Microstructural Randomness and Scaling in Mechanics of Materials

Download Microstructural Randomness and Scaling in Mechanics of Materials PDF Online Free

Author :
Publisher : CRC Press
ISBN 13 : 1420010271
Total Pages : 500 pages
Book Rating : 4.4/5 (2 download)

DOWNLOAD NOW!


Book Synopsis Microstructural Randomness and Scaling in Mechanics of Materials by : Martin Ostoja-Starzewski

Download or read book Microstructural Randomness and Scaling in Mechanics of Materials written by Martin Ostoja-Starzewski and published by CRC Press. This book was released on 2007-08-13 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: An area at the intersection of solid mechanics, materials science, and stochastic mathematics, mechanics of materials often necessitates a stochastic approach to grasp the effects of spatial randomness. Using this approach, Microstructural Randomness and Scaling in Mechanics of Materials explores numerous stochastic models and methods used in the m

Mathematical Methods And Models In Composites (Second Edition)

Download Mathematical Methods And Models In Composites (Second Edition) PDF Online Free

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

DOWNLOAD NOW!


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.

Mathematical Methods And Models In Composites

Download Mathematical Methods And Models In Composites PDF Online Free

Author :
Publisher : World Scientific
ISBN 13 : 178326411X
Total Pages : 521 pages
Book Rating : 4.7/5 (832 download)

DOWNLOAD NOW!


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 521 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.

Handbook Of Porous Materials: Synthesis, Properties, Modeling And Key Applications (In 4 Volumes)

Download Handbook Of Porous Materials: Synthesis, Properties, Modeling And Key Applications (In 4 Volumes) PDF Online Free

Author :
Publisher : World Scientific
ISBN 13 : 9811223246
Total Pages : 1495 pages
Book Rating : 4.8/5 (112 download)

DOWNLOAD NOW!


Book Synopsis Handbook Of Porous Materials: Synthesis, Properties, Modeling And Key Applications (In 4 Volumes) by :

Download or read book Handbook Of Porous Materials: Synthesis, Properties, Modeling And Key Applications (In 4 Volumes) written by and published by World Scientific. This book was released on 2020-10-20 with total page 1495 pages. Available in PDF, EPUB and Kindle. Book excerpt: This four-volume handbook gives a state-of-the-art overview of porous materials, from synthesis and characterization and simulation all the way to manufacturing and industrial applications. The editors, coming from academia and industry, are known for their didactic skills as well as their technical expertise. Coordinating the efforts of 37 expert authors in 14 chapters, they construct the story of porous carbons, ceramics, zeolites and polymers from varied viewpoints: surface and colloidal science, materials science, chemical engineering, and energy engineering. Volumes 1 and 2 cover the fundamentals of preparation, characterisation, and simulation of porous materials. Working from the fundamentals all the way to the practicalities of industrial production processes, the subjects include hierarchical materials, in situ and operando characterisation using NMR, X-Ray scattering and tomography, state-of-the-art molecular simulations of adsorption and diffusion in crystalline nanoporous materials, as well as the emerging areas of bio-artificing and drug delivery. Volume 3 focuses on porous materials in industrial separation applications, including adsorption separation, membrane separation, and osmotic distillation. Finally, and highly relevant to tomorrow's energy challenges, Volume 4 explains the energy engineering aspects of applying porous materials in supercapacitors, fuel cells, batteries, electrolysers and sub-surface energy applications.The text contains many high-quality colourful illustrations and examples, as well as thousands of up-to-date references to peer-reviewed articles, reports and websites for further reading. This comprehensive and well-written handbook is a must-have reference for universities, research groups and companies working with porous materials.Related Link(s)

Hydro-Environmental Analysis

Download Hydro-Environmental Analysis PDF Online Free

Author :
Publisher : CRC Press
ISBN 13 : 1138000868
Total Pages : 5742 pages
Book Rating : 4.1/5 (38 download)

DOWNLOAD NOW!


