CERTAIN OF
PROBLEMS
VISCOELASTIC V.
V.
OF
STRENGTH
LONG-TERM
MEDIA
Moskvitin
UDC539.4
A limiting condition of long-term strength of the type of the nonlinear criterion of A. A. II'yushtn [i] is proposed. The criterion considered is a generalization of the well-known limiting condition of Bailey. Certain problems of using this criterion in viscoelastic media are discussed. I . We denote by
w=
co(t) the damage function (t is the time) for which the conditions (0) - - o;
~o (t,) : : I
(1)
a r e s a t i s f i e d . H e r e t , is the t i m e up to the o c c u r r e n c e of the l i m i t i n g s t a t e (the life) f o r a n a r b i t r a r y of v a r i a t i o n of t h e s t r e s s s t a t e with t i m e . We a s s e s s t h e s t r e s s s t a t e by a c e r t a i n e q u i v a l e n t s t r e s s a e ( t ) w h i c h is a n i n v a r i a n t q u a n t i t y . l a r g e s t p r i n c i p a l s t r e s s crt, the s t r e s s i n t e n s i t y / 3 ~in~-~ ( T
\1/2 S,,Siq '
s.j : crrl-- a6ij,
3(~ = a u
law The
(2)
o r the m o r e g e n e r a l e x p r e s s i o n s ~
+
(1--~,)2 4-)~cr t/2,
(3) ~
3a
1 q-k
= -T-- (1 - - X) -4- - - - T - crin (n = c o n s t )
c a n be t a k e n a s a e . The choice of ae in each particular case is dictated by the condition of long-term strength for a timeinvariant stress state. We assume that from life tests for the given material the condition t o = t o (4) has been determined.
(4)
H e r e t o is the t i m e up to f r a c t u r e ; (rOe is t h e e q u i v a l e n t s t r e s s not d e p e n d i n g on t i m e .
We i n t r o d u c e a k i n e t i c e q u a t i o n f o r the f u n c t i o n co(t). In c o n t r a s t to t h e e x i s t i n g a p p r o a c h e s [2-4], we a s s u m e t h a t the r a t e of d a m a g e a c c u m u l a t i o n d e p e n d s on c0(t), a n d a l s o on a c e r t a i n f u n c t i o n f2(t) ~o (t) __ ~ (co (t), ~ (t)).
(5)
dt
T h e f u n c t i o n ~(t) is d e t e r m i n e d by t h e law of v a r i a t i o n of the s t r e s s with t i m e . At the s a m e t i m e , it is a s s u m e d t h a t the q u a n t i t y d w / d t is i n f l u e n c e d not only by t h e e q u i v a l e n t s t r e s s cre( t) at the i n s t a n t t, but a l s o by t h e s t r e s s e s w h i c h e x i s t e d d u r i n g the t i m e 0 -<~"-
(6)
f~ (t) = ~ F (t - - x) cpz(oe (x)) dx 0 ( r (ere) is an unknown f u n c t i o n ) .
M. V. L o m o n o s o v M o s c o w State U n i v e r s i t y . T r a n s l a t e d f r o m P r o b l e m y P r o c h n o s t i , 58, F e b r u a r y , 1971. O r i g i n a l a r t i c l e s u b m i t t e d O c t o b e r 16, 1970.
No. 2, pp. 5 5 -
9 1971 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission o[ the publisher. A copy o[ this article is available [rom the publisher for $15.00.
172
0,20
F(m,~
~20
oc.3
o, lo
9
o./o o,o5
0
Fig. I
Fig. 2
F i g . 1.
G r a p h of the function f(m, c~) f o r c e r t a i n values of the a r g u m e n t s .
F i g . 2.
G r a p h of the function ~ ( m , ~) f o r c e r t a i n v a l u e s of the a r g u m e n t s .
R e p r e s e n t i n g as usual the function ~(co, ~2) in the f o r m of a product, and noting that ~2(t) is given by ae y e t unknown functions F and ~t, we w r i t e the kinetic equation (5) in the following f o r m : d0~
dt -- f (co) ~ (t).
I n t e g r a t i n g this equation and b e a r i n g in mind the f i r s t condition (1), we obtain do, _ t (co) 0
~(~)d~.
(7)
o
T a k i n g into a c c o u n t the r e l a t i o n (6) and s a t i s f y i n g the second condition (1), we t r a n s f o r m Eq. (7) into to 0
0
where I
W2( % ) - %(%)
A = : ~ do}
We r e p r e s e n t the influence function F(t--T) in the following f o r m : F (t - - ~) = F o (t - - ~)" (F o, n = const). H e r e , with the a i m of t r a n s f o r m i n g the double i n t e g r a l in (8), we use the w e l l - k n o w n D i r i c h l e t f o r m u l a t
~,
t
S S
0
0
0
As a r e s u l t , Eq. (8) is w r i t t e n in the f o r m t.
1 = ~ (t, - ~ ) m (% (~)) dr,
(9)
o
where (%) - - ~
% (~),
m
I +
To d e t e r m i n e the function (P(~e), we use the t e s t r e s u l t s (4). (9) that
n.
