INTRODUCTION TO LARGE DEFORMATION AND NON-LINEAR DYNAMIC ANALYSIS OF SOLIDS BY THE FINITE ELEMENT METHOD
Prof. J.P. PONTHOT Aerospace Laboratory/LTAS-Computational Mechanics University of Liège Liège, Belgium
TABLE OF CONTENTS
CHAPTER I: INTRODUCTION - GOALS - MATHEMATICAL PRELIMINARIES
CHAPTER II: KINEMATICS IN LARGE DEFORMATION
- Body, Motion, Configuration - Lagrangian and Eulerian coordinates - Deformation gradient - Polar decomposition - Composition of deformation - Small displacements and small strains - Strain tensors - Velocity gradient, strain rate and spin tensor
CHAPTER III: STRESS TENSORS & MOMENTUM BALANCE - The Cauchy stress tensor - The Piola stress tensor - The second Piola-Kirchhoff stress tensor - Momentum balance equations (linear and angular) in the current and reference configurations - Comparison of the stress tensors
CHAPTER IV: PRINCIPLE OF VIRTUAL WORK (PVW)
- PVW on the current configuration - PVW on the reference configuration - Conjugated stress and strain tensors
CHAPTER V: FINITE DEFORMATION CONSTITUTIVE THEORY
GENERAL PRINCIPLES
- Axioms of constitutive theory - Change of frame/Change in observer - Invariance under a change in observer: The Objectivity Principle HYPERELASTIC MATERIAL MODELS Typical rubber behavior Finite elasticity & Objectivity Principle Finite elasticity : thermodynamical point of view Hyperelasticity : rubber & foams models Neo-Hookean material Incompressibility, quasi-incompressibility and implications Anisotropic material models
HYPOELASTIC MATERIAL MODELS - Objective Rates & examples - Corotational rates - Elasto-plastic materials . yield function . strain rate decomposition into its elastic and plastic part . flow rule . the von Mises criterion - Elastic-viscoplastic materials . similarities with elastic-plastic materials . the Perzyna model
CHAPTER VI: CLASSICAL FORMALISMS FOR LARGE DEFORMATION ANALYSIS
- Total Lagrangian formulation - Updated Lagrangian formulation - Cauchy stress/Lagrangian Mesh formulation - Choice of a formulation
CHAPTER VII: FINITE ELEMENTS FOR LARGE DEFORMATION ANALYSIS
- Discretization by the Finite Element Method - Discretization of the equilibrium equations - Bulk elements and shell elements - Choice of a finite element type and degree - GAUSS integration for finite elements - Incompressibility condition: implications - Selective & total reduced integration
CHAPTER VIII: NUMERICAL TECHNIQUES TO INTEGRATE HYPOELASTIC ELASTO-VISCOPLASTIC CONSTITUTIVE EQUATIONS
- Choice of an integration path - Incremental Objectivity - Final Rotation Path - Numerical integration of constitutive equations: mean normal & radial return algorithm - Numerical solution for non-linear equilibrium equations - Tangent stiffness matrix and consistent tangent stiffness matrix
CHAPTER IX:
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TIME INTEGRATION METHODS
Introduction Quasi-static case Explicit integration : the central difference algorithm Explicit integration: numerical tricks to speed up the calculation Implicit integration: Newmark's family . The Newmark method . The Hilber-Hughes-Taylor (HHT) -method . The Wood-Bossack-Zienkiewicz (WBZ) -method . The Chung-Hulbert (CH) method - Implicit integration: Energy Conserving Schemes
CHAPTER X: FRICTIONAL CONTACT
- Physical description of the phenomenon - Normal contact & penetration - Constitutive equation for frictional contact - Elastic-plastic analogy - Radial return for numerical integration of the constitutive equation - Frictional contact and the finite element method
CHAPTER XI:
ARBITRARY LAGRANGIAN EULERIAN FORMULATION
-Classical formulations -ALE formulation of Continuum Mechanics equations -The Mesh Management problem -Data convection between meshes -Examples and applications
CHAPTER XII:
DAMAGE AND FRACTURE
-Phenomenology of damage -How to take into account damage in your model -Classical damage theories I: the Lemaitre approach -Classical damage theories II: the Gurson approach -Numerical methods to take damage into account -Parameter identification of damage constitutive models -Fracture: numerical techniques for element erosion -Coupling damage and fracture -Applications