A state-of-the-art review of applicable constitutive models with selection of two for detailed comparison with a wide range of experimental tests was conducted. The experimental matrix contained uniaxial and biaxial tensile, creep, stress relaxation, and cyclic fatigue tests at temperatures to 1093 C and strain rates from .0000001 to .001/sec. Some nonisothermal cycles will also be run. The constitutive models will be incorporated into the MARC finite element structural analysis program with a demonstration computation made for advanced turbine blade configuration. In the code development work, particular emphasis is being placed on developing efficient integration algorithms for the highly nonlinear and stiff constitutive equations. Another area of emphasis is the appropriate and efficient methodology for determing constitutive constants from a minimum extent of experimental data

Lindholm, U. S.

English

The objective is to develop a unified constitutive model for finite element structural analysis of turbine engine hot-section components. This effort constitutes a different approach for non-linear finite-element computer codes which have heretofore been based on classical inelastic methods. The unified constitutive theory to be developed will avoid the simplifying assumptions of classical theory and should more accurately represent the behavior of superalloy materials under cyclic loading conditions and high temperature environments. During the first two years of the program, extensive experimental correlations were made with two representative unified models. The experiments were both uniaxial and biaxial at temperatures up to 1093 C (2000 F). In addition, the unified models were adopted to the MARC finite element code and used for stress analysis of notched bar and turbine blade geometries

Lindholm, Ulric S.

