Time Derivative of Convected Coordinates

Having defined strain tensors in convected coordinates, we now describe the rate-of-strain (or rate-of-deformation) tensor. This may be obtained by taking the derivative of a strain tensor with time, with the convected coordinates held constant. Such a derivative is commonly referred to as the material derivative, which may be considered as the time rate of change as seen by an observer in a convected coordinate system. Using the notation D/Dr for the substantial (material) time derivative, we have from Eq. (2.86) 

Since every material point always has the same convected coordinate position at all times, regardless of the extent of deformation of the medium, the relative coordinate displacements between any two points must be constant, so that any change in the actual distance between the points must be reflected by a change in the metric v-j. That is, if the distance between two points d changes with time, the convected metric v-j must change accordingly with time since, by definition, the convected coordinates f of a material point are independent of time. Therefore Eq. (2.99) may be rewritten with the aid of Eq. (2.86) as 

Similarly, for the contravariant convected strain tensor we have 

It should be remembered that the metric tensor v j and in a convected coordinate system are related to the metric tensors gij and g J in a spatial coordinate system by Eqs. (2.84) and (2.85). 

Therefore, we can write the following general rule of coordinate transformation of a second-order tensor  

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