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DYNAMIC CRACK PROPAGATION IN ELASTIC SOLID
J. Lee, H. Liebowitz
To cite this version:
J. Lee, H. Liebowitz. DYNAMIC CRACK PROPAGATION IN ELASTIC SOLID. Journal de Physique Colloques, 1985, 46 (C5), pp.C5-257-C5-269. �10.1051/jphyscol:1985534�. �jpa-00224765�
J O U R N A L D E PHYSIQUE
Colloque C5, suppldment au n08, Tome 46, aoDt 1985 page C5-25 7
D Y N A M I C CRACK PROPAGATION I N ELASTIC SOLID
J.D. Lee and H . ~ i e b o w i t z '
Productivity Center, Department of Mechanical! Engineering, U n i v e r s i t y o f Minnesota, MinneapoZis, Minnesota 55455, U . S.A .
' ~ c h o o l ! o f Engineering and Applied Science, The George Washington U n i v e r s i t y , Washington, D.C. 20052, U.S.A.
Resum6 - On analyse l e probl6me de l a propagation dynamique de fissure dans un solide 6lastique f i n i en u t i l i s a n t l a combinaison de l a m6thode par 616- ments f i n i s e t de lfanalyse modale. On prouve, de fagon analytique e t num6- rique, que l a vitesse de relaxation de 116nergie t o t a l e e s t 6gale Zi c e l l e d e lf6nergie locale. La vitesse de relaxation de lr6nergie e s t employee en tant que c r i t s r e de fracture, de t e l l e sorte que l a vitesse de l a fissure estd6- termin6e en fonction du temps, pour un chargement dynamique donn6.
Abstract - The dynamic crack propagation i n f i n i t e e l a s t i c solid i s analyzed by using the combination of f i n i t e element method and modal analysis. I t i s proved analytically and numerically that the global energy release r a t e equals the local energy release rate. Energy release r a t e i s employed as a fracture criterion so that the crack speed i s determined as function of time for a given dynamic loading.
I - INTRODUCTION
Considerable attention has been given i n recent years t o fracture problems of dynamically loaded structures with rapidly running crack [ l ,2,3] . In these situa- tions, the i n e r t i a force has t o be included in the equation of equilibrium and the kinetic energy has t o be taken into consideration in the energy conservation law.
Moreover, r e a l i s t i c a l l y speaking, the crack speed - even i n the simple cases of self-similar crack propagation - should be considered as an additional unknown function of t h e , and, hence, a fracture criterion regarding the crack speed is needed t o perform the analysis of dynamic crack propagation.
Freund solved the problems of crack propagation i n an i n f i n i t e e l a s t i c solid analytically [4,5,6] and also considered the energy flux into the extending crack t i p [ 7 ] . In Freund's work, the crack speed i s considered t o be a-given function of time. Kishimoto, Aoki, and Sakata formulated a path-independent J-integral t o take into account of the existence of a fracture process region and the effects of plas- t i c deformations, body forces, thermal s t r a i n s and i n e r t i a of material 81. They also performed f i n i t e element analyses of dynamic crack problems using i -integral as the fracture criterion 19,101 . Achenbach, Kminen, and Popelar obtained the asymptotic solutions a t the crack t i p f o r f a s t fracture of e l a s t i c - p l a s t i c solid
Ill]. Atluri e t a l . 112-151 presented a moving-singular-element procedure f o r the dynamic analysis of crack propagation in f i n i t e bodies. Kobayashi e t a l . [16-211 experimentally determined the dynamic fracture toughness, KID: VS the crack speed, g, relations f o r various specimens and also employed the K A relation as the dynamic fracture critarion i n the f i n i t e element analyses. I D ~ a r n e s , Ahmad , and Kanninen took a dual J/CTOA approach i n which a c r i t i c a l 3 value governs the crack i n i t i a t i o n and a constant crack t i p opening angle, CTOA, controls the rapid crack propagation 1221. Very comprehensive survey in the f i e l d of crack propagation has been given by Kanninen e t a l . 1231.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985534
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In t h i s paper, the dynamic crack propagation i n f i n i t e e l a s t i c solid is analyzed by using the combination of f i n i t e element method and modal analysis. Two types 'of problems are solved, namely, f o r a given dynamic loading and crack speed as func- tions of time, the stresses, the displacements , and the fracture parameters deriv- able from those, such as the energy release r a t e , can be obtained; and, on the other hand, i f the energy release r a t e is postulated t o be a constant during the entire process of dynamic crack propagation, then the crack speed can be determined as function of time f o r a given dynamic loading. I t i s noticed t h a t the method of modal analysis provides the exact solutions with respect t o the governing matrix equations which are derived by using the f i n i t e element analysis. Therefore, it is proved analytically and numerically that the global energy release r a t e equals the local energy release r a t e , which indicates the v a l i d i t y of the formulation and the corresponding computer programming.
