Special Issue "Analytical and Numerical Methods for Linear and Nonlinear Analysis of Structures at Macro, Micro and Nano Scale"

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

Deadline for manuscript submissions: closed (30 April 2022) | Viewed by 5099

Special Issue Editors

Dr. Krzysztof Kamil Żur
E-Mail Website
Guest Editor
Department of Mechanical Engineering, Bialystok University of Technology, Wiejska 45C, 15-351 Bialystok, Poland
Interests: applied mathematics; linear and non-linear mechanics of composite structures at macro, micro, and nano scale; non-local continuum mechanics; smart materials and structures; composite materials
Special Issues, Collections and Topics in MDPI journals
Dr. Jinseok Kim
E-Mail Website
Guest Editor
Department of Mechanical and Aerospace Engineering, Western Michigan University, Kalamazoo, MI 49008-5343, USA
Interests: computational solid mechanics; finite element method; plates and shell theories; non-local mechanics; composite materials; smart materials
Prof. Dr. J. N. Reddy
grade E-Mail Website
Guest Editor
J. Mike Walker ’66 Department of Mechanical Engineering, Texas A & M University, College Station, TX 77843-3123, USA
Interests: linear and nonlinear finite element analysis; variational methods; plate and shell theories; composite structures; numerical heat transfer; computational fluid dynamics; non-local theories; applied mathematics

Special Issue Information

Dear Colleagues,

The mathematical models of physical phenomena are based on the fundamental scientific laws of physics.  Mathematical models consist of a combination of algebraic and differential (sometimes even integral) equations. Mathematical models of structural elements (e.g., beams, plates, and shells) based on continuum assumption require a proper treatment of the kinematic, kinetic, and constitutive issues accounting for possible sources of non-local and non-classical continuum mechanics concepts and solving associated boundary value problems. The development of mathematical models and their solutions by analytical and numerical methods have been the focus of many researchers. In particular, the mechanical response of ultrasmall structures has received a great deal of attention because of their wide applications in high-tech devices, such as nanoelectromechanical and microelectromechanical systems.

This special issue is aimed at collecting high-quality papers on the latest developments, techniques, and approaches for the modeling and simulation of the mechanical behavior of structures at macro, micro and nano scales. Advanced accurate numerical and analytical methods to solve PDEs are of high interest. The vibrational response, buckling instability, wave propagation analysis, and static deformation of macro, micro and nano scales’ structural components are covered in this special issue.

Dr. Krzysztof Kamil Żur
Dr. Jinseok Kim
Prof. Dr. J. N. Reddy
Guest Editors

Manuscript Submission Information

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

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • linear and nonlinear boundary/initial value problems
  • continuum mechanics
  • elasticity theories
  • micro/nanostructured systems
  • macroscale structures
  • mechanical response
  • discretization methods
  • numerical analysis
  • analytical methods
  • multifield loads
  • partial differential equations
  • verification and validation
  • fast numerical algorithms

Published Papers (9 papers)

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Editorial

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Editorial
Special Issue of Mathematics: Analytical and Numerical Methods for Linear and Nonlinear Analysis of Structures at Macro, Micro and Nano Scale
Mathematics 2022, 10(13), 2215; https://doi.org/10.3390/math10132215 - 24 Jun 2022
Viewed by 194
Abstract
The mathematical models of physical phenomena are based on the fundamental scientific laws of physics [...] Full article

