Vibrations of Linear Piezostructures
3. Review of Continuum Mechanics
Since piezoelectric materials exhibit coupling between their mechanical and electrical properties, in this chapter we review elementary principles from the foundations of continuum mechanics and linear elasticity. We present the definition of the stress vector τ in a continuum, the corresponding definition of the stress tensor T := Tijgi ⊗ gj, Cauchy’s formula that relates τ and T, and the equilibrium equations in Section 3.1. The definition of the linear strain tensor S ∶= Sijgi ⊗ gj is given in Section 3.2. The definition of the strain energy density, as well as some example calculations of strain energy for specific common structural elements, is the topic of Section 3.3. Generalized Hooke’s law is presented in Section 3.4, which specifies the constitutive laws that are employed in linear elasticity. Finally, a summary of the initial-boundary value problem that underlies the formulation of mechanics for linearly elastic materials is presented in Section 3.5.