NASA Contractor Report 175015

Anisotropic Constitutive Model for Nickel

Base Single Crystal Alloys: Development

and Finite Element Implementation

(HAS A-CF-175.0 15) ANISOTBOPIC CONSTITUTIVE N86-21952

MODEL FOE NICKEL BASE SINGLE;CSYSIA L ALLOYS:

DEVELOPMENT AND.FINIT E ELEHEKT . .

IHPLEMENTATION Final Report^ (Cincinnati Onclas

>JJ-ni5..).__L.1.3.Q.jLe_HC _a07./«f->'0| _ CSCL 20K 63/39^ Q5770_

/ j

L.T. Dame and D.C. Stouffer

University of Cincinnati

-Cincinnati, Ohio

March 1986

Prepared for

Lewis Research Center

Under Grant NAG 3-511

NASA

National Aeronautics and

Space Administration
TABLE OF CONTENTS

PACE

CHAPTER I INTRODUCTION 1

1.1 Alloy Chemistry and Structure 2

1.2 Mechanical Response of the Alloy 5

1.3 Constitutive Modeling History 10

1.3.1 Continuum Mechanics Approach 10

1.3.2 Crystallographic Approach . 15

1.4 Scope of the Present Study 16

CHAPTER II DEVELOPMENT OF AN ANISOTROPIC INELASTIC

CONSTITUTIVE EQUATION 18

2.1 Observed Deformation Characteristics 18

2.1.1 Tensile Response 18

2.1.2 Creep Response 19

2.2 Metallurgical Models 22

2.3 Kinematics 24

2.4 Inelastic Constitutive Equations 26

2.4.1 Octahedral Flow Equation 27

2.4.2 Cube Flow Equation 33
PACE

CHAPTER III EXPERIMENTAL DATA AND MATERIAL CONSTANTS 35

3.1 Rene N4 Composition A Response 36

3.2 Rene N4 VF317 Response 41

3.3 Derivation of the Material Constants 44

3.3.1 Flow Equation Constants 46

3.3.2 Evaluation of Orientation Factors 49

3.3-3 Evaluation of the Hardening Parameters. 52

3.3.4 Tensile/Creep Coupling Terms 55

CHAPTER IV FINITE ELEMENT IMPLEMENTATION .... 58

4.1 Initial Strain Method 58

4.2 Linear Load History 60

4.3 Integration of the Constitutive Equations .... 62

4.4 Iteration Procedure 63

4.5 Dynamic Time Incrementing 64

4.5.1 Stress Increment Criterion 66

4.5.2 Inelastic Strain Increment Criterion... 66

4.5.3 Rate of Change of the Inelastic

Strain Rate 67

4.6 Convergence Criteria 67

CHAPTER V COMPARISON OF THE CALCULATED AND

EXPERIMENTAL RESULTS 69

5.1 Comparison of Experimental and Calculated

Response for Rene N4, Composition A 69

5.1.1 Orientation Dependence and

Tension/Compression Asymmetry 70

5.1.2 Cyclic Response 70

5.1.3 Strain Rate Sensitivity 74

11
5.2 Comparison of Experimental and Calculated

Response for Rene N4 VF317, Composition B 75

5.2.1 Orientation Dependent Stress

Strain Craves 75

5.2.2 Creep Response 78

5.2.3 Stress Relaxation 78

CHAPTER VI DISCUSSION AMD SUMMARY 84

6.1 Constitutive Model Development 84

6.1.1 Coupling of Creep and High Strain

Rate Response 84

6.1.2 Nonisothermal Response 85

6.1.3 Thermal Recovery 86

6.1.4 Latent Hardening 86

6.1.5 Cyclic Hardening 88

6.1.6 Other Response Characteristics 88

6.2 Summary of Data Base Requirements 89

6.3 Conclusion . 92

APPENDIX A EK2NSPACE APPROACH 93

A.1 Cubic Materials 98

A.2 Transverse Isotropic Materials 99

A.3 Incompressible Orthotropic Materials 101

A.4 Conditions Necessary for Stationary

Eigenvectors 103

A.4.1 Transverse Isotropic Materials 103

A. 4.2 Orthotropic Materials 104

APPENDIX B DEVELOPMENT OF THE KINEMATIC EQUATIONS 108

REFERENCES 119

111
CHAPTER I

INTRODUCTION

Nickel base single crystal superalloys, because of their superior

high- temperature properties, have attracted considerable interest for

use in gas turbine engines. The initial development of these alloys as

turbine blade and vane materials began in the 1960's at Pratt and

Whitney Aircraft with the alloy PWA 1409 which is the single crystal

version of Mar-M200, [1]-[5]. Since that time a variety of similar

alloys have been developed, both by Pratt and Whitney and by other

aircraft engine companies. These include the Pratt and Whitney alloy

PWA H»80, the AiResearch alloy Mar-M2i»7 and the General Electric alloy

Rene-N4, which is the alloy of interest in this research.

During the manufacture of single crystal turbine blades the [001 ]

crystal orientation is the natural growth direction. The [001] axis is

parallel to the span of the blade which is also in the direction of the

centrifugal loading. At this orientation considerable improvement in

creep and fatigue resistance is obtained over conventionally cast

turbine blades. There is some variation in the [001] orientation from

one blade to another due to the manufacturing process. The transverse

orientation of the crystal is generally not controlled, but it can be

controlled by using a seed crystal. At the present time it is not

possible to adequately evaluate the benefits of controlling the

transverse orientation.

One of the basic needs is the development of better mechanical

analysis methods for use in the design of single crystal engin e

1
components. Specifically, it is necessary to develop a model to predict

the inelastic constitutive behavior of the material and to incorporate

the model into a general purpose finite element code. The purpose of

this research is to satisfy this need. Presented in this study is a

constitutive model for Rene NU at 760°C and the implementation of this

model in a nonlinear three dimensional finite element code.

1.1 Alloy Chemistry and Structure

Rene N4 was developed by General Electric specifically for use as

a turbine blade and vane alloy. The nominal chemical compositions of

Rene N4 and PWA 1U80 are presented in Table I, along with the two

specific compositions examined in this study, References [6] and [68].

The elements C,B, and Zr which are typically included in polycrystalline

hig h temperature , nickel base superalloys for*th e purpose of

strengthening the grain boundaries are omitted.

All of the modern single crystal materials are two phase alloys

with a large volume fraction of Y' phase. The Y' precipitates have L1_

type crystal structure and are interspersed in a coherent face centered

cubic Y solid solution, Figure 1. The strength of the alloy is a

functio n of the Y' size and the percentage of Y', [7], [15].

Experimentally it has been determined that the peak creep strength is

achieved with a volume fraction of Y1 of about 60J. In high volume

fractio n Y' alloys deformatio n must occur by shearing of the

precipitates. Much of the behavior of the alloy can be explained on

2
this basis. As a result, considerable research has been devoted to

crystals with L1 2 structure, [8] - [14].

TABLE 1 .

Composition and Heat Treatment of Rene N4 and PWA 1180

Element, WT. %

A1 Ti Ta Nb Cr Mo W Co C B Zr

*RENE N4

NOMINAL

c

RENE N4

COMPOSITION A

d

RENE N4

COMPOSITION B

b

PWA-1 480

3.7

3.77

3.6

5.0

4.2

4.24

4.6

1.5

4.0

3.96

4.1

12

0.5

0.5

0.6

-

9.25

9.26

8.7

10

1.5

1.60

1.6

-

6.0

5.88

6.0

4.0

7.5

7.53

7.4

5.0

-

-

-

-

-

-

-

-

-

-

-

-

aHeat Treatment: 1260°C/2h/ga s quench + 1080°C/th/air cool

900°C/l6h/air cool.

bHeat Treatment: 1288°C/4h/air cool * 1079°C/4h + 871°C/32h.

Actual composition for specimens reported in References [6], C373.

Actual composition for specimens reported in Reference [68] and

designated in VF 317.
" '"•

i'^lK •' Ji3-ii/->=!.- i^«*^ 's. -

a) Dendritic Structure at Low Magnification (100X)

b) Gamma Prim Morphology at High Magnification (10,OOOX)

FIGURE 1. MICROSTRUCTURE OF FULLY HEAT-TREATED RENE' N4

4
1.2 Mechanical Response of the Alloy

Elastic strains are associated with crystal lattice distortions

and are fully recoverable whereas inelastic strains result from the

movement of atoms within the lattice and are not recoverable. The

effects of inelastic strains on the elastic properties are assumed to be

negligible.

Rene Ntt as well as other single crystal alloys, exhibits cubic

symmetr y in the elastic range. Reference [78], There are three

independent elastic constants and in the principal material directions

the elastic constitutive equation is written as

/e \ 11

6

22

E

33

6

3T

<

6

12>

-

>

I - I - I ° ° ° "

- I E - I '° ° °

-I -i i o o -o

Q 0 0 ^ 0 0

1

000 0 ^ — 0

0 0 000 ^

<

'•J

"22

°33

°23

°31

,

a

12>

(1.1)

The components of the stress and elastic strain tensors are denoted by

a., and E... Since the elastic modulus, E, shear modulus, G, and

Poisson's ratio, v, are independent elastic constants the terms of the

compliance matrix depend on orientation. Rotation of the elastic

compliance to another orientation can fully populate the matrix. For

example, rotation about the 3 axis gives shear/normal coupling,
Lekhnitski coefficients and variations In the rotated moduli E', G' and

v' as shown in Figure 2.

The inelastic response of single crystal materials is quit e

different from the behavior of polycrystalline nickel base superalloys.

The yield stress of single crystal alloys is a function of the material

orientation relative to the direction of the applied stress. They also

exhibit a significant tension/compression asymmetry in yielding. Figure

3, [15] shows the yield stress of PWA 1 U80 at 593°C as a function of

orientation along the [001] - [011] boundar y of the standar d

sterographi c triangl e fo r tensio n an d compression . Th e

tension/compression asymmetry is negligible near the [111] orientation.

Further, the orientation dependence and tens ion/com press ion asymmetry

decrease as a function of increasing temperature above a critica l

temperature. As an example, the variation of yield stress for PWA 1480

at three orientations in tension and compression for a range of

temperatures is shown in Figure 4, [15]. In creep, at lower strain

rates, single crystal alloys also exhibit orientation dependence even

though the deformation mechanisms are expected to be different.

Other observed behavior is similar to that seen in isotropic

nickel base superalloys. These.materials exhibit both strain rate

sensitivity and cyclic hardening. Shown in Figure 5, [17] is the cyclic

stress strain response for PWA 1U80, 871°C, at different strain rates in

the [111] orientation. An initial stress strain hysteresis loop as
OLLVa S.NOSSIOd

H

CO

§

as

•§

o

-«3

<n

u

tu

ce

aIU

a

o

.

o

(C

u.o

IU

o

O

o

e»

8

smnaow

(O

UJ

Ui

oc

oIU

a

.10 <

* Ho

cc

u.o

UI

a

e

<o

6

«•

o

s

o

IO

cu

6

SlN3IOIdd300 IMS1INHX31
1200

Q.

I

o

UJ

<r

m 1000

ui

>

*

Ol

d

800

COMPRESSION

0

[001]

15 30

ANGLE FROM (001), OEG.

45

[011]

FIGURE 3. YIELD STRENGTH OF PWA 1480 at 593°C (1100°F) WITH

ORIENTATION ALONG THE [001] - [Oil] BOUNDARY OF THE

STANDARD STEREOGRAPHIC TRIANGLE. (REFERENCE [15])

S. 1200

o

UJ

DC

« 800

a

CM

6

400.

TENSION COMPRESSION

1200-

400

0 400 800 0 400 800

TEMPERATURE, *C TEMPERATURE. "C

FIGURE 4. YIELD STRENGTH OF PWA 1480 AS A FUNCTION OF TEMPERATURE FOR

THE [001], [Oil] and [111] ORIENTATIONS IN TENSION AND

COMPRESSION. (REFERENCE [ 15 ])

8
§MS§H04o — •

flO >— 3s< z 3 wCU OSw

ft. te, HV) enMenwi£Htowat u
compared to a loop after cyclic hardening is shown in Figure 6, [18],

for PWA 1U80 at 760°C in the [1231 orientation.

1.3 Constitutive Modeling History

1.3.1 Continuum Mechanics Approach

A number of investigators have proposed anisotropic plasticity and

creep theories. The majority of these are based on a mathematical form

similar to yield surface plasticity theory for isotropic materials, [19]

- [29]. Material anisotropy is introduced by including an aniaotropy

matrix in the yield function and kinematic hardening is modeled by

utilizing a back stress tensor or a displacement of the yield surface.

Isotropic hardening is accounted for with a change in yield surface

size. A recent modification of this approach involves using a unifie d

constitutive equation while retaining the basic mathematical structure

for the flow law but without an explicit yield surface [38], [U9], in

order to demonstrate the general features of the approach it is useful

to summarize one of these theories as an example.

Lee, Zaverl, and Shin have developed an anisotropic plasticity and

creep theory that successfully predicts many of the features of the

inelastic response of Zircaloy, [23], [2U], [26], and [30]. The yield

function is an extension of Hill's work [28], and is given by

in a six dimensional stress space, a.. The anisotropy matri x M. .

describes the variation of yield stress with orientation. The parameter

k defines an effective yield surface size and the back stress a.

