Complex Variable Methods in Elasticity

A. H. England

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486432304
• eBook ISBN: 9780486153414
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The plane strain and generalized plane stress boundary value problems of linear elasticity are the focus of this graduate-level text, which formulates and solves these problems by employing complex variable theory. The text presents detailed descriptions of the three basic methods that rely on series representation, Cauchy integral representation, and the solution via continuation. Its five-part treatment covers functions of a complex variable, the basic equations of two-dimensional elasticity, plane and half-plane problems, regions with circular boundaries, and regions with curvilinear boundaries. Worked examples and sets of problems appear throughout the text. 1971 edition. 26 figures.

1

FUNCTIONS OF A COMPLEX VARIABLE

In this chapter some of the basic definitions and properties of functions of a complex variable are stated as a preliminary to their use in later sections. It is hoped that sufficient detail has been included to enable readers to resolve points of difficulty without frequent recourse to the standard texts on this subject.^{7, 28, 43, 44}

1.1 Basic definitions

In the following it will be assumed that all definitions refer to curves and regions lying entirely in the complex plane.

An arc is a continuous non-intersecting line which has a continuously varying tangent except at a finite number of points. A contour is a simple closed arc, for example an ellipse.

We shall refer to an open connected set in the plane as a region. When a region which we denote by S⁺ has one or more non-intersecting contours as its boundary, the positive sense of description of each contour is taken to be that for which the region S⁺ lies to the left. For example when S⁺ is bounded internally by the contours C₁, C₂, . . ., C_n and externally by the contour C₀, then C₁, C₂, . . ., C_n have a clockwise sense of description and C₀ anticlockwise. This is illustrated in Figure 1.1 for the case n = 2. We denote the open set exterior to S⁺ and the bounding contours by S⁻ so that on moving in the positive sense along a bounding contour, S⁺ lies to the left and S⁻ to the right. In general S⁺ is a multiply connected region, being simply connected only when S⁺ is bounded by a single contour C₀.

Figure 1.1

1.2 Complex functions

Let S be an arbitrary point set in the complex plane, if to each point z₀ = x₀ + iy₀ of S there corresponds a complex number u(x₀, y₀) + iv(x₀, y₀) we say that a complex function θ(z) has been defined on S. The value of the function is

at the point z = x + iy where u, v are real functions of the variables x, y. We note that a specific functional dependence on z rather than say

is not assumed by this notation. For example

is a complex function.

In view of the relati...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Contents
• Introduction
• 1 Functions of a Complex Variable
• 2 The Basic Equations of Two-Dimensional Elasticity
• 3 Plane and Half-Plane Problems
• 4 Regions with Circular Boundaries
• 5 Regions with Curvilinear Boundaries
• Bibliography
• Summary of Formulae
• Answers to the Examples
• Index