to mobile dislocation effects, was not included in M A T M O D BSSOL because the improvements in accuracy were judged to be insufficient to justify the added complexity of the equations [2]. One technologically important aspect of solid solution strengthening is its effect on ductility. Alloying often produces a low or negative strain rate sensitivity, m, at intermediate temperatures and strain rates, leading to poor ductility [17, 18]. At high temperatures or low strain rates, however, m may reach values as large as 0.3. Failure strains in tension as large as 175% have been reported for A1-Mg alloys deformed in this regime [18]. Although M A T M O D - B S S O L does not directly address the problem of failure, it does provide information that can be used within a suitable failure model to predict ductility. Specifically, both m and the strain hardening rate can be calculated from the results of M A T M O D - B S S O L simulations. STRESS

(0 '''1

....

I ....

(ksl) I ....

I ....

I ....

Pure AI

0.3 z

curve are therefore independent predictions of the model. Predictions of the amount of primary creep straint as a function of stress are shown to compare favorably with data for pure aluminum in Fig. 6(a). Figure 6(b) presents a similar comparison for A1-5% Mg. M A T M O D - B S S O L correctly predicts that the amount of primary creep is much smaller for the alloy than for pure aluminum. This behavior is predicted because the M A T M O D - B S S O L material constants governing the amount of primary creep are determined by fitting constant strain rate tensile data at high temperature. For pure aluminum deformed at constant strain rate, the steady state stress, a,s, is reached at large strain, for example 7-20% at 600 K for various strain rates [15]. For A1-5% Mg, however, a,s at high temperatures is reached after strains of only 1-3% [2]. Physically one would expect a correlation between the amount of transient deformation for strain rate and stress controlled tests, since the underlying dislocation mechanisms are the same. Based on the predictions in Fig. 6 it is clear that M A T M O D - B S S O L has captured this aspect of nonelastic deformation. One feature of the primary creep (and high temperature tensile) data for A I - M g alloys that is not predicted by M A T M O D - B S S O L , however, is the inverted nature of the transient [16].

Or) n LU

0.2

LO O~ 0 >-

0.1 n~ n

~. / X

o DATA Or mBELI"G - - - - .~.O0-SSSOL ~ED]CTIO"

,.,I

0.0

....

20

I .... I .... 40 60 STRESS (HPa)

I .... 80

I tO0

(b) i

0.05

AI- 5% Mg

i

o Data of Mills

0.04 [] MATMOD-BSSOL Prediction O. O

0.03

O

O

0

> , 0.02 ¢=

[3

E

=.. O.

[] 0.01

0.00

m 10

I

I

I

20

30

40

i

I 50

60

Stress (MPa)

tThe primary creep strain for pure AI is defined as follows. A straight line is drawn through the strain vs time curve in the steady state regime and extrapolated to zero time. The point at which this line intersects the strain axis is the primary creep strain. For the A1 5 Mg data the inverted primary curve makes this procedure impossible, so for the alloy the primary creep strain is simply defined as the strain at which steady state creep begins.

Fig. 6. (a) The MATMOD-BSSOL prediction of the amount of primary creep strain for pure aluminum as a function of the applied stress is compared with the data of Ahlquist and Nix [16], Gibeling [32], and Sherby et al. [37]. (b) Comparison of the primary creep strain data of Mills [36] for A1-5% Mg with the MATMOD-BSSOL predictions for two different applied stresses.

HENSHALL and MILLER:

2123

UNIFIED CONSTITUTIVE EQUATIONS~II

Focusing on the strain rate sensitivity, Fig. 7 presents the MATMOD-BSSOL predictions of the "constant structure" strain rate sensitivity, r~, at a strain of 2% as a function of temperature for A1-5.8% Mg. The constant structure strain rate sensitivity is presented because it is automatically computed during numerical integration, and is defined as ~N

0.20

....

w ~_~~_ ~ o.15 ~ to to 0.z0 ~ ~ ~

~z

z

I ....

