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.