10
CO

fcj

as

11
describes the Initial tension/ compression strength differential and

subsequent kinematic hardening. All of the parameters, k, a , and M

are functions of the plastic strain increments. The flow rule,

*J-i?si

 (K3)

do

is a generalization of the Prandtl-Reuss equations for isotropic

material s. The effective stress o is defined by

o

2

 - Mij^j-a^Co -a ) (1.4 )

and the effective plastic strain increment is defined by

(deP) - M* de* de^ (1.5)

»

where M. . is the generalized inverse of M. . and the generalize d

devlatoric stress is given by

S

i * 3 MiJ

 (aJ-°J}

 * (K6 )

To extend the theory into the creep regime an effective creep

-c

strain rate e is assumed to be a function of the effectiv e stress o,

define d by Equation (1.U) , as well as the temperature and shear

resistance T. The shear resistance is assumed to be proportional to k.

The flow Equation (1.3) is modified slightly to become

(1.7)

12
where c. are the creep strain rate components and the total inelastic

strain is then the sum of the creep and plastic strain components. A

method for the evaluation of material parameters has been established

and the calculated and experimental response of Zircaloy 2 has been

presented in References [23] and [26], The determination of the yield

function behavior requires load reversal tests and monotonic tension and

compression tests in the three principal directions of anisotropy. The

rate dependent material parameters are evaluated from load relaxation

tests.

A unified model that falls within the continuum mechanics approach

was proposed by Stouffer and Bodner [38], for initial or deformation

induced anisotropy. The approach, as summarized in Appendix A, was

investigated as part of the study for application to Rene NU. The model

is based on transforming the anisotropic flow equation, when written in

a six dimensional space, into the eigenspace of the compliance (or

stiffness) matrix. This results in a system of six uncoupled scalar

flow equations in the compliance eigenvalues which has some advantage

for computation. The major disadvantage is that the transformation

between the eigenspace and physical space depends upon the evolution of

material properties arising from the deformation and is generally

unknown. However, as presented in Appendix A, it is possible to

determine this transformation exactly for various symmetry classes

including cubic symmetry. The results in Appendix A show that once a

specific transformation is established; for example, for cubic symmetry,

13
the response will be forced to follow this symmetry class exactly for

all future deformations. Unfortunately, this condition appears to be

too restrictive. Relaxation of the symmetry restriction leads to

parameters in the eigenspace that are not physically motivated and

difficult to determine experimentally. Thus, it was decided to abandon

this eigenspace approach in favor of an approach wit h materia l

parameters that are motivated by the physics of the deformation and

directly related to experiments.

The most significant shortcoming of the continuum mechanics

approach as described in both of the above examples is that the actual

deformation mechanisms are included in the theory only very crudely if

at all. This limits the predictive capability of the model outside the

range for which it has been calibrated. Another serious problem is

modeling the plasticity, creep and fatigue interactions. Lee, Zaverl,

and Shih have made an attempt to correct this deficiency by assuming a

relationship between the shear resistance in the creep equation and the

yiel d surfac e size in the yield function. Finally, although not

discussed above, the current theories do not contai n method s of

including temperature history effects. Current technology is based on

evaluating model parameters from isothermal data at differen t

temperatures and then interpolating between temperatures to determine

the response. This does not account for a change in deformatio n

mechanism due to a temperature excursion.

The principal argument in favor of a continuum mechanics approach

is the relative numerical simplicity. For the purpose of implementation

14
in a finite element code it is somewhat less complicated and generally

requires less calculation than a crystallograpic approach.

1.3-2 Crystallographic Approach

Early developments in this approach are attributed to Taylor [31],

Bishop and Hill [32], [33] and Bishop [31*]. The application of the

Crystallographic approach to single crystal nickel base superalloys

began with the work of Paslay, Wells, Leverant and Burck [35], [36], and

more recently by Shah [8] .to the f phase of these alloys.

The Crystallographic approach is based on identifying the active

slip planes and slip directions. The shear stresses are computed on

each of the slip planes from the applied stress. The slip-deformatio n

is computed on each slip system and the macroscopic inelastic strain

rates or strain increments are then the sum of the contributions of the

individual slip systems. This approach is computationally intensive

since there are a number of slip systems to be considered at each point

in the body. Furthermore the response at the Crystallographic level is

not necessarily easy to determine.

The classical constitutive assumption is Schmid's law in which it

is assumed that the slip on a particular system is a functio n of the

resolved shear stress on the slip plane in the direction of slip. This

assumption, however, does not apply for L1_ crystals in general or for

single crystal nickel base superalloys. More recently, Pope and others

[7]-[12], [14]-[16], have proposed an extension to Schmid's law that

includes three components of stress. This approach, which is discussed

15
in Chapter 2, appears to be successful in correlating the deformation

mechanism with several of the macroscopic anisotropic effects.

The principal advantage of this method is that a significant

portion of the model is based on the. physics of the deformation

mechanisms. Presumably, this will enhance the predictive capability of

the model. Furthermore, as additional information is obtained about

deformation mechanisms at different temperatures and strain rates the

local constitutiv e models can be modified to accommodate the new

knowledge. -The major disadvantage of this-approach is that knowledge of

the metallurgy and the interface with mechanics must be understood. A

less important objection is the additional difficult y in numerically

implementing crystallographic models in finite element codes and the

increased computational requirements.

1.U Scope of the Present Study

The purpose of this research is to provid e a tool for the

mechanical design and analysis of single crystal turbine blades and

vanes. This objective is achieved through the developmen t of an

appropriate constitutive model and its implementation in a general

purpose three dimensional nonlinear finite element code.

The important features of the nonlinear constitutive response are

to be modeled. These include the orientation dependent stress/strain

behavior and tension/compression asymmetry. The constitutive model must

also predict cyclic and creep response. Because of the extremel y

limited data base, some important aspects of the material response have

not been modeled in this study. The most important aspects of the

16
response that need further investigation are the nonisothermal behavior

and the response of the material in a multiaxial stress state. Also of

concern and not currently included in the constitutive model is the

cyclic softening/hardening behavior and latent hardenin g due to

intersecting slip systems.

T h e constitutiv e mode l ha s bee n develope d usin g th e

crystallpgraphic approach. The orientation and tension/compression

yield asymmetry can be accounted for using the metallurgical models

developed in References [7]-[12], [T»]-[16].' The development of a

constitutive model from metallurgical concepts should provide a basis

for further development of both the mechanics and metallurgy associated

with single crystal technology.

The intent in developing a finite element program is to produce a

code that is general enough to model a wide range of geometries and load

histories, flexible enough to easily modify the local constitutive

equations and at the same time efficient. The need to model general

three dimensional geometries lead to the choice of the twenty noded

isoparametric solid element. The code efficientl y and accurately

integrate s the constitutive equations over piecewise linear load

histories using an initia l strain metho d wit h a dynami c tim e

incrementing procedure. Because of the intended application, small

displacement and small strain measures are utilized.

17
CHAPTER II

DEVELOPMENT OF AN ANISOTROPIC INELASTIC CONSTITUTIVE EQUATION

The philosophy in developing the constitutive model is to produce

a system of equations that is as simple as possible while still

describing the important aspects of the observed material response. As

a result of an extremely limited data base, certain effects that could

have been included are omitted rather than engaging in speculation. The

effort in this report is primarily directed toward modeling isothermal

tensile and creep response with some hypothetical extensions to strain

rate effects, stress relaxation and initial cyclic response.

2.1 Observed Deformation Characteristics

Slip trace studies of single crystal alloys indicate that one or

more types of slip may occur under different temperature, orientation and

strain rate conditions. These include: A. (Octahedral Slip) slip on the

four octahedral planes in the three directions similar to the [T 0 1 ]

direction (see Figure B2, Appendix B); B. (Octahedral Slip) slip on the

octahedral planes in the three directions similar to the [T 2 l"]

direction; and C. (Cube Slip) slip on the three cube planes in the two

directions similar to the [T 0 1] direction. The slip condition s

occurrin g durin g creep and tensile tests are examined for use in

development of the model.

2.1.1 Tensile Response

Recent metallurgical research in the behavior of nickel base

single crystal superalloys indicates that the tensile response of these

18
alloys is controlled to a large extent by the behavior of the Y' phase

alone, [7], [15] and [16]. Thus, the response of L._ ordered alloys and

Y/Y' single crystals are discussed together.

In uniaxial tests the critical resolved shear stress (CRSS), or

component of stress on the slip plane in the direction of slip required

for yielding, is approximately constant or increases slightly up to a

critical temperature and is a function of orientation. Below this

critical temperature slip occurs primarily on the octahedral (111)[7 0

1] slip systems and significan t orientatio n dependenc e and

tension/compressio n asymmetr y is observed. Above the critical

temperature there is a sharp drop in the CRSS similar to the yield

stress in Figure 1, cube slip, (010)[1 0 1] becomes more prominent, and

the tens ion/compression asymmetry is reduced, [16]. There are two

exceptions to this behavior. First, test specimens near the [111]

orientation exhibit cube slip at all temperatures. Second, loading near

the [001 ] orientation is thought to produce only octahedral slip at

all temperatures since the resolved shear stress on the cube planes is

zero.

2.1.2 Creep Response

The understanding of the creep properties and the active slip

systems during creep in Y/Y 1 alloys is much less complet.e than the

tensile response. Only a limited number of studies have been reported

in the literature; however, a few important observations can be included

in this review.

19
Unlike the tensile yield response, the creep properties of ordered

L-2 crystals are much different than the creep response of the Y/Y'

alloys C69]. The creep rate of Y' is significantly higher than that of

Y/Y1 alloys under identical conditions.

Some nickel base superalloy single crystals exhibit an incubation

period in addition to the usual primary, secondary and tertiary creep,

see Figure 7b [7]. The incubation period and primar y creep are

attributed to (111 ) [T 2 T] slip. In single crystal Rene NU at

temperatures from 760 F to 1150 C the response did not include an

incubation period similar to that in Mar-M200, see Figure 7 and 11.

In secondary creep at temperatures above the critical temperature

slip is inferred to occur by (111) [T 0 1] slip [8], However, cube slip

is considered to be present near the [111] orientation and absent near

the [001 ] orientatio n simila r to the tensile response. These

deformation modes have been verified by transmission electron microscopy

(TEM) studies for PWA 144M , [63] and in addition dodecahedral slip,

(011 ) planes, wa s observed for loading in the (001 ) and (011 )

directions. In this study a tens ion/compress ion asymmetry was observed

and typical response showed a creep rate that is higher in tension than

compression for a creep in the [1 1 0] direction. This response is

similar to the tension/compression asymmetry observed in the yield

stress.

20
100 200

TIME (hrs)

Q.

LU

LU

CC

O

i 1 2 3 4 5

TIME (hrs)

FIGURE 7. CREEP CURVES OF MAR-M200 CRYSTALS AT 760°C AND 690 MPa.

(REFERENCE 7)

21
2.2 Metallurgical Models

A major step in understanding this behavior was made by Takeuchi

and Kuramot o [10] in their study of Ni-Ga single crystals. They

proposed that the increase in CESS up to the critical temperature and

non Schmid's law behavior is a result of cross slip of screw dislocation

segments from the octahedral to cube planes. The cross slipped segments

pin the dislocations and therefore increase the flow stress. They

further proposed that the cross slip mechanism is thermally activated

and is driven by the resolved shear stress in the (010)[7 0 1] system.

However, their model did not fully, explain the tension/compression

asymmetry or the orientation dependence observed in other L1_ alloys.

An improvement in the Takeuchi-Kuramoto model was proposed by

Lall, Chin and Pope in [11]. In their theory the octahedral | [T 0 1 ]

dislocation is an extended dislocation consisting of two Shockley

partial dislocation pairs, -I [2 1 1 ] + •§ [7 T 2]. In order to cross

slip the pair must constrict into a single •= [1 01] dislocation. The

constriction is aided by a shear stress on the (111.) plane in the

[121 ] direction. It is important to note that a shear stress in the

opposite direction extends the dislocation pair and thus inhibits:cross

slip. This effect is generally referred to as the "core width effect "

and gives rise to the tens ion/compress ion asymmetry observed in these

alloys. Lall, Chin and Pope also proposed that the change in flow

22
stress AT,^. on the octahedral plane in the [T 0 1 ] direction, for

example, from a reference state at 0 K is given by

At (in ) - A exp -]L , (2.1)

where A is a constant, T is temperature, and k is Boltzmann's constant.

The parameter H is a function of the resolved shear stress on the (111 )

plane in the [T 2 1] direction, TI, and the resolved shear stress on the

(010) plane in the [T 0 1 ] direction, T . The stress components T. and

ty are shown in Figures B1 and B2. By expanding H in a Taylor series

about the reference condition H , Equation (2.1) becomes

- H + V T + V T + ...

A T (111) ' Ae 'P — kT ^ • (2'2)

where V. and V are constants.

In the orientations and at temperatures where cube slip is

dominant the orientation dependence and tension/compression asymmetry is

reduced considerably. Therefore, it is believed that Schmid's law can

be used to relate the slip rate to the resolved shear stress on the cube

planes.

A major effort in this work is to recast the above results, both

experimental and theoretical, into the form of a constitutive equation

involving state variables that characterize the change in material

properties due to the deformation history. These results are valuable

23
for determining the appropriate independent variables and functional

forms.

2.3 Kinematics

It is necessary to develop the kinematic equations that relate the

shear stresses on the crystallograpic planes and in the directions of

interest to the stress tensor relative to the principal material axes.

The constitutive equations are applied at the crystallographic level and

it is therefore necessary to also develop the relationship between the

crystallographic shear strain rates and the -macroscopic inelastic strain

rate tensor. The required stress and infinitesimal strain relations

were derived by Bishop in [31] and are summarized below. The sign

convention used in this research and the details of the calculations are

presented in Appendix B.

