Finite elasticity is the study of elastic responses of solid materials under reversible deformations. When geometrical changes are small, such that the deformations are not detectable by the human eye, their configuration can be considered as fixed, and any changes in their geometry can be neglected. This so-called “small strain” assumption is at the basis of the classical theory of linear elasticity, which is successfully used in structural mechanics and many other engineering areas. However, many modern applications (involving soft solids, inflatable structures, polymers and synthetic rubbers) and biological structures (such as plants and vital organs) involve large deformations. In the framework of large deformations, finite elasticity covers the simplest case where internal forces only depend on the present deformation of the body and not on its history (i.e., it excludes plasticity, viscosity, and damage). Because of the large deformations involved, the mathematical models used in finite elasticity are typically nonlinear, and the numerical solution of the resulting equations requires careful approximation strategies and powerful algorithms.

The purpose of this course is to provide a general overview of three-dimensional finite elasticity in the setting of analytical quasi-static theory. Since the main mathematical formulations are given in terms of tensors, the course starts with a summary of the relevant tensor algebra and analysis.

• Describe the general model of nonlinear elastic deformation in the reference (Lagrangian. material) and current (Eulerian, spatial) system of references.
• Formulate general boundary value problems for static and quasi-equilibrated motion of finite elasticity.
• Solve analytically some classical problems involving specific isotropic materials and simple geometries, including: the Rivlin cube, simple or generalised shear of a cuboid, bending of a rectangular block, extension and inflation of a cylindrical tube, inflation of a spherical shell, cavitation of spheres.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Ability to relate mathematical theory to specific analytical computations by employing basic tools of calculus, differential equations and linear algebra.

1.     Introduction

        1.1.   Tensors and tensor fields

2.     Deformation and Motion

        2.1.   Kinematics

        2.2.   Deformation and strain

        2.3.   Examples of deformations

        2.4.   Equilibrium equations

        2.5.   Forces and stresses

3.     Material Constitutive Models

        3.1.   Materials with internal constraints

        3.2.   Hyperelastic materials

        3.3.   Nonlinear elastic parameters

        3.4.   Linear elastic approximation

        3.5.   Stochastic hyperelastic models

4.     Boundary Value Problems

        4.1.   Explicit solutions of particular problems

        4.2.   Infinitesimal strain superposed on large strain deformation

        4.3.   Quasi-equilibrated motion