Bases of asymptotic theory of beams, plates and shells are stated. The comparison with classic theory is conducted. New classes of thin bodies problems, for which hypotheses of classic theory are not applicable, are considered. By the asymptotic method effective solutions of these problems are obtained. The effectiveness of the asymptotic method for finding solutions of as static, as well as dynamic problems of beams, plates and shells is shown.
For a plate subject to stress boundary condition, the deformation determined by the Reissner–Mindlin plate bending model could be bending dominated, transverse shear dominated, or neither (intermediate), depending on the load. We show that the Reissner–Mindlin model has a wider range of applicability than the Kirchhoff–Love model, but it does not always converge to the elasticity theory. In the case of bending domination, both the two models are accurate. In the case of transverse shear domination, the...
We study the 3-D elasticity problem in the case of a non-symmetric heterogeneous rod. The asymptotic expansion of the solution is constructed. The coercitivity of the homogenized equation is proved. Estimates are derived for the difference between the truncated series and the exact solution.
Crack propagation in anisotropic materials is a persistent problem. A general concept to predict crack growth is the energy principle: A crack can only grow, if energy is released. We study the change of potential energy caused by a propagating crack in a fully three-dimensional solid consisting of an anisotropic material. Based on methods of asymptotic analysis (method of matched asymptotic expansions) we give a formula for the decrease in potential energy if a smooth inner crack grows along a...