Book Synopsis Hydro-Environmental Analysis by : James L. Martin

Download or read book Hydro-Environmental Analysis written by James L. Martin and published by CRC Press. This book was released on 2013-12-04 with total page 5742 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing on fundamental principles, Hydro-Environmental Analysis: Freshwater Environments presents in-depth information about freshwater environments and how they are influenced by regulation. It provides a holistic approach, exploring the factors that impact water quality and quantity, and the regulations, policy and management methods that are necessary to maintain this vital resource. It offers a historical viewpoint as well as an overview and foundation of the physical, chemical, and biological characteristics affecting the management of freshwater environments. The book concentrates on broad and general concepts, providing an interdisciplinary foundation. The author covers the methods of measurement and classification; chemical, physical, and biological characteristics; indicators of ecological health; and management and restoration. He also considers common indicators of environmental health; characteristics and operations of regulatory control structures; applicable laws and regulations; and restoration methods. The text delves into rivers and streams in the first half and lakes and reservoirs in the second half. Each section centers on the characteristics of those systems and methods of classification, and then moves on to discuss the physical, chemical, and biological characteristics of each. In the section on lakes and reservoirs, it examines the characteristics and operations of regulatory structures, and presents the methods commonly used to assess the environmental health or integrity of these water bodies. It also introduces considerations for restoration, and presents two unique aquatic environments: wetlands and reservoir tailwaters. Written from an engineering perspective, the book is an ideal introduction to the aquatic and limnological sciences for students of environmental science, as well as students of environmental engineering. It also serves as a reference for engineers and scientists involved in the management, regulation, or restoration of freshwater environments.

Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures

Download Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures PDF Online Free

Author :
Publisher : CRC Press
ISBN 13 : 1315884887
Total Pages : 1112 pages
Book Rating : 4.3/5 (158 download)

DOWNLOAD NOW!


Book Synopsis Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures by : George Deodatis

Download or read book Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures written by George Deodatis and published by CRC Press. This book was released on 2014-02-10 with total page 1112 pages. Available in PDF, EPUB and Kindle. Book excerpt: Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures contains the plenary lectures and papers presented at the 11th International Conference on STRUCTURAL SAFETY AND RELIABILITY (ICOSSAR2013, New York, NY, USA, 16-20 June 2013), and covers major aspects of safety, reliability, risk and life-cycle performance of str

Multiscale Modeling and Uncertainty Quantification of Materials and Structures

Download Multiscale Modeling and Uncertainty Quantification of Materials and Structures PDF Online Free

Author :
Publisher : Springer
ISBN 13 : 3319063316
Total Pages : 303 pages
Book Rating : 4.3/5 (19 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Modeling and Uncertainty Quantification of Materials and Structures by : Manolis Papadrakakis

Download or read book Multiscale Modeling and Uncertainty Quantification of Materials and Structures written by Manolis Papadrakakis and published by Springer. This book was released on 2014-07-02 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the proceedings of the IUTAM Symposium on Multiscale Modeling and Uncertainty Quantification of Materials and Structures that was held at Santorini, Greece, September 9 – 11, 2013. It consists of 20 chapters which are divided in five thematic topics: Damage and fracture, homogenization, inverse problems–identification, multiscale stochastic mechanics and stochastic dynamics. Over the last few years, the intense research activity at micro scale and nano scale reflected the need to account for disparate levels of uncertainty from various sources and across scales. As even over-refined deterministic approaches are not able to account for this issue, an efficient blending of stochastic and multiscale methodologies is required to provide a rational framework for the analysis and design of materials and structures. The purpose of this IUTAM Symposium was to promote achievements in uncertainty quantification combined with multiscale modeling and to encourage research and development in this growing field with the aim of improving the safety and reliability of engineered materials and structures. Special emphasis was placed on multiscale material modeling and simulation as well as on the multiscale analysis and uncertainty quantification of fracture mechanics of heterogeneous media. The homogenization of two-phase random media was also thoroughly examined in several presentations. Various topics of multiscale stochastic mechanics, such as identification of material models, scale coupling, modeling of random microstructures, analysis of CNT-reinforced composites and stochastic finite elements, have been analyzed and discussed. A large number of papers were finally devoted to innovative methods in stochastic dynamics.

Structural Interfaces and Attachments in Biology

Download Structural Interfaces and Attachments in Biology PDF Online Free

Author :
Publisher : Springer Science & Business Media
ISBN 13 : 1461433177
Total Pages : 385 pages
Book Rating : 4.4/5 (614 download)

DOWNLOAD NOW!


