Special Issue "Applied Mathematics and Solid Mechanics"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 May 2021).

Special Issue Editors

Prof. Dr. Eduard-Marius Craciun
E-Mail Website
Guest Editor
Fac. of Mech. Ind. and Maritime Eng. Bd. Mamaia 124, University of Constanta, 900527, Constanta, Romania
Interests: applied mathematics and mechanics; solid mechanics; fracture
Prof. Dr. Marin Marin
E-Mail Website
Co-Guest Editor
Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania
Interests: differential equations; partial differential equations; equations of evolution; integral equations; mixed initial-boundary value problems for PDE; termoelasticity; media with microstretch; environments goals; nonlinear problems
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Special Issue Information

Dear Colleagues,

The present Special Issue covers most areas of Applied Mathematics in the theory of classical and non-classical solid mechanics and the purpose is to gather articles reflecting the latest developments in these fields, including theoretical, numerical/computational, and experimental aspects.

The topics of interest for publication include but are not limited to the boundary value problems, the study of mixed problems for generalized continua and recent developments in the field of mathematical modeling for fracture mechanics problems.

All interested researchers are kindly invited to contribute to this Special Issue with their original research articles, short communications, and review articles.

Prof. Dr. Eduard-Marius Craciun
Prof. Dr. Marin Marin
Guest Editors

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Keywords

  • Solids mechanics
  • Fracture mechanics
  • Mathematical approaches
  • Mathematical models
  • Asymptotic analysis
  • Composites
  • Piezoelectricity
  • Mixed problems in generalized continua
  • Biomechanics

Published Papers (16 papers)

