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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.

Topics covered

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

Graduate

Course material

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

I | Background and Motivation | ||
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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 | ||
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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 | ||
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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 | ||
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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

A list of courses from the same subject at a lower difficulty level.

Suggested follow-up courses

A list of courses from the same subject at a higher difficulty level.

Related resources

Distribution license

Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License