The relative rate of displacement of a point located at x due to

• a

a unifor m shear strain rate Y parallel to the a plane in the S

direction is given by

u. - I YaB nVl?B (2.3)

J

 J J

n ftfi where n. is the unit normal to the a plane and I. is the unit vector in

the B direction on the a plane (no summation of repeated indices).

Summing for all planes and slip directions gives

1

 aBJ

 J J i

Applying the linear strain displacement relation

24
e

ij

1

2

£1,.i

(2.5)

gives the desired relationship between the crystallographic shear strain

*afi

rates Y , and global strain rates e.,

eA - I ^ Y°BU°6nj + *jBni) • (2.6)

aB

The stress vector o. on a plane a is given by

o - I a n° , (2.7)

J i J

where o. is the applied stress tensor in the principal material axes.

The local shear stress component in the B direction on the a plane is

given by

T

08

 - I o.lf . (2.8)

J

 J J

Combining the above two equations gives

a8

[ a., n? i

1J J

(2.9)

the transformation between the global applie d and local stress

components.

In summary, the constitutive model is based on separating the

total global strain at a point in a body into elastic and inelastic

components. The elastic strains are computed using Equation (1.1 ) and

the applie d stress tensor. Inelasti c strain s are obtained by

integrating the inelastic strain rate with time. The inelastic strain

25
rat e is calculated by summing the contributions from each of the

crystallographic slip systems using Equation (2.6). The inelastic shear

strain rate on each of the slip systems is computed from the inelastic

constitutive equations which are developed in the remainder of this

chapter. In addition, the inelastic shear strain rates on the active

slip systems are a function of the local shear stresses which are

computed from Equation (2.9).

2.1 Inelastic Constitutive Equations

There are at least two separate flow equations required to

describe the tensile response of single crystals, i.e., cube slip and

octrahedral slip in the [T 0 1 ] direction. In addition dodecahedral

slip and octahedral slip in the [T T 2] direction have been reported in

creep for some single crystal superalloys. Since no incubation period

has been observed for Rene Nl in creep at any temperature, it appears

that octahedral slip in the [T T 2] directions may not be present. Creep

deformation is attributed to the immediate operation of (111) [7 0 T] or

(001 )[T 0 1] slip depending on orientation. Further, dodecahedral slip

has not been observed for Rene Nl in a recent study of tensile and

fatigue response, [6], [37]. Thus in this work it is assumed that only

cube slip and octahedral slip in the [7 0 1] directions are active, and

the model will be developed on this basis.

In addition, ther e appear to be at least two deformatio n

mechanisms present. At the higher strain rates, such as in tensile

tests, deformation occurs by dislocations cutting the Y? precipitate .

26
The deformation rate depends on the size and volume fraction of Y'. The

orientation dependence of the CRSS at temperatures below the critical

temperatur e depends on the core width effect and the cross slip

mechanism. The creep strength of Y/Y1 alloys appears to be controlled

by the Y/Y' interfaces in the material. This explains the difference in

creep behavior of Yf and Y/Yf alloys. The activation energy during

creep is much higher than expected from a vacancy diffusion process.

Tien, et al [70], proposed that during steady state creep the presence

of Y' particles leads to a dislocation network that exhausts the vacancy

concentration. Deformation occurs by the less favorable process of

emitting and diffusing interstitials, which is consistent with the

higher activation energies. This model has been supported by many

observations as summarized in [7].

The modeling effort should include both the tensile and creep

mechanisms. In addition , it appears there should be a strong

interaction effect since the dislocation network and Y' particles are

important in both cases. Unfortunately, there is very little or no

evidence available for the development of a model to characterize these

interaction effects. In the next two subsections a constitutive

equation for cube and octahedral slip is proposed.

2.1.1 Octahedral Flow Equation

The octahedral slip system is active for both tensile and creep

deformation, but the deformation mechanisms are different as discussed

above. The model for octahedral slip is proposed as the sum of two

components i.e.,

27
•

 <2-'°>

The first component is used to characterize dislocation cutting of the

Y' particles and the second is motivated by the interstitial emission

and diffusion mechanism. The two terms must be coupled to characterize

the effect of the dislocation network on both mechanisms. Equation

•(2.10) is typical of classical models involving the use of both plastic

and creep strain components. The origin of this approach is clearly

b^sed on the physical motivation of two or rore deformation mechanisms.

The thrust of the unified strain measure is to eliminate the need to

identify the coupling between the two mechanisms. Unfortunately, many

of the unified models are reasonable for modeling plasticity or creep

but are not completely adequate for both. In the present study a

. a

coupling parameter is proposed in combination with the flow laws for Y

• _g

and Y2 that appears to be an improvement.

* aft The functional form of the octahedral flow equations for Y is

simila r to the exponentia l form developed by Bodner et. al. for

isotropic materials. This form has been used with success for a number

of materials including isotropic nickel base superalloys at high

temperatures, [39], [10], [50]=[60]. The Bodner flow equation was taken

from an expression for mobile dislocation velocity proposed by Oilman,

[61], [62] and is expected to capture the essential features of the

28
response for Rene N4 at the higher strain rates below the critical

temperature. The form of the equation used in this study is

OCT

,00

exp

081

(2.11)

where Z is a state variable introduced to characterize the resistance

to Inelastic flow. The parameters D. and n. are used to characterize

the limiting strain rate and the strain rat e sensitivity . The

n ft

octahedral flow equation and state variable Z. must also characterize

the orientation and tension/compression asymmetry properties. Since the

form of the flow equation is similar to the Lall, Chin and Pope model

their results can easily be incorporated into Equation (2.11).

The functional form of the octahedral flow equation for Y^L. is

taken in a form similar to Equation (2.11), i.e.

- °

£1

ctBi

(2.12)

This form is selected because it is consistent with representations used

to characterize thermally driven diffusion processes as well as mobile

dislocation velocities. Further, maintaining a similar structure allows

for the same interpretation of the state variables Z° and Z° .which

should aid in establishing coupling between the two terms.

Combining Equations (2.10), (2.11) and (2.12) gives

29
* D2exp

"! .

(2.13)

1 -1 -1 The constants D and D_ are chosen to be 10 sec and 1 sec to allow

uncoupling at high and low strain rates. The first term is negligible

during creep and the second term is negligible during high rate tensile

tests. Both terms are active at intermediate values of stress or strain

rate. The constants n. and n_ characterize the strain rate sensitivity

and are determined independently. This is consistent with the observed

response of .single crystal alloys [7]. The state variables Z° and Z°8

include work hardening arising from the development of a dislocation

microstructure and include core width and cross slip effects.

fl

Th§ flow resistance at high strain rates, Z , is assumed to be

similar to the Lall, Chin and Pope result, Equation (2.2), and is given

by

aft aB _ „v

«B aB

n 12 (2.11)

/* fl /^ fl

where ^ is the initial value of Z°D, Z (0) - 0 and V and V12 are

rt A rt A constant^. The parameter Z is a measure of work hardening and T. and

aB

are the shear stress components associated with the core widt h

effec t and the cross slip mechanism previously discussed. The shear

stress T is on the same octahedral plane and is perpendicular to T .

30
M Q M Q

The sign convention used for T and T. is given in Appendix B. The

n ft shear stress i- is the magnitude of the resolved shear stress on the

a o

cube plane in the same direct!on as t . The increase in flow

n ft resistance due to work hardening Z , is given by the evolution equation

Z

aB - M^W^Z06)!08?08

 ' (2.15)

where M- and W are material constants. The functional form of the

n fl * n ft evolution equation involving inelastic work rate T Y rather than

Y

a alone is very similar to the form used in the Bodner model for

isotropic materials. A major difference is that there is no recovery

term included in Equation (2.15). Also absent is an explicit function

to model the effect of hardening due to intersecting slip systems. Even

though some theoretical work has been done on latent hardening due to

intersecting slip systems, [6U]-[67], data to support the inclusion in

the constitutive model currently do not exist for Rene N4.

The expression for the state variable Z? at low strain rates is

similar in form to Equation (2.14) and is given by

(2,16)

31
where V_ and V_ are material parameters associated with the shear

stress components, T° and T° , and J2 is the second invariant of the

deviatoric stress tensor. Equations (2.1*4) and (2.16) are coupled since

rt ft the same wor k hardening term Z appears in both equations. The

parameters a and $? define the development of the dislocation structure

(hardening) at the stress-strain rate levels associated with creep.

Equation (2.16) is valid for primary and secondary creep only. A

thermal recovery term may also be necessary at high temperature.

Tertiary creep is neglected because it occurs at strain levels beyond

the range of interest for this study.

It is possible to include a component of kinematic hardening into

n Q

the model. Two values are assigned to each state variable, Z+ and Z_ ,

corresponding to the direction of slip. Equation (2.15 ) is used to

• a

calculate Z* in the direction of slip. In the opposite direction the

d U

* afl hardening rate, Z , is given by

where q is a material parameter. Isotropic hardening occurs when q = 1

and kinematic hardening occurs when q - - 1. This form is similar to

the system used by Bodner et al [38] for tens ion/ compress ion response of

isotropic materials,

32
2.4.2 Cube Flow Equation

Cube slip occurs in specimens where the loading is near the

[111 ] material direction and becomes increasingly important at high

temperature. Furthermore, Schmid's law is a good approximation for cube

slip.

• g

The inelastic shear strain rate, X,tT__, on an a cube plane in the

LUob

B direction is formulated with two terms similar to Equation (2.10) for

octahedral slip and is given by

exp

z--3

exp

|T«« I

. (2.18)

Once again the constants D. and D^ are the limiting strain rates and are

chosen to be 10 sec and 1 sec to allow separation of the two terms.

The constants n, and nj. reflect the strain rate sensitivity of the

n ft n ft material in cube slip, and Z_ and Z^ are the state variables that

include work hardening.

The evolution equations that account for the work hardening are

similar to that for octahedral slip except that the orientation factors

are excluded. These can be summarized as

T

a8 , , -oB

and

,08 bZ

(2.19)

(2.20)

where the evolution of Z is given by

33
Z°8

 - M3(W3 - Za6)Ta6T°8 , (2.21)

n fl where Z (0) - 0 and VL and M, are determined from the high rate data.

The parameters <K and b relate the hardening in the high rate and low

rate response for cube slip.

34
CHAPTER III

EXPERIMENTAL DATA AND MATERIAL CONSTANTS

Moat of the experimental data available for Rene N4 are from

tensile and creep testa with the specimens oriented in the [001]

material direction. Unfortunately, this is not particularly useful when

developing a constitutive model for a material that exhibits orientation

dependence and tension/compression asymmetry. The two temperatures at

which the best data base for Rene N4 is currently available are 760°C

and 980 C. In this study the temperature is limited to 760 C because

the orientation dependence and tension/compression asymmetry are more

predominant and the repeatability of the data is better.

At 760 the stress-strain response for several orientations in both

tension and compression, and fatigue response was determined by Gabb,

Gayda, Miner, and Voigt [6], [37]. Also available at 760°C are tensile

stress-strain and creep data at three orientations for an earlier

version of Rene N4, designated as VF317, Reference [68]. The slightly

different chemistry for VF317 may be the source of differences in the

observed response. The tensile response reported by .Gabb, Gayda, Minor

and Voigt for specimens oriented in the [001] was about -30 percent

weaker than VF317 in tension. Sjlnce the response characteristics are

significantly different for the two data sets it was not possible to

develop a single set of material constants for the constitutive model.

The Gabb, Gayda, Miner and Voigt data were used to develop the

constants and test the model for octahedral and cube slip at high strain

35
rates In tension, compression and fatigue. The VF317 data were used to

develop the constants for octahedral slip at high strain rates in

tension and low stress levels in creep.

3.1 Rene NU Composition A Response

The tensile data reported by Gabb, Gayda, Miner and Voigt,

Reference [6], consist of seven tests at three orientations. The

specimens had a cylindrical gage section 19 mm long by 4.7 mm in

diameter. The tests were run at a constant crosshead rate with an

-4 -1

Initial strain rate of about 2 x 10 sec . Data recorded were load

and crosshead displacement. Plastic strain was estimated using the

offset from the elastic loading line and the specimen gage length. The

resulting stress strain curves are shown in Figure 8.

It was confirmed by TEM analysis that slip in the [001] and

[011 ] specimens was in the octahedral system while cube slip was

observed with orientations near the [T 1 1] orientation. Tensile axis

rotation was observed In the specimens tested to failure. The

variability of the yield stress, within a crystal was small; however, the

variability between crystals was as large as 23?. This difference which

is attributed to the orientation and tension/compression asymmetry is

typical of other single crystals near this temperature. However, at

980°C the orientation and asymmetry properties are much less important

and Schmid's law appears to be applicable, Reference [6]. Stresses

i

beyond yield increased initially and flattened out for the [001 ] and

[T 1 2] specimens. The [011 ] specimen exhibited decreasing loads

36
WiOSwovOazw

ofi zo&. iwf4•w00Wo: §

'ssaais

37
sharp serrations to about 5% strain. The serrated yielding, which is

observed in PWA 1180 and Mar M200, is attributed to the operation of a

small number of slip planes, typically one, two or four.

The fatigue response at 760 C reported in Part II of Reference

[37], consisted of the monotonic yield points for specimens at six

orientations in tension and compression and first cycle hysteresis loops

for three orientations. The specimens had a cylindrical gage section

15mm long by 5mm in diameter. The control waveform was sinusoidal with

a frequency of .1 Hz.

The tensile and eompressive yield stresses at several orientations

are shown in Table 2. The [001 ] specimen had the greatest initial

yield in tension and the [011 ] specimen was strongest in compression.

