ME676A

NONLINEAR FINITE ELEMENT METHOD IN SOLID MECHANICS



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, Nonlinear Finite Element Analysis: Essentials (Volume 1), John Wiley & Sons, Ltd.

M. A. Crisfield, Nonlinear Finite Element Analysis: Advanced topics (Volume 2), John Wiley & Sons, Ltd.
Prerequisites:
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 nonlinear finite element computations – comparisons using examples.

2

Variational calculus – formulating linear and nonlinear 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, NeoHookean 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.

