By Arthur S. Lodge
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Extra resources for Body Tensor Fields in Continuum Mechanics. With Applications to Polymer Rheology
For the body vector άξ, we use a boldface type for d as well as for ξ to emphasize the fact that d\ is a single entity [defined by (27)] and is not the difference of two vectors ξ : The coordinate matrices £ , £ , . . for a particle do not satisfy the transformation law for a contravariant vector and so we cannot define a body vector ξ whose components equal the coordinates of P. 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.
2(12) takes the form (12) (ά*)2 = άξγαξ = άξγαξ. 4(12)]. The body metric tensor field γ'(P, i) varies with time (unless the motion is rigid); the space metric tensor field g (Q) is independent of time. , the change of metric) for any two states t and t'. The covariant strain-rate body tensor γ [which occurs in (20) when n = 1 ] is defined as a limit and the «th-strain-rate tensor [which occurs in (20)] is defined by successive applications of this process. Equations (20) and (21) involve differences of body tensors at a given particle P; these differences are themselves body tensors (of the same kind) at P because body tensors of given kind at any given particle form a linear space.
2 BODY AND SPACE MANIFOLDS 21 after we have defined the metric tensor. The vl simply represent the rate of change of place coordinates for a given particle. We could (in the interests of symmetry between the body manifold and the space manifold) also consider the derivatives οφ\χ, t)/dt, which describe the rate of change of particle coordinates at a given place, but we have no use for them. (9) Definition A metric for a geometric manifold is a correspondence be tween unordered pairs of neighboring points and positive real numbers ds.