Both specimens displayed significant tension/compression asymmetry while

the [T 1 1], [023 ] and [2 3 6] specimens displayed very little or

none.

The initial hysteresis loops for [0 0 1], [0 1 1 ] and [T 1 1]

orientations are shown in Figure 9. The [01 1], [T U 5], [23 6]

and [02 3] specimens had serrated flow characteristics. All specimens

displayed slight hardening with continued cycling which increased with

increasing strain range but was generally less than 10$. For all tests

the response stabilized well before half life.

38
•Reference 1371

956

905

6U2

817

253

7H7

695

170

£0231

T\H

716

166

12361

792

39
uoo

oo

EdZ2COsen r-~» enu u tc u w z H W en ec ^4 Btl w u en o ^ 5Hen&

fr. to

40
3.2 Rene Ml VF317 Response

The tensile data for Rene M VF317 at 760°C consist of three

specimens at three orientations. The specimens had a Codep coated

cylindrical gage section 18mm long by 3.3mm in diameter. The tensile

-5 -1

tests were run at a constant strain rate of 8.333 x 10 sec to

inelastic strains of .2%. At .2$ yield the extensometers were removed

and the specimens were pulled to failure at a constant head rate of

.02117 mm/sec.

The response to approximately 1$ strain, as shown in Figure 10,

displays considerable orientation dependence. The stress increased

about 30? between the .02$ yield stress and ultimate stress for all

specimens. Although no TEM work was done to determine active slip

systems it is reasonable to assume that cube slip is not active for

these orientations.

Tensile creep rupture tests were performed on Rene NM VF317 at

760° for the same three orientations. The 3.175 mm diameter specimens

were Codep coated. A total of seven creep tests at three orientations

were conducted and the results are shown in Figure 11. Two of the test

at 620 MPa were terminated prior to failure. The reduction in area

ranged from 17$ to 48$.

41
1000

(0

o.

CO

(0

UJ

cc

H-

(0

500 H

[0 342 940]

0 0.5 1.0

STRAIN, %

FIGURE 10. TENSILE RESPONSE OF RENE N4 VF 317 AT 760°C.

1.5

42
I I

O

.O

CM

O '

*

LU

o

O

o

°0

>

W

2

W

as

Cd

g

S

DC

O

O

CM

% 'Nivais

43.
Not e that the specimens In the [0 3^2 940] orientation were

stronger in the tensile tests and in creep than expected when compared

to the other data set or to PWA 1 M80 (Figure 3). The specimens at this

orientation were nearly as strong as the specimens in the [001 ]

orientation. The reason for this anomaly is not known.

3.3 Derivation of the Material Constants

The octahedral slip constants are derived from tests where cube

slip is not present, for example, the [001 ] and [011 ] orientations.

Conversely, the cube slip constants are derived from tests where

octahedral slip Is not present i.e., [1 1 1] orientation. In addition,

the first term in the octahedral or cube flow equation is negligible in

creep and the second term is negligible at high strain rates. When a

choice of orientations is available for developing constants the data

from the [001 ] orientation is used since it is the primary loading

direction in turbine blade applications.

Since the constitutive model is developed on the individual slip

systems it is essential to know the relationship between the applied

stress tensor and local shear stresses and the relationship between

local slip rates and the global strain rate tensor. To simplify this

task the kinematic equations in Appendix B were implemente d in a

computer code. For example, the shear stress components from an applied

stress of 100 MPa in each of four directions is shown in Table 3. The

local stress in every slip direction on each of the octahedral and cube

planes is shown. The notation is described in Appendix B. The stresses

in the (111 ) [T 2 T] system are used only for determining T. in the

44
gH

g

Wl

< i

A ^^

feg

ofc

o u

•- OS

M

feQ

r— i

CO 0

CO aCd C7>

ce

t- CM

co *r

pn

0

tt

^J

 O *_ 1 ?" 1 ^^J

5u o

0. 2

<e «£

z <— i

«* ^«

»•

O£ M»

%Z

Cb

*

eo •— >

%~

ee o

&•• 1 1 r* <—>

en

J rn

O O

C3 0

J t_i

•

pn

s03

?

• CO

^^ CO

O bl

a- ce

e* (-

CO

CM

a- as

pn «;

oi

l_i CO

CO

p-i ce

«- to

,Z, cc

I

CO

r-i OC

ioi_J OS

C

1

O

CO

CQu

1—1 ce

O CO

o

i—i ce

^

S

CO

z

0

WH Hu

u

OS

3

g

3

a.

u

S

CO

a.

I-H

cTvoovoinoo^ocjvovoin

vococo pn a- co oo oo \o co pna-

^-CMOOOpncMCOCM^CMONp n

ey>^-f-CMeo^-h-'-cr>'-cMco

a-pn.-<M i pn»-pna-pncM l I 1 til l

t- t- i~- t~ r- t-

. CM CM CM CM CM CM

oooot-c^t-oe^r-t~o

CM CM CM CM CM CM

in in in in

S S S3

ooooooooooo o

sr a sf sr

I I I I

inin in in inminin

CMCM CM CM CMCMCMC M

CO CO GO CO CO CO GO CO

ooooooooooo o

a'a1 it sf aa

-

a

-

a

>

1 III

^fMPn—cvjpn^CMPn^cMp n

.__,-CMCMCMPnpnpna-a-ar— l

O

1^-

i^j

J^

f-vocoe>>^a-cOvo^a-t^« -

pn^-a-t^ocMa-^-POPvit^o

t~p-a-oopna-a-t«rc^aeopn

e— covorg^-covocot— oooj^ -

TpnJTCMpnia-pn i ICMp n

I I i t

t"» pn pn pn f*» pn pn pn f*-

a; t^ t»- r- a- t-- f- f- .aooo'-miftin»-ininin^-

i " *~ "~ i "~ *" *~.T

o o o o o o

r- a- t- c~ a- t^

in *- in in »- in

pnt>-pnooopnt-pnooo

CM a- CM cv a- CM

OOOOOOOOOOO O

t-f-a- l^-a- t^a- t^t-l~t— ainin'-m — in'-inininin^

pn pn t*» pn t*- pn r** pn pn pn pn t»

CMCMa-CMa-CVa-CMCMCMCMa -

II I l (il l

— CMPn«-CMpn.-cMPn.-CMpn

^^.»-cMCMCMpnpnpnaa-at— i

CM

i_i

^»

O^ O^ CTV CT%

(- ^ P ?^

O O CM CM CM CM

CM CM CM CM

1

O O O

a- a- ab•f ~ - o t-~

 o ^ e~- o

—r ^r ^T

in in in in

in in in in

pn pn pn pn

o o in in in in

pn PO pn rn

00000 0

~- CM •- CM — CM

_ ,_ CM CM pn pn

r—i

O

i_j

O

o

to

45
model, T_ is determined from the(001)[1 0 1] slip system and T is

determined for either the (111)[ 1 0 1] or (001)[T 0 1] slip system

dependening on the active system.

For the purpose of developing material constants it is also

important to choose test orientations where the local stress and strain

rate on each of the slip planes can be determined. For example, a

load in the [001 ] direction will produce equal stresses on eight of

the (111 ) [T 0 1] slip systems and no stress on the other (111 ) [7 0 1]

systems or the cube systems. It is expected that slip on all eight

systems will be the same. By comparison, a load on a [0 342 940]

specimen will have shear stresses in all of the (111 ) [7 0 1] slip

directions. Further, it is expected that these shear stress will

produce slip on more than one system and the slip rates are not expected

to be equal. Thus, the tests in the [001] , [01 1] and [111 ] are

preferred to evaluating constants due to the uniformity of stress and

strain rate in the octahedral and cube systems. Table 3 also shows why

only octahedral slip is found near the [001 ] orientation (no stress on

the cube planes) and only cube slip is found near the [111 ] direction

(dominant stress on the cube planes).

3.3.1 Flow Equation Constants

The octahedral and cube slip systems are assumed to have limitin g

14 -1

strain rates, D. and D_, of 10 sec in high strein rate tests and the

limiting strain rate in creep, D , is chosen to be 1 sec" . Neithe r

46
data set contains creep tests near the [111] orientation that would

activate cube slip. As a result, it is not possible to evaluate any of

the constants associated with cube slip in creep and D^ is set to zero

H _i

for all of the numerical exercises. The value 10 sec has been

accepted as a constant for many materials except under extreme loading

rates. The value D_ - 1 sec was chosen since creep rates are about

-4

10 less than tensile rates.

The constants n., n_, and n_ are all evaluated using essentially

the same method. To compute n. two or more tensile tests at different

strain rates with the specimens in the [001] orientation are necessary.

Two or more creep tests with specimens in the [001] orientation and at

different stress levels are required to determine n_. The evaluation of

n_ can be accomplished with two or more tensile tests at different

strain rates with specimens oriented in the [111] direction. The

tensile tests are loaded under constant strain rate conditions to

failure and the saturated values of stress are determined . The

secondary creep rates are used from the creep tests.

At high strain rates where the contribution of the creep rate term

is negligible, Equation (2.11) may be rewritten as

in l-^ \ n i} (3.1)

_aS „ otS

» n. in Z. - n. in T

47
ft ft where Z is the fully saturated value at that strain rate. The [001]

* n ft n ft orientation has eight equally active octahedral systems and Y and T

are easily calculated from the applied stress and the measured inelastic

strain rate. When In

n ft is plotted against in T the slope

Q

of the line will be - n. . Once n. is known the saturated value of Z.

can be calculated from Equation (3-D or the ordinate of the line.

n ft The constants n« and Z_ are calculated in a similar manner from

the secondary creep rates while neglecting the contribution from the

high strain rate term. The values for n_ and nj. and the saturated

/) fc/j

values Z_ and Z^ are obtained from tensile and creep tests using [111]

oriented specimens. The parameters n., n_, n_ and n^ are independent of

orientation and are also assumed constant for constant temperature. The

Q f*Q

values Z , and Z_ depend on orientation.

Since the required tensile tests were not available for either of

the current data sets, the value for n1 was estimated from an analysis

of PWA-1480 data at 1600°F and 1800°F. The value for n. was assumed to

be the same. Data were available for the calculation of n2 for the

Rene NU VF317 chemistry. Obviously the strain rat e sensitivit y

48
predicte d by the current set of materia l constants is at best an

estimate.

•3'3.2 Evaluation of Orientation Factors

The orientation dependence and tension/compression asymmetry of

the material in octahedral slip is characterized by the constants V

and V at tensile strain rates and V and V in creep. The constants

can be evaluated from any three tensile tests at constant strain rate

(or any three creep tests) as long as the orientation s and/o r

tension/compression sense is different and cube slip is not involved.

Best results, however, are obtained- if the orientations are not close

together and both tensile and compressive data are used. Tests close

together tend to magnify the experimental variability. In addition , it

is very helpful if the active slip systems are equally stressed as

explained earlier. An optimum set of tests is probably [001] in tension

and compression and [011] in either tension or compression. This choice

of orientation is well away from the cube slip regime, there is a large

variation in T. and T_, and the model is calibrated for the orientation

of greatest interest.

For the high strain rate tests where the contribution of the creep

Q

rate term is negligible the saturated value for Z for the three tests

can be calculated. Rearranging Equation (2.11) gives

Z1-'

1 1 0 6 T , (3.2)

49
where Y - an d

0

are the values when the inelastic strain rate and

stress become constant. Letting

S. - Z •*•<!>.• constant (3.3)

Equation (2.14) can be written for each of the three tests, i.e.

} - S -V (r06) + V

V

1 11l

 1 /1 1

Since T°° °

B

 and T^ are known from Table 3 or equivalent and Z° can be

computed from Equatio n (3.2 ) for each test, the constants S , V and

V . can be determined.

The calculation of the orientation factors V_ and V for the

creep regime is similar. Neglecting the contribution of the high strain

rate term, Equation (2.12) can be rearranged to give

,08

'

 m

(3.5)

where Yn_T is the value during secondary-creep. Once again letting

(3.6)

Equation (2.16) can be written for each of the three tests, i.e.

50
aB v / 06

Since T? t T° and J_ are known and Z_ can be computed from Equation

(3.5) for each teat, the constants S-, V-., and V.. can be calculated.

For the Rene NM VF317 data set, the tensile stress strain curves

o

for the three orientations were used to obtain the constants S , V.. ,

and V _. These -tests were not optimal for this calculation since the

strain rate was changed before the stress level was saturated and none

of the tests were in compression. The constants S-, V , and V were

calculated for the secondary creep rates- for three specimens in

different orientations. These tests also were not optimal since none

were in compression.

The Rene N4 data of Gabb, Gayda, Miner and Voigt contains only two

stress-strain curves at a constant rate that activated octahedral slip.

Recall that the test in the [Tl2] orientation activated cube slip. This

is not sufficient to establish the orientation factors. Alternatively,

the data set contains yield stresses for the initial quarter cycle of

the fatigue loops for specimens in several orientations. Since the

51
control wave form is sinusoidal, the total strain range varies from test.

to test and the strain rate varies during the loading quarter cycle, it

is not possible to determine the orientation factors as described above.

Thus, to test the orientation and tension/compression aspects of

the constitutive model, some assumptions were made. It was assumed that

the yield stress variation in the sinusoidal first quarter cycle is

representative of the saturated stress in constant strain rate tensile

-U

tests. Furthermore, a strain rate of 2 x 10 was assumed . The

constants were calculated from [011] tension and [001] tension and

compression yield stresses from the first cycle. It is obvious that

this procedure will not generate the actual constants, but it does

provide a means for exercising the constitutive model with the data that

are available.

