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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 21491

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Prof. Dr. Eduard-Marius Craciun

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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
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Marin Marin

E-Mail Website
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
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The present Special Issue covers most areas of applied mathematics in the theory of continuum 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 study of mixed problems for continua, the recent developments in the field of mathematical modelling for fracture mechanics problems, boundary value problems, etc.

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

continuum mechanics
fracture mechanics
mathematical models
asymptotic analysis
composite materials
piezoelectricity
generalized continua

Published Papers (14 papers)

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Research

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25 pages, 24986 KiB

Open AccessArticle

A Reconstruction Approach for Concurrent Multiscale Topology Optimization Based on Direct FE² Method

by Ang Zhao, Vincent Beng Chye Tan, Pei Li, Kui Liu and Zhendong Hu

Mathematics 2023, 11(12), 2779; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122779 - 20 Jun 2023

Viewed by 1051

Abstract

The rapid development of material science is increasing the demand for the multiscale design of materials. The concurrent multiscale topology optimization based on the Direct FE² method can greatly improve computational efficiency, but it may lead to the checkerboard problem. In order to solve the checkerboard problem and reconstruct the results of the Direct FE² model, this paper proposes a filtering-based reconstruction method. This solution is of great significance for the practical application of multiscale topology optimization, as it not only solves the checkerboard problem but also provides the optimized full model based on interpolation. The filtering method effectively eliminates the checkerboard pattern in the results by smoothing the element densities. The reconstruction method restores the smoothness of the optimized structure by interpolating between the filtered densities. This method is highly effective in solving the checkerboard problem, as demonstrated in our numerical examples. The results show that the proposed algorithm produces feasible and stable results. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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16 pages, 4492 KiB

Open AccessArticle

Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

by Aly R. Seadawy, Syed T. R. Rizvi and Hanadi Zahed

Mathematics 2023, 11(5), 1074; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051074 - 21 Feb 2023

Cited by 7 | Viewed by 971

Abstract

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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23 pages, 6347 KiB

Open AccessArticle

Accuracy of Variational Formulation to Model the Thermomechanical Problem and to Predict Failure in Metallic Materials

by Lotfi Ben Said and Mondher Wali

Mathematics 2022, 10(19), 3555; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193555 - 29 Sep 2022

Cited by 8 | Viewed by 1016

Abstract

The main purpose of this study is to develop a variational formulation for predicting structure behavior and accounting for damage mechanics in metallic materials. Mechanical and coupled thermomechanical models are used to predict failure in manufacturing processes. Ductile failure is accompanied by a significant amount of plastic deformation in metallic structural components. Finite element simulation of damage evolution in ductile solids is presented in this paper. Uncoupled models are implemented in a finite element model simulating deep drawing as well as cutting processes. Based on the Johnson–Cook model, the effect of deformation on the evolution of flow stress is described. The combined effect of strain, strain rate, and temperature on plasticity and damage behavior in cutting processes is considered. The accuracy of these models is verified when predicting ductile damage in forming and cutting processes. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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24 pages, 12345 KiB

Open AccessArticle

Numerical Computation of Hybrid Carbon Nanotubes Flow over a Stretching/Shrinking Vertical Cylinder in Presence of Thermal Radiation and Hydromagnetic

by Nur Adilah Liyana Aladdin, Norfifah Bachok, Haliza Rosali, Nadihah Wahi, Nor Aliza Abd Rahmin and Norihan Md Arifin

Mathematics 2022, 10(19), 3551; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193551 - 29 Sep 2022

Cited by 6 | Viewed by 1164

Abstract

The discovery of hybrid carbon nanotubes shows the tendency toward the improvement of heat transfer performance in comparison to various classical fluids. This paper expands the novelty in utilizing the hybrid carbon nanotubes over vertical stretching/shrinking cylinder in presence of hydromagnetic and thermal radiation. It is essential to analyze the hydromagnetic due to its high potential capability especially in drug and gene release, hyperthermia effects as well as cell separation and manipulation in bio-medical field. The investigation on thermal radiation effect is added in this current study as it enhances the rate of heat transfer. To initiate this problem, partial differential equations (PDE) for the hybrid nanofluid flow with relevant boundary conditions (BCs) is set up and transformed into an ordinary differential equation (ODE). Adopting the similarity solutions and numerically solved using bvp4c (MATLAB). Findings on the variation of local Nusselt number, skin friction coefficient, shear stress and local heat flux having the effects of magnetic,