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Research

Article
Quasistatic Porous-Thermoelastic Problems: An a Priori Error Analysis
Mathematics 2021, 9(12), 1436; https://doi.org/10.3390/math9121436 - 20 Jun 2021
Viewed by 194
Abstract
In this paper, we deal with the numerical approximation of some porous-thermoelastic problems. Since the inertial effects are assumed to be negligible, the resulting motion equations are quasistatic. Then, by using the finite element method and the implicit Euler scheme, a fully discrete [...] Read more.
In this paper, we deal with the numerical approximation of some porous-thermoelastic problems. Since the inertial effects are assumed to be negligible, the resulting motion equations are quasistatic. Then, by using the finite element method and the implicit Euler scheme, a fully discrete approximation is introduced. We prove a discrete stability property and a main error estimates result, from which we conclude the linear convergence under appropriate regularity conditions on the continuous solution. Finally, several numerical simulations are shown to demonstrate the accuracy of the approximation, the behavior of the solution and the decay of the discrete energy. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
Mutual Influence of Geometric Parameters and Mechanical Properties on Thermal Stresses in Composite Laminated Plates with Rectangular Holes
Mathematics 2021, 9(4), 311; https://doi.org/10.3390/math9040311 - 04 Feb 2021
Viewed by 339
Abstract
In this research, the mutual influence of the mechanical properties and geometric parameters on thermal stress distribution in symmetric composite plates with a quasi-rectangular hole subjected to uniform heat flux is examined analytically using the complex variable technique. The analytical solution is obtained [...] Read more.
In this research, the mutual influence of the mechanical properties and geometric parameters on thermal stress distribution in symmetric composite plates with a quasi-rectangular hole subjected to uniform heat flux is examined analytically using the complex variable technique. The analytical solution is obtained based on the thermo-elastic theory and the Lekhnitskii’s method. Furthermore, by employing a suitable mapping function, the solution of symmetric laminates containing a circular hole is extended to the quasi-rectangular hole. The effect of important parameters including the stacking sequence of laminates, the angular position, the bluntness, and the aspect ratio of the hole and the flux angle in the stacking sequence of [45/−45]s for composite materials are examined in relation to the thermal stress distribution. The thermal insulated state and Neumann boundary conditions at the hole edge are taken into account. It is found out that the hole rotation angles and heat flux angle play key roles in obtaining the optimum thermal stress distribution around the hole. The present analytical method can well investigate the interaction of effective parameters on symmetric multilayer composites under heat flux. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
A Simple-FSDT-Based Isogeometric Method for Piezoelectric Functionally Graded Plates
Mathematics 2020, 8(12), 2177; https://doi.org/10.3390/math8122177 - 06 Dec 2020
Cited by 4 | Viewed by 575
Abstract
An efficient isogeometric analysis method (IGA) based on a simple first-order shear deformation theory is presented to study free vibration, static bending response, dynamic response, and active control of functionally graded plates (FGPs) integrated with piezoelectric layers. Based on the neutral surface, isogeometric [...] Read more.
An efficient isogeometric analysis method (IGA) based on a simple first-order shear deformation theory is presented to study free vibration, static bending response, dynamic response, and active control of functionally graded plates (FGPs) integrated with piezoelectric layers. Based on the neutral surface, isogeometric finite element motion equations of piezoelectric functionally graded plates (PFGPs) are derived using the linear piezoelectric constitutive equation and Hamilton’s principle. The convergence and accuracy of the method for PFGPs with various mechanical and electrical boundary conditions have been investigated via free vibration analysis. In the dynamic analysis, both time-varying mechanical and electrical loads are involved. A closed-loop control method, including displacement feedback control and velocity feedback control, is applied to the static bending control and the dynamic vibration control analysis. The numerical results obtained are accurate and reliable through comparisons with various numerical and analytical examples. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
Topology Optimization of Elastoplastic Behavior Conditions by Selectively Suppressing Plastic Work
Mathematics 2020, 8(11), 2062; https://doi.org/10.3390/math8112062 - 19 Nov 2020
Viewed by 575
Abstract
This work conducted topology optimization with an implicit analysis of elastoplastic constitutive equation in order to design supporting structures for unexpected heavy loading conditions. In this topology optimization model, plastic work was extracted from strain energy and selectively employed in the objective function [...] Read more.
This work conducted topology optimization with an implicit analysis of elastoplastic constitutive equation in order to design supporting structures for unexpected heavy loading conditions. In this topology optimization model, plastic work was extracted from strain energy and selectively employed in the objective function according to deformation mode. While strain energy was minimized in elastic deformation areas, in elastoplastic deformation areas, the plastic work was minimized for the purpose of suppressing plastic deformation. This method can focus on suppressing plastic strain in the plastic deformation zone with maintaining elastic stiffness in the elastic deformation zone. These formulations were implemented into MATLAB and applied to three optimization problems. The elastoplastic optimization results were compared to pure elastic design results. The comparison showed that structures designed with accounting for plastic deformation had a reinforced area where plastic deformation occurs. Finally, a finite element analysis was conducted to compare the mechanical performances of structures with respect to the design method. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
