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 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.

Crack-Tip, Stress-Intensity Factors for Plane Extension and Plate Bending Problems

1962 ◽  
Vol 29 (2) ◽  
pp. 306-312 ◽  
Author(s):  
G. C. Sih ◽  
P. C. Paris ◽  
F. Erdogan
Keyword(s):  
Stress Intensity ◽  
Complex Variable ◽  
Stress Intensity Factors ◽  
Plate Bending ◽  
Intensity Factors ◽  
Variable Approach ◽  
Crack Tips ◽  
Arbitrary Plane ◽  
Fracture Theory ◽  
Bending Problems

A complex variable method for evaluating the strength of stress singularities at crack tips in plane problems and plate bending problems is derived. The results of these evaluations give Irwin’s stress-intensity factors for plane problems and analogous quantities for bending problems, a form familiar to the practitioner of “fracture mechanics.” The methods derived are integrated with the complex variable approach of Muskhelishvili to obtain the stress-intensity factors for various basic examples applicable to the extension and bending of plates with through-the-thickness cracks. The results suggest the possibility of extension of the Griffith-Irwin fracture theory to arbitrary plane extensional and/or bending problems in plates.

scholarly journals ELEMENT DKT_CST FOR ANALYSIS OF LAMINATED ANISOTROPIC PLATES/ELEMENTAS DKT_CST SLUOKSNIUOTŲ ANIZOTROPINIŲ PLOKŠTELIŲ ANALIZEI

2000 ◽  
Vol 6 (5) ◽  
pp. 351-356
Author(s):  
Edvard Michnevič ◽  
Rimantas Belevičius
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Triangular Element ◽  
Plate Bending ◽  
Anisotropic Plates ◽  
Numerical Examples ◽  
Out Of Plane ◽  
Bending Problems ◽  
Slab Bending ◽  
Plate Bending Problem

The new finite element of multilayered built up with an arbitrary series of layers plate for plate bending problem is formulated on the ground of widely used, effective finite element Discrete Kirchhof Theory (DKT). The material of each layer is supposed to be different and orthotopic. Triangular element has 6 d.o.f.'s at each of 3 nodal points: 3 displacements and 2 rotations about co-ordinate axes. The 6th fictitious rotation about axis perpendicular to the element is also introduced due to numerical requirements. The element takes into account all the in-plane/out-of-plane effects except the shear. The element could find an application in the slab bending problems or in the plate, where the shear influence could be neglected, bending problems. The numerical examples are presented. Present solutions are compared with available analytical and numerical solutions.

A Unified Formalism of Two-Dimensional Anisotropic Elasticity, Piezoelectricity and Unsymmetric Laminated Plates

2005 ◽  
Vol 72 (3) ◽  
pp. 422-431
Author(s):  
Wan-Lee Yin
Keyword(s):  
General Solution ◽  
Constitutive Relations ◽  
Anisotropic Elasticity ◽  
Complex Variables ◽  
Laminated Plates ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Eigenvalues And Eigenvectors ◽  
Infinite Domain ◽  
Anisotropic Plates

A unified formalism is presented for theoretical analysis of plane anisotropic elasticity and piezoelectricity, unsymmetric anisotropic plates, and other two-dimensional problems of continua with linear constitutive relations. Complex variables are used to reduce the governing differential equations to algebraic equations. The constitutive relation then yields an eigenrelation, which is easily solved explicitly for the material eigenvalues and eigenvectors. The latter have polynomial expressions in terms of the eigenvalues. When the eigenvectors are combined after multiplication by arbitrary analytic functions containing the corresponding eigenvalues, one obtains the two-dimensional general solution. Important results, including the orthogonality of the eigenvectors, the expressions of the pseudometrics and the intrinsic tensors, are established here for nondegenerate materials, including the case of all distinct eigenvalues. Green’s functions of the infinite domain, and of the semi-infinite domain with interior or edge singularities, are determined explicitly for the most general types of point loads and discontinuities (dislocations).

scholarly journals THE LAYERED ELEMENT DKT_CST FOR ANALYSIS OF ANISOTROPIC PLATES

2007 ◽  
Vol 13 (1) ◽  
pp. 41-46
Author(s):  
Edvard Michnevič
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Triangular Element ◽  
Plate Bending ◽  
Anisotropic Plates ◽  
Numerical Examples ◽  
Out Of Plane ◽  
Bending Problems ◽  
Slab Bending ◽  
Plate Bending Problem

The new finite element of multilayered built up with an arbitrary series of layers plate for plate bending problem is formulated on the ground of widely used, effective finite element Discrete Kirchhof Theory (DKT). The material of each layer is supposed to be different and orthotropic. Triangular element has 6 d.o.f.’s at each of 3 nodal points: 3 displacements and 3 rotations about co‐ordinate axes. The element takes into account all the in‐plane/out‐of‐plane effects except for shear. The element could find an application in the slab bending problems or in the plate, where the shear influence could be neglected, bending problems. Numerical examples are presented. Present solutions are compared with available analytical and numerical solutions.

Discussion on the conformal mapping of a half-plane onto a unit disk in anisotropic elasticity and related applications

2020 ◽  
Vol 26 (1) ◽  
pp. 110-117
Author(s):  
Ming Dai ◽  
Jian Hua
Keyword(s):  
Thin Films ◽  
Conformal Mapping ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Variable ◽  
Plane Deformation ◽  
Anisotropic Elasticity ◽  
Two Dimensional ◽  
Plane Shear ◽  
Anisotropic Elastic

The conformal mapping, which transforms a half-plane into a unit disk, has been used widely in studies involving an isotropic elastic half-plane under anti-plane shear or plane deformation. However, very little attention has been paid to the possibility of utilizing this mapping in the study of an anisotropic elastic half-plane under the same deformation. In this paper, we discuss a general case of an arbitrarily located anisotropic elastic half-plane that corresponds to several affine counterparts (resulting from corresponding complex variable formalism). We show that this mapping is indeed applicable to each of the affine half-planes if and only if the key parameters in the mapping satisfy simple geometrical conditions. In addition, we introduce the application of this mapping with the corresponding geometrical conditions to the related study of anisotropic thin films under two-dimensional deformation.

Sign in / Sign up

Export Citation Format

Share Document