Discussion of the published manuscript lazopoulos K.A., karaoulanis D., lazopoulos Α.Κ. (2016) «on fractional modelling of viscoelastic mechanical systems», mechanics research communications, 78, pp. 1–5

The paper [20] attempts to solve a one dimensional model of the Zener type viscoelastic model (see Fig. 1 of the paper) replacing the classical time derivatives with a new fractional time derivative that was presented earlier by two of the authors (see ref. [1], [9] and [10] cited in the work). In addition, the authors present the classical solution of what is called in the paper “conventional viscoelastic model” and the solution due to Caputo fractional derivative, which is the solution of what the authors call “existing fractional order derivative viscoelastic models”. The paper seems to have several problems, typos and mistakes that will be discussed as follows     

A boundary element investigation of extrudate swell

Numerical studies of die swell have until now dealt perimarily with the Maxwell or Oldroyd-B viscoelastic models. However, these models exhibit features that often make them unsuitable for numerical work. Furthermore, they are not realistic representations of actual viscoelastic fluids. In this report a comparison is made between the behaviour of a variety of different viscoelastic models when applied to the die swell problem. A wide range of elongational and shear behaviour is exhibited by the models examined. Both types of behaviour are shown to be important in the die swell problem, and the observed swelling is related to these characteristics of the models.     

丙烯酸弹性体的率相关分数阶黏弹性模型研究

丙烯酸弹性体VHB 4910作为一种重要的介电弹性体,在软体机器人、致动器、俘能器和智能隔振器等领域有很好的应用前景.但材料的非线性黏弹性对其力学行为有显著影响.近来分数阶模型在复杂材料的建模中取得了成功.本文基于分数阶有限变形Kelvin-Voigt流变学模型建立弹性体的三维张量本构,并进一步推导单向拉伸情况下的本构...     

Stress retardation versus stress relaxation in linear viscoelasticity

We present a preliminary examination of a new approach to a long-standing problem in non-Newtonian fluid mechanics. First, we summarize how a general implicit functional relation between stress and rate of strain of a continuum with memory is reduced to the well-known linear differential constitutive relations that account for “relaxation” and “retardation.” Then, we show that relaxation and retardation are asymptotically equivalent for small Deborah numbers, whence causal pure relaxation models necessarily correspond to ill-posed pure retardation models. We suggest that this dichotomy could be a possible way to reconcile the discrepancy between the theory of and certain experiments on viscoelastic liquids that are conjectured to exhibit only stress retardation.     

A numerical shock profile in a slowly relaxing medium

A numerical solution of the equation which governs the shock solution in a viscoelastic medium displaying stress relaxation is obtained. This equation has been found to be the one which defines the lowest-order approximation to the inner solution of a “generalized Burgers' equation”. The case when the stress relaxation function is algebraically decaying is considered and the results are shown to be consistent with the expected behaviour of the solution.     

Controllability of viscoelastic stresses for nonlinear Maxwell models

We investigate a class of models for viscoelastic fluids, in which the elastic stress is determined by a conformation tensor, and the conformation tensor is linked to the velocity field by a system of ordinary differential equations. We study the question which values of the conformation tensor can be reached in a homogeneous flow, subject to a given initial condition and arbitrary velocity fields. This problem is a special “easy” case for the question of controllability of viscoelastic flows. For a class of models, we show that constraints on the values of the conformation tensor are given by lower and/or upper bounds on its determinant. The behavior of seemingly similar models, e.g. the PTT, Giesekus and Peterlin dumbbell models, turns out to be surprisingly different.     

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter \(\nu > -1\). Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order 1/2) for short times and of a standard Maxwell model for long times.     

三种分形和分数阶导数阻尼振动模型的比较研究

标准的整数阶导数方程不能准确描述粘弹性材料的记忆性参考文献[1]和阻尼的分数次幂频率依赖[2],因此分形导数、分数阶导数及正定分数阶导数被用于描述粘弹性介质中的阻尼振动.该文通过分析模型和数值模拟,比较了三种模型描述的振动过程.结果显示,当p小于约O.75或大于约1.9时(p为非整数阶导数的阶数),分形导数模型衰减最快;当P大于约0.75且小于约1.9时,正定分数阶导数模型衰减最快,衰减最慢的分别为分数阶导数模型(p<1)和分形导数模型(p>1).且正定分数阶导数模型衰减快于分数阶导数模型,当p接近2时,两种模型较为相近.     

Optimal control of a class of fractional heat diffusion systems

In this paper, a solution procedure for a class of optimal control problems involving distributed parameter systems described by a generalized, fractional-order heat equation is presented. The first step in the proposed procedure is to represent the original fractional distributed parameter model as an equivalent system of fractional-order ordinary differential equations. In the second step, the necessity for solving fractional Euler–Lagrange equations is avoided completely by suitable transformation of the obtained model to a classical, although infinite-dimensional, state-space form. It is shown, however, that relatively small number of state variables are sufficient for accurate computations. The main feature of the proposed approach is that results of the classical optimal control theory can be used directly. In particular, the well-known “linear-quadratic” (LQR) and “Bang-Bang” regulators can be designed. The proposed procedure is illustrated by a numerical example.     

