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Mathematics
Special Issues
Generalized Continuum Models and Higher-Order Partial Differential Equations
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Generalized Continuum Models and Higher-Order Partial Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 5291

Special Issue Editors

Dr. Sergei Khakalo

E-Mail Website1 Website2
Guest Editor
Department of Civil Engineering, Aalto University, 02150 Espoo, Finland
Interests: applied mechanics and mathematics; classical and generalized continuum mechanics; strain gradient elasticity; strain gradient plasticity; numerical methods; FEM; IGA; computational homogenization; variational homogenization; lattice structures
Dr. Emilio Barchiesi

E-Mail Website
Guest Editor
National School of Engineers of Brest, University of Western Brittany, 29280 Plouzané, France
Interests: asymptotic homogenization; fracture and damage mechanics; mechanics of particulate media; gradient continua; mechanical metamaterials; eaves in nonlinear media; stability analysis

Special Issue Information

Dear Colleagues,

In the framework of continuum description of condensed matter, various physical phenomena, e.g., thermal conduction and deformation, are described in terms of non-stationary partial differential equations (PDEs). In particular, most natural materials are modelled within classical continuum mechanics governed by second-order PDEs, which results in size-independent constitutive behavior.

For the structures or mechanical metamaterials with highly noticeable, architected microstructure, the significance of micro-scale mechanisms in influencing macro-scale size-dependent material behaviors is nowadays largely recognized in the context of mechanics. At the continuum level, this requires incorporation of additional (besides displacements field) degrees of freedom and higher gradients of the kinematic and/or state variables, resulting in (non-)linear higher-order PDEs.

In relation to microstructured metamaterials and generalized continuum mechanics, this Special Issue collects papers covering (but not limited to) the following topics:

  • Material non-linearity, e.g., plasticity and damage phenomena;
  • Higher-order phase-field models for brittle and ductile fracture;
  • Large deformation aspects and elastic local instabilities;
  • Wave propagation phenomena;
  • Frequency and amplitude band gaps;
  • Higher-order computational and variational asymptotic homogenization schemes;
  • Development and formulation of dimensionally reduced models for beam-, plate-, and shell-structural elements;
  • Analytical and numerical methods and material parameter analysis.

Dr. Sergei Khakalo
Dr. Emilio Barchiesi
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • elasticity
  • plasticity
  • fracture
  • wave propagation
  • mechanical metamaterials
  • microstructure
  • homogenization
  • numerical methods

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Published Papers (3 papers)

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Research

10 pages, 285 KiB  
Article
Strong Ellipticity and Infinitesimal Stability within Nth-Order Gradient Elasticity
by Victor A. Eremeyev
Mathematics 2023, 11(4), 1024; https://doi.org/10.3390/math11041024 - 17 Feb 2023
Cited by 7 | Viewed by 1487
Abstract
We formulate a series of strong ellipticity inequalities for equilibrium equations of the gradient elasticity up to the Nth order. Within this model of a continuum, there exists a deformation energy introduced as an objective function of deformation gradients up to the [...] Read more.
We formulate a series of strong ellipticity inequalities for equilibrium equations of the gradient elasticity up to the Nth order. Within this model of a continuum, there exists a deformation energy introduced as an objective function of deformation gradients up to the Nth order. As a result, the equilibrium equations constitute a system of 2N-order nonlinear partial differential equations (PDEs). Using these inequalities for a boundary-value problem with the Dirichlet boundary conditions, we prove the positive definiteness of the second variation of the functional of the total energy. In other words, we establish sufficient conditions for infinitesimal instability. Here, we restrict ourselves to a particular class of deformations which includes affine deformations. Full article
(This article belongs to the Special Issue Generalized Continuum Models and Higher-Order Partial Differential Equations)
11 pages, 365 KiB  
Article
A Weakly Nonlinear Dynamic Problem for a Model of the Thermoelastic Medium Absorbing a Part of the Acoustic Spectrum
by Mikhail Babenkov and Ekaterina Podolskaya
Mathematics 2022, 10(21), 4142; https://doi.org/10.3390/math10214142 - 6 Nov 2022
Viewed by 1212
Abstract
We consider a dynamic problem with a short laser impact on a semi-opaque insulated layer with free borders, accounting for the selective absorption of the acoustic spectrum regions by the media. The behavior of the material is modeled by the extended coupled thermoelasticity [...] Read more.
We consider a dynamic problem with a short laser impact on a semi-opaque insulated layer with free borders, accounting for the selective absorption of the acoustic spectrum regions by the media. The behavior of the material is modeled by the extended coupled thermoelasticity formulated in the previous work of the series. Following the experimental results, we introduce a weakly nonlinear correction to the thermal expansion coefficient. Thus, we aim to level out the inability of classical thermoelasticity (CTE) to correctly describe the deformation processes in a solid under a high-frequency impact, yet staying within the framework of linear models. The parameters of the system of novel equations can be tuned to fit the experimentally measured data, i.e., the frequency-dependent attenuation coefficient. The series solutions of the extended thermoelasticity problem are compared with those obtained within CTE. In contrast to CTE and in accordance with experiments, the model allows for the simultaneous existence of positive and negative extrema for stress over time. Full article
(This article belongs to the Special Issue Generalized Continuum Models and Higher-Order Partial Differential Equations)
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21 pages, 981 KiB  
Article
Uniform Asymptotics of Solutions of Second-Order Differential Equations with Meromorphic Coefficients in a Neighborhood of Singular Points and Their Applications
by Maria V. Korovina and Hovik A. Matevossian
Mathematics 2022, 10(14), 2465; https://doi.org/10.3390/math10142465 - 15 Jul 2022
Cited by 5 | Viewed by 1555
Abstract
In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood [...] Read more.
In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood of a singular point and apply the results obtained to the equations of mathematical physics. The main results related to the construction of uniform asymptotics are obtained using resurgent analysis methods applied to differential equations with irregular singularities. These results allow us to construct asymptotics for any second-order equations with meromorphic coefficients—that is, with an arbitrary order of degeneracy. This also allows one to determine the type of a singular point and highlight the cases where the point is non-singular or regular. Full article
(This article belongs to the Special Issue Generalized Continuum Models and Higher-Order Partial Differential Equations)
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