Short term strain rate change, creep and relaxation tests were performed in an MTS computer controlled servohydraulic testing machine. Aging and recovery were found to be insignificant for test times not exceeding 30 hrs. The material functions and constants of the theory were identified from results of strain rate change tests. Numerical integration of the theory for relaxation and creep tests showed good predictive capabilities of the viscoplasticity theory based on overstress

Krempl, E.

Lu, H.

Yao, D.

English

NASA Technical Reports Server

N88-21499 
THE VISCOPLASTICITY THEORY BASED Oh’ OVERSTRESS 
APPLIED TO THE MODELING OF A N I C K E L  BASE SUPERALLOY AT 815°C 
E. Krempl, H .  Lu and D. Yao 
Department of Mechanical Engineering, 
Aeronaut ica l  Engineering and Mechanics 
Rensselaer  Polytechnic  I n s t i t u t e  
Troy, NY 12180-3590 
Short-tenn s t r a i n  ra te  change, c reep  and r e l axa t ion  tests were performed i n  an MTS 
computer-controlled servohydraul ic  t e s t i n g  machine. Aging and recovery were found t o  
be i n s i g n i f i c a n t  for test  times no t  exceeding t h i r t y  hours .  The m a t e r i a l  func t ions  
and cons t an t s  of t h e  theory  were i d e n t i f i e d  from results of s t r a i n  r a t e  change tes t s ,  
Numerical i n t e g r a t i o n  of t h e  theory  f o r  r e l a x a t i o n  and creep  tes ts  showed good pre-  
d i c t i v e  c a p a b i l i t i e s  of t he  v i s c o p l a s t i c i t y  theory based on o v e r s t r e s s .  
In t roduc t ion  
Advanced materials are being developed f o r  use i n  high temperature  gas t u r b i n e  
a p p l i c a t i o n s .  To f u l l y  u t i l i z e  t h e  c a p a b i l i t y  of t hese  new materia-1s t h e i r  deforma- 
t i o n  p r o p e r t i e s  and t h e i r  c reep  and f a t i g u e  f r a c t u r e  c h a r a c t e r i s t i c s  m u s t  be de t e r -  
mined by s u i t a b l e  experiments .  The experimental  f i nd ings  must be analyzed,  i dea l i zed  
and t r a n s l a t e d  i n t o  c o n s t i t u t i v e  equat ions  f o r  use i n  stress a n a l y s i s  and l i f e  pred ic-  
t i o n .  Only when t h e s e  i n g r e d i e n t s  t oge the r  with appropr i a t e  computat ional  t o o l s  a r e  
a v a i l a b l e  can d u r a b i l i t y  a n a l y s i s  be performed i n  t h e  design s t a g e  long before  a compo- 
nent  i s  being b u i l t .  This paper  c o n t r i b u t e s  t o  t he  design methodology and r e p o r t s  on 
an experimental  i n v e s t i g a t i o n  of t h e  deformation behavior  of a n icke l -base  supera l loy  
a t  815°C and on i t s  mathematical  modeling us ing  the  v i s c o p l a s t i c i t y  theory based on 
o v e r s t r e s s .  
Tes t ing  Method and Test Materials 
All  tests were performed i n  an MTS ax ia l - to r s ion  servohydraul ic  mechanical t e s t i n g  
machine wi th  an MTS 4 6 3  Data/Control processor  f o r  computer c o n t r o l .  
furnace  w a s  used t o  h e a t  t h e  specimens. S t r a i n  measurement on t h e  gage s e c t i o n  was 
done with an MTS high-temperature u n i a x i a l  extensometer.  S t r a i n  and load con t ro l l ed  
tests were performed a t  815°C. 
( a t  room temperature)  and on t h e  uniformity of t he  temperature  a long t h e  gape l eng th .  
For t h e  du ra t ion  of t h e  tes ts  t h e  temperature  of the  gage s e c t i o n  s tayed  wi th in  2 ° C  
of t h e  nominal temperature .  
An MTS c lamshel l  
