A STUDY ON PATH OF DYNAMIC CRACK PROPAGATION

HAL Id: jpa-00224760

https://hal.archives-ouvertes.fr/jpa-00224760

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A STUDY ON PATH OF DYNAMIC CRACK PROPAGATION

K. Fujimoto, T. Shioya

To cite this version:

K. Fujimoto, T. Shioya. A STUDY ON PATH OF DYNAMIC CRACK PROPAGATION. Journal de

Physique Colloques, 1985, 46 (C5), pp.C5-233-C5-238. �10.1051/jphyscol:1985530�. �jpa-00224760�

JOURNAL DE PHYSIQUE

Colloque C5, supplCrnent au n08, Tome 46, aoQt 1985 page C5-233

A STUDY ON P A T H OF D Y N A M I C CRACK PROPAGATION

K. Fujimoto and T. Shioya

Department of Aeronautics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan

RBsumk - La t r a j e c t o i r e d'une f i s s u r e en cours de propagation e s t k t u d i k e 2

l ' a i d e de l a m6thode de 1 1 6 n e r g i e t o t a l e . La p o s i t i o n e t l a v i t e s s e de l ' e x t r g - mitk de f i s s u r e s o n t considkrkes comme des paramstres g 6 n 6 r a l i s 6 s dans 1'6qua- t i o n de mouvement de Lagrange. C e t t e d e r n i s r e e s t r 6 s o l u e de fason 2 o b t e n i r l a t r a j e c t o i r e t o t a l e de l a f i s s u r e dans l e c a s de l a f i s s u r a t i o n en mode I dans une plaque b r i d g e s u r l e cGt6. Le r e s u l t a t montre que dans l e domaine des p e t i t e s v i t e s s e s , l a f i s s u r e s'approche de l a l i g n e mkdiane de l a plaque ; cependant, e l l e s 1 6 l o i g n e du c e n t r e l o r s q u ' e l l e s'approche de l a v i t e s s e des ondes de Rayleigh. Des t r a v a u x expgrimentaux o n t 6 t 6 men&.

A b s t r a c t - The p a t h of a running crack i s s t u d i e d by a t o t a l energy method.

The Lagrange's e q u a t i o n of motion i s used i n which t h e crack t i p p o s i t i o n and i t s v e l o c i t y a r e regarded a s t h e g e n e r a l i z e d c o o r d i n a t e and v e l o c i t y . The e q u a t i o n i s solved t o o b t a i n t h e e n t i r e crack p a t h i n t h e case of Mode I crack i n f i x e d s i d e d p l a t e . The r e s u l t shows t h a t i n t h e low v e l o c i t y r a n g e , t h e crack approaches t o t h e c e n t e r l i n e of t h e p l a t e , however, i t moves away from t h e c e n t e r a s i t approaches t o t h e Rayleigh wave v e l o c i t y . Related experiment was conducted.

I - INTRODUCTION

The p r e d i c t i o n of a crack propagation p a t h i s one of t h e most important problems i n dynamic behaviour of m a t e r i a l s . T y p i c a l s t u d i e s i n t h e p a s t about t h i s problem have been aimed f o r a n a l y z i n g t h e s t r e s s f i e l d n e a r t h e c r a c k t i p and determining t h e d i - r e c t i o n of t h e crack p a t h . A s t h e c r i t e r i o n o f crack d i r e c t i o n , some i d e a s a r e pro- posed such t h a t t h e crack s t a r t s a t i t s t i p i n t h e p l a n e p e r p e n d i c u l a r t o t h e d i r e c - t i o n o f maximum t a n g e n t i a l s t r e s s (Erdogan and S i h / l / ) , o r it propagates t o t h e d i - r e c t i o n of minimum s t r a i n energy ( S i h / 2 / ) . I n t h e p r e s e n t work, t h e concept o f t o - t a l energy b a l a n c e method ( G r i f f i t h / 3 / , Mott / 4 / ) i s extended t o t h e crack p a t h problem by a d a p t i n g t h e Lagrange's e q u a t i o n . The Lagrange's e q u a t i o n of motion i s d e s c r i b e d a s ,

where L ( = P - U ) i s t h e Lagrangian, T i s t h e k i n e t i c energy, U i s t h e p o t e n t i a l ener- gy, qi and qi a r e t h e g e n e r a l i z e d c o o r d i n a t e and t h e g e n e r a l i z e d v e l o c i t y , and N . i s t h e non-conservative f o r c e .

