New PDF release: Body Tensor Fields in Continuum Mechanics. With Applications

By Arthur S. Lodge

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Extra resources for Body Tensor Fields in Continuum Mechanics. With Applications to Polymer Rheology

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These vectors form a basis for contravariant space vectors at Q ; for any such vector u(Q), we have (36) »(Q) = u'WftfiS, Q), r where [v (x)] is the matrix representing v(Q) in S. »! + a2v2)S = αγνχ + a2v2 = LQ\S, S^a^ + a2ϋ2), where vl and v2 are any two vectors at Q and vx and v2 are their representative matrices in 5. An important example of a contravariant space vector at Q is the displace­ ment vector (38) dx = dx'biiS, Q), which is determined by any two neighboring places Q and Q1 ; dx1 denotes their coordinate differences in S.

It is clear that body tensor fields are more con­ venient than general space tensor fields for this purpose. (dh) = ( g - d . ) · - Y " HP, 0 · (grada), (24) since grad σ9 a body vector normal to the material surface σ(ξ) = c, is independent of time. This equation gives an interpretation to the contravariant nth strain-rate body tensor, which may be compared with the corre­ sponding interpretation given by (20) for the covariant nth strain-rate body tensor. , Δγ is additive). We claim that these are the simplest properties which one can reasonably require of a mathematical description of strain.

Y " HP, 0 · (grada), (24) since grad σ9 a body vector normal to the material surface σ(ξ) = c, is independent of time. This equation gives an interpretation to the contravariant nth strain-rate body tensor, which may be compared with the corre­ sponding interpretation given by (20) for the covariant nth strain-rate body tensor. , Δγ is additive). We claim that these are the simplest properties which one can reasonably require of a mathematical description of strain. As far as we are aware, there is no space tensor field (general or Cartesian) which has all these proper­ ties.

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