A Note on the Solutions for a Higher-Order Convective Cahn–Hilliard-Type Equation
Mathematics 2020, 8(10), 1835; https://doi.org/10.3390/math8101835 - 19 Oct 2020
Cited by 1 | Viewed by 519
Abstract
The higher-order convective Cahn-Hilliard equation describes the evolution of crystal surfaces faceting through surface electromigration, the growing surface faceting, and the evolution of dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we study the H3 solutions of the Cauchy problem and prove, under different assumptions on the constants appearing in the equation and on the mean of the initial datum, that they are well-posed. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
Article
The Effect of a Hyperbolic Two-Temperature Model with and without Energy Dissipation in a Semiconductor Material
Mathematics 2020, 8(10), 1711; https://doi.org/10.3390/math8101711 - 04 Oct 2020
Viewed by 399
Abstract
In this work, the new model of photothermal and elastic waves, with and without energy dissipation, under a hyperbolic two-temperature model, is used to compute the displacement, carrier density, thermodynamic temperature, conductive temperature and stress in a semiconductor medium. The medium is considered [...] Read more.
In this work, the new model of photothermal and elastic waves, with and without energy dissipation, under a hyperbolic two-temperature model, is used to compute the displacement, carrier density, thermodynamic temperature, conductive temperature and stress in a semiconductor medium. The medium is considered in the presence of the coupling of plasma and thermoelastic waves. To get the complete analytical expressions of the main physical fields, Laplace transforms and the eigenvalue scheme are used. The outcomes are presented graphically to display the differences between the classical two-temperature theory and the new hyperbolic two-temperature theory, with and without energy dissipation. Based on the numerical results, the hyperbolic two-temperature thermoelastic theory offers a finite speed of mechanical waves and propagation of thermal waves. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
Fractional-Order Thermoelastic Wave Assessment in a Two-Dimensional Fiber-Reinforced Anisotropic Material
Mathematics 2020, 8(9), 1609; https://doi.org/10.3390/math8091609 - 18 Sep 2020
Viewed by 483
Abstract
The present work is aimed at studying the effect of fractional order and thermal relaxation time on an unbounded fiber-reinforced medium. In the context of generalized thermoelasticity theory, the fractional time derivative and the thermal relaxation times are employed to study the thermophysical [...] Read more.
The present work is aimed at studying the effect of fractional order and thermal relaxation time on an unbounded fiber-reinforced medium. In the context of generalized thermoelasticity theory, the fractional time derivative and the thermal relaxation times are employed to study the thermophysical quantities. The techniques of Fourier and Laplace transformations are used to present the problem exact solutions in the transformed domain by the eigenvalue approach. The inversions of the Fourier-Laplace transforms hold analytical and numerically. The numerical outcomes for the fiber-reinforced material are presented and graphically depicted. A comparison of the results for different theories under the fractional time derivative is presented. The properties of the fiber-reinforced material with the fractional derivative act to reduce the magnitudes of the variables considered, which can be significant in some practical applications and can be easily considered and accurately evaluated. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
MHD Radiative Blood Flow Embracing Gold Particles via a Slippery Sheet through an Erratic Heat Sink/Source
Mathematics 2020, 8(9), 1597; https://doi.org/10.3390/math8091597 - 16 Sep 2020
Cited by 3 | Viewed by 535
Abstract
Cancer remains one of the world’s leading healthcare issues, and attempts continue not only to find new medicines but also to find better ways of distributing medications. It is harmful and lethal to most of its patients. The need to selectively deliver cytotoxic [...] Read more.
Cancer remains one of the world’s leading healthcare issues, and attempts continue not only to find new medicines but also to find better ways of distributing medications. It is harmful and lethal to most of its patients. The need to selectively deliver cytotoxic agents to cancer cells, to enhance protection and efficacy, has prompted the implementation of nanotechnology in medicine. The latest findings have found that gold nanomaterials can heal and conquer it because the material is studied such as gold (atomic number 79) which produces a large amount of heat and contribute to the therapy of malignant tumors. The purpose of the present study is to research the consequence of heat transport through blood flow (Casson model) that contains gold particles in a slippery shrinking/stretching curved surface. The mathematical modeling of Casson nanofluid containing gold nanomaterials towards the slippery curved shrinking/stretching surface is simplified by utilizing suitable transformation. Numerical dual solutions for the temperature and velocity fields are calculated by using bvp4c methodology in MATLAB. Impacts of related parameters are investigated in the temperature and velocity distribution. The results indicate that the suction parameter accelerates the velocity in the upper branch solution and decelerates it in the lower branch solution, while the temperature diminishes in both solutions. In addition, the Casson parameter shrinks the thickness of the velocity boundary-layer owing to rapid enhancement in the plastic dynamics’ viscosity. Moreover, the nanoparticle volume fraction accelerates the viscosity of blood as well as the thermal conductivity. Thus, findings suggested that gold nanomaterials are useful for drug moving and delivery mechanisms since the velocity boundary is regulated by the volume fraction parameter. Gold nanomaterials also raise the temperature field, so that cancer cells can be destroyed. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
