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Modelling of Viscoelastic Materials and Mechanical Behavior

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A special issue of Materials (ISSN 1996-1944). This special issue belongs to the section "Materials Physics".

Deadline for manuscript submissions: closed (20 September 2023) | Viewed by 20526

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Prof. Dr. Angelo Morro

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DIBRIS, University of Genoa, 16145 Genoa, Italy
Interests: mathematical modeling in mechanics and electromagnetism of continuous media
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Dear Colleagues,

Viscoelasticity denotes the joint property of elasticity and viscosity and hence describes materials with both fluid and solid properties at the same time. Well-known accounts of viscoelasticity can be traced back to Maxwell, Boltzmann, and Kelvin. While Maxwell and Kelvin models describe the viscoelastic behavior via first-order differential equations relating stress and strain (rheological models), in the Boltzmann theory the stress is determined by a functional of the past history of the strain. Physically these schemes are the prototypes on the basis of the current models of viscoelastic materials.

This Special Issue is devoted to recent advances in the modeling of viscoelastic materials, possibly interacting with electromagnetic fields and temperature fields, along with mathematical properties of the solution to associated evolution problems. The following are some topics to be investigated in this issue:

The modeling of viscoelastic materials is developed within the domain of materials with fading memory. The model is based on the classical linear functional for the stress-strain constitutive relation; to account for aging properties the kernel is allowed to depend explicitly on time. The thermodynamic analysis yields a set of properties characterizing the functional for both aging and non-aging materials. Likewise, a rate-type (Maxwell) model is shown to account for hysteresis effects in viscoelasticity. Further, viscoelastic materials are considered with a singular kernel.

Interaction of deformation with the temperature field is investigated for a nonlinear viscoelastic beam with different conditions at the boundary; existence and uniqueness of the solution are proved along with an exponential decay property.

More involved models of viscoelastic materials are considered by accounting for the effects of magnetic or electric fields. Basic schemes for the modeling of such materials may contain rate-type equations (as with the Cattaneo–Maxwell law for the heat flux) or fading memory functionals (as with the Boltzmann model) or possibly with the two types of constitutive relations. Rate-type equations for the magnetization are considered in the light of some customary evolution equations in the physical literature.

Prof. Dr. Angelo Morro
Guest Editor

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Keywords

Constitutive relations
Materials with memory
Rate-type equations
Thermodynamic consistency
Relaxation and creep
Aging
Hysteresis
Existence and uniqueness

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Published Papers (11 papers)

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Research

20 pages, 5681 KiB

Open AccessArticle

A Thermo-Electro-Viscoelastic Model for Dielectric Elastomers

by Bao Qin, Zheng Zhong and Tong-Yi Zhang

Materials 2023, 16(17), 5917; https://doi.org/10.3390/ma16175917 - 29 Aug 2023

Viewed by 2015

Abstract

Dielectric elastomers (DEs) are a class of electro-active polymers (EAPs) that can deform under electric stimuli and have great application potential in bionic robots, biomedical devices, energy harvesters, and many other areas due to their outstanding deformation abilities. It has been found that stretching rate, temperature, and electric field have significant effects on the stress-strain relations of DEs, which may result in the failure of DEs in their applications. Thus, this paper aims to develop a thermo-electro-viscoelastic model for DEs at finite deformation and simulate the highly nonlinear stress-strain relations of DEs under various thermo-electro-mechanical loading conditions. To do so, a thermodynamically consistent continuum theoretical framework is developed for thermo-electro-mechanically coupling problems, and then specific constitutive equations are given to describe the thermo-electro-viscoelastic behaviors of DEs. Furthermore, the present model is fitted with the experimental data of VHB4905 to determine a temperature-dependent function of the equilibrium modulus. A comparison of the nonlinear loading-unloading curves between the model prediction and the experimental data of VHB4905 at various thermo-electro-mechanical loading conditions verifies the present model and shows its ability to simulate the thermo-electro-viscoelastic behaviors of DEs. Simultaneously, the results reveal the softening phenomena and the instant pre-stretch induced by temperature and the electric field, respectively. This work is conducive to analyzing the failure of DEs in functionalities and structures from theoretical aspects at various thermo-electro-mechanical conditions. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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19 pages, 4596 KiB

