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# Computational Continuum Mechanics

## Overview

Continuum mechanics as a full-fledged course is a very interesting but a challenging subject. Usually,its application within the nonlinear finite element codes is not clear to the student. Computational continuum mechanics tries to bridge this gap. Hence, it can be treated as an applied version of continuum mechanics course. It assumes no prior exposure to continuum mechanics. The course starts with sufficient introduction to tensors, kinematics, and kinetics. Then, the course applies these concepts to set up the constitutive relations for nonlinear finite element analysis of a simple hyperelastic material. This is followed by the linearization of the weak form of the equilibrium equations followed by discretization to obtain the finite element equations, in particular, the tangent matrices and residual vectors is discussed. Finally, the Newton-Raphson solution procedure is discussed along with line search and arc length methods to enhance the solution procedure. INTENDED AUDIENCE :Masters student and research scholarsPREREQUISITES : Introduction to Solid Mechanics I and II, A undergraduate course in Engineering Mathematics. Exposure to undergraduate course on numerical methods will be an added advantage.INDUSTRIES SUPPORT :VSSC, ISRO, Siemens India Limited, Ansys India or any firm involved in R&D involving finite element analysis

## Syllabus

Week 1:Introduction – origins of nonlinearityWeek 2:Mathematical Preliminaries -1: Tensors and tensor algebraWeek 3:Mathematical Preliminaries -2: Linearization and directional derivative, Tensor analysisWeek 4:Kinematics – 1: Deformation gradient, Polar decomposition, Area and volume change
Week 5:Kinematics – 2: Linearized kinematics, Material time derivative,Rate of deformation and spin tensorWeek 6:Kinetics – 1 : Cauchy stress tensor, Equilibrium equations,Principle of virtual workWeek 7:Kinetics – 2 : Work conjugacy, Different stress tensors, Stress ratesWeek 8:Hyperelasticity - 1: Lagrangian and Eulerian elasticity tensor
Week 9:Hyperelasticity - 2: Isotropic hyperelasticity, Compressible Neo-Hookean materialWeek 10:Linearization : Linearization of internal virtual work, Linearization of external virtual workWeek 11:Discretization: Discretization of Linearized equilibrium equations – material and geometric tangent matricesWeek 12:Solution Procedure: Newton-Raphson procedure, Line search and Arc length method

### Taught by

Prof. Sachin Singh Gautam

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