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## Über dieses Buch

by the author to the English edition The book aims to present a powerful new tool of computational mechanics, complex variable boundary integral equations (CV-BIE). The book is conceived as a continuation of the classical monograph by N. I. Muskhelishvili into the computer era. Two years have passed since the Russian edition of the present book. We have seen growing interest in numerical simulation of media with internal structure, and have evidence of the potential of the new methods. The evidence was especially clear in problems relating to multiple grains, blocks, cracks, inclusions and voids. This prompted me, when preparing the English edition, to place more emphasis on such topics. The other change was inspired by Professor Graham Gladwell. It was he who urged me to abridge the chain of formulae and to increase the number of examples. Now the reader will find more examples showing the potential and advantages of the analysis. The first chapter of the book contains a simple exposition of the theory of real variable potentials, including the hypersingular potential and the hypersingular equations. This makes up for the absence of such exposition in current textbooks, and reveals important links between the real variable BIE and the complex variable counterparts. The chapter may also help readers who are learning or lecturing on the boundary element method.

## Inhaltsverzeichnis

### Introduction

Abstract
K. F. Gauss called complex variables “a wonderful golden source”. In his letter to Bessel he wrote (Gauss [1]): “Analysis ... loses in beauty and value when imaginary quantities are neglected”.

### Chapter 1. Real Potentials of Elasticity Theory

Abstract
Consider the elastostatics equations for a region D, finite or infinite:
(1.1)
, where σ ij are components of a stress tensor in the global co-ordinates which are assumed to be Cartesian. Here we use Einstein’s summation rule. The components of the stress tensor are connected to the components of the strain tensor by Hooke’s law:
(1.2)
, where c ijkl = c ijlk = c klij are elastic constants; ε kl are the components of the strain tensor:
(1.3)
, u k are the components of the displacement vector u. Substitution of (1.3) into (1.2) and the result into (1.1) leads to the complete system of elasticity theory in terms of displacements:
(1.4)
.

### Chapter 2. Singular Solutions and Potentials in Complex Form

Abstract
In this Chapter we will follow the procedure described in Chapter 1. In fact, we will simply rewrite the equations in complex variable form. Such a rewriting gives significant benefit for the theoretical analysis and computations. We will need some results on complex variables and functions of complex variables that are given in the first section.

### Chapter 3. Complex Integral Equations of the Indirect Approach

Abstract
The potentials (13.7) and (13.9) satisfy plane elasticity equations for displacements. Hence, they may serve to represent displacement fields. The potentials (13.8) and (13.10) satisfy elasticity equations for stresses; they serve to find stresses corresponding to displacements (13.7) or (13.9). The potentials (13.11) and (13.12) satisfy equations for the resultant force. They may serve to find the resultant force corresponding to (13.7) or (13.9). However, before using these formulae for points within a considered region, we need to choose the density in the potential so that its limiting values satisfy prescribed boundary conditions. These conditions, given in real variables by (1.17)–(1.19), have the same form in the complex variables: it is sufficient to refer displacements, traction and their discontinuities to complex vectors introduced by formulae (11.6), (11.10). When using the resultant force, we obtain it by integration of traction in accordance with (11.17). In this case, there appear constants, sometimes unknown, which express the value of the resultant force at the initial point of each isolated part of the contour.

### Chapter 4. Complex Integral Equations of the Direct Approach

Abstract
The common procedure for deriving the equations of the direct approach is as follows: reciprocal (Betti’s) formula — Somigliana’s identities — integral equations. Certainly, we may try to shorten this way by starting directly from the real Somigliana’s formulae for displacements (8.6) and substituting the complex expressions for kernels and densities into it. In practice this is longer than following the whole chain. Thus we choose to take all the steps. In addition this will provide us with the complex variable form of Betti’s formula, which is important in itself.

### Chapter 5. Functions of Kolosov-Muskhelishvili and Holomorphicity Theorems

Abstract
It is well known (see, e. g. Muskhelishvili [5]) that the equations of plane elasticity can be reduced to the bi-harmonic equation for Airy’s stress function. Hence, two harmonic functions are sufficient to represent a general solution. This means that the general solution in complex variables is represented by two holomorphic functions of the complex argument z = x + iy. The classical Goursat formula gives the connection between the real Airy function and these two holomorphic functions. By taking displacements and stresses expressed in terms of Airy’s function and using Goursat’s formula, we obtain the displacements and stresses expressed in terms of these two complex functions. This idea was first employed by Kolosov [1] and comprehensively exploited by Muskhelishvili [5].