AI-5.8Z Hq - 2.0Z

I ....

I ....

HATHOD-BSSOL PREDICTION

/ /

- 1.28 X 10-45 -1

Serrated Yleldznq Observed

o.os

C)

m=

to

[ a / E -- (RA + Rs)]

0.00

....

I

×

t~[a/E --

(R A-I- RB)]

(3)

where dN is the effective nonelastic strain rate and [ a / E - ( R A + R a ) ] is the effective moduluscompensated back stress-corrected stress. Although is numerically different from the commonly measured values at constant strain rate or stress, it exhibits the same qualitative trends as a function of temperature and strain. Figure 7 shows that the predicted strain rate sensitivity is close to zero at low temperatures, producing nearly rate-independent "plasticity". At intermediate temperatures n~ is negative. Negative values of rh are produced in the equations by F~ol, which increases in this regime as the temperature increases. As discussed in Part I of this paper, the product of Fp and Fso1 represents the interaction between mobile solutes and dislocations. Since this interaction is generally accepted to be the cause for negative strain rate sensitivities [19-21], MATMOD-BSSOL properly represents the physics underlying the macroscopic behavior. Further, the quantitative accuracy of the predictions can be assessed by comparing the temperature regime for which MATMOD-BSSOL predicts negative strain rate sensitivities with that for which the data exhibit serrated yielding. The fact that serrations in the stress-strain curve occur only when m is negative has been well established [20]. The data of Henshall and Miller [8] for 2% strain show that serrated yielding occurs for temperatures approximately the same as those for which th is negative in Fig. 7. The MATMOD-BSSOL predictions shown in Fig. 7 also demonstrate that the strain rate sensitivity increases rapidly at high temperature, eventually becoming much larger than its low temperature value. The high temperature constant structure strain rate sensitivities shown in Fig. 7, however, are considerably lower than those that would be expected based on measurements of the steady state stress exponent in the Class I regime. The steady state value of rn at high temperatures is given by m~ = 1In

(4)

where n is the steady state stress exponent. In the Class I regime rns~ should therefore have a value of approximately 0.3. The discrepancy between

I .... 200

I ....

400 T£MPERATURE [K)

I .... 000

800

Fig. 7. The MATMOD-BSSOL prediction of the constant structure strain rate sensitivity at a strain of 2% as a function of temperature for A1-5.8% Mg. The temperature range for which serrated yielding was observed experimentally by Henshall and Miller [8] at 2% strain is also shown. this value and those shown in Fig. 7 is due to the numerical differences between th and the experimentally measured mss at constant strain rate, as verified by strain rate change simulations performed at constant temperature in the Class I regime [2]. The results of these simulations were used to compute mss from equation (5) d(log trss) rns~= d(log d~s)"

(5)

The value of rn~s predicted by MATMOD-BSSOL was approximately 0.3, as expected. 3.2. C o m p l e x loading histories

The ultimate goal in developing an advanced constitutive model such as MATMOD-BSSOL is to be able to predict material response to complex loading histories. In general, these histories may include large changes in temperature, strain rate, or stress, as well as cyclic loadings of varying amplitude. The remaining predictions presented in this paper demonstrate the capability of MATMOD-BSSOL to produce reasonably accurate predictions for a variety of complex loadings. The ability of MATMOD-BSSOL to simulate cyclic deformation has been described in detail elsewhere [2, 22, 23] and so will only be discussed briefly here. Because of the importance of cyclic saturation in the development of the equations [1], the MATMOD-BSSOL prediction of the room temperature cyclic stress-strain curve for pure aluminum is presented in Fig. 8. For each nonelastic strain amplitude, both the data of Ziaai [24] and the MATMOD-BSSOL calculations were produced by cycling until a saturated hysteresis loop was achieved. The peak saturated tensile stress was then measured and plotted as a function of the applied nonelastic strain amplitude. The steady state flow stress predicted by the model for the same temperature and strain rate as that used in the cyclic case is indicated by the dashed line in Fig. 8. The level of this line in

2124

HENSHALL and MILLER: ....

