# C5.2 Elasticity and Plasticity - Material for the year 2020-2021

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Familiarity will be assumed with the Part A course options: *A2: Metric Spaces and Complex Analysis*, *A1: Differential Equations 1*, *A6 Differential Equations 2* and *ASO: Calculus of Variations*. A basic understanding of stress tensors from either *B5.3 Viscous Flow* or *C5.1 Solid Mechanics* will also be required. The following courses are also helpful: *B5.1 Techniques of Applied Mathematics*, *B5.2 Applied Partial Differential Equations*, *C5.5 Perturbation Methods*, and *C5.6 Applied Complex Variables*.

16 lectures

### Assessment type:

- Written Examination

The course starts with a rapid overview of mathematical models for basic solid mechanics. Benchmark solutions are derived for static problems and wave propagation in linear elastic materials. It is then shown how these results can be used as a basis for practically useful problems involving thin beams and plates. Simple geometrically nonlinear models are then introduced to explain buckling, fracture and contact. Models for yield and plasticity are then discussed, both microscopically and macroscopically.

By the end of this course, students will be able:

- to formulate and solve problems in linear elasticity, including models for elastic waves, mode conversion and mode III cracks;
- to derive models for thin elastic beams and to describe buckling behaviour using weakly nonlinear asymptotic analysis;
- to derive, analyse and solve linear complementarity problems describing smooth contact of elastic strings and membranes;
- to formulate, analyse and solve free boundary problems for linear perfectly plastic yield of granular materials and metals in simple geometries.

Review of tensors, conservation laws, Navier equations. Antiplane strain, torsion, plane strain. Elastic wave propagation, Rayleigh waves. Ad hoc approximations for thin materials; simple bifurcation theory and buckling. Simple mixed boundary value problems, brittle fracture and smooth contact. Perfect plasticity theories for granular materials and metals.

- P. D. Howell, G. Kozyreff and J. R. Ockendon,
*Applied Solid Mechanics*(Cambridge University Press, 2008) - S. P. Timoshenko and J. N. Goodier,
*Theory of Elasticity*(McGraw-Hill, 1970) - L.D. Landau and E.M. Lifshitz,
*Theory of Elasticity*(Pergamon Press, 1986)