EGM 6934 Nonlinear Finite Element Methods
Spring 1997
1. Graduate Catalog Description: Prerequisite: EGM 6611 Continuum Mechanics I, EGM 6351 Finite Element Methods, equivalent courses, or consent of instructor (see below). Nonlinear finite element methods applied to solid mechanics problems, with material and/or geometric nonlinearities. Calculus of variations, Hu-Washizu mixed principle. Nonlinear elasticity, plasticity. Large deformation, large overall motion. Solution methods for nonlinear problems. Linearization, consistent tangent. Geometrically-exact structures. Contact problems.
2. Textbook: None required.
3. References: Zienkiewicz, O.C., and Taylor, R.L. [1991], The Finite Element Method, Vol.2: Solid and Fluid Mechanics, Dynamics and Non-linearity, McGraw-Hill. Additional references will be given in class.
4. Instructor: Dr. L. Vu-Quoc, 219A Aero Bldg.
Tel: (904) 392-6227.
E-mail: vql@ufl.edu
5. Prerequisites by topics: First graduate courses in Continuum Mechanics and FEM.
6. Goals: Provide students with a firm grasp of what are involved in solving solid mechanics problems having material nonlinearity and/or geometric nonlinearity, whereas EGM 6351 FEMs and EGM 6352 Advanced FEMs focus primarily on the foundation of the FEMs and linear PDEs in many areas of engineering. Formulate computational algorithms for nonlinear elastic, plastic materials, for large deformation and large overall motion, and for solving nonlinear algebraic equations.
7. Topics: Directional derivative. Elasticity. Calculus of variations, fundamental lemma, Euler-Lagrange equation, Vainberg theorem, weak form, Hu-Wahizu mixed principle, FE discretization. Material nonlinearity: Nonlinear elasticity, simple damage model, plasticity. Basic algorithm for nonlinear problems. Operator splitting: elasto-plastic problems, return mappings, tangent operator, stability analysis. Nearly incompressible problems: Hu-Washizu principle, B-bar procedure. Geometric nonlinearity: Review of continuum mechanics, kinematics of large deformation. Geometrically-exact beam: Large deformation, large overall motion, mechanical power in terms of first Piola-Kirchhoff stress tensor, stress resultants, stress couples, objective rates, nonlinear strain measures, equations of motion, tangent material stiffness, tangent geometric stiffness. Contact mechanics: Geometric constraint, linearization.
8. Projects: To be announced in class.
9. Computer support required: None.
10. Engineering applications: Solid mechanics: nonlinear elasticity, plasticity, large deformation, large overall motion, contact mechanics.