Reformation processes effect on stress time-to- fracture behavior of solid

Hsiao, C. C.

Moghe, S. R.

Von Schmeling, H. H. K.

English

NASA Technical Reports Server

Technical Report NO. 105 
THEORETICAL CONSIDERATION OF TKE INFLUENCE O F  REYORMING 
PROCESSES ON THE FRACTURE STRENGTH OF SOLIDS 
H. H. Kausch von Schmeling, S. R. Moghe and C. C. Hsiao 
Un ive r s i ty  of Minnesota, Minneapolis, Minnesota 
March 1966 
https://ntrs.nasa.gov/search.jsp?R=19660016830 2020-03-16T19:56:10+00:00Z
Theore t i ca l  Consideration of the Inf luence  of Reforming 
Processes  on the Frac ture  S t r eng th  of Sol ids"  
H. H. Kausch von Schmeling, S. R. Moghe and C. C. Hsiao 
Un ive r s i ty  of Minnesota, Minneapolis, m n n e s ~ t a  
The macroscopic s t r e n g t h  of an o r i en ted  medium i s  
i n v e s t i g a t e d  under the assumption tha t  processes  of 
breaking  and reforming of microscopic components i n  the 
medium may occur. The d i f f e r e n t i a l  equat ion governing 
the t i m e  dependent v a r i a t i o n  of the number of load  
ca r ry ing  components i s  considered. Emphasis i s  placed 
on the  s tudy  of the e f f e c t  of the reformation processes  
on the stress t ime-to-fracture  behavior.  It i s  suggested 
tha t  under small loads  these processes  may be r e spons ib l e  
f o r  the f r e q u e n t l y  observed phenomena of the d e v i a t i o n  
rrom l i n e a r i t y  of the s t r e n g t h  and logari thm of t i m e -  
t o - f r a c t u r e  r e l a t i o n s h i p .  An at tempt  i s  a l s o  made t o  
compare t h e o r e t i c a l  r e s u l t s  with some repor ted  experi- 
mental data. 
*Supported i n  part  by t h e  National Aeronautics and Space 
Adminis t ra t ion 
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Recently the mechanical p r o p e r t i e s  of o r i en ted  s o l i d s  
have been c a r e f u l l y  analyzed. The mathematical model used 
i s  a system of o r i en ted  l i n e a r  elements embedded i n  m arbi- 
t r a r y  domain. The time-dependent s t r e n g t h  p r o p e r t i e s  were 
a l s o  s tud ied  w i t h  good success . However, a complete 
understanding of f r a c t u r e  behavior under e i ther  la rge  o r  
small simple t e n s i l e  stress i s  no t  y e t  obtained. The p resen t  
paper i s  an at%empt toward t h i s  end. Emphasis i s  placed upon 
the i n t e r p r e t a t i o n  of the inf luence  of molecular reformation 
processes  on s t r eng th .  
For s i m p l i c i t y  we consider  a p e r f e c t l y  or ien ted  s o l i d  
composed of a system of l i n e a r  microscopic molecular elements. 
L e t  f be the r e l a t i v e  number of unbroken elements. S t r e s s  
concent ra t ion  i n  any area sha l l  be neglected.  I n  genera l  
f w i l l  be a f f e c t e d  by processes  of reformation and breakage 
of molecular f o r c e s  r ep resen t ing  the k i n e t i c  na tu re  of the 
system and thus  l ead ing  t o  the fo l lowing  equation: 
1-6 
(1 - f)Kr - f% df d t  7 -
where Kr and $ are the  r a t e s  of reformation and breakage 
r e s p e c t i v e l y ,  
o f  breakage may be expressed i n  t h e  fo l lowing  manner: 
According t o  the r e a c t i o n  ra te  theory  the ra te  
where 
are material cons tan ts ,  F i s  the f o r c e  appl ied  on each 
element, and k and T are Boltzmann's cons tan t  and abso lu te  
temperature r e spec t ive ly .  Similarly,  the rate of reformation 
i s  given by 
i s  the frequency of molecular motion, Ub  and B, 
If the system i s  subjec ted  t o  a cons tan t  axial  t e n s i o n  o0 
. 
3 
t hen  F i s  p ropor t iona l  t o  ao/f. 
q ( t )  may be i n t e r p r e t e d  a s  the average stress on t h e  un- 
