Approximation of residual thermal stresses in flat sheet molded from chemical hardening material

Levitsky, M.

Shaffer, B. W.

English

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https://ntrs.nasa.gov/search.jsp?R=19710008097 2020-03-12T00:21:47+00:00Z
Repor t  No. S-70-5 
NEW YORK U N L W R S I T Y  
Xew York, New York 
RESIDUAL STRESSES DEVELOPED I N  MOLDING A FLAT SHEET 
OF A CHEPIICALLY-HARDEMIWG RELATIVELY INCOMPRESSIBLE PIATERIAL 
Myron Levi tsky 
A s s i s t a n t  P ro fe s so r  
College of t h e  C i ty  of New York 
Assoc ia te  Research S c i e n t i s t  
New York U n i v e r s i t y  
and 
Bernard W. S h a f f e r  
P ro fe s so r  of Mechanical Engineer ing 
New York U n i v e r s i t y  
Prepared f o r  The Off ice  of U n i v e r s i t y  A f f a i r s  
Nat iona l  Aeronaut ics  and Space Adminis t ra t ion 
under Research Grant  NGL-33-016-067. 
September 1970 
mSIDUAL ST-SSES DEWLOPED I ; J  FJIOLDING A F L A T  SmET 
O F  A Cm\i i ICALLY-FEARDENING mLATIWLP I;';ICOb$ilPRESSIBm YjIATERIAE 
Myron Levitsky and Bernard W. Shaffer  
Abstract  
--
Residual thermal s t r e s s e s  i n  a  f l a t  shee t  molded from a 
chemically hardening mate r i a l  a r e  approximated by an expression 
having a  simple closed form represen ta t ion  whenever the  shear  
modulus i s  much smal ler  than the  bulk modulus i n  the f u l l y  
hardened m a t e r i a l ,  The r e s u l t  i s  used t o  evalua te  the hard- 
ening r a t e  which y i e l d s  maximum r e s i d u a l  s t r e s s e s  i n  the  
i n t e r i o r  of a  s h e e t .  A comparison between numerical r e s u l t s  
obtained from the  exact  and approximate s o l u t i o n s  t o  the  
problem show t h a t  the more e a s i l y  obtained l a t t e r  r e s u l t s  
r ep resen t  a  conservat ive es t imate  of the  exac t  s t r e s s  
d i s t r i b u t i o n .  
In t roduct ion  
A s  p a r t  of a c u r r e n t  i n v e s t i g a t i o n  devoted t o  t h e  
determinat ion of r e s i d u a l  thermal s t r e s s e s  i n  ma te r i a l  under- 
going an exothermic r e a c t i o n ,  [1'21* i t  was found t h a t  during 
the molding process  the s o l e  non-vanishing r e s i d u a l  s t r e s s  
component i s  p a r a l l e l  t o  the  wal l  su r face .  Unfortunately,  the 
mathematical expression f o r  t h i s  s t r e s s  component took the form 
of an  i n f i n i t e  s e r i e s  i n  which the  c o e f f i c i e n t s  themselves were 
i n f i n i t e  s e r i e s .  Although a  s o l u t i o n  can be r e a d i l y  obtained 
i n  numerical form, the  expression i s  not  amenable t o  f u r t h e r  
i n v e s t i g a t i o n  i n  an a n a l y t i c a l  form. 
It i s  observed t h a t  a  u s e f u l  mathematical s i m p l i f i c a t i o n  
i s  o f t en  made i n  s tudying the behavior of time dependent ma te r i a l s  
by considering the ma te r i a l  t o  be incompressible,  [334959 1 
Perhaps the same advantage can be obtained in the  investigation 
of the molding p rocess ,  However, instead of s t a r t i n g  the  
Superscr ip t  numbers i n  sauared bra.ckets r e f e r  t o  the references 
l i s t e d  i n  the Bibliography. 
analysis under the assumption of an incompressible material 
a cornpasable result may be obtained by viewing the incompressible 
material as the limft of a material whose shear modulus is 
much smaller  than i t s  bulk modulus, i n  which case use can be 
made of the previously derived s o l u t i o n .  Furthermcre, i t  w i l l  
be shown t h a t  the  expression developed f o r  the  r e s i d u a l  s t r e s s e s  
i n  the wal l  may then be approximated by an equat ion which can 
be w r i t t e n  i n  closed form i n  terms of elementary func t ions .  
