Mathematics

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9 pages, 603 KiB

Open AccessArticle

A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams

by Snježana Maksimović and Aleksandar Borković

Mathematics 2021, 9(15), 1778; https://doi.org/10.3390/math9151778 - 27 Jul 2021

Cited by 1 | Viewed by 2172

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of

L^{2} (R)

space and [...] Read more.

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of

L^{2} (R)

space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper. Full article

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

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20 pages, 3332 KiB

Open AccessArticle

A Two-Dimensional Thermoelasticity Solution for Bimodular Material Beams under the Combination Action of Thermal and Mechanical Loads

by Si-Rui Wen, Xiao-Ting He, Hao Chang and Jun-Yi Sun

Mathematics 2021, 9(13), 1556; https://doi.org/10.3390/math9131556 - 2 Jul 2021

Cited by 5 | Viewed by 2167

Abstract

A typical characteristic of bimodular material beams is that when bending, the neutral layer of the beam does not coincide with its geometric middle surface since the mechanical properties of materials in tension and compression are different. In the classical theory of elasticity, [...] Read more.

A typical characteristic of bimodular material beams is that when bending, the neutral layer of the beam does not coincide with its geometric middle surface since the mechanical properties of materials in tension and compression are different. In the classical theory of elasticity, however, this characteristic has not been considered. In this study, a bimodular simply-supported beam under the combination action of thermal and mechanical loads is theoretically analyzed. First, a simplified mechanical model concerning the neutral layer is established. Based on this mechanical model, Duhamel’s theorem is used to transform the thermoelastical problem into a pure elasticity problem with imaginary body force and surface force. In solving the governing equation expressed in terms of displacement, a special solution of the displacement equation is found first, and then by utilizing the stress function method based on subarea in tension and compression, a supplement solution for the displacement governing equation without the thermal effect is derived. Lastly, the special solution and supplement solution are superimposed to satisfy boundary conditions, thus obtaining a two-dimensional thermoelasticity solution. In addition, the bimodular effect and temperature effect on the thermoelasticity solution are illustrated by computational examples. Full article

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

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13 pages, 2768 KiB

Open AccessArticle

A Study of Yielding and Plasticity of Rapid Prototyped ABS

by Dan-Andrei Șerban, Cosmin Marșavina, Alexandru Viorel Coșa, George Belgiu and Radu Negru

Mathematics 2021, 9(13), 1495; https://doi.org/10.3390/math9131495 - 25 Jun 2021

Cited by 8 | Viewed by 1971

Abstract

In this article, the yielding and plastic flow of a rapid-prototyped ABS compound was investigated for various plane stress states. The experimental procedures consisted of multiaxial tests performed on an Arcan device on specimens manufactured through photopolymerization. Numerical analyses were employed in order [...] Read more.

In this article, the yielding and plastic flow of a rapid-prototyped ABS compound was investigated for various plane stress states. The experimental procedures consisted of multiaxial tests performed on an Arcan device on specimens manufactured through photopolymerization. Numerical analyses were employed in order to determine the yield points for each stress state configuration. The results were used for the calibration of the Hosford yield criterion and flow potential. Numerical analyses performed on identical specimen models and test configurations yielded results that are in accordance with the experimental data. Full article

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

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13 pages, 509 KiB

Open AccessArticle

Quasistatic Porous-Thermoelastic Problems: An a Priori Error Analysis

by Jacobo Baldonedo, José R. Fernández and José A. López-Campos

Mathematics 2021, 9(12), 1436; https://doi.org/10.3390/math9121436 - 20 Jun 2021

Viewed by 1558

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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24 pages, 11965 KiB

Open AccessArticle

A Simple-FSDT-Based Isogeometric Method for Piezoelectric Functionally Graded Plates

by Tao Liu, Chaodong Li, Chao Wang, Joel Weijia Lai and Kang Hao Cheong

Mathematics 2020, 8(12), 2177; https://doi.org/10.3390/math8122177 - 6 Dec 2020

Cited by 18 | Viewed by 3132

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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18 pages, 15816 KiB

Open AccessArticle

Topology Optimization of Elastoplastic Behavior Conditions by Selectively Suppressing Plastic Work

by Eun-Ho Lee and Tae-Hyun Kim

Mathematics 2020, 8(11), 2062; https://doi.org/10.3390/math8112062 - 19 Nov 2020

Cited by 5 | Viewed by 3534

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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31 pages, 417 KiB

Open AccessArticle

A Note on the Solutions for a Higher-Order Convective Cahn–Hilliard-Type Equation

by Giuseppe Maria Coclite and Lorenzo di Ruvo

Mathematics 2020, 8(10), 1835; https://doi.org/10.3390/math8101835 - 19 Oct 2020

Cited by 8 | Viewed by 1912

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

H^{3}

solutions of the Cauchy [...] Read more.

