A brief description of the implementation in MARK, the general purpose finite element structural analysis code, of two viscoplastic models developed by Robinson is given. One model is for isotropic materials and the other is for metal matrix composites. Also presented are analytical results obtained for hot section components using these models

Arya, V. K.

English

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F I N I T E  ELEMENT IMPLEMENTATION OF VISCOPLASTIC MODELS 
V . K .  Arya* 
NASA Lewis Research Center 
Cleveland, Ohio  
The need for an accurate and realistic description of inelastic, high- 
temperature behavior of materials has attracted the attention of numerous 
investigators in recent years. As a result of their concerted efforts, a number of 
constitutive models called viscoplastic models has emerged. These models consider 
all the inelastic strain (including creep, relaxation, etc.) as a unified quantity 
and, in general, provide a better description of high-temperature inelastic behavior 
of materials. Since these models incorporate as much material science as possible, 
their mathematical structure is very complex. To use these models effectively and 
reliably for the analysis and design of turbine engine hot section components, we 
must demonstrate their feasibility by incorporating the models in nonlinear finite 
element structural analysis codes and perform nonlinear stress/life analyses for 
these components. 
With this objective in mind, the NASA Lewis Research Center has mounted an 
in-house effort to implement some of the more commonly used viscoplastic models into 
the general purpose finite element structural analysis code - MARC. The intent is to 
provide the designer and structural analyst with a tool for realistic and rational 
analysis and designs of hot section components. 
This paper gives a brief description of the implementation in MARC of two 
viscoplastic models developed by Robinson (refs. 1 and 2 ) .  One model is for 
isotropic materials and the other for metal matrix composites. Also presented are 
analytical results obtained for hot section components using these models. Future 
work is also discussed. 
VISCOPLASTIC MODELS 
Robinson's models for isotropic materials and metal matrix composites are based 
on the concept of a flow potential. The flow and growth laws for the internal 
variables are derived from this flow potential. The material behavior is elastic for 
all the stress states within the potential, and it is viscoplastic for all the stress 
states outside the flow potential. The displacements (strains) are assumed to be 
small, and the total strain is assumed to be the sum of elastic and inelastic compo- 
nents. The models include two internal state variables to account for the kinematic 
and isotropic hardenings. 
A brief description of the constitutive equations of these two models and the 
values of constants utilized in the numerical computations are given below. 
*NASA-NRC Resident Research Associate. 
335 
https://ntrs.nasa.gov/search.jsp?R=19890007957 2020-03-20T03:31:06+00:00Z
Isotropic Model 
The constitutive equations for the isotropic model developed by Robinson 
(ref. 1) are 
Flow law. - 
Evolution law. - 
a ij 2ph0Eij - ro ; G > Go and Sijaij 5 0. 
where 
1 s i j = o  - - s  6 ij 3 kk ij 
Cij = S - a 
ij ij 
1 J2 = - 1. .I 2 ij ji 
1 I = - a  a 2 2 ij ij 
H 
G 
h = -  
P 
H - -  h o -  P 
GO 
F = r?) - 1 
G = )rr;/k2 
r = RG m-P 
ro = RGO m-13 
In the above equations, Oij denotes the stress, Eij the inelastic strain 
rate, 6ij the Kronecker delta, and Cij the effective stress. Repeated 
subscripts imply summation. IC is the drag stress and aij is the back stress. 
Numerical values of constants. - The values of constants are for 2-1/4Cr-LMo 
I steel. 
p = 3 . 6 1 ~ 1 0 ~  11 = p exp(81) 
n = 4  m = 7.73 
336 
p = 1.5 R = 9.0~10-8 exp(e2) 
H = 1.37~10-~ Go = 0.14 
and 
e2 = 40 ooo (A - e )  
Here, 0 is the absolute temperature in degrees Kelvin. 
Metal Matrix Composite Model 
The constitutive equations for this model, taken from reference 2, are 
summarized as follows: 
Flow law. - 
Fn = - I -  
ij 2p ij ' 
Evolution law. - 
& = % Eij - RGrn-% ij ' 
G ij 
where 
r = Eij - S[dkdiCjk + djdkCki - 41 d.d.1 - 4<13(3d.d - S..) 
ij 3 1 1  1 j 13 
1 2 = d d C E  -413 2
i j jk ki 
1 I3 = - d.d.1.. 
2 1 J J 1  
1 I; = 5 aijaij 
I ' = d d a  a 12 2 i j jk ki - 413 
1 I' = - d.d.a.. 3 2 1 J J 1  
337 
Here, di denotes a unit vector along the preferred direction at a point of the 
material, and w and q denote the anisotropic parameters. 