11. GOVERNING EQUATIONS
A center-cracked specimen of length ZR, width 2w, and crack size 2a(t) i s shown i n Fig. 1, the crack i s assumed t o propagate along x-axis, and hence, only the f i r s t quadrant of the specimen i s t o be analyzed. The f i n i t e element mesh of the f i r s t quadrant of the specimen with 90 4-node isoparametric elements and 109 nodal points i s shown i n Fig. 2.
Y
o = o
X Y
I--- 2 w
+
The boundary conditions may be specified as
Fig. 1 - The geometry and loading conditionofacenter- cracked specimen.
Fig. 2 - Finite element mesh of the 1 s t quadrant NE = 90, NP = 109, NDOF = 218.
I t i s noticed t h a t the crack size is a function of time and there is no a p r i o r i assumption made about the crack propagation speed.
The v i r t u a l work equation i s sinply
/ a.. Geij dv 11 + / p iii 6ui dv = / a.. n. 6ui ds ,
11 3 (5)
where the surface integration accounts f o r a l l the specified surface traction in- cluding the release of stresses in the wake of the advancing crack t i p . Following a standard procedure [24,25], the governing f i n i t e element equations can be obtained i n the following matrix form
where M, K are the symmetric mass matrix and s t i f f n e s s matrix, respectively; u i s the noaal-point displacement vector; F is the nodal point force vector includyng the release of crack t i p force as the crack t i p advances. In t h i s paper, the dis- tributed mass system i s adopted, therefore, the mass matrix has the same band width as the s t i f f n e s s matrix.
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According t o the f i n i t e element mesh shown i n Fig. 2 , the number of degrees of free- dom i s 218 before the displacement specified boundary conditions are imposed; and the band width i s 30. I t i s recognized t h a t t h i s mesh i s a very coarse one and hence, the numerical r e s u l t obtained i n t h i s paper i s only meaningful qualitatively.
111. MODAL ANALYSIS
After the displacement specified boundary conditions being specified, eqn. (6) can be solved by using the method of modal analysis as follows. The c h a r a c t e r i s t i c equation of eqn. (6) i s
which yields N natural frequencies wi and N corresponding orthonormalized eigen- vectors Et, i.e. ,
where N i s the number of net degrees of freedom; and Z: stands f o r the transpose of Z . Now, the displacement vector g can be written a?'
-1 '
and f i ( t ) is governed by the following ordinary d i f f e r e n t i a l equation
f i + 0; f i = L; E(t) (12)
I t i s straightforward t o obtain the following:
f i ( t ) = Ai sinwit + Bi C O S W ~ ~ + f t F(t) sinwi(t-T) d~ ,
0 - (13)
f i ( t ) = wi(Ai cosoit - Bi sinw. t ) 1 + 2; J j ( ~ ) t coswi ( t - T ) d~ ,
0 (14)
fi ( t ) = - W: ( A ~ sinwit + Bi cosw. t ) - o. Z* ft F (r) sinwi (f - r ) dr + L$F-(t) , (15)
1 1 -1
0 -
where the unknown parameters, A. and B . , are t o be determined by the i n i t i a l condi- tions. It should be emphasizedlat t h i & time t h a t the method of modal analysis pro- vides the exact solution with respect t o the governing equation (6).
Now, assume the system i s i n i t i a l l y a t r e s t and the crack s t a r t s t o propagate a t t = tl, then the solutions f o r t &(O,tl) are
f i ( t ) = 2; zi (t)/w i y (16)
fi ( t ) = 2; gi ( t ) , (17)
Zi ( t ) = - wi LT si ( t ) + -1 Z* ~ " ( t ) - , (18)
t a
S. ( t ) = / F (T) sinwi ( t -.r) d~ ,
-1 (19)
0
and Fa(t) is the external applied loading. Right a t t = t , the nodal point d i s - placements , velocities, and accelerations can be obtained ks
and then the nodal point force vector a t t = tl can be calculated as
Let the crack t i p force a t t = t be denoted by FC(t ) which i s a scalar. Suppose the crack t i p advances t o the n e i t nodal point i n th& time interval (tl,t2) , i n other words, the crack propagation speed i s
where h i s the distance between the current crack t i p and the next crack t i p ; and A t = t 2 - tl . Assume t h a t the crack t i p force i s released as
In order t o obtain the solutions f o r t ~ ( t ,t ) , one has t o solve the characteris- t i c equation again for a new s e t of eigenv&le8, w . , and eigenvectors, Li , simply be- cause there i s an extra degree of freedom - the mhement of the current crack t i p in the y-direction - emerging. Then, f o r t € ( t l 7 t 2 ) , it i s straightforward t o obtain
t a
f i ( t ) = ui Ai coswi(t-tl) - wi Bi sinw. 1( t - t 1) + Z? I F (T) coswi(t-T) d-r tl
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t a
f i ( t ) = - w i Ai sinwi (t-tl) - W? 1 B. 1 ( t - t l ) - wi L! f c (T) sinwi (<.c) d~
t.
