非线性弹性动力学Hamilton型变分原理的革新——非传统Hamilton型变分原理

An Innovation for the Hamilton-type Variational Principles in Nonlinear Elastodynamics —Unconventional Hamilton-type Variational Principles

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出的一条简单而统一的新途径,系统地建立了几何非线性弹性动力学的各类非传统Hamilton型变分原理。而这种非传统Hamilton型变分原理能反映几何非线性弹性动力学初值—边值问题的全部特征,因此它是对Hamilton变分原理的重要革新。文中给出一个重要的积分关系式,可以认为,在力学上它是几何非线性动力学的广义虚功原理的表式。从该式出发,不仅能得到几何非线性动力学的虚功原理,而且通过所给出的一系列广义Legendre变换,还能系统地成对导出几何非线性弹性动力学的5类变量、3类变量、2类变量和1类变量非传统Hamilton型变分原理的互补泛函,以及相空间非传统Hamilton型变分原理的泛函。同时,通过这条新途径还能清楚地阐明这些原理之间的内在联系。

According to the basic idea of classical yin_yang complementarity and modern dual_complementarity, in a new, simple and unified way proposed by Luo, the unconventional Hamilton_type variational principles for geometrically nonlinear elastodynamics can be established systematically. The unconventional Hamilton_type variational principle can fully characterize the initial_boundary_value problem of geometrically nonlinear elastodynamics. This new variational principle is equivalent to the above problem of this dynamics, so that it is an important innovation for the Hamilton variational principle. An important integral relation is given, which can be considered as the expression of the generalized principle of virtual work for geometrically nonlinear dynamics. Based on this relation, it is possible not only to obtain the principle of virtual work in geometrically nonlinear dynamics, but also to derive systematically the complementary functionals for five_field, three_field, two_field and one_field unconventional Hamilton_type variational principles, and the functional for the unconventional Hamilton_type variational principle in phase space by the generalized Legendre transformations given here. Furthermore, with this new approach, the intrinsic relationship among various principles can be explained clearly.