Research

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Article
One- and Two-Dimensional Analytical Solutions of Thermal Stress for Bimodular Functionally Graded Beams under Arbitrary Temperature Rise Modes
Mathematics 2022, 10(10), 1756; https://doi.org/10.3390/math10101756 - 21 May 2022
Cited by 1 | Viewed by 324
Abstract
In this study, we analytically solved the thermal stress problem of a bimodular functionally graded bending beam under arbitrary temperature rise modes. First, based on the strain suppression method in a one-dimensional case, we obtained the thermal stress of a bimodular functionally graded [...] Read more.
In this study, we analytically solved the thermal stress problem of a bimodular functionally graded bending beam under arbitrary temperature rise modes. First, based on the strain suppression method in a one-dimensional case, we obtained the thermal stress of a bimodular functionally graded beam subjected to bending moment under arbitrary temperature rise modes. Using the stress function method based on compatibility conditions, we also derived two-dimensional thermoelasticity solutions for the same problem under pure bending and lateral-force bending, respectively. During the solving, the number of unknown integration constants is doubled due to the introduction of bimodular effect; thus, the determination for these constants depends not only on the boundary conditions, but also on the continuity conditions at the neutral layer. The comparisons indicate that the one- and two-dimensional thermal stress solutions are consistent in essence, with a slight difference in the axial stress, which exactly reflects the distinctions of one- and two-dimensional problems. In addition, the temperature rise modes in this study are not explicitly indicated, which further expands the applicability of the solutions obtained. The originality of this work is that the one- and two-dimensional thermal stress solutions for bimodular functionally graded beams are derived for the first time. The results obtained in this study may serve as a theoretical reference for the analysis and design of beam-like structures with obvious bimodular functionally graded properties in a thermal environment. Full article
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Article
Influences of Boundary Temperature and Angular Velocity on Thermo-Elastic Characteristics of a Functionally Graded Circular Disk Subjected to Contact Forces
Mathematics 2022, 10(9), 1518; https://doi.org/10.3390/math10091518 - 02 May 2022
Cited by 1 | Viewed by 310
Abstract
The behaviors of functionally graded (FG) engineering structures are influenced by various parameters, such as the boundary temperature, the angular velocity, variations in the thickness, the weight of the structure, and the loading state. The thermo-elastic characteristics of FG rotating circular disks under [...] Read more.
The behaviors of functionally graded (FG) engineering structures are influenced by various parameters, such as the boundary temperature, the angular velocity, variations in the thickness, the weight of the structure, and the loading state. The thermo-elastic characteristics of FG rotating circular disks under the loading of contact forces were investigated. Hooke’s law in plane stress problems was applied to derive a pair of partial differential equations and a finite volume method was developed due to the complexity of the governing equations. The thermo-elastic characteristics of the FG rotating disks were investigated according to the variations in their outer boundary temperature and angular velocity. The increase in the outer boundary temperature caused crack generation at the inner surface of the circular disk and on the opposite side to the loading point. The increase in the angular velocity caused unstable thermo-elastic behaviors near the area of the outer boundary surface, especially at 0.7<(ra)/(ba) < 0.9, and may have led to crack generation at the outer surface of the rotating disk. These results may be applied to the design of functionally graded circular cutters or grinding disks undergoing contact forces to produce proper and reliable thermo-elastic characteristics for practical applications. Full article
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Article
Revisiting the Boundary Value Problem for Uniformly Transversely Loaded Hollow Annular Membrane Structures: Improvement of the Out-of-Plane Equilibrium Equation
Mathematics 2022, 10(8), 1305; https://doi.org/10.3390/math10081305 - 14 Apr 2022
Cited by 1 | Viewed by 295
Abstract
In a previous work by the same authors, a hollow annular membrane structure loaded transversely and uniformly was proposed, and its closed-form solution was presented; its anticipated use is for designing elastic shells of revolution. Since the height–span ratio of shells of revolution [...] Read more.
In a previous work by the same authors, a hollow annular membrane structure loaded transversely and uniformly was proposed, and its closed-form solution was presented; its anticipated use is for designing elastic shells of revolution. Since the height–span ratio of shells of revolution is generally desired to be as large as possible, to meet the need for high interior space, especially for the as-small-as-possible horizontal thrust at the base of shells of revolution, the closed-form solution should be able to cover annular membranes with a large deflection–outer radius ratio. However, the previously presented closed-form solution cannot meet such an ability requirement, because the previous out-of-plane equilibrium equation used the assumption of a small deflection–outer radius ratio. In this study, the out-of-plane equilibrium equation is re-established without the assumption of a small deflection–outer radius ratio, and a new and more refined closed-form solution is presented. The new closed-form solution is numerically discussed regarding its convergence and effectiveness, and compared with the old one. The new and old closed-form solutions agree quite closely for lightly loaded cases but diverge as the load intensifies. Differences in deflections, especially in stresses, are very significant when the deflection–outer radius ratio exceeds 1/4, indicating that, in this case, the new closed-form solution should be adopted in preference to the old one. Full article
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Article
Mathematical Modeling and Analytical Solution of Thermoelastic Stability Problem of Functionally Graded Nanocomposite Cylinders within Different Theories
Mathematics 2022, 10(7), 1081; https://doi.org/10.3390/math10071081 - 28 Mar 2022
Cited by 1 | Viewed by 427
Abstract
