Non-Classical Memory Kernels in Linear Viscoelasticity

Abstract

In linear viscoelasticity a large variety of regular kernels have been classically employed, depend- ing on the mechanical properties of the materials to be modeled. Nevertheless, new viscoelastic materials, such as viscoelastic gels, have been recently discovered and their mechanical behavior requires convolution integral with singular kernels to be described. On the other hand, when the natural/artificial aging of the viscoelastic material has to be taken into account, time dependent kernels are needed. The aim of this chapter is to present a collection of non-standard viscoelastic kernels, with special emphasis on singular and time-dependent kernels, and discuss their ability to reproduce experimental behavior when applied to real materials. As an application, we study some magneto rheological elastomers where viscoelastic and magnetic effects are coupled

Response functions in linear viscoelastic constitutive equations and related fractional operators

“…To complete this section, we address the reader's attention to the analysis of Carillo and Giorgi [29] where clearly the physical reasons to use singular power-law kernels are explained: the response of the material may be for short-time or as long tail (for long times) modelled by time-dependent power-law functions, thus allowing the application of the classical fractional derivatives with memory kernels which are unbounded at the origin in contrast to the regular memory kernels. Hence, if the power-law is not exhibited by the material response functions, then it is inadequate to apply the power-law memory kernels.…”

“…Which approximation function would fit the experimental points depends on the type of materials and would be established by simple trial error processes. We already commented the power-law relaxation function which is related to the so-called power-law materials such as new viscoelastic polymers, bio-inspired materials or natural materials as mineralized tissues such as bones, ligaments and tendons (see the detailed comments in [29]) and is also related to short-time spectra of nonlinear viscoelastic materials (refer the previous sections). Now, we stress that the attention on two possibilities relevant to the fractional modelling by non-singular kernels: (i) discrete spectrum approximation by Prony series and (ii) discontinuous approximation by the Mittag-Leffler function.…”

Linear Viscoelastic Responses: The Prony Decomposition Naturally Leads Into the Caputo-Fabrizio Fractional Operator

“…Since the power-law relaxation is not the main problem discussed in this work, at the end of this point, we refer to the study of Carillo and Giorgi [56] (section 4 in this chapter) where it is clearly demonstrated that the response of the material may be for short-time or as long tail (for long times) modeled by timedependent power-law function. Here, we may repeat again (see the comments in Hristov [14], too) that, if the power-law is not exhibited by the material response functions then it is inadequate to apply the power-law memory kernels.…”

A 3-dimensional singular kernel problem in viscoelasticity: an existence result

“…To take into account a wider class of materials, as specified for instance in [12,14], the regularity assumptions on G are relaxed. In particular the request (5) 2 is removed, that is, we replace ( 5) with (10).…”

“…The interest on viscoelastic materials and their mechanical response to external actions is testified by a wide literature which goes from pioneering works due to Boltzmann [5] and Volterra [40] to a the theoretical study developed to understand and model new materials also artificially devised. An overview on models which refer to materials requiring a non-classical memory kernel is provided in [14] and in the references therein. Specifically, the model adopted throughout is due to Giorgi and Morro [26], and, later, reconsidered by Gentili [25].…”