## Chapter 7

### Solid mechanics

The equations of linear elasticity are presented. The Euler-Bernoulli
beam equation is derived, elastic waves are discussed: P, S, and
Rayleigh waves. The complex variable formulation of planar cracks is
presented, and the derivation of the singular integral equation
relating crack pressure to crack extension. Plasticity is discussed
via the Prandtl-Reuss equations, and an application (out of Hill's
book) given to the elastic-plastic torsion of a cylinder. Soil
mechanics, poroelasticity, and consolidation are
described. Viscoelasticity is modelled for nonlinear strains via the
introduction of the corotational Jaumann derivatives of tensors; the
co-deformational models are also described*. An application to a
simple shear flow of a corotational Jeffreys fluid is described, and
the development of normal stresses is demonstrated, as well as the
nonlinear flow law which in fact results from a constitutively
*linear* flow law.

*Apparently, the two codeformational derivatives of the trace of the
deviatoric stress tensor are equal to its material derivative plus or
minus the viscous dissipation. It seems to me that this precludes the
codeformational derivatives from properly describing the motion of an
incompressible medium, where the trace of this stress tensor is
zero. I don't understand the matter fully.