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Mathematics
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Mathematical Problems in Materials Science
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Mathematical Problems in Materials Science

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

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 15995

Special Issue Editor

Prof. Dr. David Bassir

E-Mail Website
Guest Editor
1. Centre de mathématiques et de leurs applications, CNRS, ENS Paris-Saclay, Université Paris-Saclay, 94235 Cachan, CEDEX, France
2. UBFC, UTBM, UMR–CNRS 5060, Rue de Leupe, 90010 Belfort, CEDEX, France
Interests: scientific research activities are related to simulation and optimisation of materials and structures using advanced decision strategies based on heuristic methods; artificial neural networks; artificial intelligence to solve industrials; modelling at different scales macro to nano scales
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nowadays, material science has arrived into a new Era, where mathematics is a real challenge to get to the next level of understanding and predicting the material behaviours. These challenges are often focusing on the building of numerical models for materials at different scales from Macro, Meso to Nano scales that can help to predict the material responses efficiently. Metamodels often need to be updated to fit the experimental analysis and testing. Many approaches combined to finite element analysis are used at this stage to solve the parameter identification problems for instance. Among these methods : neural network, optimization methods either based on gradient or heuristic methods, artificial intelligence, deep learning, hybrid et multiscale analysis, inverse problem, image processing etc.

The purpose of this Special Issue is to gather a collection of articles presenting recent research dealing with Mathematical Problems in Materials Science. The contribution should underline the mathematical problem and the solution. Industrial applications that show the mathematical aspects in Materials Science are also welcome.

Prof. Dr. David Bassir
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

  • Material science
  • Applied mathematics
  • Modelling
  • Simulation
  • Parameter identification & Inverse problem
  • Multiscale analysis
  • Optimization
  • Artificial intelligent
  • Additive manufacturing
  • Composites materials
  • Bio-inspired methods
  • Characterisation
  • Computational materials science
  • Stochastic optimization techniques
  • Genetic algorithm
  • Surface response & Metamodel
  • Particle swarm method
  • Image processing
  • Mathematical problems in materials science
  • Structural mechanics

Published Papers (7 papers)