Open AccessArticle

The Thermal Stress Problem of Bimodular Curved Beams under the Action of End-Side Concentrated Shear Force

by Xiao-Ting He, Xin Wang, Meng-Qiao Zhang and Jun-Yi Sun

Materials 2023, 16(15), 5221; https://doi.org/10.3390/ma16155221 - 25 Jul 2023

Viewed by 1069

Abstract

A bimodular material is a kind of material that presents two elastic moduli in tension and compression. In classical thermoelasticity, however, the bimodular material is rarely considered due to its complexity in analysis. In fact, almost all materials will present, more or less, bimodular characteristics, and in some cases, the mechanical properties of materials cannot be fully utilized simply by ignoring the bimodular characteristics. In this study, the thermal stress problem of bimodular curved beams under the action of end-side concentrated shear force is analytically and numerically investigated, in which the temperature rise modes in a thermal environment are considered arbitrary. Using the stress function method based on compatibility conditions, a two-dimensional solution of thermoelasticity of the bimodular curved beam subjected to end-side concentrated shear force was obtained. The results show that the solution for a bimodular curved beam with a thermal effect can be reduced to that of a bimodular curved beam without a thermal effect. At the same time, the numerical simulation for the problem verifies the correctness of the theoretical solution. The results may serve as a theoretical reference for the refined analysis and optimization of curved beams in a thermal environment. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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20 pages, 1658 KiB

Open AccessArticle

Modelling of Electro-Viscoelastic Materials through Rate Equations

by Claudio Giorgi and Angelo Morro

Materials 2023, 16(10), 3661; https://doi.org/10.3390/ma16103661 - 11 May 2023

Cited by 2 | Viewed by 1031

Abstract

Models of dielectric solids subject to large deformations are established by following a thermodynamic approach. The models are quite general in that they account for viscoelastic properties and allow electric and thermal conduction. A preliminary analysis is devoted to the selection of fields for the polarization and the electric field; the appropriate fields are required to comply with the balance of angular momentum and to enjoy the Euclidean invariance. Next, the thermodynamic restrictions for the constitutive equations are investigated using a wide set of variables allowing for the joint properties of viscoelastic solids, electric and heat conductors, dielectrics with memory, and hysteretic ferroelectrics. Particular attention is devoted to models for soft ferroelectrics, such as BTS ceramics. The advantage of this approach is that a few constitutive parameters provide a good fit of material behaviour. A dependence on the gradient of the electric field is also considered. The generality and the accuracy of the models are improved by means of two features. The entropy production is regarded as a constitutive property per se, while the consequences of the thermodynamic inequalities are made explicit by means of representation formulae. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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34 pages, 6846 KiB

Open AccessArticle

Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor

by Anna Stankiewicz

Materials 2023, 16(9), 3565; https://doi.org/10.3390/ma16093565 - 6 May 2023

Cited by 6 | Viewed by 1384

Abstract

The viscoelastic relaxation spectrum is vital for constitutive models and for insight into the mechanical properties of materials, since, from the relaxation spectrum, other material functions used to describe rheological properties can be uniquely determined. The spectrum is not directly accessible via measurement and must be recovered from relaxation stress or oscillatory shear data. This paper deals with the problem of the recovery of the relaxation time spectrum of linear viscoelastic material from discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation test. A two-level identification scheme is proposed. In the lower level, the regularized least-square identification combined with generalized cross-validation is used to find the optimal model with an arbitrary time-scale factor. Next, in the upper level, the optimal time-scale factor is determined to provide the best fit of the relaxation modulus to experiment data. The relaxation time spectrum is approximated by a finite series of power–exponential basis functions. The related model of the relaxation modulus is proved to be given by compact analytical formulas as the products of power of time and the modified Bessel functions of the second kind. The proposed approach merges the technique of an expansion of a function into a series of independent basis functions with the least-squares regularized identification and the optimal choice of the time-scale factor. Optimality conditions, approximation error, convergence, noise robustness and model smoothness are studied analytically. Applicability ranges are numerically examined. These studies have proved that using a developed model and algorithm, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and noise robust; small model errors are obtained for the optimal time-scale factors. The complete scheme of the hierarchical computations is outlined, which can be easily implemented in available computing environments. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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14 pages, 888 KiB

