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A special issue of Materials (ISSN 1996-1944).

Deadline for manuscript submissions: closed (30 June 2020) | Viewed by 38462

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Prof. Dr. Tomasz Strek

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Guest Editor

Institute of Applied Mechanics, Poznan University of Technology, 60-965 Poznan, Poland
Interests: computational mechanics; auxetic; smart materials; finite element analysis; modeling and simulation
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Dr. Hubert Jopek

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Institute of Applied Mechanics, Poznan University of Technology, Poznan, Poland
Interests: mechanics of materials; metamaterials; and smart materials modeling; auxetics; composites; topological optimization; computational methods in mechanics
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Prof. Dr. Zoran Ren

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Guest Editor

Mechanical Engineering, University of Maribor, 2000 Maribor, Slovenia
Interests: advanced computational modeling and simulation; high-performance computing; impact and high velocity phenomena; material strain-rate dependency; advanced geometrical characterization and analysis; computational materials engineering of cellular and multi-functional materials
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

This Special Issue of Materials is devoted to analytical and computational methods in material and mechanical engineering. Among others, the following topics are the main fields of interest of this Issue: linear and non-linear behavior elasticity and plasticity models; materials with anomalous characteristics; metamaterials; auxetic cellular materials; porous materials; functionally graded materials, fatigue of materials; the topological optimization of structures; heat transfer in materials and structures; as well as other topics related to computational methods in materials, mechanics, and engineering.

We invite you to submit research articles on the latest research work in these areas, with an emphasis on applications in all areas of materials and mechanics engineering.

Prof. Tomasz Strek
Dr. Hubert Jopek
Prof. Zoran Ren
Guest Editors

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. Materials 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

Computational materials engineering
Auxetic cellular materials
Material strain-rate dependency
Advanced geometrical characterization and analysis of porous materials
Solid and structural mechanics
Fluid mechanics
Mechanics of materials
Heat transfer
Dynamics
Biomechanics
Molecular dynamics.

Published Papers (14 papers)

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Research

20 pages, 2836 KiB

Open AccessArticle

Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation

by Jarosław Jędrysiak and Magda Kaźmierczak-Sobińska

Materials 2020, 13(18), 4031; https://doi.org/10.3390/ma13184031 - 11 Sep 2020

Cited by 7 | Viewed by 1519

Abstract

In this paper, the problem of the stability of functionally graded thin plates with a microstructure is presented. To analyse this problem and take into consideration the effect of microstructure, tolerance modelling is used. The tolerance averaging technique allows us to replace the equation with non-continuous, tolerance-periodic, highly oscillating coefficients of the system of differential equations with slowly-varying coefficients, which describes also the effect of the microstructure. As an example, the buckling of a microstructured functionally graded plate band on a foundation is investigated. To obtain results, the tolerance model and the asymptotic model combined together with the Ritz method are used. It is shown that the tolerance model allows us to take into account the effect of microstructure on critical forces. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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16 pages, 919 KiB

Open AccessArticle

Tolerance Modelling of Vibrations and Stability for Periodic Slender Visco-Elastic Beams on a Foundation with Damping. Revisiting

by Jarosław Jędrysiak

Materials 2020, 13(18), 3939; https://doi.org/10.3390/ma13183939 - 06 Sep 2020

Cited by 9 | Viewed by 1635

Abstract

The mathematical modelling of certain problems of vibrations and stability for periodic slender visco-elastic beams is presented in this note. To consider these problems and take into account the effect of the microstructure, the tolerance modelling approach is proposed. Using this technique, the equation with non-continuous, periodic, highly oscillating coefficients is replaced by a system of differential equations with constant coefficients. Moreover, these governing equations describe the effect of the microstructure on the overall behavior of the beams under consideration. The tolerance modelling can lead to equations of two different tolerance models—the standard and the general, under weakened assumptions. This averaging tolerance method was assessed by comparison with the asymptotic homogenization, the governing equations of which omit this effect. My considerations were limited to proposing and presenting only mathematical models describing investigated beams. In a simple analytical example, the application of the presented average models is shown. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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21 pages, 1476 KiB

Open AccessArticle

Numerical Study of CH₄ Generation and Transport in XLPE-Insulated Cables in Continuous Vulcanization

by Mohd Fuad Anwari Che Ruslan, Dong Joon Youn, Roshan Aarons, Yabin Sun and Shuyu Sun

Materials 2020, 13(13), 2978; https://doi.org/10.3390/ma13132978 - 03 Jul 2020

Cited by 3 | Viewed by 2356

Abstract

In this work, we apply a computational diffusion model based on Fick’s laws to study the generation and transport of methane (CH

_{4}

) during the production of a cross-linked polyethylene (XLPE) insulated cable. The model takes into account the heating process in [...] Read more.

