Memory decay rates of viscoelastic solids: not too slow, but not too fast either

Abstract

Fading memory is a distinguishing characteristic of viscoelastic solids. Its assessment is often achieved by measuring the stress due to harmonic strain histories at different frequencies: from the experimental point of view, the storage and loss moduli are, hence, introduced.On the other side, the mathematical modeling of viscoelastic materials is usually based on the consideration of a kernel function whose decay rate is sufficiently fast. For several different solid materials, we have collated experimental evidence showing an high sensitivity to frequency variations of both the storage and loss moduli. By contrast, we prove that the commonly employed viscoelastic kernels (Prony series, continuous kernel, etc.) cannot reproduce this experimental behavior, as the resulting frequency sensitivity of the storage modulus is always zero when assessed at low frequency. This leads to identification problems of the material parameters which are strongly ill conditioned. However,we identify the specific kernel property which is responsible for this misbehavior: the long-term material memory must not decrease too fast. Some viscoelastic kernels, showing the correct memory’s rate of decay, are introduced and their improved ability to match the experimental data analyzed

Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity

“…A chief example is β(t) = 1 Γ(α) 1 t 1−α , α ∈ (0, 1). For more details and examples see [7] and references therein. This is the reason for introducing kernels of Mittag-Leffler type or fractional operators.…”

Non-Classical Memory Kernels in Linear Viscoelasticity

“…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 [1]. 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][27][28][29][30][31], and especially concerning applications of fractional calculus to the theory of viscoelasticity and the study of new bio-inspired materials [15,[32][33][34][35].…”

“…Starting from the rheological model of a standard viscoelastic solid, whose kernel involves a single exponential, a large variety of regular kernels have been classically employed: discrete and continuous Prony series, completely monotonic functions, etc. Recently, new viscoelastic materials, such as viscoelastic gels, have been described by virtue of convolution integral with singular kernels: for instance, fractional and hypergeometric kernels [1]. On the other hand, when the natural/artificial aging of the viscoelastic material has to be taken into account, timedependent kernels are needed.…”

Some remarks on the model of rigid heat conductor with memory: Unbounded heat relaxation function

“…The term singular kernel problem is introduced to refer to this model to stress that (1.9) is characterised by a kernel unbounded at t = 0. The interest in singular kernel problems is well known both under the analytical viewpoint [28,30,31,34,12] as well as in connection to the study of innovative materials [17] or materials whose response is changed due to aging processes [7] or models devised to describe bio-materials [23]. As a special case, the current interest is on models in which the singular kernel is represented by a fractional derivative term [25,32,22].…”