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 ,