where ZC i s the component associated with of the current crack t i p in the i - t h eigenve&tor. From eqns . (27,28) the nodal2oint displacements and velocities, a t t = tl, are obtained as
which should match the values obtained from eqns. (21,22) i . e . ,
and hence B. and A. are determined. And the same procedure i s applicable f o r the subsequent Erack p*opagation.
IV. ENERGY RELEASE RATE
The t o t a l s t r a i n energy and the t o t a l kinetic energy a t any time t can be obtained as
where U* and G* are the transposes of u and G, respectively. The t o t a l work done by theextern51 applied loading a t timE t is-
The global energy release r a t e , G i s defined as g'
G g = ( I ? - O - k ) / i .
Suppose the crack t i p advances from X a t tl t o X + &a t it 2 = tl + A t , then the
average global energy release rate becomes
+ $ p 2 ) - f l ( t l ) ] /Ax .
l (38)
The local energy release r a t e , G&, i s defined t o be the negative of the rate of work done by the release of crack t i p force divided by the crack speed, i . e . ,
I t can be proven analytically that GE = GRwhich i s nothing but the statement of conservation of energy. Also, it has been proven numerically - by using the compu- t e r program developed f o r t h i s work - t h a t GP = G , . This demonstrates the validity of the method of modal analysis, and of the ;orresponding computer program as well.
In the following, f o r i l l u s t r a t i v e purpose, l e t the attention be focused on a special class of problems which can be described as: The material and geometric parameters are
E = 10300 ksi.
v = 0.33
p = 0.097542 lb/in. 3 R = 12 in.
w = 6 i n . a(0) = l in.
The loading and the crack size as functions of time are shown i n Fig. 3 , and can be specified as
u2(y,t) = 0 ,
where a i s the magnitude of the applied loading; v i s the crack speed; t i s called the r i s e time. I t i s seen t h a t , f o r t h i s class of problems, the crack &mains stationary and propagates with constant speed before and a f t e r r i s e time, respectively; and fixed loading condition i s imposed a f t e r the crack s t a r t s t o propagate. Also, the plane s t r e s s condition i s assumed. The numerical results -
energy release r a t e vs crack size - are shown in Fig. 4 and Fig. 5 for
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f --- --- A p p l e d S t r e s s d ( t )
-. -. -. -. -. -. C r a c k S i z e a ( t ) C r a c k s p e e d v ( t )
Fig.
t r : R i r e T i m e
3 - Applied s t r e s s , crack size, crack speed vs.
G
t
' i m e t time.
0 1 2 3 4 5 6
C r a c k S i z e ( i n c h )
Fig. 4 - Energy release r a t e vs.crack size.
5
Applied Stress o = 20 ksi
- 4
.- C
.
n ol 0 3
- 0 W 0
m CL
w 2
m 0
7 W CT
a m
: 1 /
Y
0 , a
0 1 2 3 4 S 6
C r a c k Size ( i n c h )
Fig. 5 - Energy release r a t e vs crack size.
v = 0.1 inch/microsecond and v = 0.0115 inch/microsecond, respectively. I t i s seen t h a t the general trend i s : the energy release r a t e increases as crack t i p advances
- t h i s i s due t o the finiteness of the specimen. Also, the energy release r a t e i s larger f o r smaller crack speed - it i s in qualitative agreement with Freund's analytical solution of i n f i n i t e wide specimen [4] .
V. THE INVERSE PROBLEM
I t is really straightforward t o perform the modal analysis of dynamic crack propaga- tion in e l a s t i c solid i f the motion of the crack t i p as function of time i s pre- scribed. However, in a r e a l i s t i c situation, when and how does the crack t i p advance should be the objective, not the starting point, of the analysis. That is why cer- t a i n fracture c r i t e r i a are needed. Indeed, numerous fracture c r i t e r i a have been proposed [23].
