Research On The Quasi-variational Principles Of Non-conservative System And Their Application

Non-conservative system refers to that where work input is dependent on the loading path during displacement and deformation when loads are imposed. It can be seen everywhere in practical engineering and most of the mechanical problems belong to this category. It is called follower force system if non-conservative force changes with the deformed configuration in a typical non-conservative system. Then the corresponding force is called follower force. In present paper non-conservative system is specified as follower force system.The variational integral method is introduced to establish the variational principle for non-conservative system in linear elastic theory. Since stationary (extreme) value for a functional doesn't exist in a non-conservative system, variational expressions (or they may be called variational equations) are adopted and such variational principle is called quasi-variational principle.Firstly quasi-variational principle for elasto-static non-conservative system is studied. Virtual work principle and quasi-potential energy principle for non-conservative system in elasto-statics are deduced. Different expressions for quasi-potential energy principle are discussed. Virtual work principle is applicable to both conservative system and non-conservative system with follower force. Complementary virtual work principle and quasi-complementary energy principle in elasticity are deduced. Another expression for quasi-complementary energy principle is given. The first and the second types generalized quasi-potential energy principle and quasi-complementary energy principle which with two kinds of variables are established. Also total generalized quasi-variational principle with three kinds of variables, which includes generalized quasi-complementary energy principle and quasi-potential energy principle with three kinds of variables, is deduced. Generalized quasi-variational principle which indicates constitutive relation and equilibrium equations is obtained. Another kind which indicates constitutive relation (in different form) and geometric condition is given. A typical example is given. Computational method by which the internal force anddeformation may be obtained simultaneously is set up by quasi-complementary energy principle with two kinds of variables in elasto-statics. The stability is also discussed.Secondly, the concept of quasi-stationary value condition is given initially and derivation method for quasi-stationary value condition is introduced with quasi-potential energy principle as an example. As for the categoricalness of conditions for the quasi-variational principle, two kinds of intrinsic meaning are expounded as follows: first, prior conditions and quasi-stationary value conditions for quasi-variational principle in elasto-statics form a proper differential equation team; second, the total equations of elasticity may be obtained from prior conditions, supplementary conditions and the laws indicated for the quasi-variational principle in elasticity. With the application of categoricalness of conditions for the quasi-variational principle, quasi-stationary value conditions for four kinds of quasi-variational principles are investigated. According to different quasi-stationary value conditions, quasi-variational principles are classified. Another two kinds of quasi-variational principles are supplemented for the elasto-statics.Thirdly, for the initial value problem of non-conservative system in elasto-dynamics, quasi-Hamilton principle and quasi-complementary Hamilton principle are established. The first and the second types generalized quasi-potential energy principle and quasi-complementary energy principle which with two kinds of variables are given. Also generalized quasi-variational principle with three kinds of variables is deduced. Generalized quasi-variational principle which indicates constitutive relation and dynamic equilibrium equations is obtained. Another kind which indicates constitutive relation (in different form) and geometric conditions is given. Also a typical example is given. The equations of motion of the wings of a plane are established based on quasi-Hamilton principle, then the stability is discussed.Fourthly, for the initial value problem of non-conservative system in elasto-dynamics, quasi-potential energy principle and quasi-complementary energy principle in convolutional forms are established. The generalized quasi-variational principles in convolutional forms with two kinds of variables are given.Also that with three kinds of variables in convolutional form is deduced. Generalized quasi-variational principle in a convolutional form which indicates constitutive relation and dynamic equilibrium equations is obtained. Another kind in a convolutional form which indicates constitutive relation and geometric conditions is given. Generalized quasi-variational principle in convolutional forms which indicate constitutive relations in different forms are derived.