M,

curvature, ϒ, thermal radiation, Nr, mixed convection parameter,

λ

as well as volume fraction of nanoparticles,

φ

are demonstrated and elaborated in detail. Moreover, the research reveals that duality of solutions occurs when the buoyance force is in opposing flow with respect to the fluid motion,

λ < 0

, as well as shrinking area,

ε < 0

. The occurrence of magnetic reduces the heat transfer as well as skin friction coefficient. In addition, the skin friction coefficient and local Nusselt number tend to improve as volume fraction of nanoparticles and curvature are increased. In contrast, the low of thermal radiation enhance the heat transfer. Indeed, the consequences of using hybrid carbon nanotubes help intensify the skin friction coefficient and Nusselt number compared to SWCNT nanofluid and MWCNT nanofluid. These crucial findings may benefit the scientists and academicians hence giving an add-on value to their expertise. A stability analysis must be performed since there exists a non-unique solution throughout the computation. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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22 pages, 3504 KiB

Open AccessArticle

Solution of the Thermoelastic Problem for a Two-Dimensional Curved Beam with Bimodular Effects

by Xiao-Ting He, Meng-Qiao Zhang, Bo Pang and Jun-Yi Sun

Mathematics 2022, 10(16), 3002; https://0-doi-org.brum.beds.ac.uk/10.3390/math10163002 - 19 Aug 2022

Cited by 4 | Viewed by 1204

Abstract

In classical thermoelasticity, the bimodular effect of materials is rarely considered. However, all materials will present, in essence, different properties in tension and compression, more or less. The bimodular effect is generally ignored only for simple analysis. In this study, we theoretically analyze a two-dimensional curved beam with a bimodular effect and under mechanical and thermal loads. We first establish a simplified model on a subarea in tension and compression. On the basis of this model, we adopt the Duhamel similarity theorem to change the initial thermoelastic problem as an elasticity problem without the thermal effect. The superposition of the special solution and supplement solution of the Lamé displacement equation enables us to satisfy the boundary conditions and stress continuity conditions of the bimodular curved beam, thus obtaining a two-dimensional thermoelastic solution. The results indicate that the solution obtained can reduce to bimodular curved beam problems without thermal loads and to the classical Golovin solution. In addition, the bimodular effect on thermal stresses is discussed under linear and non-linear temperature rise modes. Specially, when the compressive modulus is far greater than the tensile modulus, a large compressive stress will occur at the inner edge of the curved beam, which should be paid with more attention in the design of the curved beams in a thermal environment. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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16 pages, 6663 KiB

Open AccessArticle

Generalized Thermo-Diffusion Interaction in an Elastic Medium under Temperature Dependent Diffusivity and Thermal Conductivity

by Aatef Hobiny and Ibrahim Abbas

Mathematics 2022, 10(15), 2773; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152773 - 04 Aug 2022

Cited by 6 | Viewed by 1133

Abstract

The purpose of this work is to investigate, within the context of extended thermo-diffusion theory, the transient thermo-diffusion responses for a half-space with variable thermal conductivity and diffusivity. The half-bounding space’s surface is traction-free and exposed to a time-dependent thermal shock, but the chemical potential is believed to be a known function of time. Because the nonlinear equations are complicated, the finite element technique is applied to solve these equations. Numerical outcomes are produced and graphically illustrated. The effects of varying thermal conductivity and diffusivity on the response are studied using parameter studies. Using the results of this study, researchers hope to understand better how thermo-mechanical fields interact in real materials. By ignoring the new parameter, a comparison of numerical results and analytical cases is produced, and the behavior of physical quantities for numerical solutions is studied to ensure that the proposed technique is accurate. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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32 pages, 7868 KiB

Open AccessArticle

Queueing Theory-Based Mathematical Models Applied to Enterprise Organization and Industrial Production Optimization

by Laurentiu Rece, Sorin Vlase, Daniel Ciuiu, Giorgian Neculoiu, Stefan Mocanu and Arina Modrea

Mathematics 2022, 10(14), 2520; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142520 - 20 Jul 2022

