Plane Elastic Behavior and Stress Intensity Factor for an InhomogeneousInfinite Medium with a Griffith Crack

摘要

In recent years, inhomogeneous materials such as functionally graded materials (FGMs) have been developed and collected the technical interests as new functional and intelligent materials that are adaptable to super high temperature environments. For instance, FGMs of thermal stress relief type are the inhomogeneous media which have arbitrary graded compositional profiles of ceramics and metal for securement in heat resistance and material strength. In general, it seems to be difficult to develop elastic problems for a medium with arbitrary inhomogeneous material properties analytically. As one of methods of analytical development for inhomogeneous medium in which material properties are changed along one direction, the method based on the assumption that such inhomogeneous medium can be treated as multilayered homogeneous media has been developed [1-5]. The other is the method of analytical development based on the assumption that the inhomogeneity in material properties can be expressed by elementary functions with respect to one coordinate variable. For such inhomogeneous medium, an analytical development for the polar coordinate system has already been proposed by M. K. Kassir [6, 7] under the assumption that the shear modulus of elasticity is changed arbitrarily with power product form of axial coordinate variable. He has solved several kinds of boundary value problems in an axisymmetrical or a three dimensional state with a single displacement function governed by a differential equation with second order. However, the fundamental equation system proposed is not sufficient to solve the boundary value problems. Under the circumstances above-mentioned, one of authors has attempted to reconsider the fundamental system of equations in an axisymmetrical or a three dimensional state of isothermal elastic and thermo-elastic problems, and have successfully established the fundamental equation system making use of two kinds of displacement functions [8-11], which are represented by two kinds of differential equations with second order. And we have treated several kinds of boundary value problems and have proved appropriateness of the method of analytical development which we have proposed. Furthermore, we have attempted to establish the fundamental system of equations in plane problems [12] for inhomogeneous medium in which material property is changed along one direction. As an example, we solve analytically a singular stress problem for an inhomogeneous infinite medium with a Griffith crack under the condition that uniformly distributed loads are acted on the crack surfaces. And we carry out numerical calculations and examine the effect of inhomogeneous material property on the distributions of components of displacement and stress, and stress intensity factor at the tip of a crack.