Nonlinear theory of electroelastic and magnetoelastic interactions.

*(English)*Zbl 1291.78002
New York, NY: Springer (ISBN 978-1-4614-9595-6/hbk; 978-1-4614-9596-3/ebook). xi, 313 p. (2014).

This is an excellent book devoted to a unified theory of nonlinear interactions of electroelastic and magnetoelastic fields. The presentation in this book is informative and useful for mathematicians interested in applications of the nonlinear PDE and boundary value problems. After Chapter 1 “Introduction”, the authors review in Chapter 2 “Electromagnetic theory” the basic equations of electromagnetism from the Lorentz force and Coulomb’s law to Maxwell’s equations and the effects of polarization and magnetization. Boundary conditions and integral representations are discussed. Chapter 3 “Nonlinear elasticity background” contains foundations of continuum mechanics (kinematics, balance laws, mechanical stress and elastic constitutive laws). This chapter is ended by boundary value problems corresponding the considered phenomena. Chapter 4 “Nonlinear electroelastic interactions” focuses on some of the different ways in which the mechanical equations can be written in the presence of electromechanical interactions. Particular attention is payed to isotropic materials. The authors use the Lagrangian formulations to present compact systems of equations. Chapter 5 “Electroelastic boundary value problems” contains application of the theory to boundary value problems taking into account the coupling of electric with elastic deformations. Chapter 6 “Nonlinear magnetoelastic interactions” is analogous to Chapter 4 and Chapter 7 “Magnetoelastic boundary value problems” to Chapter 5. In Chapter 8 “Variational formulations in electroelasticity and magnetoelasticity”, equations are treated in variational forms by use of the scalar and vector potential functions. In Chapter 9 “Incremental equations”, the equations of continuum electrodynamics in Lagrangian form provide the linearized equations associated with “small” incremental motions and electromagnetic fields. Chapter 10 “Electroelastic stability” illustrates the application of the equations governing the incremental behaviour of electroelastic solids in the analysis of stability of the half-space and of the finite thickness plate. In Chapter 11 “Magnetoelastic wave propagation”, incremental motions are considered superimposed on a static deformed configuration of magnetoelastic material based on the quasimagnetostatic approximation and applied to wave propagation. The surface waves are discussed on a deformed half-plane for various orientations of the underlying magnetic field. Useful formulas from vector and tensor calculus are summarized in the appendix.

Reviewer: Vladimir Mityushev (Kraków)

##### MSC:

78-02 | Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

78A25 | Electromagnetic theory, general |

74F15 | Electromagnetic effects in solid mechanics |

35Q61 | Maxwell equations |

74B20 | Nonlinear elasticity |