I t  i s  noted, of course,  t h a t  i n  ma te r i a l s  f o r  which the 
shear  modulus i s  much smal ler  than the bulk modulus, Poissonls  
r a t i o  approaches one h a l f .  The l i m i t i n g  value i s  the  same a s  
Po i s son t s  r a t i o  i n  an incompressible m a t e r i a l .  
Reduction of the Residual S t r e s s  t o  Closed Form 
It has been shown [ * I  t h a t  when a ma te r i a l  i s  poured or  
i n j e c t e d  between two p a r a l l e l  planes which comprise the mold, 
and hardens a s  the r e s u l t  of an exothermic chemical r e a c t i o n  
while the mold su r faces  a r e  maintained a t  the  i n i t i a l  ambient 
temperature,  the  r e s i d u a l  s t r e s s  component p a r a l l e l  t o  the  wa l l  
su r face  i s  given by the expression 
where the parameter q was defined 
or as a function of PoissonDs Ratio p measured in the fully 
hardened s t a t e  
The bulk modulus i s  K, Go i s  t he  s h e a r  modulus of' t h e  i ' u l l y  
hardened m a t e r i a l ,  and a t  i s  t h e  c o e f f i c i e n t  of the rmal  
expansion.  The term 3 * i s  c a l l e d  t h e  r e a c t i o a  r a t e  paramete r  
and t h e  f a c t o r  H/pc measures t h e  maximum p c s s i b l e  r i s e  i n  
temperature  of t h e  r e a c t i n g  m a t e r i a l  i n  terms of t he  vo lumet r ic  
h e a t  of r e a c t i o n  H , t h e  m a t e r i a l  d e n s i t y  e , and t h e  s p e c i f i c  
h e a t  c . The x coord ina t e  i s  measured from one of t h e  s u r -  
f a c e s  of t h e  s h e e t  whose t h i c k n e s s  has  been normalized t o  u n i t y .  
The r a t i o  of s h e a r  modulus t o  bu lk  modulus may be ex-  
p ressed  [61 
When Go i s  smal l  compared t o  K , P o i s s o n l s  r a t i o  t ends  
towards one-ha l f ,  and t h e  parameter  approaches z e r c .  Under 
t h e s e  c i rcumstances ,  t h e  i n t e r n a l  s e r i e s  of Eaua t ion  (1) may 
be s i m p l i f i e d  by r e t a i n i n g  on ly  t h e  f i r s t  term, and n e g l e c t i n g  
h ighe r  powers of q , s o  t h a t  exp re s s ion  f o r  r e s i d u a l  s t r e s s e s  
can be w r i t t e n  
Q) 2  - nn s i n  n ~ x  y - ~ K ~ I ( H / ~ C ) ( $ )  Z, - n = 1 , 3 , 5 . .  . (5) 
n  ( n n ) 2 [ l + ( ~ / 4 ) 2 1  
The q u a n t i t i e s  ( l+p) / ( l - -w)  and q a r e  shown by Equatj-ons 
( 2 )  and ( 3 )  t o  be e x p r e s s i b l e  i n  terms of Go and K , s o  t h a t  
Equation ( 5 )  may be w r i t t e n  
Under the  cond i t i on  t h a t  Go/K<cl,  
and 
M u l t i p l i c a t i o n  of t h e  preced ing  q u a n t i t i e s ,  and r e t e n t i c n  of 
the  l i n e a r  term i n  t h e  r a t i o  G,/K shows t h a t  Equat ion ( 6 )  
can be expressed a s  
0)  
rg2  s i n  nnx 
= 1 2  G ~ ~ ' ( H / ~ C )  E L - 2 g2 P n  n n [ ~ ~ + ( n r r ) ~ ]  (nn)  2 [ 3  2 + ( n ~ ) ~ ]  (9) 
Elsewhere i t  has  been shown from a  s t u d y  of t he  Four i e r  
S e r i e s  r e p r e s e n t a t i o n  of' some hyperbol ic  f u n c t i o n s ,  [7y81  t h a t  
Q) 
C s i n  n  Q - --[I 31 - cash [ a ( n P - Q )  I] 0 c Q < a 2 2  cosh an/2 - - 
n n ( a + n )  4a2 
n  = 1,3 ,5 .  . .  . 
and 
CO 
1 1 
- 
a n  -1 
7 2 2 7  n  n  ( a  tn ) a  tanh - 2 J  ' n = l j 3 , 5 . .  . . 
