Mathematical Methods And Models In Composites

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Mathematical Methods And Models In Composites

Author : Vladislav Mantic
Publisher : World Scientific
ISBN 13 : 178326411X
Total Pages : 520 pages
Book Rating : 4.7/5 (832 download)

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Book Synopsis Mathematical Methods And Models In Composites by : Vladislav Mantic

Download or read book Mathematical Methods And Models In Composites written by Vladislav Mantic and published by World Scientific. This book was released on 2013-10-25 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a representative selection of the most relevant, innovative, and useful mathematical methods and models applied to the analysis and characterization of composites and their behaviour on micro-, meso-, and macroscale. It establishes the fundamentals for meaningful and accurate theoretical and computer modelling of these materials in the future. Although the book is primarily concerned with fibre-reinforced composites, which have ever-increasing applications in fields such as aerospace, many of the results presented can be applied to other kinds of composites. The topics covered include: scaling and homogenization procedures in composite structures, thin plate and wave solutions in anisotropic materials, laminated structures, instabilities, fracture and damage analysis of composites, and highly efficient methods for simulation of composites manufacturing. The results presented are useful in the design, fabrication, testing, and industrial applications of composite components and structures. The book is written by well-known experts in different areas of applied mathematics, physics, and composite engineering and is an essential source of reference for graduate and doctoral students, as well as researchers. It is also suitable for non-experts in composites who wish to have an overview of both the mathematical methods and models used in this area and the related open problems requiring further research.

Mathematical Methods And Models In Composites (Second Edition)

Author : Vladislav Mantic
Publisher : World Scientific
ISBN 13 : 1800611897
Total Pages : 731 pages
Book Rating : 4.8/5 (6 download)

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Book Synopsis Mathematical Methods And Models In Composites (Second Edition) by : Vladislav Mantic

Download or read book Mathematical Methods And Models In Composites (Second Edition) written by Vladislav Mantic and published by World Scientific. This book was released on 2023-03-10 with total page 731 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical Methods and Models in Composites (Second Edition) provides an in-depth treatment of modern and rigorous mathematical methods and models applied to composites modeling on the micro-, meso-, and macro scale. There has been a steady growth in the diversity of such methods and models that are used in the analysis and characterization of composites, their behavior, and their associated phenomena and processes. This second edition expands upon the success of the first edition, and has been substantially revised and updated.Written by well-known experts in different areas of applied mathematics, physics, and composite engineering, this book is mainly focused on continuous fiber reinforced composites and their ever increasing range of applications (for example, in the aerospace industry), though it also covers other kind of composites. The chapters cover a range of topics including, but not limited to: scaling and homogenization procedures in composites, thin plate and wave solutions in anisotropic materials, laminated structures, fiber-reinforced nonlinearly elastic solids, buckling and postbuckling, fracture and damage analysis of composites, and highly efficient methods for simulation of composites manufacturing such as resin transfer molding. The results presented are useful for the design, fabrication, testing and industrial applications of composite components and structures.This book is an essential reference for graduate and doctoral students, as well as researchers in mathematics, physics and composite engineering. Explanations and references in the book are sufficiently detailed so as to provide the necessary background to further investigate the fascinating subject of composites modeling and explore relevant research literature. It is also suitable for non-experts who wish to have an overview of the mathematical methods and models used for composites, and of the open problems in this area that require further research.

Topics in the Mathematical Modelling of Composite Materials

Author : Andrej V. Cherkaev
Publisher : Springer
ISBN 13 : 3319971840
Total Pages : 317 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis Topics in the Mathematical Modelling of Composite Materials by : Andrej V. Cherkaev

