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

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By Arthur S. Lodge

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Example text

Hence the vectors form a linear space, as stated. The dimen­ sionality follows from the fact that A3xl is three-dimensional and {Θ(Ρ)} is isomorphic to A3xi [see also (23)]. (20) Definition The tangent base vectors ß,(ß, P) at particle P for a body coordinate system B are three contravariant body vectors at P whose repre­ sentative matrices in B are öi9 where δ, = [1, 0, 0], (21) δ2 = [0, 1, 0], δ3 = [0,0, 1]. , the elements of the representative matrix) of the vector ß, in B. We use a superscript, as in 0', to label components of a contravariant vector; thus Θ has components Θ1 in B, where 0 = [0r] is the representative matrix in B.

Some additional operations and definitions will be postponed to Chapter 5, after we have given the application to shear flow in Chapter 3. The wording in (41a) admits two different interpretations. , 01 in (11)]. , space vectors or body vectors) associated with points of the manifold, with one object to one point. 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 ^ .

Any covariant space vector i#(Q) at Q can be written in the form (22) I I ( Q ) = «,(JC)*'(S, Q), 2 36 THE BODY M E T R I C TENSOR FIELD where [ur(x)] = u is the representative matrix of w(Q) in S. The scalar product of M(Q) and v(Q) is defined by the equation II(Q) · v(Q) = (23) UiixMx), where v(Q) is a contravariant space vector at Q having representative matrix [v\x)] in S. A space scalar field s is a mapping s\2l-^0l which assigns to each place Q a unique real number s(Q) ; the transformation law is of the form (24) j(Q) = s(x) = s(x), where s(x) and s(3c) are the component functions of the scalar field in coordi­ nate systems S and S, respectively.

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