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Mathematics
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Applied Mathematics and Mechanics
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Applied Mathematics and Mechanics

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 April 2016) | Viewed by 28400

Special Issue Editor

Dr. Reza Abedi

E-Mail Website
Guest Editor
Department of Mechanical Aerospace and Biomedical Engineering, University of Tennessee Space Institute, Tullahoma, TN 37388, USA
Interests: spacetime discontinuous Galerkin; computational mechanics; fracture mechanics; computational electromagnetics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Advances in technology and materials science have required constitutive modeling of modern materials and the formulation of the computational tools necessary for their analysis. For example, many new designs, such as microelectromechanical and nanoelectromechanical systems (MEMS and NEMS), smart materials, and multi-functional materials, are inherently multiphysical and require rigorous constitutive modeling. Successful experimental demonstrations of negative electrical permittivity, magnetic permeability, effective elastic moduli, and mass density in the so-called metamaterials and extreme solids are other examples that emphasize the importance of classical applied mechanics fields, such as continuum mechanics, in recent years. Of particular importance have been multiscale and homogenization approaches, given the role of specific microstructural designs on the response of modern materials. There has also been a greater emphasis on non-deterministic approaches, given the higher sensitivity of the aforementioned materials to design deviations and the importance of stochastic distribution on small scale features in overall response (for example, in fracture mechanics and turbulence).  Such advances have in turn necessitated the formulation of computational methods capable of the efficient and accurate rendering of these material models. Multiscale and high-order methods, rigorous analysis of numerical errors and efficiency, homogenization schemes, and efficient approaches for the solution of stochastic partial differential equations are but a few of the relevant topics.

Dr. Reza Abedi
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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 2600 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

  • Solid mechanics
  • Fluid mechanics
  • Thermal mechanics
  • Thermodynamics
  • Fracture mechanics
  • Continuum mechanics
  • Constitutive models for modern materials
  • Multiphysics problems
  • Homogenization
  • Multiscale methods
  • Stochastic partial differential equations
  • Computational mechanics including error and efficiency analysis
  • Finite element methods

Published Papers (6 papers)

