Page 368 - Biaxial Multiaxial Fatigue and Fracture

352 c. CAL~ R. CITARELLA AND M. PERRELLA

and attain a value of [16]. At the end, high resolution accuracy at the comer may not be
worth pursuing, in any case, because the geometry where the crack intersects the surface is
expected to adjust under loading so as to ameliorate the stronger singularity [ 171.

The incremental direction. The crack growth direction and SIF equivalent are computed by the
minimum strain energy density criterion. The criterion for three dimensional problems can be
found in [18]. The explicit expression of strain energy density around the crack front can be
written as:

dW s(0) +0(1)
-=-
dV r.cosq5

where S(8) is given by
S(0) = a,, - K: + 2a,, - K, K, + a22 Ki + aY3 . Ki!
(9)

and

a,, = -. 1 (3 - 4v - cos@). (1 + COSO)
16rp
1
aI2 = - -sine. (cos0 - 1 + 4v)
8v
a22 = -. 1 [4(1- v) (1 - cos@) + (3 cos 0 - 1). (1 + cos 011
1 6n,

in which pstands for the shear modulus of elasticity and v is the Poisson ratio. S/cos(
represents the amplitude of intensity of the strain energy density field and it varies with the
angle 4 and 0. It is apparent that the minimum of S/cosb always occurs in the normal plane of
the crack front curve, namely +=O. S is known as strain energy density factor and plays a
similar role to the stress intensity factor.
The theory is based on three hypotheses:
1. The direction of the crack growth at any point along the crack front is toward the region
with the minimum value of strain energy density factor S as compared with other regions on
the same spherical surface surrounding the point.
2. Crack extension occurs when the strain energy density factor in the region determined by
hypothesis S = &in, reaches a critical value, say SCr.
3. The length, r,, of the initial crack extension is assumed to be proportional to Sm,, such that
Smi,,/r0 remains constant along the new crack front.
It can be seen that the minimum strain energy density criterion can be used both in two and
three dimensions. Note that the direction evaluated by the criterion in three dimensional cases is
insensitive to KII~. The crack growth direction angle 8, is obtained by minimising the strain
energy density factor S(8) 6f Eq.(9) with respect to 8. The minimum strain density factor S(0,)