Download or read book Topics in the Mathematical Modelling of Composite Materials written by Andrej V. Cherkaev and published by Springer. This book was released on 2018-09-18 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the past several decades, we have witnessed a renaissance of theoretical work on the macroscopic behavior of microscopically heterogeneous materials. This activity brings together a number of related themes, including: (1) the use of weak convergence as a rigorous yet general language for the discussion of macroscopic behavior; (2) interest in new types of questions, particularly the "G-closure problem," motivated in large part by applications of optimal control theory to structural optimization; (3) the introduction of new methods for bounding effective moduli, including one based on "compensated compactness"; and (4) the identification of deep links between the analysis of microstructures and the multidimensional calculus of variations. This work has implications for many physical problems involving optimal design, composite materials, and coherent phase transitions. As a result, it has received attention and support from numerous scientific communities, including engineering, materials science, and physics, as well as mathematics. There is by now an extensive literature in this area. But for various reasons certain fundamental papers were never properly published, circulating instead as mimeographed notes or preprints. Other work appeared in poorly distributed conference proceedings volumes. Still other work was published in standard books or journals, but written in Russian or French. The net effect is a sort of "gap" in the literature, which has made the subject unnecessarily difficult for newcomers to penetrate. The present, softcover reprint is designed to make this classic text available to a wider audience. "Summarizes some of the fundamental results achieved and offers new perspectives in the mechanics of composite and micromechanics... Will become a classic in the two fields." —Applied Mechanics Review

Advances in Mathematical Modelling of Composite Materials

Author : K Z Markov
Publisher : World Scientific
ISBN 13 : 9814502448
Total Pages : 304 pages
Book Rating : 4.8/5 (145 download)

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Book Synopsis Advances in Mathematical Modelling of Composite Materials by : K Z Markov

Download or read book Advances in Mathematical Modelling of Composite Materials written by K Z Markov and published by World Scientific. This book was released on 1994-05-18 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains papers of leading experts in the modern continuum theory of composite materials. The papers expose in detail the newest ideas, approaches, results and perspectives in this broadly interdisciplinary field ranging from pure and applied mathematics, mechanics, physics and materials science. The emphasis is on mathematical modelling and model analysis of the mechanical behaviour and strength of composites, including methods of predicting effective macroscopic properties (dielectric, elastic, nonlinear, inelastic, plastic and thermoplastic) from known microstructures. Contents:Effective Field Method in Mechanics of Matrix Composite Materials (S K Kanaun & V M Levin)Functional Series and Hashin-Shtrikman Type Bounds on the Effective Properties of Random Media (K Z Markov & K D Zvyatkov)Bounds for the Effective Properties of Nonlinear Composite Materials (D R S Talbot)On the Modelling of the Inelastic Thermomechanical Behaviour and the Failure of Fibre-Reinforced Composites — A Unified Approach (K P Herrmann & I Mihovsky)Modelling of Elastic and Inelastic Behaviour of Composites (O B Pedersen & B Johannesson)Random Structure Models for Composite Media and Fracture Statistics (D Jeulin) Readership: Applied mathematicians. keywords:Continuum Theory;Composite Materials;Macroscopic Properties

Approximate Models of Mechanics of Composites

Author : Igor V. Andrianov
Publisher : CRC Press
ISBN 13 : 1000890309
Total Pages : 441 pages
Book Rating : 4.0/5 (8 download)

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Book Synopsis Approximate Models of Mechanics of Composites by : Igor V. Andrianov

Download or read book Approximate Models of Mechanics of Composites written by Igor V. Andrianov and published by CRC Press. This book was released on 2024-06-20 with total page 441 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approximate Models of Mechanics of Composites: An Asymptotic Approach is an essential guide to constructing asymptotic models and mathematical methods to correctly identify the mechanical behavior of composites. It provides methodology for predicting and evaluating composite behavior in various structures, leading to accurate mathematical and physical assessments. The book estimates the error of approximations through comparing asymptotic solutions with the results of numerical and analytical solutions to gain a holistic view of the data. The authors have developed asymptotic models based on mathematical and physical rigorous approaches, which include three-phase models of fibrous composites, a modernized three-phase composite model with cylindrical inclusions, and models of two-dimensional composites of hexagonal structure. Also covered are two-phase models of composites related to the Maxwell formula and a percolation transition model for elastic problems based on the self-consistency method and Padé approximations. By obtaining analytical expressions to effectively characterize composite materials, their physical and geometric parameters can be accurately assessed. This book suits engineers and students working in material science, mechanical engineering, physics, and mathematics, as well as composite materials in industries such as construction, transport, aerospace, and chemical engineering.

