Yh. Chen et N. Hasebe, EXPLICIT FORMULATIONS OF THE J-INTEGRAL CONSIDERING HIGHER-ORDER SINGULAR TERMS IN EIGENFUNCTION EXPANSION FORMS - PART I - ANALYTICAL TREATMENTS, International journal of fracture, 85(1), 1997, pp. 11-34
A semi-infinite crack with a nonlinear zone around the crack tip is st
udied in detail in the following four cases: (i) antiplane shear defor
mation, (ii) plane deformation, (iii) plane anisotropic deformation wi
th purely imaginary characteristic roots, and (iv) interface between d
issimilar solids, respectively. The complete Williams eigenfunction ex
pansion forms including both positive and negative powers of a distanc
e from the crack tip in each of the four cases are considered, which c
ould be used to describe the elastic state in an annulus around the no
nlinear zone. Explicit formulations of the path independent J-integral
are presented by utilizing the differential property and the so-calle
d pseudo-orthogonal property of the complete Williams expansion forms
in each of the four cases. It is shown that the complete Williams expa
nsion forms in every case mentioned above have two kinds of contributi
ons to the J-integral. The first one is similar to the traditional one
arising from the well-known r(-1/2) singularity (or r(-1/2+i epsilon)
singularity for the interface crack). The second one is a summation i
nduced from the interaction of higher order singular terms and nonsing
ular terms of the expansion forms. Although the coefficients of the co
mplete Williams expansion forms-in each of the four cases should be de
termined not only by the prescribed outer boundary conditions but by s
ome specific material model for the nonlinear zone surrounding the cra
ck tip, once they are determined by whatever method, the J-integral co
uld be calculated by using the formulations derived in this paper with
out any difficulties. It is found also that the elastic T-term acting
parallel to the crack plane has no direct interaction with the higher
order singular terms such that has no direct effect to the J-integral,
although the presence of the T-term will dramatically affect the size
of the nonlinear zone and in this way affect the coefficients of the
higher order singular terms and in turn the values of the J-integral.
The present results in the four cases support the conclusion derived b
y Hui and Ruina (1995) that the nonsingular terms and the higher order
singular terms in the complete Williams expansion forms are of equal
importance. Thus, in order to improve and to confirm the small scale y
ielding description, not only the nonsingular terms, but also the high
er order singular terms should be determined for a given crack configu
ration, body geometry, loading conditions and the prescribed material
model in the nonlinear zone.