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THE STEADILY GROWING, ELASTIC-PLASTIC CRACK TIP IN A FINITE ELEMENT TREATMENT

Hans Andersson The Lund Institute of Technology Fack 725, S-220 07 Lund 7, Sweden tel: 046/12 46 O0

The point of instability of a ductile crack tip with stable growth resembles the limiting case of quasi-static, steady state growth, rather than the idealized case of a stationary crack tip around which the plastic enclave grows without irreversible effects. Particularly this should be the case in small scale yielding. Then the crack opening displacement, or the strain and stress field, of a static, elastic-plastic crack tip is of no significance for the fracture criterion describing conditions for continued growth.

Recently a perfectly plastic solution to a steadily growing crack tip in antiplane strain has been presented by Chitaley and McClintock [I], where a finite crack opening is obtained giving a finite energy release to the crack tip. Rice [2] has shown that a growing elastic- plastic crack tip should absorb all the energy from the surrounding elastic areas in the continuum-plastic region.

This has stimulated an investigation of steadily moving elastic- plastic crack tips with finite element techniques, for the antiplane strain situation and for plane stress. The crack tip region, with the elastic singular stress field as a boundary has been simulated by a finite element mesh. Crack growth is modelled by simultaneous relaxation of the forces at the crack tip node, and redistribution of the boundary forces in an incremental representation (see Fig. i). After a "step of growth" the mesh is moved to the right, so that the next step is performed with the crack tip in the same position as the foregoing one, thus enabling an infinite process. Convergence to a steady state is obtained after a growth of about the length of the plastic zone in the static case, at constant stress-intensity factor in the elastic boundary field. Note that no fracture criterion is involved, a certain load, giving a static plastic-zone, is imposed, and then the propagation process is forced upon the system.

The f i n i t e e lement program i s a mod i f i ed v e r s i o n of a program c o n s t r u c t e d and use& p r e v i o u s l y by the w r i t e r (Andersson [3] , [ 4 ] ) . I t enab les the d e s c r i p t i o n o f an i s o t r o p i c a l l y ha rden ing m a t e r i a l , i n c l u d i n g e l a s t i c u n l o a d i n g in an i n c r e m e n t a l p l a s t i c i t y f o r m u l a t i o n . The same mesh and e lement t ype ( t r i a n g u l a r , c o n s t a n t s t r a i n ) i s used f o r bo th the p l ane s t r e s s and the a n t i p l a n e s t r a i n case . Thus an e s t i m a t e of the accuracy may be o b t a i n e d from comparisons wi th the t h e o r e t i c a l s o l u t i o n f o r the s t a t i c crack t i p in a n t i p l a n e s t r a i n by Rice [5].

The J-integral ~u

J = I (WedY - T ~ ds) (1)

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describes the total energy release t o the region inside the contour, at steady state Conditions, if W is taken as the elastic strain

e energy

e

1j

W e = I ~kldekl (2) o

A measure of the energy reaching the crack t i p i s obta ined from the r e l a x a t i o n of the crack t i p node (see Fig. 1)

u 1 7. F(v)dv (,3) =~-

o

The r e s u l t s show tha t the d i s c r e t e process gives an accura te d e s c r i p t i o n of the energy t r a n s f e r only fo r m a t e r i a l s with cons iderab le hardening. The macro-sca le r e s u l t s , f o r i n s t ance p l a s t i c zone shapes, are given sa t i s fac tor i ly also in n e a r l y non-hardening cases . Compari- son of existing theoretical results with the present solutions show an agreement within, typically, 5 percent.

As examples are shown the p l a s t i c zones f o r d i f f e r e n t hardenings in the plane s t r e s s case (Fig. 2) and the p ropor t ion of r e l e a s e d energy that is transmitted to the crack tip as a functioning of the hardening (Fig. 3). The fact that a secondary plastic zone occurs only for very low hardening and that the plastic zone completely surrounds the crack tip for strongly hardening materials should stimulate the search for steady-state solutions in terms of hardening material models where the crack tip is embedded in the primary plastic zone.

REFERENCES

[1] A. D. Ch i ta ley and F. A. McClintock, Journal of the Mechanics and Physics of Solids 19 (1971) 147-163.

[2] J. R. Rice, Proceedings, First International Congress on Fracture 1 (1965) A18.

[3] H. Andersson, Journal of the Mechanics and Physics of Solids 20 (1972) 33-51.

[4] H. Andersson, Journal of the Mechanics and Physics of Solids 21 (1973) ( to appear) .

[5] J . R. Rice, Journal of Applied Mechanics 34 (1967) 287-298.

13 February 1973

Int Journ of Fracture 9 (1973)

W

233

Figure I.

///iiI-"~-------" / [ ] Primary plastic zone "-,\

~ ' ' I ~ [ ] Elastic wake

/ / /

v !

Sketch o f r eg ion ana lyzed wi th f i n i t e e lement mesh. P" r e p r e s e n t s t he e x t e r n a l boundary c o n d i t i o n g iven by the e l a s t i c s i n g u l a r f i e l d . F i s t he r e l a x i n g noda l f o r c e g iv ing the crack opening v f o r a new crack t i p a d i s t a n c e A f u r t h e r ahead. The con tour r i s used f o r e v a l u a t i o n o f t he J - i n t e g r a i .

/ ~ / ~ ¢ " ' ~ ~ , ErIE = 114

/( \ 1 •

ET/E = 11100

Figure 2. Plastic zones in plane stress for two relationships be- tween the tangent modulus E T and the Young's modulus E in a linearly hardening material. Dashed curves represent the static and solid curves the growing crack case.

Figure 3.

1.0

0.5

2y/J

\ \ \

In E/ET

R e l a t i o n s h i p between energy r e l e a s e d to t he crack t i p r e g i o n , J , and energy t r a n s f e r r e d t o the f r a c t u r e p rocess at the very crack tip, 27.

Int Journ of Fracture 9 (1973)