I n o r d e r t o o b t a i n t h e Lagrange's e q u a t i o n with a few parameters, t h e p o t e n t i a l ener- gy and t h e k i n e t i c energy o f t h e system.are d e s c r i b e d a s f u n c t i o n s of t h e crack t i p p o s i t i o n (X, Y) and i t s v e l o c i t i e s ( X , Y) which a r e regarded a s t h e g e n e r a l i z e d coor- d i n a t e s and t h e g e n e r a l i z e d v e l o c i t i e s i n t h e e q u a t i o n . The non-conservative f o r c e s i n t h e Lagrange's e q u a t i o n a r e d e r i v e d from energy d i s s i p a t i o n of t h e system, i . e . , t h e s u r f a c e energy of t h e crack i n t h e p r e s e n t c a s e .

By s o l v i n g t h e Lagrange's e q u a t i o n of motion, t h e e n t i r e crack p a t h i s c a l c u l a t e d .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985530

C5-234 JOURNAL DE PHYSIQUE

A s examples, o f f - c e n t e r c r a c k s running i n a f i x e d s i d e d and i n f i n i t e l y long p l a t e a r e taken. The a u t h o r s p r e v i o u s l y analyzed t h e c a s e of Mode I11 running crack a s t h e s i m p l e s t example and o b t a i n e d t h e r e s u l t t h a t t h e crack t i p moves towards t h e c e n t e r l i n e i n t h e slow e x t e n s i o n v e l o c i t y , while it moves away from t h e c e n t e r l i n e a s t h e v e l o c i t y approaches t o t h e e l a s t i c wave v e l o c i t y / 5 , 6 / . I n t h e p r e s e n t paper, t h e case of Mode I c r a c k i s analyzed s i n c e it i s more p r a c t i c a l c a s e , though i t i n v o l v e s more complex t e c h n i q u e i n t h e a n a l y s i s a r i s i n g from t h e presence of two independent e l a s t i c waves.

I1

-

A s e m i - i n f i n i t e b r i t t l e crack i s considered t o be running i n an i n f i n i t e l y long p l a t e a s shown i n Fig. 1. The width of t h e p l a t e , t h e p o s i t i o n of t h e crack from t h e cen- t e r l i n e ( e c c e n t r i c i t y ) and t h e crack propagation v e l o c i t y a r e denoted by h , d and V, r e s p e c t i v e l y . The m a t e r i a l o f t h e p l a t e i s assumed t o be homogeneous and i s o t r o p i c with t h e s h e a r modulus G , t h e P o i s s o n ' s r a t i o V and t h e mass d e n s i t y p . The Carte- s i a n c o o r d i n a t e s (X, Y, Z ) and ( X , y, 2 ) a r e taken such t h a t (X, Y, Z) is f i x e d t o t h e m a t e r i a l and ( X , y , z ) i s moving i n t h e X - d i r e c t i o n with t h e c r a c k propagation v e l o c i t y V, i . e . ,

X = X -

v t ,

The displacements ( U , v , W ) i n (X, Y, Z) components a r e f i x e d a t t h e b o t h s i d e s of t h e p l a t e , i. e . ,

The o b j e c t of t h i s s t u d y i s t o f i n d t h e c r a c k p a t h and s o t h e d e f l e c t i o n of t h e crack path should a l s o be determined by t h e a n a l y s i s . However, i n c a l c u l a t i n g t h e e n e r g i e s of t h e system, t h e e l a s t i c f i e l d i n which t h e crack i s p a r a l l e l t o t h e s i d e s of t h e p l a t e i s employed. This assumption i s r e a s o n a b l e because t h e h i g h l y deformed r e g i o n i s c o n c e n t r a t e d n e a r t h e crack t i p s o d e r i v a t i v e s of t h e t o t a l e n e r g i e s w i t h r e s p e c t t o t i p p o s i t i o n a t each i n s t a n t o f time t h e n r e p r e s e n t t h e change i n t h e energy of t h e system.

Fig. 1

-

The t r a c t i o n f r e e c o n d i t i o n on t h e crack s u r f a c e i s ,

The e l a s t i c f i e l d h a s been c a l c u l a t e d by use of t h e method o f c o n t i n u o u s l y d i s t r i b - u t e d d i s l o c a t i o n s model under t h e boundary c o n d i t i o n s ( 3 ) and ( 4 ) w i t h t h e assumption of c o n s t a n t crack v e l o c i t y V /7/.

