Asymptotic behaviour at a tip of a rigid line inclusion in linearized elasticity

Abstract

Key words Linearized elasticity, singularities at a tip of rigid line inclusion, delamination, invariant integral, Irwin's formula.We consider an asymptotic behaviour of a solution near a tip of a rigid line inclusion in two dimensional homogeneous isotropic linearized elasticity. By means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution around there. Furthermore, we give expressions of the invariant integral and the Irwin's formula.

Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body

“…In spite of one-to-one correspondence between the spaces H(Ω 0 ) and H(Ω ε ), the set of admissible displacements K 0 (Ω 0 ) is not mapped into the set of admissible displacements K ε (Ω ε ) of perturbed problem under mapping (11). There is no such correspondence even for a rectilinear rigid inclusion, see [14,36]. At first, it is connected with the fact that the unit normal vector to γ 0 is not mapped into the unit normal vector to the perturbed crack γ ε .…”

“…By using asymptotic formulas (14), describe the functions belonging to the set K ε (Ω 0 ). By ν ε , denote the image of the normal vector ν ε to γ ε defined on γ 0 .…”

“…Let us note the following papers: [4,8,[14][15][16]19,35,36] presenting the investigation of different models of elastic bodies with rigid inclusions and cracks with both linear and nonlinear boundary conditions on the crack faces.…”

Contacting crack faces within the context of bodies exhibiting limiting strains

“…The inner product on Sym(R d×d ) assigns σ : ε = ∑ d i,j=1 σ ij ε ij and is associated with the matrix norm ∥σ∥ = √ σ : σ. Our aim is to specify a class of functions F in (9), which exhibit limiting small strain behavior and guarantee solvability of the problem (1)- (8).…”

“…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]. Numerical methods suitable for this class of nonlinear crack problems were developed in [14].…”