# C5.1 Solid Mechanics - Material for the year 2019-2020

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 Prelims Linear Algebra course, is useful preparation); and (ii) some 3D calculus (mainly differentiation of vector-valued functions of several variables).

16 lectures

### Assessment type:

- Written Examination

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 continuum mechanics and in particular, elasticity theory, the central model of solid mechanics, and give some interesting applications to the behaviour of materials. The course is useful preparation for C5.2 Elasticity and Plasticity. Taken together the two courses will provide a broad overview of modern solid mechanics, with a variety of approaches.

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.

- Introduction: one-dimensional elasticity

Kinematics, dynamics, balance equations, applications - Kinematics

Lagrangian and Eulerian descriptions of motion, deformations, vectors and tensors, derivatives of vector and tensor fields, deformation gradients, transformation of volume, surface, line elements, polar decomposition theorems, strain tensors. Examples of deformations - Dynamics

Balance laws of continuum mechanics (conservation of mass, linear momentum, angular momentum). Stress tensors, Energy balance, Constitutive equations for fluids and solids. - Nonlinear elasticity

Nonlinear elasticity (frame indifference, constitutive equations, material symmetries, isotropy). Exact solutions in elastostatics: Incompressibility and models of rubber. Universal deformations for compressible materials. Exact solutions for incompressible materials, e.g. the Rivlin cube, simple shear, inflation of a balloon. - Linear Elasticity

Linear elasticity as a linearization of nonlinear elasticity. Compatibility conditions. Plne-strain, plane stress solutions, planar waves in elasto-dynamics.

- Ray Ogden,
*Nonlinear Elastic Deformations*, (Dover, 1997). - M. E. Gurtin,
*A introduction to continuum mechanics*, (Academic Press, 1982).

- A. Goriely,
*The Mathematics and Mechanics of Biological Growth*, Chapters 4 and 11 (Springer, 2017) - S. S. Antman,
*Nonlinear Problems of Elasticity*, vol 107 of Applied Mathematical Sciences (Springer, 2015). - P. G. Ciarlet,
*Mathematical Elasticity*, Studies in Mathematics and its Applications; v. 20, 27, 29 (North--Holland, 1988).