3.3.3 Evaluation of the Hardening Parameters

The evaluation of the constants in the state variable evolution

Equations, (2.15) and (2.21), for both the octahedral and cube slip

systems proceeds in a similar way. The evaluation of M., W and <)>. for

octahedral slip and M_, W, and $, for cube slip i? based using the high

strain rate tensile data.

For the octahedral system, Equation (2.15 ) can be integrated to

give

Z

aB

 - W1 [1 - exp .(- M^S08)] . (3-8)

Combining Equations (2.14), (3.3) and (3.8) yields

52
Mji"6

?*8 - in [S, - V11T«

B+ V12xf . (3.9)

The quantity In is log linear in the local

A * O n

rate of inelastic working, T° T° . The parameter Z, can be evaluated

using Equation (3.2) and stress-strain test data since T , Y , D and

n.. are all known. By plotting Equation (3.9) for the [001] tensile

data, Figure 12, the constants W and M are obtained. Notice that at

T

a8Yafl

 - 0, In

obtained, * can be calculated from Equation (3-3).

Once W

In cube slip the calculation of M_, W_ and <fr is identical to the

procedur e for octahedral slip. Since the creep strain rate is

negligible at high strain rates the cube flow Equation (2.18), becomes

n

3

exp (3.10)

a8i

for a high strain rate test with the specimen loaded in the [111]

orientation. Rearranging Equation (3.10) gives

3 T (3.11)

and at saturation

Z_ - <fr_ * W_ - constant (3.12)

53
u

oo

CO 1

tdZCd as•o03 COOSwsga- §s1gcda:oM

54
Integrating the evolution Equation (2.21) results in

Z°6 - W3 [1 - exp (- M3T

a6Ya8)] . (3.13)

Using Equations (2.19) and (3.13) gives

ln(W3) - M3T

a6-raS - in [Z°B - *3] . (3.11)

0

The parameter Z, can be evaluated using Equation (3.11) and the stress

0 * Q

strain test data since T , Y , D_ and n, are all known. A plot

similar to Figure 12 is used to find W_ and M_. Using the saturated

n ft

value of Z_ , Equation (3-12) is used to find $,.

Unfortunately, a constant strain rate tensile test with loading in

the [111] direction was not available and M_ and W_ were assumed to be

the same as for octahedral slip for the numerical exercises.

3.3-1* Tensile/Creep Coupling Terms

The tensile/creep coupling constants, a and $_ for octahedral slip

and b and $j. for cube slip can be calculated from the tensile constants

and creep response. The constants a and $_ are calculated for an

orientation where octahedral slip is dominant, such as [001]. The

calculation of b and <K requires data for an orientation where cube slip

is dominant, such as [111].

O

The value of Z2 can be obtained from Equation (3.5) for the

primary and secondary creep rates. At the beginning of the creep test

55
ft prior to any hardening, the hardening parameter Z will be zero and

Equation (2.16) reduces to

.

n ft

so $2 can be calculated. During secondary creep Z is saturated and

equal to VL , Equation (2.16) becomes

and a may be calculated.

The constants for cube slip in creep may be similarly calculated

using the initial creep strain rate and the secondary creep strain rate.

The data were not available in the current data base for the calculation

of b and ^.

A summary of all the constants for both data sets is presented in

Table 4. There are a total of 22 constants in addition to the 3 elastic

constants. However, D.. , D2, D_ and D^ are assumed known, thus there are

effectively 18 constants required to model the inelastic response.

56
CONSTANT

TABLE U - MATERIAL CONSTANTS

VF317

GABB, GAYDA, MINER

VOIGT DATA

D1

n1

D

2

n

2

*1

V

11

V,.

12

M

1

W1

a

*2

V

2

V..,

22

D

3

n_

10000 SEC*1

2.02

1 SEC"1

.708

959.1 MPA

3.7454

3.5799

5.656 MPA"1

177.9 Mpa

52.52

5626 MPa

20070 MPa

18690 MPa

_

____

10000 SEC*1

2.02

••••.

1109 MPA

.5752

-.5799

8.194 MPA"1

437.5 MPA

— —

— -—

— _

10000 sec"1

2.02

n

M

3

w.

1198 MPA

MPa

437.5 MPa

-1

The elastic constants at 760 C are E - 100.0 GPa, G. - 96.5 GPa and

v - .38.

57
CHAPTER IV

FINITE ELEMENT IMPLEMENTATION

The finite element implementation of the constitutive model was

done using a twenty noded isoparametric solid element. This element was

chosen because it is possible to model almost any three dimensional

geometry and also allows for any orientation of the material principal

axes. Order two Gaussian integration was used both for stiffness

generation and the calculation of body forces.

The ability to model piecewise linear load histories was also

included in the finit e element code. This capability is particularly

useful when modeling stress strain tests or fatigue loops and also for

certain analysis applications. Since the inelastic strain rate could be

expected to change dramatically during a linear load history it is

important to include a dynamic time incrementing procedure. The current

study is restricted to isothermal conditions, consequently the finit e

element code is limited to steady- state thermal conditions during a load

case. The term "load case" is used to denote a time period for which

the initial and final loads and boundary conditions are defined and vary

linearly between the end points.

U.1 Initial Strain Method

The initia l strain metho d is an efficien t techniqu e for

incorporating time dependent constitutive models in nonlinear finite

element codes [71], [72], It is an economical technique because it is

necessary to reform the stiffness matrix only for temperature changes.

58
The derivation of the nodal equilibrium equations follows directly

from the principle of virtual work as described below. Element strains

T

{e }, in terms of nodal displacement {x} are given by

U

T) - CB]{x} , (U.I)

where [B] is the element strain displacement matrix. Since the total

E>

strains are the sum of the elastic strains {e } and the inelastic

strains {e1} it follows that

{e1} - UE} * {e1} . (H.2 )

The stresses are related to the elastic strains by

la} - CE]{eE} , (4.3)

where [E] is the matrix of elastic constants. Defining {f} as the

A

element nodal forces and {x} as the associated virtual displacements and

applying the method of virtual work yields

(x}T (fJ - /v{e>T{ff} dV , (4.4)

A A

where {e} are the strains associated with the displacements {x}. The

integration is over the volume of the element. Substituting Equations

(4.1), (1.2) and (4.3) into (4.4) gives

{x)T{f} - / v (CB]{x}) T[E]({e} - {e1}) dV . (4.5)

Since {x} is arbitrary and independent of the integration (4.5) becomes

{f} + {f1} - Ck]{x) , (4.6 )

where the inelastic pseudo force {f } is defined as

{f1} » /..CB^CEIU1} dV , (4.7)

59
and the elastic stiffness matrix [k] is defined by

Clc] - /y CB]T[E][B] dV . (4.8)

When the elemental equations are assembled the global equilibrium

equation

CK]{dT} - {F} * {F1} (1.9)

T

is obtained. The matrix [K] is the global stiffness matrix, {d } is the

total displacement vector, {F} is the vector of applied thermomechanical

forces, and (*} is the vector of inelastic pseudo forces.

The calculation of the inelastic strain rates and the stat e

variable evolution rate is accomplished in the constitutive subroutines.

The integration over the element volume is generally performe d

numerically and in this study order two Gaussian quadrature was used.

The total Inelastic force vector is then assembled by summing the

contributions from all of the elements.

4.2 Linear Load History

In order to incorporate linear load histories into this scheme the

total displacement vector is decomposed into elastic and inelastic

components i.e., let

{d1} - {dE} * {d1} . (4.10)

The vector {d } is the displacement due to applied thermomechanical

forces and {d } is the displacement due to the inelastic pseudo-forces.

60
These displacement vectors can be calculated using

and

(dE) -

{d1} - CKD'V1} . (M.12 )

The elastic displacements are obtained for the initial and fina l

thermomechanical loads in the load case. Assuming a linear variation

the elastic displacements at any time in the load case are given by

CdE} - {dE}Q + (—-|-)({dE}f - {dE}Q) . (H.13)

f o

E "* E The vectors {d } and {d }., are the elastic displacements due to the

initial and final applied thennomechanical forces. The current time in

the load case is t, and t and t_ are the initial and final times in the

o r

load case. The displacements due to the inelastic strains at any time

during the load case are given by

{d1} - {d1} + I {Ad1} . (U.iU )

TIME

INCREMENTS

The vector {d } is the vector of displacements due to inelastic strains

at the beginning of the load case, and {Ad } is a displacement increment

due to the inelastic strains during a time step. The increment in

displacements {Ad } due to the change in inelastic strains {Ae } during

a time step is computed using

{Ad1} - CK]"1{AF1} . (J».15 )

61
The inelastic pseudo force increment {AF } is calculated from

UF1} - I /.CB^CEHAe 1. } dV , (4.16)

ELEMENTS

where {Ae } is the change in inelastic strain during the time increment.

4.3 Integration of the Constitutive Equations

Although the integration of the constitutive equations to obtain

the inelastic strain increment could be achieved by any numbe r of

schemes, [4.3], [4.4], the second order Adams-Moulton predictorcorrector method lends itself readily to the required equilibrium

iteration. The global inelastic strain and the state variables (written

as Z. for convenience) are integrated using

I 'Wl5 I I

Ue1} -- 2

 ({£ }i-l * U }i)

 ' (U'

1 7 )

and

(t -t ) .

Zj_ --

 1

 2* (Z1H* Zt) * Zi_1 . (4.18)

The vectors {e }. and {e' }._1 are the inelastic strain rates at times

t. and t._. and Z. and Z, . are the state variable at times t. and t._. .

Since the Inelastic response of the material is computed on the

crystallographic planes, there is a state variable for each slip system

at every integration point. The inelastic strain rate is computed by

summing the contribution from each of the active slip systems. The

inelastic strain rate on each of the active slip systems is a function

of the stress state and the state variable for each system as described

in Chapter II.

62
M. 14 Iteration Procedure

At the beginning of a load case the initial and final elastic

displacements are computed using Equation (4.11) and the displacements

due to prior inelastic strains are computed using Equation (U.12) . The

total strains at the beginning of the load case are recovered for each

integration point and the elastic strains are computed from

U

E

} - UT} - UaAT} - U1} , (U.19 )

*7* f* /(.T

where {e } are the total strains and {e } are the thermal strains.

The stresses are computed, transformed to the material axis system and

Equation (2.9) is used to find the shear stresses on each of the slip

systems.

Using the current values of the state variables and the stress

state allows the calculation of the Initial values of the shear strain

rates and state variable evolution rates on each of the slip systems.

Equation (2.6) is then used to compute the macroscopic inelastic strain

rates. Before entering the time loop an initial time increment is

computed and the inelastic strain increments are estimated using a

forward Euler integration formula. From the estimated inelastic strain

increments an initial estimate is made for the inelastic pseudo force

increment using Equation (U.16). The usual technique employed with the

initial strain method is to assume that the incremental inelastic force

{AF }, the corresponding displacements {Ad }, and che inelastic strain

increments {Ae } are all zero on the first iteration of a time step.

The stability of the method can be improved considerably when a forward

Euler integration of the inelastic strain rates is used to make an

63
estimate of Ue }, {AF }, and {Ad } on the first iteration. This method

results in an initial estimate which is much closer to the solution. In

sample cases the overall number of iterations was reduced by more than

one half.

The procedure during a time increment is to estimate the solution

on the first iteration using a forward Euler scheme as outlined above.

Then displacements, strains, stresses, inelastic strain rates, and state

variable evolution rates are computed at the end of the time increment.

The inelastic strains and state variables are integrated over the time

increment and an improved inelastic force is computed. The procedure is

repeated until convergence is achieved at the end of the time increment.

Figure 13 summarizes the logic.

M.5 Dynamic Time Incrementing

In a computer code that allows a linear variation of loads with

time a dynamic time incrementing scheme is very desirable since large

excursions in stress and inelastic strain rate are to be expected. The

procedure used to compute the time increments requires a certain amount

of Initial experimentation to determine appropriate time step control

parameters. However, once this has been done the procedure works quite

well and is a tremendous improvement in economy over a constant time

increment.

The time increment is based on three separate time step control

criteria. These are the maximum stress increment, maximum inelastic

strain increment, and maximum rate of change of the Inelastic strain

rate. The minimum time step calculated from the three criteria is the

value used. Since the calculations are based on values readily

64
FIGURE 13. FLOW CHART OF FINITE ELEMENT SOLUTION PROCEDURE

BEGIN LOAD CASE,COMPUTE INITIALS AND

le'} ,AND t FOR EVERY ELEMENT

i

BEGIN TIME LOOP

BEGIN ITERATION LOOP

]dr},=

BEGIN LOOP OVER ELEMENTS

BEGIN LOOP OVER INTEGATION POINTS

CONSTITUTIVE EQUATION COMPUTATION

Z..A2. {E'

NO

YES

UPDATE |C'l. Z .etc ; COMPUTE NEW At

NO

END OF LOAD

CASE ?

65
available from the previous time step, little computational effort is

required.

M.5.1 Stress Increment Criterion

A maximum stress increment criterion is used to control the time

increment during primarily elastic excursions. This criterion is

necessary to prevent overshoot of the point where significant inelastic

strain rates begin. The calculation for the time increment is given by

k -iMA X

where At. , is the previous time increment, (Ac . ..).,.... *3 tne

change in effectiv e stress for all integration points during the

previous time increment, and Aa_ N C is the maximum desired stress

increment. The value for AoTMr is program input and will vary somewhat,

depending on material constants. Typical values are about 15 MPa.

