By Arthur S. Lodge

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Extra info for Body Tensor Fields in Continuum Mechanics. With Applications to Polymer Rheology

Sample text

There are similar results for other kinds of tensor. Suppose we have a body coordinate system B which is instantaneously orthogonal at time t. Then the base vectors ß f (£, P) are orthogonal at t and so are the base vectors P'(B, P). , by dividing them by their magnitudes at time t, which are given by (4).

15) Definition Given any material surface whose equation in an arbitrary body coordinate system B is σ(ξ) = c, where c is a constant. A normal to the surface at P is a covariant body vector v(P) at P whose component matrix in B is λ[δσ/δξί'] where λ is any scalar. From the analysis used in proving (12), it follows that the component matrix in any other body coordinate sys­ tem B is X[döldlr]. The justification for calling v a normal lies in the equation (16) ν(Ρ)·

The d in άζι in (27), on the other hand, does denote a difference of coordinates. To use a symbol d\ (instead of <ίξ) might therefore be misleading. In the example of d\, we have a graphic illustration of the fact that a contravariant body vector is an entity different from every Cartesian vector of elementary vector analysis, for a given Cartesian vector has a unique magni­ tude and direction (in space); the body vector