Special Issue "Applied Mathematics and Mechanics"

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

Deadline for manuscript submissions: closed (30 April 2016)

Special Issue Editor

Guest Editor
Prof. Dr. Reza Abedi

Mechanical, Aerospace & Biomedical Engineering, University of Tennessee Knoxville Space Institute, 411 B. H. Goethert Parkway, MS 21, Tullahoma, TN 37388-9700, USA
Website | E-Mail
Interests: Fluid; thermal; solid; fracture mechanics; constitutive models for modern materials; thermodynamics; homogenization theory; multiscale and stochastic approaches; computational mechanics

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

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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

Open AccessArticle
Analysis of Dynamics in Multiphysics Modelling of Active Faults
Mathematics 2016, 4(4), 57; https://doi.org/10.3390/math4040057
Received: 30 April 2016 / Revised: 9 September 2016 / Accepted: 14 September 2016 / Published: 22 September 2016
Cited by 2 | PDF Full-text (6292 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle
Exponential Energy Decay of Solutions for a Transmission Problem With Viscoelastic Term and Delay
Mathematics 2016, 4(2), 42; https://doi.org/10.3390/math4020042
Received: 7 January 2016 / Revised: 3 June 2016 / Accepted: 6 June 2016 / Published: 9 June 2016
Cited by 1 | PDF Full-text (270 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle
Stagnation-Point Flow towards a Stretching Vertical Sheet with Slip Effects
Mathematics 2016, 4(2), 27; https://doi.org/10.3390/math4020027
Received: 23 February 2016 / Revised: 11 April 2016 / Accepted: 18 April 2016 / Published: 21 April 2016
Cited by 6 | PDF Full-text (1187 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle
Solution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method
Mathematics 2016, 4(1), 11; https://doi.org/10.3390/math4010011
Received: 14 December 2015 / Revised: 19 February 2016 / Accepted: 24 February 2016 / Published: 2 March 2016
Cited by 1 | PDF Full-text (1104 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle
A Note on Burg’s Modified Entropy in Statistical Mechanics
Mathematics 2016, 4(1), 10; https://doi.org/10.3390/math4010010
Received: 3 December 2015 / Revised: 2 February 2016 / Accepted: 16 February 2016 / Published: 27 February 2016
Cited by 2 | PDF Full-text (1093 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle
Microtubules Nonlinear Models Dynamics Investigations through the exp(−Φ(ξ))-Expansion Method Implementation
Mathematics 2016, 4(1), 6; https://doi.org/10.3390/math4010006
Received: 24 November 2015 / Revised: 18 January 2016 / Accepted: 20 January 2016 / Published: 4 February 2016
Cited by 9 | PDF Full-text (2933 KB) | HTML Full-text | XML Full-text
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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