broken elements. Using + ( t )  and equat ions (1) - ( 3 )  one 
can ob ta in  the fo l lowing  expression: 
Let oo/f = $(t)  where 
where % and 0 
o as w e l l  as o t h e r  q u a n t i t i e s  independent of s t r e s s .  
Here f3 and 'y are p o s i t i v e  constants .  Upon i n t e g r a t i o n  
we ob ta in  the  time-dependent s t r e n g t h  behavior f o r  any 
given cons tan t  stress o0:  
are as soc ia t ed  r e s p e c t i v e l y  w i t h  (la and r 
t. 
where q0 1 s  the molecular stress a t  t i m e  t = 0. ?ho i s  
equal  t o  o0 i f  w e  assume t h a t  a l l  t h e  elements are un- 
broken i i d t i a l l y .  
breakage, It i s  conceivable t h a t  f r a c t u r e  will occur f o r  
any system i f  7(/ reaches a c e r t a i n  l i m i t i n g  value 9 For 
an e l a s t i c  system t h i s  i s  a l s o  equiva len t  to f r a c t u r e  a t  
some l i m i t i n g  value of s t r a i n .  Other c r i t e r i a  can  be based 
on maximum energy (mechanical) e i t h e r  f o r  an  i n d i v i d u a l  
element or  f o r  the e n t i r e  system. Reluctance t o  assume 
any of t h e s e  o r  similar c r i t e r i a  w i l l  l e ave  us  no chodce 
b u t  t o  use the condi t ion  f = 0 o r  qm = 00 a t  fracture which 
seems t o  be the most n a t u r a l  one t o  assume. 
?hh i s  the  molecular stress a t  t i m e  of 
b m" 
For f u r t h e r  cons idera t ion  l e t  u s  use the fol lowing 
abbrev ia t ion  : 
R 
- (-- 9 l)&xp[-(y + p)+ ]  = @ ( @ )  
b o 0 
The func t ion  @(ll/) has a maximum a t  9 given by 
g 
4 
The corresponding maximum value @ i s :  g 
For values  of f$ e 1  a f a i r l y  good approximation towards 
the s o l u t i o n  of (5) can b e  obtained. I n  t h i s  case t h e  
corresponding high and moderate appl ied  s t r e s s e s  are 
considered. The e f f e c t  o f  reformation i s  seen to be 
very small. The smallness o f  @ i n  t h i s  region can be 
a s s e r t e d  by (8 )  s i n c e  Boo i s  an appropr i a t e  l a r g e  value.  
Then t o  a f irst  approximation (1 - $)-I 
reg ion  can be defined by @ < where can be chosen 
according to t h e  requi red  accuracy. For = 0.1, say,  
t h e  accuracy of the approximation w i l l  be b e t t e r  than 
+ - 1%. 
r e s u l t s  can be obtained i n  eva lua t ing  (5) :  
g,-. 
1 + @. This  
g C 
Therefore,  us ing  th i s  approximation f a i r l y  good 
I n t e g r a t i o n  of ( 9 )  gives :  
Fu r the r  approximation us ing  the follow 
E i ( - z )  f o r  l a r g e  values  o f  z t h a t  
ng p r o p e r t i e s  of 
e - Z  1 -Ei(-z) t -(1 z - -1 z (11) 
and omi t t ing  h igher  order  b u t  s m a l l  t e r m s  conta in ing  
reduces equat ion ( 1 0 )  to: 
5 
which i s  a s o l u t i o n  of equation (5 )  f o r  high and moderate 
app l i ed  s t r e s s e s .  Hcwever, i f  approaches qb then lnS;8tb 
goes t o  minus i n f i n i t y  and (12) can be developed i n t o  the 
fol lowing form: 
0 
When t h e  appl ied  s t r e s s e s  are s m a l l ,  the  corresponding 
condi t ion  i s  tha t  @(@) > #c, then i t  i s  found necessary t o  
eva lua te  (5)  by numerical  methods. 
i s  chosen so  t h a t  @(q > $'a) < @c i s  sat isf ied,  then (5 )  'a 
can be separated i n t o  two i n t e g r a l s :  
I n  the case t ha t  i f  a 
T h e  f i r s t  i n t e g r a l  on t h e  r i g h t  hand s ide can be i n t e g r a t e d  
numerically g iv ing  U b t a ,  and t h e  second one may be evaluated 
using ( 1 2 )  g iv ing  nb( tb  - t 1. a 
Thus t h e  complete time-dependent f r a c t u r e  behavior 
under simple t ens ion  may be roughly divided i n t o  t h r e e  
por t ions .  The  lower po r t ion  i s  for t h e  case when oo i s  
comparable t o  qb. I n  t h i s  case t h e  time-to-break tb i s  