Separa te  tests checked on t h e  extensometer c a l i b r a t i o n  
The nickel-base supe ra l loy  test m a t e r i a l  was donated by AVCO Lycoming and was 
de l ive red  i n  the  form of 1 9  mm coupons. The machined specimens had a gage s e c t i o n  
diameter of about 6 .5  nun. They were t e s t e d  i n  the  as-received cond i t ion  a t  8 1 5 O C .  
Experiment a1 Resu 1 t s 
E f f e c t s  of s t r a i n  r a t e  
Figure 1 shows t h e  r e s u l t s  of t h r e e  tests:  two were subjec ted  t o  continuous 
s t r a i n i n g  a t  
s t r a i n  r a t e s .  The s i g n i f i c a n t  i n f luence  of s t r a i n  r a t e  i s  ev iden t .  These and o the r  
s t r a i n  r a t e  change tests revea led  no d i s c e r n i b l e  s t r a i n  r a t e  h i s t o r y  e f f e c t .  
s t r a i n  ra te  change tes t s  i n  F ig .  1 and o t h e r s  i n d i c a t e  t h a t  t h e r e  i s  a unique s t r e s s  
l e v e l  a s soc ia t ed  with each s t r a i n  r a t e  which w i l l  be reached a f t e r  a t r a n s i e n t  per iod 
1 
and s-’, r e s p e c t i v e l y ;  the  o the r  t o  a s e q u e n t i a l  decrease i n  
The 
https://ntrs.nasa.gov/search.jsp?R=19880012115 2020-03-20T06:23:59+00:00Z
-2- 
i r r e s p e c t i v e  of t h e  p r i o r  h i s t o r y .  The stress l e v e l s  were determined from the  tes ts  
and are l i s t e d  i n  Table 1 toge the r  wi th  t h e  stress l e v e l  d i f f e r e n c e s  r e l a t i v e  t o  the  
s t r e s s  corresponding t o  10" s-I. 
The tests i n  Fig.  1 show good r e p r o d u c i b i l i t y  which w a s  a l s o  found w i t h  o t h e r s .  
An except ion  i s  t h e  stress l e v e l  a t  A t  1.1 percent  
s t r a i n ,  specimen 1 5  su f fe red  a temperature v a r i a t i o n  of about 4°C which may have caused 
the  drop-off a t  po in t  A.  
found i n  o t h e r  experiments.  Premature cracking i s  probably the  cause f o r  t he  decrease 
i n  stress l e v e l  of specimen # 7  before  unloading s t a r t e d ,  
s-l f o r  specimens 1/12 and 1115. 
The f i l l e d  t r i a n g l e  i n d i c a t e s  t h e  stress l e v e l  a t  lo- '  s-' 
Aging and recovery 
To a s c e r t a i n  whether t hese  phenomena have a s i g n i f i c a n t  i n f luence  on the  deformation 
behavior  two specimens were sub jec t ed  t o  a 3 and 33 hours  hold pe r iod ,  r e spec t ive ly ,  a t  
ze ro  load a f t e r  loading  t o  1 percent  s t r a i n  and subsequent unloading had been accomplished. 
It i s  seen from Fig.  2 t h a t  t h e  s t r e s s - s t r a i n  curves  be fo re  and a f t e r  t he  rest per iod a r e  
wi th in  t h e  normal s c a t t e r  of t h e  test r e s u l t s .  This  can be e a s i l y  a sce r t a ined  by compar- 
i n g  t h e  responses  of t h e  two specimens and t h e  r e s u l t s  of F igs .  1 and 2 .  
The tes t s  show t h e r e f o r e  t h a t  recovery and aging are i n s i g n i f i c a n t  f o r  t h i s  ma te r i a l  
f o r  times less than 33 hours  a t  815OC. This r e s u l t  w a s  very s u r p r i s i n g  t o  t h e  au tho r s .  
Modeling. The V i s c o p l a s t i c i t y  Theory Based on Overstress (VBO) 
The model and i t s  material func t ions  
In  t h e  p re sen t  v e r s i o n  of VBO recovery and aging are no t  included.  In  t h e  un iax ia l  
s t a t e  of stress t h e  theory c o n s i s t s  of two coupled,  nonl inear .  o rd inary  d i f f e r e n t i a l  
equat ions  which con ta in  two p o s i t i v e ,  decreas ing  m a t e r i a l  func t ions ;  t he  v i s c o s i t y  func- 