Open AccessArticle

A Magneto-Viscoelasticity Problem with Aging

by Sandra Carillo and Claudio Giorgi

Materials 2022, 15(21), 7810; https://doi.org/10.3390/ma15217810 - 5 Nov 2022

Cited by 1 | Viewed by 1231

Abstract

This study addresses a magneto-viscoelasticity problem, considering the one-dimensional case. The system under investigation is given by the coupling a non-linear partial differential equation with a linear integro-differential equation. The system models a viscoelastic body whose mechanical behavior, described by the linear integro-differential equation, is also influenced by an external magnetic field. The model here investigated aims to consider the concomitance of three different effects: viscoelasticity, aging and magnetization. In particular, the viscoelastic behavior is represented via an integro-differential equation whose kernel characterizes the properties of the material. In a viscoelastic material subject to the effects of aging, all changes in the response to deformation are due not only to the intrinsic memory of the material but also to deterioration with the age of the material itself. Thus, the relaxation function is not assumed to depend on the two times, present and past, via their difference, but to depend on both the present and past times as two independent variables. The sensibility to an external magnetic field is modeled by a non-linear partial differential equation taking its origin in the Landau–Lifschitz magnetic model. This investigation is part of a long-term research project aiming to provide new insight in the study of materials with memory and, in particular, viscoelastic materials. Specifically, the classical model of viscoelastic body introduced by Boltzmann represents the fundamental base from which a variety of generalizations have been considered in the literature. In particular, the effects on the viscoelastic body due to interaction with an external magnetic field are studied. The new aspect under investigation is the combined presence of the external magnetic field with the effect of aging. Indeed, the coupling of viscoelasticity, which takes into account the deterioration of the material with time, with the presence of an external magnetic field, was never considered in previous research. An existence and uniqueness result is proved under suitable regularity assumptions. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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28 pages, 492 KiB

Open AccessArticle

Viscoelastic and Electromagnetic Materials with Nonlinear Memory

by Claudio Giorgi and John Murrough Golden

Materials 2022, 15(19), 6804; https://doi.org/10.3390/ma15196804 - 30 Sep 2022

Cited by 1 | Viewed by 1883

Abstract

A method is presented for generating free energies relating to nonlinear constitutive equations with memory from known free energies associated with hereditary linear theories. Some applications to viscoelastic solids and hereditary electrical conductors are presented. These new free energies are then used to [...] Read more.

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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19 pages, 475 KiB

Open AccessArticle

Magneto-Viscoelastic Materials: Memory Functionals and Rate Equations

by Claudio Giorgi and Angelo Morro

Materials 2022, 15(19), 6699; https://doi.org/10.3390/ma15196699 - 27 Sep 2022

Cited by 3 | Viewed by 1265

Abstract

The properties of viscoelastic solids subject to a magnetic field are modelled within two thermodynamically consistent approaches that are typical of models with a non-instantaneous response. One is based on memory functionals: the reversible changes are described by the instantaneous response, while the dissipativity is expressed by the dependence on histories. The other approach involves objective rate equations. While memory functionals lead to the difficulty of determining thermodynamically consistent free energy functionals, rate equations result in a simpler scheme. The greater simplicity allows the discovery of, in particular, models of magneto-hyperelastic materials, magneto-hypoelastic materials, and various forms of magneto-viscoelastic behaviour. The novelty of the procedure is based on two features: a representation formula, originating from the entropy inequality, and the use of the entropy production as a constitutive function. Relations with other approaches in the literature are examined in detail. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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14 pages, 3838 KiB

