Dynamics of Continuum

Abstract

In this chapter we study the dynamics of a continuum. We use Newton’s second law to derive the momentum equation. We show that the stresses inside a continuum are completely defined by the stress tensor. We use the angular momentum equation to prove that the second-order stress tensor is symmetric. Then we derive the momentum equation in differential form in both the Eulerian as well as in Lagrangian variables. We obtain that in the Eulerian variables the product of the continuum acceleration and the density is equal to the sum of the divergence of the stress tensor and the body force, which is an external force exerted on the continuum, like the force of gravity. To derive the momentum in Lagrangian variables we introduce the nominal stress tensor, also called the Piola–Kirchhoff tensor. This tensor is not symmetric and is expressed in terms of the stress tensor and deformation-gradient tensor. Then the product of the continuum acceleration and the density taken at the initial time is equal to the sum of the divergence of the nominal stress tensor and the body force. At the end of this chapter we consider the kinematic and dynamic boundary conditions that can be imposed on the continuum.