STRESS STATE ANALYSIS OF SEMI-INFINITE LAYER WITH CYLINDRICAL CAVITIES

Abstract

The subject matter of the article is the stress-strain state of a semi-bounded isotropic elastic layer with N longitudinal cylindrical cavities under prescribed displacements on all surfaces of the body. The goal is to develop an analytical-numerical methodology for solving the spatial elasticity problem for the semi-bounded layer with cylindrical cavities under kinematic boundary conditions and to investigate the stress state for four qualitatively distinct variants of boundary conditions differing in the parity of the prescribed displacement components with respect to the coordinate z. The tasks to be solved are as follows: to construct a solution of the Lamé equations in Cartesian and local cylindrical coordinate systems; to reduce the problem for the semi-bounded layer to an equivalent problem for an infinite layer; to satisfy the boundary conditions on all surfaces and reduce the problem to an infinite system of linear algebraic equations; to obtain numerical stress distributions on the layer boundaries and cavity surfaces; to verify the methodology against known solutions. The methods used are the generalized Fourier method with addition theorems for transition between coordinate systems; the mirror reflection method, whereby assigning even or odd displacements on the horizontal boundaries automatically enforces «smooth wall» or «free face» conditions on the plane z = 0; and the reduction method for solving the resulting infinite system of linear algebraic equations of the second kind. Conclusions. Even-parity loadings give rise to nearly the full set of stress components in the cross-section of the cavity, whereas odd-parity loadings produce a predominantly antiplane stress state. Normal displacement produces stress concentration on the cavity surface 3.5–4 times greater than the tangential displacement of the same amplitude. An oscillatory stress distribution near the discontinuity points of the prescribed displacements is of a physical nature and is independent of the truncation order of the system. Scientific novelty. For the first time, the spatial elasticity problem for a semi-bounded layer with N cylindrical cavities under kinematic boundary conditions has been solved. The combination of the generalized Fourier method with the mirror reflection method is extended to a semi-bounded body with cylindrical inhomogeneity. The methodology is applicable to an arbitrary number of cavities and may serve as a benchmark for verification of finite element solutions.