I ....

I ....

UNIFIED CONSTITUTIVE EQUATIONS--II

I 15

3 I00a. • DATA UTILIZED IN GETTING CONSTANTS

-~

Oi u~

to

E

tn

o~

+ +o

+

) 25 -<~

0

o DATA OF Z I A A I MATMOD-BSSOL PREDICIION

. . . .

I

,

5

Pure AI 2gO K 6 - 7 x I0-4=- I

--

,

I

. . . .

0.05 0.1 NONELASTIC STRAIN AMPLITUDE

v-

I

0

0.15

Fig. 8. The MATMOD-BSSOL prediction of the cyclic stress-strain curve for pure aluminum at 296 K is compared with the data of Ziaai [24]. The solid circle represents the data point used in evaluating some of the material constants in the model. The dashed line represents the monotonic steady state flow stress predicted by MATMOD-BSSOL for the temperature and strain rate given in the figure. relation to the data demonstrates the important fact that the cyclically saturated flow stress is significantly lower than the monotonic steady state flow stress (not to be confused with the monotonic stress-strain curve). Some of the M A T M O D - B S S O L material constants were evaluated from the saturation stress at a strain amplitude of 1.8%, which is indicated by the solid circle in Fig. 8. The overall level of the cyclic stress-strain curve computed by M A T M O D BSSOL is therefore not independent of the data. The calculated slope and curvature, however, are independent predictions of the model. These predictions are more accurate than those of previous versions of M A T M O D . M A T M O D - B S S O L is capable of properly predicting the shape of the cyclic stress strain curve because of the state variable interactions in the structure evolution equations, as described in Part I of this paper and elsewhere [2, 22, 23]. Through these interactions the model properly represents the physical processes leading to cyclic saturation in pure aluminum. A sudden change in the applied stress during creep is another example of complex loading. Figure 9(a) presents the M A T M O D - B S S O L predictions of the stress-change data of Sherby et al. [25]. The steady state creep rates computed by the model at large strains are not independent predictions because steady state data were used to determine some of the material constants. The predicted strain rates during all of the transients, not just those following the stress change, are independent of the data. The transient prior to the stress change (a = 27.6 MPa) and that in which the stress was not changed (a = 13.8 MPa) are both accurately predicted. The predicted transient following the stress change has the proper curvature, but is somewhat too rapid. Considering the breadth of phenomena covered by M A T M O D - B S S O L , however, this prediction is reasonably accurate. The reason that M A T M O D - B S S O L predicts such a rapid transient is that RA, which dominates the stress,

evolves very quickly following the stress change, as shown in Fig. 9(b). The evolution of Fp following the stress change is also quite rapid and contributes to the error in the prediction. It is noteworthy that qualitatively similar predictions are produced for stress-increases or changes in the applied strain rate [2]. In all cases the model over predicts the rate of the transients following a change in the boundary conditions, but the results are generally reasonable. It has also been shown that M A T M O D - B S S O L predicts strain-enhanced recovery and strain softening of pure aluminum due to sudden decreases in the applied strain rate at high temperatures [2]. Since the ability of the earlier model M A T M O D - 4 V to predict these behaviors was considered to be a major improvement to M A T M O D , o) ....

100

I ....

I ....

27.0MP~a

~= i o - 3 =<

I ....

- MATMOD-BSSOL -- -- -- PREDICTIONS

(4 ksz)

z

I ....

o, o Da~aoF Sherby, Tr0zera, end Dorn

0

<

~,

m

10"6

-

0 0 ~

~Q-

-e

J 12 ke]]

o z

Pure A| 478 K

10 " 9

....

I .... O.l

I ....

I ....

I ....

0.2 0.3 NONELASTIC STRAIN

0.4

0.5

(b) 0

1.2 1.0

Pure A t 478 K

o.e (~

0.6 0.4

.