NASA Technical Reports Server

N88- 11171
CONSTITUTIVE MODELING FOR ISOTROPIC MATERIALS*
Ulric S. Lindholm
Southwest Research Instititute
INTRODUCTION
The objective of the present program is to develop a unified constitutive
model for finite-element structural analysis of turbine engine hot section
components. This effort constitutes a different approach for non-linear
finite-element computer codes which have heretofore been based on classical inelastic
methods. The unified constitutive theory to be developed will avoid the simplifying
assumptions of classical theory and should more accurately represent the behavior
of superalloy materials under cyclic loading conditions and high-temperature
environments. This class of constitutive theory is characterized by the use of
kinetic equations and internal variables with appropriate evolution equations for
treating all aspects of inelastic deformation including plasticity, creep, and
stress relaxation. Model development is directed toward isotropic, cast nickel-base
alloys used for air-cooled turbine blades and vanes. Recent studies have shown
that this approach is particularly suited for determining the cyclic behavior of
superalloy-type blade and vane materials and is entirely compatible with
three-dimensional inelastic finite-element formulations. More efficient and accurate
inelastic analysis of hot section components--turbine blades, turbine vanes,
combustor liners, and seals--fabricated from "age hardenable" isotropic superalloy
materials will be realized as the result of these developments.
During the first two years of the program, extensive experimental correlations
have been made with two representative unified models. The experiments are both
uniaxial and biaxial at temperatures up to I093°C (2000°F). In addition, the unified
models have been adapted to the MARC finite element structural code and used for
stress analysis of notched bar and turbine blade geometries.
CONSTITUTIVE MODEL SELECTION
A literature survey was conducted to assess the state-of-the-art of time-
temperature dependent elastic-viscoplastic constitutive theories which are based
on the unified approach. As reported earlier (ref. l), the review identified more
than ten such unified theories which are shown to satisfy the uniqueness and stabil-
ity criteria imposed by Drucker's postulate and Ponter's inequalities. These
theories are compared on the basis of the types of flow law, kinetic equation,
evolution equation of the internal variables, and treatment of temperature
dependence.
As a result of the literature survey, the models of Bodner and Partom (ref.
2) and of Walker (ref. 3) were selected for further study. These two models are
representative of the class of unified models considered in the review process
but differ significantly in the choice of particular functional forms for the basic
flow law, the kinetic relationship, the parameter used as a measure of hardening,
and the evolution equations for the internal variables describing work hardening.
Thus, a direct comparison between these two models and the experiments should illus-
trate well the consequences of a wide range in constitutive modeling approach. It
*Work done under NASA Contract NAS3-23925.
303
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is also significant that both models have already found significant application
to analysis of gas turbine materials and to hot section components. Therefore,
they are further along in their development and evaluation than most of the other
comparable models.
F_TERIAL SELECTION
Two materials widely used in gas turbines are being evaluated with the constitu-
tive models: PWA alloy BI9OO+Hf and MAR-M247. Tests completed to date have been
with BI9OO+Hf. Casting configurations and pour and mold temperatures were selected
to assure a grain size of ASTM No. 1 to 2 in the gage section and a ¥_ size of
0.9 _m in the fully heat treated condition for the BI9OO+Hf.
EVALUATION OF UNIFIED MODELS
The testing program included uniaxial and biaxial (tension-torsion) isothermal
specimens in monotonic, creep, stress relaxation, and cyclic loading sequences
encompassing wide variations in strain rate and temperature. Various thermo-mechani-
cal fatigue cycles were investigated also. In each case, the measured material