Comparison of the  preced ing  equa t ions  wi th  Enuation (9) indicates 
t h a t  the two series which, appear i n  Eaua t i on  (9) can be w r i t t e n  
in terms of hyperbolle functions, The ~esult GT t h i s  s u b s t i -  
t u t i o n  pe rmi t s  us t o  put the exp re s s ion  for r e s i d u a l  s t r e s s e s  
i n t o  t h e  form 
The l a t t e r  expression i s  appl icable  a s  long a s  the  r a t i o  G,/K 
i s  much l e s s  than u n i t y .  
Discussion of the  Resul t s  
- - 
One ins tance  of the  advantage of the approximate s o l u t i o n  
i n  closed form, obtained when the  r a t i o  of shear  t o  bulk 
modulus i s  much l e s s  than u n i t y ,  a r i s e s  when the a p p l i c a t i o n  
r e q u i r e s  f u r t h e r  mathematical a n a l y s i s ,  a t a sk  no t  e a s i l y  
accomplished with the  complete s o l u t i o n .  
It was observed i n  the  d i scuss ion  of the  numerical r e s u l t s  
por t ion  of a previous paper [ 21  t h a t  the r e s i d u a l  s t r e s s  com- 
ponent p a r a l l e l  t o  the  wa l l  sur face  of the m a t e r i a l  being 
molded reaches a maximum t e n s i l e  value a t  the cen te r  of the 
shee t  a t  a r a t e  of hardening which i s  a s  y e t  undetermined but 
can be measured by the  r e a c t i o n  r a t e  parameter J * . Such a 
s t a t e  of s t r e s s  may e i t h e r  be d e s i r a b l e  or undes i rable ,  depend- 
ing  upon the  u l t ima te  use of the  s l a b  being molded. The hard- 
ening r a t e  which produces the l a r g e s t  t e n s i l e  s t r e s s e s  can of 
course be evaluated from the exact  s o l u t i o n  numerically.  I t  
i s  much s impler ,  however, t o  est imate the value of the  r e a c t i o n  
r a t e  parameter a t  which the  maximum t e n s i l e  s t r e s s  i n  the  i n t e r i c r  
from the newly obtained approximate s o l u t i o n ,  
A t  t h e  c e n t e r  plane of the sheet x I/2 , where the  
maxlmuxn t ens - i l e  s"i-,ress ks known to occur ,  Equation (12) shows 
that the resldual stress component parallel to the mold  
surfaces is given by the expression 
0 ' 2 -d d -' 3afG,(~/pc)L7 tanh - - sech 1 x.1/2 2 
I n  order  t o  determine the r e a c t i o n  r a t e  parameter 3 , which 
r e s u l t s  i n  the  maximum t e n s i l e  s t r e s s ,  take the d e r i v a t i v e  
of t h e  preceding expression with r e s p e c t  t o  3 , and s e t  the  
r e s u l t  equal  t o  zero  t o  f ind  
r 2  $ 1  2 $  1 -4 0 = ~ C ~ ~ G , ( H / ~ C ) ~ ; - ~  tanh + - sech - + s i n h  sech2 (14)  
-4 2 
The des i red  r e a c t i o n  r a t e  parameter r e q u i r e s  the s o l u t i o n  of 
the  t ranscendenta l  equat ion 
9 1 + $ s i n h  - - s i n h  = 0 
2 iJ 
It can be found numerically t h a t  3 approximately equal  t o  
5.5, or  $ = 30 s a t i s f i e s  t h e  previous equation f o r  4 , 
and r e p r e s e n t s  the r e a c t i o n  r a t e  parameter which y i e l d s  a  
maximum r e s i d u a l  t e n s i l e  s t r e s s  a t  the  center  p lane .  