-

o.,75

~=

o.125fO.lOOL

.

.

.

.

.

.

.

.

.

-~

~o o.15o10.075

~-

0.050~

-

,

i

....

,

....

5" 2.5I (~) x

tL'~

2.0 1.5 1.0 0.5

;

t

0.1

0.2

2.5 ~ x ~

2.0

~

1.0

1.5 -

0

-

.

,

,

.

i

0.3

. . . . 0.4

NoneLastic s t r a i n

Fig. 9. (a) The effect of a decrease in the applied stress on the nonelastic strain rate of pure aluminum deformed at 478 K. The circles and squares represent the data of Sherby et aL [25], and the lines represent the MATMOD-BSSOL predictions. The circles and the dashed line represent the behavior at a constant stress of 13.8MPa (2ksi). The squares and the solid line represent the behavior for a stress decrease from 27.6 MPa (4 ksi) to 13.8 MPa (2 ksi). (b) The behavior of the MATMOD-BSSOL structure variables during the stress decrease prediction shown in (a).

2125

HENSHALL and MILLER: UNIFIED CONSTITUTIVE EQUATIONS--If

it is important that MATMOD-BSSOL retain these capabilities. One important feature of MATMOD-BSSOL, and previous versions of MATMOD, is that the low temperature "plastic" strain rate is not artificially separated from the high temperature "creep" strain rate. This "unified" approach enables the model to predict the interactions between creep and plasticity that occur when a material is deformed over a wide range of temperature. Variable temperature loading histories are common, for example in gas turbine and nuclear reactor components, so it is important that the model be capable of handling this type of behavior. The ability of MATMOD-BSSOL to predict creep-plasticity interactions is demonstrated in Fig. 10. The open circles and the lowest line correspond to deformation of annealed pure aluminum +at 298 K and the constant strain rate shown in the figure. The other symbols and the upper two lines correspond to constant strain rate deformation at 298 K following a 35% prestrain under constant stress creep conditions at 478 K. MATMOD-BSSOL properly predicts the effect of the prior creep history on the flow stress at low temperature. In particular, the prior creep deformation increases the flow stress without significantly changing the strain hardening rate, which is consistent with the data. The reason that MATMOD-BSSOL is capable of predicting creep-plasticity interaction is that the same state variables are involved in both "creep" and "plastic" deformation. The high temperature prestrain causes the structure variables to reach values that are larger than those for an annealed material. The increased values of the structure variables then cause the low temperature flow stress to be larger than that for the annealed material. Finally, Fig. 11 compares the MATMOD-BSSOL predictions of the stress relaxation behavior of pure aluminum at 296K with the data of Hart and

8°L

I....

. . . .

L

6or~

I

. . . .

DATA OF SHERBY 8. BORN

|0

• 2.8 x 10-3= -1

F

O []

~

03 03

I . . . .

Pure AI 208 K

0

[] r I D

0¢

40

W

6 ~

S

m 03 bJ

03

1478 K! 20

/'o 0

.... °o

O 4 k s t TO E: - 0,35 ¢' 2 k,= TO £ 0.35 o ANNEALED -NATMOO-BSSOL -

--

2

I .... I,,,,1 ..... o 0.02 0.04 0.06 o.o8 TOTAL STRAIN

Fig. 10. MATMOD-BSSOL predictions of the effects of prior creep straining at 478 K on the stress-strain curve for

pure aluminum at 298 K. Data of Sherby and Dorn [38].

......I ....... 1 ....... 1 ....... 1 ....... 1 ........I ........I " 80

50

Pure AI 298K

3=

~

~x

.~

, o

¢

l*

x e

1o 9 8

o

7

4c

6

t,.o

~.,

4

o 0~{ PRESTRAIN -..... ,I ...... J

.......

J ,,,;,,,~

Mm.OO-BSSOL ....... 1 ........ I ........