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Research

6292 KiB  
Article
Analysis of Dynamics in Multiphysics Modelling of Active Faults
by Sotiris Alevizos, Thomas Poulet, Manolis Veveakis and Klaus Regenauer-Lieb
Mathematics 2016, 4(4), 57; https://doi.org/10.3390/math4040057 - 22 Sep 2016
Cited by 3 | Viewed by 4309
Abstract
Instabilities in Geomechanics appear on multiple scales involving multiple physical processes. They appear often as planar features of localised deformation (faults), which can be relatively stable creep or display rich dynamics, sometimes culminating in earthquakes. To study those features, we propose a fundamental [...] Read more.
Instabilities in Geomechanics appear on multiple scales involving multiple physical processes. They appear often as planar features of localised deformation (faults), which can be relatively stable creep or display rich dynamics, sometimes culminating in earthquakes. To study those features, we propose a fundamental physics-based approach that overcomes the current limitations of statistical rule-based methods and allows a physical understanding of the nucleation and temporal evolution of such faults. In particular, we formulate the coupling between temperature and pressure evolution in the faults through their multiphysics energetic process(es). We analyse their multiple steady states using numerical continuation methods and characterise their transient dynamics by studying the time-dependent problem near the critical Hopf points. We find that the global system can be characterised by a homoclinic bifurcation that depends on the two main dimensionless groups of the underlying physical system. The Gruntfest number determines the onset of the localisation phenomenon, while the dynamics are mainly controlled by the Lewis number, which is the ratio of energy diffusion over mass diffusion. Here, we show that the Lewis number is the critical parameter for dynamics of the system as it controls the time evolution of the system for a given energy supply (Gruntfest number). Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics)
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270 KiB  
Article
Exponential Energy Decay of Solutions for a Transmission Problem With Viscoelastic Term and Delay
by Danhua Wang, Gang Li and Biqing Zhu
Mathematics 2016, 4(2), 42; https://doi.org/10.3390/math4020042 - 09 Jun 2016
Cited by 5 | Viewed by 4039
Abstract
In our previous work (Journal of Nonlinear Science and Applications 9: 1202–1215, 2016), we studied the well-posedness and general decay rate for a transmission problem in a bounded domain with a viscoelastic term and a delay term. In this paper, we continue to [...] Read more.
In our previous work (Journal of Nonlinear Science and Applications 9: 1202–1215, 2016), we studied the well-posedness and general decay rate for a transmission problem in a bounded domain with a viscoelastic term and a delay term. In this paper, we continue to study the similar problem but without the frictional damping term. The main difficulty arises since we have no frictional damping term to control the delay term in the estimate of the energy decay. By introducing suitable energy and Lyapunov functionals, we establish an exponential decay result for the energy. Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics)
1187 KiB  
Article
Stagnation-Point Flow towards a Stretching Vertical Sheet with Slip Effects
by Khairy Zaimi and Anuar Ishak
Mathematics 2016, 4(2), 27; https://doi.org/10.3390/math4020027 - 21 Apr 2016
Cited by 24 | Viewed by 4208
Abstract
The effects of partial slip on stagnation-point flow and heat transfer due to a stretching vertical sheet is investigated. Using a similarity transformation, the governing partial differential equations are reduced into a system of nonlinear ordinary differential equations. The resulting equations are solved [...] Read more.
The effects of partial slip on stagnation-point flow and heat transfer due to a stretching vertical sheet is investigated. Using a similarity transformation, the governing partial differential equations are reduced into a system of nonlinear ordinary differential equations. The resulting equations are solved numerically using a shooting method. The effect of slip and buoyancy parameters on the velocity, temperature, skin friction coefficient and the local Nusselt number are graphically presented and discussed. It is found that dual solutions exist in a certain range of slip and buoyancy parameters. The skin friction coefficient decreases while the Nusselt number increases as the slip parameter increases. Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics)
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1104 KiB  
Article
Solution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method
by H. M. Abdelhafez
Mathematics 2016, 4(1), 11; https://doi.org/10.3390/math4010011 - 02 Mar 2016
Cited by 6 | Viewed by 5115
Abstract
The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate [...] Read more.
The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obtained using the fourth-order Runge-Kutta numerical solution method. A comparison of the result by the numerical Runge-Kutta fourth-order accuracy method is compared with the result by the MDTM and plotted in a long time domain. Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics)
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1093 KiB  
Article
A Note on Burg’s Modified Entropy in Statistical Mechanics
by Amritansu Ray and S. K. Majumder
Mathematics 2016, 4(1), 10; https://doi.org/10.3390/math4010010 - 27 Feb 2016
Cited by 2 | Viewed by 4574
Abstract
Burg’s entropy plays an important role in this age of information euphoria, particularly in understanding the emergent behavior of a complex system such as statistical mechanics. For discrete or continuous variable, maximization of Burg’s Entropy subject to its only natural and mean constraint [...] Read more.
Burg’s entropy plays an important role in this age of information euphoria, particularly in understanding the emergent behavior of a complex system such as statistical mechanics. For discrete or continuous variable, maximization of Burg’s Entropy subject to its only natural and mean constraint always provide us a positive density function though the Entropy is always negative. On the other hand, Burg’s modified entropy is a better measure than the standard Burg’s entropy measure since this is always positive and there is no computational problem for small probabilistic values. Moreover, the maximum value of Burg’s modified entropy increases with the number of possible outcomes. In this paper, a premium has been put on the fact that if Burg’s modified entropy is used instead of conventional Burg’s entropy in a maximum entropy probability density (MEPD) function, the result yields a better approximation of the probability distribution. An important lemma in basic algebra and a suitable example with tables and graphs in statistical mechanics have been given to illustrate the whole idea appropriately. Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics)
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2933 KiB  
Article
Microtubules Nonlinear Models Dynamics Investigations through the exp(−Φ(ξ))-Expansion Method Implementation
by Nur Alam and Fethi Bin Muhammad Belgacem
Mathematics 2016, 4(1), 6; https://doi.org/10.3390/math4010006 - 04 Feb 2016
Cited by 39 | Viewed by 5048
Abstract
In this research article, we present exact solutions with parameters for two nonlinear model partial differential equations(PDEs) describing microtubules, by implementing the exp(−Φ(ξ))-Expansion Method. The considered models, describing highly nonlinear dynamics of microtubules, can be reduced to nonlinear ordinary differential equations. While the [...] Read more.
In this research article, we present exact solutions with parameters for two nonlinear model partial differential equations(PDEs) describing microtubules, by implementing the exp(−Φ(ξ))-Expansion Method. The considered models, describing highly nonlinear dynamics of microtubules, can be reduced to nonlinear ordinary differential equations. While the first PDE describes the longitudinal model of nonlinear dynamics of microtubules, the second one describes the nonlinear model of dynamics of radial dislocations in microtubules. The acquired solutions are then graphically presented, and their distinct properties are enumerated in respect to the corresponding dynamic behavior of the microtubules they model. Various patterns, including but not limited to regular, singular kink-like, as well as periodicity exhibiting ones, are detected. Being the method of choice herein, the exp(−Φ(ξ))-Expansion Method not disappointing in the least, is found and declared highly efficient. Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics)
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