Cited by 5 | Viewed by 1986

Abstract

In the paper, a new method was presented using queueing theory models in order to ensure an optimal production department size, optimized production costs and optimal provision. Queueing/waiting mathematical models represent the development matrix for an experimental algorithm and implicitly numerical approach, both successfully applied (and confirmed in practice) in a production section design for a real industrial engineering unit with discussed method technological flow and equipment schemes compatibility. The total costs for a queueing system with S servers depend on the number of servers. The problem of minimizing cost in terms of S was the main aim of the paper. In order to solve it, we estimated all the variables of the system that influence the cost using the Monte Carlo method. For a Jackson queueing network, the involved linear system has good properties such that it can be solved by iterative methods such as Jacobi and Gauss–Seidel. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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16 pages, 2985 KiB

Open AccessArticle

A Mathematical Study of a Semiconducting Thermoelastic Rotating Solid Cylinder with Modified Moore–Gibson–Thompson Heat Transfer under the Hall Effect

by Iqbal Kaur, Kulvinder Singh and Eduard-Marius Craciun

Mathematics 2022, 10(14), 2386; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142386 - 07 Jul 2022

Cited by 10 | Viewed by 1199

Abstract

This research aims to investigate photo-thermoelastic interactions in a rotating infinite semiconducting solid cylinder under a high magnetic field acting along its axis with the Hall current effect. The boundary surface is subjected to a variable heat flux generated by an exponential laser pulse. The governing equations are expressed using a new photo-thermoelastic model generalized in the Moore–Gibson–Thompson photo-thermal (MGTPT) heat transfer model for a semiconducting medium. The Moore–Gibson–Thompson (MGT) equation is obtained by introducing a thermal relaxation parameter into the Green–Naghdi (GN III) model. The Laplace transform is utilized to determine the mathematical expressions for the components of displacement, carrier density, temperature field, and thermal stresses in the transformed domain. The numerical inversion technique is used to obtain the expressions in the physical domain. The impacts of thermal relaxations, different theories of thermoelasticity, the Hall current, and rotation on the displacement, temperature, thermal stresses, and carrier density are represented graphically using MATLAB software. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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10 pages, 1193 KiB

Open AccessArticle

Generalized Thermoelastic Interaction in a Half-Space under a Nonlocal Thermoelastic Model

by Ibrahim Abbas, Aatef Hobiny, Sorin Vlase and Marin Marin

Mathematics 2022, 10(13), 2168; https://0-doi-org.brum.beds.ac.uk/10.3390/math10132168 - 22 Jun 2022

Cited by 4 | Viewed by 1511

Abstract

In the current article, the nonlocal thermoelastic theory is used to discuss the wave propagation in unbounded thermoelastic materials. Due to the inclusion of relaxation time in thermal conduction formulation and the equations of motion, this model was developed using Lord and Shulman’s generalized thermoelastic model. The theory of the nonlocal continuum proposed by Eringen is used to obtain this model. The integral transforms of the Laplace transform methods used to generate an analytical solution for physical variables are utilized to produce the analytical solutions for the thermal stress, displacement, and temperature distribution. The effects of nonlocal parameters and relaxation time on the wave propagation distributions of physical fields for material are visually shown and explored. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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21 pages, 4578 KiB

Open AccessArticle

Thermoelastic Plane Waves in Materials with a Microstructure Based on Micropolar Thermoelasticity with Two Temperature and Higher Order Time Derivatives

by Ahmed E. Abouelregal, Marin Marin and Fahad Alsharari

Mathematics 2022, 10(9), 1552; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091552 - 05 May 2022

Cited by 7 | Viewed by 1396

Abstract

The study of the effect of the microstructure is important and is most evident in elastic vibrations of high frequency and short-wave duration. In addition to deformation caused by temperature and acting forces, the theory of micropolar thermoelasticity is applied to investigate the microstructure of materials when the vibration of their atoms or molecules is increased. This paper addresses a two-dimensional problem involving a thermoelastic micro-polar half-space with a traction-free surface and a known conductive temperature at the medium surface. The problem is treated in the framework of the concept of two-temperature thermoelasticity with a higher-order time derivative and phase delays, which takes into consideration the impact of microscopic structures in non-simple materials. The normal mode technique was applied to find the analytical formulas for thermal stresses, displacements, micro-rotation, temperature changes, and coupled stress. The numerical results are graphed, and the effect of the discrepancy indicator and higher-order temporal derivatives is examined. There are also some exceptional cases that are covered. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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18 pages, 300 KiB

