The concept of stress is introduced in terms of interatomic forces acting through a plane, and in the Cauchy sense of a force per unit area on a plane in a continuum. Normal stresses and shear stresses are defined. Invariants of the stress tensor are derived and the von Mises shear stress is expressed in terms of them. The conditions for mechanical equilibrium in a continuum are derived, one of which leads to the stress tensor being symmetric. Stress is also shown to be the functional derivative of the elastic energy with respect to strain,which enables the stress tensor to be derived in models of interatomic forces. Adiabatic and isothermal stresses are distinguished thermodynamically and anharmonicity of atomic interactions is identified as the reason for their differences. Problems set 2 containsfour problems, one of which is based on Noll’s insightful analysis of stress and mechanical equilibrium.
On Certain Translation Invariant Properties of Interior Transmission Spectra and Their Doppler’s Effect
