Asymptotic solution for crack in orthotropic material and strain gradient elasticity

The asymptotic solution for the displacement and stress fields at the tip of the crack in homogeneous orthotropic material governed by the strain gradient elasticity based on the simplified Mindlin’s Form II gradient theory is derived. The Lekhnitskii-Eshelby-Stroh (LES) formalism is applied and generalized to the fracture problem in the strain gradient elasticity for an orthotropic material. The expressions for the displacements and stress functions are derived. According to the LES formalism, the Cauchy type stresses and double stresses are expressed as the partial derivatives of the stress functions. The stress singularity exponent is evaluated from the condition of the stress-free crack faces at its tip and consequently it is derived the expression for the -integral. The particular components of the asymptotic solutions for the displacements and the Cauchy type stresses are discussed as well as the strong singular auxiliary force tractions in front of the crack. The influence of the material symmetry orientation with respect to the crack plane orientation is shown. • The generalization of the Lekhnitskii-Eshelby-Stroh formalism to the fracture problem of an orthotropic material in the strain gradient elasticity based on the Mindlin’s Form II gradient theory. • The derivation of the asymptotic solution for the displacement and stress fields at the tip of the crack in homogeneous orthotropic material governed by the strain gradient elasticity. • The evaluation and derivation of the stress singularity exponent and the -integral from the stress and strain filed prevailing in the near crack tip region. • The discussion of the influence of the material symmetry orientation with respect to the crack plane orientation.