Elastic-viscoelastic correspondence

A theory of thermoviscoelasticity that includes the temperature dependence of the relaxation or retardation functions is necessarily nonlinear, and consequently the elastic-viscoelastic correspondence principle is not applicable. Nevertheless, a linear theory of thermoviscoelasticity can be developed in the framework of rational thermodynamics with further constitutive assumptions (Ref. 5, Chap. 3 see also Ref. 10). 

Equations (17.20) are Laplace transforms of the equations of viscoelastic beams and can be considered a direct consequence of the elastic-viscoelastic correspondence principle. The second, third, and fourth derivatives of the deflection, respectively, determine the forces moment, the shear stresses, and the external forces per unit length. The sign on the right-hand side of Eqs. (17.20) depends on the sense in which the direction of the strain is taken. 

The Phenomenology of the Linear Theory of Viscoelasticity. One of the powers of the linear viscoelasticity theory is that it is predictive. The constitutive law that comes from Boltzmann superposition theory requires simply that the material functions discussed above be known for a given material. Then, for an arbitrary stress or deformation history, the material response can be obtained. In addition, the elastic-viscoelastic correspondence principle can be used so that boundary value problems such as beam bending, for which an elastic solution exists, can be solved for linear viscoelastic materials as well. Both of these subjects are treated in this section. 

And integration of the hereditary integrals for strain and deflection gives the solution to any applied history of the moment M t). A note of caution, however, arises for mixed conditions in which the interface between the stress and the displacement boundaries is not constant. In such cases the elastic-viscoelastic correspondence principle is not applicable and the solutions become more difficult (21). 

Summary Non-stationary random vibrations of polvcfonally shaped slightly damped Kirchhoff-plates are presented. The frequency response function of the undamped structure is calculated by an advanced bound-ary-integral equation method with Green s functions of finite domains. Subsequently, light hysteretic damping is built in by applying the quadrature type of elastic-viscoelastic correspondence. ... 

Having evaluated the frequency response function F(v) of the deflection of the undamped plate, light damping is built in by an alternative quadrature type of the elastic-viscoelastic correspondence principle, /4/, /5/. 

ZIEGLER, F, The elastic-viscoelastic correspondence in case of numerically determined discrete elastic response spectra. ZAMM 63 (1983), T 135-137. 

The fact that (2.3.9) and (2.3.13) are manifestly correct is the essential justification of (2.3.5) and (2.3.6). The arguments leading up to (2.3.9) and (2.3.13) could strictly have been omitted. However, they provide a link between reasoning based on formal elastic-viscoelastic correspondence as incorporated in (2.1.10), (2.1.11) and (2.3.1) and the methodology which is developed below. 

Recently, Matadi Boumbimba et al. [12] proposed a temperature- and frequency-dependent version of the rule of mixtures to describe the viscoelastic response, in terms of storage modulus, of PMMA/Cloisite 20A and SOB. In the present work, to predict the effective viscoelastic response of polymer-based nanocomposites, the elastic-viscoelastic correspondence principle [11] is applied to our micromechanical model. The two implicit equations (5) become ... 

The elastic-viscoelastic correspondence says that the viscoelastic solution can be generated from the elastic one using a Boltzmann hereditary integral (hereditary for past dependent) with respect to past time from f = 0 to the present time if = t. This is illustrated as... 

Using the elastic-viscoelastic correspondence with hereditary Boltzmann Integral allows for the general solution to be found as (Section 6.4.5.S, Shimizu, Yanagimoto, and Sakai, 1999 Sakai, 2002 Fischer-Cripps, 2004b Cheng and Cheng, 2005 Oyen, 2005 Bernard et ai, 2010)... 

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