CHAPTER 2KINEMATICS
In this chapter, the general kinematic equations for the continuum are developed. The kinematic analysis presented in this chapter is purely geometric and does not involve any force analysis. The continuum is assumed to undergo an arbitrary displacement and no simplifying assumptions are made except when special cases are discussed. Recall that in the special case of an unconstrained three-dimensional rigid-body motion, six independent coordinates are required in order to describe arbitrary rigid-body translation and rotation displacements. The general displacement of an infinitesimal material volume on a deformable body, on the other hand, can be described in terms of 12 independent variables: 3 translation parameters, 3 rigid-body rotation parameters, and 6 deformation parameters. One can visualize these modes of displacements by considering an infinitesimal cube that may undergo an arbitrary displacement. The cube can be translated in three independent orthogonal directions (translation degrees of freedom), rotated as a rigid body about three orthogonal axes, and subjected to six independent modes of deformation. These deformation modes are elongations or contractions in three different directions and three shear deformation modes.
It is shown in this chapter that the rotations and the deformations can be completely described using the matrix of the position vector gradients, which in general has nine independent elements. This fact can be mathematically ...