New Analytical Model Used in Finite Element Analysis of Solids Mechanics
Mathematics 2020, 8(9), 1401; https://doi.org/10.3390/math8091401 - 21 Aug 2020
Cited by 1 | Viewed by 589
Abstract
In classical mechanics, determining the governing equations of motion using finite element analysis (FEA) of an elastic multibody system (MBS) leads to a system of second order differential equations. To integrate this, it must be transformed into a system of first-order equations. However, [...] Read more.
In classical mechanics, determining the governing equations of motion using finite element analysis (FEA) of an elastic multibody system (MBS) leads to a system of second order differential equations. To integrate this, it must be transformed into a system of first-order equations. However, this can also be achieved directly and naturally if Hamilton’s equations are used. The paper presents this useful alternative formalism used in conjunction with the finite element method for MBSs. The motion equations in the very general case of a three-dimensional motion of an elastic solid are obtained. To illustrate the method, two examples are presented. A comparison between the integration times in the two cases presents another possible advantage of applying this method. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
A Study on Thermoelastic Interaction in a Poroelastic Medium with and without Energy Dissipation
Mathematics 2020, 8(8), 1286; https://doi.org/10.3390/math8081286 - 04 Aug 2020
Viewed by 418
Abstract
In the current work, a new generalized model of heat conduction has been constructed taking into account the influence of porosity on a poro-thermoelastic medium using the finite element method (FEM). The governing equations are presented in the context of the Green and [...] Read more.
In the current work, a new generalized model of heat conduction has been constructed taking into account the influence of porosity on a poro-thermoelastic medium using the finite element method (FEM). The governing equations are presented in the context of the Green and Naghdi (G-N) type III theory with and without energy dissipations. The finite element scheme has been adopted to present the solutions due to the complex formulations of this problem. The effects of porosity on poro-thermoelastic material are investigated. The numerical results for stresses, temperatures, and displacements for the solid and the fluid are graphically presented. This work provides future investigators with insight regarding details of non-simple poro-thermoelasticity with different phases. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
The Effects of Variable Thermal Conductivity in Semiconductor Materials Photogenerated by a Focused Thermal Shock
Mathematics 2020, 8(8), 1230; https://doi.org/10.3390/math8081230 - 27 Jul 2020
Cited by 2 | Viewed by 539
Abstract
In this work, the generalized photo-thermo-elastic model with variable thermal conductivity is presented to estimates the variations of temperature, the carrier density, the stress and the displacement in a semiconductor material. The effects of variable thermal conductivity under photo-thermal transport process is investigated [...] Read more.
In this work, the generalized photo-thermo-elastic model with variable thermal conductivity is presented to estimates the variations of temperature, the carrier density, the stress and the displacement in a semiconductor material. The effects of variable thermal conductivity under photo-thermal transport process is investigated by using the coupled model of thermoelastic and plasma wave. The surface of medium is loaded by uniform unit step temperature. Easily, the analytical solutions in the domain of Laplace are obtained. By using Laplace transforms with the eigenvalue scheme, the fields studied are obtained analytically and presented graphically. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
The Size-Dependent Thermoelastic Vibrations of Nanobeams Subjected to Harmonic Excitation and Rectified Sine Wave Heating
Mathematics 2020, 8(7), 1128; https://doi.org/10.3390/math8071128 - 10 Jul 2020
Cited by 6 | Viewed by 627
Abstract
In this article, a nonlocal thermoelastic model that illustrates the vibrations of nanobeams is introduced. Based on the nonlocal elasticity theory proposed by Eringen and generalized thermoelasticity, the equations that govern the nonlocal nanobeams are derived. The structure of the nanobeam is under [...] Read more.
In this article, a nonlocal thermoelastic model that illustrates the vibrations of nanobeams is introduced. Based on the nonlocal elasticity theory proposed by Eringen and generalized thermoelasticity, the equations that govern the nonlocal nanobeams are derived. The structure of the nanobeam is under a harmonic external force and temperature change in the form of rectified sine wave heating. The nonlocal model includes the nonlocal parameter (length-scale) that can have the effect of the small-scale. Utilizing the technique of Laplace transform, the analytical expressions for the studied fields are reached. The effects of angular frequency and nonlocal parameters, as well as the external excitation on the response of the nanobeam are carefully examined. It is found that length-scale and external force have significant effects on the variation of the distributions of the physical variables. Some of the obtained numerical results are compared with the known literature, in which they are well proven. It is hoped that the obtained results will be valuable in micro/nano electro-mechanical systems, especially in the manufacture and design of actuators and electro-elastic sensors. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
