By Arthur S. Lodge
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A part of the Biomedical Engineering sequence, this booklet examines the administration of scientific know-how, together with scientific engineering, moral matters and destiny instructions
Extra info for Body Tensor Fields in Continuum Mechanics. With Applications to Polymer Rheology
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)·ξ = 0, which is valid when ί/ξ is the contravariant vector determined by any two neighboring particles P and Vx in the given surface.
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 ξ has a unique direction (in the material) but does not have a unique magnitude.
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