I0"8

O-g 10-8 10-7 10-5 10-4 NONELASTIC STRAIN RATE [s "~)

I ,

3

10"3

Fig. 11. MATMOD-BSSOL predictions of the stress relaxation behavior of pure aluminum at 296K following

prestrains of 6 and 14%. Data of Hart and Solomon [26].

Solomon [26] for two different prestrains. Prestraining was performed at a total strain rate of 10 -3 s -l, and was followed by holding the total strain constant and allowing the stress to relax. In the M A T M O D BSSOL calculations prestraining was stopped when the total strain was equal to that at which Hart and Solomon began their relaxation experiments. The overall predicted stresses are low because the MATMOD-BSSOL material constants for pure aluminum cause the predicted flow stress to be below the experimental value in this temperature and strain regime. Although MATMOD-BSSOL predicts that stress relaxation does occur, the amount is under predicted for both prestrains, particularly at low strain rates. In addition, the concave-downward shapes of the experimental stress relaxation curves are not duplicated. An interesting implication of these M A T M O D BSSOL predictions concerns the fundamental nature of stress relaxation. Hart and Solomon [26] proposed that data of the type shown in Fig. 11 represent constant structure behavior. In the M A T M O D BSSOL predictions none of the structure variables undergo any recovery during relaxation [2], Thus, the shape of the predicted curves results from the exponential form of the kinetic equation, which represents deformation at constant structure. Since this equation, and the material constants for pure A1, were derived directly from constant structure creep data, the errors in the predictions suggest that the data in Fig. 11 do not truly represent constant structure behavior. Regardless of these considerations, the failure of M A T M O D - B S S O L to properly predict the shape of the stress relaxation curve suggests that some improvements to the model are needed. This conclusion is supported by the fact that M A T M O D BSSOL does not predict that the stress relaxation curves formed by prestraining to different stress or strain levels can be translated along the log ~ and log a axes to form a single "master" curve. Hart and co-workers [26, 27] have demonstrated the existence

2126

HENSHALL and MILLER:

UNIFIED CONSTITUTIVE EQUATIONS--II

of such a curve for pure aluminum and several other materials. 4. DISCUSSION

4. I. Breadth As discussed in Part I of this paper, in order to model complex deformation histories the constitutive equations must simulate a broad range of phenomena. Collectively, the previous versions of MATMOD simulate as broad a range of behavior as any other constitutive model currently available [28]. MATMOD-BSSOL combines most of the capabilities of these previous models in one relatively simple set of equations, and therefore exhibits excellent breadth. The simulations and independent predictions presented in Sections 2 and 3 demonstrate some of the behavior covered by MATMOD-BSSOL. To summarize these capabilities and to mention others that have been demonstrated elsewhere [2, 6, 29], Fig. 12 lists the phenomena modeled by the MATMODBSSOL equations. Note that MATMOD-BSSOL is the first version of MATMOD capable of modeling both solute strengthening and complex strain softening behavior.

4.2. Accuracy When assessing the accuracy of MATMODBSSOL two things must be kept in mind. The first is that in a complex model designed to handle a broad range of phenomena, some inaccuracies for particular simple histories must be expected. Of course, if one is interested in simulating only very limited behavior, a simple (probably empirical) constitutive model with high accuracy would be more appropriate than a model such as MATMOD-BSSOL. The inaccuracies in MATMOD-BSSOL simulations are largely due to compromises made during evaluation of the material constants [2], and to the variation in the properties used to evaluate the material constants for materials taken from different sources. Independent predictions may also appear to be in error because of differences in the properties of the material used to determine the material constants and that used to perform the experiments against which the predictions are compared. As discussed in Part I, accurately modeling the effects that solutes have on mechanical behavior over wide ranges in temperature, nonelastic strain rate, and nonelastic strain was instrumental in developing

