﻿ 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

International Journal of Mechanics Research
Vol.04 No.04(2015), Article ID:16383,5 pages
10.12677/IJM.2015.44009

Strain Energy Density of Generalized Stoney Model with Cylindrical Coordinate

Jia Li1*, Junjie Shi2, Huizhao Liu1

1School of Science, Hebei University of Technology, Tianjin

2State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, Research Center for Wide-Gap Semiconductors, School of Physics, Peking University, Beijing

Received: Oct. 30th, 2015; accepted: Nov. 17th, 2015; published: Nov. 20th, 2015

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.

Keywords:Strain and Stress, Mechanics, Stoney Model, Shear Strain, Strain Energy

1河北工业大学理学院，天津

2北京大学物理学院，宽禁带半导体研究中心，人工微结构和介观物理国家重点实验室，北京

1. 引言

2. 应变能函数

2.1. Stoney模型的应变能函数

(1)

(2)

(3)

(4)

Stoney模型假设z方向应力为零，且不存在剪切应变，这样则(3)式则变为

(5)

(6)

(7)

(8)

(9)

2.2. Stoney模型应变能函数的推广

(10)

(11)

(12)

(12)式即为体系内任一点处的应变能密度。

(13)

(14)

W的含义是膜内任意一点的应变能密度，即单位体积的应变能，如果想得到整个体系的应变能，则需要对整个体系进行积分。即表达为如下形式

(15)

3. 结论

Strain Energy Density of Generalized Stoney Model with Cylindrical Coordinate[J]. 力学研究, 2015, 04(04): 71-75. http://dx.doi.org/10.12677/IJM.2015.44009

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2. 2. 李佳, 史俊杰, 吴洁君, 刘辉召, 齐浩然. GaN-蓝宝石异质厚膜体系界面应力特性研究[J]. 力学研究, 2014(3): 55-64.

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4. 4. 杨海波, 曹建国, 李洪波编著. 弹性与塑性力学简明交城[M]. 北京: 清华大学出版社, 2011.

5. NOTES

*通讯作者。