extremely s h o r t  and t h e  curve i s  nea r ly  v e r t i c a l  as i t  
should be i n  view of the assumptions made. The middle 
p o r t i o n  i s  the major p a r t  of t h e  curve which corresponds 
t o  moderate and r e l a t i v e l y  high appl ied  stresses. Within 
t h i s  reg ion  t h e  logari thm of l i f e t i m e  i s  almost l i n e a r l y  
r e l a t e d  t o  the appl ied  stress. T h i s  can be v e r i f i e d  by 
d i f f e r e n t i a t i n g  equat ion ( 1 2 )  w i t h  r e spec t  t o  Bo- as fol lows:  
U 
d ( lhbtb 1 exp ( -Boo 1 1 
~-= Ei(-(300)g -'-  
0 0 Boo 
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Since  1/Bo0 i s  small i n  t h i s  region and v a r i e s  only slowly 
the s lope  of t h e  curve i s  almost cons tan t .  
The upper po r t ion  r e f l e c t s  the importance of the re- 
formation mechanism i n  the  molecular motion. 
corresponds t o  the a p p l i c a t i o n  of only mall s t r e s s e s .  A s  
a r e s u l t  the r o l e  of the molecular reformation processes  
becomes i n c r e a s i n g l y  s i g n i f i c a n t .  It c o n t r i b u t e s  t o  the 
i n c r e a s e  i n  t i m e  r e q u i r e d  f o r  rup tu r ing  a l l  t h e  elements, 
and t h u s  the curve d e v i a t e s  from t h e  l i n e a r  r e l a t i o n s h i p  
between l n t b  and oo as v e r i f i e d  by (14) .  
small stresses appl ied ,  one may expect t h e  t ime-to-break 
to be i n f i n i t y .  It can b e  seen by r e f e r r i n g  t o  (5 ) ,  ( 6 )  
and (8)  t h a t  tb goes t o  i n f i n i t y  as @ approaches 1. 
condi t ion  f o r  i n f i n i t e  l ifetime i s  $I L 1. T h i s  condi t ion 
can be sat isf ied by a range of  small values  of u 
Corresponding t o  $I = 1 the re  e x i s t s  a stress ae which i s  
the  l a r g e s t  of these values  obta inable  from the fol lowing 
expression:  
T h i s  p a r t  
I n  f a c t ,  f o r  very 
So the 
g 
0' 
g 
At the  stress l e v e l  of oe, t he  t ime-to-break tb approaches 
i n f i n i t y .  
For the sake of i l l u s t r a t i n g  q u a l i t a t i v e l y  the  com- 
p l e t e  t ime dependent s t r e n g t h  behavior,  Fig.  1 i s  p l o t t e d  
wi th  assumed values  of 2 f o r  @ae and of 30 f o r  Bo,. 
m u s t  be  pointed out,  however, t h a t  the l i fe t ime tb a t  any 
stress Poo < 25 w i l l  show no change i f  a l a r g e r  maximum 
stress i s  aPsumed i n s t e a d  of @om = 30. So t h e  l i fe t ime 
i n  the major s t r e s s  reg ion  i s  not  a f f e c t e d  by t h e  assumed 
values  of the  maximum stress. T h i s  r e l i e v e s  most of t h e  
d i f f i c u l t i e s  t ha t  had been apparent i n  de f in ing  om. 
r e s u l t s ,  a t tempts  were made t o  f i t  the t h e o r e t i c a l  curves 
w i t h  some a v a i l a b l e  experimental  data. I n  curve f i t t i n g  
t h e  fol lowing experimental  q u a n t i t i e s  have proved t o  be 
It 
T n  o rde r  tG ShZV! t h e  v a l i d i t y  Gf t h e  z m z e p t  and 
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u s e f u l :  
(1) The s lope  b = -d( log tb) /da of t he  l i n e a r  p o r t i o n  
of t h e  experimental  curve.  
( 2 )  The stress value (J where a dev ia t ion  from l i n e a r -  
i t y  between l o g  tb and a beconies s i g n i f i c a n t .  
(3)  The stress va lue  oe where l o g  tb approaches i n f i n i t y .  
A r e l a t i o n s h i p  between these experimental  va lues  and 
the t h e o r e t i c a l  parameters yJ fa and a /a  
by the condi t ion  tha t  a t  some stress axthe s lope  b of the  
experimental  curve and t h e  first d e r i v a t i v e  of the t h e o r e t i c a l  
f u n c t i o n  a r e  equal,  i . e .  
m 
pan b e  e s t a b l i s h e d  r b 
