弹性膜结构动力学的各类非传统Hamilton型变分原理

UNCONVENTIONAL HAMILTON-TYPE VARIATIONAL PRINCIPLES FOR NONLINEAR ELASTODYNAMICS OF MEMBRANE STRUCTURES

根据古典阴阳互补和现代对偶互补的基本思想,通过罗恩早已提出的一条简单而统一的新途径,系统地建立了弹性膜结构动力学的各类非传统Hamilton型变分原理.这种新的非传统Hamilton型变分原理能反映这种动力学初值-边值问题的全部特征.文中首先给出膜结构动力学的广义虚功原理的表式,然后从该式出发,不仅能得到膜结构动力学的虚功原理,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弹性膜结构动力学的5类变量(pα,pβ,pγ,vα,vβ,vγ,Nα,Nβ,Sαβ,εα,εβ,εαβ,u,v,w)、4类变量(pα,pβ,pγ,Nα,Nβ,Sαβ,εα,εβ,εαβ,u,v,w)、3类变量(Nα,Nβ,Sαβ,εα,εβ,εαβ,u,v,w)和2类变量(Nα,Nβ,Sαβ,u,v,w)非传统Hamilton型变分原理的互补泛函、以及相空间(pα,pβ,pγ,vα,u,v,w)非传统Hamilton型变分原理的泛函与1类变量(u,v,w)非传统Hamilton型变分原理势能形式的泛函.同时,通过这条新途径还能清楚地阐明这些原理的内在联系.

According to the basic idea of classical yin-yang complementarity and modem dual-complementarity, in a new, simple and unified way proposed by Luo, the unconventional Hamilton-type variational principles for geometrically nonlinear elastodynamics of membrane structures can be established systematically. The unconventional Hamilton-type variational principle can fully characterize the initial-boundary-value problem of geometrically nonlinear elastodynamics. An important integral relation is given, which can be considered as the generalized principle of virtual work for dynamics of membrane structures in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work for dynamics of membrane structures, but also to derive systematically the complementary functionals for five-field （Pα,Pβ,pγ,Vα,Vβ,Vγ,Nα,Nβ,Sαβ,εα,εβ,εαβ,u,v,w）, four-field （Pα,Pβ,pγ,Vα,Vβ,Vγ,Nα,Nβ,Sαβ,εα,εβ,εαβ,u,v,w） ,three-field （Nα,Nβ,Sαβ,εα,εβ,εαβ,u,v,w）and two-field （Nα,Nβ,Sαβ,u,v,w） unconventional Hamilton-type variational principles. And the functional for the unconventional Hamilton-type variational principle in phase space（Pα,Pβ,pγ,u,v,w） and the potential energy functional for one-field （u,v,w） unconventional Hamilton-type variational principle for geometrically nonlinear elastodynamics of membrane structures are obtained by the generalized Legendre transformation. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.