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

Lurie Lurie,Volkov-Bogorodsky Volkov-Bogorodsky,Vasiliev Vasiliev

doi:10.3390/mca24040093

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.

Highlights

The problem of singularities in the theory of elasticity and in fracture mechanics is widely discussed in the related scientific literature [1,2,3]
Only non-singular solutions have been constructed in gradient fracture mechanics for test problems corresponding to cracks of Mode III [7]
We propose 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 representation in a symmetric form, convenient for applications

Summary

Introduction

The problem of singularities in the theory of elasticity and in fracture mechanics is widely discussed in the related scientific literature [1,2,3]. The general solution of problems of the plane theory of elasticity can be expressed using two analytical functions in the form of Equation (5), and in equivalent form using one harmonic function f = F1(w) + F2(w) (i.e., through Papkovich–Neuber complex potential):. Consider again the Papkovich–Neuber representation (see [27]), which allows us to present a general solution to the problem of the theory of elasticity; that is, both displacement vector u and their corresponding stresses σ(u) on a surface element with a normal vector n, through only one harmonic vector potential f (in accordance with Theorem 1):. Strain–stress state for Mode I, II, and III singular cracks can be described on the basis of the Papkovich–Neuber formulas with the aid of one complex potential with a fractional degree These forms are convenient for constructing a generalized gradient regular solution. The field of local stresses will describe the non-singular crack solutions

Regular Gradient Solutions in Crack Mechanics

Conclusions