文档介绍：The Energy Principles
and the Variational Calculus
Chapter 10
Elasticity
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第十章能量原理与变分法
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§10-1 The Specific Energy of Deformation and Strain
Energy of Elastic Body
§10-3 The Variational Method of Displacement
§10-4 The Variational Equations of Stress and the
Variational Method of Stress
§10-2 The Variational Equation of Displacement and
The Principle of Minimum Potential Energy
The Basic Solution of The Temperature Stress Problems
Chapter 10 The Energy
Principles and theVariational Calculus
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第十章能量原理与变分法
§10-1 弹性体的变形比能与形变势能
§10-3 位移变分法
§10-4 应力变分方程与应力变分方法
§10-2 位移变分方程与极小势能原理
能量原理与变分法
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The problems of the elastic theory need a series of partial differential equations. It always meet with difficulties in math. So we have to seek for approximate solution. The approximate solution of variational calculus is a method mon use. In math, variational problems are the problems to solve the limit of the functional. In elastic mechanics, the functional is the energy(work) of elasticity problems. And the variational calculus is to calculate the extremum of the energy(work). We can yield the solution of the elastic problems when calculating the extremum. The direct method of variation problems facilitate us to obtain the approximate solution.
We will give the expressions which calculate the potential energy of strain at first in this chapter. We will mainly introduce the variational method of displacement and that of stress through the relation of work and energy.
The Basic Solution of The Temperature Stress Problems
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弹性理论问题需要解一系列偏微分方程组,并满足边界条件,这在数学上往往遇到困难。因此需要寻求近似的解法。变分法的近似解法是常用的一种方法。在数学上,变分问题是求泛函的极限问题。在弹性力学里,泛函就是弹性问题中的能量(功),变分法是求能量(功)的极值,在求极值时得到弹性问题的解,变分问题的直接法使我们比较方便地得到近似解。
本章首先给出计算形变势能的表达式。利用功与能的关系,主要介绍了位移变分法和应力变分法。
能量原理与变分法
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§10-1 The Specific Energy of
Deformation and Strain Energy of Elastic Body
The specific energy expressed by ponents of stress
I. The specific energy of deformation
Under plex stress conditions, suppose that the