Numerical values of constants. - In the absence of experimental data, the 
following values of the constants are assumed to carry out the numerical 
computations: 
1.1 = 3.60~10~ n = 4  
m = 7.73 13 = 0.75 
Go= 0.005 Kt = 0.82 
R = 8.87~10-~ w = 7  
IMPLEMENTATION IN MARC 
I The user subroutine HYPELA in the MARC code provides the user with an efficient 
way of implementing and integrating the general nonlinear constitutive relationships 
of Robinson's model. The details of this implementation may be found in reference 3 .  
To avoid the numerical instabilities encountered due to the "stiff" and 
"discontinuous" nature of constitutive equations, the use of "smoothing functions" 
(or "spline functions") is made. 
defined in references 1 and 3 .  
The spline functions used in the computations are 
RESULTS 
Numerical results using the Robinson's models are depicted in figures 1 to 18. 
Figures 1 to 9 exhibit the results for the isotropic model, whereas figures 10 to 18 
show the results for the metal matrix composite model. 
Isotropic Model 
The cyclic thermal and mechanical loads used to generate the uniaxial hysteresis 
loops are shown in figure 1. Three different strain rates for isothermal, non- 
isothermal in-phase, and nonisothermal out-of-phase loadings are shown in figures 2 
to 4, respectively. Figure 5 shows excellent agreement between experiment and 
predictions, and verifies the correct finite element implementation of the model. I 
The results for the multiaxial problem of a thick internally pressurized cylin- 
der are exhibited in figures 6 to 9. The analytical results shown in these figures 
were obtained using the expressions given in reference 3. 
results agree very well and establish the feasibility of performing a nonlinear 
structural analysis using the viscoplastic model. 
The MARC and analytical 
338 
Metal Matrix Composite Model 
The results of uniaxial calculations using Robinson's metal matrix composite 
model are depicted in figures 10 to 18. These figures show the hysteresis loops, 
creep curves, and relaxation curves generated with the finite element implementation 
of the model. The implementation clearly shows all the expected characteristics 
(e.g., strain-rate sensitivity, creep, relaxation, etc.) and the effect of preferred 
orientations (anisotropy) of the composites. The experimental work to determine the 
values of constants in the model for a metal matrix composite material is in progress 
at Lewis. 
CONCLUSIONS AND FUTURE WORK 
The viscoplastic models for the isotropic material and metal matrix composites 
developed by Robinson have successfully been implemented in MARC. The results 
obtained (in particular using the isotropic model) support the feasibility of 
performing nonlinear structural analyses using these models. These models will 
provide for more realistic and rational designs of gas turbine engine hot section 
components. 
The implementation of other viscoplastic models in MARC - such as (1) visco- 
plastic models for nickel-base superalloys, (2) viscoplastic models for single 
crystal alloys, and ( 3 )  viscoplastic models for metal matrix composites including 
damage - is in progress. The implementation of these models and their results, 
uniaxial and multiaxial predictions, will be reported subsequently. 
REFERENCES 
1. Robinson, D.N.; and Swindeman, R.W.: Unified Creep-Plasticity Constitutive 
Equations for 2-1/4Cr-lMo Steel at Elevated Temperature. ORNL TM-8444, 
Oct. 1982. 
2. Robinson, D.N.; Duffy, S.F.; and Ellis, J.R.: A Viscoplastic Constitutive 
Theory for Metal Matrix Composites at High Temperature. 
1986. 
NASA CR-179530, 
3 .  Arya, V.K.; and Kaufman, A.: Finite Element Implementation of Robinson's 
Unified Viscoplastic Model and Its Application to Some Uniaxial and 
Multiaxial Problems. NASA TM-89891, 1987. 
