An Introduction to Nonlinear Elasticity

Dr. Harish Narayanan, Mechanics Academy

Nonlinear elasticity theory plays a fundamental role in modeling the mechanical response of many polymeric and biological materials. This course begins with an overview of continuum mechanics theory and proceeds to specialise the material to the needs of modelling nonlinear elastic materials. It concludes with a look at how the finite element method can be used to solve interesting practical problems.

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Topics covered
Finite Element Method, Nonlinear Elasticity, Tensor Calculus, Time-stepping schemes, Material Modelling, Programming, FEniCS Software
Course level


Course material

13 lecture videos • 7 simulation demos • 7 practice exercises • 5 code challenges

I Background and Motivation
1 Who is this course for, and what can you expect to learn from it?
2 Pre-course exercise on tensor algebra and calculus
3 Pre-course exercise on basic Python programming
4 The broad applicablity of nonlinear field theories
5 A selection of simulations from realistic applications
II An overview of nonlinear elasticity theory
6 Setting up the overall problem within continuum mechanics Watch lecture
7 Visualizations of different deformation gradients
8 Further exploring kinematics
9 The basic equation we need to solve
10 Recalling Gauss’ divergence theorem
11 Different measures of stress and strain
12 How do we account for different materials?
13 The impact of material models on structure deformations
14 With respect to what are invariants... invariant?
15 Modelling of anisotropic and incompressibile materials
16 The state-of-the-art in modelling passive heart tissue
III Numerical methods and high-level implementation
17 A brief overview of the finite element method
18 From Poisson to nonlinear Poisson
19 Weak formulation of the (static) balance of linear momentum
20 Algorithms for stepping through time
21 Time step size and stability
22 Stability region of an explicit scheme
23 Mixed-field formulations for incompressible materials
24 How good are two and three field formulations handling locking?
IV Programming and Applications
25 A tutorial introduction to the FEniCS Project
26 Recalling important syntax
27 Implementing the weak form for the heat equation
28 Playing with Dirichlet and Neumann boundary conditions
29 A recap of all that we have learnt, leading up to the final implementation
30 A boundary value problem with the St. Venant Kirchhoff model
31 Reproducing the myocardium model shown earlier
32 Where can you go from here?
Suggested prerequisites

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Suggested follow-up courses

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Distribution license

Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License