Modeling Damage, Fatigue and Failure of Composite Materials

Author : Ramesh Talreja
Publisher : Elsevier
ISBN 13 : 0443184887
Total Pages : 620 pages
Book Rating : 4.4/5 (431 download)

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Book Synopsis Modeling Damage, Fatigue and Failure of Composite Materials by : Ramesh Talreja

Download or read book Modeling Damage, Fatigue and Failure of Composite Materials written by Ramesh Talreja and published by Elsevier. This book was released on 2023-09-29 with total page 620 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modeling Damage, Fatigue and Failure of Composite Materials, Second Edition provides the latest research in the field of composite materials, an area that has attracted a wealth of research, with significant interest in the areas of damage, fatigue, and failure. The book is fully updated, and is a comprehensive source of physics-based models for the analysis of progressive and critical failure phenomena in composite materials. It focuses on materials modeling while also reviewing treatments for analyzing failure in composite structures. Sections review damage development in composite materials such as generic damage and damage accumulation in textile composites and under multiaxial loading. Part Two focuses on the modeling of failure mechanisms in composite materials, with attention given to fiber/matrix cracking and debonding, compression failure, and delamination fracture. Final sections examine the modeling of damage and materials response in composite materials, including micro-level and multi-scale approaches, the failure analysis of composite materials and joints, and the applications of predictive failure models. Provides a comprehensive source of physics-based models for the analysis of progressive and critical failure phenomena in composite materials Assesses failure and life prediction in composite materials Discusses the applications of predictive failure models such as computational approaches to failure analysis Covers further developments in computational analyses and experimental techniques, along with new applications in aerospace, automotive, and energy (wind turbine blades) fields Covers delamination and thermoplastic-based composites

The Theory of Composites

Author : Graeme W. Milton
Publisher : SIAM
ISBN 13 : 1611977487
Total Pages : 761 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis The Theory of Composites by : Graeme W. Milton

Download or read book The Theory of Composites written by Graeme W. Milton and published by SIAM. This book was released on 2022-12-07 with total page 761 pages. Available in PDF, EPUB and Kindle. Book excerpt: Composites have been studied for more than 150 years, and interest in their properties has been growing. This classic volume provides the foundations for understanding a broad range of composite properties, including electrical, magnetic, electromagnetic, elastic and viscoelastic, piezoelectric, thermal, fluid flow through porous materials, thermoelectric, pyroelectric, magnetoelectric, and conduction in the presence of a magnetic field (Hall effect). Exact solutions of the PDEs in model geometries provide one avenue of understanding composites; other avenues include microstructure-independent exact relations satisfied by effective moduli, for which the general theory is reviewed; approximation formulae for effective moduli; and series expansions for the fields and effective moduli that are the basis of numerical methods for computing these fields and moduli. The range of properties that composites can exhibit can be explored either through the model geometries or through microstructure-independent bounds on the properties. These bounds are obtained through variational principles, analytic methods, and Hilbert space approaches. Most interesting is when the properties of the composite are unlike those of the constituent materials, and there has been an explosion of interest in such composites, now known as metamaterials. The Theory of Composites surveys these aspects, among others, and complements the new body of literature that has emerged since the book was written. It remains relevant today by providing historical background, a compendium of numerous results, and through elucidating many of the tools still used today in the analysis of composite properties. This book is intended for applied mathematicians, physicists, and electrical and mechanical engineers. It will also be of interest to graduate students.

