The Griffith formula and the Rice–Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains

Abstract

As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R2 is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.

On derivative of energy functional for elastic bodies with cracks and unilateral conditions

“…In conclusion, note that the formulae similar to (45) were obtained for isotropic twoand three-dimensional cracked bodies with conditions (5) at the crack faces provided that the perturbation <I>f of the domain describes the crack length change [1,2],…”

“…In the works [1,2] the appropriate technique of finding derivatives of the energy functional with respect to the crack length for unilateral boundary conditions is proposed, which can be used for a wide class of the unilateral problems. Qualitative properties of solutions (solution existence, solution regularity, dependence of solutions on parameters, etc.)…”

Contacting crack faces within the context of bodies exhibiting limiting strains

“…These equations are endowed with the homogeneous Dirichlet condition (4), the Neumanntype condition (5), and conditions (6)- (8) at the crack. The boundary traction σ n is continuous across the crack according to (6) and has zero tangential component by (7). The non-penetration inequality [[u n ]] ≥ 0 enforces the complementarity conditions (8) involving the normal stress [5] for further details.…”

“…In order to avoid the other drawback of the linearized theory, namely, the possibility of penetration between the crack faces, nonlinear crack problems subject to nonpenetration conditions have been previously established within the framework of the variational theory [5][6][7]. The principal challenge here is finding singular solutions at the crack tip [8,9] and obtaining a formula for the energy release rate [10][11][12], which is relevant to brittle as well as quasi-brittle materials fracturing [13].…”

Crack on the boundary of a thin elastic inclusion inside an elastic body

“…To model correctly the physical behavior mutual penetration should be forbidden, and this is achieved by means of inequality type boundary conditions. A theory with non-penetration conditions at the crack faces has been developed in the last years: the book [4] contains many results on crack models with the non-penetration conditions, the elastic behavior of bodies with cracks and inequality type boundary conditions is analyzed also in [5]. In our work we will indeed rely on some results belonging to this piece of literature.…”