Complex states of stress and strain are introduced into components during service in engineering applications. It follows that analysis of such components requires material descriptions, or constitutive theories, which reflect the tensorial nature of stress and strain. For applications involving stress levels above yield, the situation is more complex in that material response is both nonlinear and history dependent. This has led to the development of viscoplastic constitutive theories which introduce time by expressing the flow and evolutionary equation in the form of time derivatives. Models were developed here which can be used to analyze high temperature components manufactured from advanced composite materials. In parallel with these studies, effort was directed at developing multiaxial testing techniques to verify the various theories. Recent progress in the development of constitutive theories from both the theoretical and experimental viewpoints are outlined. One important aspect is that material descriptions for advanced composite materials which can be implemented in general purpose finite element codes and used for practical design are verified

Ellis, J. R.

English

NASA Technical Reports Server

N88-2 386 :
BIAXIAL EXPERIMENTS SUPPORTING THE DEVELOPMENT OF
CONSTITUTIVE THEORIES FOR ADVANCED HIGH-TEMPERATURE MATERIALS
J.R. Ellis
Structural Mechanics Branch
NASA Lewis Research Center
ABSTRACT
In most engineering applications, complex states of stress and strain are
introduced into components during service. It follows that analysis of such
components requires material descriptions, or constitutive theories, which
reflect the tensorial nature of stress and strain. In most applications
involving low homologous temperatures and stress levels below yield, material
response is linear and elastic. These material characteristics lead to sim-
ple constitutive relationships involving two material constants.
For applications involving stress levels above yield, the situation is more
complex in that material response is both nonlinear and history dependent.
Plasticity theories have been developed to model these features by using
flow laws and evolutionary laws expressed in differential form. At elevated
temperatures, the situation is more complicated still in that material
response is time dependent. This has led to the development of viscoplastic
constitutive theories which introduce time by expressing the flow and evolu-
tionary equation in the form of time derivatives. Most recently, attempts
have been made to extend viscoplastic theories to transversely isotropic
materials. The intent here is to develop models which can be used to analyze
high-temperature components manufactured from advanced composite materials.
In parallel with the theoretical studies, considerable effort has been
directed at developing multiaxial testing techniques to verify the various
theories. Thiswork faces two technical challenges. The first is identi-
fying the type of experiment best suited to verifying particular theories.
In the case of plasticity, yield surface testing techniques were developed
and used successfully to support the theoretical effort. A similar approach
has recently been proposed to support development of viscoplastic theories.
In this case, the time-dependent characteristics of elevated temperature
response are captured by using inelastic strain rate as a measure of inelas-
tic state and by determining flow surfaces. It is interesting to note that
this approach can be extended without difficulty to experimental studies of
composite materials.
The second challenge is being able to make stress and strain measurements
to high levels of precision under biaxial loading conditions. Presently,
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https://ntrs.nasa.gov/search.jsp?R=19880013002 2020-03-20T06:33:43+00:00Z
tension-torsion load cells are fully developed, and so stress measurements
do not present muchdifficulty. In contrast, the systems available for meas-
uring biaxial strains need further development. At room and intermediate
temperatures, foil strain gage rosettes have been used with somesuccess in
yield surface studies. Attempts to extend this approach to elevated tempera-
tures have proved less successful because of the unreliability of high-
temperature strain gages whenused in biaxial stress fields. This has led
to the development of a number of mechanical extensometers designed specifi-
cally for use in flow surface determinations at elevated temperature. Again,
this instrumentation can be used to characterize advanced composite materials
provided these materials can be fabricated in the form of thin-walled tubes.
The primary aim of this paper is to outl_ne recent progress in the develop-
ment of constitutive theories both from the theoretical and experimental
viewpoints. One important aspect of this work is that it is leading to veri-
fied material descriptions for advanced composite materials which can be
implemented in general purpose finite element codes and used for practical
design.