From which i s  determined: 
1 B = b lnlO - - 
Ox 
If  the experimental  and t h e o r e t i c a l  curves  correspond i n  
t h e i r  approach t o  i n f i n i t e  l i fe t ime t, a t  LT , equat ion  
(15) n u s t  be sa t i s f ied .  T h l s  means t ha t  
U e 
(18) 
r 
b 
R 
- I n ( @  + Y)ae + 1 n r  - (Y + B)oe = o 
A s  f o r  the stress value an i t  i s  found t o  be convenient t o  
assume tha t  an corresponds with $I 
r e l a t i o n :  
= 0.08 l ead ing  t o  a 
g 
n 
-1n(p + y)on + I n  - (8 + y)oe  = - 2.5 
Because t h e  va lues  f o r  an and ae cannot be c l e a r l y  d e f i n e d ,  
t h e  method of l e a s t  squares  was used  f o r  ob ta in ing  the  best  
n 
a 0  
f i t .  
From (18) and ( 1 9 )  l n g  
b 
and y can b e  obtained as 
fol lows:  
8 
s2 2.5-111 ( ) 2.5-1n(an/oe) 
+ I n  O u n’e - 1  
l n r =  1 + 
b U C J  n ’ e - 1  
Y 
A s  y i s  considered 
s t r i c t e d  i n  (21).  
p, y and ar/’r$, can 
2 .5 -h  ( on/oe ) 
- 8  - 
e On-O 
p o s i t i v e ,  t h e  va lue  of un/ae i s  re- 
By ( 1 7 ) ,  ( 2 0 ) ,  and (21 )  t h e  parameters  
be ca l cu la t ed  and thus  t h e  func t ion  
f o r  tb determined on t h e  basis of t h e  known experimental  
in format ion .  T r i a l  runs  were c a r r i e d  out with a CDC 1604 
computer. Fig. 2 shows the f i t t e d  curve t o g e t h e r  wi th  
the experimental  d a t a  f o r  organic  g l a s s .  S ince  t h e  model 
used i s  a f a i r l y  gene ra l  one, e f f o r t s  have a l s o  been made 
7 
t o  f i t  a curve f o r  aluminum t e s t e d  a t  a n  e l eva ted  temperature ,  7 
Fo r  organic  g l a s s  a t  7 O o C ,  as shown i n  Fig.  2 t h e  
fo l lowing  set  of parameters was used t o  c o n s t r u c t  t h e  b e s t  
r e p r e s e n t a t i o n :  
A (1) 8 = 2.27 X 10  -2 cm 2 /kp, y =I 0,2 X 10 -2 em 2 /kp, 
( 2 )  fir/fib = 1 7 1 8 ,  Bon = 6.5, Boe = 4,5, and ( 5 )  lnRb = 
1 .  T I  h r n  u / n 1  = -13.24, 
,l’% - 
For alunTnum a t  40OoC.  a s  p l o t t e d  i n  Fig. 3, t h e  
fo l lowing  s e t  of parameters proved to be bes t :  
(1) @ = 7.0 X 1 0  - 2  ern 2 /kp, y = 0.2 X 1 0  -2 cm 2 /kp, 
(2) = 16.5, 80 = 2.8 ,  Boe = 1.398, and ( 3 )  lnDb = n 
-13.72. 
From t h e s e  r e s u l t s  the e f f e c t  of re format ion  processes  
i n  t h e  molecular motion i s  c l e a r l y  brought  out.  
be s t a t e d  aga in  t h a t  t h e  dev ia t ion  of t h e  l i n e a r l y  r e l a t e d  
l n t b  and o0 f o r  small s t r e s s e s  may be  a t t r i b u t e d  t o  t h i s  
reformation mechanism. 
It may 
1. 
2. 
3 .  
40 
5. 
60 
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REFERENCES 
C. C. Hsiao, J. Appl. Phys, 30, 1492-97 (1959)  
0. Lo Anderson a. D, A. S t u a r t ,  Ind. Eng. Chem. 46, 154 
(1954) 
-
F. Bueche, J. Appl. Phys, 28, 784 (1957) 
B, D. Coleman a. D, W. Marquardt, J. Appl. Phys. 29, 
109 (1958) 
-
n -1 
3.  N. .murkov, ~nys. Cnem. 215, 18s ( i 9 6 o j  
C. C. Hsiao, S.  R. Moghe a H. H. Kausch von Schmeling 
( t o  be publ i shed)  
S .  N. Zhurkov, "Kinet ic  Concept on t h e  S t r eng th  of 
Sol ids" ,  presented a t  the  I n t e r n a t i o n a l  Conference 
on Frac ture ,  Sendai,  Japan, Sept.  12-17, 1965, w i l l  
appear i n  t h e  Proceedings and I n t e r n a t i o n a l  Journa l  
of F rac tu re  Mechanics. 
-
It i s  a p l easu re  t o  acknowledge t h e  a s s i s t a n c e  i n  
drawing t h e  f i g u r e s  by T. C. Chang and typing of t h e  
manuscript  by Penelope Boyer. 
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Theoretical Consideration of the Influence of Reforming Processes on the Fracture Strength of Solids Technical Report No. 105

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