有限变形弹性动力学的非传统简化Gurtin型变分原理——文献[1]的续篇

UNCONVENTIONAL SIMPLIFIED GURTIN-TYPE VARIATIONAL PRINCIPLES FOR FINITE DEFORMATION ELASTODYNAMICS——CONTINUATION OF THE REFERENCE [1]

根据古典阴阳互补和现代对偶互补的基本思想 ,通过作者提出的一条简单而统一的新途径 ,建立了有限变形弹性动力学的另一种单卷积形式的变分原理—各类非传统简化Gurtin型变分原理 .首先给出一个以卷积表示的关系式 ,在力学上它是有限变形动力学的广义虚功原理的另一种表式 .然后从该式出发 ,不仅能得到有限变形动力学另一种形式的虚功原理 ,而且通过文中所给出的一系列广义Legendre变换 ,还能成对导出 5类变量、3类变量、2类变量和 1类变量非传统简化Gurtin型变分原理的互补泛函 .同时 ,通过这条新途径还能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 simple and unified new way proposed by Luo, the unconventional simplified Gurtin-type variational prinicples for finite deformation elastodynamics can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as another expression of the generalized principle of virtual work in finite deformation dynamics . Based on this relation, it is possible not only to obtain another expression of the principle of virtual work in finite deformation dynamics, but also to derive systematically the complementary functionals for five-field, three-field, two-field and one-field unconventional simplified Gurtin-type variational principles by the generalized Legendre transformations given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.