Functional Analysis and Applications Seminar 

25/02/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
Roland Duduchava, A. Razmadze Mathematical Institute, Academy of Sciences, Tbilisi, Georgia

Interface Cracks in Anisotropic Composite Materials

The linear model equations of elasticity give rise to oscillatory solutions in some vicinity of interface crack fronts. In this presentation we apply the Wiener-Hopf method which yields the asymptotic behaviour of the elastic fields and, in addition, criteria to prevent oscillatory solutions. The exponents of the asymptotic expansions are found as eigenvalues of the symbol of corresponding boundary pseudodifferential equations. The method works for three-dimensional anisotropic bodies and we demonstrate it for the example of two anisotropic bodies, one of which is bounded and the other one is its exterior complement. The common boundary is a smooth surface. On one part of this surface, called the interface, the bodies are bounded, while on the complementary part there occurs a crack. By applying the potential method, the problem is reduced to an equivalent system of boundary pseudo-differential equations (BPE) on the interface with the stress vector as unknown. The BPEs are defined via Poincaré-Steklov operators. We prove the unique solvability of these BPEs and write a full asymptotic expansion of the solution near the crack front. The resulting asymptotic expansion for the stress field has a singularity of order $-1/2$ at the boundary if written as a function of the distance to the boundary and the parameter along the boundary. In the general case such solution has logarithmic (so called "shadow") singularities and oscillate. Both of them deteriorate the solution and prevent the decoupling of three important modes. If these singularities eliminate, the asymptotic simplifies significantly and all three modes decouple automatically. In the joint work with M. Costabel and M. Dauge there were found conditions preventing the appearance of logarithmic terms in the asymptotic. In the joint work with A.M. Sändig and W.L. Wendland there was found a criterion preventing oscillation of the solutions. We investigate more detailed the interface crack problem for isotropic bodies. We present a simple criterion in terms of shear moduli and the Poisson ratios and give a rigorous justification for the three-dimensional case. The same result was already formulated by Williams and Ting for two-dimensional bodies without rigorous justification. Some explicit results are available for transversally isotropic bodies as well (O. Chkadua).