Edge effects in a sandwich plate: a plane problem

Abstract

Edge effects in a rectangular sandwich plate with isotropic components are studied. The mathematical model is represented by the homogeneous equations of linear elasticity, which is indicative of an approximate approach in edge-effect theory. The initial equations are reduced to inhomogeneous ones and an exact problem is formulated. Approximate solutions are found by the mesh method. Discrete problems are based on the concept of base scheme. The mesh equations are written in an explicit form and then solved using a computation optimization procedure. As an example, edge-effect zones in a real composite are analyzed.Keywords: edge effect, exact approach, free strains, sandwich plate, base scheme Introduction. Edge effects in deformable media can be studied using either exact or approximate approach. The exact approach employs exact mathematical and mechanical models and an exact (quantitative) criterion of edge-effect zones [3]. Other approaches are approximate.For the problem under consideration, the exact approach is characterized by the inhomogeneous equations of linear elasticity (exact mathematical model), a piecewise-homogeneous medium model (exact mechanical model), and a quantitative (exact) criterion of edge-effect zones. The exact approach allows determining, with prescribed accuracy, the boundaries of stress-concentration zones and analyzing the geometry and stress state within these zones.Almost all approximate approaches in edge-effect theory employ homogeneous (eigen)solutions, which are used in the case of self-balanced loading [1,2,4,8]. These solutions do not allow analyzing the stress state within the edge-effect zones.The criterion of edge effect is decrease in the surface load at a point of interest by a given factor. Since quantitative and qualitative characteristics of loading are disregarded, this criterion is called approximate (qualitative). Eigensolutions allow us to estimate quantitatively (to admit or reject) Saint Venant's principle.The present paper analyzes edge effects in a sandwich plate under the conditions of uniaxial loading and plane strain. The governing equations describe a plane homogeneous problem of elasticity for piecewise-homogeneous media and imply, as mentioned above, an approximate approach in edge-effect theory.We will set up inhomogeneous equations of elasticity. To derive inhomogeneous equations and to formulate an exact problem for edge effects, we will use the concept of free strains. The inhomogeneous boundary-value problem will be solved approximately by the mesh method based on the concept of base scheme [5,[9][10][11]. Discrete problems will be formulated and methods for their solution selected. Since the stresses σ ij may exhibit singularity near the free edge [2], we will use a computation optimization procedure to improve the accuracy of solving the mesh equations. An example will be given.1. Problem Formulation. Consider a three-layer rectangular plate with isotropic layers. The face layers are identical, which results in the geometrical symmetry of the s...

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