弹性体表面分层压缩失稳问题的数学弹性力学解

采用数学弹性力学的稳定平衡方程并结合富氏积分变换的方法研究了含表面平行裂纹的弹性体在压缩载荷下的表面分层失稳问题。导出了一级显式的精确齐次奇异积分方程组,然后.通过Gauss-Chebyshev积分公式,得到一组齐次代数方程组,从而求出临界压缩载荷。并将结果与经典的材料力学梁板稳定的研究方法所得结果进行了比较,指出经典方法误差太大而不适于求解此问题。最后,利用数学弹性力学解求出的等效弹性支承常数给出一个简单精确的临界压缩载荷计算公式。

A Solution for the Compressive Stability of a Sufrace Delamination in Elastic Solids Based on Mathematical Theory of Elasticity

In this paper, the compressive stability of a surface delamination of a elastic solid containing a crack parallel to the surface is investigated based on the stability equilibrium equations derived from mathematical theory of elasticity, the basic equations are reduced to a system of homogeneous singular integral equations by means of Fourier integral transform. The integral equations are solved numerically by the use of Gauss-Chebyshev integral formulae. Numerical results for critical compressive stress are presented with various geometrical parameters, and are compared with those obtained from classical theory of beam-plate stability by the strength of materials. The comparison of both results indicates that the results from classical theory are much large than hose from mathematical theory of elasticity, hence the classical theory is not suitable for this kind of stability problems. Finally. a simple but accurate approximate formula for the estimation of the critical compressive stress is proposed based on the equivalent elastic support coefficient obtained from present paper and the Euler formula from classical theory for the case of elastic supported ends.