7R20. Continuum Models for Phase Transitions and Twinning in Crystals. Applied Mathematics, Volume 19. - M Pitteri and G Zanzotto (Dept of Math Methods and Models for Appl Sci, Univ of Padova, Italy). Chapman and Hall/CRC, Boca Raton FL. 2003. 385 pp. ISBN 0-8493-0327-3. $99.95.

Reviewed by CA Rossit (Dept de Ingenieria, Univ Nacional del Sur, Avenida Alem 1253, Bahia Blanca, 8000, Argentina).

This nicely written textbook for graduate students and researchers is an extremely valuable reference treatise for anyone undertaking research in these areas, basically the nonlinear thermoelastic theories for twinning and phase transitions in crystalline materials. The field goes from very deep theoretical issues in mathematical physics to very important problems in material science.

As pointed out by the authors, several definitions and remarks on the phenomenon of twinning are given by many distinguished researchers. However, they have selected the standard viewpoint stating that “a twin is a polycrystalline edifice built up of two or more homogeneous portions of the same crystal species in juxtaposition and oriented with respect to one another according to well defined laws.” Twins are generally classified according to their physical origin. Hence growth, deformation, and transformation twins are often distinguished.

The authors present fundamental experimental facts regarding twins originated during growth or along phase transformations or by mechanical deformation, along with early geometric theories, in Chapter 1.

Chapter 2 reviews basic concepts for the treatment of the subject. The molecular model of a simple (mono—atomic) lattice is introduced in Chapter 3 in order to describe periodicity of crystalline solids.

Chapter 4 deals with weak-transformation neighborhoods and variants, while Chapter 5 is concerned with explicit variant structures.

In Chapter 6, titled Energetics, a nonconvex energy function per unit cell of the lattice is introduced, following concepts developed by Cauchy. This molecular appa ratus is connected to elasticity theory using a hypothesis due to Cauchy and improved by Born (“Born rule”).

Chapter 7 depicts some results regarding bifurcation patterns that are possible during solid-state phase transitions involving changes of symmetry in simple lattices. This approach was first introduced by Landau.

The concept of mechanical twinning deformations is introduced in Chapter 8, which ends up with a synthetic discussion, leading to treating the Born rule with caution. This leads to interesting applications in the Earth Sciences as shown by Zanzotto in an earlier contribution.

Transformation twinning is studied in Chapter 9 while Chapter 10 deals with the modeling of various complicated microstructures often found in crystals of shape memory alloys.

Multilattice is introduced in Chapter 11, and the relevant variables are defined and studied with particular application to the twinning mode in hexagonal metals and the twinning mode of β-tin.

The authors greatly succeed in achieving their stated aims and stimulate deep interest in the subject matter. This reviewer welcomes the appearance of Continuum Models for Phase Transitions and Twinning in Crystals in the world scientific literature.