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Modelling of Viscoelastic Materials and Mechanical Behavior (2nd Edition)
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Modelling of Viscoelastic Materials and Mechanical Behavior (2nd Edition)

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

Deadline for manuscript submissions: 10 April 2025 | Viewed by 1835

Special Issue Editor

Prof. Dr. Angelo Morro

E-Mail Website
Guest Editor
DIBRIS, University of Genoa, 16145 Genoa, Italy
Interests: mathematical modeling in mechanics and electromagnetism of continuous media
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Viscoelasticity denotes the joint properties 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 linking 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 forming the basis of 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 Special 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 relationship; 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.

The interaction of deformation with the temperature field is investigated for a nonlinear viscoelastic beam with different conditions at the boundary; the existence and uniqueness of the solution are proven 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 heat flux) or fading memory functionals (as with the Boltzmann model) or possibly with two types of constitutive relations. Rate-type equations for the magnetization are considered in light of some customary evolution equations in the physical literature.

Prof. Dr. Angelo Morro
Guest Editor

Manuscript Submission Information

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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 (2 papers)

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Research

33 pages, 8447 KiB  
Article
Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials
by Anna Stankiewicz
Materials 2024, 17(19), 4870; https://doi.org/10.3390/ma17194870 - 3 Oct 2024
Viewed by 581
Abstract
Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining [...] Read more.
Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined. Full article
(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior (2nd Edition))
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37 pages, 22300 KiB  
Article
Sampling Points-Independent Identification of the Fractional Maxwell Model of Viscoelastic Materials Based on Stress Relaxation Experiment Data
by Anna Stankiewicz
Materials 2024, 17(7), 1527; https://doi.org/10.3390/ma17071527 - 27 Mar 2024
Cited by 2 | Viewed by 856
Abstract
Considerable development has been observed in the area of applying fractional-order rheological models to describe the viscoelastic properties of miscellaneous materials in the last few decades together with the increasingly stronger adoption of fractional calculus. The fractional Maxwell model is the best-known non-integer-order [...] Read more.
Considerable development has been observed in the area of applying fractional-order rheological models to describe the viscoelastic properties of miscellaneous materials in the last few decades together with the increasingly stronger adoption of fractional calculus. The fractional Maxwell model is the best-known non-integer-order rheological model. A weighted least-square approximation problem of the relaxation modulus by the fractional Maxwell model is considered when only the time measurements of the relaxation modulus corrupted by additive noises are accessible for identification. This study was dedicated to the determination of the model, optimal in the sense of the integral square weighted model quality index, which does not depend on the particular sampling points applied in the stress relaxation experiment. It is proved that even when the real description of the material relaxation modulus is entirely unknown, the optimal fractional Maxwell model parameters can be recovered from the relaxation modulus measurements recorded for sampling time points selected randomly according to respective randomization. The identified model is a strongly consistent estimate of the desired optimal model. The exponential convergence rate is demonstrated both by the stochastic convergence analysis and by the numerical studies. A simple scheme for the optimal model identification is given. Numerical studies are presented for the materials described by the short relaxation times of the unimodal Gauss-like relaxation spectrum and the long relaxation times of the Baumgaertel, Schausberger and Winter spectrum. These studies have shown that the appropriate randomization introduced in the selection of sampling points guarantees that the sequence of the optimal fractional Maxwell model parameters asymptotically converge to parameters independent of these sampling points. The robustness of the identified model to the measurement disturbances was demonstrated by analytical analysis and numerical studies. Full article
(This article belongs to the Special Issue Modelling of Viscoelastic Materials and Mechanical Behavior (2nd Edition))
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