Body Tensor Fields in Continuum Mechanics. With Applications by Arthur S. Lodge

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3(28) illustrates two configurations of a deforming body to emphasize the rather unfamiliar fact that d\ is independent of time. 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 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. In general, a given particle P will occupy different places Q and Qr at different times t and t'; the space tensors of given kind at Q and at Q' form two disjoint linear spaces and cannot be added or compared by any elementary process.

5(19)]. 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.