t i o n  k c o n t r o l s  t h e  ra te  dependence and t h e  shape func t ion  J I  i n f luences  the  shape of t h e  
knee of t h e  s t r e s s - s t r a i n  curve.  In  add i t ion ,  t h e  e l a s t i c  modulus E ,  t h e  tangent  modulus 
E t  and t h e  asymptot ic  va lue  A of t h e  equi l ibr ium stress g e n t e r  i n t o  t h e  theory .  The 
equ i l ib r ium stress is a t t a i n e d  i n  t h e  l i m i t  as rates approach zero .  The d i f f e r e n c e  
a - g = x  between t h e  stress 0 and g i s  c a l l e d  t h e  o v e r s t r e s s  and k and J I  a r e  only func t ions  
of x. The equa t ions  are 
d c / d t  = dcel /dt  + dcin/dt = da/d t /E  + (a-g) / (Ek[x])  (1) 
where square  b racke t s  fo l lowing  a symbol denote  " func t ion  of ."  Under cons t an t  s t r a i n  
ra te  loading ,  t h e  system of equa t ions  permi ts  asymptot ic  s o l u t i o n s  given by 
{da/dc} = {dg/dEj = Et 
{XI  = (E-Et) k [ { x I ] d ~ / d t  
{g-EtE) = A dE/dt /  IdEin/dt I 
I where b races  denote  asymptot ic  va lues .  Details of t h e  theory are found i n  [1 ,2 ] .  
The asymptotic s o l u t i o n s  are a l g e b r a i c  equat ions  and are used i n  i d e n t i f y i n g  the  
material cons t an t s  and func t ions .  The procedure e x p l o i t s  (4)  t o  compute t h e  stress 
2 
-3- 
l e v e l  d i f f e r e n c e s  a t  d i f f e r e n t  s t r a i n  rates so t h a t  t h e  d a t a  of Table 1 can be used. 
D e t a i l s  can be found i n  [ 3 ] .  
l e a s t  square  a n a l y s i s  i s  performed t o  determine t h e  c o n s t a n t s  of t h e  shape func t ion ,  
The m a t e r i a l  func t ions  and c o n s t a n t s  obtained by t h i s  method a r e  given i n  Table 2 .  
The g-curve i s  obtained by e x t r a p o l a t i o n  and a nonl inear  
P r e d i c t i o n s  of t h e  theory 
With t h e  material c o n s t a n t s  determined from s t r a i n  r a t e  change tests, t h e  theory 
was app l i ed  t o  p r e d i c t  t h e  outcome of r e l a x a t i o n  and creep  t e s t s .  F igure  3 shows the  
resul ts  of a s t r a i n  r a t e  change test followed by r e l a x a t i o n  tests of 1024 s dura t ion .  
The t r i a n g l e s  r e p r e s e n t  experimental  r e s u l t s ;  t h e  cont inuous l i n e ,  t h e  p r e d i c t i o n s  of 
t h e  theory  as obta ined  by i n t e g r a t i n g  (1) and (2)  numer ica l ly  us ing  t h e  IMSL rou t ine  
DGEAR. The p r e d i c t i o n  f o r  incremental  c reep  tests is dep ic t ed  i n  Fig. 4. The creep 
pe r iods  l a s t  700 s except a t  t h e  h ighes t  stress l e v e l  where c reep  was t e m i n a t e d  a f t e r  
300 s. The experimental  r e s u l t s  a r e  aga in  given as t r i a n g l e s .  
The discrepancy between p r e d i c t i o n s  and experiment i n  F ig .  3 i s  mainly due t o  an 
ove rp red ic t ion  of r e l a x a t i o n  i n  t h e  f i r s t  r e l a x a t i o n  per iod  a a ' .  
f o r  t h i s  d i f f e r e n c e  and t h e  t h e o r e t i c a l  curves  are t r a n s l a t e d  upwards so t h a t  they 
co inc ide  a t  t h e  end of t h e  f i r s t  r e l a x a t i o n  pe r iod ,  t h e  subsequent p r e d i c t i o n s  a r e  very 
reasonable .  A s i m i l a r  observa t ion  holds f o r  t he  stress vs time r e l a x a t i o n  c u m e s .  
If allowance i s  made 
It is  seen t h a t  bo th  t h e  theory  and t h e  experiment do no t  show creep  a t  s t r e s s  
levels a and b in Fig.  4. 