Book Synopsis Structural Interfaces and Attachments in Biology by : Stavros Thomopoulos

Download or read book Structural Interfaces and Attachments in Biology written by Stavros Thomopoulos and published by Springer Science & Business Media. This book was released on 2012-10-05 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: Attachment of dissimilar materials in engineering and surgical practice is a perennial challenge. Bimaterial attachment sites are common locations for injury, repeated injury, and mechanical failure. Nature presents several highly effective solutions to the challenge of bimaterial attachment that differ from those found in engineering practice. Structural Interfaces and Attachments in Biology describes the attachment of dissimilar materials from multiple perspectives. The text will simultaneously elucidate natural bimaterial attachments and outline engineering principles underlying successful attachments to the communities of tissue engineers and surgeons. Included an in-depth analysis of the biology of attachments in the body and mechanisms by which robust attachments are formed, a review of current concepts of attaching dissimilar materials in surgical practice and a discussion of bioengineering approaches that are currently being developed.

Multiscale Modeling of Heterogeneous Structures

Download Multiscale Modeling of Heterogeneous Structures PDF Online Free

Author :
Publisher : Springer
ISBN 13 : 3319654632
Total Pages : 374 pages
Book Rating : 4.3/5 (196 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Modeling of Heterogeneous Structures by : Jurica Sorić

Download or read book Multiscale Modeling of Heterogeneous Structures written by Jurica Sorić and published by Springer. This book was released on 2017-11-30 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an overview of multiscale approaches and homogenization procedures as well as damage evaluation and crack initiation, and addresses recent advances in the analysis and discretization of heterogeneous materials. It also highlights the state of the art in this research area with respect to different computational methods, software development and applications to engineering structures. The first part focuses on defects in composite materials including their numerical and experimental investigations; elastic as well as elastoplastic constitutive models are considered, where the modeling has been performed at macro- and micro levels. The second part is devoted to novel computational schemes applied on different scales and discusses the validation of numerical results. The third part discusses gradient enhanced modeling, in particular quasi-brittle and ductile damage, using the gradient enhanced approach. The final part addresses thermoplasticity, solid-liquid mixtures and ferroelectric models. The contents are based on the international workshop “Multiscale Modeling of Heterogeneous Structures” (MUMO 2016), held in Dubrovnik, Croatia in September 2016.

Multiscale Theory of Composites and Random Media

Download Multiscale Theory of Composites and Random Media PDF Online Free

Author :
Publisher : CRC Press
ISBN 13 : 0429894376
Total Pages : 257 pages
Book Rating : 4.4/5 (298 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Theory of Composites and Random Media by : Xi Frank Xu

Download or read book Multiscale Theory of Composites and Random Media written by Xi Frank Xu and published by CRC Press. This book was released on 2018-09-21 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book to introduce Green-function-based multiscale theory and the corresponding finite element method, which are readily applicable to composites and random media. The methodology is considered to be the one that most effectively tackles the uncertainty of stress propagation in complex heterogeneities of random media, and which presents multiscale theory from distinctive scale separation and scale-coupling viewpoints. Deliberately taking a multiscale perspective, it covers scale separation and then scale coupling. Both micromechanics and novel scale-coupling mechanics are described in relation to variational principles and bounds, as well as in the emerging topics on percolation and scale-coupling computation. It gives detail on the different bounds encountered, covering classical second and third order, new fourth order, and innovative ellipsoidal variations. Green-function-based multiscale theory is addressed to applications in solid mechanics and transport of complex media ranging from micro- and nano-composites, polycrystals, soils, rocks, cementitious materials, to biological materials. It is useful as a graduate textbook in civil and mechanical engineering and as a reference.

Theory and Modeling of Polymer Nanocomposites

Download Theory and Modeling of Polymer Nanocomposites PDF Online Free

Author :
Publisher : Springer Nature
ISBN 13 : 3030604438
Total Pages : 330 pages
Book Rating : 4.0/5 (36 download)

DOWNLOAD NOW!