2-38
POSTER PRESENTATION
BIAXIAL EXPERIMENTS SUPPORTING DEVELOPMENT OF PLASTICITY THEORIES
Yield surface experiments have been used extensively to investigate multiax-
ial deformation behavior at room and intermediate temperatures. In these
experiments, tubular specimens are loaded under computer control at fixed
ratios of torsional stress and axial stress. The specimens are instrumented
with strain gage rosettes which provide decoupled measures of axial strain
and tensorial shear strain. During the initial stages of loading, straight
line relationships are established between stress and strain for both compo-
nents of loading. The stress levels are increased proportionately until the
stress-strain response deviates a predetermined amount from linear. These
deviations, or small offset definitions of yield, are kept small, 25 pc or
less, to avoid changing the material's state significantly. After achieving
the target offset, the specimen is unloaded, and the procedure is repeated
for stress ratios in all quadrants of tension-torsion stress space. The com-
binations of torsional stress and axial stress giving the required yield con-
dition are noted at the termination of each probe and used to construct yield
surfaces.
INDIVIDUAL LOADING PROBES ARE
TERMINATED WHEN A TARGET VALUE OF
EQUIVALENT INELASTIC STRAIN _P IS
REACHED
TORSION TENSION
PROBING SEQUENCE USED TO DETERMINE
YIELD SURFACES IN TENSION/TORSION
STRESS SPACE
TORSION
°12 TENSION/TORSION
°'12 _ °'11_ 7 ____l___'_ TENSION
O..Ol( e12 ._0 _]_1 _ (11 _111(_,4_ ._
[(¢.),,s,,
PROBINOSEOUENCENUMBER I0"_I _
CD-88-33305
2-39
PLASTICITY THEORIES FOR INITIALLY ISOTROPIC MATERIALS
The aim of plasticity theories is to provide mathematical descriptions of
material response at stress levels above yield. Here, material response is
both nonlinear and history dependent. In the case of initially isotropic
materials, material descriptions should not be directional. This requirement
can be met by using as many as three invariants to introduce stress into the
theory. Classically, the single invariant used for this purpose is the J2,
or Von Mises, form of equivalent stress. One approach adopted in treating
history-dependent material response is to use an internal state variable
_ij which provides a measure of inelastic state. Evolutionary equations are
used to relate changes in =ij to incremental changes of inelastic strain
(Prager) or effective deviatoric stress (Ziegler). It is important to note
that these theories can only be verified through careful experimentation.
af dO.kI (DRUCKER)
FLOW LAW: dc_ = ,_ a--_ii aOkl
POSSIBLE EVOLUTIONARY EQUATIONS: d(xij = C d_E
d(xij = (Sij- cxij)dp,
POSSIBLE YIELD FUNCTIONS: f(J2) "" J2 - K
f(J2,J3) = J2 +/_ J32/3 - K
(PRAGER)
(ZIEGLER)
(VON MISES)
(DRUCKER)
WHERE
J2 = 1/2 SijSij;
Sij = o'ij - 113 Okk _ij;
AND c, _, AND p. ARE MATERIAL CONSTANTS.
J3 = 1/3 SijSjkSki
_ij- INTERNAL STATE VARIABLE
C0-88-33306
2-40
COMPARISONFEXPERIMENTALYIELD SURFACESWITHTHEORETICALPREDICTIONS
The yield surface concept described earlier provides a convenient meansof
interpreting plasticity theories. Adopting this approach, the yield function
describes the shape of the yield surface, and the evolutionary equation
describes the translation of the yield surface in stress space. Interpreta-
tion of the data can be simplified further by using modified stress space
a12 versus aliA/3. This is because the Von Mises yield function plots as
a circle in this stress space. Theoretical and experimental yield surfaces
for a series of nonproportional loadings beyond initial yield are shown here.
