Investigation of the behavior of a mixed-mode crack in a functionally graded magneto–electro-elastic material by use of the non-local theory

• Eringen's non-local theory has been extended to investigate the mixed-mode crack problem in a functionally graded magneto-electro-elastic medium.
• The mixed-boundary value problem is solved for the electric displacement, stress and magnetic fields by converting the governing equations into a system of four integral equations.
• In contrast to classical theories, the calculated stresses, electric displacements and magnetic fields are devoid of singularities at the crack tips.
• The maximum stress hypothesis can potentially be used as a fracture criteria for functionally graded magneto-electro-elastic materials with mixed mode cracks.

In this paper, we consider the problem of a mixed-mode crack embedded in an infinite medium made of a functionally graded magneto–electro-elastic material (FGMEEM) with the crack surfaces subjected to magneto–electro-mechanical loadings. Eringen’s non-local theory of elasticity is applied to obtain the governing magneto–electro-elastic equations. To make the analysis tractable, it is assumed that the magneto–electro-elastic material properties vary exponentially along a perpendicular plane to the crack. Using Fourier transform, the resulting mixed-boundary value problem is converted into four integral equations, in which the unknown variables are the jumps of mechanical displacements, electric and magnetic potentials across the crack surfaces. To solve the integral equations, the jumps of displacements and electric and magnetic potential across crack surfaces are directly expanded in a series of Jacobi polynomials and the resulting equations are solved using the Schmidt method. Unlike classical magnetic, electric and elasticity solutions, it is found that no mechanical stress, electric displacement and magnetic flux singularities are present at the crack tips. This enables the use of the maximum stress as a fracture criterion. The primary objective of this study is to investigate the effects of crack length, material gradient parameter describing functionally graded materials and lattice parameter on the mechanical stress, magnetic flux and electric displacement field near crack tips.