Deformation, Kinematics and Dynamics

Abstract

This chapter will systematically discuss the differential geometry, kinematics and dynamics of deformation in continuous media. To discuss deformation geometry, the deformation gradients will be introduced in the local curvilinear coordinate system, and the Green and Cauchy strain tensors will be presented. The length and angle changes will be discussed through Green and Cauchy strain tensors. The velocity gradient will be introduced for discussion of the kinematics, and the material derivatives of deformation gradient, infinitesimal line element, area and volume in the deformed configuration will be presented. The Cauchy stress and couple stress tensors will be defined to discuss the dynamics of continuous media, and the local balances for the Cauchy momentum and angular momentum will be discussed. Piola-Kirchhoff stress tensors will be presented and the Boussinesq and Kirchhoff local balance of momentum will be discussed. The local principles of the energy conservation will be discussed by the virtual work principle. This chapter will present an important foundation of continuum mechanics. From such a foundation, one can further understand other approximate existing theories in deformable body and fluids.