Synopsis for C6.1a: Solid Mechanics

[This course will run if teaching resources allow]

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit. OSS paper code 2466.

Recommended Prerequisites

There are no formal prerequisites. In particular it is not necessary to have taken any courses in fluid mechanics, though having done so provides some background in the use of similar concepts. Use is made of (i) elementary linear algebra in (e.g., eigenvalues, eigenvectors and diagonalization of symmetric matrices, and revision of this material, for example from the Mods Linear Algebra course, is useful preparation); and (ii) some 3D calculus (mainly differentiation of vector-valued functions of several variables). All necessary material is summarized in the course.

Overview

Solid mechanics is a vital ingredient of materials science and engineering, and is playing an increasing role in biology. It has a rich mathematical structure. The aim of the course is to derive the basic equations of elasticity theory, the central model of solid mechanics, and give some interesting applications to the behaviour of materials.The course is useful preparation for C6.1b Elasticity and Plasticity. Taken together the two courses will provide a broad overview of modern solid mechanics, with a variety of approaches.

Learning Outcomes

Students will learn basic techniques of modern continuum mechanics, such as kinematics of deformation, stress, constitutive equations and the relation between nonlinear and linearized models.The emphasis on the course is on the structure of the models, but some applications are also discussed.

Synopsis

Kinematics: Lagrangian and Eulerian descriptions of motion, deformation gradient, invertibility

Analysis of strain: polar decomposition, stretch tensors, Cauchy–Green tensors Stress Principle: forces in continuum mechanics, balance of forces, Cauchy stress tensor, the Piola–Kirchhoff stress

Constitutive Models: stress-strain relations, hyperelasticity and stored energy function, boundary value problems, the variational problem, frame indifference, material symmetry, isotropic materials

Further topics: incompressible elasticity, linearized elasticity and the shape-memory effect in crystalline solids.

1. O. Gonzales and A. Stuart, A first course in continuum mechanics, (Cambridge University Press, 2008).
2. M. E. Gurtin, A introduction to continuum mechanics, (Academic Press, 1981).

Further Reading

1. P. G. Ciarlet, Mathematical Elasticity. Vol. I Three-dimensional Elasticity, (North-Holland, 1988)
2. S. S. Antman, Nonlinear Problems of Elasticity, (Springer, 1995)
3. J. E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity, Prentice–Hall, 1983