MechSE professor Martin Ostoja-Starzewski, together with his Ph.D. student Amirhossein Amiri-Hezaveh and a colleague at Mälardalen University in Sweden, co-authored a book: Random Fields of Piezoelectricity and Piezomagnetism, 2020. It simultaneously came out in the SpringerBriefs in Applied Sciences and Technology and the SpringerBriefs in Mathematical Methods.
The research reported in this book has been motivated by a major challenge in mechanics and physics of spatially inhomogeneous materials: to develop physically correct tensor-valued random field models that may serve as inputs to stochastic partial differential equations (SPDE) and stochastic finite elements (SFE). It is well known in turbulence theory that a statistically homogeneous and isotropic velocity field requires two scalar random fields. However, SPDE and SFE conventionally rely on a single scalar-valued random field for tensorial rank 2, 3, or 4 constitutive models such as, respectively, conductivity, piezoelectricity, or elasticity. Thus, the main purpose of this book is to provide a complete description of piezoelectricity (and, by analogy) piezomagnetism of random media in terms of second-order wide-sense homogeneous and isotropic tensor-valued random fields.
Within the restriction to linear responses, the book begins with elements of continuum mechanics of electromagnetic solids and relevant variational principles. The governing equations are written in terms of either a displacement approach or a (much less known) stress approach; the first exemplified by the Navier equation of elastodynamics, while the second by the Ignaczak equation of elastodynamics. On this basis, the book presents a development of second-order statistically homogeneous and isotropic rank-3 tensor-valued random fields. Working from the standpoint of invariance of physical laws with respect to the choice of a coordinate system, the spatial domain representations, as well as their spectra, are given in full detail for the orthotropic, tetragonal, and cubic crystal systems.Using group representation theory as the foundation, the derivations are done in terms of the complete description of the one- and two-point correlation tensors of the above class of fields as well as the spectral expansions of the fields in terms of stochastic integrals.
The target audience primarily comprises researchers in theoretical mechanics, statistical physics, and spatial statistics. The book will also benefit graduate students in engineering, science, and mathematics.