ME676A
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NON-LINEAR FINITE ELEMENT METHOD IN SOLID MECHANICS
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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:
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Ted Belytschko, Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, Ltd.
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K. J. Bathe, Finite Element Procedures. Prentice – Hall Ltd.
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M. A. Crisfield, Non-linear Finite Element Analysis: Essentials (Volume 1), John Wiley & Sons, Ltd.
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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.
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No. of lectures
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Lecture Details
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1
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Introductory lecture: review of Linear Finite Element Methods, presentation of course content.
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1
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Demonstration lecture on Abaqus – installation and running the software, geometric modelling, writing user subroutine – UMAT.
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3
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Review of continuum Mechanics: Tensor algebra & Calculus, Kinematics
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2
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Review of continuum Mechanics: Stress measures.
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2
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Review of continuum Mechanics: Clausius Duhem inequality.
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2
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Review of continuum Mechanics: Objectivity with examples, objective rates used in non-linear finite element computations – comparisons using examples.
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2
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Variational calculus – formulating linear and non-linear mechanics problems, Introduction to Directional derivative.
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2
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Directional derivative – variation of various stress and strain measures, Introduction to Linearization
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2
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Introduction to Total and Updated Lagrangian formulations – derivation of weak forms, Solution methods – Newton Raphson method and variants.
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2
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Updated Lagrangian formulation: Discretized FE equations using IsoParametric formulation
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1
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Restrictions on the constitutive equations imposed by frame indifference and thermodynamics
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5
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Constitutive equations for hyperelasticity (with and without incompressibility), rate dependent and independent plasticity in metals and Crystal plasticity.
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2
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Linearization of constitutive equations to be used in weak forms
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5
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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.
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2
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Convergence measures, rate of convergence, Patch test
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1
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Geometric and material stiffness matrices – discussion on rank, deficiency and implementation details.
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2
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Discussion of techniques to deal with incompressibility condition
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3
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Review Gauss Quadrature, Reduced integration, Locking issues.
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