It was previous ly  shown ['I t h a t  the  parameter d i s  
2 equal t o  QoL / l l H  , where Q, denotes the i n i t i a l  r a t e  of 
volumetric h e a t  genera t ion  i n  the  hardening mate r i a l ,  L i s  
the thickness  of the  wal l ,  K i s  the  thermal d i f f u s i v i t y  of  
the w a l l  m a t e r i a l ,  and H i s  the volumetric hea t  of r e a c t i o n  
Thus i t  i s  seen t h a t  the value of Q ' i s  determined not  only 
by the intrinsic speed of the reaction, b u t  also by the thermal 
properties and dimensions of the slab, Hence adjustment of 
the r e a c t i o n  r a t e  parameter i n  order  t o  alter the tensile 
residual thermal stress may be accomplished by changing either 
t h e  chemical constants, the t h e ~ m a l  constants, or t he  wall 
thickness s e p a r a t e l y  o r  i n  a s u i t a b l e  combinat icn .  
The accuracy  of Equat ion (12)  a s  an approximat ion t o  t h e  
e x a c t  s e r i e s  s o l u t i o n  of Equat ion (1) may be s een  by t h e  ccm- 
p a r i s o n  of Fig. 1, where s t r e s s  r ep re sen t ed  i n  d imens ion less  
form by t h e  v a r i a b l e  o / a t  ( ~ / p c ) K  i s  shown a s  a  f u n c t i o n  
of X/L , t h e  d imens ion less  coo rd ina t e  measured from one 
s u r f a c e  of t h e  s h e e t  f o r  two r e p r e s e n t a t i v e  va lues  of t h e  
r e a c t i o n  r a t e  paramete r .  It i s  observed t h a t  t h e  accuracy  
of t h e  c losed  form approximation dec rease s  a s  t h e  r a t i o  of 
s h e a r  t o  bu lk  modulus i n c r e a s e s ,  which i s  t o  be expec ted .  
The two s o l u t i o n s  a r e  i n  f a i r  q u a l i t a t i v e  agreement when 
Go/K i s  approximately  o n e - f i f t h .  It i s  a l s o  observed t h a t  
t h e  magnitude of t h e  s t r e s s  v a r i a b l e  determined by t h e  
approximate r e l a t i o n  of Equat ion (12 )  exceeds t h a t  g iven  by 
the  e x a c t  s o l u t i o n  of Equat ion ( l ) ,  s o  t h a t  t h e  c losed  form 
approximat ion t u r n s  ou t  t o  p rov ide  a  conse rva t ive  e s t i m a t e  
of t h e  r e s i d u a l  s t r e s s e s ,  
Conclusion 
I t  has  been shown t h a t  t h e  p r e v i o u s l y  der ived  exp re s s ion  
f o r  t h e  r e s i d u a l  s t r e s s  d i s t r i b u t i o n  which a r i s e s  i n  t h e  mold- 
i n g  of a  s l a b  from a n  exothermic chemica l ly  hardening m a t e r i a l  
may be approximated by a s imple  c losed  form r e s u l t  when t h e  
r a t i o  of shear t o  bu lk  modulus of t h e  m a t e r i a l  Go/K i s  s m a l l .  
For descriptive purposes such a material  has been called 
approximately incompressive because P o i s s o n k  Ra t io  1s t hen  
in the neighborhood of one-half, Such p r o p e r t i e s  are 
characteristic of many thermosetting plastics, 
The closed form is found $0 give approximate results 
that are easily determined and represent a conservative 
es t ima te  of the  a c t u a l  r e s i d u a l  s t r e s s  d i s t r i b u t i o n .  
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DIMENSIONLESS THICKNESS 
EXACT SOLUTION 
APPROXIMATE SOLUTION 
Fig. ?. 
A Comparison of =act  and Approximate S o P u t l ~ n s  
for the D i s t r i b u t i o n  s f  Residual Thermal Stress, 


Residual stresses developed in molding a flat sheet of a chemically-hardening relatively incompressible material

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