有限位移理论线弹性力学二类和三类混合变量的变分原理及其应用

Abstract

Variational principles for dual and triple mixed variables of linear elasticity with finite displacements were proposed. Considering the variation of prescribed boundary conditions and using the reciprocal theorem of finite displacements played the key and bridging roles in derivation of the above variational principles. First, in view of the variation of the prescribed geometrical boundary conditions and based on the reciprocal theorem, the principle of minimum potential energy with dual mixed variables was derived. In a similar way, the principle of stationary complementary energy with dual mixed variables was also given. Then the relation between the strain energy density and the complementary energy density was applied to the above 2 principles, and the variational principle with triple mixed variables was deduced. In turn, the principles of virtual work and virtual complementary work with dual and triple mixed variables were directly given. Meantime, the generalized variational principles were derived with the Lagrangian multiplier method. Through an example the Lagrangian multiplier method in certain cases was proved to be ineffective. The semiinverse method for construction of the functionals for generalized variational principles was also introduced. Finally, a cantilever beam with large deflection was calculated by means of the principle of minimum potential energy for dual mixed variables.