The experiments were conducted at 20 °C on 9Cr-IMo steel and used a 25-_c-
offset definition of yield (Ellis, 1985). As is typical for most structural
alloys, the Von Mises yield function gives a close representation of initial
yield behavior and a poor representation of subsequent yield behavior. Also,
Prager and Ziegler forms of evolutionary equation provide reasonable approxi-
mations of yield surface translation in stress space.
TRANSLATION OF YIELD SURFACES RESULTING FROM NONPROPORTIONAL
LOADINGS BEYOND INITIAL YIELD
IDEALIZED EXPERIMENTAL
I
I
\
¢
\
Ol2 /r(_
I
r . °-N-"_. "
./ \V-,\
,- \ _. '\\
t I!/-, ,l';
_" Ii_ Jl °11 Iv°
\ ;;:Ji
,-\__-4.%4>
. % •
=2s ,,
CD-88-33307
2-41
ELEVATED-TEMPERATUREBIAXIAL TESTINGSUPPORTINGTHEDEVELOPMENT
OFVISCOPLASTICONSTITUTIVETHEORIES
Attempts to extend yield surface testing techniques to elevated temperatures
have proved less than successful. Onemajor difficulty has been the unrelia-
bility of high-temperature strain gages when used in biaxial stress fields.
This led to the development of a number of biaxial extensometers which have
recently shownpromise in probing-type experiments (Ellis, 1983). Another
difficulty is that the traditional concept of yield breaks downat elevated
temperatures whenmaterial response becomestime dependent. Oneexperimental
approach developed to resolve this difficulty is to use inelastic strain rate
cP as a measure of change in inelastic state. By adoptin_ this approach,
flow surfaces can be determined for particular values of ¢P and used to
guide the development of viscoplastic constitutive theories.
INDIVIDUAL LOADING PROBES ARE
TERMINATED WHEN A TARGET VALUE OF
EQUIVALENT INELASTIC STRAIN RATE _P
IS REACHED
TORSION
°12 y
0 _12
= ;12- ;1 2G
AND _P: [(_1P2)2 + 314(_Pl)2] 1/2
PROBING SEQUENCE USED TO DETERMINE
FLOW SURFACES IN TENSION/TORSION
STRESS SPACE
TORSION
o110L_N _11 °12 I TENSION/TORSION__
Q PROBINGSEQUENCE
NUMBER CD4e-333oe
2-42
A VISCOPLASTICONSTITUTIVETHEORYFORINITIALLY ISOTROPICMATERIALS
In addition to treating the complexities of plasticity, viscoplastic consti-
tutive theories attempt to model the time-dependent features of material
response at elevated temperature. These features include creep, relaxation,
recovery, and rate dependence. A convenient method of introducing time into
the material description is to express the flow and evolutionary equations
in the form of time derivatives. The majority of viscoplastic theories use
two internal state variables, =ij and K, to treat the complexities of
elevated temperature behavior (Freed, 1988). As in the case of plasticity,
stress is introduced into the material description by means of invariants.
Again, the Von Mises form of equivalent stress is used in the majority of
these models for initially isotropic materials. The validity of this
approach remains to be verified experimentally.
FLOW LAW: eij = #(T) Z E2
H K
1
Z;ij_ _ -#(T) r(K)
WHERE ZIE2), rlK), AND O(T) ARE MATERIAL FUNCTIONS; H AND h ARE
MATERIAL PARAMETERS; AND
3 1
Eii = Sii- (zij Sij = _ _ril - 2 akk 6ij
]_2 = [_ ]_ij_ij] 112
CD-88-33309
2-43
DETERMINATIONF FLOWSURFACESAT ELEVATEDTEMPERATURES
The feasibility of using flow surfaces to investigate multiaxial response at
elevated temperatures has been demonstrated in a series of experiments con-
ducted on type-316 stainless steel at 20 and 650 °C (Battiste and Ball,
1986). As indicated earlier, the main challenge in conducting these experi-
ments is strain measurement. Obvious requirements in probing-type experi-
ments are linearity and high resolution. A requirement for minimum crosstalk
is unique to biaxial testing. This problem arises when loading in the axial
sense produces apparent torsional strains and vice versa. Further complica-
tions arise in conducting these experiments at elevated temperatures. The
problem here is that the electrical noise produced by most heating systems
causes problems with test system control and resolution. As demonstrated by
the results shown here, mechanical extensometers have been developed which
meet most of the requirements discussed previously. Further improvements are
necessary, however, before some of the more fundamental theoretical assump-
tions can be tested with a high degree of confidence.
