Crack Paths 2009

where B’ is the compatibility matrix evaluated at point C .

The effective normal and

shear stresses, 0'

TM, acting on the samecrack plane but producedby the modified e,n ’

stress tensors o and o are expressed as follows: rel,n rel,s ’

G e m: (“rel,n Ts,n : ( 6 r e l , s By imposing that 0',” I oc(uc) and re,” : rc(uc), the correction factors sn, ss can be

determined by writing the corrected stress state at point C :

S :(¢'-i)-i_a,(u,): _ (not)

8 :(¢'-r)-j_T,(u,):1_

rcotc)

(15)

"

[(c;B'-s,)-i]-i

[(c;B'-s,)-i]-i’

S

[(c;B'-6,)-i]-j

[(c;B'-6,)-i]-j

and the stress tensor correction (12) becomes

6,, I 6 _ c; ]VS(N - (6,’, + 6m)“: 6 _ C: ]VS(N -6,,)]

(16)

The above stress-based formulation of the discontinuous displacement field can be

reinterpreted by a variational approach considering Eq. (5). Fromeq. (16), we have:

(V1611 *)6,,,dQ I (V1611 *){6 _ c; [VS(N - (aW_,))]}dQ

I (on *bdQ + (on *tdF I

o

o

o

r,

(as *’ B’cdQ I (as *’ B’C B - SWdQ+ (a; *’ N’ bdQ + (as *’ N’ tdF

(17)

o

o

o

r,

Since the variation of the displacementfield is arbitrary, w eobtain:

jB’c Bans I IN’ bdQ + IN’ or + (BoB - SWdQ i.e.

o

o

r,

o

(18)

s I K*1(r1 + xsm)

F r o meqs (15), w ehave:

_

0,01,)

_ _

_

tc(uc)

_ ‘

‘

:

6W’s_l[(C:B'-6u)-i]-i

1) MK) 6”+{[(C:B'-8v).i].j

1} M") at

(19)

A

B

I +N(x)[A(Q,6,oC(uC)) - a, + B(Q,6,r,(uC)) - 6,]

and equation (18) can be rewritten as follows:

jB'cBdoaI IN’ bdQ + ( NtdF + ( B oB -N[A(Q,6,oE(uC)) - a, + B(Q,a,1,(u,)) - 5,1619

9

o

r,

Q

or a I K1[r, _ jB’cB -N[A(Q,6,a,(u,))-a,

+ B(Q,6,t£(uC)) - 6,1619]

(20)

Q

It can be observed as the last espression in eq.(l9) is similar to eq.(3). Thediscontinuity

vector 6W’, I 6M + 6H must be evaluated by an iterative process summarised in eq.

(19).

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