Basic Principles of Deformed Objects with Methods of Analytical Mechanics

Analytical mechanics is the most fundamental discipline in this field. The basic principles of analytical mechanics should also be applicable to deformed objects. However, the virtual displacement principle proposed by analytical mechanics is only applicable to particle systems and rigid body systems, and not to general deformed objects. In this study, the basic principle, which includes the virtual displacement principle and d’Alembert–Lagrange principle (also called the virtual displacement principle of dynamics), of general deformed objects (such as, elastic, plastic, elasto-plastic, and flexible objects) is derived using analytical mechanics. First of all, according to the method of analytical mechanics, the external force, internal force, constraint reaction force and elastic recovery force of the deformed object system under the equilibrium state are analyzed, and the concepts of virtual displacement, ideal constraint and virtual work are introduced, and the virtual displacement principle (also called virtual work principle) of deformed objects is proposed; secondly, vector form, coordinate component form and generalized coordinate form of generalized virtual displacement principle of deformed object are given; thirdly, Introduce inertial force and use analytical mechanics to derive the d’Alembert–Lagrange principle of dynamic systems; fourthly, as application of the principle, the virtual displacement principle of deformed objects in plane polar coordinate system, space cylindrical coordinate system and spherical coordinate system are given; fifthly, the constitutive relationship between the gravitational strain of elastic–plastic materials was introduced, and an example of the application of the d'Alembert–Lagrange principle in elastic–plastic objects was given; finally, a brief conclusion is drawn. This study unifies the virtual displacement principle of elastic objects, plastic, elastoplastics, deformed object systems and rigid object systems using the basic analytical mechanics method. This is a basic principle for dealing with the static problems of deformed objects. This work also lays the foundation for further study of the dynamics of deformed object systems.