A New Approach to Non-Singular Plane Cracks Theory in Gradient Elasticity

Abstract

A non-local solution is obtained here in the theory of cracks, which depends on the scale parameter in the non-local theory of elasticity. The gradient solution is constructed as a regular solution of the inhomogeneous Helmholtz equation, where the function on the right side of the Helmholtz equation is a singular classical solution. An assertion is proved that allows us to propose a new solution for displacements and stresses at the crack tip through the vector harmonic potential, which determines by the Papkovich–Neuber representation. One of the goals of this work is a definition of a new representation of the solution of the plane problem of the theory of elasticity through the complex-valued harmonic potentials included in the Papkovich–Neuber relations represented in a symmetric form, which is convenient for applications. It is shown here that this new representation of the solution for the mechanics of cracks can be written through one harmonic complex-valued potential. The explicit potential value is found by comparing the new solution with the classical representation of the singular solution at the crack tip constructed using the complex potentials of Kolosov–Muskhelishvili. A generalized solution of the singular problem of fracture mechanics is reduced to a non-singular stress concentration problem, which allows one to implement a new concept of non-singular fracture mechanics, where the scale parameter along with ultimate stresses determines the fracture criterion and is determined by experiments.

Higher-order asymptotic crack-tip fields in simplified strain gradient elasticity

“…From representation (25), it can be seen that within SGE we have the possibility to avoid the classical part of PN solution. Namely, we can avoid the terms r k from (25) by the appropriate choice of the constants a, b, c and d in ( 22), ( 23) since the modified Bessel functions have the same asymptotic behavior I k (r) ∼ O(r k ) around r = 0 (for discussion, see also [35,50]).…”

“…It can be shown, that the resulting eigen equation within SGE has the following form (similar result has been obtained in Refs. [8,9,19]): R(r, k, ν) sin 2 2kπ = 0 (35) where R(r, k, κ) is the function of radial coordinate r, order k and Poisson's ratio ν, which explicit form does not presented here for brevity and which does not provide additional roots for this equation. Relation (35) determines the following possible eigenvalues of the order k for the terms that can be obtained using PN solution for the in-plane crack problems:…”

“…Several fracture criteria have been suggested within SGE with respect to the surface tractions (that remain singular at the crack tip) [8,30] and with respect to the regularized stresses and strains [20,[31][32][33][34][35]. Experimental validation of these criteria and identification of SGE material constants for brittle and quasi-brittle materials with different types of cracks and sharp notches have been performed in Refs.…”

Steady-State Crack Growth in Nanostructured Quasi-Brittle Materials Governed by Second Gradient Elastodynamics

“…From (14), it is seen that if the Lamé potentials have the order r n , then the displacement field will have the order r n−1 . Therefore, all terms with n < 1 should be excluded from the displacement solution or combined with each other to provide their vanishing when r → 0 (corresponding discussion within SGET; see [37,38]). The strain energy density within SGET is evaluated accounting for the quadratic form of the strain gradients [5].…”

The Strain Gradient Viscoelasticity Full Field Solution of Mode-III Crack Problem

“…Li and Wang [33] studied a mode III crack in an elastic layer on a substrate and solved it with both volumetric and surface strain gradient are taken into account. Many other studies also solved different crack problems using strain gradient theories [34][35][36][37][38]. Meanwhile, the viscoelastic effect in the crack problem has been studied.…”