Stroh Formalism and Rayleigh Waves

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 equations of elasticity are complicated, because they constitute a system and, particularly for the anisotropic cases, inherit many parameters from the elasticity tensor. The Stroh formalism reveals simple structures hidden in the equations of anisotropic elasticity and provides a systematic approach to these equations. This exposition is divided into three chapters. Chapter 1 gives a succinct introduction to the Stroh formalism so that the reader could 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, for long a topic of utmost importance in nondestructive evaluation, seismology, and materials science. There we discuss existence, uniqueness, phase velocity, polarization, and perturbation of Rayleigh waves through the Stroh formalism.

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The Table of Contents and Index are also provided as Electronic Supplementary Material for online readers at doi:      

二维各向异性弹性力学的Stroh公式及其推广

Stroh公式是解决各向异性弹性体二维变形的强有力的工具。该公式形式上简洁优美、应用上方便有力。它的推广形式也日益在混合边值问题、压电材料、非对称弯曲板、三维各向异性弹性体、稳态运动、多自由度系统振动等方面得到广泛应用。本文简要介绍Stroh公式及其推广形式的研究状况与进展。     

Stroh formulation for a generally constrained and pre-stressed elastic material

The Stroh formalism is essentially a spatial Hamiltonian formulation and has been recognized to be a powerful tool for solving elasticity problems involving generally anisotropic elastic materials for which conventional methods developed for isotropic materials become intractable. In this paper we develop the Stroh/Hamiltonian formulation for a generally constrained and prestressed elastic material. We derive the corresponding integral representation for the surface-impedance tensor and explain how it can be used, together with a matrix Riccati equation, to calculate the surface-wave speed. The proposed algorithm can deal with any form of constraint, pre-stress, and direction of wave propagation. As an illustration, previously known results are reproduced for surface waves in a pre-stressed incompressible elastic material and an unstressed inextensible fibre-reinforced composite, and an additional example is included analyzing the effects of pre-stress upon surface waves in an inextensible material.     

含刚性线夹杂及裂纹的各向异性压电材料耦合场分析

采用各向异性弹性力学中Ｓｔｒｏｈ方法对含刚性线夹杂及裂纹的无限大各向异性压电材料耦合的弹性场和电场进行了分析。并得到夹杂和基体界面间耦合场的实型显函表达式及夹杂尖端的１／２阶奇异性。     

Proof of the Strong Eshelby Conjecture for Plane and Anti-plane Anisotropic Inclusion Problems

Based on the Stroh formalism for anisotropic elasticity and the complex variable function method, we prove in this paper that the strong Eshelby conjecture holds for simply-connected anisotropic inclusion problems under plane or anti-plane deformation. The interfaces can be either perfect or dislocation-like. For these inclusion problems, if the induced stress field inside the inclusion is uniform for a single uniform eigenstrain, the inclusion is of the elliptic shape. Thanks to the generality of the proof method, we obtain also alternative proofs of the strong Eshelby conjecture for isotropic inclusion problems, which are given in the Appendix.     

Symmetric Representation of Stress and Strain in the Stroh Formalism and Physical Meaning of the Tensors L, S, L(θ) and S(θ)

The Stroh formalism for two-dimensional deformation of an anisotropic elastic material does not give the stress _ij explicitly in a symmetric form. It does not give an explicit expression for the strain _ij at al. Mantic and Paris [1] have recently derived an explicit symmetric representation of stress. We present here a new and elementary derivation that is more straight forward and transparent. The derivation does not require consideration of the surface traction or the normalization of the Stroh eigenvectors. The new derivation also provides an explicit symmetric representation of strain. Moreover, it allows us to deduce two of the three Barnett–Lothe tensors L, S [2] and the associated tensors L ( ), S ( ) [3], resulting in a physical interpretation of these tensors and the component ( L S )₂₁.     