Book Synopsis Theory and Modeling of Polymer Nanocomposites by : Valeriy V. Ginzburg

Download or read book Theory and Modeling of Polymer Nanocomposites written by Valeriy V. Ginzburg and published by Springer Nature. This book was released on 2020-12-16 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edited volume brings together the state of the art in polymer nanocomposite theory and modeling, creating a roadmap for scientists and engineers seeking to design new advanced materials. The book opens with a review of molecular and mesoscale models predicting equilibrium and non-equilibrium nanoscale structure of hybrid materials as a function of composition and, especially, filler types. Subsequent chapters cover the methods and analyses used for describing the dynamics of nanocomposites and their mechanical and physical properties. Dedicated chapters present best practices for predicting materials properties of practical interest, including thermal and electrical conductivity, optical properties, barrier properties, and flammability. Each chapter is written by leading academic and industrial scientists working in each respective sub-field. The overview of modeling methodology combined with detailed examples of property predictions for specific systems will make this book useful for academic and industrial practitioners alike.

Materials with Internal Structure

Download Materials with Internal Structure PDF Online Free

Author :
Publisher : Springer
ISBN 13 : 3319214942
Total Pages : 135 pages
Book Rating : 4.3/5 (192 download)

DOWNLOAD NOW!


Book Synopsis Materials with Internal Structure by : Patrizia Trovalusci

Download or read book Materials with Internal Structure written by Patrizia Trovalusci and published by Springer. This book was released on 2015-10-17 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents a series of concise papers by researchers specialized in various fields of continuum and computational mechanics and of material science. The focus is on principles and strategies for multiscale modeling and simulation of complex heterogeneous materials, with periodic or random microstructure, subjected to various types of mechanical, thermal, chemical loadings and environmental effects. A wide overview of complex behavior of materials (plasticity, damage, fracture, growth, etc.) is provided. Among various approaches, attention is given to advanced non-classical continua modeling which, provided by constitutive characterization for the internal and external actions (in particular boundary conditions), is a very powerful frame for the gross mechanical description of complex material behaviors, able to circumvent the restrictions of classical coarse–graining multiscale approaches.

Advances and Challenges in Space-time Modelling of Natural Events

Download Advances and Challenges in Space-time Modelling of Natural Events PDF Online Free

Author :
Publisher : Springer Science & Business Media
ISBN 13 : 3642170862
Total Pages : 263 pages
Book Rating : 4.6/5 (421 download)

DOWNLOAD NOW!


Book Synopsis Advances and Challenges in Space-time Modelling of Natural Events by : Emilio Porcu

Download or read book Advances and Challenges in Space-time Modelling of Natural Events written by Emilio Porcu and published by Springer Science & Business Media. This book was released on 2012-01-05 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book arises as the natural continuation of the International Spring School "Advances and Challenges in Space-Time modelling of Natural Events," which took place in Toledo (Spain) in March 2010. This Spring School above all focused on young researchers (Master students, PhD students and post-doctoral researchers) in academics, extra-university research and the industry who are interested in learning about recent developments, new methods and applications in spatial statistics and related areas, and in exchanging ideas and findings with colleagues.

Uncertainty Quantification in Multiscale Materials Modeling

Download Uncertainty Quantification in Multiscale Materials Modeling PDF Online Free

Author :
Publisher : Woodhead Publishing
ISBN 13 : 008102942X
Total Pages : 606 pages
Book Rating : 4.0/5 (81 download)

DOWNLOAD NOW!


Book Synopsis Uncertainty Quantification in Multiscale Materials Modeling by : Yan Wang

Download or read book Uncertainty Quantification in Multiscale Materials Modeling written by Yan Wang and published by Woodhead Publishing. This book was released on 2020-03-10 with total page 606 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty Quantification in Multiscale Materials Modeling provides a complete overview of uncertainty quantification (UQ) in computational materials science. It provides practical tools and methods along with examples of their application to problems in materials modeling. UQ methods are applied to various multiscale models ranging from the nanoscale to macroscale. This book presents a thorough synthesis of the state-of-the-art in UQ methods for materials modeling, including Bayesian inference, surrogate modeling, random fields, interval analysis, and sensitivity analysis, providing insight into the unique characteristics of models framed at each scale, as well as common issues in modeling across scales. - Synthesizes available UQ methods for materials modeling - Provides practical tools and examples for problem solving in modeling material behavior across various length scales - Demonstrates UQ in density functional theory, molecular dynamics, kinetic Monte Carlo, phase field, finite element method, multiscale modeling, and to support decision making in materials design - Covers quantum, atomistic, mesoscale, and engineering structure-level modeling and simulation

Finite Element Methods for Structures with Large Stochastic Variations

Download Finite Element Methods for Structures with Large Stochastic Variations PDF Online Free

Author :
Publisher : Oxford University Press, USA
ISBN 13 : 9780198526315
Total Pages : 282 pages
Book Rating : 4.5/5 (263 download)

DOWNLOAD NOW!


