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

2008 ◽  
Vol 45 (16) ◽  
pp. 4460-4473 ◽  
Author(s):  
Chyanbin Hwu
Keyword(s):  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Explicit Expressions

The Stroh Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
General Solution ◽  
Elastic Body ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Stroh's Formalism ◽  
Elasticity Problems ◽  
Anisotropic Elastic ◽  
Stroh’S Formalism

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of Eshelby, Read, and Shockley (1953). We therefore present the latter first. Not all results presented in this chapter are due to Stroh (1958, 1962). Nevertheless we name the sextic formalism after Stroh because he laid the foundations for researchers who followed him. The derivation of Stroh's formalism is rather simple and straightforward. The general solution resembles that obtained by the Lekhnitskii formalism. However, the resemblance between the two formalisms stops there. As we will see in the rest of the book, the Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. The possibility of extending the formalism to three-dimensional deformations is explored in Chapter 15.

Stroh-Like Complex Variable Formalism for the Bending Theory of Anisotropic Plates

2003 ◽  
Vol 70 (5) ◽  
pp. 696-707 ◽  
Author(s):  
C. Hwu
Keyword(s):  
Complex Variable ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Plate Bending ◽  
Two Dimensional ◽  
New Formalism ◽  
Anisotropic Plates ◽  
Present Formalism ◽  
Bending Problems ◽  
Almost All

Based upon the knowledge of the Stroh formalism and the Lekhnitskii formalism for two-dimensional anisotropic elasticity as well as the complex variable formalism developed by Lekhnitskii for plate bending problems, in this paper a Stroh-like formalism for the bending theory of anisotropic plates is established. The key feature that makes the Stroh formalism more attractive than the Lekhnitskii formalism is that the former possesses the eigenrelation that relates the eigenmodes of stress functions and displacements to the material properties. To retain this special feature, the associated eigenrelation and orthogonality relation have also been obtained for the present formalism. By intentional rearrangement, this new formalism and its associated relations look almost the same as those for the two-dimensional problems. Therefore, almost all the techniques developed for the two-dimensional problems can now be applied to the plate bending problems. Thus, many unsolved plate bending problems can now be solved if their corresponding two-dimensional problems have been solved successfully. To illustrate this benefit, two simple examples are shown in this paper. They are anisotropic plates containing elliptic holes or inclusions subjected to out-of-plane bending moments. The results are simple, exact and general. Note that the anisotropic plates treated in this paper consider only the homogeneous anisotropic plates. If a composite laminate is considered, it should be a symmetric laminate to avoid the coupling between stretching and bending behaviors.

Explicit Expressions of the Fundamental Elasticity Matrices of Stroh-Like Formalism for Symmetric/Unsymmetric Laminates

2002 ◽  
Vol 18 (3) ◽  
pp. 109-118 ◽  
Author(s):  
M.C. Hsieh ◽  
Chyanbin Hwu
Keyword(s):  
Composite Laminates ◽  
Stroh Formalism ◽  
Plate Bending ◽  
Two Dimensional ◽  
Real Form ◽  
The Real ◽  
Unsymmetric Laminates ◽  
Bending Problems ◽  
Explicit Expressions

AbstractBased upon our recent development of Stroh-like forma lism for symmetric/unsymmetric laminates, most of the relations for bending problems can be organized into the forms of Stroh formalism for two-dimensional problems. Through the use of Stroh-like formalism, the fundamental elasticity matrices Ni, S, H and L appear frequently in the real form solutions of plate bending problems. Therefore, the determination of these matrices becomes important in the analysis of plate bending problems. In this paper, by following the approach for two-dimensional problems, we obtain the explicit expressions of the fundamental elasticity matrices for symmetric and unsymmetric laminates, which are all expressed in terms of the extensional, bending and coupling stiffnesses of the composite laminates.