1. General "plasticity",including: (a) essentiallyelastic behavior followed by gradual yielding (b) strain rate sensitivity,includingnegative values at intermediate temperatures, and large positive values at high temperatures (c) temperature sensitivity,includingpeaks and plateaus for alloys 2. Steady state "creep", including: (a) power-law and power-law breakdown (b) Class I behavior with a constant stress exponent approximatelyequal to 3 (c) change in stress exponent from 3 to 5 correspondingto the change from Class I to Class II behavior (d) dependenceof the steady state creep rate on solute concentration 3. Primary creep, including: (a) the stress dependenceof the primary creep strain 4. Cyclic stress-strain behavior,including: (a) Bauschingereffect (b) cyclic hardening and softening (c) shakedown to a saturated condition of constant stress or strain amplitude (d) hysteresis loop asymmetry 5. Recovery, including: (a) static recovery at high temperatures (b) dynamic recovery 6. Strain softening, including: (a) unidirectionalstrain softening (b) directional strain softening (c) cyclic strain softening 7. Complex histories, including: (a) stress changes (b) strain rate changes (c) temperaturechanges (d) creep-plasticityinteraction (e) yield stress plateaus in cold-workedmaterials (f) multiaxialdeformation 8. Precipitationstrengthening 9. Interactionsof all the above Fig. 12. Summary of the phenomena modeled by the MATMOD-BSSOL constitutive equations.

HENSHALL and MILLER: UNIFIED CONSTITUTIVE EQUATIONS---II the equati6ns. As shown in Figs 1-5, M A T M O D BS~;OL simulates these effects very well. Most notably, the complex stress-temperature behavior of A1-Mg alloys is simulated with no more than a 15% error in stress for temperatures between 0.08 Tm and 0.8 Tm and nonelastic strains from 0.02 to 20%. Class I and Class II steady state creep behavior, including the effects of varying solute content, are also simulated accurately. Independent predictions in which the boundary conditions do not change during the deformation history are consistent with the data, for example those shown in Figs 6 and 7. MATMOD-BSSOL predictions in which the boundary conditions change during deformation are qualitatively correct and, in most cases, are reasonably accurate quantitatively. As shown in Fig. 9, however, the transients predicted by MATMOD-BSSOL immediately following a change in boundary conditions are invariably too rapid. These errors are caused by the rapid evolution and domination of the short range back stress variable, R A. This appears to be a general deficiency of the model and deserves more attention in future work. The stress relaxation predictions shown in Fig. 11 demonstrate another area in which improvements to the model are warranted. Although there are some problem areas, the results presented in this paper show that, in general, the accuracy of MATMOD-BSSOL is good and qualitatively correct results are obtained in almost every case. The internal behavior of the model underlying the macroscopic responses has also been shown to be consistent with the physical processes believed to control nonelastic deformation. It is this consistency between deformation physics and internal behavior of the model that provides the capability to successfully predict nonelastic deformation under complex thermo-mechanical loading conditions.

4.3. Simplicity and Manageability In order to accurately model complex deformation histories a broad range of behavior must be simulated, leading to complicated equations and a large number of material constants. The user of the model, however, desires simplicity. A major concern in this respect is evaluation of the material constants, which is currently the most problematic aspect of implementing advanced constitutive equations in structural analyses. One obvious strategy in addressing this problem is to minimize the number of material constants (without seriously sacrificing accuracy or breadth of the model). This is the strategy that was used in developing MATMOD-BSSOL, as discussed in Part I of this paper. For a given number of material constants, however, there are three other factors affecting the effort needed to determine their values. The first is the number of experiments that must be performed in order to evaluate all of the material constants. Table 1 summarizes the experiments needed to evalu-