M.5.2 Inelastic Strain Increment Criterion

The inelastic strain increment criterion controls the time step

whe n the stress and inelasti c strai n rates are not changin g

significantly. This is given by

The maximum change in effective inelastic strain for all integration

points during the previous time increment is (Ae. .)..,.. and AeTN_ is the

maximum desired inelastic strain increment. The value for AeT.._ is INC

program input and typical values are about .000100.

66
4.5.3 Rate of Change of the Inelastic Strain Rate

This criterion controls the time increment when the inelastic

strain rate is changing rapidly such as in the "knee" of a stress strain

curve. The quantity e is a measure of how close the initial forward

Euler estimation is to the final converged solution. The backward

difference formula

is used to estimate e.. The maximum value of e for all integration

points (e ) x is used to estimate the next time step using

\

2AVi

*4 *^— . (U.23)

(O'i'MAX

The parameter e is the maximum desired error by which the initia l

forward Euler estimation is in error. The value for e is program input

and typical values are about .01. Equation (4.23 ) is derived simply

from taking the difference between an Euler integration scheme and the

more accurate second order Adams-Moulton method.

U.6 Convergence Criteria

Convergence is required at every integration point. Two separate

criteria must be satisfied for convergence. First, the difference in

the effective inelastic strain increment from subsequent iterations must

be less than a prescribed value, i.e.

67
Second the change in effective stress from subsequent iterations must

also be less than a prescribed value, i.e.

< 6 . (4.25)

For all of the calculations done in this study the convergenc e

tolerances were set at one percent of the maximum desired effective

stress increment and one percent of the maximum desired inelastic strain

increment used in computing the time step.

68
CHAPTER V

COMPARISON OF THE CALCULATED AND EXPERIMENTAL RESULTS

All of the numerical calculations were performed using the finite

element code described in the previous chapter. Since all of the

calculations were for uniaxial tests it was not necessary to model the

actual specimen geometry. However, to test the force and stiffness

assembly routines a model with two twenty noded bricks was used to

simulate the specimen response. The calculations were performed to

compare with test results and also to exercise the constitutive model

and computer code under conditions for which no data are available.

5.1 Comparison of Experimental and Calculated Response for Rene N^,

Composition A

The calculations were performed using the constants derived from

the data presented by Gabb, Cayda, Miner and Voigt in References [6] and

[37]. The constants are for octahedral and cube slip at high strain

rate (Table U). The features of the model that are exercised are the

orientation dependence, tension/compression asymmetry, strain rate

dependence and isotropic vs kinematic hardening for hysteresis loops.

Because of the assumptions required in the derivation of the material

constants due to the limited data base, most of the exercises for this

data set are intended to demonstrate features of the constitutive model

rather than correlation with experimental data.

69
5.1.1 Orientation Dependence and Tension/Compression Asymmetry

Recall that in Chapte r III the constants were derived by

hypothesizing that the .02* yield stress for the first quarter cycle of

sinusoidal loading was representative of the orientation dependence and

tension/compression asymmetry for constant strain rate tests. The

octahedral constants were derived using the [001] tension and

compression tests and the [011] tension data. The cube slip constants

were derived using the [Tl1] tension data. The stress response was

calculated for these tests and for the tests at other orientations. The

calculated results are compared to the experimental results in Table 5.

All of the predicted results are within 6% of the experimental values

and withi n the confidenc e level for the experiments. It can be

concluded that the constitutive model successfully predicts the

orientation dependence and tension/compression asymmetry in octahedral

slip. It can also be seen that the octahedral and cube flow equations

produce good results when simultaneously activated.

5.1.2 Cyclic Response

The first 1.25 cycles of the fatigue response presented in

Reference [37] are compared to the predicted results usin g both

isotropic hardening and kinematic hardening (Figures 14 and 15). The

stress ranges for the kinematic hardening assumption are generally

better than for the isotropic hardening assumption.

70
TABLE 5. COMPARISON OF MONTONIC HELD STRESSES (REFERENCE [37])

AND CALCULATED SATURATION STRESSES WITH OCTAHEDRAL AND

CUBE SLIP CONSTITUTIVE MODELS ACTIVE, RENE N4, 760°C

TENSION OR .02% YIELD CALCULATED

ORIENTATION COMPRESSION STRESS (REF [37])

(MPa)

[001]*

[001]*

[011]*

[011]

[Tn]*

[ni]

[023]

[023]

[236]

[236]

[T45]

[T45]

T

C

T

C

T

C

T

C

T

C

T

C

956

-818

748

-905

817

-842

695

-747

716

-714

656

-792

SATURATION STRESS j ERROR

(MPa) (*)

956

-819

752

-865

827

^828

705

-741

725

-752

692

-763

0

.1

.5

4.4

1.3

1.7

1.5

.9

1.2

5.3

5.5

3.6

* CONSTANTS WERE DERIVED USING THIS DATA

71
72
OO

wMa;wo-awCi 0- OS3o

73
There is a significant difference in the shapes of the curves.

The use of a sinusoidal wave form has a tendency to produce flat curves

but the predicted response even shows a stress drop near the peak strain

points. Probably the most significant variation between the predicted

and experimental results is seen in the response of the specimen in the

[Tn] orientation after the first 1/1 cycle. These differences can at

least partially be attributed to the assumptions that were required to

derive the constants. Recall, that the values for n.. and n. wer e

estimated from PWA 1180 data at 871°c and 982°c. This may not be

representative of Rene N4 at 760 c. In addition , the hardenin g

constants were estimated from Rene Nl VF317. These extrapolations could

cause significant errors in the predicted hardening and strain rate

sensitivity.

A slight mismatch in elastic modulus is seen between the predicted

and experimenta l results for the specimen oriented in the [011]

direction. The elastic moduli used in the analysis are the nominal

values and are entered in the principal material directions. A slight

variation in elastic modulus from the nominal value is not surprising.

5.1.3 Strain Rate Sensitivity

In order to demonstrate the strain rat e sensitivit y of the

constitutive model, first cycle fatigue loops were calculated for two

different orientations and three different rates using the sinusoidal

total strain history. The results of this exercise are shown in Figure

16. Qualitatively, it can be stated that the observed trend is as

74
expected. Since no data are available, quantitative comparisons are not

possible.

5.2 Comparison of Experimental and Calculated Response for Rene NU

VF3T7, Composition B

The calculations were performed using the constants derived from

the Rene MM VF317 data, Reference [68]. The constants are for

octahedral slip at high strain rate and creep (Table M). The features

of the model that are exercised are orientation dependent stress strain

curves, orientation dependent creep response and stress relaxation

behavior.

5.2.1 Orientation Dependent Stress Strain Curves

The stress strain response was calculated for a constant strain

rate of 8.333 x 10 sec for three orientations. The calculated

response compared to the experimental data is shown in Figure 17.

The orientation constants were determined using all three curves so the

variation in the yield stress levels is reproduced well. The hardening

parameters were calculated from the [001] curve only.

Notice that there is a significant mismatch in the elastic modulus

for the predicted and experimental response for the [0 3^2 9^0]

orientation. Recall the observed VF317 response does not agree with the

PWA 1 U80 response shown in Figure 3 because the yield stress for the

[0 3^2 9»»0] specimen, 20° from [001] in the [011] direction, is far

above the expected level. The difference in the experimental and

calculated elastic modulus suggests that the specimens in this data set

designated as being in the [0 342 9^0] direction are probably much

75
to

§CO

.0

vO

w

OS

o

76
FIGURE 17. PREDICTED AND EXPERIMENTAL STRESS STRAIN CURVES FOR

RENE N4 VF317 AT 760°C.

1000-

(0

a.

5

en

v>

UJ

cc

H

500-

[0 342 940]

EXPERIMENTAL DATA

CALCULATED RESPONSE

0.5 1.0

STRAIN, %

1.5

77
closer to the [001 ] direction. This inconsistency is further verified

through an example compression calculation. The predicted saturation

stress response for a [001 ] specimen in compression was less than a

third of the corresponding tensile [001] response. This is not

realistic, especially since the model worked very well for the previous

tension/compression exercises

5.2.2 Creep Response

The tensile creep curves for seven specimens at three orientations

were calculated and are compared with experimental data in Figures 18-

20. The experimental data arefrom creep rupture tests which usually do

not yield reliable data at the beginning of a test. Usually this type

of test produces significant variation from test to test due to poor

alignment and strain measuring systems as well as inherent differences

from specimen to specimen. The correlation between the predicted and

actual response for this type of test must be considered to be

excellent.

5.2.3 Stress Relaxation

Although no stress relaxation data are available the stress

relaxation response for specimens oriented in the [001] and [011 ]

directions was calculated. Both predictions are for a constant

displacement boundary condition with an initial stress level of 758 MPa.

The qualitative behavior is as expected with the stress in the [011]

oriented specimen relaxing faster than in the [001] oriented specimen,

Figure 21. The strain rates in the [001 ] oriented specimen are in the

same range as in the creep tests. However, the strain rates in the

78
o°0§MHU9MOiI

Pt, wHIWw8BSuMPK

d33dO

79
% 'Nivyis daaao

80
§

Cti OSo-»ONw15en o to vO > •*• CdtaPSwenz2Mbl OSWUOS

UJ

81
oovO<rz

Ct] wasCOWoM33enCOwasHenOwOSa,

O

- O

CSJ

rH

(MWOSo

1 oo00

iOoCO

oo

•«t

Ia0CN

'ssaais

82
[011] oriented specimen span the range between the tensile and creep

data. Although the model behaved as expected it would be interesting to

compare it to actual data in this intermediate strain rate range.

83
CHAPTER VI

DISCUSSION AND SUMMARY

The constitutive model as developed in this report is physically

based and is capable of modeling most of the relevant material behavior

at constant temperature. However, consideration must be given to future

development needs. In addition, it is useful to specify the data base

required to full y characterize the materia l parameters in the

constitutive model as it now exists.

6,1 Constitutive Model Development

In the short term it should be possible to utilize the current

constitutive model and finite element code as a viable mechanical

analysis tool with little further development. For the future a number

of potential refinements in the constitutive model are anticipated as

outlined below.

6.1.1 Coupling of Creep and High Strain Rate Response

The work hardening in the flow equation for each slip system is

n ft characterized by a state variable Z . This state variable is a measure

of flow resistance at high strain rates and in creep. In the current

data base this approach appeared satisfactory, but questions remain for

load histories in which both high strain rates and creep occur. To

evaluate the coupling as it is formulated in this study requires a high

strain rate loading followed by creep and/or a creep test followed by a

tensile test.

Since different deformation mechanisms are postulated for high

strain rates and creep it is reasonable to assume that two wor k

84
hardening state variables may be necessary. If this is the case, it is

logical that coupling would occur in the state variable evolution

equations.

6.1.2 Nonisothermal Response

The most obvious problem in modeling the nonisothermal response of

Rene N4 is the lack of a data base sufficient to derive the material

parameters over a range of temperatures. A simple nonisothermal

implementation can be accomplished in the same manner as an isotropic

formulation. The material constants are interpolated for temperature,

and temperature rate and history effects are neglected.

Although the model has not been calibrated for any other

temperatures it is possible to propose a method for interpolating some

of the material parameters. In the flow Equations (2.13) and (2.18) the

parameters D., D_, D, and D^ are interpreted as limiting strain rates

and are expected to be constant with temperature. The strain rate

sensitivity parameters n. , n., n_ and n^. can be represented by an

equation of the form

n - | * b (6.1)

as proposed by Bodner in Reference [53L where a and b are constants and

T is the absolute temperature . The thermal dependenc e of the

orientation parameters ^ , $2, V^, V , V and V__ should be in the

form of an Arhennius function, (Equation (2.2)). The variation of the

material parameters in the state variable evolution Equations (2.15) and

85
(2.21) with temperature is unknown and would be postulated from the

data.

The validity of this approach needs to be Investigated . A

reasonable approach would be to compare predictions of the current model

with nonisothermal experiments. It is anticipated that the state

variable evolution Equations (2.15) and (2.21) may require a thermal

rate term to include thermal history effects.

6.1.3 Thermal Recovery

The treatment of secondary creep as a balance between strain

hardening and recovery was originally proposed by Orowan in 191*?,

Reference [76], Nearly all of the recently developed unifie d

constitutive models include a hardening term and a recovery term in the

work hardening state variable evolution equations. This form has been

successful in modeling the inelastic response of many isotropic metals.

In this formulation an adequate data base for evaluating a recovery term

did not exist and the term was not included. Also at the temperature

studied, recovery is probably not as importan t as at highe r

temperatures. Future work should include the investigation of a thermal

recovery term in the state variable evolution equations.

6.1.H Latent Hardening

The work hardening state variable evolution Equations (2.15) and

(2.21) include "self hardening" only. That is, the work hardening on a

slip system is assumed to be effected only by slip on that system. The

hardening of inactive slip systems by intersection with active slip

systems or "latent hardening" is generally considered to be an important

86
part of the theoretical basis for hardening in single crystal

plasticity, References [6Jl]-[67]. In a recent review, Reference [77],

the ratios of latent hardening to self hardening for single crystal

aluminum and copper are reported in the range of 1 to 2.

There are two methods for determining the effects of latent

hardening mentioned in the literature. The first is to plastically

deform a large crystal oriented for single slip, cut the crystal into

specimens oriented to activate previously latent systems and compare the

response of the initially latent systems to that in the original test.