### Chapter 6. Complex Variable Integral Equations

Abstract
The Kolosov-Muskhelishvili formulae (25.1), (25.2) show that two functions φ(z) and Ψ(z), holomorphic in a simply connected region, define the complex displacements and the resultant force. On the other hand, the holomorphicity theorems are formulated in terms of the limiting values of such functions. Thus we may obtain CV-BIE by substituting the limiting values of φ(z) and Ψ(z) into the holomorphicity conditions and using the connection between these limiting values and the physical values expressed by the boundary conditions (25.10), (25.11) (or, in more general form, by (25.14), (25.15)). Alternatively, we may start from equations (25.6), (25.7), employing the functions φ(z) and Ψ(z) and use the boundary conditions (25.12), (25.13) (or, in more general case (25.16), (25.17)). The latter choice leads to equations that follow from the equations of the former after differentiation with respect to the free variable. It involves lengthier derivation; we prefer the first choice.

### Chapter 7. Periodic Problems

Abstract
Consider an infinite plane with periodically repeated holes and/or cracks (Figure 8). Later we will comment on how to extend the results to inclusions and blocky systems.

### Chapter 8. Doubly Periodic Problems

Abstract
We are interested in doubly periodic problems for two reasons. First, these problems arise in practice when dealing with perforated plates, regular composites and regular structures in solids and constructions. Secondly, as is especially important today, they arise from a general tendency of modern physics and applied science to account for details of the internal structure of a medium. A new branch of science, termed micromechanics, is swiftly growing up (see, e.g. Kemeny and Cook [1], Liu et al. [1], Napier and Pierce [1], Li and Wisnow [1], Tashkinov et al. [1], Dobroskok et al. [1]). Numerical simulation of doubly (in 2D) or triply (in 3D) periodic systems provides a unique opportunity to develop this science (Linkov and Koshelev [2]). It allows us to complement or substitute expensive and limited physical experiments by numerical ones. This approach has three advantages: (i) the cyclic constants of a displacement field provide rigorous definition of average (macroscopic) strains for prescribed average stresses, (ii) the ability to account for at least two hierarchical levels of structure, that of whole cells which may interact on their boundaries and that of internal elements of cells (grains, cracks, voids, etc.), (iii) calculations involve only a restricted area represented by the main cell. In 2D, the approach is strongly supported by the CV-BIE presented in this chapter.

### Chapter 9. Problems for Bonded Half-Panes and Circular Inclusion

Abstract
Problems concerning cracks, holes, blocks and/or inclusions near the interface of media with different properties, in particular near the free or attached boundary of a body, are of great interest in material science, fracture mechanics and geomechanics. In this chapter we present general methods for reducing these problems to CV-BEM (Linkov [10]). We employ a device presented in the next paragraph. It works if the K-M functions are known for the whole plane: then standard transformations lead to K-M functions for bonded half planes.

### Chapter 10. Complex Hypersingular and Finite-Part Integrals

Abstract
We want to justify the operations used when deriving equations, first of all hypersingular equations. What do we need to prove? In our discussion and derivations of hypersingular integrals, we employed the following formal elements:
a)
we assumed that there is a definition of divergent integrals that endows them with the usual properties of integrals, specifically with linearity and additivity;

b)
we used differentiation under the integral sign;

c)
we used integration by parts;

d)
we used formulae of Sokhotski-Plemelj type that connect direct values of divergent integrals with their limiting values.

### Chapter 11. Complex Variable Hypersingular Integral Equations (CVH-BIE)

Abstract
As noted in chapter 10, hypersingular integrals can be obtained by differentiation of singular integrals and/or integration by parts. This gives a connection between complex variable hypersingular and singular equations. Hence, we can construct a theory of complex variable hypersingular equations (CVH-BIE) by employing the well-established theory of singular integral equations (SIE). We use such an approach in the theoretical discussion in this chapter. Paragraphs 49–51 reproduce the results briefly presented in papers by Linkov and Mogilevskaya (Linkov and Mogilevskaya [4], Linkov, Zoubkov and Mogilevskaya [1]). The general case of the plane elasticity problem for a set of open and closed contours is presented in § 52 for the first time.

### Chapter 12. Complex Variable Bondary Element Method (CV-BEM)

Abstract
The boundary element method (BEM) is connected with elements of the boundary of a region. Boundary integral equations (BIE) serve as its basis. The essence of the method lies in a specific approach to solving such equations. The BEM uses additivity of integrals: an integral over a whole boundary (a surface in 3D or a contour in 2D) is equal to the sum of integrals over elements of which the boundary is composed or into which it is divided. So, first of all, the boundary is represented by a set of such elements termed boundary elements. This procedure is called discretization of a boundary.

### Chapter 13. Numerical Experiments Using CV-BEM

Abstract
In this chapter we present numerical results that show the accuracy and capability of the CV-BEM. In numerical examples we reveal the importance of using a complete set of basis functions and accounting for the asymptotic behavior of functions at singular points, in particular, by employing tip elements. Under a reasonably chosen approximation, a solution for displacements and SIF normally has at least three correct significant digits even for a rather small number of boundary elements. It is sufficient to follow the simple general rules discussed below.