response was correlated with the two unified constitutive models. A small subset
of the data was used to determine the constants for each model; about eight to
ten constants are required. Also, definitive procedures were developed to determine
each constant. The data correlations presented are based on the models with material
constants determined from tensile or tensile and a few fatigue cycles. In this
sense the results are predictive.
Some representative results are shown in Figures I-4 for tensile, creep, cyclic,
and TMF load histories. Over 60 different loading histories were examined with
generally good correlation using both models.
Correlation of local deformation at the root of a notched round tensile specimen
was not as good as for the homogenously-stressed test specimens. This was attributed
to specimen material inhomogeneity and to accuracy of the material model in the
near yield (small inelastic strain) region.
CONCLUSIONS
The results of this program provide strong evidence for the applicability
of the unified constitutive equation approach to describe the strongly non-linear,
time- and temperature-dependent response of metals to arbitrary load or deformation
histories. As a minimum, the two models studied were correlative in that they
demonstrated reasonable agreement with all the experimental data generated over
a very wide range of temperature, deformation rate, and load cycle. This was accom-
plished with a fixed model and fixed set of material constants. In a broader sense,
the models were predictive in that the functional representations and the material
constants were obtained from a small subset of the total tests performed. Even
better overall correlation could have been achieved if all the data were used to
optimize the functions and constants in each model.
A major criticism of the unified models has been the apparent (and often real)
difficulty in defining and experimentally determining the multiple material constants
3o4
employed in the models. We believe we have made significant progress in this regard.
The different terms in the constitutive equations are very interactive so that
it is essential to have a physical insight into the meaning of each function and
its associated constants. Procedures developed delineate a deterministic method
for evaluating the constitutive constants in a sequential manner from a relatively
small material test data base.
While the correlative and predictive capacity of the unified models has been
demonstrated, their practical application will depend on their adaptability in
efficient computational algorithms when incorporated into finite element or other
numerical methods for structural analysis. Preliminary use with the MARC finite
element code indicates that improvement is needed in the numerical integration
procedures used with these stiff, nonlinear equations. The computation time for
an equivalent problem is approximately the same order of magnitude as when using
classical, time-independent plasticity formulations.
REFERENCES
I. Lindholm, U. S.; Chan, K. S.; Bodner, S. R.; Weber, R. M.; Walker, K. P.; and
Cassenti, B. N.: Constitutive Modeling for Isotropic Materials. Annual Report
NASA CR-174718, 1984.
2. Bodner, S. R.; and Partom, Y.: ASME J. of Applied Mechanics, vol. 42, 1975,
p. 385.
3. Walker, K. P.; NASA Contract Report NASA CR-165533, 1981.
1000
8OO
600
p
G
_, qO0
200
0 o
; + I
B1900 ÷ Hf
• 8.3x10"Ssec "I
I ; I_ Walker
.2 ._ .6 .8 1.0
Strain, %
='E.NSIL_C_ = ._ATEOAT_
S_.e_cy SteTS8Creeo Rat=-(s=-c"!)
Figure 1
COMPARISON OF EXPERIMENTAL AND CALCULATED
STRESS-STRAIN CURVES OF BlgOO+Hf AT THREE
TEMPERATURES
Figure 2
THE CALCULATED AND EXPERIMENTAL
RESULTS OF CONSTANT LOAD
CREEP TESTS
305
400
3O0
200
100
0
-IOO -_
-200 i
.8
------ Experiment
Bodner-Part.oe Pqode1
.... Walker Model
. _ _ _ _ _ _--" -_.=,_
J
I I I I I I I
-. -.4 -.2 0 .2 .4 .6
Stratn,
Figure 3
CONPARISON OF THE EXPERINENTAL AND CALCULATED
STABLE HYSTERESIS LOOPS OF B1900+Hf at 1093°C
1200
8OO
_, 400
_ 0
i
m
-400 -
-800
BI_ + Hf
THF Cycle 8
(Strain out-of-phase
with temperature) /_
Znd Cycle _cold
_'_" _ Experiment
_/ _-- Bodner-Partom
hot Model
---- Walker Hodel
I I I I I I I
-.4 -.2 .o .2 .4 .6 .8
Hechanical Strain, %
Figure 4
THERNOI_CHANICAL CYCLIC DATA COHPARED WI_ THE
WALKER AND THE BODNER-PARTON MODEL PREDICTION
FOR THE OUT-OF-PHASE THF CYCLE
306


