Stroh Formalism and Rayleigh Waves

ISBN-13: 9781402063886 / Angielski / Twarda / 2007 / 159 str.

Kazumi Tanuma

Opis

Roger Fosdick The Journal of Elasticity: The Physical and Mathematical Science of Solids invites expository articles from time-to-time in order to collect results in areas of research that have made significant impact on the field of solid mechanics and that show continued interest in present-day research and thinking. The Stroh formalism is one such area, especially as it impacts upon the analysis of anisotropic media, that has been generalized from linear elastostatics to cover linear theories of piezoelectro-elasticity and megneto-elasticity as well as elastic wave phenomena. In this work, Kazumi Tanuma presents the essential elements of the Stroh formalism, its major theorems with proofs and applications, in a research-level textbook form. The presentation is self-contained and the development is clear and concise. Professor Tanuma s efforts to produce a straightforward, accurate and readable account of this subject are clearly evident. He has organized the subject with a common logical thread throughout and he has given basis for not only the importance of this area of research, but, most importantly, for understanding its meaning and significance."

Spis treści

Preface

Chapter 1: The Stroh Formalism for Static Elasticity

Section 1.1: Basic Elasticity

Section 1.2: Stroh's Eigenvalue Problem

Section 1.3: Rotational Invariance of Stroh Eigenvector in Reference Plane

Section 1.4: Forms of Basic Solutions When Stroh's Eigenvalue Problem is Degenerate

Section 1.5: Rotational Dependence When Stroh's Eigenvalue Problem is Degenerate

Section 1.6: Angular Average of Stroh's Eigenvalue Problem: Integral Formalism

Section 1.7: Surface Impedance Tensor

Section 1.8: Examples

Subsection 1.8.1: Isotropic Media

Subsection 1.8.2: Transversely Isotropic Media

Section 1.9: Justification of the Solutions in the Stroh Formalism

Section 1.10: Comments and References

Section 1.11: Exercises

Chapter 2: Applications in Static Elasticity

Section 2.1: Fundamental Solutions

Subsection 2.1.1: Fundamental Solution in the Stroh Formalism

Subsection 2.1.2: Formulas for Fundamental Solutions: Examples

Section 2.2: Piezoelectricity

Subsection 2.2.1: Basic Theory

Subsection 2.2.2: Extension of the Stroh Formalism

Subsection 2.2.3: Surface Impedance Tensor of Piezoelectricity

Subsection 2.2.4: Formula for Surface Impedance Tensor of Piezoelectricity: Example

Section 2.3: Inverse Boundary Value Problem

Subsection 2.3.1: Dirichlet to Neumann map

Subsection 2.3.2: Reconstruction of Elasticity Tensor

Subsubsection 2.3.2.1: Reconstruction of Surface Impedance Tensor from Localized Dirichlet to Neumann Map

Subsubsection 2.3.2.2: Reconstruction of Elasticity Tensor from Surface Impedance Tensor

Section 2.4: Comments and References

Section 2.5: Exercises

Chapter 3:  Rayleigh waves in the Stroh formalism

Section 3.1: The Stroh Formalism for Dynamic Elasticity

Section 3.2: Basic Theorems and Integral Formalism

Section 3.3: Rayleigh Waves in Elastic Half-space

Section 3.4: Rayleigh Waves in Isotropic Elasticity

Section 3.5: Rayleigh Waves in Weakly Anisotropic Elastic Media

Section 3.6: Rayleigh Waves in Anisotropic Elasticity

Subsection 3.6.1: Limiting Wave Solution

Subsection 3.6.2: Existence Criterion Based on S_3

Subsection 3.6.3: Existence Criterion Based on Z

Subsection 3.6.4: Existence Criterion Based on Slowness Sections

Section 3.7: Comments and References

Section 3.8: Exercises

 

 

 

Blurb

The Stroh formalism is a powerful and elegant mathematical method developed for the analysis of the equations of anisotropic elasticity. The purpose of this exposition is to introduce the essence of this formalism and demonstrate its effectiveness in both static and dynamic elasticity.

The exposition is divided into three chapters. Chapter 1 gives a succinct introduction to the Stroh formalism so that the reader can grasp the essentials as quickly as possible. In Chapter 2 several important topics in static elasticity, which include fundamental solutions, piezoelectricity, and inverse boundary value problems, are studied on the basis of the Stroh formalism. Chapter 3 is devoted to Rayleigh waves, which has long been a topic of the utmost importance in nondestructive evaluation, seismology, and materials science. Here existence, uniqueness, phase velocity, polarization, and perturbation of Rayleigh waves are discussed through the Stroh formalism.

This work will appeal to students and researchers in applied mathematics, mechanics, and engineering science.

Reprinted from the Journal of Elasticity, Vol. 89:1-3, 2007.