ENERGY PRINCIPLES AND FINITE ELEMENT INCREMENTAL FORMULATION FOR GEOMETRIC NONLINEAR PROBLEMS

The numerical methods for geometric nonlinear problems are important topics in computational mechanics course. In this paper, the measurements of large deformation are generalized and developed, and a unified derivation and expression of the energy principles and the finite element solution formulations for the geometric nonlinear problems is established. The reference configurations of the deformable body are distinguished as the unknown and known equilibrium configurations. The relationships among some principles and formulations are clarified, including the principles of the virtual displacement, the virtual power, and the incremental virtual displacement, the total Lagrangian formulation and the updated Lagrangian formation, the solution strategies of linearization followed by element discretization and element discretization followed by linearization, and the constitutive equations with the total, incremental and rate variables. This paper provides a clear and uniform expression of the energy principles and the finite element solution formulations for the geometric nonlinear problems, which is convenient for teaching, learning, understanding and application.