FLOWSURFACESFORSMALLVALUESOFEQUIVALENTINELASTICSTRAINRATE
CANBEDETERMINEDIN.POST-TESTANALYSISOFTHEEXPERIMENTALDATA
_P-
LOADING SEQUENCE16
14 v+"/ I
0 I I
,,,j_ t
0._1-,.. M.Tr'- I
.......... IZ-- 
_- 012
0 I I
TIME c=_e*-_,o
PREUMINARYFLOWSURFACESDETERMINEDFORTYPE-316STAINLESSSTEELAT20 "C
0.12
/,,,o _ .-.o----
f f/ fl "_//, I/
I J _ , I
_LOADING
SEQUENCE 16
L ,'-ii .
"0__.,.,.., ,,-_p
:._.._ '-_,_._. .
I , I 1,1
' 77
/ /
O'll
CD-M- 33311
2-44
EXTENSION OF BIAXIAL TESTING AND CONTINUUM THEORIES TO
ADVANCED COMPOSITE MATERIALS
It has been recognized for some time that the thin-walled tube and tension-
torsion loading provide an ideal means of investigating deformation behavior
in composite materials. One advantage of this approach is that properties
can be determined for compressive loadings without too much difficulty. Fur-
ther, shear properties can be obtained with minimum ambiguity by using the
torsional component of loading. Thus, it appears that the high-temperature
instrumentation and experimental techniques developed for isotropic materials
can be used to advantage in developing deformation theories for advanced
composite materials.
APPLICATIONOF CONTINUUM THEORIESTO STRUCTUREDMATERIALSREOUIRIEI
IDENTIFICATIONOF A CONTINUUM ELEMENT(O) THAT IS SMALL COMPAREDTO
CHARACTERISTICSTRUCTURALDIMENSIONSBUT LARGECOMPAREDTO
CELLSIZE DIMENSIONS
I---,---I
-L
o® c ®+
D
-<<1
L
C
-<<1
D
C0-le-33312
MEASURESOF ANISOTROPY, _ AND ._,CANBE OBTAINEDBY DETERMININGFLOW
SURFACESFORTUBES REINFORCEDIN THELONGITUDINALANDTRANSVERSESENSES
TORSION
TENSION
o fiTr
otl1"t
_=--
_cl7c
_'(_ TORSION
CD-8_-33313
2-45
VISCOPLASTIC CONSTITUTIVE THEORY FOR TRANSVERSELY ISOTROPIC MATERIALS
Procedures are well established for extending continuum mechanics concepts to
composite materials. Here, the challenge is to incorporate directionality
into the material description without violating the laws of mechanics. A
viscoplastic constitutive theory developed by using this approach is shown
below for the case of transversely isotropic materials (Robinson et al.,
1982). This form of directionality was incorporated into the theory by spec-
ifying a strong direction, di. This increased the number of invariants
initially involved in the theoretical development to seven. This number sub-
sequently, was reduced to three by tailoring the theory to match assumed
material response. The strength of the continuum ,mechanics approach is that
it makes use of more than three decades of progress in the field of visco-
plasticity. This almost assures that the theory will prove successful in
predicting time dependence and history dependence in composite structures
involving a single strong direction. Whether the theory can be extended to
predict the response of complex composite structures is open to question.
