Kinematics in Moving Convected Coordinates

The primary thrust of this section is to prepare ourselves in order to be able to write the kinematic quantities defined in a moving coordinate system via the transformation rules in terms of the Cartesian components in the fixed coordinate system. This is necessary, as will be discussed in the next chapter, for transforming a constitutive equation, which was first written in a moving coordinate system, into a fixed coordinates so that it can be used in conjunction with the equations of continuity, motion, and energy that are normally written in the fixed coordinates. 

In describing the kinematics of a deformable body, instead of using a coordinate system fixed in space, it is convenient to use a coordinate system embedded in the moving object. This is frequently referred to as a convected coordinate system, and was first introduced by Oldroyd (1950). Any measure of deformation (strain) defined relative to such a coordinate system always refers to the same element of materials, and therefore should be independent of the local rate of translation or rotation. As will be shown in this section, if they are going to be useful, all kinematic variables defined in terms of the convected coordinates must be transformed to a fixed coordinate system as all physical measurements are made relative to the fixed coordinate system. 

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