-->is positive semi-definite, absolutely continuous and summable on ℝ+. Then, for every
Besides, it is differentiable for all
for every compact set
(M3) For every fixed
for every compact set
(M4) There exists a nonnegative scalar function
for every (
According to (M2), the
Borrowing from the scalar case,
Finally, an integration by part of Eq. (30) yields
provided that Eq. (2) holds for
As an advantage, within Eq. (34),
In order to stress the aging effects, we might assume the following
(M5) There exist three functions,
Accordingly, the stress-strain relation (34) may be rewritten into the form (22). The aging factors
So far, we restrict our attention to scrutinize stress-strain relations in the form (30). In particular, for isotropic materials takes the special form
In the sequel, we scrutinize the special isotropic vector-valued kernel
as in the rheological Wiechert-Maxwell model devised in Section 3.1. We first prove that properties (M1)-(M4) hold provided that some additional restrictions are imposed on the material functions
(M1) Starting from (C1), it is quite trivial to prove this property.
(M2) By virtue of Eq. (35) and (C1),
Hence (M2) is fulfilled.
(M3) It is obviously true as
(M4) In order to prove this property we need more restrictive conditions. Since
a sufficient condition to ensure (M4) is given by
In order to prove these inequalities, we now assume
and for every
It is apparent that
When non-aging material parameters are involved, Eq. (37) reduces to
We present here some special expressions of material functions
For simplicity, we restrict our attention to a single Maxwell element. Letting
so that (A1) and (A2) hold true. Condition (38) is fulfilled for all
so that (A1) and (A2) hold true. On the other hand, condition (38) reduces to
which is equivalent to
and is fulfilled for all
We now derive the motion equation related to the time-dependent viscoelastic stress-strain relation (32) and we examine its compatibility with thermodynamics. The displacement field
In order to introduce the initial boundary value problem for this equation, we have to take in mind that it is not invariant under time shift.
Consistent with linear viscoelasticity, we restrict attention to isothermal processes, namely those where the temperature is constant and uniform. Hence, the local form of the second law inequality reduces to the
Accordingly, we take the dissipation inequality in the form
In materials with memory, the motion equations are required to rule both the displacement instantaneous value
Now, we introduce a time-dependent free energy density borrowing its expression from the Graffi’s single-integral quadratic form (see  and references therein). Let
For ease in writing, hereafter the dependence on
Then, by virtue of (42), and some integration by parts, we obtain
and, taking into account (41),
In summary, we end up with
Owing to (M4), this yields
The study of singular kernel problems is motivated by the modeling of new materials and, in particular, of the mechanical behavior of some new viscoelastic polymers and bio-inspired materials. As noticed in , the appropriate way to handle the response of certain time-dependent systems exhibiting long tail memories is to account for power laws, both for creep and relaxation, leading to the occurrence of fractional hereditariness. Another example encountered in natural materials is mineralized tissues as bones, ligaments, and tendons. They exhibit a marked power-law time-dependent behavior under applied loads (see e.g. ), since the high stiffness of the crystals in such tissues is combined with the exceptional hereditariness of the collagen protein-based matrix. In all these cases, we are forced to abandon the regularity assumptions (8) and assume the memory kernels obey Eq. (6) and are unbounded at the origin.
The idea of singular kernels to model particular cases of viscoelastic behaviors was introduced by Boltzmann  in the nineteenth century. The fast growth of polymer science motivated further developments of viscoelasticity in the middle of the twentieth century [16, 17], but a Volterra-type integro-differential equation with a regular kernel (typically, a finite sum of exponentials) was preferred to the Boltzmann approach in the modeling of the mechanical response [5, 18]. Later, however, many authors addressed their interest to singular kernel problems, both under the analytical as well as the model point of view [19–24], and their thermodynamical admissibility was analyzed in . In modern viscoelasticity, it is a central problem to understand how to model the memory kernels, and it should be argued as far as possible on physical grounds. So, the first question to answer to is why do we consider singular kernel models. More recently, new viscoelastic materials, such as viscoelastic gels, have been discovered and their mechanical properties are well described by virtue of convolution integral with singular kernels: for instance, fractional and hypergeometric kernels . This applicative interest gave rise to a wide research activity concerning singular kernel problems, both in rigid thermodynamics with memory as well as in viscoelasticity (see, for instance, [26–31], and especially concerning applications of fractional calculus to the theory of viscoelasticity and the study of new bio-inspired materials [15, 32–35]. A recent book  provides an overview on this subject. In this framework, Fabrizio  analyzes the connection between Volterra and fractional derivatives models and shows how experimental results motivate us to adopt, as in this present article, less restrictive functional requirements on the kernel representing the relaxation modulus.
