Research On The Quasi-variational Principles In Nonlinear Non-conservative Elasticity

Nonlinear mechanics is a subject to research geometric and physical nonlinearity, which exists in nature widely. The systems are usually non-conservative for nonlinear problems, especially for large displacement problems. Non-conservative system is that the work input is dependent on the loading path during the process of displacement and deformation-making when loads are imposed. There is a typical non-conservative system called accompanying force system, in which the force changes with the deformation of the object. In present paper non-conservative system is specified as accompanying force system. .Started from the nonlinear basic equations, the variational integral method was generalized to nonlinear non-conservative elasticity, and the quasi-variational principles in nonlinear non-conservative elasticity were studied.First of all, the quasi-variational principles in nonlinear non-conservative elastostatics 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 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 principles and quasi-complementary energy principles with three kinds of variables were established. The concept of quasi-stationary value condition in nonlinear non-conservative elasticity was proposed, and the quasi-variational principles were examined by deriving their quasi-stationary value conditions. The equilibrium equation of nonlinear Leipholz bar was gained by quasi-potential energy principle, and its buckling property was researched. The first type generalized quasi-potential energy principle with two kinds of variables for large deflection rectangular sheet which bore accompanying force was established, and its quasi-stationary value conditions were obtained.Secondly, the quasi-variational principles of time boundary value problem in nonlinear non-conservative elastodynamics were studied. The quasi-Hamilton principle, quasi-complementary Hamilton principle and their other forms were gained. The first and second types generalized quasi-Hamilton principles and quasi-complementary Hamilton principles with two kinds of variables, and generalized quasi-Hamilton principle and quasi-complementary Hamilton principle with three kinds of variables were established. The motion equation of nonlinear Leipholz bar was gained by quasi-Hamilton principle, and its dynamical property was researched. The generalized quasi-Hamilton principle with three kinds of variables for large deflection rectangular sheet which bore accompanying force was established, and all of its basic equations were gained by deducing its quasi-stationary value conditions.Thirdly, the quasi-variational principles of time initial value problem in nonlinear non-conservative elastodynamics were studied. Using restricted variation, variational integral method, transformation of naught addition and convolution knowledge, the convolutional quasi-potential energy principle, quasi-complementary energy principle, the first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables, and the generalized quasi-potential energy principles and quasi-complementary energy principles with three kinds of variables were established.Fourthly, taking the time boundary value problem in nonlinear non-conservative elastodynamics for example, it was studied the method of deducing quasi-stationary value condition of classical quasi-variational principles by the operation of Lagrange Multiplier not joining variational, and establishing generalized quasi-variational principles by the operation of Multiplier joining variational.Fifthly, taking nonlinear non-conservative elastostatics for example, it was worked out the quasi-variational principles and generalized quasi-variational principles which were applicable for the calculation of finite element method.Furthermore, it was concluded that the meaning of the quasi-variational principles in nonlinear non-conservative elasticity established in the paper was very abundant. It could be degenerated to nonlinear conservative elasticity, linear non-conservative elasticity and linear conservative elasticity, so the corresponding variational principles or quasi-variational principles were derived.