Department of Mathematics

MA7011 Applied Nonlinear Elasticity

Subject Knowledge

Aims

Learning Outcomes

Methods

Assessment

Subject Skills

Aims

To develop students' abilities to synthesise and build upon knowledge acquired in the previous years.

To provide a bridge between abstract mathematical concepts and meaningful engineering applications.

Learning Outcomes

Methods

Assessment

Explanation of Pre-requisites

Course Description

Solid mechanics deals with the motions and deformations experienced by solid bodies under the action of external agents; a special branch of this field is the theory of elasticity which is built on the assumption that no permanent deformations are allowed. Rubber is an archetypal elastic material but most metals behave linearly when subjected to sufficiently weak loads. The official birth of the theory of elasticity took place in when the British scientist Robert Hooke discovered that the deformation of any spring is proportional to the normal tension applied to it. This was the starting point for an extremely vast amount of research carried out in the centuries that followed. Historically, among the most prominent contributors to this branch of mechanics were Cauchy, Lagrange, Saint-Venant, Kirchhoff and Airy, to name but a few. Until the beginning of the twentieth century most elasticity theories were linear. The technological advances in the early part of the last century, however, had stimulated a great deal of interest in the mechanics of rubber-like materials. It was Ronald Rivlin in the mid-1940's who, in a series of seminal papers laid out the seeds from which the finite theory of elasticity has blossomed ever since.

The main aim of the theory of elasticity is to provide a description of deformed bodies in terms of certain preferential configurations. The easiest way to achieve this requires the introduction of the concept of tensor, which then allows us to represent compactly a great deal of physical information. Also this helps establish the intrinsic form of various balance laws, in particular, the balance of mass, linear momentum, angular momentum and energy in a moving body.

The primary objective of this course is to illustrates the theory of elasticity with a pervading emphasis on nonlinear aspects and the interplay between mathematics and engineering sciences.

Syllabus

Tensors, the divergence theorem, material time derivatives.

Principles of linear and angular momentum, the stress tensor, Piola-Kirchhoff tensors, Cauchy stress, principal stresses, stress invariants.

Constitutive equations, material symmetry, hyper-elasticity and hypo-elasticity, incompressibility, forms of the strain-energy function.

Homogeneous deformations, inverse methods, simple shear, pure homogeneous deformations, non-homogeneous deformations, simple torsion, extension and torsion.

Reading list

Recommended:

P. Chadwick, Continuum Mechanics, Dover Publications, 1999.

R. W. Ogden, Non-linear Elastic Deformations, Dover Publications, 1997.

M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981.

G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineers, John Wiley and Sons, 2000.

Resources

Module Evaluation

Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
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