Elasticity 0:10 The Energy principles and the Variational Calculus
1 Elasticity
弹单生力 第十章分活
2
Chapter 10 The Energy Principles and the variational calculus 8 10-1 The Specific Energy of Deformation and Strain Energy 0 of Elastic body 8 10-2 The Variational Equation of Displacement and The Principle of minimum Potential Energy 8 10-3 The Variational Method of Displacement 810-4 The Variational Equations of Stress and the Variational Method of stress
3 §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 Chapter 10 The Energy Principles and theVariational Calculus
熊量原与分法 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 >§10-2位移变分方程与极小势能原理 §10-3位移变分法 p」§10-4应力变分方程与应力变分方法
4 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
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 in common 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 s 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
5 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 in common 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 problemsfacilitate 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
熊量原与分法 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
6 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
8 10-1 The Specific Energy of Deformation and Strain Energy of Elastic Body I. The specific energy of deformation Under the complex stress conditions. suppose that the elastic body is imposed all the six components of stress ax,O, O,,Ty,Tzx,Exy According to the principle of conservation of energy, e value of the strain energy has no relation to the sequence of the forces imposed on the elastic body, but is completely decided by the final value of the stress and strain Thus we obtain the strain energy density or specific energy of the elastic body U1=1(0+0,,+0+17+27n+) The specific energy expressed by the components of stress 1(++0)1201+0+0)++=++ 2E 7
7 §10-1 The Specific Energy of Deformation and Strain Energy of Elastic Body ( ) ( ) ( )( ) 2 2 2 2 2 2 1 2 2 1 2 1 x y z y z z x x y yz z x xy E U = + + − + + + + + + The specific energy expressed by the components of stress I. The specific energy of deformation ( ) U x x y y z z yz yz z x z x xy xy = + + + + + 2 1 1 Under the complex stress conditions, suppose that the elastic body is imposed all the six components of stress . According to the principle of conservation of energy, the value of the strain energy has no relation to the sequence of the forces imposed on the elastic body,but is completely decided by the final value of the stress and strain. Thus we obtain the strain energy density or specific energy of the elastic body: x y z yz zx xy , , , ,
熊量原与分法 §10-1弹性体的变形比能与形变势能 变形比能 在复杂应力状态下,设弹性体受有全部六个应力 分量σxσσx。根据能量守恒定理,形变 势能的多少与弹性体受力的次序无关,而完全确定于 应力及形变的最终大小。从而有弹性体的形变势能密 度或比能: U1=;E1+0,6,+02+x+rxyx+rny 比能用应力分量表示 U20o+0+)2+a+a)+2+)2+x2+
8 §10-1 弹性体的变形比能与形变势能 一 变形比能 在复杂应力状态下，设弹性体受有全部六个应力 分量 。根据能量守恒定理，形变 势能的多少与弹性体受力的次序无关，而完全确定于 应力及形变的最终大小。从而有弹性体的形变势能密 度或比能： x y z yz zx xy , , , , , ( ) U x x y y z z yz yz z x z x xy xy = + + + + + 2 1 1 ( ) ( ) ( )( ) 2 2 2 2 2 2 1 2 2 1 2 1 x y z y z z x x y yz z x xy E U = + + − + + + + + + 比能用应力分量表示
The specific energy expressed by the components of strain E U71= 20+01-2 e+(2++E)+,V=+y=+y Where e=8.,t8. Therefore, we have the partial differentiation of specific energy by components of stress 少=eU 0U7 aU Y-r? aT 二X 9
9 The specific energy expressed by the components of strain: ( ) ( ) ( ) + + + + + + + − = 2 2 2 2 2 2 2 1 2 1 2 1 1 2 x y z yz z x xy e E U Where x y z e = + + Therefore, we have the partial differentiation of specific energy by components of stress z z y y x x U U U = = = 1 1 1 , , xy xy z x z x yz yz U U U = = = 1 1 1 ,
熊量原与分法 比能用应变分量表示 E e+8+84+8 20+0)1-2a +r=r +yn 2 其中e=E.+E.+E 因此,我们有比能对应力分量的偏导 aU aU 6 =8 aU =ry2? 01 au =r HEr T 10
10 比能用应变分量表示 ( ) ( ) ( ) + + + + + + + − = 2 2 2 2 2 2 2 1 2 1 2 1 1 2 x y z yz z x xy e E U 其中 x y z e = + + 因此，我们有比能对应力分量的偏导 z z y y x x U U U = = = 1 1 1 , , xy xy z x z x yz yz U U U = = = 1 1 1 ,