FLOW LAW: ;_ - I(F)Fil
EVOLUTIONARYLAW: _ii = h (G) _ - (G)1Eli
WHERE
f(F), h(G), AND _(G) AREMATERIALFUNCTIONS,AND
-'[ ]F=K2 ]1 4(4w2 ]3 -1
1
Pij = Eij- ,E[dkdiEjk+ djdkEki- 2_didj] - _ _']o(3didj - _i|)
G=K- _ I;+_+4(4w2__ 3
' 1 I
_'ij = aii - E[dkdiajk+ djdkaki- 21odidi]- [_ ]"Io(3didi - 8ii)
7/2- 1 4(o02- 1)
E= z?2 _'= 4w2_ 1
7/= KLIKT oo= YL/YT
I1 = J:,-I+_I 3 12 =1-I3
I3 = _o)2 J2= _ Eij [:ij
CC_-3_114
I = didi'EjkEki Io= didiEji
NOTETHATTHEPRIMEDINVARIANTSIN G AND "n'ij AREOBTAINEDBYREPLACING
2_ij BY ai] IN THE ABOVEEXPRESSIONS.
2-46
CI_33315
IMPLEMENTATION OF THE VISCOPLASTIC CONSTITUTIVE THEORY FOR TRANSVERSELY
ISOTROPIC MATERIALS IN THE MARC FINITE ELEMENT CODE
The subject viscoplastic model has been implemented in the MARC finite ele-
ment code and exercised in a number of trial calculations (Arya, 1987).
The material properties used in these calculations were similar to those
of a metal matrix composite material, tungsten-copper, at about 800 °F.
Initially, the analyses were limited to uniaxial stress states. Under these
conditions, it was shown that the model can treat cyclic plasticity, rate
dependence, and creep for a range of fiber orientations. More recently, a
thick-walled cylinder was analyzed to determine performance in calculations
involving multiaxial stress states. As indicated below, the theory was
successful in predicting creep-like behavior which was highly dependent on
fiber orientation. For example, creep in the hoop direction can be seen to
be negligible for the case of _ = 90.
THE SUBJECT CONSTITUTIVE THEORY HAS
BEEN IMPLEMENTED IN THE MARC FINITE
ELEMENT CODE AND USED TO ANALYZE A
THICK-WALLED CYLINDER WITH A RANGE OF
FIBER ORIENTATIONS (_))
WHEN LOADED UNDER INTERNAL PRESSURE,
=4000 psi, THE MATERIAL EXHIBITED
CREEP-LIKE BEHAVIOR WHICH WAS HIGHLY
DEPENDENT ON FIBER ORIENTATION
0 r
Z
ri = 0.16 in
0
o= 0.24 in.
TEMPERATURE800 "F
_--0
4D- 30
4D=N
- 4D=_
TIME
CD-88-33316
2-47
REFERENCES
Arya, V.K., 1987, "Finite Element Implementation of Viscoplastic Models," Tur-
bine Engine Hot Section Technology 1987, NASA CP-2493, pp. 335-348.
Battiste, R.L., and Ball S.J., 1986, "Determination of Surfaces of Constant
Inelastic Strain Rate at Elevated Temperature," Turbine Engine Hot Section
Technology 1986, NASA CP-2444, pp. 307-325.
Ellis, J.R., 1985, "An Experimental Study of Biaxial Yield in Modified 9Cr-IMo
Steel at Room Temperature," NASA CR-175012.
Ellis, J.R., 1983, "A Multiaxial Extensometer for Measuring Axial, Torsional,
and Diametral Strains at E}evated Temperature," ORNL- TM-8760.
Freed, A.D., 1988, "Structure of a Viscoplastic Theory," NASA TM-I00794.
Robinson, D.N., Duffy, S.F., and Ellis, J.R., 1987, "A Viscoplastic Constitu-
tive Theory for Metal Matrix Composites at High Temperature," Thermal
Stress_ Material Deformation and Thermo-Mechanical Fatigue, ASME PVP
123, H. Sehitoglu and S.Y. Zamrik, eds., ASME, pp. 49-56.
2-48


Biaxial experiments supporting the development of constitutive theories for advanced high-temperature materials

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