The second method involves measuring the crystal axis rotations in a

tension test. During finite straining the slip direction rotates toward

the loading direction and eventually a second slip system, or

"conjugate" system, becomes equally stressed. If the latent hardening

of the conjugate system is identical to the self hardening of the

primary system it will slip equally under the same stress. If the

latent hardening on the conjugate system is greater than the self

hardening on the primary system equal inelastic straining will not be

achieved until the stress on the conjugate system is greater than that

on the primary system.

Since Rene NM is a high temperature alloy and the specimens are

coated to prevent environmental degradation, the testing procedures

outlined above would be further complicated. It seems that a viable

alternative testing procedure for investigating latent hardening would

be to run a combination of proportional and nonproportional tensiontorsion tests.

87
For small strains and proportional loading, latent hardening will

have little or no effect on the material's response or the predictive

capability of the constitutive model. For nonproportional loading or

large strains the effect of latent hardening for Rene MM is probably

important but cannot be investigated with the current data base.

Extension of the present theory to large strains and/or nonproportional

loading will require consideration of latent hardening effects.

6.1.5 Cyclic Hardening

Nickel base single crystal alloys experience cyclic hardening

resulting in up to a 10J increase in stress between initial and

stabilized loops in a strain range controlled test, see Figure 6.

Because of the limited data base this behavior has not been modeled.

One way to model this effect is to include an additional term in the

state variable hardness equation. A commonly used measure correlated

with cyclic hardening is accumulated inelastic work which could be the

independent variable in the cyclic hardening term.

6.1.6 Other Response Characteristics

There are a number of other response characteristics not modeled

in this study. For example , slip bursts are observed at some

orientations but have not been considered analytically. This effect is

of little importance and in fact would present serious numerica l

problems if these effects were included in the constitutive model.

Anelastic recovery, or negative creep strain rates at a positive

stress following a stress drop, are commonly observed in nickel base

superalloys. For isotropic materials this behavior is usually modeled

88
by including a back stress tensor. A similar approach could be used for

single crystal materials by including a back stress term on each slip

system. In addition, the back stress model is important for isotropic

modeling because it allows the direction of the devlatoric stress and

inelastic strain rate vectors to be different and to vary as a function

of the deformation. This effect has been observed and documented in the

literature for a variety of multiaxial loading conditions. The current

formulation does not force the stress and inelastic strain rate vectors

to be parallel; however, there are no multiaxial tests of nickel base

single crystal alloys to evaluate this effect.

Tertiary creep is not predicted by the current model. The usual

technique for isotropic materials to account for tertiary creep is by

Includin g a damage measure in the constitutive model. A similar

approach could be used for Rene MM by including another state variable

and evolution equation. However, since tertiary creep occurs at strains

above 2% to 3%, there is little need to model tertiary creep for turbine

blade and vane applications.

6.2 Summary of Data Base Requirements

For the nonlinea r finit e element code wit h the Ren e NU

constitutive model to be a useful tool it is necessary to develop the

data base to generate the required material parameters. In addition, it

is necessary to evaluate the model at other temperatures. The

development of the data base can also be used for refining the

constitutive model. In fact, part of the value of the constitutive

89
model is to provid e a framewor k for alloy development and for

understanding experimental results.

The temperature environment for turbine blades and vanes varies

from ambient up to about 1200°C. Inelastic behavior during operation

would occur mostly at temperatures over 550 C. The test data for the

inelastic response of the material should therefore span the range from

about 550°C to about 1200 C. Considering the variation in response with

temperature, it is desireable to obtain data at about 100 C intervals.

A hypothetical test matrix at a single temperature is presented in Table

6. The matrix is based on the tests required to determine the material

parameters as discussed in Chapter III. The data base must be chosen to

activate the octahedral and cube slip systems separately. Further, the

tests should fully span the space of the sterographic triangle in both

tension and compression. The [001] orientation is favored (half of the

tests are in this orientation ) since this direction is coincident with

the radial direction of the blade or vane. In addition, tests should be

run at other orientations to verify the model. The matrix in Table 6 is

only for the model as presented in this report. The testing should also

include a number of fatigue tests to evaluate cyclic hardening and

further verify the model. It is not expected that the full test matrix

should be run at all temperatures; however, the full matrix should be

run at least one temperature above and below the critical temperature,

about 800°C, since the deformation mechanisms appear to be different.

90
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91
6.3 Conclusion

The constitutive model developed in this study successfully

predicts an important part of the observed material behavior. A method

has been developed for the derivation of material constants and no

special or difficult tests are required.

The implementation of the constitutive model in a nonlinear finite

element code has been accomplished. It is computationall y mor e

intensive than comparable rate dependent isotroplc models or rate

independent anisotropic models, but It is not significantl y mor e

expensive. Some of the features in the finite element code enhance its

usability and decrease its cost.

The goal of developing a practical design tool for Rene N4 gas

turbine engine components has substantially been achieved.

92
APPENDIX A

ElffiNSPACE APPROACH

In the early stages of this research a considerable effort was made

to further develop and apply the unified constitutive theory proposed by

Stouffer and Bodner in 1979 [38]. The theory as proposed included

initial or deformation induced anisotropy.

The form of the deformation rate equation is an extension of the

Prandtl-Reuss flow law to anisotropic inelasticity. The rate of

inelastic deformation tensor is related to the deviatoric stress tensor

by a fourth order linear transformation whose components are functions

of the stress and state variables. The constitutive equations are an

extension of earlier work by Bodner and Partom [39], [^0] and do not

employ the use of a yield criteria or separate loading and unloading

conditions. The state variables are introduced to characterize the work

hardening of the material due to inelastic deformation. These variables

are shown to transform as a fourth order tensor and are a central part

of the anisotropic model.

The tensor ial structure of the constitutive theory is typical of

most anisotropic constitutive models, namely

where e. . is the strain or strain rate, o , is the Cauchy or deviatoric

stress and * is a fourth order tensor that characterizes all of the

material properties. The components of X. ' . may be constants as in

linear elasticity or may be functions of stress and the previous history

93
as required by the particular choice of material and deformatio n

mechanisms. Frequently Equation (1.1) is written in a six dimensional

vector space as

e

a -

 XaB°B U

-

2)

Where a, B - 1,2,... 6 and e.. and a., are written as vectors. In

Reference [38] these vectors were defined as o. - a .., o_ - o_2 o- -

o__, <jj. - S2~a--, a_ - J2~a~^, a, - S2~o and similarly for the strain

tensor e. ..

The coupling between stress and deformation is obvious in Equation

(A. 2). That is a single stress, a. for example, will produce six

components of deformation. One approach to modeling is to transform

Equation (A. 2) into diagonal matrix; that is

« * »

with no sum on a, e and o are the transformed variables and A are

a a aa

the eigenvalues of X 0. This approach has the advantage that the stress

<XD

and deformation are completely uncoupled; that is, one component of

stress produces only one component of deformation. This could be

particularly convenient in computational and experimental work.

However, the major complication is that the eigenvectors are expected to

be functions of \ a and, in general depend on the deformation history. .

94 c
The philosophy behind the proposed approach is to specify the

eigenvalues and eigenvectors rather than the components of X directly.

QD

In Ref. [38]t it was assumed that there existed a class of materials for

which the eigenvectors are Invariant.

Let us begin by calculating the eigenvalues and eigenvectors for an

inelastic compressible isotropic material with a constitutive equation

as described by Equation (A.2). Designate e as the inelastic strain

rate and a as the Cauchy stress. The tensor X . is assumed to be

a do

symmetric, and since the components are real, then X . will have real

Cto

eigenvalues. Ric e [11], Ponter and Leckie [42], Ponter [^31 and

Abuelfoutouh [HI] have shown that the strain rate can be expressed as

the derivative of a potential function with respect to the stress for

history dependent polycryatallinematerials. This result can be used to

establish symmetry in \ a. The constitutive equation for an isotropic

OB

material in a six dimensional vector space can be written as

V

*2

e

3

V

e

5

A

•

>

"ll *12

 X

12 ° ° °

x12 xn x12 o o o

X

12

 X12 X11 ° ' . ° °

o o o xn-x12 o o

0 0 0 0 ;11~X12 °

o o o o o x

n"

A

i2

x

^l"

°2,

ff3

°»

°5

^

(A.4)

95
using two independent material functions A and A . The eigenvalues

are

0 An *A, 2 0

0 0 0 A^-)

OOO O

/ *A

'3

*

\

(A.5)

where e and o are the transformed strain rate and stress vectors in

ot a

* T the eigenvector basis. The eigen configuration is defined as A - Q \Q

where Q is the proper orthogonal transformation that can be written as

the matrix of the components of the eigenvectors. For this case the

eigenvalues are not all distinct and the six eigenvectors are not

unique. The only unique eigenvector is in the one direction. The other

fiv e eigenvectors are arbitrary , however, they should form an

orthonormal basis. A typical orthogonal transformation, written with

eigenvectors arranged as the rows of the matrix, can be defined as

96
1 1 1 o

1 -1 o

0 0

0 0

0 0

0

0

0

1

0

0

0

1

0

(A.6)

Using Q, the components of the transformed strain rate and stress

vectors can be calculated in terms of the untranafonned components. The

eigenvector (1//3, 1//3, 1//3, 0, 0, 0) is the hydrostatic line in

stress space and the other two normal stress eigenvectors are in the

deviatoric stress plane. Budiansky and O'Connel [^5] recently developed

a method to evaluate the properties of isotroplc polycrystalline

material using this transformation.

97
A.1 Cubic Materials

Anisotropic materials with cubic symmetry have three independent

material parameters A^, A-2 and A^. The constitutive equation, [8],

can be written as

e

1

6

2

CH

e

5

,

6

6,

>'

At1 A A 0 0 0 0 11 i^ ^<i

A 1 2 A n A 1 2 0 0 0 0

A1 2 A1 2 An 0 0 0 0

0 0 0 A44 0 0 0

0 0 0 0 0 A M 0

0 0 0 0 0 0 \^

' i

a,

1

°2

°3

°4

a

5

Q _

Q ,

(A.7)

in the principal directions of the material. This reduces to isotropic

response when a relationship exists between the normal and shear

components; i.e. A^ - \1 1 - A1 2 -

The analysis for the eigenvalues and eigenvectors is identical to

*

that for isotropic materials. The eigenvalues A , are A,, * 2A,_, act IT 12

~

 X

12'

for

to 6» respectively.

98
A. 2 Transverse Isotropic Materials

Since a specific application is to metal inelasticity, it is

reasonable to impose the condition of Incompressibility on the material

response. This requires that *« - 0 in Equation (A.1) for arbitrary

values of stress history and current loading. This result was derived

by Olzak and Urbanowski [46] and Hill [47] for the general case of

plastic flow with an anisotropic yield criterion. Introducing the

devlatoric stress components, s. - a.. - (1/3) a 6.. , into Equation

ij ij mm ij

(A.1) and imposing the incompressibility and symmetry conditions

' W - ° (A-

8)

gives

E

iJ ' V*3kt * XiJkp(1/3)Vki • X S {A'

9)

Thus, for incompressible inelastic material response let e and s

denote the inelastic strain rate and deviatori c stress vectors

respectively, in a six dimensiona l space. In this space, the

restriction of incompressibility can be written as six scalar equations

A22 * *32 " ° and X14 * X25 * X35 " ° (A.10)

A23 * X33 * ° *16 * A26 * A36

 = ° '

99
The incompresaibility condition specified in Equation (A. 10) can be

incorporated into the representation for a transverse isotropic

materials [48] to yield

'

E

r

E

2

E

3

e

u

E

5

e6

-

~ ^

U12*A13) . -A12 -A13r oo o

-A12 (A1 2 *AU ) -A13 0 0 0

-A13 -A13 2A1 3 0 0 0

o o o AMM o o

0 0 0 0 A^ 0

0 0 00 0 (2 A +A '

/ \

<

3

1

3

2

3

3

3

4

3

5

,

3

6>

JA.11)

Equation (A.11 ) was written with the coordinate axes in the threedimensional space oriented parallel to the principa l axes of the

orthotropy of the body. In this case, coupling does not exist between

the shear and normal components of the compliance matrix X . Again

cttJ

applying an elementary analysis gives three unique eigenvalues in the

form

/ * \

E

2

E

3

*

E

5

/I

>

0 000 0 0

0 (2A1 2+A ) 00 0 0

0 0 3A13 0 0 0

0 0 0 A44 0 0

0 0 0 0 A^ 0

0 000 0 (2A1 2 ~A )

<

f *\

3

1

4

3

l

*

S

5

.

3

l

(A.12)

100
The eigenvectors can be arranged to give the transformation specified by

Equation (A.6). The eigenvectors for isotropic materials were defined

to coincide wit h the eigenvectors for incompressible transverse

isotropic materials.

A.3 Incompressible Orthotropic Materials

The material matrix

ncompressibility can

(O

62

£3

£5

A

m

>

"

(il2*JLl3)

-x12 (x1

"

X13

0

0

0

* for orthotropic

be written as

-A12 -X

2*X23) -A23

"

X23 (X13*X23J

0 0 J

0 0

0 0

materials with th

0

0

0

U»

0

0

0

0

0

0

X55

0

0"

0

0

0

0

X66

•<

f3^

S

2

33

3U

35

L

36j

(A.13)

where e and s are again the components of the inelastic strain rate

a a •

and deviatoric stress vectors.

Solution of Equation (A.13) for the eigenvalues yields six distinct

eigenvalues, henc e th e eigenvectors ar e all unique . Th e

»

incompressibility condition requires X. - 0 and the first eigenvector e.

- (1//3, 1//3~t 1/^J, 0, 0, 0) aa before. However, the second two

eigenvalues cannot be directly calculated; i.e.