creep  s t r a i n  bu t  o v e r p r e d i c t s  i t  a t  stress l e v e l s  e and f .  
A t  t h e  next two stress l e v e l s  t h e  theory  unde rp red ic t s  t h e  
The f i t t i n g  of t h e  m a t e r i a l  f u n c t i o n s  was only done once on t h e  b a s i s  of t h e  s t r a i n  
r a t e  change test r e s u l t s ,  
t h e  theory  o v e r p r e d i c t s  a t  high i n e l a s t i c  s t r a i n  rates. Since underpredic t ion  is 
observed a t  t h e  small stress l e v e l s  i n  Fig.  4, it i s  reasonable  t o  assume t h a t  an 
op t imiza t ion  of t h e  m a t e r i a l  f u n c t i o n s  i s  p o s s i b l e  so as t o  improve t h e  p r e d i c t i o n s ,  
This op t imiza t ion  and o t h e r  tests a r e  planned f o r  t h e  f u t u r e .  
Both t h e  c reep  and t h e  r e l a x a t i o n  experiments  s u g g e s t  t h a t  
Ac kn owled p e n  t 
This i n v e s t i g a t i o n  was made p o s s i b l e  by a g r a n t  from AVCO Lycming.  D r .  S a i t  &soy 
provided l i a i s o n  and h e l p f u l  sugges t ions .  
References 
[ l ]  Krempl, E., J. J. McMahon and D. Yao, " V i s c o p l a s t i c i t y  Theory Based on Overstress 
wi th  a D i f f e r e n t i a l  Growth Law f o r  t h e  Equi l ibr ium Stress," i n  p r i n t  Mechanics of 
Materials. 
[ 2 ]  Yao, D., and E. Krempl, ' V i s c o p l a s t i c i t y  Based on Overstress, The P red ic t ion  of 
Monotonic and Cycl ic  P ropor t iona l  and Nonproportional Loading Pa ths  of an 
Aluminum Alloy,"  Int .  J. of P Z a s t i c i t y ,  1, 259-274 (1985). 
(31 Krempl, E., H. Lu and D. Yao, Rensselaer  Poly technic  I n s t i t u t e  Report KMZ 85-4, 
November 1985. 
3 
TABLE 1 
Averaged Flow Stress Levels and Stress Level Differences 
for ClOl at 815OC 
Strain Rate Stress Level Stress Level 
-1 at 1.2% Difference S 
MPa MPa 
966 
834 
721 
638.4 
554.4 
132 
0 
-113 
-195.6 
-279.6 
TABLE 2 
THE DETERMINED MATERIAL CONSTANTS AND FUNCTIONS 
.- 
Material Constants Shape Modulus 
E = 156620 MPa $[XI = c1 + (C2-cl)exp(-c31xl) 
Et = 267 MPa 
A = 421.7 MPa 
Viscosity Function 
cl, c2 and x in units of MPa; 
c =62500, c =150000 
3 
1 2 
c in units of MP~-'; - - 
-k 
k[x] = kl(l +e) 
2 
kl = 3150000 s 
k2 = 186 MPa 
kg = 9.96 
c =0.0387 3 
4 
E O i  
I :20 
0.90 
S 
T 
R 
E 
S s 0.60  
M 
P 
A 
0 . 3 0  
0 .00  
A 
0 .10 0.15 0.20 0.05 
E-0 1 
0 .00  
AXIAL STRAIN. 
I 
Figure 1. Stress-strain curves of constant strain-rate test 
(solid lines) and strain-rate change tests (triangles). 
Filled triangle denotes the approximate stress level at 
lxlO-' per second. 
I I 
800 
; IOo 
ul 
3 
5 
yI 400 
200 
' 
c rrl 
I I I t 
STRAW X 
0.4 0.8 L2 LI 2.0 
Figure 3. Prediction of strain-rate change and relaxation 
tests. Triangles represent experimental results. Strain- 
rate history: 
cc' 0. 
'time - 1024 s. 
oa lo-'. aa' 0. a'b bb' 0 .  b'c lo-', 
Relaxation All values are given In units of s-l .  
E 03 
1.20 
0.90 
S 
T 
R 
E 
S s 0.60 
M 
P 
A 
0 .30 
0.0E 
E 
- 4  
1x10 
Figure 2. Stress-strain curves of specimens #E(-) and #lo(*). 
Hold periods of 3 and 33 hrs at A. 
eoo 
STRAIN X 
0 
Figure 4. Incremental creep tests at various stress levels. 
Between creep periods the stress rate is 2.8 MF'a/s. 
no creep is observed at the two stress levels below 300 MPa. 
Creep periods are 700 s except at level 4. 
experimental points. 
Almost 
Circles represent 


The viscoplasticity theory based on overstress applied to the modeling of a nickel base superalloy at 815 C

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