薄板理论的正交关系及其变分原理

AN ORTHOGONALITY RELATIONSHIP FOR THIN PLATE THEORY ANDITS VARIATIONAL PRINCIPLE^1

利用平面弹性与板弯曲的相似性理论,将弹性力学新正交关系中构造对偶向量的思路推广到各向同性薄板弹性弯曲问题,由混合变量求解法直接得到对偶微分方程并推导了对应的变分原理.所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点.发现了两个独立的、对称的正交关系,利用薄板弹性弯曲理论的积分形式证明了这种正交关系的成立.在恰当选择对偶向量后,弹性力学的新正交关系可以推广到各向同性薄板弹性弯曲理论.

While Hamiltonian system was led to solution of elastic theory a new systematic methodology for theory of elasticity was established and a symplectic orthogonality relationship was presented (Zhong Wanxie, 1995). For two-dimensional theory of elasticity a new dual vector and a new dual differential matrix were presented by putting the old dual vector in a new order. It was discovered for isotropic materials that the symplectic orthogonality relationship may be decomposed into 2 independently and symmetrically orthogonal sub-relationships (Luo Jianhui et al., 2002). The new orthogonality relationship includes the symplectic orthogonality relationship. The new orthogonal relationship was generalized into three-dimensional elasticity problems in which a direction of coordinate is an orthogonal direction of materials (Luo Jianhui et al., 2003). The research of a systematic methodology for bending theory of thin and thick plate has also been noticed. Some conclusion of the systematic methodology for bending theory of Reissner-Mindlin thick plate was obtained (Luo Jianhui et al., 2004). Firstly, the Hamiltonian dual differential equations for thick plates were derived. Then, the functional expressions of Hamiltonian variational principle were obtained by using the variable substitution and multiplier method. At last, the new orthogonality relationship of thick plate theory was proposed. But the new orthogonality relationship of thick plate theory can not be degenerated into thin plate theory. Therefore it is necessary to research the new orthogonality relationship of thin plate theory. Based on the analogy between plate bending problems and plane elasticity problems Hamiltonian system was applied to thin plate bending problems and its symplectic orthogonality relationship was presented (Zhong Wanxie et al., 1999). For thin plate bending theory a new dual vector is presented while the dual vectors based on the analogy are put in a new order. A variational principle based on the new dual vector is proposed and also demonstrated by a new method. The principal diagonal sub-matrixes of the dual differential matrix are zero matrixes. As a result of the peculiarity of the dual differential matrix it is discovered that the orthogonality relationship of thin plate bending theory based on the analogy may be decomposed into 2 orthogonal sub-relationships. Based on the integral form (Luo Jianhui et al., 2002) of the systematic methodology for elasticity, the new orthogonal relationship is demonstrated. The new orthogonality relationship of theory of elasticity is generalized into anisotropic thin plate bending theory. The theoretical achievements of the Hamiltonian system for thin plates provide new effective tools for the research on analytical and finite element solutions of thin plates.