In this work, we apply a computational diffusion model based on Fick’s laws to study the generation and transport of methane (CH

_{4}

) during the production of a cross-linked polyethylene (XLPE) insulated cable. The model takes into account the heating process in a curing tube where most of the cross-linking reaction occurs and the subsequent two-stage cooling process, with water and air as the cooling media. For the calculation of CH

_{4}

generation, the model considers the effect of temperature on the cross-linking reaction selectivity. The cross-linking reaction selectivity is a measure of the preference of cumyloxy to proceed either with a hydrogen abstraction reaction, which produces cumyl alcohol, or with a

β

-scission reaction, which produces acetophenone and CH

_{4}

. The simulation results show that, during cable production, a significant amount of CH

_{4}

is generated in the XLPE layer, which diffuses out of the cable and into the conductor part of the cable. Therefore, the diffusion pattern becomes a non-uniform radial distribution of CH

_{4}

at the cable take-up point, which corresponds well with existing experimental data. Using the model, we perform a series of parametric studies to determine the effect of the cable production conditions, such as the curing temperature, line speed, and cooling water flow rate, on CH

_{4}

generation and transport during cable production. The results show that the curing temperature has the largest impact on the amount of CH

_{4}

generated and its distribution within the cable. We found that under similar curing and cooling conditions, varying the line speed induces a notable effect on the CH

_{4}

transport within the cable, while the cooling water flow rate had no significant impact. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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12 pages, 3757 KiB

Open AccessFeature PaperArticle

An Experimental and Computational Study of the High-Velocity Impact of Low-Density Aluminum Foam

by Matej Borovinšek, Matej Vesenjak, Kazuyuki Hokamoto and Zoran Ren

Materials 2020, 13(8), 1949; https://doi.org/10.3390/ma13081949 - 21 Apr 2020

Cited by 7 | Viewed by 2016

Abstract

The study presents the results of an experimental and computational study of the high-velocity impact of low-density aluminum foam into a rigid wall. It is shown that the aluminum foam samples deformed before hitting the rigid wall because of the high inertial forces during the acceleration. During the impact, the samples deformed only in the region contacting the rigid wall due to the high impact velocity; the inertial effects dominated the deformation process. However, the engineering stress–strain relationship retains its typical plateau shape until the densification strain. The experimental tests were successfully reproduced with parametric computer simulations using the LS-DYNA explicit finite element code. A unique computational lattice-type model was used, which can reproduce the randomness of the irregular, open-cell structure of aluminum foams. Parametric computer simulations of twenty different aluminum foam sample models with randomly generated irregular lattice structures were carried out at different acceleration levels to obtain representative statistical results. The high strain-rate sensitivity of low-density aluminum foam was also observed. A comparison of experimental and computational results during aluminum foam sample impact shows very similar deformation behavior. The computational model correctly represents the real impact conditions of low-density aluminum foam and can be recommended for use in similar high-velocity impact investigations. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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19 pages, 7235 KiB

Open AccessArticle

Mathematical Modeling of Breast Tumor Destruction Using Fast Heating during Radiofrequency Ablation

by Marek Paruch

Materials 2020, 13(1), 136; https://doi.org/10.3390/ma13010136 - 28 Dec 2019

Cited by 38 | Viewed by 3868

Abstract

In oncology, hyperthermia is understood as a planned, controlled technique of heating cancerous changes in order to destroy their cells or stop their growth. In clinical practice, hyperthermia is used in combination with radiotherapy, chemotherapy, or immunological therapy. During the hyperthermia, the tissue is typically exposed to a temperature in the range of 40–45 °C, the exception is thermoablation, during which the temperatures reach much higher values. Thermoablation is characterized by the use of high temperatures up to 90 °C. The electrode using the radiofrequency is inserted into the central area of the tumor. Interstitial thermoablation is used to treat, among others, breast and brain cancer. The therapy consists of inducing coagulation necrosis in an area that is heated to very high temperatures. Mathematical modeling is based on the use of a coupled thermo-electric model, in which the electric field is described by means of the Laplace equation, while the temperature field is based on the Pennes equation. Coupling occurs at the level of the additional source function in the Pennes equation. The temperature field obtained in this way makes it possible to calculate the Arrhenius integral as a determinant of the destruction of biological tissue. As a result of numerical calculations regarding the temperature field and the Arrhenius integral, it can be concluded that, with the help of numerical tools and mathematical modeling, one can simulate the process of destroying cancerous tissue. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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16 pages, 3727 KiB