The Lekhnitskii Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Symmetry Plane ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Engineering Community ◽  
Anisotropic Elastic

A two-dimensional deformation means that the displacements ui, (i= 1,2,3) or the stresses σij depend on x1 and x2 only. Among several formalisms for two-dimensional deformations of anisotropic elastic materials the Lekhnitskii (1950, 1957) formalism is the oldest, and has been extensively employed by the engineering community. The Lekhnitskii formalism essentially generalizes the Muskhelishvili (1953) approach for solving two-dimensional deformations of isotropic elastic materials. The formalism begins with the stresses and assumes that they depend on x1 and x2 only. The Stroh formalism, to be introduced in the next chapter, starts with the displacements and assumes that they depend on x1 and x2 only. Therefore the Lekhnitskii formalism is in terms of the reduced elastic compliances while the Stroh formalism is in terms of the elastic stiffnesses. It should be noted that Green and Zerna (1960) also proposed a formalism for two-dimensional deformations of anisotropic elastic materials. Their approach however is limited to materials that possess a symmetry plane at x3=0. The derivations presented below do not follow exactly those of Lekhnitskii.

ISING MODEL IN THE MAGNETIC FIELD iπkT/2

1988 ◽  
Vol 02 (03n04) ◽  
pp. 471-481 ◽  
Author(s):  
K. Y. LIN ◽  
F. Y. WU
Keyword(s):  
Magnetic Field ◽  
Free Energy ◽  
Ising Model ◽  
Zero Field ◽  
Two Dimensional ◽  
Anisotropic Interactions ◽  
The Magnetic Field ◽  
Explicit Expressions

It is shown that the free energy and the magnetization of an Ising model in the magnetic field H = iπkT/2 can be obtained directly from corresponding expressions of these quantities in zero field, provided that the latter are known for sufficiently anisotropic interactions. Using this approach we derive explicit expressions of the free energy and the magnetization at H = iπkT/2 for a number of two-dimensional lattices.

scholarly journals Conservation laws in anisotropic elasticity I. Basic framework

1993 ◽  
Vol 443 (1917) ◽  
pp. 139-151 ◽  
Keyword(s):  
Conservation Laws ◽  
General Procedure ◽  
Anisotropic Elasticity ◽  
Two Dimensional ◽  
Real Form ◽  
Elastic Materials ◽  
Complete Class ◽  
First Order ◽  
Cauchy's Theorem ◽  
Anisotropic Elastic

A complete class of first order conservation laws for two dimensional deformations in general anisotropic elastic materials is derived. The derivations are based on Stroh’s formalism for anisotropic elasticity. The general procedure proposed by P. J. Olver for the construction of conservation integrals is followed. It is shown that the conservation laws are intimately connected with Cauchy’s theorem for complex analytic functions. Real-form conservation laws that are valid for degenerate or non-degenerate materials are given.

Singularities at Grain Triple Junctions in Two-Dimensional Polycrystals With Cubic and Orthotropic Grains

1996 ◽  
Vol 63 (2) ◽  
pp. 295-300 ◽  
Author(s):  
R. C. Picu ◽  
V. Gupta
Keyword(s):  
Grain Boundary ◽  
Plane Strain ◽  
Anisotropic Elasticity ◽  
Symmetric Case ◽  
Elastic Plane ◽  
Two Dimensional ◽  
Triple Junctions ◽  
Stress Singularities ◽  
Plane Strain Deformation ◽  
Singular Configuration

Stress singularities at grain triple junctions are evaluated for various asymmetric grain boundary configurations and random orientations of cubic and orthotropic grains. The analysis is limited to elastic plane-strain deformation and carried out using the Eshelby-Stroh formalism for anisotropic elasticity. For both the cubic and orthotropic grains, the most singular configuration corresponds to the fully symmetric case with grain boundaries 120 deg apart, and with symmetric orientations of the material axes. The magnitude of the singularities are obtained for several engineering polycrystals.

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