5R22. Heterogeneous Media: Micromechanics Modeling Methods and Simulations. (Modeling and Simulation in Science, Engineering and Technology Series.) - Edited by K Markov (Fac of Math and Informatics, Univ of Sofia, St Klimentohridski, Sofia, BG-1164, Bulgaria) and L Preziosi (Dept di Matematica, Politecnico di Torino, Torino, I-10129, Italy). Birkhauser Boston, Cambridge MA. 2000. 477 pp. ISBN 0-8176-4083-5. $79.95.
Reviewed by GC Gaunaurd (Code AMSRL-SE-RU, Army Res Lab, 2800 Powder Mill Rd, Adelphi MD 20783-1197).
The editors have put together five articles by different authors and organized them into the present book, all dealing with some aspect of the title subject. The editors have also authored or co-authored some of the chapter/articles, which average close to 100 pages each. The titles of the five chapters are as follows: 1) Elementary Micromechanics of Heterogeneous Media, by K Markov, 2) Diffusion-Absorption and Flow Processes in Disordered Porous Media, by S Torquato, 3) Self-Consistent Methods in the Problem of Wave Propagation through Heterogeneous Media, by S Kanaun, 4) Deformable Porous Media and Composite Manufacturing, by A Farina and L Preziosi, and finally, 5) Micromechanics of Poroelastic Rocks, by R Zimmerman.
These chapters all attempt to present models and mathematical tools to predict the overall macro-reaction of a medium structure taking into account the medium’s micro-structure. These models are tested or realistic examples, explicit results are extracted in analytic or numerical form, and then a comparison with the experimental findings is performed. The degree of coincidence between prediction and experimental observation is a test of the model accuracy. The authors and editors have attempted to unify all the languages and ways of thinking about these problems present in many disciplines such as solid and fluid mechanics, solid state physics, biomechanics, etc. It is hard to summarize these chapters in the brief paragraphs of this review, but our attempt follows.
Markov’s first chapter reviews the introductory ideas related to homogenization. It replaces heterogeneous media by homogeneous ones which macroscopically behave in the same way by having some gross effective properties. These are related in a complex way to the medium’s internal structure. There is a large historical background here over the last 200 years and many references (180) are cited, although ten times that amount could have also been easily mentioned. The examples and references cover all the above-mentioned fields and concern conductivity, elasticity, polycrystals, etc. The next four chapters are more specialized.
Torquato’s second chapter deals with rigorous methods to estimate the effective properties associated with two types of processes present in random porous media: diffusion-absorption and flow-phenomena. The first one examines the trapping constant or mean survival time, and also diffusion-relaxation times. The second process deals with fluid permeability and viscous relaxation times. Several topics are presented and some of the most interesting include rigorous bounds on the effective properties in terms of correlation functions and cross-property relations that link diffusion to flow properties. It cites about 60 representative references.
Kanaun’s third chapter considers the evaluation of mean wave fields and the effective dynamic properties of random composites with microstructure. It is the only chapter dealing with dynamic (ie, frequency-dependent) properties. The main emphasis goes to two of the main self-consistent schemes: the effective field and the effective medium approaches. An example includes monochromatic electromagnetic (EM) waves propagating through particulate composites, and there is some discussion on the sources of possible inaccuracies of these methods and of ways to overcome them. About 40 references are cited for the case selected. This is the area of greatest interest to this reviewer. There is a very large body of knowledge on this topic in the Acoustics literature that is unfortunately not mentioned.
The fourth chapter by Farina and Preziosi reviews appropriate models to mathematically describe porous media. This leads to applications in composite materials manufacturing, a key issue in the real production technologies of composites. Many industries demand special advanced materials satisfying strict requirements and lower cost, and these requirements involve a mixture of various properties that should act synergistically to satisfy the needs of the application. About 150 references are cited and discussed to some extent. The presentation is quite theoretical, and some of the sources cited range from early works of Darcy, Eringen, and Truesdel, to more recent ones by the authors, still in press.
The final fifth chapter by Zimmerman reviews the poroelactisity of rock-like media. The emphasis here goes to the micro-scale deformation of the pore space. A non-linear (and later a linearized) theory of the deformation of porous media under hydro-static loading is presented. The seminal work of M Biot is discussed at length and also some of its later extensions, including works of T Plona and of the author of the present chapter himself. There are about 100 references cited.
This is not a textbook. It is a reference book for the practitioner. However, the editors express the wish that perhaps some of these chapters could be used in graduate courses for PhD students in applied and industrial mathematics. There are few figures, but many references. Heterogeneous Media: Micromechanics Modeling Methods and Simulations is recommended for graduate students and institutional libraries, as well as for the design practitioner in the field.
