Convolution integrals, linear viscoelasticity

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

In particular, the LS-Dyna finite viscoelastic relationship [175] takes into accotmt rate effects through linear viscoelasticity by a convolution integral. The model corresponds to a Maxwell fluid consisting of dampers and springs in series. The Abaqus FEA model is reminiscent of, and similar to, a well-established model of finite viscoelasticity, namely the Pipkin-Rogers model [161]. This model, with an appropriate choice of the constitutive parameters, reduces to the Fung (QLV) model [173, 177]. 

In Chapter 6 it was shown that linear viscoelastic materials could be represented by the hereditary convolution integrals. 

From a mathematical point of view, the reaction force experienced by a linear viscoelastic device at rest for f < 0 can be expressed in the time domain through the following convolution integral ... 

The general approach to discussing linear viscoelasticity comes from the Boltzmann superposition principle represented as a convolution integral. For the shear stress as a function of shear strain, one obtains... 

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