Open AccessArticle

Compressive Behaviour of Closed-Cell Aluminium Foam at Different Strain Rates

by Nejc Novak, Matej Vesenjak, Isabel Duarte, Shigeru Tanaka, Kazuyuki Hokamoto, Lovre Krstulović-Opara, Baoqiao Guo, Pengwan Chen and Zoran Ren

Materials 2019, 12(24), 4108; https://doi.org/10.3390/ma12244108 - 09 Dec 2019

Cited by 28 | Viewed by 3182

Abstract

Closed-cell aluminium foams were fabricated and characterised at different strain rates. Quasi-static and high strain rate experimental compression testing was performed using a universal servo-hydraulic testing machine and powder gun. The experimental results show a large influence of strain rate hardening on mechanical properties, which contributes to significant quasi-linear enhancement of energy absorption capabilities at high strain rates. The results of experimental testing were further used for the determination of critical deformation velocities and validation of the proposed computational model. A simple computational model with homogenised crushable foam material model shows good correlation between the experimental and computational results at analysed strain rates. The computational model offers efficient (simple, fast and accurate) analysis of high strain rate deformation behaviour of a closed-cell aluminium foam at different loading velocities. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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24 pages, 6283 KiB

Open AccessArticle

Three-Dimensional Digital Reconstruction of Ti₂AlC Ceramic Foams Produced by the Gelcast Method

by Christos S. Stiapis, Eugene D. Skouras and Vasilis N. Burganos

Materials 2019, 12(24), 4085; https://doi.org/10.3390/ma12244085 - 06 Dec 2019

Cited by 4 | Viewed by 2467

Abstract

A digital reconstruction technique is presented that generates three-dimensional (3D) digital representations of ceramic foams created by the foam-gelcasting technique. The reconstruction process uses information that is directly extracted from Scanning Electron Microscopy (SEM) images and offers a 3D representation of the physical sample accounting for the typically large pore cavities and interconnecting windows that are formed during the preparation process. Contrary to typical tessellation-based foam treatments, a spherical representation of the pores and the pore windows of the foams is assumed and a novel hybrid algorithm that combines a variation of Lubachevsky-type and Random Close Packing of Hard Spheres (RCPHS) algorithms has been developed to obtain near-optimum solutions to the packing problem of the spheres that represent the pores. Numerical simulations are performed directly on the 3D reconstructed foams to determine their gas permeability. The model predictions are compared with experimental gas permeability data that were obtained for the physical samples. The pore wall thickness can be treated as the single fitting parameter in the entire reconstruction process, although it is shown that images of sufficient resolution could eliminate the need even for that. The foams that are produced by this method yield quantitatively similar pressure drops with experiments for various superficial velocity values, with a very small deviation in the range of 1.7–2.8%. The proposed methodology could be utilized for the prediction of the permeability and transport properties of complex foamy porous structures, similar to the gelcast-type of foams, from a single SEM image of the foam sample without resorting to serial tomography or other structural information, thus saving considerable time and effort from experimental work. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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15 pages, 4644 KiB

Open AccessArticle

Effects of Servo Tensile Test Parameters on Mechanical Properties of Medium-Mn Steel

by Xuemin Chi and Shuo Han

Materials 2019, 12(22), 3793; https://doi.org/10.3390/ma12223793 - 19 Nov 2019

Cited by 4 | Viewed by 2467

Abstract

As a new type of third-generation automotive steel with high strength and plasticity, medium-Mn steel (MMnS) has been widely used in automotive industries for its excellent properties. In recent years, servo stamping technology for high-strength metal forming is a hot topic due to its good performance in forming under complex processing conditions, and servo parameters determine the forming quality. In this paper, experiments considering tensile speed and position where speed changes (PSC) were carried out on MMnS to investigate the influences of tensile parameters on mechanical properties including strength and total elongation (TE). The results show that PSC does not significantly impact total elongation. Initial tensile speed (ITS) and final tensile speed (FTS) significantly impact the total elongation. The interaction between all tensile parameters can impact total elongation. Two artificial neural networks, back propagation neural network (BPNN) and radial basis function neural network (RBFNN), were used to establish analytical models. The results of supplemental experiment and residual analysis were conducted to verify the accuracy of the analytical models. The BPNN has a better performance and the analytical model shows that with the increase of PSC, it has a slight impact on the changes of optimal and minimum total elongation, but the combinations of tensile parameters to obtain total elongations higher than 40% change significantly. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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21 pages, 5468 KiB