I n Figure 2 , examples of t h e c a l c u l a t e d c r a c k opening displacements v ( X ) and u C ( x ) a r e shown v a r y i n g t h e crack v e l o c i t y V and t h e e c c e n t r i c i t y d. A noteworthy f e a t u r e i n t h e f i g u r e i s t h a t t h e crack i s over-expanded n e a r t h e t i p , and t h i s e f f e c t i s en- hanced a s t h e crack v e l o c i t y becomes h i g h e r . I t is q u i t e n a t u r a l t h a t t h e r e l a t i v e displacement between t h e upper and lower s u r f a c e s of t h e c r a c k has t h e s l i d i n g compo- n e n t u c ( x ) , i. e . , Mode I1 component when t h e e c c e n t r i c i t y o f t h e c r a c k i s non-zero.

The p o t e n t i a l energy U ( X , Y, V ) of t h e system is expressed a s ,

where U ₀ i s c o n s t a n t and t h e second term i n r h s . r e p r e s e n t s t h e d e c r e a s e o f t h e ener- gy due t o t h e crack e x t e n s i o n i n t h e X - d i r e c t i o n . K = 3 -4V f o r p l a n e s t r a i n , and K = ( 3 - v ) / ( l + v ) f o r p l a n e s t r e s s . The r e l a t i v e p o t e n t i a l energy E(Y, V ) which does n o t depend on X i s expressed a s ,

r 0

The k i n e t i c energy T(Y, V ) i s expressed a s , h / 2

T Y , V) = V

1: I

2

a

- h / 2

Fig. 2 - Crack opening displacements, vc(x) and u c ( x ) .

JOURNAL DE PHYSIQUE

Fig. 4 - K i n e t i c energy, T ( Y , V ) .

F i g . 3 - R e l a t i v e p o t e n t i a l energy, E ( Y , V ) .

The c a l c u l a t e d E ( Y , V ) and T ( Y , V ) i n c a s e o f K = 2 . 2 a r e shown i n Fig. 3 and F i g . 4 , r e s p e c t i v e l y , which a r e used i n t h e f o l l o w i n g a n a l y s i s of t h e crack p a t h .

I11 - ANALYSIS OF CRACK PATH

The g e n e r a l Lagrange's e q u a t i o n (1) i s r e w r i t t e n i n t h e p r e s e n t case a s ,

The,non-conservative f o r c e s NX and N due t o t h e s u r f a c e energy a c t a g a i n s t t h e c r a c k Y

e x t e n s i o n , t a n g e n t i a l l y t o t h e crack p a t h , s o t h a t ,

i I.

N X = - - -

(

i 2

The c a l c u l a t e d

?

The e n t i r e crack p a t h i s c a l c u l a t e d n u m e r i c a l l y by s o l v i n g Eqs. ( 1 0 ) a s an i n i t i a l valued problem. Examples a r e shown i n Fig. 5. F i g u r e 5 - ( a ) i s t h e c a s e when t h e s p e c i f i c s u r f a c e energy does n o t depend on t h e crack v e l o c i t y . The c r a c k t i p t e n d s t o approach t o t h e c e n t e r l i n e d u r i n g t h e slow p r o p a g a t i o n s t a g e , however, it t u r n s towards t h e s i d e a f t e r a c c e l e r a t i n g t o high v e l o c i t y . The l i m i t i n g propagation ve- l o c i t y i s t h e Rayleigh wave v e l o c i t y cR. F i g u r e s 5-(b) and ( c ) a r e t h e c a s e s when t h e s u r f a c e energy i s l i n e a r l y p r o p o r t i o n a l t o t h e c r a c k v e l o c i t y . In t h i s c a s e , t h e r e e x i s t s a t e r m i n a l c r a c k v e l o c i t y V a t which t h e r e l e a s e d p o t e n t i a l energy i s

t

i n e q u i l i b r i u m w i t h t h e consumed s u r f a c e energy. F i g u r e 5-(b) i s t h e c a s e when V <

t 0.7c, and Fig. 5 - ( c ) i s t h e c a s e when V > 0 . 7 ~ . These f i g u r e s s u g g e s t t h a t t h e

t

c r a c k t i p moves towards t h e s i d e only i f t h e a p p l i e d p o t e n t i a l energy i s l a r g e enough t o make t h e t e r m i n a l v e l o c i t y V exceed 0 . 7 ~ .

t

CENTER L I N E X-DIRECTION

( a ) y = 0 . 5

r 0 .

CENTER LINE

X-DIRECTION (b) y ( v ) = 2 r O v / c .

-

CENTER L I N E X-DIRECTION

( C ) Y ( V ) = r 0 v / c .