Constitutive modeling for isotropic materials

8 7 - 1 1 2 1 1
CONSTITUTIVE MODELING FOR ISOTROPIC MATERIALS
Ulric S. Lindholm
Southwest Research Institute
INTRODUCTION
This report presents results of the first year of effort on a program with the
objective to develop a unified constitutive model for finite-element structural anal-
ysis of turbine engine hot section components. The program is a joint effort be-
tween Southwest Research Institute and Pratt & Whitney Aircraft.
The initial two year program includes a state-of-the-art review of applicable
constitutive models with selection of two for detailed comparison with a wide range
of experimental test. The experimental matrix contains uniaxial and biaxial ten-
sile, creep, stress relaxation and cyclic fatigue tests at temperatures to I093°C
and strain rates from 10-7 to 10-3 sec-l. Some non-isothermal TMF cycles will be
run also. The constitutive models will be incorporated into the MARC finite ele-
ment structural analysis program with a demonstration computation made for an ad-
vanced turbine blade configuration. In the code development work, particular empha-
sis is being placed on developing efficient integration algorithms for the highly
non-linear and stiff constitutive equations. Another area of emphasis is the appro-
priate and efficient methodolgy for determining constitutive constants from a min-
imum extent of experimental data.
CONSTITUTIVE MODELS
An extensive review of currently available unified constitutive models was
made from which a review is given in references 1 and 2. In a "unified" theory, the
inelastic strain rate term, _P, is considered to include all strains that are not
elastic; i.e., the difference between the total strain and the elastic strain,
_ge. Thus, unified implies that all aspects of inelastic behavior such as plastic
flow, creep and stress relaxation are included in the single function, _P, and are
simply representative response characteristics for different loading histories. In
such theories, inelastic behavior may be described with or without the use of a
yield function or concept of plastic potential. Those chosen here for further study
do not employ a yield criteria and are based on internal variables to describe
"yielding" and strain or work hardening behavior.
Two particular constitutive models were chosen for detailed study and compar-
ison with experimental data. These were developed by Bodner and Partom (B-P)
(ref. 3) and by Walker (WK) (ref. 4). Both models had considerable prior application
to high-temperature alloys used in gas turbine components. Most unified models are
of the basic form
_-xI
X 2
= f (_P,T) (i)
285 PRE_I,_iG PAGE BLANK i_r FM.IIII&D
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here o, _P and T are stress, inelastic strain rate and temperature, respectively.
Generally_ two independent internal variables are used, _i a tensor quantity de-
scribing directional material hardening (often referred to as a back stress, equi-
librium stress, or kinematic hardening variable) and X 2 a scalar measure of the
magnitude of isotropic hardening. The evolutionary eqHations for both internal
variables are usually of the hardening-recovery form,
X'I = h(Xi) _ + r(Xi'T) (2)
where M is a physical measure of hardening and h and r are hardening and recovery
functions.
The WK model uses a power law for the kinetic term, f(_P,T), and plastic
strain as the measure of hardening M. The B-P model uses an exponential form in
the kinetic term and plastic work for M. The other major difference between the
two models is that B-P avoids the use of the back stress _i and encorporates both
isotropic and directional hardening in a partitioning of X 2. For a more detailed
comparison see references 1 and 2.
EXPERIMENTAL PROGRAM
An extensive test program is underway to generate a comprehensive set of data
which is to be compared with model predictions. A cast nickel base alloy, BI9OO+Hf,
with grain size of ASTM No. 1 to 2 is used for the specimens shown in figure i. As
indicated, tensile, creep, isothermal cyclic, thermomechanical cyclic and biaxial
(tension-torsion) tests are being performed. To date the tensile, creep and iso-
thermal cyclic tests are complete. Sample results, including correlations with the
B-P model, are given in figures 2-5 (correlations with the WK model are being gen-
erated also but were not available as of this writing). All model correlations are
made with a single set of material constants.
Figure 2 shows the correlation for tensile curve at three temperatures. In
the process of determining the constants associated with work hardening in the B-P
model, the experimental hardening data was plotted as in figure 3 where y = do/dWp
do/o dEP. Analysis shows that the data at each strain rate can be closely approxl-
mated as the sum of two linear curves, whose slopes and intercepts yield the co-
efficients for the isotropic and kinematic hardening terms. Thus, it appears pos-
sible to predict cyclic behavior from monotonic stress-strain curves. This con-
clusion needs further verification but holds potential for reducing the testing