339 
CYCLIC THERMAL AND MECHANICAL LOADINGS 
USED TO GENERATE HYSTERESIS LOOPS 
BY FINITE ELEMENT CODE-MARC 
I SOTHERMAL LOAD1 NG 
800 
NONISOTHERMAL (IN-PHASE) LOADINGS 
TEMPERATURE, OF w, STRAIN p+ 
480 - .002 
NONISOTHERMAL (OUT-OF-PHASE) LOADINGS 
800 STRAIN “‘1 
480 TIME - .002 TIME 
CD-87-29222 
Figure 1 
HYSTERESIS LOOPS AT DIFFERENT STRAIN RATES 
USING ROBINSON’S MODEL-ISOTHERMAL CASE, 800 OF 
40 r 
STRESS, 
KSI 
-40 
-.0020 -.0015 -.0010 -.OW5 0 .0005 ,0010 .0015 .0020 
STRA I N 
CD-87-29223 
Figure 2 
340 
HYSTERESIS LOOPS AT DIFFERENT STRAIN RATES 
USING ROBINSON’S MODEL-NONISOTHERMAL CASE 
I I 
IN-PHASE THERMAL AND MECHANICAL LOADINGS 
I 
30 r 
STRESS. 
KS I 
20 - 
10 - 
STRAIN RATE, 
STRAIN/MIN 
0.0004 
.004 
.04 
.ob05 .;lo .Oil5 . d 2 0  
- ---- --- 
-5?O020 -.do15 -.dolo -.6005 0 
STRAIN 
CD-87-29224 
Figure 3 
HYSTERESIS LOOPS AT DIFFERENT STRAIN RATES 
USING ROBINSON’S MODEL-NONISOTHERMAL CASE 
OUT-OF-PHASE THERMAL AND ECHANICAL LOADINGS 
STRESS. 
KS I 
50 ,-- 
40 STRAIN RATE, - 
STRAI N/MI N 
0.0004 - 
30- -___ --- 
20 - 
10 - 
Figure 4 
341 
COMPARISON OF MARC AND EXPERIMENTAL 
HYSTERESIS LOOPS AT 1000 OF 
HOOP 
STRESS. 
K S I  
30 r 
9 -  
r T I E ,  
o v  A .0060 
HR 
0 0.0007 
U 0 .5500 
v 
8 -  o 
U V 0 20.00 
v v  V STEADY STATE ( = 2000 HR) 
ELASTIC SOLUTION V 
A A d  
v v  
V 0 0 0  0 
0 f i e 8  0 0 0  
7 -  A '2, 
d 
B B 8 Q Q  :v8v! A 
6 -  Q o  A 
A A V V 
V v Q  ' A  
A V 
A 
' O .  
$ 9 ,  
V 
V 
5 1  
STRESS, 
K S I  
EXPERIENT STRAIN RATE. 
STRAIN/HIN 
.004 
.04 
- ----- 
-.4 -.2 0 .2 .4 -30 
CD-87-29226 
STRAIN. PERCENT 
Figure 5 
'f 16 .17 .18 .19 .20 .21 -22 -23 -24 .25 
RADIUS. I N .  CD-87-29227 
Figure 6 
342 
COMPARISON OF MARC AND ANALYTICAL RESULTS 
FOR STRAIN DISTRIBUTION IN AN INTERNALLY 
PRESSURIZED THICK CYLINDER USING ROBINSON'S MODEL 
STRAIN 
3.0 
M 
h 
2.6 
2.2 h 
NARC 
0 
0 
0 
A 
RADIUS, I N .  
TIME, 
HR 
0.0007 
.5500 
STEADY STATE 
(=2000) 
TINE, 
20.00 
HR 
0.0007 
,5500 
20.00 
STEADY STATE 
(-2000) 
CD-87-29228 
Figure 7 
COMPARISON OF MARC AND ANALYTICAL RESULTS 
FOR STRESS DISTRIBUTION IN AN INTERNALLY 
PRESSURIZED THICK CYLINDER USING 
ROBINSON'S MODEL 
HOOP 
STRESS, 
K S I  
8"'f".o 
7m8t 0.0 
7.4 
7.0 
6.6 
6.2 
5.8 
5.4 
5.0 
+ 
+ o  
8 O  
A 
a'A 
Q 
e A % A  
b 
b, A A  
AA 
@O 
I 
,16 .17 ,18 .19 .20 ,21 .22 .23 .29 -25 
RADIUS. I N .  