A state space formalism for anisotropic elasticity. Part I: Rectilinear anisotropy

A state space formalism for anisotropic elasticity including the thermal effect is developed. A salient feature of the formalism is that it bridges the compliance-based and stiffness-based formalisms in a natural way. The displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation, yet it is simple and clear. This is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three-dimensional equations of anisotropic elasticity into a compact state equation and an output equation. The homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of Stroh. Extension, twisting, bending, temperature change and body forces are accounted for through the particular solution. Based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner.     

Stroh-like formalism for the coupled stretching–bending analysis of composite laminates

Due to the two-dimensional nature of thin plates, the lamination theory considering the composite laminates with in-plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation. By following the steps of Stroh formalism for two-dimensional linear anisotropic elasticity, a displacement complex variable formalism developed by the other researchers was introduced and re-derived in a different but more Stroh-like way. In addition, a brand-new mixed formalism (mixed use of displacements and stresses as basic functions) is established to compensate the displacement formalism. In order to transfer all the related formulae and mathematical techniques of the Stroh formalism to these two formalisms, the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of Stroh formalism. Moreover, by using the presently developed mixed formalism, the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates. Furthermore, letting the coupling stiffness vanish, the formalism has been reduced to the case of symmetric laminates and checked by a recently developed Stroh-like formalism for the plate bending problems. The comparison between Stroh formalism for two-dimensional problem, Stroh-like formalism for plate bending problem, displacement formalism and mixed formalism is then made at the end of this paper, and through their connection some useful relations are obtained.     

Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.     

Symmetrical Representation of Stresses in the Stroh Formalism and its Application to a Dislocation and a Dislocation Dipole in an Anissotropic Elastic Medium

Symmetrical stress representation in the Stroh formalism for anisotropic elastic bodies is introduced and the range of its applicability is analysed. By making use of this stress representation new formulae for influence functions giving stresses in an infinite anisotropic medium subjected to a straight dislocation and a straight dislocation dipole are derived. The advantage of the new formulae is that they explicitly show the symmetrical structure of these influence functions not referred to previously. Relations of these influence functions to influence functions giving stresses and Airy stress function due to a straight wedge disclination, whose explicit expressions are also introduced, are derived. Application of these results in computation of stresses by the hypersingular and regularized Somigliana stress identities is discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Some explicit expressions of extended Stroh formalism for two-dimensional piezoelectric anisotropic elasticity

Since the extended Stroh formalism for two-dimensional piezoelectric anisotropic elasticity preserves essential features of Stroh formalism for pure elastic materials, it becomes important to get the corresponding explicit expressions of some important matrices frequently appeared in Stroh formalism. In this paper, explicit expressions are obtained for the fundamental matrix N, material eigenvector matrices A and B, and Barnett–Lothe tensors L, S and H. Although the explicit expressions are presented under the generalized plane strain and short circuit condition, by suitable replacement of the material constants they are still valid for the other two-dimensional states. To provide a clear picture of these expressions, two typical examples are presented, which are piezoelectric ceramics with two different poling axes.     

Surface waves in a rotating anisotropic elastic half-space

The Stroh formalism for surface waves in an anisotropic elastic half-space is extended to the case when the half-space rotates about an axis with a constant rotation rate. The sextic equation for the Stroh eigenvalues, the eigenvectors, the orthogonality and closure relations are obtained. The Barnett–Lothe tensors are no longer real, but two of them are Hermitian. Taziev’s equation is generalized and used to derive the polarization vector and the secular equation without computing the Stroh eigenvalues and eigenvectors. An alternative derivation using the method of first integrals by Mozhaev and Destrade yields new invariants that relate the displacement and stress and are independent of the depth from the free surface. Explicit expression of the polarization vector and the secular equation for monoclinic materials with the symmetry plane at x₃ = 0 with the rotation about the x₃-axis obtained by Destrade is re-examined, and new results are presented. Also presented is the one-component surface wave in the rotating half-space.     

