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

By Arthur S. Lodge

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Coordinate sys­ tems which are congruent at one instant will not, in general, be congruent at another instant. 2, we defined quantities of interest in connection with the flow or deformation of a continuous material: vl describes the velocity relative to space, άξι describes a material direction as defined by two neigh­ boring particles, and γ^(ξ, t) describes the metric (or shape) as a function of time t. The values of all these quantities, however, depend not only on the properties described (as we wish) but also on the choice of coordinate system (S for v\ and B for άξι and y 0 ), which is arbitrary.

This ambiguity is a disadvantage of the tensor notation (not present when the components are used) which is not, however, very serious when one deals with first- and second-rank tensors only. The convention (vii) is obtained if one forms Θ: Ψ by contracting Θ · Ψ. (16) Problem Prove that the quantities given by contraction in the list (15) are quantities of the kinds stated on the right. , that the reciprocal is a unique inverse. Prove also that (ii) δ · Φ = Φ · δ = Φ. There are similar results and definitions for space tensors of second rank which can all be obtained from the above body tensor results by making the obvious changes.

3(5) possessed by transformation matrices A? *(£, 5), . . 3(12), which ensures that the vector has a significance independent of the choice of any particular coordinate system from the set ^ . 3(6) shows that the matrices Ä P (£, B), . . also possess the same group property; it follows that we can define another kind of vector (called a covariant vector) if we simply replace Af1 in the transformation law (11) by Ä P . (1) Definition Let φ be any 3 x 1 column matrix associated with any given body coordinate system E and particle P.

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