科研之友
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摘要
Numerous solutions of the plane theory of elasticity, derived over the past hundred years, provide a very good approximation to the in-plane stress components. However, an accurate evaluation of the out-of-plane (or transverse) stresses is often impossible as the three-dimensional stress field may not follow either plane stress or plane strain assumptions, specifically, near the edges formed by notches, cracks and other stress concentrators. For relatively simple geometries and loading conditions, the 3D stress and displacement fields can be analytically evaluated using the first-order plate theory. This theory can be considered as an elementary extension of the plane theory of elasticity and it retains the simplicity of the two-dimensional formulation. However, the derivation of exact solutions within this theory is far more difficult. In this paper, we present the governing equations of the first-order plate theory in arbitrary curvilinear coordinates, and based on these equations we develop a simplified method for the evaluation of the out-of-plane displacements and stress components. Several examples are considered to demonstrate the effectiveness and the accuracy of the new method. Due to simplicity it can be applied to investigate many 3D elastic problems involving sharp wedges and plate geometries weakened by cracks, notches and other stress concentrators. All these problems currently represent a great challenge for analytical approaches.
- 出版日期2017-7