Perturbation Formulas for Polarization Ratio and Phase Shift of Rayleigh Waves in Prestressed Anisotropic Media

This paper completes an earlier study (Tanuma and Man, Journal of Elasticity, 85, 21–37, 2006) where we derive a first-order perturbation formula for the phase velocity of Rayleigh waves that propagate along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a suitably-chosen, comparative, unstressed and isotropic state be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, here we derive first-order perturbation formulas for the changes in polarization ratio and phase shift of Rayleigh waves from their respective comparative isotropic value. Examples are given, which show that the perturbation formulas for phase velocity and polarization ratio can serve as a starting point for investigations on the possible advantages of using Rayleigh-wave polarization, as compared with using wave speed, for acoustoelastic measurement of stress.      

Cosserat (micropolar) elasticity in Stroh form

The theory of defects in Cosserat continua is sketched out in strict analogy to the theory of line defects in anisotropic elasticity (Stroh theory). This rewrite of the second order equilibrium equations of elasticity in a 3-dimensional space as first order equations in a 6-dimensional space is analogous to replacing the Laplace equation by the Riemann–Cauchy equations. For generalized plane strain of anisotropic micropolar (Cosserat) elasticity one obtains a 14-dimensional coupled linear system of differential equations of first order and for plane strain of anisotropic micropolar (Cosserat) elasticity we obtain a 6-dimensional coupled linear system of differential equations of first order.     

Explicit solutions for the coupled stretching–bending problems of holes in composite laminates

Although the classical lamination theory was developed long time ago, it is still not easy to apply this theory to find the analytical solutions for the curvilinear boundary value problems especially when the stretching and bending are coupled each other. To overcome the difficulties, recently we developed a Stroh-like formalism for the general composite laminates. By using this formalism, most of the relations for the coupled stretching–bending problems can be organized into the forms of Stroh formalism for two-dimensional anisotropic elasticity problems. With this newly developed Stroh-like formalism, it becomes easier to obtain an analytical solution for the coupled stretching–bending problems of holes in composite laminates. Because the Stroh-like formalism is a complex variable formalism, the analytical solutions for the whole field are expressed in complex form. Through the use of some identities derived in this paper, the resultant forces and moments around the hole boundary are obtained explicitly in real form. Due to the lack of analytical solutions for the general cases, the comparison is made with the existing analytical solutions for some special cases. In addition, to show the generality of our analytical solutions, several numerical examples are presented to discuss the coupling effect of the laminates and the shape effect of the holes.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

Transient motion of an anisotropic elastic half-space due to a buried line source

A method to deal with the two-dimensional transient problem of a line force or dislocation in an anisotropic elastic half-space is developed. The proposed formulation is similar to Stroh’s formalism for anisotropic elastostatics in that the two-dimensional anisotropic elastodynamic problem is cast into a six-dimensional eigenvalue problem and the solution is expressed in terms of the eigenvalues and eigenvectors. An analytic solution is obtained without performing integral transforms. Numerical examples are presented for a silicon half-space subjected to a line force or dislocation.     

General coupled solution of anisotropic piezoelectric materials with an elliptic inclusion

In this investigation, the Stroh formalism is used to develop a general solution for an infinite, anisotropic piezoelectric medium with an elliptic inclusion. The coupled elastic and electric fields both inside the inclusion and on the interface of the inclusion and matrix are given. The project supported by the National Natural Science Foundation of China     

The Stroh formalism and the reciprocity properties of reflection-transmission problems in crystal piezo-acoustics

The paper is concerned with the Helmholtz-Rayleigh reciprocity, which implies invariance of mode-into-mode transformation with respect to interchange of incident mode and reflected or transmitted mode. This concept is considered for a wide range of acoustic reflection-transmission problems in anisotropic piezoelectric media. Resorting to the ideas of the Stroh formalism and casting the wave solutions of a boundary problem into a self-orthogonal and complete set, we develop the common approach which allows us to prove the reciprocity properties in a similar fashion for reflection-transmission for various boundary conditions.     

Coupled fields of two-dimensional anisotropic magneto-electro-elastic solids with an elliptical inclusion

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