F o r ~e =a~ = c o n s t it follows f r o m Eq.
173
1 = ,p(o~t 0'+m':' ~ e t l " ~'" ~
'
while Eq. (9) itself assumes the f o r m (~~ 1 = 1 -~- m
t, .t" ( t , - - q0 m
dT t01+m(oe (T)) "
(10)
0
The r e l a t i o n (10) is the s o u g h t - f o r condition of l o n g - t e r m s t r e n g t h , and it g i v e s the t g n e up to f r a c t u r e , t , , f o r a g i v e n f u n c t i o n ae(t) and the function t0(cr~ d e t e r m i n e d e x p e r i m e n t a l l y . The r e l a t i o n (10) a l s o c o n t a i n s the c o n s t a n t m, found by s u p p l e m e n t a r y t e s t s . F o r the sake of e x p l i c i t n e s s , let the limiting s t r e s s of a c y l i n d r i c a l t e s t p i e e e , s t r e t c h e d with a c o n s t a n t r a t e of change of s t r e s s a, be known. H e r e ~e(~) = av, aB = ~ * (in the c a s e of s i m p l e t e n s i o n a e c o i n c i d e s with the axial s t r e s s ) , and f r o m (10) we obtain the r e q u i r e d equation f o r d e t e r m i n i n g the c o n s t a n t m :
1 -{- m :
o
tl-l-m(~r~)
o
In a s i m i l a r m a n n e r r e s u l t s of o t h e r e x p e r i m e n t s c a n be used to d e t e r m i n e the c o n s t a n t m . F i n a l l y , we note that f o r m=O the w e l l - k n o w n Bailey c r i t e r i o n of l o n g - t e r m s t r e n g t h t, 1=
to (
T))
follows f r o m the limiting condition (10). 9 T h u s the effect of the loading h i s t o r y in the limiting condition of l o n g - t e r m s t r e n g t h can be t a k e n into account by introducing the single constant m . 2. W e now consider certain particular cases of the life law (4). First to and ~~e be connected by the power relation
to [~~ ----B
(~, B = const)
(12)
widely used in the l i t e r a t u r e . F o r this Zq. (10) is t r a n s f o r m e d into t,
Bl+m I+m
(13)
o
E q u a t i o n (11), d e t e r m i n i n g the c o n s t a n t m, a s s u m e s the f o r m
B+I-Frn i m = ~ ( t , - - ~)~'~d~ or, after a transformation with o-B = bt, taken into account, B I-Fro 1+ m
=
O.~,-.Fm)(l+a) ~,+m
t (m, ~).
(14)
Here l
f (m, a) ~--.I(I - ~)m~ad~ = r 0 + m) r (I + ~) r(2+m+~) ' 0
(~ = r
(15)
(1 + ,n))
where r is a g a m m a function. It is obvious that f(0, c~)=1/(1+ ~). The graph of the function f(m, ~) is shown in Fig. 1 for several values of the arguments. Equation (14) is transformed into a form that is m o r e convenient to apply if the time up to fracture, tOB, for the constant stress o']3is known from (12). Here from (14) it follows that
174
tou t-':- = (p (m, m),
(16)
where t . = orB/d, while the function ~P(m, ~) is connected with the function (15) by the relation I
(17)
(p (m, o0---- [(1 + m) f (m, [g)]l+m
The graph of the function ~O(m, ~) is shown in Fig. 2. To find the required constant m f o r a given value of tOB/~., when ~x is known, it is sufficient to draw a horizontal straight line on the graph f r o m the origin of the coordinates. The point of i n t e r s e c t i o n of this straight line with the curve of ~P(m, o0 c o r r e s p o n d i n g to the given value of ~ d e t e r m i n e s the sought-for constant m. Let now the life be d e t e r m i n e d by the well-known formula of Zhurkov [ 5]
Ts-_coost, In this case Eq. (10) a s s u m e s the f o r m I.
Dl+r n
9 1 -t- m ": . I (t, - - ~)mexp (Yl% (~'))d%
0
while the equation for determining the constant m is ~dd D I+m
~ [~a
~ra
.
o or
, + m ~
=
(l - - ~)mexp (~g) dg,
0 l-t-m '~s :-~
kT s
~ffB.