Investigation of the thermoviscoelastic material behaviour of adhesive bonds close to the glass transition temperature

In the present contribution, a phenomenological material model is presented describing the viscoelastic behaviour of polyurethane bonds under finite deformations. The model includes the temperature dependency of the material behaviour close to the glass transition temperature. Experimentally observed size effects can also be described. It is known from previous investigations that the bonds show an effective behaviour that is influenced by the layer thickness. The physical reason for this behaviour can be found in the formation of boundary layers in the polymer which are located close to the substrate. In these boundary layers, the material behaviour differs from the bulk behaviour. The effective properties of bonds of different thickness are experimentally investigated in isothermal shear tests performed at different temperatures. In order to model the observed effects, a previously developed approach of viscoelasticity is extended including the temperature dependency. Furthermore, a supplementary parameter field is introduced, taking the formation of the boundary layers into account. This field is determined by a partial differential equation in form of a reaction–diffusion equation containing the boundary conditions. The local properties of the polymer are linked to the distribution of this additional field. It is shown that the model parameters can be identified and that the model is able to describe the complex experimentally observed behaviour.     

On the derivation of structural models with general thermomechanical prestress

The vibrating behaviour of thin structures is affected by prestress states. Hence, the effects of thermal prestress are important research subjects in view of ambient vibration monitoring of civil structures. The interaction between prestress, geometrically non-linear behaviour, as well as damping and its coupling with the aforementioned phenomena has to be taken into account for a comprehensive understanding of the structural behaviour. Since the literature on this subject lacks a clear procedure to derive models of thin prestressed and damped structures from 3D continuum mechanics, this paper presents a new derivation of models for thin structures accounting for generic prestress, moderate rotations and viscous damping. Although inspired by classical approaches, the proposed procedure is quite different, because of (i) the definition of a modified Hu–Washizu (H-W) functional, accounting for stress constraints associated with Lagrange multipliers, in order to derive lower-dimensional models in a convenient way; (ii) an original definition of a (mechanical and thermal) strain measure and a rotation measure enabling one to identify the main terms in the strain energy and to derive a cascade of lower-dimensional models (iii) a new definition of “strain–rotation domains” providing a clear interpretation of the classical assumptions of “small perturbations” and “small strains and moderate rotations”; (iv) the introduction of a pseudo-potential with stress constraints to account for viscous damping. The proposed procedure is applied to thin beams.     

层状分数阶黏弹性饱和地基与梁共同作用的时效研究

饱和地基与梁共同作用问题的研究在力学领域及工程界都具有重要意义.采用分数阶Merchant模型研究饱和地基的流变固结,该模型比常用整数阶黏弹性模型更能精确反映地基的时变特征.基于层状正交各向异性黏弹性饱和地基的固结解答,采用有限元法与边界元法耦合的方法,研究梁与分数阶黏弹性饱和地基的共同作用问题.依据Timoshenk...     

Fractional relaxation and the time-temperature superposition principle

Relaxation processes in complex systems like polymers or other viscoelastic materials can be described by equations containing fractional differential or integral operators. In order to give a physical motivation for fractional order equations, the fractional relaxation is discussed in the framework of statistical mechanics. We show that fractional relaxation represents a special type of a non-Markovian process. Assuming a separation condition and the validity of the thermo-rheological principle, stating that a change of the temperature only influences the time scale but not the rheological functional form, it is shown that a fractional operator equation for the underlying relaxation process results.     

An initial value problem arising in mechanics

We study an initial value problem for a system consisting of an integer order and a distributed-order fractional differential equation describing forced oscillations of a body attached to a free end of a light viscoelastic rod. An explicit form of a solution for a class of linear viscoelastic solids is given in terms of a convolution integral. Restrictions on storage and loss moduli following from the second law of thermodynamics play the crucial role in establishing the form of the solution. Some previous results are shown to be special cases of the present analysis.     

Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading

The stability of a viscoelastic plate strip, subjected to an axial load with the Kelvin–Voigt fractional order constitutive relationship is studied. Based on the classical plate theory, the structural formulation of the plate is obtained by using the Newton’s second law and the aerodynamic force due to the fluid flow is evaluated by piston theory. The Galerkin method is employed to discretize the equation of motion into a set of ordinary differential equations. To determine the stability margin of plate the obtained set of ordinary differential equations are solved using the Laplace transform method. The effects of variation of the governing parameters such as axial force, retardation time, fractional order and boundary conditions on the stability margin of fractional viscoelastic panel are investigated and finally some conclusions are outlined.     

Constitutive equations based on the transient network concept

A consitutive theory for polymetric liquids based on the transient network concept is developed, following Wiegel and Jongschaap. The Phan-Thien-Tanner equation is shown to follow from the general theory with two critical assumption, one of these is quasi-equilibrium of the internal structure, which preludes consistency of application in ”fast flows“. The Marrucci model can be made consistent with the general format with a small change in the kinetic equation that can be deduced from asymptotic behaviour and leaves the steady viscometric behavior unchanged. The simplest genaral formulation requires the linear viscoelastic spectrum and two parameters; the latter cannot be determined uniquely from steady viscometric flow data.     

A “constrained-chain” network model for viscoelastic fluids

A specific case of the general “constrained-chain” strain-dependent integral viscoelastic model is evaluated with steady simple shear data, stress growth and relaxation data, and steady elongational viscosity data. The model is qualitatively correct and quantitatively reasonable in its predictions and, on balance, compares favourably with strain-dependent, strain-rate-dependent, and stress-dependent models from the current literature. Specific model comparisons are made to demonstrate the effect of the “constrained-chain” or finite extensibility feature of the model.