Kane’s Method-Based Simulation and Modeling Robots with Elastic Elements, Using Finite Element Method
Mathematics 2020, 8(5), 805; https://doi.org/10.3390/math8050805 - 15 May 2020
Cited by 2 | Viewed by 781
Abstract
The Lagrange’s equation remains the most used method by researchers to determine the finite element motion equations in the case of elasto-dynamic analysis of a multibody system (MBS). However, applying this method requires the calculation of the kinetic energy of an element and [...] Read more.
The Lagrange’s equation remains the most used method by researchers to determine the finite element motion equations in the case of elasto-dynamic analysis of a multibody system (MBS). However, applying this method requires the calculation of the kinetic energy of an element and then a series of differentiations that involve a great computational effort. The last decade has shown an increased interest of researchers in the study of multibody systems (MBS) using alternative analytical methods, aiming to simplify the description of the model and the solution of the systems of obtained equations. The method of Kane’s equations is one possibility to do this and, in the paper, we applied this method in the study of a MBS applying finite element analysis (FEA). The number of operations involved is lower than in the case of Lagrange’s equations and Kane’s equations are little used previously in conjunction with the finite element method (FEM). Results are obtained regardless of the type of finite element used. The shape functions will determine the final form of the matrix coefficients in the equations. The results are applied in the case of a planar mechanism with two degrees of freedom. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
A New Solution to Well-Known Hencky Problem: Improvement of In-Plane Equilibrium Equation
Mathematics 2020, 8(5), 653; https://doi.org/10.3390/math8050653 - 25 Apr 2020
Cited by 3 | Viewed by 559
Abstract
In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, [...] Read more.
In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, the so-called small rotation angle assumption of the membrane is given up when establishing the out-of-plane equilibrium equation, and the in-plane equilibrium equation is, for the first time, improved by considering the effect of the deflection on the equilibrium between the radial and circumferential stress. Furthermore, the resulting nonlinear differential equation is successfully solved by using the power series method, and a new closed-form solution of the problem is finally presented. The conducted numerical example indicates that the closed-form solution presented here has a higher computational accuracy in comparison with the existing solutions of the well-known Hencky problem, especially when the deflection of the membrane is relatively large. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
Optimum Design of Infinite Perforated Orthotropic and Isotropic Plates
Mathematics 2020, 8(4), 569; https://doi.org/10.3390/math8040569 - 11 Apr 2020
Cited by 3 | Viewed by 694
Abstract
In this study, an attempt was made to introduce the optimal values of effective parameters on the stress distribution around a circular/elliptical/quasi-square cutout in the perforated orthotropic plate under in-plane loadings. To achieve this goal, Lekhnitskii’s complex variable approach and Particle Swarm Optimization [...] Read more.
In this study, an attempt was made to introduce the optimal values of effective parameters on the stress distribution around a circular/elliptical/quasi-square cutout in the perforated orthotropic plate under in-plane loadings. To achieve this goal, Lekhnitskii’s complex variable approach and Particle Swarm Optimization (PSO) method were used. This analytical method is based on using the complex variable method in the analysis of two-dimensional problems. The Tsai–Hill criterion and Stress Concentration Factor (SCF) are taken as objective functions and the fiber angle, bluntness, aspect ratio of cutout, the rotation angle of cutout, load angle, and material properties are considered as design variables. The results show that the PSO algorithm is able to predict the optimal value of each effective parameter. In addition, these parameters have significant effects on stress distribution around the cutouts and the load-bearing capacity of structures can be increased by appropriate selection of the effective design variables. The main innovation of this study is the use of PSO algorithm to determine the optimal design variables to increase the strength of the perforated plates. Finite element method (FEM) was employed to examine the results of the present analytical solution. The results obtained by the present solution are in accordance with numerical results. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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Article
Some Results in Green–Lindsay Thermoelasticity of Bodies with Dipolar Structure
Mathematics 2020, 8(4), 497; https://doi.org/10.3390/math8040497 - 02 Apr 2020
Cited by 10 | Viewed by 679
Abstract
The main concern of this study is an extension of some results, proposed by Green and Lindsay in the classical theory of elasticity, in order to cover the theory of thermoelasticity for dipolar bodies. For dynamical mixed problem we prove a reciprocal theorem, [...] Read more.
The main concern of this study is an extension of some results, proposed by Green and Lindsay in the classical theory of elasticity, in order to cover the theory of thermoelasticity for dipolar bodies. For dynamical mixed problem we prove a reciprocal theorem, in the general case of an anisotropic thermoelastic body. Furthermore, in this general context we have proven a result regarding the uniqueness of the solution of the mixed problem in the dynamical case. We must emphasize that these fundamental results are obtained under conditions that are not very restrictive. Full article
(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)
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