On the Crack Quasi-Static Growth

The theoretical model of quasi-static crack growth in the elastic-plastic material under load variation in a wide range. Small-scale yielding is principal assumption and main restriction of proposed theory. The model of crack growth provides for continues and interrelated both the crack propagation and plastic deformation development. The nonlinear first-order differential equation describes the quasi-static process of crack growth. In dimensionless form this equation invariant in respect to geometrical configuration and material. The critical size of the plastic zone is proposed as the characteristics of material resistance which is directly connected with the fracture toughness, but more convenient in practical applications of invariant equation. The demonstration of solution is performed for the double cantilever beam that widely used as the standard (DCB) sample for measurement of the mode-I interlaminar fracture toughness. he short analysis of some properties of solution of the invariant equation and its application is done.

Modification of gravity by a spherically symmetric wormhole

We examine corrections to the standard Newton's law due to the presence of a spherically symmetric wormhole. We show that the Newtonian potential can be decomposed into two terms. The first term does not depend on the wormhole metric and corresponds to the standard law. It obeys to the conformal invariant equation. The second term explicitly depends on the wormhole metric and corresponds to corrections. We show that such corrections can be described in terms of a polarization energy density.

Construction of dipole type singular solutions for a biharmonic equation with critical Sobolev exponent

In this paper, we construct positive weak solutions of a fourth order conformally invariant equation on S