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Research

16 pages, 6841 KiB  
Article
Mathematical Modeling and Analysis Using Nondimensionalization Technique of the Solidification of a Splat of Variable Section
by Juan Francisco Sánchez-Pérez, Guillermo Jorde-Cerezo, Adrián Fernández-Roiz and José Andrés Moreno-Nicolás
Mathematics 2023, 11(14), 3174; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143174 - 19 Jul 2023
Cited by 1 | Viewed by 844
Abstract
In this work, the solidification and cooling process of an irregularly shaped splat is modeled using the network simulation method. The procedure for its implementation, which uses the finite difference method and optimized circuit analysis algorithms, allows the precise incorporation and assessment of [...] Read more.
In this work, the solidification and cooling process of an irregularly shaped splat is modeled using the network simulation method. The procedure for its implementation, which uses the finite difference method and optimized circuit analysis algorithms, allows the precise incorporation and assessment of the effect of certain conditions in the thermal process, offering its specificity and high performance in numerical simulation. It should be noted, on the one hand, that the geometry used for the simulation has been obtained from experimental splat data visualized using an electron microscope and, on the other hand, that the model implements both the phase change phenomenon and the variability of the material properties with temperature. Finally, the study of the physical behavior of the problem is carried out using the mathematical technique of nondimensionalization, allowing the interpretation of the results obtained by simulation, where the formation of horizontal bars and columns that maintain the structure of the splat while the solidification process is taking place stands out. It is worth highlighting the obtaining, among others, of two monomials. The first relates the phenomenon of radiation to conduction and is equivalent to the Nusselt number with convection, and the second relates the solidification time with the Stefan number. Full article
(This article belongs to the Special Issue Mathematical Problems in Materials Science)
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12 pages, 314 KiB  
Article
Solving Nonlinear Boundary Value Problems Using the Higher Order Haar Wavelet Method
by Mart Ratas, Jüri Majak and Andrus Salupere
Mathematics 2021, 9(21), 2809; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212809 - 05 Nov 2021
Cited by 14 | Viewed by 2165
Abstract
The current study is focused on development and adaption of the higher order Haar wavelet method for solving nonlinear ordinary differential equations. The proposed approach is implemented on two sample problems—the Riccati and the Liénard equations. The convergence and accuracy of the proposed [...] Read more.
The current study is focused on development and adaption of the higher order Haar wavelet method for solving nonlinear ordinary differential equations. The proposed approach is implemented on two sample problems—the Riccati and the Liénard equations. The convergence and accuracy of the proposed higher order Haar wavelet method are compared with the widely used Haar wavelet method. The comparison of numerical results with exact solutions is performed. The complexity issues of the higher order Haar wavelet method are discussed. Full article
(This article belongs to the Special Issue Mathematical Problems in Materials Science)
27 pages, 2580 KiB  
Article
Large Deformation Problem of Bimodular Functionally-Graded Thin Circular Plates Subjected to Transversely Uniformly-Distributed Load: Perturbation Solution without Small-Rotation-Angle Assumption
by Xue Li, Xiao-Ting He, Jie-Chuan Ai and Jun-Yi Sun
Mathematics 2021, 9(18), 2317; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182317 - 18 Sep 2021
Cited by 4 | Viewed by 1528
Abstract
In this study, the large deformation problem of a functionally-graded thin circular plate subjected to transversely uniformly-distributed load and with different moduli in tension and compression (bimodular property) is theoretically analyzed, in which the small-rotation-angle assumption, commonly used in the classical Föppl–von Kármán [...] Read more.
In this study, the large deformation problem of a functionally-graded thin circular plate subjected to transversely uniformly-distributed load and with different moduli in tension and compression (bimodular property) is theoretically analyzed, in which the small-rotation-angle assumption, commonly used in the classical Föppl–von Kármán equations of large deflection problems, is abandoned. First, based on the mechanical model on the neutral layer, the bimodular functionally-graded property of materials is modeled as two different exponential functions in the tensile and compressive zones. Thus, the governing equations of the large deformation problem are established and improved, in which the equation of equilibrium is derived without the common small-rotation-angle assumption. Taking the central deflection as a perturbation parameter, the perturbation method is used to solve the governing equations, thus the perturbation solutions of deflection and stress are obtained under different boundary constraints and the regression of the solution is satisfied. Results indicate that the perturbation solutions presented in this study have higher computational accuracy in comparison with the existing perturbation solutions with small-rotation-angle assumption. Specially, the computational accuracies of external load and yield stress are improved by 17.22% and 28.79% at most, respectively, by the numerical examples. In addition, the small-rotation-angle assumption has a great influence on the yield stress at the center of the bimodular functionally-graded circular plate. Full article
(This article belongs to the Special Issue Mathematical Problems in Materials Science)
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16 pages, 3850 KiB  
Article
A Refined Theory for Bending Vibratory Analysis of Thick Functionally Graded Beams
by Youssef Boutahar, Nadhir Lebaal and David Bassir
Mathematics 2021, 9(12), 1422; https://0-doi-org.brum.beds.ac.uk/10.3390/math9121422 - 18 Jun 2021
Cited by 7 | Viewed by 1879
Abstract
A refined beam theory that takes the thickness-stretching into account is presented in this study for the bending vibratory behavior analysis of thick functionally graded (FG) beams. In this theory, the number of unknowns is reduced to four instead of five in the [...] Read more.
A refined beam theory that takes the thickness-stretching into account is presented in this study for the bending vibratory behavior analysis of thick functionally graded (FG) beams. In this theory, the number of unknowns is reduced to four instead of five in the other approaches. Transverse displacement is expressed through a hyperbolic function and subdivided into bending, shear, and thickness-stretching components. The number of unknowns is reduced, which involves a decrease in the number of the governing equation. The boundary conditions at the top and bottom FG beam faces are satisfied without any shear correction factor. According to a distribution law, effective characteristics of FG beam material change continuously in the thickness direction depending on the constituent’s volume proportion. Equations of motion are obtained from Hamilton’s principle and are solved by assuming the Navier’s solution type, for the case of a supported FG beam that is transversely loaded. The numerical results obtained are exposed and analyzed in detail to verify the validity of the current theory and prove the influence of the material composition, geometry, and shear deformation on the vibratory responses of FG beams, showing the impact of normal deformation on these responses which is neglected in most of the beam theories. The obtained results are compared with those predicted by other beam theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of FG beams. Full article