Book Synopsis Finite Element Methods for Structures with Large Stochastic Variations by : Isaac Elishakoff

Download or read book Finite Element Methods for Structures with Large Stochastic Variations written by Isaac Elishakoff and published by Oxford University Press, USA. This book was released on 2003 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: The finite element method (FEM) can be successfully applied to various field problems in solid mechanics, fluid mechanics and electrical engineering. This text discusses finite element methods for structures with large stochastic variations.

Stochastic Systems

Download Stochastic Systems PDF Online Free

Author :
Publisher : Springer Science & Business Media
ISBN 13 : 1447123271
Total Pages : 534 pages
Book Rating : 4.4/5 (471 download)

DOWNLOAD NOW!


Book Synopsis Stochastic Systems by : Mircea Grigoriu

Download or read book Stochastic Systems written by Mircea Grigoriu and published by Springer Science & Business Media. This book was released on 2012-05-15 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty is an inherent feature of both properties of physical systems and the inputs to these systems that needs to be quantified for cost effective and reliable designs. The states of these systems satisfy equations with random entries, referred to as stochastic equations, so that they are random functions of time and/or space. The solution of stochastic equations poses notable technical difficulties that are frequently circumvented by heuristic assumptions at the expense of accuracy and rigor. The main objective of Stochastic Systems is to promoting the development of accurate and efficient methods for solving stochastic equations and to foster interactions between engineers, scientists, and mathematicians. To achieve these objectives Stochastic Systems presents: A clear and brief review of essential concepts on probability theory, random functions, stochastic calculus, Monte Carlo simulation, and functional analysis Probabilistic models for random variables and functions needed to formulate stochastic equations describing realistic problems in engineering and applied sciences Practical methods for quantifying the uncertain parameters in the definition of stochastic equations, solving approximately these equations, and assessing the accuracy of approximate solutions Stochastic Systems provides key information for researchers, graduate students, and engineers who are interested in the formulation and solution of stochastic problems encountered in a broad range of disciplines. Numerous examples are used to clarify and illustrate theoretical concepts and methods for solving stochastic equations. The extensive bibliography and index at the end of the book constitute an ideal resource for both theoreticians and practitioners.

Lecture Notes on Composite Materials

Download Lecture Notes on Composite Materials PDF Online Free

Author :
Publisher : Springer Science & Business Media
ISBN 13 : 1402087721
Total Pages : 237 pages
Book Rating : 4.4/5 (2 download)

DOWNLOAD NOW!


Book Synopsis Lecture Notes on Composite Materials by : Tomasz Sadowski

Download or read book Lecture Notes on Composite Materials written by Tomasz Sadowski and published by Springer Science & Business Media. This book was released on 2008-12-11 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: Composite materials are heterogeneous by nature, and are intended to be, since only the combination of different constituent materials can give them the desired combination of low weight, stiffness and strength. At present, the knowledge has advanced to a level that materials can be tailored to exhibit certain, required properties. At the same time, the fact that these materials are composed of various, sometimes very different constituents, make their mechanical behaviour complex. This observation holds with respect to the deformation behaviour, but especially with respect to the failure behaviour, where complicated and unconventional failure modes have been observed. It is a challenge to develop predictive methods that can capture this complex mechanical behaviour, either using analytical tools, or using numerical me- ods, the ?nite element method being the most widespread among the latter. In this respect, developments have gone fast over the past decade. Indeed, we have seen a paradigm shift in computational approaches to (composite) ma- rial behaviour. Where only a decade ago it was still customary to carry out analyses of deformation and failure at a macroscopic level of observation only – one may call this a phenomenological approach – nowadays this approach is being progressively replaced by multiscale methods. In such methods it is r- ognized a priori that the overall behaviour is highly dependent on local details and ?aws.