Mathematical Modelling and Biomechanics of the Brain

Author : Corina Drapaca
Publisher : Springer Nature
ISBN 13 : 1493998102
Total Pages : 155 pages
Book Rating : 4.4/5 (939 download)

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Book Synopsis Mathematical Modelling and Biomechanics of the Brain by : Corina Drapaca

Download or read book Mathematical Modelling and Biomechanics of the Brain written by Corina Drapaca and published by Springer Nature. This book was released on 2019-09-06 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph aims to provide a rigorous yet accessible presentation of some fundamental concepts used in modeling brain mechanics and give a glimpse of the insights and advances that have arisen as a result of the nascent interaction of the mathematical and neurosurgical sciences. It begins with some historical perspective and a brief synopsis of the biomedical/biological manifestations of the clinical conditions/diseases considered. Each chapter proceeds with a discussion of the various mathematical models of the problems considered, starting with the simplest models and proceeding to more complex models where necessary. A detailed list of relevant references is provided at the end of each chapter. With the beginning research student in mind, the chapters have been crafted to be as self-contained as possible while addressing different clinical conditions and diseases. The book is intended as a brief introduction to both theoreticians and experimentalists interested in brain mechanics, with directions and guidance for further reading, for those who wish to pursue particular topics in greater depth. It can also be used as a complementary textbook in a graduate level course for neuroscientists and neuroengineers.

The Structural Integrity of Carbon Fiber Composites

Author : Peter W. R Beaumont
Publisher : Springer
ISBN 13 : 3319461206
Total Pages : 969 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis The Structural Integrity of Carbon Fiber Composites by : Peter W. R Beaumont

Download or read book The Structural Integrity of Carbon Fiber Composites written by Peter W. R Beaumont and published by Springer. This book was released on 2016-11-26 with total page 969 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings together a diverse compilation of inter-disciplinary chapters on fundamental aspects of carbon fiber composite materials and multi-functional composite structures: including synthesis, characterization, and evaluation from the nano-structure to structure meters in length. The content and focus of contributions under the umbrella of structural integrity of composite materials embraces topics at the forefront of composite materials science and technology, the disciplines of mechanics, and development of a new predictive design methodology of the safe operation of engineering structures from cradle to grave. Multi-authored papers on multi-scale modelling of problems in material design and predicting the safe performance of engineering structure illustrate the inter-disciplinary nature of the subject. The book examines topics such as Stochastic micro-mechanics theory and application for advanced composite systems Construction of the evaluation process for structural integrity of material and structure Nano- and meso-mechanics modelling of structure evolution during the accumulation of damage Statistical meso-mechanics of composite materials Hierarchical analysis including "age-aware," high-fidelity simulation and virtual mechanical testing of composite structures right up to the point of failure. The volume is ideal for scientists, engineers, and students interested in carbon fiber composite materials, and other composite material systems.

Multi-scale Simulation of Composite Materials

Author : Stefan Diebels
Publisher : Springer
ISBN 13 : 366257957X
Total Pages : 178 pages
Book Rating : 4.6/5 (625 download)

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Book Synopsis Multi-scale Simulation of Composite Materials by : Stefan Diebels

Download or read book Multi-scale Simulation of Composite Materials written by Stefan Diebels and published by Springer. This book was released on 2019-02-01 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: Due to their high stiffness and strength and their good processing properties short fibre reinforced thermoplastics are well-established construction materials. Up to now, simulation of engineering parts consisting of short fibre reinforced thermoplastics has often been based on macroscopic phenomenological models, but deformations, damage and failure of composite materials strongly depend on their microstructure. The typical modes of failure of short fibre thermoplastics enriched with glass fibres are matrix failure, rupture of fibres and delamination, and pure macroscopic consideration is not sufficient to predict those effects. The typical predictive phenomenological models are complex and only available for very special failures. A quantitative prediction on how failure will change depending on the content and orientation of the fibres is generally not possible, and the direct involvement of the above effects in a numerical simulation requires multi-scale modelling. One the one hand, this makes it possible to take into account the properties of the matrix material and the fibre material, the microstructure of the composite in terms of fibre content, fibre orientation and shape as well as the properties of the interface between fibres and matrix. On the other hand, the multi-scale approach links these local properties to the global behaviour and forms the basis for the dimensioning and design of engineering components. Furthermore, multi-scale numerical simulations are required to allow efficient solution of the models when investigating three-dimensional problems of dimensioning engineering parts. Bringing together mathematical modelling, materials mechanics, numerical methods and experimental engineering, this book provides a unique overview of multi-scale modelling approaches, multi-scale simulations and experimental investigations of short fibre reinforced thermoplastics. The first chapters focus on two principal subjects: the mathematical and mechanical models governing composite properties and damage description. The subsequent chapters present numerical algorithms based on the Finite Element Method and the Boundary Element Method, both of which make explicit use of the composite’s microstructure. Further, the results of the numerical simulations are shown and compared to experimental results. Lastly, the book investigates deformation and failure of composite materials experimentally, explaining the applied methods and presenting the results for different volume fractions of fibres. This book is a valuable resource for applied mathematics, theoretical and experimental mechanical engineers as well as engineers in industry dealing with modelling and simulation of short fibre reinforced composites.