1 K i l G v 0 2 Fig. 5 - C a l c u l a t e d c r a c k p a t h s . (

r 0

-

2 K - l h

JOURNAL DE PHYSIQUE

I V - EXPERIMENT ON CRACK PATH

Experiment on t h e p a t h o f r u n n i n g c r a c k i n a b r i t t l e m a t e r i a l h a s been conducted.

T e s t specimen o f PElMA p l a t e was f i x e d t o t h e g r i p o f t h e u n i v e r s a l t e s t i n g machine and t h e c r a c k p r o p a g a t i o n was i n i t i a t e d by a n impact a t t h e pre-notched p o s i t i o n i n one end o f t h e p l a t e . An example o f t h e c r a c k p a t h i s shown i n F i g . 6 . The measured t e r m i n a l c r a c k v e l o c i t y i s a b o u t 550 m / s which i s a b o u t 0 . 4 o f t h e s h e a r wave v e l o c i - t y o f t h e m a t e r i a l . The f a c t t h a t t h e c r a c k moves towards t h e c e n t e r l i n e i n t h i s r a n g e o f v e l o c i t y i s c o n s i s t e n t w i t h t h e a n a l y t i c a l r e s u l t . I t i s p r a c t i c a l l y d i f f i - c u l t t o o b t a i n t h e c r a c k v e l o c i t y more t h a n 0 . 7 c , s o t h e tendency o f approaching t h e s i d e i n t h e h i g h v e l o c i t y r a n g e h a s n o t been confirmed by t h e experiment.

P 5 cm t h i c k n e s s : 3 mm

F i g . 6 - Observed c r a c k p a t h i n f i x e d s i d e d p l a t e o f PMMA.

V - DISCUSSIONS AND CONCLUSIONS

The p a t h o f dynamic c r a c k p r o p a g a t i o n is s t u d i e d by t o t a l energy concept u s i n g Lag- r a n g e ' s e q u a t i o n of motion. The a n a l y s i s o f t h e Mode I r u n n i n g c r a c k i n f i x e d s i d e d p l a t e shows t h a t a t t h e low c r a c k v e l o c i t y , t h e c r a c k approaches t o t h e c e n t e r l i n e of t h e p l a t e , w h i l e it moves away from t h e c e n t e r l i n e a t h i g h v e l o c i t y . I n t h e p r e s e n t a n a l y s i s , t h e b r a n c h i n g of t h e c r a c k p a t h i s n o t c o n s i d e r e d . I t i s a l s o a s - sumed t h a t t h e s l o p e o f t h e c r a c k p a t h i s n o t s t e e p . For t h e c a s e when t h e s t e e p n e s s of t h e p a t h cannot b e n e g l e c t e d , t h e c a l c u l a t e d p a t h may n o t c o i n c i d e w i t h t h e a c t u a l one, b u t t h e above tendency s h o u l d b e t a k e n r a t h e r q u a l i t a t i v e l y . I n t h e a c t u a l b r i t t l e f r a c t u r e p r o c e s s , b r a n c h i n g of t h e c r a c k p a t h h a s o f t e n b e e n observed a t h i g h v e l o c i t y p r o p a g a t i o n . The p r e s e n t a n a l y s i s may n o t a p p l y d i r e c t l y t o s u c h c a s e s .

REFERENCES

/l/ Erdogan, F. and S i h , G . C . , Trans. ASME, J. Basic Engineering,

85

/ 2 / S i h , G . C . , i n Mechanics of Fracture, Vol. 1, S i h , G . C . ( e d . ) , Noordhoff, Leyden (19 73) I n t r o d u c t o r y C h a p t e r .

/3/ G r i f f i t h , A . A . , PhiZ. Trans. Roy. Soc. London,

A221

/ 4 / Mott, N. F . , Engineering,

165

/ 5 / Shioya, T. and Fujimoto, K . , Trans. Japan Soc. Aero. Space Sci.,

5

/6/ Shioya, T. and Fujimoto, K . , i n MathematicaZ ModeZZing i n Science and TeehnoZogy, t h e Fourth I n t e r n a t i o n a l Conference, Zurich ( 1 9 8 3 ) , Avula, X . J . R . e t a l . ( e d . ), Pergamon, New York (1984) 513.

/7/ Fujimoto, K. and Shioya, T . , Proc. 28th Japan Gong. MateriaZs Research, t h e Soci- e t y o f M a t e r i a l s S c i e n c e , Kyoto (1985).