(This article belongs to the Special Issue Mathematical Problems in Materials Science)
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26 pages, 3238 KiB  
Article
Closed-Form Solution for Circular Membranes under In-Plane Radial Stretching or Compressing and Out-of-Plane Gas Pressure Loading
by Bin-Bin Shi, Jun-Yi Sun, Ting-Kai Huang and Xiao-Ting He
Mathematics 2021, 9(11), 1238; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111238 - 28 May 2021
Cited by 5 | Viewed by 2880
Abstract
The large deflection phenomenon of an initially flat circular membrane under out-of-plane gas pressure loading is usually involved in many technical applications, such as the pressure blister or bulge tests, where a uniform in-plane stress is often present in the initially flat circular [...] Read more.
The large deflection phenomenon of an initially flat circular membrane under out-of-plane gas pressure loading is usually involved in many technical applications, such as the pressure blister or bulge tests, where a uniform in-plane stress is often present in the initially flat circular membrane before deflection. However, there is still a lack of an effective closed-form solution for the large deflection problem with initial uniform in-plane stress. In this study, the problem is formulated and is solved analytically. The initial uniform in-plane stress is first modelled by stretching or compressing an initially flat, stress-free circular membrane radially in the plane in which the initially flat circular membrane is located, and based on this, the boundary conditions, under which the large deflection problem of an initially flat circular membrane under in-plane radial stretching or compressing and out-of-plane gas pressure loading can be solved, are determined. Therefore, the closed-form solution presented in this paper can be applied to the case where the initially flat circular membrane may, or may not, have a uniform in-plane stress before deflection, and the in-plane stress can be either tensile or compressive. The numerical example conducted shows that the closed-form solution presented has satisfactory convergence. Full article
(This article belongs to the Special Issue Mathematical Problems in Materials Science)
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18 pages, 2870 KiB  
Article
Method to Determine the Constitutive Permeability Parameters of Non-Linear Consolidation Models by Means of the Oedometer Test
by Gonzalo García-Ros and Iván Alhama
Mathematics 2020, 8(12), 2237; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122237 - 17 Dec 2020
Cited by 5 | Viewed by 2463
Abstract
This paper presents an easy-to-apply methodology that allows obtaining the permeability index and the initial hydraulic conductivity of clayey soils, basic constitutive parameters in non-linear models of consolidation, based on the laboratory oedometer test. For this, the data of the void ratio, compressibility [...] Read more.
This paper presents an easy-to-apply methodology that allows obtaining the permeability index and the initial hydraulic conductivity of clayey soils, basic constitutive parameters in non-linear models of consolidation, based on the laboratory oedometer test. For this, the data of the void ratio, compressibility index and characteristic consolidation time are taken from the test and, as an inverse problem, the constitutive permeability parameters sought are determined by applying the universal solutions of the characteristic time for a general non-linear consolidation model with constitutive relations void ratio-effective soil stress and hydraulic conductivity-void ratio of logarithmic type. The application protocol of the inverse problem is described in detail and illustrated by a series of applications carried out on real laboratory data belonging to two different soils. The influence that errors in laboratory parameter measurements can have on the final values of the permeability index and initial hydraulic conductivity is studied, showing the maximum deviations that may appear and, by last, the precision of the results obtained. Full article
(This article belongs to the Special Issue Mathematical Problems in Materials Science)
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24 pages, 4085 KiB  
Article
Statistical Approach for the Design of Structural Self-Compacting Concrete with Fine Recycled Concrete Aggregate
by Víctor Revilla-Cuesta, Marta Skaf, Ana B. Espinosa, Amaia Santamaría and Vanesa Ortega-López
Mathematics 2020, 8(12), 2190; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122190 - 09 Dec 2020
Cited by 28 | Viewed by 2205
Abstract
The compressive strength of recycled concrete is acknowledged to be largely conditioned by the incorporation ratio of Recycled Concrete Aggregate (RCA), although that ratio needs to be carefully assessed to optimize the design of structural applications. In this study, Self-Compacting Concrete (SCC) mixes [...] Read more.
The compressive strength of recycled concrete is acknowledged to be largely conditioned by the incorporation ratio of Recycled Concrete Aggregate (RCA), although that ratio needs to be carefully assessed to optimize the design of structural applications. In this study, Self-Compacting Concrete (SCC) mixes containing 100% coarse RCA and variable amounts, between 0% and 100%, of fine RCA were manufactured and their compressive strengths were tested in the laboratory for a statistical analysis of their strength variations, which exhibited robustness and normality according to the common statistical procedures. The results of the confidence intervals, the one-factor ANalysis Of VAriance (ANOVA), and the Kruskal–Wallis test showed that an increase in fine RCA content did not necessarily result in a significant decrease in strength, although the addition of fine RCA delayed the development of the final strength. The statistical models presented in this research can be used to define the optimum incorporation ratio that would produce the highest compressive strength. Furthermore, the multiple regression models offered accurate estimations of compressive strength, considering the interaction between the incorporation ratio of fine RCA and the curing age of concrete that the two-factor ANOVA revealed. Lastly, the probability distribution predictions, obtained through a log-likelihood analysis, fitted the results better than the predictions based on current standards, which clearly underestimated the compressive strength of SCC manufactured with fine RCA and require adjustment to take full advantage of these recycled materials. This analysis could be carried out on any type of waste and concrete, which would allow one to evaluate the same aspects as in this research and ensure that the use of recycled concrete maximizes both sustainability and strength. Full article
(This article belongs to the Special Issue Mathematical Problems in Materials Science)
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