Description of loadings and screenings of cracks with the aid of universal weight functions

Abstract

The stress intensity vector K i is defined as the limiting behaviour of the stress near the tip of a crack, the stress components being proportional to r-1/2 for any external loading. Internal stresses caused by dislocations show the same power dependence at the crack tip; the stress intensity associated with a loading can thus be screened (or amplified) by a plastic zone. Since for any particular specimen and crack geometry the stress intensity vector must be a functional of the loading and screening which are of vectorial character (lines of forces fi or dislocations with a Burgers vector bi) one can define two tensorial weight functions, one for screenings, Dsi(x, a), and one for forces, F~i(x, a), so that the stress intensity K s can be found by integration over the product of weight functions and dislocation or force density. In order to find the weight functions the displacement field and the Airy stress vector must be known for some, completely arbitrary, loading or screening.

Weight function method for interface cracks in anisotropic bimaterials

“…This relation is consistent with those of Kirchner [14] and Sham and Zhou [16]. By comparison, it is seen that the interface weight function formulation is completely analogous to its homogeneous counterpart except for a different moduli matrix, e.g.…”

“…The similarity between (12)(13)(14)(15)(16) and (28)(29) is striking. The vector potential function ~(z) is apparently analogous to the isotropic scalar potential ~b(z).…”

“…[12,13]) for arbitrary geometry. The weight function method has also been studied for cracks in a homogeneous solid with anisotropic elastic properties [14][15][16].…”

Higher Order Weight Functions in Fracture Mechanics of Multimaterials

“…The weight functions play the role of influence functions for stress intensity factors, since the weight function value at a point situated inside the body or at its surface (including crack faces) is equal to the stress intensity factor, which is due to the unit concentrated force applied at this point. The weight function based functionals can be constructed not only for external forces but also for the dislocation distributions described by the dislocation density tensor, as it was shown by Kirchner [14]. The objective of the weight function theory is not to compute complete stress distributions in cracked bodies for an arbitrary loading, but to express only one parameter K characterizing the strength of the near-tip stress field as a functional (weighted average) of the loading.…”

Calculation of the stress intensity factor for arbitrary finite cracked body by using the boundary weight function method

“…A finite element implementation of the variational principle is given and this leads to a unified approach in the numerical computations of weight functions for all three fracture modes. Two-dimensional weight functions in arbitrary anisotropic solids under mixed boundary conditions are analyzed by Sham The weight function method has also been studied for cracks in a homogeneous solid with anisotropic elastic properties by Kirchner [17], An [18] and Gao [19].…”