Open AccessArticle

Creep-Based Reliability Evaluation of Turbine Blade-Tip Clearance with Novel Neural Network Regression

by Chun-Yi Zhang, Jing-Shan Wei, Ze Wang, Zhe-Shan Yuan, Cheng-Wei Fei and Cheng Lu

Materials 2019, 12(21), 3552; https://doi.org/10.3390/ma12213552 - 29 Oct 2019

Cited by 18 | Viewed by 2793

Abstract

To reveal the effect of high-temperature creep on the blade-tip radial running clearance of aeroengine high-pressure turbines, a distributed collaborative generalized regression extremum neural network is proposed by absorbing the heuristic thoughts of distributed collaborative response surface method and the generalized extremum neural network, in order to improve the reliability analysis of blade-tip clearance with creep behavior in terms of modeling precision and simulation efficiency. In this method, the generalized extremum neural network was used to handle the transients by simplifying the response process as one extremum and to address the strong nonlinearity by means of its nonlinear mapping ability. The distributed collaborative response surface method was applied to handle multi-object multi-discipline analysis, by decomposing one “big” model with hyperparameters and high nonlinearity into a series of “small” sub-models with few parameters and low nonlinearity. Based on the developed method, the blade-tip clearance reliability analysis of an aeroengine high-pressure turbine was performed subject to the creep behaviors of structural materials, by considering the randomness of influencing parameters such as gas temperature, rotational speed, material parameters, convective heat transfer coefficient, and so forth. It was found that the reliability degree of the clearance is 0.9909 when the allowable value is 2.2 mm, and the creep deformation of the clearance presents a normal distribution with a mean of 1.9829 mm and a standard deviation of 0.07539 mm. Based on a comparison of the methods, it is demonstrated that the proposed method requires a computing time of 1.201 s and has a computational accuracy of 99.929% over 10⁴ simulations, which are improvements of 70.5% and 1.23%, respectively, relative to the distributed collaborative response surface method. Meanwhile, the high efficiency and high precision of the presented approach become more obvious with the increasing simulations. The efforts of this study provide a promising approach to improve the dynamic reliability analysis of complex structures. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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15 pages, 2613 KiB

Open AccessArticle

Inverse Model for the Control of Induction Heat Treatments

by Mohammad Zhian Asadzadeh, Peter Raninger, Petri Prevedel, Werner Ecker and Manfred Mücke

Materials 2019, 12(17), 2826; https://doi.org/10.3390/ma12172826 - 02 Sep 2019

Cited by 10 | Viewed by 2372

Abstract

In this work, we present and test an approach based on an inverse model applicable to the control of induction heat treatments. The inverse model is comprised of a simplified analytical forward model trained with experiments to predict and control the temperature of a location in a cylindrical sample starting from any initial temperature. We solve the coupled nonlinear electromagnetic-thermal problem, which contains a temperature dependent parameter

α

to correct the electromagnetic field on the surface of a cylinder, and as a result effectively the modeled temperature elsewhere in the sample. A calibrated model to the measurement data applied with the process information such as the operating power level, current, frequency, and temperature provides the basic ingredients to construct an inverse model toolbox, which finally enables us to conduct experiments with more specific goals. The input set values of the power supply, i.e., the power levels in the test rig control system, are determined within an iterative framework to reach specific target temperatures in prescribed times. We verify the concept on an induction heating test rig and provide two examples to illustrate the approach. The advantages of the method lie in its simplicity, computationally cost effectiveness and independence of a prior knowledge of the internal structure of power supplies. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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12 pages, 2626 KiB

Open AccessArticle

Influence of Strengthening Material Behavior and Geometry Parameters on Mechanical Behavior of Biaxial Cruciform Specimen for Envelope Material

by Zhipeng Qu, Houdi Xiao, Mingyun Lv, Xihe Wang, Pengfei Wang and Lei Xu

Materials 2019, 12(17), 2680; https://doi.org/10.3390/ma12172680 - 22 Aug 2019

Cited by 3 | Viewed by 2059

Abstract

The stratospheric airship envelope material is operated in biaxial stress, so it is necessary to study the in-plane biaxial tensile strength. In this paper, a theoretical model is developed to evaluate the mechanical properties of in-plane biaxial specimens. Being applied to the finite element analysis, the theoretical model is employed to evaluate the influence of strengthening material behavior (E*) and geometry parameters on the mechanical behavior in the central. The follows results are drawn: (i) smaller the length of the central region (L_cen), E* and larger the central region corner radius (r) contribute to smaller coefficient of variation (CV); (ii) smaller L_cen and larger E* contribute to smaller stress concentration factor (k), k in the limit state of r is larger than that in other conditions. (iii) The CV and k under stress ratio of 1:1 are smaller than those under other stress ratios. The study can provide a useful reference for the design of biaxial specimens. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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25 pages, 21086 KiB