Open AccessArticle

Two-Scale Homogenization of Piezoelectric Perforated Structures

by Houari Mechkour

Mathematics 2022, 10(9), 1455; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091455 - 26 Apr 2022

Cited by 4 | Viewed by 1306

Abstract

We are interested in the homogenization of the elastic-electric coupling equation with rapidly oscillating coefficients, in a periodically perforated piezoelectric body. The holes, whose size are supposed to tend to zero, are periodically distributed. We give a new approach, based on the two-scale convergence, and we justify the two first terms in the usual asymptotic development of the solution. A two-scale homogenized system is obtained as the limit of the periodic problem, and explicit formulae of elastic, piezoelectric and dielectric homogenized coefficients are reported. In the static limit, the method provides homogenized electroelastic coefficients coinciding with those deducted from alternative approaches. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

10 pages, 1454 KiB

Open AccessArticle

Effects of the Nonlocal Thermoelastic Model in a Thermoelastic Nanoscale Material

by Tareq Saeed and Ibrahim Abbas

Mathematics 2022, 10(2), 284; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020284 - 17 Jan 2022

Cited by 11 | Viewed by 1728

Abstract

In this work, a novel nonlocal model without energy dissipations is presented to investigate the impacts of the nonlocal thermoelastic parameters in a nanoscale material by the eigenvalue approach. The basic equations are applied under the Green and Naghdi model without energy dissipations. To obtain this model, the theory of the non-local continuum suggested by Eringen is applied. The Laplace transformation technique is used for the basic formulations to obtain the analytical solutions of the thermal stress, the displacement, and the temperature during the nanoscale thermo-electric medium. The inverse of the Laplace transformation is used with the numerical technique to obtain the complete solutions of the studying fields in the time–space domains. The main physical fields are displayed graphically and theoretically discussed under the influence of nonlocal parameters. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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14 pages, 6600 KiB

Open AccessArticle

Surface Roughness Investigation of Poly-Jet 3D Printing

by Nectarios Vidakis, Markos Petousis, Nikolaos Vaxevanidis and John Kechagias

Mathematics 2020, 8(10), 1758; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101758 - 13 Oct 2020

Cited by 36 | Viewed by 3862

Abstract

An experimental investigation of the surface quality of the Poly-Jet 3D printing (PJ-3DP) process is presented. PJ-3DP is an additive manufacturing process, which uses jetted photopolymer droplets, which are immediately cured with ultraviolet lamps, to build physical models, layer-by-layer. This method is fast and accurate due to the mechanism it uses for the deposition of layers as well as the 16 microns of layer thickness used. Τo characterize the surface quality of PJ-3DP printed parts, an experiment was designed and the results were analyzed to identify the impact of the deposition angle and blade mechanism motion onto the surface roughness. First, linear regression models were extracted for the prediction of surface quality parameters, such as the average surface roughness (Ra) and the total height of the profile (Rt) in the X and Y directions. Then, a Feed Forward Back Propagation Neural Network (FFBP-NN) was proposed for increasing the prediction performance of the surface roughness parameters Ra and Rt. These two models were compared with the reported ones in the literature; it was revealed that both performed better, leading to more accurate surface roughness predictions, whilst the NN model resulted in the best predictions, in particular for the Ra parameter. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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Review

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15 pages, 308 KiB

Open AccessReview

Two-Dimensional Equivalent Models in the Analysis of a Multibody Elastic System Using the Finite Element Analysis

by Maria Luminita Scutaru and Sorin Vlase

Mathematics 2023, 11(19), 4149; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194149 - 02 Oct 2023

Viewed by 647

Abstract

Analytical mechanics provides methods for analyzing multibody systems with mathematically equivalent elastic elements. The paper analyzes several of these models, highlighting the advantages and disadvantages offered by each of these methods. The main methods used by the researchers are described in a unitary form, presenting the methods of obtaining the evolution equations in each of these cases, mentioning the strengths and weaknesses of each method. The equations of Lagrange, Gibbs–Appell, Kane, Maggi, and Hamilton are analyzed for the particular case of two-dimensional systems, which present certain particularities that facilitate the analysis. Full article

(This article belongs to the Special Issue Applied Mathematics and Continuum Mechanics)

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