2127

ate the MATMOD-BSSOL material constants. The number of experiments is, of course, closely related to the number of constants, but the relationship is not necessarily one-to-one. Naturally, it is desirable to perform the minimum number of experiments. One must bear in mind, however, that as the breadth of the model increases, a larger number of experiments will generally be needed to characterize the material. For example, if the model covers cyclic deformation then some cyclic experiments will be required to evaluate the material constants. The second factor is the ease of performing the experiments. Simple, standard experiments are more desirable than experiments that require complicated loading paths, high-resolution strain measurements, a high level of expertise, or specialized equipment. One advantage of standard tests is that the data are often available in the literature, so that no additional experiments need to be performed. Table 1 shows that, except for the constant structure tests, all the experiments needed to determine the M A T M O D BSSOL material constants are standard experiments and are easily performed. The final aspect of evaluating the material constants is the ease of data analysis. It would be desirable to use standard analysis techniques, for example multiple linear regression, to evaluate the material constants from experimental data. Unfortunately, such techniques have not been used in determining most of the MATMOD-BSSOL material constants. The major stumbling block is that the structure variable interactions discussed in Part I cause complex inter-dependencies to exist between the material constants. Thus, rather tedious trialand-error comparisons between simulations and the experimental data have been used to evaluate the MATMOD-BSSOL constants. In the future, application of artificial intelligence and expert systems techniques may be helpful in automating this process, thus greatly reducing the effort required to use the model. 5. SUMMARY AND CONCLUSIONS MATMOD-BSSOL is the most recent and complete member of the M A T M O D family of unified constitutive equations. Using A I - M g as a model system, computer calculations using M A T M O D BSSOL have been compared with experimental data. Based on these comparisons the model has been assessed in terms of its breadth, accuracy, and manageability with the following conclusions: 1. MATMOD-BSSOL combines most of the capabilities of previous versions of M A T M O D in one comparatively simple set of equations. The model therefore exhibits excellent breadth, which is required for simulating complex deformation histories. 2. The accuracy of the model is generally good, although some areas for improvement have been discussed. Solute strengthening phenomena, particularly

2128

HENSHALL and MILLER:

UNIFIED CONSTITUTIVE EQUATIONS--II

plateaus and peaks in the curve of stress vs temperature and Class I steady state creep behavior, are accurately simulated by M A T M O D - B S S O L . The ability of M A T M O D - B S S O L to accurately simulate these behaviors stems largely from modeling the effects of solute concentration on the evolution of the short range back stress. Independent predictions using the model have also been shown to be consistent with a wide variety of mechanical test data. Specifically, the capability of M A T M O D - B S S O L to model transient creep, the temperature dependence of the strain rate sensitivity, creep-plasticity interaction, cyclic saturation, and changes in applied stress have been demonstrated. The ability of M A T M O D BSSOL to properly model these (and other) aspects of nonelastic deformation stems from the consistency between the internal behavior of the model and the physical processes believed to control nonelastic flow in metals. 3. The number of material constants present in M A T M O D - B S S O L is significantly less than in previous versions of M A T M O D having similar capabilities. In addition, almost all of the constants can be determined from simple creep, tensile and cyclic experiments. Additional effort is required, however, in automating the process of evaluating the material constants from the mechanical test data. Acknowledgements--The authors wish to thank Dr Toshi Tanaka for his assistance in numerical integration of the equations. The financial support of the U.S. Department of Energy under grant No. DE-FG03-84ER45119 is gratefully acknowledged. REFERENCES

1. G.A. Henshall and A. K. Miller, Acta metall, mater. 38, 2101 (1990). 2. G. A. Henshall, Ph.D. dissertation, Stanford Univ. (1987). 3. A. K. Miller and T. G. Tanaka, J. Engng Mater. Tech. 110, 205 (1988). 4. T. G. Tanaka and A. K. Miller, Int. J. Num. Meth. Engng. 26, 2457 (1988). 5. K. P. Walker, NASA Report CR-165533 (1981). 6. R. O. Adebanjo, Ph.D. dissertation, Stanford Univ. (1987). 7. A. K. Miller, T. G. Tanaka and R. O. Adebanjo, Application of the M A T M O D - B S S O L Constitutive Equations to the Single Crystal Nickel-Base Superalloy, CMSX-3, Report to the Garrett Turbine Engine Company (1988). 8. G. A. Henshall and A. K. Miller, Acta metall. 37, 2693 (1989). 9. T. G. Tanaka, Ph.D. dissertation, Stanford Univ. (1983).