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

H^{3}

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)

11 pages, 3316 KiB

Open AccessArticle

The Effect of a Hyperbolic Two-Temperature Model with and without Energy Dissipation in a Semiconductor Material

by Faris Alzahrani and Ibrahim Abbas

Mathematics 2020, 8(10), 1711; https://doi.org/10.3390/math8101711 - 4 Oct 2020

Cited by 5 | Viewed by 2192

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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14 pages, 3431 KiB

Open AccessArticle

Fractional-Order Thermoelastic Wave Assessment in a Two-Dimensional Fiber-Reinforced Anisotropic Material

by Samah Horrigue and Ibrahim A. Abbas

Mathematics 2020, 8(9), 1609; https://doi.org/10.3390/math8091609 - 18 Sep 2020

Cited by 14 | Viewed by 2269

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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23 pages, 4791 KiB

Open AccessArticle

MHD Radiative Blood Flow Embracing Gold Particles via a Slippery Sheet through an Erratic Heat Sink/Source

by Umair Khan, Anum Shafiq, Aurang Zaib, El-Sayed M. Sherif and Dumitru Baleanu

Mathematics 2020, 8(9), 1597; https://doi.org/10.3390/math8091597 - 16 Sep 2020

Cited by 17 | Viewed by 2344

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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10 pages, 3117 KiB

Open AccessArticle

A Study on Thermoelastic Interaction in a Poroelastic Medium with and without Energy Dissipation

by Tareq Saeed

Mathematics 2020, 8(8), 1286; https://doi.org/10.3390/math8081286 - 4 Aug 2020

Cited by 4 | Viewed by 1966

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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9 pages, 1270 KiB

Open AccessArticle

The Effects of Variable Thermal Conductivity in Semiconductor Materials Photogenerated by a Focused Thermal Shock

by Faris Alzahrani

Mathematics 2020, 8(8), 1230; https://doi.org/10.3390/math8081230 - 27 Jul 2020

Cited by 10 | Viewed by 2756

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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13 pages, 3552 KiB

Open AccessArticle

The Size-Dependent Thermoelastic Vibrations of Nanobeams Subjected to Harmonic Excitation and Rectified Sine Wave Heating

by Ahmed E. Abouelregal and Marin Marin

Mathematics 2020, 8(7), 1128; https://doi.org/10.3390/math8071128 - 10 Jul 2020

Cited by 96 | Viewed by 3140

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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21 pages, 1918 KiB

Open AccessArticle

Kane’s Method-Based Simulation and Modeling Robots with Elastic Elements, Using Finite Element Method

by Sorin Vlase, Iuliu Negrean, Marin Marin and Silviu Năstac

Mathematics 2020, 8(5), 805; https://doi.org/10.3390/math8050805 - 15 May 2020

Cited by 23 | Viewed by 6144

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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19 pages, 3852 KiB

Open AccessArticle

A New Solution to Well-Known Hencky Problem: Improvement of In-Plane Equilibrium Equation

by Xue Li, Jun-Yi Sun, Zhi-Hang Zhao, Shou-Zhen Li and Xiao-Ting He

Mathematics 2020, 8(5), 653; https://doi.org/10.3390/math8050653 - 25 Apr 2020

Cited by 9 | Viewed by 2550

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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23 pages, 2423 KiB

Open AccessArticle

Optimum Design of Infinite Perforated Orthotropic and Isotropic Plates

by Mohammad Jafari, Seyed Ahmad Mahmodzade Hoseyni, Holm Altenbach and Eduard-Marius Craciun

Mathematics 2020, 8(4), 569; https://doi.org/10.3390/math8040569 - 11 Apr 2020

Cited by 9 | Viewed by 2704

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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12 pages, 262 KiB

Open AccessEditor’s ChoiceArticle

Some Results in Green–Lindsay Thermoelasticity of Bodies with Dipolar Structure

by Marin Marin, Eduard M. Craciun and Nicolae Pop

Mathematics 2020, 8(4), 497; https://doi.org/10.3390/math8040497 - 2 Apr 2020

Cited by 28 | Viewed by 2651

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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