Open AccessArticle

Studies on the Mechanical Models and Behaviors for the Stamp/Film Interface in Microtransfer Printing

by Mengjie Wu, Yuyan Zhang, Xin Dai and Ling Jiang

Materials 2022, 15(17), 5915; https://doi.org/10.3390/ma15175915 - 26 Aug 2022

Viewed by 1328

Abstract

The adhesion/delamination characteristics at the stamp/film interface are critical for the efficiency of film microtransfer printing technology. To predict and regulate the interface mechanical behaviors, finite element models based on the J-integral, Virtual Crack Closure Technology (VCCT), and the cohesive zone method (CZM) were established and compared. Then, the effects of pulling speed and interface parameters on the pull-off force, which is used to characterize the interface adhesion strength, were investigated. Comparisons between the simulation results and previous experimental results demonstrated that the model based on the CZM was more applicable than the models based on the J-integral and VCCT in analyzing the adhesion/delamination behaviors of the stamp/film interface. Furthermore, the increase in pulling speed could enlarge the pull-off force for the viscoelastic stamp/film interface, while it had no influence on the pull-off force for the elastic stamp/film interface. In addition, a larger normal strength and normal fracture energy resulted in a larger pull-off force, which was beneficial to the realization of the picking-up process in microtransfer printing. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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32 pages, 16458 KiB

Open AccessArticle

A Numerical Model for Investigating the Effect of Viscoelasticity on the Partial Slip Solution

by Dongze Wang, Gregory de Boer and Ali Ghanbarzadeh

Materials 2022, 15(15), 5182; https://doi.org/10.3390/ma15155182 - 26 Jul 2022

Cited by 7 | Viewed by 2235

Abstract

To investigate the effects of viscoelasticity on the stick-slip behaviour, a new model reproducing the partial slip of viscoelastic materials under fully coupled conditions is developed in this paper. The ratio of retardation time to relaxation time is employed to characterize the rheological property of a viscoelastic material. It is found that materials with higher ratios exhibit more fluid-like behaviours while those with lower ratios perform more like solid. As long as the contact input (load or displacement) is constant, the stick ratio (ratio of stick area to contacting area) is found to be insensitive to the viscoelasticity of materials. However, the separation pattern of the stick and slip regions varies with time when different contact phenomena (creep or stress relaxation) are encountered in the lateral and normal directions. The transition process from partial slip to gross sliding of viscoelastic materials, unlike the elastic response, tends to be abrupt when fully coupled conditions between shear tractions and pressure are introduced. When identical contact parameters are specified for different viscoelastic materials, the more fluid-like material always experiences a quicker transition from partial slip to gross sliding. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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16 pages, 318 KiB

Open AccessArticle

Thermodynamic Restrictions in Linear Viscoelasticity

by Angelo Morro

Materials 2022, 15(8), 2706; https://doi.org/10.3390/ma15082706 - 7 Apr 2022

Cited by 3 | Viewed by 1461

Abstract

The thermodynamic consistency of linear viscoelastic models is investigated. First, the classical Boltzmann law of stress–strain is considered. The kernel (Boltzmann function) is shown to be consistent only if the half-range sine transform is negative definite. The existence of free-energy functionals is shown to place further restrictions. Next, the Boltzmann function is examined in the unbounded power law form. The consistency is found to hold if the stress functional involves the strain history, not the strain–rate history. The stress is next taken to be given by a fractional order derivative of the strain. In addition to the constitutive equations involving strain–rate histories, finding a free-energy functional, consistent with the second law, seems to be an open problem. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

23 pages, 396 KiB

Open AccessEditor’s ChoiceArticle

Nonlinear Models of Thermo-Viscoelastic Materials

by Claudio Giorgi and Angelo Morro

Materials 2021, 14(24), 7617; https://doi.org/10.3390/ma14247617 - 10 Dec 2021

Cited by 11 | Viewed by 2564

Abstract

The paper develops a general scheme for viscoelastic materials, where the constitutive properties are described by means of measures of strain, stress, heat flux, and their time derivatives. The constitutive functions are required to be consistent with the second law of thermodynamics. Indeed, a new view is associated with the second law: the non-negative expression of the entropy production is set equal to a further constitutive function. The introduction of the entropy production as a constitutive function allows for a much wider range of models. Within this range, a scheme to obtain nonlinear models of thermo-viscoelastic materials subject to large deformations is established. Notably, the Kelvin–Voigt, Maxwell, Burgers, and Oldroyd-B viscoelastic models, along with the Maxwell–Cattaneo heat conduction, are obtained as special cases. The scheme allows also for modelling the visco-plastic materials, such as the Prandtl–Reuss work-hardening function and the Bingham–Norton fluid. Full article

(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior)

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