Homogenisation: Averaging processes in periodic media. Mathematical problems in the mechanics of composite materials. Transl. from the Russian by D. Leites.

*(English)*Zbl 0692.73012
Mathematics and Its Applications: Soviet Series, 36. Dordrecht etc.: Kluwer Academic Publishers. xxxv, 366 p. Dfl. 285.00; $139.00; £94.00 (1989).

This volume is the English edition of a book first published in the USSR in 1984 (Zbl 0607.73009), it is based on lectures held by the authors at Moscow University. The required knowledge to understand this book is that of an applied mathematician PHD student. It treats materials consisting of alternating compounds having different properties. Mathematical modelling is performed considering composites having periodic structures. In the studied problems the geometric periodic structure is always given, so the media are fixed or in the framework of small perturbation theory. The studies are relevant to elasticity theory, heat conduction, filtration in porous media, skeletal structures... Media with large deformation processes, as suspensions for example, are not considered.

In composite media having finely periodic structures, fields and processes are described by partial differential equations with rapidly oscillating coefficients. The purpose is to obtain equations whose coefficients are not rapidly oscillating while their solutions are close to those of the original equations. These new equations are called the “averaged equations” and their coefficients the “effective coefficients” of the composite material; in literature they are also called “homogenized equations” and “homogenized coefficients”. To obtain this result the asymptotic averaging method is used. At first, the solutions of the equations are investigated using double scale asymptotic expansions with respect to the small parameter \(\epsilon\) which is the ratio of the period size to the macroscopic characteristic length (dimension of the specimen or wavelength). The coefficients of the asymptotic expansions depend both on the macroscopic variables \(x_ i\) and the microscopic variables \(\xi_ i=x_ i/\epsilon\), and are periodic functions in \(\xi\). The variation with respect to the microscopic variables of these coefficients is determined by a recurrent chain of differential equations in \(\xi\), x being a parameter. The averaged equations connect macroscopic quantities (generally means over period), they arise naturally in the asymptotic process, often as the first necessary and sufficient solvability condition for equations verified by the terms of the asymptotic expansions.

In Chapter 1 the concept of generalized solution is introduced in order to treat the equations with discontinuous coefficients. This definition of the solution by means of a integral identity is also called in the literature the variational formulation of a problem. Theorems which prove existence and uniqueness of solutions of basic equations of mathematical physics in the Sobolev space \(W^ 1_ 2\) are formulated. The Chapter 2 is devoted to the concept of asymptotic expansion of a function, and formal asymptotic solution of a problem. To illustrate the averaging method a simple unidimensional problem is studied. Chapter 3 deals with layered media. It investigates different problems stationary or nonstationary, linear or nonlinear in elasticity, viscoelasticity, heat conduction. In some of these studies, mathematical justification of the asymptotics is given in detail.

The next chapters concern spatially many dimensional problems. In Chapter 4, partial differential equations which describe physical processes in composite materials are considered, as stationary thermal field in a composite (with justification of the asymptotic expansion) and in a porous medium, as equations of elasticity in all situations (linear or nonlinear, stationary or nonstationary). Averaging of Stokes and Navier- Stokes equations is also performed, Darcy’s law is obtained in the linear case, the nonlinear case is as a matter of fact treated only in the case of small nonlinearity. Chapter 5 considers formal procedures of averaging general nonlinear differential equations and linear operator equations. Chapter 6 studies properties of the effective coefficients: symmetry, convexity, use of the possible symmetry of the periodicity domain that make the search of effective coefficients much easier.

Chapter 7 deals with composite materials where some physical characteristics (elastic or thermal) of the first component (the reinforcement) differ greatly from those of the second component (the matrix). Such composites have two parameters, \(\epsilon\) as previously \((\epsilon \ll 1)\) and a physical parameter \(\omega\) which is equal to the ratio of the physical property of the reinforcement characteristic to that of the matrix \((\omega \gg 1)\). In this chapter it is supposed that \(\omega \epsilon^ 2\ll 1\). In chapter 8, a method is proposed for calculating processes in skeletal or latticed structures which is based on averaging the partial differential equations describing these processes. Explicit formulas to calculate the macroscopic characteristics of this type of structure are given. Chapter 9 studies the boundary effects that may occur near the boundary of a composite. This may be useful in investigating failures of composite material which often start from the boundary. Formal asymptotic expansions taking into account the contact conditions on the boundary line are sought for some simple problems.

