Synopsis for C6.1b: Elasticity and Plasticity

Level: M-level \hfill Method of Assessment: Written examination.
Weight: Half-unit. OSS paper code 2467.

Recommended Prerequisites

Familiarity with classical and fluid mechanics (Part A) and simple perturbation theory (Part C course) will be useful. The complementary Part C course, Solid Mechanics, will be especially useful.

Overview

The course gives a rapid review of mathematical models for basic solid mechanics. Starting with one-dimensional systems, the structure of theory of continuum mechanics is explained then generalised to three dimensions. The general theory of continuum materials is reviewed and different limits will be explained. There will be a special emphasis on elastic and hyper-elastic materials. Reduction from exact elasticity to linear elasticity will be considered as well as geometric reduction to rods, membranes, and shells. Statics, dynamics, and stability problems will be analysed with applications to biological, physical, and engineering systems. Given time, generalization of elasticity theories to include, fracture, contact and anelastic behaviors (plasticity, thermoealsticity, viscoelasticity, morphoelasticity) will be explained. The last part of the course will be dedicated to new challenges in the theory of elasticity/plasticity.

Synopsis

Introduction on one-dimensional systems. Kinematics, Conservation laws, Thermodynamics and Constitutive relationships. Review of tensors. Exact elasticity. Reduced theory (strings, rods, membrane, shells). Application and Analysis of systems (bifurcation theory, wave propagation, stability, buckling). Generalisation of elastic theories: 1/ Anelastic theories: thermo-elastity, elasto-plasticity, visco-elasticity, morpho-elasticity (biological systems). 2/ Fracture, contact, mixture theory and homogenisation.

1. Ray Ogden Nonlinear Elastic Deformations (Dover 1984).
2. S.S. Antman, Nonlinear problems of elasticity (Springer 2005).
3. R.M. Hill, Mathematical Theory of Plasticity (Oxford Clarendon Press, 1998).
4. A.E.H. Love, Treatise on the Mathematical Theory of Elasticity (Dover, 1944).
5. L.D. Landau and E.M. Lifshitz, Theory of Elasticity (Pergamon Press, 1986).