Open AccessArticle

The Development of a New Shock Absorbing Uniaxial Graded Auxetic Damper (UGAD)

by Hasan Al-Rifaie and Wojciech Sumelka

Materials 2019, 12(16), 2573; https://doi.org/10.3390/ma12162573 - 12 Aug 2019

Cited by 40 | Viewed by 4504

Abstract

Auxetic structures are efficient cellular materials that can absorb blast/impact energy through plastic deformation, thus protecting the structure. They are developing sacrificial solutions with light weight, high specific strength, high specific toughness and excellent energy dissipating properties, due to its negative Poison’s ratio nature. The use of auxetic and non-auxetic panels in blast resistant structures had been relatively perceived by researchers. Nonetheless, implementation of those energy dissipaters, explicitly as a uni-axial passive damper is restrained to limited studies, which highlight the potential need for further explorations. The aim of this paper is the design of a new uniaxial graded auxetic damper (UGAD) that can be used as a blast/impact/shock absorber in different scales for different structural applications. First, the geometry, material, numerical model and loading are introduced. Then, a detailed parametric study is conducted to achieve the most efficient graded auxetic system. Moreover, the designed auxetic damper is numerically tested and its static and dynamic constitutive relations are derived and validated analytically. The selection of optimum parameters was based on the ratio of the reaction force to the applied load (RFd/P) and plastic dissipation energy (PDE). The final designed UGAD contains three auxetic cores that have the same geometry, material grade (6063-T4), size and number of layers equal to eight. The cell-wall thickness t of the three auxetic cores is 1.4 mm, 1.8 mm and 2.2 mm, respectively; composing a graded auxetic system. The performance of the three auxetic cores together have led to a wide plateau region (80% of total crushing strain) and variant strength range (1–10 MPa), which in return, can justify the superior performance of the UGAD under different blast levels. Finally, the 3D printed prototype of the UGAD is presented and the possible applications are covered. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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23 pages, 4059 KiB

Open AccessArticle

Bi-Axial Buckling of Laminated Composite Plates Including Cutout and Additional Mass

by Anish, Abhay Chaubey, Ajay Kumar, Bartłomiej Kwiatkowski, Danuta Barnat-Hunek and Marcin K. Widomski

Materials 2019, 12(11), 1750; https://doi.org/10.3390/ma12111750 - 29 May 2019

Cited by 16 | Viewed by 2682

Abstract

In the presented paper, a study of bi-axial buckling of the laminated composite plate with mass variation through the cutout and additional mass is carried out using the improved shear deformation theory (ISDT). The ISDT mathematical model employs a cubic variation of thickness co-ordinates in the displacement field. A realistic parabolic distribution of transverse shear strains through the plate thickness is assumed and the use of shear correction factor is avoided. A C° finite element formulation of the mathematical model is developed to analyze the buckling behavior of laminated composite plate with cutout and additional mass. As no results based on ISDT for the considered problem of bi-axial buckling of the laminated composite plate with mass variation are available in the literature, the obtained results are validated with the data available for a laminated composite plate without cutout and additional mass. Novel results are obtained by varying geometry, boundary conditions and ply orientations. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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30 pages, 13331 KiB

Open AccessArticle

Research on the Buckling Behavior of Functionally Graded Plates with Stiffeners Based on the Third-Order Shear Deformation Theory

by Hoang-Nam Nguyen, Tran Cong Tan, Doan Trac Luat, Van-Duc Phan, Do Van Thom and Phung Van Minh

Materials 2019, 12(8), 1262; https://doi.org/10.3390/ma12081262 - 17 Apr 2019

Cited by 50 | Viewed by 3700

Abstract

This paper presents a finite element formulation to study the mechanical buckling of stiffened functionally graded material (FGM) plates. The approach is based on a third-order shear deformation theory (TSDT) introduced by Guangyu Shi. The material properties of the plate were assumed to be varied in the thickness direction by a power law distribution, but the material of the stiffener was the same as that of the one of the bottom surface where the stiffener was placed. A parametric study was carried out to highlight the effect of material distribution, the thickness-to-width ratio, and stiffener parameters on the buckling characteristics of the stiffened FGM plates. Numerical results showed that the addition of stiffener to the FGM plate could significantly reduce the weight of the FGM plate but that both the FGM plates with and without stiffener had equally high strength in the same boundary condition and compression loading. Full article

(This article belongs to the Special Issue Applied Mathematics and Computer Methods in Materials, Mechanics and Engineering )

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