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On the Foundations of Linear Isotropic Visco-Elasticity

D. R. Bland
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 250, No. 1263 (Apr. 7, 1959), pp. 524-549
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/100809
Page Count: 26
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On the Foundations of Linear Isotropic Visco-Elasticity
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Abstract

The properties of a linear visco-elastic material are developed from the hypothesis that the microscopic structure of such a material is mechanically equivalent to a network of elastic and viscous elements. The stored energy and the rate of dissipation of energy can be found for any material element at any time. For an isotropic material, each deviatoric component of strain is related solely to the corresponding deviatoric component of stress and the dilatational part of the strain solely to the dilatational part of the stress; the energies are the sum of the respective energies for the deviatoric components and for the dilatation. It is shown both that the strain can be expressed in terms of the stress increments and the creep function, and that the stress can be expressed in terms of the strain increments and the relaxation function. The complex compliances and moduli have alternating zeros and poles on the positive imaginary axis and no zeros or poles elsewhere. The stress-strain law can be expressed in operational form, Pσ = Qε , where P and Q are polynomials in d/dt with constant coefficients. The zeros of P and Q are all real and non-positive and they alternate. The energies can be expressed in terms of either the creep function and the stress at previous times, or the relaxation function and the strain at previous times, or the stress, strain and their time derivatives at a given time or, for a sinusoidal oscillation, in terms of the complex compliance and its derivative with respect to frequency. It is shown that models consisting of Voigt element in series, or of Maxwell elements in parallel, can represent the mechanical properties and the stored and dissipated energies of any visco-elastic material. The analysis can be extended to networks containing an infinite number of elements. Two examples, one of each of two different cases, are given.

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