Effect of Couple Stresses on the Stress Intensity Factors for Two Parallel Cracks in an Infinite Elastic Medium under Tension

Shouetsu Itou

doi:10.1155/2012/575891

Abstract

Stresses around two parallel cracks of equal length in an infinite elastic medium are evaluated based on the linearized couple-stress theory under uniform tension normal to the cracks. Fourier transformations are used to reduce the boundary conditions with respect to the upper crack to dual integral equations. In order to solve these equations, the differences in the displacements and in the rotation at the upper crack are expanded through a series of functions that are zero valued outside the crack. The unknown coefficients in each series are solved in order to satisfy the boundary conditions inside the crack using the Schmidt method. The stresses are expressed in terms of infinite integrals, and the stress intensity factors can be determined using the characteristics of the integrands for an infinite value of the variable of integration. Numerical calculations are carried out for selected crack configurations, and the effect of the couple stresses on the stress intensity factors is revealed.

Highlights

In the classical theory of elasticity, the differential equations of equilibrium are derived from the equilibrium of the forces for the rectangular parallelepiped element dx dy dz with respect to the rectangular coordinates x, y, z
For materials with microstructures, such as porous materials and discrete materials, the differential equations of equilibrium may be derived from a parallelepiped element, which, very small, is not infinitesimal
Ii As l/a approaches zero, KI/ p πa and KII/ p πa do not approach the corresponding value√s calculated using the classical theory of elasticity, whereas the values of M0/ pa πa approach zero, which is the value calculated by the classical theory of elasticity

Summary

Introduction

In the classical theory of elasticity, the differential equations of equilibrium are derived from the equilibrium of the forces for the rectangular parallelepiped element dx dy dz with respect to the rectangular coordinates x, y, z. A similar problem has been solved for an infinite medium containing an infinite row of spaced holes of equal diameter under tension in the linearized couplestress theory 3 In these studies 1–3 , the values of the stress concentration are shown to approach those for the corresponding classical solutions as l/r approaches zero, where 2r is the diameter of the holes. Sternberg and Muki solved the stress intensity factor around a finite crack in an infinite Cosserat medium under tension and revealed that the Mode I stress intensity factor is always larger than the corresponding value for the classical theory of elasticity 4. Gourgiotis and Georgiadis solved the Mode II and Mode III stress intensity factors for a crack in an infinite medium using the couple-stress theory and the distributed dislocation technique 9. The stress intensity factors and the couple-stress intensity factor are calculated numerically for several crack configurations

Fundamental Equations

Boundary Conditions

Analysis

Numerical Examples

Conclusion