Applied Analysis of Composite Media

Author : Piotr Drygas
Publisher : Woodhead Publishing
ISBN 13 : 0081026706
Total Pages : 372 pages
Book Rating : 4.0/5 (81 download)

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Book Synopsis Applied Analysis of Composite Media by : Piotr Drygas

Download or read book Applied Analysis of Composite Media written by Piotr Drygas and published by Woodhead Publishing. This book was released on 2019-11-05 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: Applied Analysis of Composite Media: Analytical and Computational Approaches presents formulas and techniques that can used to study 2D and 3D problems in composites and random porous media. The main strength of this book is its broad range of applications that illustrate how these techniques can be applied to investigate elasticity, viscous flow and bacterial motion in composite materials. In addition to paying attention to constructive computations, the authors have also included information on codes via a designated webpage. This book will be extremely useful for postgraduate students, academic researchers, mathematicians and industry professionals who are working in structured media. Provides a uniform, computational methodology that can be applied to the main classes of transport and elastic problems by using a combination of exact formulae, advanced simulations and asymptotic methods Includes critical phenomena in transport and elastic problems for composites and porous media Applies computational methodology to biological structures Presents computer protocols/algorithms that can be used for materials design

Topics in the Mathematical Modelling of Composite Materials

Author : Andrej V. Čerkaev
Publisher :
ISBN 13 : 9783764336622
Total Pages : 317 pages
Book Rating : 4.3/5 (366 download)

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Book Synopsis Topics in the Mathematical Modelling of Composite Materials by : Andrej V. Čerkaev

Download or read book Topics in the Mathematical Modelling of Composite Materials written by Andrej V. Čerkaev and published by . This book was released on 1997-01-01 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Asymptotical Mechanics of Composites

Author : Igor V. Andrianov
Publisher : Springer
ISBN 13 : 3319657860
Total Pages : 329 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis Asymptotical Mechanics of Composites by : Igor V. Andrianov

Download or read book Asymptotical Mechanics of Composites written by Igor V. Andrianov and published by Springer. This book was released on 2017-11-09 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book the authors show that it is possible to construct efficient computationally oriented models of multi-parameter complex systems by using asymptotic methods, which can, owing to their simplicity, be directly used for controlling processes arising in connection with composite material systems. The book focuses on this asymptotic-modeling-based approach because it allows us to define the most important out of numerous parameters describing the system, or, in other words, the asymptotic methods allow us to estimate the sensitivity of the system parameters. Further, the book addresses the construction of nonlocal and higher-order homogenized models. Local fields on the micro-level and the influence of so-called non-ideal contact between the matrix and inclusions are modeled and investigated. The book then studies composites with non-regular structure and cluster type composite conductivity, and analyzes edge effects in fiber composite materials. Transition of load from a fiber to a matrix for elastic and viscoelastic composites, various types of fiber composite fractures, and buckling of fibers in fiber-reinforced composites is also investigated. Last but not least, the book includes studies on perforated membranes, plates, and shells, as well as the asymptotic modeling of imperfect nonlinear interfaces.

