ME676A |
NON-LINEAR FINITE ELEMENT METHOD IN SOLID MECHANICS |
Credits: |
3-0-0-9 |
Review of FE techniques for linear elasticity; Review of continuum mechanics—kinematics, balance laws, stress measures, Clausius Duhem inequality, frame indifference, stress rates and constitutive equations; Introduction to directional derivatives, formulation of variational principles for nonlinear problems and linearisation; Linearisation of variational principles for nonlinear problems; Generalised Newton Raphson scheme; Applications to hyperelasticity, metal plasticity and crystal plasticity; Issues of convergence rates, measures and patch tests; Techniques for dealing with locking issues; Using and incorporating UMAT and UEL subroutines in ABAQUS.
Reference Books:
- Ted Belytschko, Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, Ltd.
- K. J. Bathe, Finite Element Procedures. Prentice – Hall Ltd.
- M. A. Crisfield, Non-linear Finite Element Analysis: Essentials (Volume 1), John Wiley & Sons, Ltd.
- M. A. Crisfield, Non-linear Finite Element Analysis: Advanced topics (Volume 2), John Wiley & Sons, Ltd.
Pre-requisites:
An introductory course on linear FEM (ME623 or equivalent) and a course on Continuum Mechanics (ME621 or equivalent). Familiarity with tensor operations will be assumed.
| No. of lectures | Lecture Details |
| 1 | Introductory lecture: review of Linear Finite Element Methods, presentation of course content. |
| 1 | Demonstration lecture on Abaqus – installation and running the software, geometric modelling, writing user subroutine – UMAT. |
| 3 | Review of continuum Mechanics: Tensor algebra & Calculus, Kinematics |
| 2 | Review of continuum Mechanics: Stress measures. |
| 2 | Review of continuum Mechanics: Clausius Duhem inequality. |
| 2 | Review of continuum Mechanics: Objectivity with examples, objective rates used in non-linear finite element computations – comparisons using examples. |
| 2 | Variational calculus – formulating linear and non-linear mechanics problems, Introduction to Directional derivative. |
| 2 | Directional derivative – variation of various stress and strain measures, Introduction to Linearization |
| 2 | Introduction to Total and Updated Lagrangian formulations – derivation of weak forms, Solution methods – Newton Raphson method and variants. |
| 2 | Updated Lagrangian formulation: Discretized FE equations using IsoParametric formulation |
| 1 | Restrictions on the constitutive equations imposed by frame indifference and thermodynamics |
| 5 | Constitutive equations for hyperelasticity (with and without incompressibility), rate dependent and independent plasticity in metals and Crystal plasticity. |
| 2 | Linearization of constitutive equations to be used in weak forms |
| 5 | Linearization of constitutive equations and FE discretisation: Example – Compressible, Neo-Hookean material (other constitutive formulations may also be taken up here), Geometric and material stiffness matrices – details of implementation, writing User subroutine UEL in Abaqus. |
| 2 | Convergence measures, rate of convergence, Patch test |
| 1 | Geometric and material stiffness matrices – discussion on rank, deficiency and implementation details. |
| 2 | Discussion of techniques to deal with incompressibility condition |
| 3 | Review Gauss Quadrature, Reduced integration, Locking issues. |