10. H. Luthy, A. K. Miller and O. D. Sherby~Acta metall. 28, 169 (1980). 11. T. C. Lowe, Ph.D. dissertation, Stanford Univ (1983). 12. P. Yavari and T. G. Langdon, Acta metall. 30, 2181 (1982). 13. F. A. Mohamed and T. G. Langdon, Acta metall. 22, 779 (1974). 14. A. K. Miller and O. D. Sherby, Acta metall. 26, 289 (1978). 15. U. F. Kocks, J. Engng Mater. Tech. 981-I, 76 (1976). 16. C. N. Ahlquist and W. D. Nix, Acta metall. 19, 373 (1971). 17. B. A. Parker, Proc. Fifth Int. Conf. on the Strength of Metals and Alloys (edited by P. Haasen, V. Gerold and G. Kostorz), pp. 899-904 (1979). 18. R. A. Ayres, Metall. Trans. 8A, 487 (1977). 19. P. G. McCormick, Acta metall. 20, 351 (1972). 20. B. A. Wilcox and A. R. Rosenfield, Mater. Sci. Engng 1, 201 (1966). 21. A. W. Sleeswyk, Acta metall. 6, 598 (1958). 22. G. A. Henshall, A. K. Miller and T. G. Tanaka, Proc. Second Int. Conf on Low Cycle Fatigue and Elasto-Plastic Behavior o f Materials (edited by K. T. Rie), pp. 184-191. Elsevier Applied Science, New York (1987). 23. G. A. Henshall and A. K. Miller, Materials and Engineering Design: The Next Decade (edited by B. F. Dyson and D. R. Hayhurst), pp. 336-344. Inst. of Metals, London (1988). 24. A. A. Ziaai-Moyyed, Ph.D. dissertation, Stanford Univ. (1981). 25. O. D. Sherby, T. A. Trozera and J. E. Dorn, Proc. A S T M 56, 789-804 (1956). 26. E. W. Hart and H. D. Solomon, Acta metall. 21, 295 (1973). 27. E. W. Hart, C.-Y. Li, H. Yamada and G. L. Wire, in Constitutive Equations in Plasticity (edited by A. S. Argon), pp. 149-197. MIT Press, Cambridge, Mass. (1975). 28. A. K. Miller, in Unified Constitutive Equations for Creep and Plasticity (edited by A. K. Miller), pp. 139-219. Elsevier Applied Science, New York (1987). 29. H. Wei, Masters thesis, Stanford Univ. (1988). 30. I. Ferriera and R. G. Stang, Mater. Sci. Engng 38, 169 (1979). 31. I. S. Servi and N. J. Grant, J. Metals 141, 909 (1951). 32. J. C. Gibeling, Ph.D. dissertation, Stanford Univ. (1979). 33. Z. Horita and T. G. Langdon, Proc. 7th Int. Conf. on the Strength o f Metals and Alloys (edited by J. McQueen, J.-P. Boilon, J. I. Dickson, J. J. Jonas and M. G. Akben) Vol. 1, pp. 767-772, Pergamon Press, Oxford (1985). 34. H. Oikawa, K. Honda and S. Ito, Mater. Sci. Engng 64, 237 (1984). 35. W. C. Oliver, Ph.D. dissertation, Stanford Univ. (1981). 36. M. J. Mills, Ph.D. dissertation, Stanford Univ. (1985). 37. O. D. Sherby, J. L. Lytton and J. E. Dorn, Acta metall. 5, 219 (1957). 38. O. D. Sherby and J. E. Dorn, Proc. Soc. Exp. Stress Analysis 12, 139 (1954).