柱坐标下推广的Stoney模型的应变能密度

Strain Energy Density of Generalized Stoney Model with Cylindrical Coordinate

摘要: 材料的应力与应变关系是理解材料力学性能的重要方面，由此可以进一步得到材料的应变能函数，从而可以得到其它一系列力学特性。Stoney模型是从薄膜基底的应力应变关系出发，首先得到应变能函数，从而推得膜内应力与曲率的关系。这里，我们推广Stoney模型，考虑z方向的应力和应变，以及和z方向相关的剪切应变，得出柱坐标下的任意一点的应变能函数，并考虑膜内面间的均匀失配应变，得到体系的应变能的积分表达式。

Abstract: The relation of strain-stress is an important aspect of understanding mechanical property of materials. According to it, we can further obtain the strain energy function of materials, and a series of other mechanical properties. The Stoney model is based on the relation of strain-stress of substrate, and the strain energy function is first obtained, and then the relation between stress in film and curvature is achieved. At the present case, we generalize the Stoney model, considering the strain and stress of z direction, and as well as shear strain related to z direction, and we obtain the strain energy function at any point with cylindrical coordinate. In addition, we obtain the integral expression of strain energy of system by considering of the in-plane uniform mismatch strain of the film.