Related Problems of Continuum Mechanics

A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: closed (31 August 2019) | Viewed by 14246

Special Issue Editors

1. Federal Research Center "Computer Science and Control", Russian Academy of Sciences, Moscow, Russia
2. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
Interests: partial differential equations & mathematical physics; elasticity system; stokes system; biharmonic (polyharmonic) equation
Special Issues, Collections and Topics in MDPI journals
Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Interests: applied mathematics; analysis and geometry and topology; shell theory; solid mechanics; solid mechanics of thin bodies; tensor calculus; differential equations

Special Issue Information

Dear Colleagues,

The Special Issue will mainly consist of selected papers presented at the
"
IX Annual International Meeting of the Georgian Mechanical Union and International Conference on Related Problems of Continuum Mechanics" (11–13 October 2018, Kutaisi, Georgia). Papers that are considered to fit the scope of the journal and are of sufficient quality, after evaluation by the reviewers, will be published free of charge.

The main topics of this Special Issue are:

  • Mechanics of Thin Bodies
  • Mathematical Modeling of Multilayered Structures
  • Applied Problems of Continuum Mechanics 
  • Related Problems of Analysis

Prof. Dr. Hovik Matevossian
Prof. Dr. Mikhail Nikabadze
Guest Editors

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Published Papers (6 papers)

