Quasi-variational Principles For Large Elastic Deformation Based On Base Forces

The problem of large elastic deformation in non-conservative systems on the base forces is a jumped-up topic. As a new description of the stress state at a point, the base forces is more simple than traditional stress tensors. Applying the base forces to the problems of large deformation, its virtue is obvious. The systems are usually non-conservative for large deformation problems. In present paper, non-conservative systems are specified as " follower force " system, in which the force changes with the deformation of the object.Started from the basic equations of large elastic deformation on the base forces, the quasi-variational principles of large elastic deformation in non-conservative systems were studied by the variational integral method and Lagrange Multiplier Method.First of all, the quasi-variational principles of large elastic deformation in non-conservative systems on the base forces were studied. The virtual work principle, quasi-potential energy principle, complementary virtual work principle and quasi-complementary energy principle were deduced. The broad applicability of virtual work and complementary virtual work principle was discussed, and the other forms of quasi-potential energy principle and quasi-complementary energy principle were introduced. The first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables were established, and so were a couple of quasi-variational principles with two kinds of variables which had precedent conditions. The generalized quasi-potential energy principle and quasi-complementary energy principle with three kinds of variables were established. The concept of quasi-stationary value condition of large elastic deformation in non-conservative systems on the base forces was proposed, and the quasi-variational principles were examined by deriving their quasi-stationary value conditions. The quasi-potential energy principle for thin-walled circular curved beams which bore follower force was established. Then equilibrium equations were gained by deducing its quasi-stationary value conditions. The first type generalized quasi-potential energy principle with two kinds of variables for large deflection rectangular sheet which bore follower force was established. Then equilibrium conditions and continuity conditions were gained by deducing its quasi-stationary value conditions.Secondly, the quasi-Hamilton variational principles of large elastic deformation time boundary value problem in non-conservative systems on the base forces were studied. The quasi-Hamilton principle, quasi-complementary Hamilton principle and their other forms were established. The first and second types generalized quasi-Hamilton principles and quasi-complementary Hamilton principles with two kinds of variables were established, and so were the quasi-Hamilton variational principles with two kinds of variables which had precedent conditions. The generalized quasi-Hamilton principle and quasi-complementary Hamilton principle with three kinds of variables were established. The quasi-Hamilton principle for large deformation overhanging beam which bore follower force was established. Then dynamical equilibrium conditions were gained by deducing its quasi-stationary value conditions. The generalized quasi-Hamilton principle with three kinds of variables for large deflection rectangular flat shell which bore follower force was established. Then all of its basic equations were gained by deducing its quasi-stationary value conditions.Thirdly, the convolutional quasi-variational principles of large elastic deformation time initial value problem in non-conservative systems on the base forces were studied. The convolutional quasi-potential energy principle and convolutional quasi-complementary energy principle of large elastic deformation in non-conservative systems on the base forces were established. The first and second types convolutional generalized quasi-potential energy principle and convolutional generalized quasi-complementary energy principle with two kinds of variables were established, and so were the convolutional quasi-variational principles with two kinds of variables which had precedent conditions. The convolutional generalized quasi-potential energy principle and quasi-complementary energy principle with three kinds of variables were established.Fourthly, according to corresponding relations between base forces Ti and the first Piola-Kirchhoff stress tensorτ,the second Piola-Kirchhoff stress tensorΣ, and according to corresponding relations between displacement gradient gi and ui,Green finite strain tensorεG, the corresponding quasi-variational principles of large elastic deformation in non-conservative systems which only consist of the first Piola-Kirchhoff stress tensorτand displacement gradient ui, were obtained. The corresponding quasi-variational principles of large elastic deformation in non-conservative systems which only consist of the second Piola-Kirchhoff stress tensorΣand Green finite strain tensorεG were obtained.Fifthly, taking elastostatics for example, it was worked out the quasi-variational principles and generalized quasi-variational principles of large elastic deformation in non-conservative systems on the base forces which were applicable for the calculation of finite element method.Furthermore, it was concluded that the meaning of the quasi-variational principles of large elastic deformation in non-conservative systems on the base forces in the paper was very abundant. They could be degenerated to the corresponding quasi-variational principles of large elastic deformation in conservative systems on the base forces.