NARC 
0 
A 
0 
0 
ANALYTICAL 
e 
A 
4 
TINE, 
HR 
0.0007 
,0060 
,5500 
STEADY STATE 
(=2000) 
TIME. 
HR 
0.0007 
.0060 
.5500 
STEADY STATE 
(=2000 ) 
CD-87-29229 
Figure 8 
343 
STRAIN DISTRIBUTION IN AN INTERNALLY 
PRESSURIZED THICK CYLINDER 
(FEM-MARC SOLUTION) USING ROBINSON'S MODEL 
2 . 8 ~ 1 0 - ~  
2.f 4 V TINE. 
HR 
0 0.5500 V 
V 0 20.00 
V V STEADY STATE ( = 2000 HR) V 
V V ELASTIC SOLUTION 1.8 - 
V 
V 
1.6 c V 
STRA I N 
1 .o 
I 0 0  
. L  
.16 -17 .18 .19 -20 -21 .22 .23 .24 -25 
RADIUS, I N  
Figure 9 
HYSTERESIS LOOPS AT DIFFERENT STRAIN 
RATES; ORIENTATION ANGLE = 00 
STRAIN 
CD-87-29231 
Figure 10 
344 
HYSTERESIS LOOPS AT DIFFERENT STRAIN 
RATES; ORIENTATION ANGLE = 300 
-50 
100 
50 
STRESS, 
KSI 
-50 
. 
STRAIN RATE, 
S T R A I N ~ I N  
0.001 
.01 
.1 I= ----- 
F  -100 
-2 -1 0 1 
STRAIN 
Figure 11 
HYSTERESIS LOOPS AT DIFFERENT 
RATES; ORIENTATION ANGLE = 
100 
50 
2 ~ 1 0 - ~  
CD-87-29232 
STRAIN 
600 
STRAIN RATE. 
STRAIN/MIN 
0.001 
.Ol 
. 1  
--- 
----- 
-2 -1 0 1 2x10-3 
-100 
STRA I N 
CD-87-29233 
Figure 12 
345 
HYSTERESIS LOOPS AT DIFFERENT STRAIN 
RATES; ORIENTATION ANGLE = 900 
1.95 
STRAIN 1.30 
-65 
0 
STRESS. 0 
KS I 
-50 
STRAIN RATE. 
STRA I N/M I N 
0.001 
-01 
.1 
--- ----- 
e
-100-2 -1 0 1 2x10-3 
STRA I N 
CD-87-29234 
Figure 13 
CREEP CURVES FOR DIFFERENT FIBER 
ORIENTATION ANGLES 
FIBER 
2. ~ O X E - ~  ORIENTATION 
Y 30 ,ro 
I I 
I 1 J 
200 1100 600 800 1000 
T I E ,  HR 
CD-87-29235 
Figure 14 
346 
RELAXATION CURVES FOR DIFFERENT FIBER 
ORIENTATION ANGLES 
loo[ 80
FIBER I ORIENTATION 
STRESS 
ANGLE, 
40 
I 
0 200 400 600 800 1000 
TIME. HR 
CD-87-29236 
Figure 15 
HYSTERESIS LOOPS FOR DIFFERENT FIBER 
ORIENTATION ANGLES; STRAIN RATE = 0 . 0 0 1 / ~ 1 ~  
STRESS. 
KS I -'x -/' ORIENTATION 
ANGLE. 
a), 
0 
30 
60 
90 
/' 
DEG 
--- ----- 
-60 VI --- 
Figure 16 
34 7 
HYSTERESIS LOOPS FOR DIFFERENT FIBER 
ORIENTATION ANGLES; STRAIN RATE = O.O~/MIN 
STRESS 
KSI 
100 
50 
- 
' 0  
ORIENTATION 
-- - 0 -__ 30 
----- 60 
90 -__ 
I I 
-2 -1 0 1 2x10-3 
ANGLE, 
a), 
e DEG -
STRAIN 
CD-87-29238 
Figure 17 
HYSTERESIS LOOPS FOR DIFFERENT FIBER 
ORIENTATION ANGLES; STRAIN RATE = O . ~ / M I N  
STRESS, 
KS I 
DEG 
STRAIN 
CD-87-29239 
Figure 18 
348 


Finite element implementation of viscoplastic models

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