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Research

14 pages, 573 KiB  
Article
A New Approach to Non-Singular Plane Cracks Theory in Gradient Elasticity
Math. Comput. Appl. 2019, 24(4), 93; https://doi.org/10.3390/mca24040093 - 26 Oct 2019
Cited by 7 | Viewed by 2041
Abstract
A non-local solution is obtained here in the theory of cracks, which depends on the scale parameter in the non-local theory of elasticity. The gradient solution is constructed as a regular solution of the inhomogeneous Helmholtz equation, where the function on the right [...] Read more.
A non-local solution is obtained here in the theory of cracks, which depends on the scale parameter in the non-local theory of elasticity. The gradient solution is constructed as a regular solution of the inhomogeneous Helmholtz equation, where the function on the right side of the Helmholtz equation is a singular classical solution. An assertion is proved that allows us to propose a new solution for displacements and stresses at the crack tip through the vector harmonic potential, which determines by the Papkovich-Neuber representation. One of the goals of this work is a definition of a new representation of the solution of the plane problem of the theory of elasticity through the complex-valued harmonic potentials included in the Papkovich-Neuber relations represented in a symmetric form, which is convenient for applications. It is shown here that this new representation of the solution for the mechanics of cracks can be written through one harmonic complex-valued potential. The explicit potential value is found by comparing the new solution with the classical representation of the singular solution at the crack tip constructed using the complex potentials of Kolosov-Muskhelishvili. A generalized solution of the singular problem of fracture mechanics is reduced to a non-singular stress concentration problem, which allows one to implement a new concept of non-singular fracture mechanics, where the scale parameter along with ultimate stresses determines the fracture criterion and is determined by experiments. Full article
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
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15 pages, 556 KiB  
Article
Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem
Math. Comput. Appl. 2019, 24(4), 88; https://doi.org/10.3390/mca24040088 - 10 Oct 2019
Cited by 3 | Viewed by 2048
Abstract
We study a variational problem for hypersurfaces in the Euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered hypersurface, which was introduced to model the [...] Read more.
We study a variational problem for hypersurfaces in the Euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered hypersurface, which was introduced to model the surface tension of a small crystal. The purpose of this paper is two-fold. First, we give uniqueness and nonuniqueness results for closed equilibria under weaker assumptions on the regularity of both considered hypersurfaces and the anisotropic surface energy density than previous works and apply the results to the anisotropic mean curvature flow. This part is an announcement of two forthcoming papers by the author. Second, we give a new uniqueness result for stable anisotropic capillary surfaces in a wedge in the three-dimensional Euclidean space. Full article
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
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11 pages, 306 KiB  
Article
Tensors in Newtonian Physics and the Foundations of Classical Continuum Mechanics
Math. Comput. Appl. 2019, 24(3), 79; https://doi.org/10.3390/mca24030079 - 03 Sep 2019
Cited by 1 | Viewed by 2209
Abstract
In the Newtonian approach to mechanics, the concepts of objective tensors of various ranks and types are introduced. The tough classification of objective tensors is given, including tensors of material and spatial types. The diagrams are constructed for non-degenerate (“analogous”) relations between tensors [...] Read more.
In the Newtonian approach to mechanics, the concepts of objective tensors of various ranks and types are introduced. The tough classification of objective tensors is given, including tensors of material and spatial types. The diagrams are constructed for non-degenerate (“analogous”) relations between tensors of one and the same (any) rank, and of various types of objectivity. Mappings expressing dependence between objective tensor processes of various ranks and types are considered. The fundamental concept of frame-independence of such mappings is introduced as being inherent to constitutive relations of various physical and mechanical properties in the Newtonian approach. The criteria are established for such frame-independence. The mathematical restrictions imposed on the frame-independent mappings by the objectivity types of connected tensors are simultaneously revealed. The absence of such restrictions is established exclusively for mappings and equations linking tensors of material types. Using this, a generalizing concept of objective differentiation of tensor processes in time, and a new concept of objective integration, are introduced. The axiomatic construction of the generalized theory of stress and strain tensors in continuum mechanics is given, which leads to the emergence of continuum classes and families of new tensor measures. The axioms are proposed and a variant of the general theory of constitutive relations of mechanical properties of continuous media is constructed, generalizing the known approaches by Ilyushin and Noll, taking into account the possible presence of internal kinematic constraints and internal body-forces in the body. The concepts of the process image and the properties of the five-dimensional Ilyushin’s isotropy are generalized on the range of finite strains. Full article
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
16 pages, 339 KiB  
Article
On the Modeling of Five-Layer Thin Prismatic Bodies
Math. Comput. Appl. 2019, 24(3), 69; https://doi.org/10.3390/mca24030069 - 11 Jul 2019
Cited by 2 | Viewed by 2113
Abstract
Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are [...] Read more.
Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the micropolar theory of K-layer thin prismatic bodies are considered. From here, we can easily get the statements of the initial-boundary value problems for the five-layer thin prismatic bodies. Full article
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
7 pages, 258 KiB  
Communication
Mixed Boundary Value Problems for the Elasticity System in Exterior Domains
Math. Comput. Appl. 2019, 24(2), 58; https://doi.org/10.3390/mca24020058 - 02 Jun 2019
Cited by 1 | Viewed by 2193
Abstract
We study the properties of solutions of the mixed Dirichlet–Robin and Neumann–Robin problems for the linear system of elasticity theory in the exterior of a compact set and the asymptotic behavior of solutions of these problems at infinity under the assumption that the [...] Read more.
We study the properties of solutions of the mixed Dirichlet–Robin and Neumann–Robin problems for the linear system of elasticity theory in the exterior of a compact set and the asymptotic behavior of solutions of these problems at infinity under the assumption that the energy integral with weight | x | a is finite for such solutions. We use the variational principle and depending on the value of the parameter a, obtain uniqueness (non-uniqueness) theorems of the mixed problems or present exact formulas for the dimension of the space of solutions. Full article
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
15 pages, 9714 KiB  
Article
Active Optics in Astronomy: Freeform Mirror for the MESSIER Telescope Proposal
Math. Comput. Appl. 2019, 24(1), 2; https://doi.org/10.3390/mca24010002 - 27 Dec 2018
Cited by 3 | Viewed by 2997
Abstract
Active optics techniques in astronomy provide high imaging quality. This paper is dedicated to highly deformable active optics that can generate non-axisymmetric aspheric surfaces—or freeform surfaces—by use of a minimum number of actuators. The aspheric mirror is obtained from a single uniform load [...] Read more.
Active optics techniques in astronomy provide high imaging quality. This paper is dedicated to highly deformable active optics that can generate non-axisymmetric aspheric surfaces—or freeform surfaces—by use of a minimum number of actuators. The aspheric mirror is obtained from a single uniform load that acts over the surface of a closed-form substrate whilst under axial reaction to its elliptical perimeter ring during spherical polishing. MESSIER space proposal is a wide-field low-central-obstruction folded-two-mirror-anastigmat or here called briefly three-mirror-anastigmat (TMA) telescope. The optical design is a folded reflective Schmidt. Basic telescope features are 36 cm aperture, f/2.5, with 1.6° × 2.6° field of view and a curved field detector allowing null distortion aberration for drift-scan observations. The freeform mirror is generated by spherical stress polishing that provides super-polished freeform surfaces after elastic relaxation. Preliminary analysis required use of the optics theory of 3rd-order aberrations and elasticity theory of thin elliptical plates. Final cross-optimizations were carried out with Zemax raytracing code and Nastran FEA elasticity code in order to determine the complete geometry of a glass ceramic Zerodur deformable substrate. Full article
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
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