required for constitutive constant determinations. Another observation from
figure 3 is that a change in strain rate does not change the hardening rate (slope)
in agreement with the separation of kinetic and hardening terms in equation (i).
The slopes do change with temperature because of thermal recovery of hardening
(eq. (2)).
Figure 4 shows the small strain (0.2%g p) flow stress over the range of tem-
perature and strain rate studied. Inflections in the curves at the intermediate
temperatures result from the influence of thermal recovery at lower rates and higher
temperatures. At low temperatures and high rates, thermal recovery is not signi-
ficant.
286
An example of initial and saturated cyclic loops at 538°C is given in figure 5.
Agreement between experiment and theory is reasonable in this and other cases exam-
ined considering the same constants are used for figures 2, 4 and 5. More complex
loops with creep or relaxation holds during a cycle will be correlated during the
second year along with cyclic biaxial data.
IMPLEMENTATION IN F.E. CODE
Both models are being implemented for use with the MARC finite element code.
The code will subsequently be used to analyze a notched tensile round test specimen
used as a benchmark experiment and also an advanced turbine blade configuration.
The latter will be a numerical demonstration only. Several numerical methods are
being studied for implementing the models in the MARC code. Integration methods
for viscoplastic theories to be examined include:
(i) Explicit Euler integration with both a fixed and self-adaptive
time step
(2) The implicit noniterative, selfcorrecting solution (NONSS)
method of Miller and Tanaka (ref. 5)
(3) Implicit integration of the integral form of the equations.
It is expected that each theory will be coded with at least two numerical inter-
gration algorithms.
SUMMARY
The work to date is encouraging with respect to the ability of unified con-
stitutive theories to predict with reasonable accuracy quite complex time and tem-
perature dependent inelastic material behavior. Also encouraging, at this point, is
the possibility of determining all necessary constitutive constants from perhaps as
little as monotonic tensile curves at several temperatures and strain rates. Such
data is generally available for alloys of interest. For implementation of the
models in finite element codes and efficient structural analysis, optimum numerical
integration schemes need further development.
.
.
1
,
5.
REFERENCES
Chan, K. S.; Bodner, S. R.; Walker, K. P.; and Lindholm, U. S.: A Survey
of Unified Constitutive Theories. Second Symposium on Nonlinear Constitutive
Relations for High Temperature Applications, Cleveland, Ohio, June 1984.
Lindholm, U. S.: Constitutive Modeling for Isotropic Materials. First
Annual Report, NASACR-174718, May 1984.
Bodner, S. R. and Partom, Y.: ASME J. of Applied Mechanics, Vol. 42,
1975, p. 385.
Walker, K. P.: NASA Contract Report NASA CR 165533, 1981.
Tanaka, T. G.: Deformation of Metals and Alloys. Ph.D. Thesis,
Stanford University, 1983.
287
<BLEND SMOOTHLY
7.00
(a) Monotonic tensile/creep specimen
0.301
DIA1.000 R. £rYP_ 0_99
BLEND SMOOTH#Y
0.5.630 : 0.0005 DIA. 2.95 _- 1.1
(b) Isothermal cyclic specimen
< .566
- DIA
:.jQ© ,< 1.50 R I'TYPI
:___ JsIO AI.oo_n. Jt aLENOSMOOTHLY
y//////////////////_ .........__ : •
1 ......
I 1._oo
8.oo
(c) Thermomechanical cyclic specimen
i
I
(aI
(bI
(c_
8.00
_, 3.000 -_
.75 RFI"YP}
1.000 OlA. I-rYP.}
DIA
8003
.75 R(TYP) _ 1.25 (,'_Y'P.}
(d) Isothermal biaxial test specimen
(d)
Figure i. Specimen Designs Utilized in Various Constitutive Tests.
288
I000 , , , ,
Q..
aJ
C,O
Figure 2.
800
600
4OO
200
bl _-.
II
0
0
B1900+Hf
Figure 3.
10-5
]
= 8_3 x sec TM
m
Bodner-Partom Theory 538°C (IO00°F)
--- Experiment
I I f I
,2 ,4 ,6 ,8
Strain, %
.0
A Comparison of the Calculated (Bodner-Partom Theory)
and the Experimental Stress-Strain Curves of
BI900+Hf at 538, 871 and 982°C.
600
500
400
3OO
200 -
I00 -
0
200 800
I I I i I
_1900 + Hf
871 °C
• x 10 -6 sec -I
F _ : 8.3 x 10-5 sec"I
r_ : 1.65 x 10-3
sec" 1
I I, I
300 400 500 500 700
Stress, o-, MPo
Work Hardening Behavior of BI900+Hf at Three
Strain Rates and 871°C.
289
1000
800
r_
600
4OO
200 B
0
10 -7
B1900+H f
SwRI Data PWA Data
• O 760°C --Bodner-Partom Theory
_ _ 0 871°C
• i-I 982°C
A AI093°C o
_ /- 760 C
O
m
m
_ T _1 I I fll _ I II1 I I III I I i,I ,
10-6 10-5 10-4 -I I0-3 i0 -2
Strain Rate, sec
I I I
I I I
i0 -I
Figure 4. Temperature and Strain-Rate Dependence of 0.2%
Offset Yield Stress for BI900+Hf.
29O
1000
8OO
60O
¢00
200
E
0
- 200
- _00
- 600
- 80O
-I000
100'3.
OOO.
600.
0::
200.
ur_
q" O. O0
-200.
-qO0.
-6uO,
-dUO.
- I O00.
I t I I
BIgOO+Hf I-2 Cycles
m
538°C
--_ = +8.3 x I0"4 sec"I / 7
__---PWA Data//
m
m
m
w
I I I 1
Strain, %
(a) Experiment
i z i i
8odner-Partom Theory
538°C
= +8.3 x 10 -4 sec "l
1
I I
,/
//
/
//
/
ed
Strain, %
(b) Bodner-Partom Theory
Figure 5. A Comparison of the Calculated (Bodner-Partom Theory) and
the Experimental Hysteresis Loops After 1-2 Cycles and at
Cyclic Saturation.
291


Constitutive modeling for isotropic materials

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