Reading of this book is generally not tedious. The methods and the results of this volume will be useful for mathematicians and engineers interested by the modelling of composite materials.

In composite media having finely periodic structures, fields and processes are described by partial differential equations with rapidly oscillating coefficients. The purpose is to obtain equations whose coefficients are not rapidly oscillating while their solutions are close to those of the original equations. These new equations are called the “averaged equations” and their coefficients the “effective coefficients” of the composite material; in literature they are also called “homogenized equations” and “homogenized coefficients”. To obtain this result the asymptotic averaging method is used. At first, the solutions of the equations are investigated using double scale asymptotic expansions with respect to the small parameter \(\epsilon\) which is the ratio of the period size to the macroscopic characteristic length (dimension of the specimen or wavelength). The coefficients of the asymptotic expansions depend both on the macroscopic variables \(x_ i\) and the microscopic variables \(\xi_ i=x_ i/\epsilon\), and are periodic functions in \(\xi\). The variation with respect to the microscopic variables of these coefficients is determined by a recurrent chain of differential equations in \(\xi\), x being a parameter. The averaged equations connect macroscopic quantities (generally means over period), they arise naturally in the asymptotic process, often as the first necessary and sufficient solvability condition for equations verified by the terms of the asymptotic expansions.

In Chapter 1 the concept of generalized solution is introduced in order to treat the equations with discontinuous coefficients. This definition of the solution by means of a integral identity is also called in the literature the variational formulation of a problem. Theorems which prove existence and uniqueness of solutions of basic equations of mathematical physics in the Sobolev space \(W^ 1_ 2\) are formulated. The Chapter 2 is devoted to the concept of asymptotic expansion of a function, and formal asymptotic solution of a problem. To illustrate the averaging method a simple unidimensional problem is studied. Chapter 3 deals with layered media. It investigates different problems stationary or nonstationary, linear or nonlinear in elasticity, viscoelasticity, heat conduction. In some of these studies, mathematical justification of the asymptotics is given in detail.

The next chapters concern spatially many dimensional problems. In Chapter 4, partial differential equations which describe physical processes in composite materials are considered, as stationary thermal field in a composite (with justification of the asymptotic expansion) and in a porous medium, as equations of elasticity in all situations (linear or nonlinear, stationary or nonstationary). Averaging of Stokes and Navier- Stokes equations is also performed, Darcy’s law is obtained in the linear case, the nonlinear case is as a matter of fact treated only in the case of small nonlinearity. Chapter 5 considers formal procedures of averaging general nonlinear differential equations and linear operator equations. Chapter 6 studies properties of the effective coefficients: symmetry, convexity, use of the possible symmetry of the periodicity domain that make the search of effective coefficients much easier.

Chapter 7 deals with composite materials where some physical characteristics (elastic or thermal) of the first component (the reinforcement) differ greatly from those of the second component (the matrix). Such composites have two parameters, \(\epsilon\) as previously \((\epsilon \ll 1)\) and a physical parameter \(\omega\) which is equal to the ratio of the physical property of the reinforcement characteristic to that of the matrix \((\omega \gg 1)\). In this chapter it is supposed that \(\omega \epsilon^ 2\ll 1\). In chapter 8, a method is proposed for calculating processes in skeletal or latticed structures which is based on averaging the partial differential equations describing these processes. Explicit formulas to calculate the macroscopic characteristics of this type of structure are given. Chapter 9 studies the boundary effects that may occur near the boundary of a composite. This may be useful in investigating failures of composite material which often start from the boundary. Formal asymptotic expansions taking into account the contact conditions on the boundary line are sought for some simple problems.

Reading of this book is generally not tedious. The methods and the results of this volume will be useful for mathematicians and engineers interested by the modelling of composite materials.

Reviewer: Th.Lévy

##### MSC:

74E05 | Inhomogeneity in solid mechanics |

74E30 | Composite and mixture properties |

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

46S30 | Constructive functional analysis |