101
2,

• 1/2 (A.14)

This further implies that the second and third eigenvectors, e? and e,,

are functions of the tensor components A..., A__ and A_ . Since the

matrix can be partitioned and analyzed as two independent three space

problems, the eigenvectors e^ , e2, and e- are all independent and can be

A

arrange d as an orthonormal basis. Since e.is fixe d for all

incompressible orthotroplc materials under any deformation history, the

A A A A

position of the e- and e, can be defined by a rotation of e. and e,

A A A

about the e. vector. Let 9 denote the position of e2 and e, relative to

the vectors (1//2, - 1//2, 0) and (1//£, 1/5" - 2//S) in the three

dimensional subspace, respectively. The first three components of the

three eigenvectors can be written as

1/^3

0 cos 9 sin 9 1

1/i/3 1

-1//2 0

0 -sin 9 cos 9 1A/6 1 A/6 -2/S5:

(A.15)

102
The vectors e_ and e~ are still in the deviatoric plane In the subspace.

A A

The angle 9 gives the history dependent position of e. and e,' and the

representation reduces to transverse isotropy when 6-0 . The

transformation matrix Q, in Equation (A. 6), becomes a function of 8 for

incompressible orthotropic materials.

The angle 8 can be evaluated in terms of the components of X in the

initial configuration. Noting that A - QAQ and that A - 0, the

*

calculation of A in terms of 9 gives

«n 2. - ta> t )

12 13 23

(A.16)

i

/TI 33 11 22

A. 4 Conditions Necessary for Stationary Eigenvectors

JU^.1 Transverse Isotropic Materials

The minimum condition for constant eigenvectors can easily be

established. If the transformation matrix Q, defined by Equation (A. 6),

is used to transform the matrix, X,

103
X -

~xn x12 x13 o

A

12

 A11 A13 °

A

13

 A13 A33 °

0 0 0 X ^

000 0

0 . 0 0 0

0

0

0

0

XM

' 0 (X

0

0

q

0

0

n-V.

(A.17)

for an arbitrary compressible transversely isotropic material, the upper

T

left quadrant of the transformed matrix [QXQ ], becomes

2

/TS

(A.18)

This implies that the off diagonal term X , must vanish if Equation

(A.18) is in the eigenspace. Thus, the condition

1/T8

-

 X11 * X12 - X13 ' A33 " ° (A.19)

is necessary to produce the constant eigenvector result.

A.U.2 Orthotropic Materials

The compliance matrix for compressible orthotropic materials has

104
nine independent material parameters (A,,, A,-, X.,, X..,, Xcc, X,,, A._,

11 22 33 ' ***• 55 oo 12

X-., X_ ). Using Q in Equation (A. 6) to transform X into the space X -

7 » * * »

QXQ , also implies that X1 2 > X_, and X_. must vanish if X is in an

eigenspace. Thus

/5x - ( x -x ) * (X-x ) - o

(A. 20)

are the necessary conditions for the eigenvectors to remain stationary.

The first two equations of (A.20) can only be satisfied if each of the

quantities in brackets are zero. Thus

X11 " X22

(A.21)

and the third equation becomes

A.« - X _ + A - A - 0 .. (A.22)

These conditions are the same as that for transverse isotropy as

expressed in Equations (A.17) and (A.19). Hence, materials with

symmetry properties defined as orthotropic can never satisfy the

stationary eigenvector condition.

The consequences of this result can be further explained by

considering an example calculation with the restriction that the

105
eigenvectors are fixed (this is the case presented in Reference C38])-

Let us examine the inelastic response of a strain hardening transverse

isotropic metal. Suppose the hardening is calculated by modifying the

eigenvalues in Equation (A.12), and let e designate the inelastic strain

.1 * * rates, e . Let X2_ and A,, be the eigenvalues after an arbitrary

deformation and transform the eigenspace into the physical space

according to Equation (A.6).

The resulting flow equation for the normal strain rates is given by

"

£r

X w >

>-

"4

 A22 *

(

'iA22

1 * 1 * 1 *

*

 X33} (" 2 A33 * 6 X33

1*1 * 1 *

I

 X33M

2

 X22 + 6

)

 " 3

 A33

3

 X33

3

 A33_

^

3

1

3

2

3

3>

(A.23)

3 33 3 33

Thus, the transverse isotropy assumption is maintained for any choice of

* »

A_9 and A_- provided the transformation Q remains constant. As a

consequence, the material hardening computed in the eigenspace will also

retai n the transvers e isotropy assumption . This result is

unacceptable for general load histories

The eigenvalues and eigenvectors must be functions of the material

hardening for general load histories. Relating these functions to

actual material response becomes a difficult, if not impossible task.

In fact, the mathematical structure bears little relationship to the

106
physical metallurgy of the problem and merely confuses the situation

rather than leading to any significant simplification.

107
APPENDIX B

DEVELOPMENT OF THE KINEMATIC EQUATIONS

This Appendix contains explicit representations for the local and

global stress and inelastic strain rate components. The sign convention

for positive slip directions on the (111) octahedral planes are shown in

Figure B1 . The positive unit normals to the four octahedral planes are

given by

n

1 - 1//3 (I + J + k)

n

2 - 1/vf (•- I + 3 - k)

n

3 - 1/VJ (I - j - k) , and

n

4— 1A/3 (- 1 - J + k) ., (B.1)

where I, J, and k are the unit vectors in the principal material

directions. The positive sign convention for the [1 0 T] directions on

the octahedral planes are:

1

11 - 1A/2 (I - k)

1

12 - \/J2 (- j + k)

1

13 - ^/J2 (I - J)

1

21 - 1//2 (I - ic)

1

22 - \/J2 (I * J.)

1

23 - 1//2 (j * k)

108
IOO

PLANE 2

010

OOl

IOO

PLANE 3 PLANE 4

OIO

OOl

OIO

IOO OOl

FIGURE Bl. OCTAHEDRAL PLANES AND SLIP DIRECTIONS

109
I

31 - 1//2 (I + j)

I

32

I

33

(I + k)

(J + k)

I

42 - 1//Z (I + io

l"3

and

- J) (B.2)

Substituting Equations (B.1) and (B.2) into (2.6) gives the inelastic

strain rate tensor due to shearing in the [1 0 l"] directions on the

octahedral planes as:

.1 .1 .1

e11 £12 e13

21 22 23

.1 .1 .1

E31 E32 e33

OCT

0 -1 1

-1 -2 0

1 0 2

OCT

OCT

2 1 0

1 0 -1

0 -1 -2

2 0 1

0 -2 1

1 -1 0

110
-2 1 0 -2 0 -1

OCT

1 0 -1

OCT

02- 1

0-1 2 -1- 1 0

0 -1 -1 20- 1

OCT

-12 0

* ;31

OCT

0 -2 -1

-1 0 -2 -1 -1 0

0 -1 1 2-1 0

OCT

-12 0

* ;33

OCT

-1 0 -1

1 0 -2 0 -1 -2

0 -1 -1 -2 -1 0

OCT

-1 -2 0 i - i 0- 1

-1 0 2 0-1 2

-2 0 1

OCT

02- 1 (8.3)

1 -1 0

111
The inelastic strain rate vector is relative to the principal material

axes. Substituting Equations (B.1) and (B.2) into (2.9) gives the shear

stress in each of the [1 0 i"] directions on the octahedral planes as:

«

1 0- 1 1 0- 1

11

T

T

23

22

T

^

T

33

U1

T

42

T

^

0 - 1 1- 1 1

1 - 1

-M

1 - 1

- 1 0 1 1 0- 1

- 1 1 00-1- 1

1 - 1 0 0- 1 - 1

1 0-1- 1 0- 1

0 - 1 1- 1 - 1

1 - 1 0- 1

22

12

- 1 1 001- 1

The stresses a., are relative to the material axes.

(B.U)

112
The positive sign convention for the [121 ] directions on the

octahedral planes are:

Z

11

 - 1//S (- I «• 2J - k)

1

12

 - 1/y£ <2i - j - k)

1

13

 - 1A# (- I - J + 2k)

1

21

 - 1//£ (I + 2J * k)

1

22

 - 1/»^ (I - J - 2k)

1

23

 - 1/^ (- 2i - J + k)

1

31

 - 1A/5" (- I + J - 2k)

1

32

 - 1//5 (21 * J * ic)

I

33

 - M& (- I - 2j * k) ,

I

41

 - i//? (- 21 + J -

(I - 2j - k) , and

I

43

 - 1//6 (i * J + 2k) (B.5)

Substituting Equations (B.1) and (B.5) into (2.9) gives the shear stress

in each of the [121 ] directions on the octahedral planes as:

113
T

12

T

t

13

21

T

22

T

t

31

,

32

T

33

T

J»2

T

«3

i

>

-1 2 - 1 1- 2 1

2-1- 1 1 1- 2

-1 -1 2-2 1

-1 2 -1 -1 -2 -1

J

22

'12

'13

J

23

-1- 1 2 2 1- 1

2-1-1- 1 1 2 o,, . (B.6)

-1- 1 2 2 - 1 1

2 -1 -1 -1 -1 -2

-1 2-1- 1 2 1

2-1- 1 1- 1 2

- 1 2- 1 1 2- 1

-1 -1 2 -2 -1 -1

These components influence the orientatio n dependent yield and

tension/compression asymmetry for octahedral slip through the core width

effect.

The sign convention for positive slip directions on the (001 )

cube planes are shown in Figure B2. The positive unit normals are given

by:

n

1

 - I ,

n - j , and

114
PLANE 1

oio

ZV,

'ooi

IOO

'OOI

PLANE 3

OIO

FIGURE B2. CUBE PLANES AND SLIP DIRECTIONS

115
n

3 - k (B.7)

The sign convention fop positive slip directions on the cube planes

are:

I

11

I

12

(J * k)

(3 - k)

1

21 - M-/2 (I * k)

1

22

 • 1A/2 (I - k)

I

31

 -

I

32

J) and

-1*J ) . (B.8)

Substituting Equations (B.7) and (B.8) into (2.1) gives the

inelastic strain rate tensor due to shearing on the cube planes as:

1 1 ~.I

•J,

.1

.1 .1 "

£

12

 E13

e

1

 e1

e

22 23

.1 .1

E

32

 £33.

•^

YCUBE

116
CUBE

0 1 -1

1 0 0

-10 0

YCUBE

0 1 0

1 0 1

0 1 0

22

CUBE

0 1 0

1 0 -1

0-1 0

TCUBE

' o o f

0 0 1

1 1 0

CUBE

0 0 -1

0 0 1

- 1 1 0

(B.9)

Substituting Equations (B.7) and (B.8) into (2.9) gives the shear stress

in each of the directions on the cube planes as:

11

T

12

21

T

22

*

31

T

32

> - 1//2

V /

"000110 ~

0 0 0 1-1 0

000 1 P 1

000 1 0- 1

0000 1 1

0000- 1 1

— — 1

<

X" ">

°22

>

O O

ff

.

ff13

a

. 23

(B.10)

117
The total inelastic strain rate tensor is the sum of the

contribution from each of the active slip systems as compute d in

Equations (B.3), (B.6), and (B.9).

118
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121
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125
1. Report No.

NASA CR-175015

2. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle

Anisotroplc Constitutive Model for Nickel Base Single

Crystal Alloys: Development and Finite Element

Implementation

5. Report Date

March 1986

6. Performing Organization Code

7. Authorfs)

L.T. Dame and D.C. Stouffer

8. Performing Organization Report No.

None

10. Work Unit No.

9. Performing Organization Name and Address

University of Cincinnati

Dept. of Aerospace Engineering and Engineering Mechanics

Cincinnati, Ohio 45221-0070

11. Contract or Grant No.

NAG 3-511

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

533-04-11

15. Supplementary Notes

Final report. Project Manager, Robert L. Thompson, Structures Division, NASA Lewis

Research Center, Cleveland, Ohio. This research was funded by the NASA Lewis HOST

Project and Wright-Patterson A1r Force Base, Dayton, Ohio (Theodore Nicholas).

16. Abstract

The purpose of this research 1s to develop a tool for the mechanical analysis of

nickel base single crystal superalloys, specifically Rene N4, used 1n gas turbine

engine components. This objective 1s achieved by developing a rate dependent an1-

sotroplc constitutive model and Implementing 1t 1n a nonlinear three dimensional

finite element code. The constitutive model 1s developed from metallurgical concepts utilizing a crystallographlc approach. A non Schmld's law formulation 1s

used to model the tension/compression asymmetry and orientation dependence 1n octahedral slip- Schmld's law 1s a good approximation to the Inelastic response of

the material 1n cube slip. The constitutive equations model the tensile behavior,

creep response and strain rate sensitivity of these alloys. Methods for deriving

the material constants from standard tests are presented. The finite element

Implementation utilizes an Initial strain method and twenty noded Isoparametric

solid elements. The ability to model plecewlse linear load histories 1s Included

1n the finite element code. The constitutive equations are accurately and economically Integrated using a second order Adams-Moulton predictor-corrector method

with a dynamic time Incrementing procedure. Computed results from the finite element code are compared with experimental data for tensile, creep and cyclic tests

at 760°C. The strain rate sensitivity and stress relaxation capabilities of the

model are evaluated. Limitations and future development needs are discussed.

17. Key Words (Suggested by Authors))

Rene N4; Fatigue; Single Crystal;

Creep; Constitutive equation;

Tensile; Mechanical properties

19. Security Classif. (of this report)

Unclassified

18. Distribution Statement

Unclassified -

STAR Category

20. Security Classif. (of this page)

Unclassified

unlimited

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21. No. of pages

129

22. Price*

A07

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