-->is positive semi-definite, absolutely continuous and summable on ℝ+. Then, for every t ∈ ℝ
Besides, it is differentiable for all τ ∈ ℝ+ and
for every compact set
(M3) For every fixed τ > 0, the map
for every compact set
(M4) There exists a nonnegative scalar function M: ℝ → ℝ+, bounded on bounded intervals, such that
for every (t, τ) ∈ ℝ × ℝ+.
According to (M2), the t-dependent relaxation function
Borrowing from the scalar case,
Finally, an integration by part of Eq. (30) yields
provided that Eq. (2) holds for E and
As an advantage, within Eq. (34),
In order to stress the aging effects, we might assume the following factorization of the memory kernel G.
(M5) There exist three functions,
Accordingly, the stress-strain relation (34) may be rewritten into the form (22). The aging factors κ and ℍ reduces to unit when non-aging materials are considered.
So far, we restrict our attention to scrutinize stress-strain relations in the form (30). In particular, for isotropic materials
where 1 is the unit second-order tensor,
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 kji and αji. Finally, we give some examples of these functions that fulfill these conditions.
(M1) Starting from (C1), it is quite trivial to prove this property.
(M2) By virtue of Eq. (35) and (C1), Gi , i=1,2, are positive and continuously differentiable with respect to t and τ. Moreover,
Hence (M2) is fulfilled.
(M3) It is obviously true as αji ∈ ℂ1(ℝ) by virtue of (C1)-(C2). In particular,
(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 t ∈ ℝ we let
It is apparent that M(t) ≥ 0 and then from (35) it follows:
When non-aging material parameters are involved, Eq. (37) reduces to ∂τ i + M i ≤ 0, i = 1, 2, which implies the exponential decay of the kernels.
We present here some special expressions of material functions kji and αji, j = 1, 2,…, N, i = 1, 2, which fulfill properties (M1)-(M4).
For simplicity, we restrict our attention to a single Maxwell element. Letting j = 1. and i = 1, 2, we choose
so that (A1) and (A2) hold true. Condition (38) is fulfilled for all t ∈ ℝ and for every choice of the parameters, so that
Otherwise, for j = 1. and i = 1,2, we can choose
where ωi, ki, ηi > 0. Accordingly,
so that (A1) and (A2) hold true. On the other hand, condition (38) reduces to
which is equivalent to
and is fulfilled for all t ∈ ℝ provided that ωi ≤ 3 Ki / ηi. If this is the case,
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 u: Ω×ℝ → ℝ3, relative to the reference configuration Ω ⊂ ℝ3, is subject to the equation of motion
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 dissipation inequality
where ρ is the mass density,
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 u(t) and its history up to t. Letting t0 ∈ ℝ be arbitrarily fixed, we define the relative displacement history ζt(x,s), with (t,s) ∈ [t0,T] × ℝ+, by
where ζt0 is the prescribed initial (relative) past history of u up to t0,
Accordingly, ζt(x,0) = 0 and the motion equation (39) becomes a system
where u : Ω×[t0, +∞) and ζt : Ω×ℝ+ → ℝ3, T ∈ [t0, +∞) are the unknown variables. Their initial conditions are prescribed at t0 ∈ ℝ as follows
Let H0 = [L2 (Ω)]3 and H1 = [H10(Ω)]3, and let denote the usual inner product in Hj, j = 0,1. For every t ≥ t0, we introduce the family of memory spaces
Theorem 1. Let ℋt = H1 × H0 ×
for some C > 0 depending only on T, t0 and the size of the initial datum ‖z‖ℋt0.
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 x is understood and not written. In addition, we assumeρ = 1. After integrating over Ω, we end up with the total free energy functional
Theorem 2. For an aging viscoelastic material, the dissipation inequality (40) is fulfilled provided that (M4) holds and
Proof. First we observe that
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
where M(t) ≥ 0, and (44) finally implies the dissipation inequality
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, G is assumed to satisfy the following functional requirements
that is, now, the relaxation function G(t) is not required to be finite at t = 0 and then Eq. (45) loses its meaning and, hence, needs to be replaced by a different one. The method to overcome this difficulty, devised in , consists in the introduction of a suitable sequence of regular problems, depending on a small parameter 0 < ε ≪ 1 which, in the limit ε → 0 reduce to the singular problem under investigation. The key steps of the approximation strategy can be sketched as follows.
Let K, termed integrated relaxation function, denote
Then, introduce the regular problems:
together with the initial and boundary conditions
then, find approximated solutions uε, 0 < ε ≪ 1,
show the existence of the limit solution
prove the uniqueness of the limit solution u which represents a weak solution admitted by the singular problem.
Note that, corresponding to each value of ε, the problem Pε is equivalent to the integral equation:
where the superscripts, in the case ε = 0, are omitted for notational simplicity. Hence, the following theorems can be proved. Here only the outlines of the proofs are given; the details are comprised in  when homogeneous Dirichlet b.c.s (50) are imposed and in  when homogeneous Neumann b.c.s are considered.
Theorem 1 Given uε solution to the integral problem Pε (51), then ∃u(t)=limε→0 uε(t) in L2(Q), Q=Ω×(0,T).
weak formulation, on introduction of test functions φ ∈ ϵ H1(Ω × (0,T) s.t. φx = 0, on ∂Ω,
consider separately the terms without ε,
the terms with uε and Kε,
prove convergence via Lebesgue’s theorem.
Furthermore, the weak solution, as stated in the following theorem, is unique.
Theorem 2 The integral problem (52) admits a unique weak solution.
Proof’s outline: The result is proved by contradiction, see  for details, assuming there are two different solution and, then, showing that such an assumption leads to a contradiction.
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 m represents the magnetization vector reads
The quantities of interest, in the general three-dimensional case, are the following ones:
where the coefficients λklmn are subject to the condition
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 time-translated approximated problems
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 a priori estimate
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 t
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.
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