Anisotropic Media with a Crack or a Rigid Line Inclusion,

Abstract

A crack, or cracks, in a material is perhaps one of the most studied problems in solid mechanics. This is due to the fact that many structural failures are related to the presence of a crack in the material. The knowledge of stress distribution near a crack tip is indispensable in a fracture mechanics analysis (Rice, 1968; Sih and Liebowitz, 1968; Sih and Chen, 1981; Kanninen and Popelar, 1985; K. C. Wu, 1989a). A crack is represented by a slit cut whose surfaces are assumed traction-free. This is a mathematical idealization. For a composite material that consists of stiff short fibers or whiskers in the matrix, we have rigid line inclusions. A rigid line inclusion is the counterpart of a crack. It is sometimes called an anticrack. The displacement at a rigid line inclusion either vanishes or has a rigid body translation and rotation. One of the puzzling problems for a crack is the one when it is located on the x1-axis that is an interface between two dissimilar materials. The displacement of the crack surfaces near the crack tips may oscillate, creating a physically unacceptable phenomenon of interpenetration of two materials. The bimaterial tensor Š introduced in Section 8.8 plays a key role in the analysis. If Š vanishes identically, there is no oscillation. If Š is nonzero, we may decompose the tractions applied on the crack surfaces into two components, one along the right null vector of Š denoted by to and the other on the right eigenplane of Š denoted by tγ . The solution associated with to is not oscillatory. It has the property that the traction on the interface x2=0 is polarized along the right null vector of Š while the crack opening displacement is polarized along the left null vector of Š. The solution associated with tγ is oscillatory. It has the property that the traction on the interface x2=0 is polarized on the right eigenplane of Š while the crack opening displacement is polarized on the left eigenplane of Š.