Analysis, Design and Optimization of Composite Structures

Author : Alexander L. Kalamkarov
Publisher : John Wiley & Sons
ISBN 13 :
Total Pages : 376 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Analysis, Design and Optimization of Composite Structures by : Alexander L. Kalamkarov

Download or read book Analysis, Design and Optimization of Composite Structures written by Alexander L. Kalamkarov and published by John Wiley & Sons. This book was released on 1997-07-07 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: Rapidly varying material and geometrical characteristics of composite materials and structures do not allow the direct study of their mechanical behavior even with the use of modern computers. This book is devoted to the mechanical design and optimization problems of composite structures, based on the previously developed asymptotic homogenization models and on the newly elaborated rigorous mathematical methods. It describes how to construct mathematically rigorous mechanical models to determine strength, stiffness, and weight minimization requirements, all important factors of design and optimization.

Technological Advancements in Construction

Author : Angela Mottaeva
Publisher : Springer Nature
ISBN 13 : 3030839176
Total Pages : 558 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Technological Advancements in Construction by : Angela Mottaeva

Download or read book Technological Advancements in Construction written by Angela Mottaeva and published by Springer Nature. This book was released on 2021-09-06 with total page 558 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book consists of original research papers in the field of Technological Advancements in Construction. It covers such topics as non-destructive testing, structural health monitoring, innovative composite materials, strengthening and rehabilitation of buildings and structures, seismic resilience of structures, thermal protection of buildings, construction and operation of buildings and structures in extreme climatic conditions, structural dynamics and vibration control, and green construction. The book contains latest information on structural mechanics of composite materials and structures, theoretical and computational modeling of new materials and structures, experimental and numerical analysis in building rehabilitation and strengthening, analytical, numerical and experimental methodologies for the analysis of multilayered structures, and advanced methods for seismic performance evaluation of building structures. The book includes original research and application papers of high academic level, where significant scientific novelty is clearly demonstrated. The book presents a valuable tool for researchers and construction professionals.

Computational Mechanics of Composite Materials

Author : Marcin Marek Kaminski
Publisher : Springer Science & Business Media
ISBN 13 : 1846280494
Total Pages : 434 pages
Book Rating : 4.8/5 (462 download)

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Book Synopsis Computational Mechanics of Composite Materials by : Marcin Marek Kaminski

Download or read book Computational Mechanics of Composite Materials written by Marcin Marek Kaminski and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational Mechanics of Composite Materials lays stress on the advantages of combining theoretical advancements in applied mathematics and mechanics with the probabilistic approach to experimental data in meeting the practical needs of engineers. Features: Programs for the probabilistic homogenisation of composite structures with finite numbers of components allow composites to be treated as homogeneous materials with simpler behaviours. Treatment of defects in the interfaces within heterogeneous materials and those arising in composite objects as a whole by stochastic modelling. New models for the reliability of composite structures. Novel numerical algorithms for effective Monte-Carlo simulation. Computational Mechanics of Composite Materials will be of interest to academic and practising civil, mechanical, electronic and aerospatial engineers, to materials scientists and to applied mathematicians requiring accurate and usable models of the behaviour of composite materials.

Multi-scale Modelling for Structures and Composites

Author : G. Panasenko
Publisher : Springer Science & Business Media
ISBN 13 : 1402029829
Total Pages : 407 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Multi-scale Modelling for Structures and Composites by : G. Panasenko

Download or read book Multi-scale Modelling for Structures and Composites written by G. Panasenko and published by Springer Science & Business Media. This book was released on 2005-06-15 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerous applications of rod structures in civil engineering, aircraft and spacecraft confirm the importance of the topic. On the other hand the majority of books on structural mechanics use some simplifying hypotheses; these hypotheses do not allow to consider some important effects, for instance the boundary layer effects near the points of junction of rods. So the question concerning the limits of applicability of structural mechanics hypotheses and the possibilities of their refinement arise. In this connection the asymptotic analysis of equations of mathematical physics, the equations of elasticity in rod structures (without these hypotheses and simplifying assumptions being imposed) is undertaken in the present book. Moreover, a lot of modern structures are made of composite materials and therefore the material of the rods is not homogeneous. This inhomogeneity of the material can generate some unexpected effects. These effects are analysed in this book. The methods of multi-scale modelling are presented by the homogenization, multi-level asymptotic analysis and the domain decomposition. These methods give an access to a new class of hybrid models combining macroscopic description with "microscopic zooms".