Compression of a composite material along a macrocrack near the surface

As is known, the methods of linear fracture mechanics cannot be used to study the compression of a material with cracks by forces directed along the planes of the cracks. This is because the component of the load acting along the crack has no effect on the stress-intensity factor or crack-opening, thus rendering classical fracture criteria irrelevant for the given case. Problems of the type just described are usually studied by an approximate approach based on the use of applied theories of beams, plates, and shells (see [1], for example). In this approach, determination of the critical loads corresponding to the beginning of fracture reduces to examination of the stability of the part of the material between the crack and the surface of the body. This interlayer is replaced in the analysis by unidimensional or two-dimensional models. Such a method does have certain shortcomings. First of all, the critical loads found in this manner depend to a significant extent on the idealized support conditions. Secondly, the method cannot be used when the relative distance between the crack and the surface of the material is large. The author of [2] proposed that problems of the type we are concerned with here be studied by a method in which the initial stage of fracture is linked with the phenomenon of local loss of stability by the material in the neighborhood of the crack. In this case, the critical loads are determined using relations from the linearized three-dimensional theory of stability [3]. Since this approach is exact, it has none of the restrictions or deficiencies of approximate methods. It should be emphasized that only the initial stage of fracture is examined here. Contact between the edges of the crack is not considered. Studies in which a linearized formulation was used to solve problems for different crack geometries were surveyed in [4, 5] and breaking loads were determined for certain material models. In [6], the focus was on an axisymmetric problem for a semiinfinite composite with a disk-shaped surface crack. Critical loads for certain specific composites w e r e reported and the validity of various approximate methods of calculation was discussed. In the present article, we use a linearized approach to determine the critical forces corresponding to nonaxisymmetric modes of instability in the initial stage of fracture of a half-space occupied by a composite material having a circular crack near its surface. The half-space is subjected to uniform biaxial compression. As in the axisymmetric problem in [6], we will study cracks the smallest dimensions of which are substantially greater than the dimensions of the structural elements of the composite. We will also examine situations in which the composite displays the properties of a piecewiseuniform medium (such as in interfacial fracture, etc.). Proceeding on the basis of these assumptions, we will use a continuum model with corrected characteristics to represent the anisotropic composite. Let a crack of radius a be located in the plane x 3 = 0 of an upper half-space x 3 _> h . For the given loading scheme, the subcritical stress-strain state that develops in the material is uniform: