An Introduction to the Phenomenological Theory of by J. Grindlay

By J. Grindlay

''In this monograph the writer describes the foundation and derivation of the macroscopic or phenomenological conception of the elastic, dielectric and thermal homes of crystals as utilized within the box of ferroelectricity. lots of the effects and ideas defined are scattered throughout the literature of this topic and this ebook offers them including their actual historical past in a single reference. The dialogue is particular to the speculation required to explain the homes of homogeneous specimens topic to low frequency fields.'' obtained it?

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F r o m eqn. ( 2 . 3 . 1 6 ) the density of mechanical work when there are no body forces is Σ [idTji/exj) dxi+Tj^{d{dxd/exj}i where δχ is an infinitesimal displacement; from eqn. ( 2 . 4 . 8 ) the density of electric work is £ · δ Ζ ) . dD dv. Consider the term in curly brackets. F o r the linear theory we retain only the leading term and write dxj dUi/dXj are the deformation gradients introduced in § 2 . 2 . 1 . t Sometimes referred to as constitutive relations. 14) í t e [ ( t ) - M Í ) - ^ where dWfdt is the rate of work done on the volume ν and ά, D and duJdXj are the time derivatives of the fields w, D and duJdXj respec­ tively.

It is easy to obtain explicit relations between the coeflScients of one set and those of another. A s an ihustration consider the relationship between cf^, Ay^, ßfj and cf^, Cj^, ε^. We denote the elements of the matrix reciprocal to the matrix [ß^j] by {ß^)"^. Thus Σ {ßiy^ßfk = ^ik ih k = l , 2 , 3). 37) Multiply both sides of eqn. 29) by (ßfi)"^ and sum over /, έ ißld-'Ei = - Σ {ßli)-'hißSß + Y {ßi)-^ßlDj. Using eqn. 38) β. ι 56 J=l The Linear Elastic Dielectric Substitute for Z), from eqn. 38) in eqn.

We may solve these equations and express any nine of the components in terms of the remaining nine. The coefficients appearing in such a set of equations are of course determined by the cf^, A/,, ßfj appearing in eqns. 29). 31) j β J β J l^d,ßTß+YerjEj. 1. The arrays of coefficients cf^, s^ß, jf^, efj, efj, ßjj are invariant under the interchange of indices a, β or /, y, whichever is appropriate. T o prove this we use the technique of Legendre transformations. F o r example, consider the quantity ρον^' = QoW—^^T^S^, F r o m eqns.

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