Introduction to Micromechanics and Nanomechanics

Citation

Silberschmidt, V.V. (2010), "Introduction to Micromechanics and Nanomechanics", Assembly Automation, Vol. 30 No. 3. https://doi.org/10.1108/aa.2010.03330cae.001

Introduction to Micromechanics and Nanomechanics

Introduction to Micromechanics and Nanomechanics

Article Type: Book review From: Assembly Automation, Volume 30, Issue 3

Shaofan Li and Gang Wang,World Scientific,Singapore,2008,$101,516 pp.,ISBN: 978-981-281-413-5,Web site: www.worldscibooks.com/engineering/6834.htm

The first author of the book links it to two monographs of his teachers – Professors Toshio Mura (Micromechanics of Defects in Solids, Kluwer Academic Publisher, 1987) and Wing Kam Liu (Nano Mechanics and Materials: Theory, Multiscale Methods and Applications, Wiley, 2005; together with Eduard G. Karpov and Harold S. Park); the latter was his PhD advisor.

The authors start with definitions of micromechanics and nanomechanics, followed by an introduction covering vectors and tensors, linear and finite elasticity, molecular dynamics and elements of lattice dynamics.

Chapter 2 introduces the Green’s function as one of the main tool for both micromechanics and nanomechanics. In the former, one of the main homogenization procedures employs the Green’s function method while the lattice Green’s function can be used in nanomechanical calculations. The chapter finishes with the Maradudin’s analysis of a screw dislocation based on the lattice Green’s function method.

The next two chapters deal with mathematical homogenization as a core area of micromechanics. Ideas and definitions of ergodicity and a representative volume element (RVE) are followed by theorems on averaged fields in an RVE and introduction of the Eshelby tensor. The theory is applied to examples of a quantum dot, screw dislocation and flat ellipsoidal crack. After dealing with objects at microscale, the authors discuss approaches used to assess effective properties of microstructured materials, limiting Chapter 4 to an effective elastic modulus of two-phase composites.

Chapter 5 introduces variational calculus together with some variational principles. Various types of bounds for effective properties – Voigt, Reuss, Hashin-Strickman – are derived based on variational principles for linear composite materials. The second half of the chapter contains a review of functional analysis and convex analysis, necessary for more complex cases, followed by the Legendre transformation and variational principles of Talbot-Willis and Ponte Castaneda for nonlinear composites.

The next chapter is based on authors’ original results for a spherical inclusion within a spherical RVE, in contrast to a classical Eshelby approach with an inclusion in an unbounded space. Two new position-dependent tensors, containing information on a volume fraction of the inclusion and boundary conditions for the RVE, are introduced to solve the problem. They are used to find bounds for a three-phase isotropic composite. A recently developed by authors variational multiscale eigenstrain method is introduced together with some examples.

A case when the second phase in a composite is presented by voids that grow and coalesce can be linked to ductile fracture. This topic is treated in Chapter 7, containing models of McClintock and Gurson as well as the Gurson-Tvergaard-Needleman scheme. A quasi-brittle type of fracture, determined by development of microcracks, is described in terms of cohesive damage.

Crystallographic defects are analysed in Chapter 8 with emphasis on dislocations that are treated using both continuum- and lattice-based formalisms. The former part covers also the popular approach of discrete dislocation dynamics, while the latter is based on the Peierls-Nabarro model (including two historical accounts of its development). The final part of the chapter deals with equilibrium, relaxation and mobility of dislocations in thin films.

In Chapter 9, an introduction into configurational mechanics using the Eshelby’s energy-momentum tensor and J-integral is followed by authors’ ideas of configurational compatibility applied to continuum theories of dislocations and disclinations as well as to ductile fracture.

Chapter 10 presents various problems – together with underpinning theories and models – at atomic and nano scales, for instance, calculating an atomistic surface energy potential.

The final chapter deals with an account for periodicity, observed at various scales, in modelling schemes. Two approaches, based, respectively, on equivalent eigenstrain and asymptotic homogenization, provide researchers with powerful tools for tackling different problems. Three examples – a periodic-structure nanowire, distribution of precipitates in a cubic lattice and masonry – are given.

A strong feature of the book, underlined by Professor Li in its preface, is detailed derivations of many relations that should assist readers in mastering these heavily mathematized theories. The book also contains multiple examples with full derivations. All chapters are ended with several exercises.

An interesting – and unexpected – addition to theoretical parts of the book is a series of mini-biographies of several important “players” in this area – Eshelby, Hill, Hashin, McClintock, Rice, Kröner and Mura.

According to its authors, the book can be used both as a textbook in micromechanics and nanomechanics for the first-year graduate students and a “guide book” for researchers interested in fundamental theories of these two areas.

Vadim V. SilberschmidtWolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, UK