Revolutionary advances in technology have led to the use of functionally graded nanocomposite structural elements that operate at high temperatures and whose properties depend on position, such as cylindrical shells designed as load-bearing elements. These advances in technology require new mathematical modeling and [...] Read more.
Revolutionary advances in technology have led to the use of functionally graded nanocomposite structural elements that operate at high temperatures and whose properties depend on position, such as cylindrical shells designed as load-bearing elements. These advances in technology require new mathematical modeling and updated numerical calculations to be performed using improved theories at design time to reliably apply such elements. The main goal of this study is to model, mathematically and within an analytical solution, the thermoelastic stability problem of composite cylinders reinforced by carbon nanotubes (CNTs) under a uniform thermal loading within the shear deformation theory (ST). The influence of transverse shear deformations is considered when forming the fundamental relations of CNT-patterned cylindrical shells and the basic partial differential equations (PDEs) are derived within the modified Donnell-type shell theory. The PDEs are solved by the Galerkin method, and the formula is found for the eigenvalue (critical temperature) of the functionally graded nanocomposite cylindrical shells. The influences of CNT patterns, volume fraction, and geometric parameters on the critical temperature within the ST are estimated by comparing the results within classical theory (CT). Full article
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Article
Analytical Solutions of Viscoelastic Nonlocal Timoshenko Beams
Mathematics 2022, 10(3), 477; https://doi.org/10.3390/math10030477 - 01 Feb 2022
Cited by 2 | Viewed by 369
Abstract
A consistent nonlocal viscoelastic beam model is proposed in this paper. Specifically, a Timoshenko bending problem, where size- and time-dependent effects cannot be neglected, is investigated. In order to inspect scale phenomena, a stress-driven nonlocal formulation is used, whereas to simulate time-dependent effects, [...] Read more.
A consistent nonlocal viscoelastic beam model is proposed in this paper. Specifically, a Timoshenko bending problem, where size- and time-dependent effects cannot be neglected, is investigated. In order to inspect scale phenomena, a stress-driven nonlocal formulation is used, whereas to simulate time-dependent effects, fractional linear viscoelasticity is considered. These two approaches are adopted to develop a new Timoshenko bending model. Analytical solutions and application samples of the proposed formulation are presented. Moreover, in order to show influences of viscoelastic and size effects on mechanical response, parametric analyses are provided. The contributed results can be useful for the design and optimization of small-scale devices exhibiting flexural behaviour. Full article
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Article
k-Version of Finite Element Method for BVPs and IVPs
Mathematics 2021, 9(12), 1333; https://doi.org/10.3390/math9121333 - 09 Jun 2021
Cited by 1 | Viewed by 583
Abstract
The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) [...] Read more.
The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work. Full article
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Article
Direct Collocation with Reproducing Kernel Approximation for Two-Phase Coupling System in a Porous Enclosure
Mathematics 2021, 9(8), 897; https://doi.org/10.3390/math9080897 - 17 Apr 2021
Cited by 2 | Viewed by 507
Abstract
The direct strong-form collocation method with reproducing kernel approximation is introduced to efficiently and effectively solve the natural convection problem within a parallelogrammic enclosure. As the convection behavior in the fluid-saturated porous media involves phase coupling, the resulting system is highly nonlinear in [...] Read more.
The direct strong-form collocation method with reproducing kernel approximation is introduced to efficiently and effectively solve the natural convection problem within a parallelogrammic enclosure. As the convection behavior in the fluid-saturated porous media involves phase coupling, the resulting system is highly nonlinear in nature. To this end, the local approximation is adopted in conjunction with Newton–Raphson method. Nevertheless, to unveil the performance of the method in the nonlinear analysis, only single thermal natural convection is of major concern herein. A unit square is designated as the model problem to investigate the parameters in the system related to the convergence; several benchmark problems are used to verify the accuracy of the approximation, in which the stability of the method is demonstrated by considering various initial conditions, disturbance of discretization, inclination, aspect ratio, and reproducing kernel support size. It is shown that a larger support size can be flexible in approximating highly irregular discretized problems. The derivation of explicit operators with two-phase variables in solving the nonlinear system using the direct collocation is carried out in detail. Full article
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Article
Trigonometric Solution for the Bending Analysis of Magneto-Electro-Elastic Strain Gradient Nonlocal Nanoplates in Hygro-Thermal Environment
Mathematics 2021, 9(5), 567; https://doi.org/10.3390/math9050567 - 07 Mar 2021
Cited by 15 | Viewed by 1030
Abstract
Nanoplates have been extensively utilized in the recent years for applications in nanoengineering as sensors and actuators. Due to their operative nanoscale, the mechanical behavior of such structures might also be influenced by inter-atomic material interactions. For these reasons, nonlocal models are usually [...] Read more.
Nanoplates have been extensively utilized in the recent years for applications in nanoengineering as sensors and actuators. Due to their operative nanoscale, the mechanical behavior of such structures might also be influenced by inter-atomic material interactions. For these reasons, nonlocal models are usually introduced for studying their mechanical behavior. Sensor technology of plate structures should be formulated with coupled mechanics where elastic, magnetic and electric fields interact among themselves. In addition, the effect of hygro-thermal environments are also considered since their presence might effect the nanoplate behavior. In this work a trigonometric approach is developed for investigating smart composite nanoplates using a strain gradient nonlocal procedure. Convergence of the present method is also reported in terms of displacements and electro-magnetic potentials. Results agree well with the literature and open novel applications in this field for further developments. Full article
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