大位移非线性弹性理论的变分原理和广义变分原理

在前文中1],作者首次提出了大位移非线性弹性力学的位能原理和余能原理,以及各种完全的和不完全的广义变分原理.但在约束条件和欧拉条件上,证明和叙述都不很明确,有时甚至把原来应该是欧拉方程的误认为是约束条件,如余能驻值原理中,应力位移关系原应是欧拉方程,但把它当作了变分约束条件.这就是说:我们把余能驻值原理约束得超过了必要的要求.还有,在所有变分原理中,应力应变关系式都是不参加变分的约束条件,亦即,他们是从已定应力导出应变或从已定应变导出应力的约束条件.这一点,在文1](1979)中,并未明确指出.本文并将用高阶拉氏乘子法,导出更一般的广义变分原理(1983)2].本文使用V.V.Novozhilov的有关非线性弹性力学的成果(1958)3].

Variational Principles and Generalized Variational Principles for Nonlinear Elasticity with Finite Dispiacemant

In a previous paper(1979), the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together withvarious complete and incomplete generalized principlesIwere studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy prindple,but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strainsfrom known stresses. This point was not clearly mentioned in the previous paper(1979). In this paper, the high order Lagrange multiplier method(1983) is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V.V. Novozhilov's results(1958) for nonlinear elasticity are used.