I f it i s assumed t h a t the energy release r a t e is kept a t a constant value during the e n t i r e process of dynamic crack propagation, then the crack speed could be found as a function of time f o r a given loading condition. For i l l u s t r a t i v e purpose, l e t the specimen be loaded t o a given level s t a t i c a l l y , i . e . , the r i s e time i s approaching infinity, then the solution a t tr is simply
fi(tr) = pi/w: , ri(tr) = f i ( t r ) = o , (42)
where
pi = L; ~ ~ ( t ~ ) . (43)
The nodal point displacements, velocities, accelerations a t t = tr are
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Using eqn. (24), the crack t i p force, FC(t ), can be calculated. A t t h i s moment, it i s convenient t o define t * = t - t thenPTfor t * ~ ( 0 , At), it i s obtained from
eqn. (27) that r 7
f . (t*) = B- coswi t * + Bi(l - coswi t*)/wl
I
Z: FC(tr) sinui t *
+ 1 1 - cosui t * - [t* - ]/At> 7
u2 i
where B. can be calculated by using eqn. (32). The energy release r a t e , eqn. (39), can be Fewritten as
I t i s further assumed that the crack s t a r t s t o propagate a t a very low speed - it i s f e l t that t h i s i s a r e a l i s t i c assumption - which means A t i s approaching i n f i n i t y ; then the c r i t i c a l value of the energy release r a t e , GC, i s obtained t o be
Then, for the subsequent crack propagation, it i s just a matter of determining the crack speed v = Ax/(t2-tl) by imposing one of the following conditions
The f o m l a s of G and GR are expressed i n eqn. (38) and eqn. (39), respectively.
g
With the same material and geometric parameters as being specified in eqn. (40), the energy release rate,_(; = Go = GP, , and the crack speed, v, as functions of the crack
0 '-
size, a ( t ) , are shown i n Fig. 6. Notice t h a t , during the e n t i r e process 'of dynamic crack propagation, the energy release r a t e i s kept a t a constant value, GC, and the applied loading i s kept a t a = 20 k s i , i.e., the fixed loading condition 1s imposed.
I t i s seen that the crack speed i s increasing as the crack t i p advances. This i s f e l t t o be qualitatively in agreement with the observed phenomenon.
VI. DISCUSSION
For the problems of dynamic crack propagation i n e l a s t i c solid, the method of modal analysis provides an exact solution t o the governing f i n i t e element equations. In t h i s paper, energy xelease r a t e is used as a fracture criterion. I t should be mentioned that the J-integral, proposed by Kishimoto, Aoki, and Sakata, could also be employed as a fracture criterion f o r crack propagating i n e l a s t i c solid.
In r e a l i t y , considerable amount of p l a s t i c i t y exists i n the neighborhood of crack t i p . A more r e a l i s t i c approach of dynamic crack propagation should incorporate
0 a
0 I 2 3 4 5 6
C r a c k S i z e ( i n c h )
Fig. 6 - Energy release r a t e , crack speed vs crack s i z e under fixed loading condition (a = 20 k s i ) .
p l a s t i c i t y into the analysis. In f a c t , one may easily perform a f i n i t e element formulation based on an elastic-plastic theory with kinematic hardening rule. Then the incremental v i r t u a l work equation, a counter part t o eqn. (S), can be written as
which leads t o the governing matrix equation of the following form:
where k i s the incremental s t i f f n e s s matrix which involves the incremental s t r e s s -
incremzntal s t r a i n relation. One may solve eqn. (50) by using the central differ- ence method. Perhaps one may s t i l l employ the method of modal analysis t o solve eqn. (50) i f the incremental step is small enough such that the D-matrix relating the incremental stresses and the incremental s t r a i n s of each element practically remains constant within that incremental step. Lee and L i i [26] have s h m the accuracy and the efficiency of using the modal analysis, the central difference, and the Newmark methods i n solving an e l a s t i c wave problem.
As fracture criterion i s concerned, i n addition t o the energy release r a t e c r i t e r i - on, the p l a s t i c energy r a t e criterion may also be employed. The p l a s t i c energy r a t e , P, i s defined as
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I I
where the integration i s performed i n the p l a s t i c zone and eii i s the p l a s t i c s t r a i n tensor. Lee, Liebmdtz, and co-workers [27- 311 have shown tgt, i n s u b c r i t i c a l crack growth, t h e dissipated p l a s t i c energy i s l i n e a r l y proportional t o the crack s i z e and discussed t h e meaning and the advantage of using t h a t relationship a s the fracture c r i t e r i o n . It i s natural t o propose t h a t , during t h e process of dynamic crack propagation, t h e p l a s t i c energy r a t e i s proportional t o , o r a function o f , crack speed. Of course, any proposed fracture c r i t e r i o n has t o be c r i t i c a l l y examined by experimental f a c t s .
ACKNOWLEDGEMENT
The author (*L) wishes t o acknowledge t h e p a r t i a l financial support f o r t h i s work through t h e Grant in Aid of Research from the Graduate School, University of Minnesota. The authors also wish t o express gratitude t o D r . Luke Lien for h i s valuable help i n computer programming.
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