Random Fields of Piezoelectricity and Piezomagnetism

Correlation Structures

Authors: Prof. Anatoliy Malyarenko, Prof. Martin Ostoja-Starzewski, Assoc. Prof. Amirhossein Amiri-Hezaveh

Publisher: Springer International Publishing

Book Series : SpringerBriefs in Applied Sciences and Technology

Part of: Springer Professional "Business + Economics & Engineering + Technology" , Springer Professional "Engineering + Technology" , Springer Professional "Business + Economics"

Table of Contents

About this book

Random fields are a necessity when formulating stochastic continuum theories. In this book, a theory of random piezoelectric and piezomagnetic materials is developed. First, elements of the continuum mechanics of electromagnetic solids are presented. Then the relevant linear governing equations are introduced, written in terms of either a displacement approach or a stress approach, along with linear variational principles. On this basis, a statistical description of second-order (statistically) homogeneous and isotropic rank-3 tensor-valued random fields is given. With a group-theoretic foundation, correlation functions and their spectral counterparts are obtained in terms of stochastic integrals with respect to certain random measures for the fields that belong to orthotropic, tetragonal, and cubic crystal systems. The target audience will primarily comprise researchers and graduate students in theoretical mechanics, statistical physics, and probability.

Frontmatter

Chapter 1. The Continuum Theory of Piezoelectricity and Piezomagnetism

Abstract

Following the motivation of this work, this chapter introduces the basic concepts of continuum mechanics and electromagnetism. Attention is then focused on linear piezoelectricity, elaborating two ways of writing the governing equations: the displacement approach and the stress approach. This leads to variational principles. The final section provides a basis for generalising piezoelectricity—and, by mathematical analogy, piezomagnetism—to random media whose description necessitates tensor-valued random fields.

Anatoliy Malyarenko, Martin Ostoja-Starzewski, Amirhossein Amiri-Hezaveh

Chapter 2. Mathematical Preliminaries

Abstract

We introduce piezoelectricity symmetry classes and define wide-sense homogeneous and isotropic random fields taking values in a linear space of piezoelectric tensors with a prescribed symmetry.

Anatoliy Malyarenko, Martin Ostoja-Starzewski, Amirhossein Amiri-Hezaveh

Chapter 3. The Choice of a Basis in the Space

Abstract

The general form of the one- and two-point correlation tensor of a homogeneous and \((K,\theta )\)-isotropic random field and the spectral expansion of such a field in terms of stochastic integrals with respect to certain random measures depend on the choice of a basis in the linear space where the field takes its values. We choose a basis for 11 different fields. It turns out that the basis depends only on the crystal system of the group K.

Anatoliy Malyarenko, Martin Ostoja-Starzewski, Amirhossein Amiri-Hezaveh

Chapter 4. Correlation Structures

Abstract

We calculate the one- and two-point correlation tensors of a homogeneous and \((K,\theta )\)-isotropic random field, where K is one of the 10 closed subgroups of the group \(\mathrm {O}(3)\) lying between the group \(D_2\) and its normaliser, \(\mathcal {O}\times Z^c_2\), and where \(\theta \) is the restriction of the natural representation of the group K in the space of all piezoelectric tensors to the 3-dimensional linear space where the group \(D_2\) acts trivially. The spectral expansion of such a field in terms of stochastic integrals with respect to certain random measures is also calculated.

Anatoliy Malyarenko, Martin Ostoja-Starzewski, Amirhossein Amiri-Hezaveh

Backmatter

Title: Random Fields of Piezoelectricity and Piezomagnetism
Authors: Prof. Anatoliy Malyarenko
Prof. Martin Ostoja-Starzewski
Assoc. Prof. Amirhossein Amiri-Hezaveh
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-60064-8
Print ISBN: 978-3-030-60063-1
DOI: https://doi.org/10.1007/978-3-030-60064-8