To start with, the one-dimensional classical viscoelasticity problem is recalled. It reads
where Ω = (0,1). When, to model the physical behavior of new materials or polymers, the regularity assumptions on the relaxation modulus are relaxed,
that is, now, the relaxation function
Then, introduce the regular problems:
together with the initial and boundary conditions
then, find approximated solutions
show the existence of the limit solution
prove the uniqueness of the limit solution
Note that, corresponding to each value of
where the superscripts, in the case
weak formulation, on introduction of test functions
consider separately the terms without
the terms with
prove convergence via Lebesgue’s theorem.
Furthermore, the weak solution, as stated in the following theorem, is unique.
As a final remark, we wish to emphasize that, since the isothermal rigid viscoelasticity model exhibits remarkable analogies, under the analytical point of view , with rigid thermodynamics with memory, then, analogous results can be obtained also in the study of singular kernel problems in such a framework .
This section is concerned about a problem in magneto-viscoelasticity, again under the assumption of a memory kernel singular at the origin. The interest in magneto-viscoelastic material finds its motivation in the growing interest in new materials such as magneto-rheological elastomers or, in general, magneto-sensitive polymeric composites (see [39–41] and references therein). The model adopted here to describe the magneto-elastic interaction is introduced in . Evolution problems in magneto-elasticity are studied in  and, later magneto-viscoelasticity problems are considered in [44, 45]. Notably, under the analytical viewpoint, when the coupling with magnetization is considered, the problem to study is modeled via a nonlinear integro-differential system while the purely viscoelastic problem is linear.
To understand the model equations, a brief introduction on the model magnetization here adopted, based on , who revisited the Gilbert magnetization model. Accordingly, when Ω ⊂ ℝ3 denotes the body configuration, the related magnetization changes according to the Landau Lifshitz equation, which, in Gilbert form, where
The quantities of interest, in the general three-dimensional case, are the following ones:
where the coefficients
Then, the following constitutive assumptions are assumed. Thus, the exchange magnetization energy is given by
Then, the magneto-elastic energy is given by
The viscoelastic energy is given by
where the tensor’s entries of
Then, the total energy of the system is given by
taking into account, further to the single magnetic and viscoelastic contribution, of the exchange energy.
The problem we are concerned about is the behavior of a viscoelastic body subject also to the presence of a magnetic field; in the one-dimensional case, it is modeled by the nonlinear system
where Ω = (0,1),
under the assumptions
Then, the following existence and uniqueness result  holds.
The proof, is based on the a priori estimate on the viscoelastic term:
A result of existence, in a three-dimensional regular magneto-viscoelasticity problem, is given in .
Now, as in the purely viscoelastic case, when the requirement
The strategy to prove the existence result , relies on the fact that the classical problem (60) as soon as the initial time is t0 = ε, for any arbitrary ε > 0, the relaxation modulus satisfies the classical regularity requirements, namely, as in subSection 4.0.1, implies that Hence, each
with the assigned initial and boundary conditions
The proof, not included here, is provided in .
consider the viscoelastic energy associated to the problem to obtain a suitable
consider the energy connected to interaction between magnetic and viscoelastic effects to obtain further suitable estimates
consider the total energy together with smooth enough initial data to estimate the energy at the generic time
introduce an appropiate weak formulation and suitable test functions
consider separately the limit process when ε → 0
As a closing remark, we can note that, under the applicative point of view as well as under the analytical one, the free energy associated to the model plays a crucial role. Indeed, the proof relies on estimates which are based on the free energies connected to the model here adopted. Specifically, the viscoelastic energy allows , also in the magneto-viscoelastic case, to prove an a priori estimate on which the subsequent results are based. This is not surprising since the connection relating free energies and evolution problems is well known; see for instance  and references therein.
S. Carillo wishes to acknowledge the partial financial support of GNFM-INDAM, INFN, and SAPIENZA Università di Roma.
© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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