《弹性力学》（双语版） 第十章 能量原理与变分法

And the partial differentiation of specific energy by components of strain OU L=Ox 08y OU -=O aa aa 二 OUAOUa0 ar II. Strain energy Since the components of stress and components of strain and then the specific energy U are all the function of coordinates, the strain energy U. of the whole elastic body is Substituting the three kinds of expressing forms of specific energy yields three kinds of integrating forms of strain energy 11

11 xy xy z x z x yz yz U U U       =   =   =   1 1 1 , , z z y y x x U U U       =   =   =   1 1 1 , , And the partial differentiation of specific energy by components of strain II. Strain energy Since the components of stress and components of strain and then the specific energy are all the function of coordinates, the strain energy . of the whole elastic body is: U1 U  U = U dxdydz 1 2 1 Substituting the three kinds of expressing forms of specific energy yields three kinds of integrating forms ofstrain energy

熊量原与分法 比能对应变分量的偏导 OU aa E dU aU dU y 2X3 -=T y y 二形变势能 由于应力分量和形变分量,进而比能U1都是位置 坐标的函数,所以整个弹性体的形变势能U为: U=∫u1ad 2 将比能的三种表达形式代入,得形变势能的三种 积分形式 12

12 xy xy z x z x yz yz U U U       =   =   =   1 1 1 , , z z y y x x U U U       =   =   =   1 1 1 , , 比能对应变分量的偏导 二 形变势能 由于应力分量和形变分量，进而比能 都是位置 坐标的函数，所以整个弹性体的形变势能 为： U1 U  U = U dxdydz 1 2 1 将比能的三种表达形式代入，得形变势能的三种 积分形式

2J0,2+0,6n+0.+y=+-xyx+rnnd U 2E ∫|+a+a)-2/o,:+0o+o)+ +20+1)(2+x2+2)dh E U 2(+A e+(a2+62+2)+2+y2+yx)otz 1-2 Substituting the geometric equations can also yield the strain energy expressed by displacement E au a aw 2(1+4)小[1-2八aya2)(a)()(a + aw a au Ow lla au dxd ydz ay d aa) a ay 13

13 x y z y u x v x w z u z v y w z w y v x u z w y v x E u U d d d 2 1 2 1 2 1 2(1 ) 1 2 2 2 2 2 2 2 2              + +      + +         + +       +          +      +         + +    + − =                             Substituting the geometric equations can also yield the strain energy expressed by displacement ( )  U = + + + + + dxdydz x x y y z z yz yz z x z x xy xy             2 1 ( ) e ( ) ( ) dxdydz E U  x y z yz z x xy       + + + + + + + − = 2 2 2 2 2 2 2 2 1 2 1 1 2          ( ) ( ) ( )( )dxdydz E U yz z x xy x y z y z z x x y 2 2 2 2 2 2 2 1 2 2 1               + + + + = + + − + + + 

熊量原与分法 U=+0,+04:+x+2+12h U 2E SS02+0 +02)-2ulo,o+0. 0,+0, ) +2(1+)乙2+x+ To dxdydz E U= 2(1+1)J01-2 e+(2+E2+E2 4/22++) 将几何方程代入,形变势能还可用位移分量来表示 E U au a aw a. an 2(+) 八ab)(a)(a)(a Ow ava aw1(av al + 一 lady lodz 2(a d 2a a) 2(a ay 14

14 x y z y u x v x w z u z v y w z w y v x u z w y v x E u U d d d 2 1 2 1 2 1 2(1 ) 1 2 2 2 2 2 2 2 2              + +      + +         + +       +          +      +         + +    + − =                             将几何方程代入，形变势能还可用位移分量来表示 ( )  U = + + + + + dxdydz x x y y z z yz yz z x z x xy xy             2 1 ( ) e ( ) ( ) dxdydz E U  x y z yz z x xy       + + + + + + + − = 2 2 2 2 2 2 2 2 1 2 1 1 2          ( ) ( ) ( )( )dxdydz E U yz z x xy x y z y z z x x y 2 2 2 2 2 2 2 1 2 2 1               + + + + = + + − + + + 

8 10-2 The Variational Equation of Displacement and The Principle of minimum Potential Energy I Variation and its properties We have learned in higher mathematics the concept of differentiation which is the increment of variable. Then what is the variation? Variation is the increment of function, often denoted by 8. Variation has the following properties 8(l+)=8a+。v u 6∫ds=JndS 15

15 I.Variation and its properties: We have learned in higher mathematics the concept of differentiation which is the increment of variable. Then what is the variation? Variation is the increment of function, often denoted by d. Variation has the following properties:   =   =   + = + udS u S u x x u u w u w δ δ d δ δ δ ( ) δ δ §10-2 The Variational Equation of Displacement and The Principle of Minimum Potential Energy