Since f o r a r b i t r a r y m the integral is not taken in the finite form, we use an e x p r e s s i o n of the function exp (-T2 x) in a s e r i e s . After this the equation for determining 1 +m -= n a s s u m e s the f o r m Dn
(a~'a)n
:~ (--I)P~'~ pl (p + n) p==i You
e-"w'= 1+ Z
"~a == k T .
nt+p,
"
The equivalent of Eq. (16) in this ease is the following relation:
[
~----~{-- | )P~rLI-~-P]nl-", ~t.a:(pl(~a,n), q)l(~a, ll)= 1 + LJ pl(p-}-n) J .o~I which can be used in the same way as Eq. (16). 3. Finally we consider the p r o b l e m of using the limiting condition (13) f o r viscoelastic bodies. As we know, for a broad c l a s s of v i s c o e l a s t i c media, including nonlinear ones [6-8], simple loading holds f o r c e r t a i n conditions, when the s t r e s s e s ~ij(x, t) are r e p r e s e n t e d in the form (~,! (x, t) = X (t)"~ i! (x),
(18)
where the f a c t o r s ~'ij(x) depend only on the x coordinates, and the function of time k(t) is given by the law of v a r i a t i o n of the external f o r c e s with time
175
F, (~,t) = ~ (t) ?, (~); R,, (x, t) = ~ (t) ~ , (x) (F i a r e body f o r c e s , while Rvi a r e s u r f a c e f o r c e s ) . i s w r i t t e n in the f o r m
The equivalent s t r e s s (re(x ,t) f o r the condition (18)
%(x, t) = x ( t ) ~ (x) and f r o m th~ e x p r e s s i o n (13) it follows that B l~-m
(1 + m) ~e(z,)
~)'x~ (~) dr,
= tI (t, - o
(19)
[~= c~(1 + m) (the point with the coordinates x . is chosen f r o m the condition that the life t . at it is the m i n i m u m ) . If, in p a r t i c u l a r , the e x t e r n a l f o r c e s v a r y at a constant r a t e with time, then },(r)=kT, k =const. H e r e the simple finite f o r m u l a I
/,+~
B_
follows f o r the life t , f r o m (19). The function ~O(m, ~) h e r e is given by Eq. (17); its graph f o r c e r t a i n v a l u e s of the a r g u m e n t s is p r e s e n t e d in Fig. 2. If the loading p r o c e s s of the body c o n s i d e r a b l y differs f r o m simple loading, then in the case of l i n e a r v i s c o e l a s t i c i t y the method p r o p o s e d by Ii'yushin [9] can be used. Since not the s t r e s s components t h e m s e l v e s , but the e x p r e s s i o n O~e(X, t), is of i n t e r e s t to us, then, using the solution-~ij of the c o r r e s p o n d ing p r o b l e m of ideal elasticity, ~ m u s t be a p p r o x i m a t e d by the function --13
e=
Co .~_ C1(o 0 _~
C1 1 ~0 §
A l+~c0o
'
H e r e w0= ( 1 - 2 v) / ( l + v) ( v is P o i s s o n ' s ratio); Co, C_l, C1, A, fl a r e quantities known as a r e s u l t of the a p p r o x i m a t i o n . The sought-for e x p r e s s i o n now is written in the f o r m l
t P
t
~(x, t) = co (x, t) + ~ ~ R (t - ~) r
t
(~, ~) + 3K ~ n ( t - ~) dc_, (x, ~) + ~ e~ (t - ~) ~ (x, ~),
U
0
0
0
where II(t), R(t), gfl(t) are functions determined experimentally [9], and K is the bulk modulus. Now the limiting condition (13) at the point x , is transformed into t.
B l + ra
t.
= ~ (t, - ~)" Co (~,, ~) ,~ + 3~, _ ~ (t, - ~)'d~ ~ n (~ - ~) r I + m J 0
1 +--~
(t,
0
x)mdr
0
(x--
(x,,
0
(~,, ~)
0
~ (t. -- x)'dx ~ g~ (x -- ~) dA (x,, ~), 0
0
f r o m which the life t . is d e t e r m i n e d . LITERATURE 1.
2. 3. 4. 5. 6.
7. 8. 9.
176
CITED
A. A. II'yushin, "On a theory of l o n g - t e r m strength, " Mekhanika T v e r d o g o Tela, No. 3 (1967). L. M. Kachanov, T h e o r y of C r e e p [in Russian], F i z m a t g i z , Moscow (1960). Yu. N. Rabotnov, "On f a i l u r e as a r e s u l t of c r e e p , P M T F , No. 2 (1963). Yu. N. Rabotnov, C r e e p of Structural E l e m e n t s [ i n R u s s i a n ] , N a u k a , M o s c o w (1966). S. N. Zhurkov, "The strength p r o b l e m of solids, " Vestnik AN SSSR, No. 11 (1957). V. V. Moskvitin, "On a s i m p l e s t p o s s i b i l i t y of accounting f o r nonlinearity in v i s c o e l a s t i c media, " Mekhanika P o l i m e r o v , No. 2 (1967). V. V. Moskvitin, "On a nonlinear model of v i s c o e l a s t i c m e d i u m taking into account the f o r m of the s t r e s s state, " Mekhanika P o l i m e r o v , No. 6 (1969). V. V. Moskvitin and V. V. Kolokol'chikov, "On the p r o b l e m of a q u a s i - l i n e a r theory of v i s c o e l a s ticity, " Mekhanika P o l i m e r o v , No. 4 (1969). A. A. II'yushin, "An e x p e r i m e n t a l method of solving an integral equation of the t h e o r y of v i s c o elasticity, " Mekhanika P o l i m e r o v , No. 4 (1969).
