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Some Basic Problems of the Mathematical Theory of Elasticity, by N.I. Muskhelishvili
Download PDF Some Basic Problems of the Mathematical Theory of Elasticity, by N.I. Muskhelishvili
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TO THE FIRST ENGLISH EDITION. In preparing this translation, I have taken the liberty of including footnotes in the main text or inserting them in small type at the appropriate places. I have also corrected minor misprints without special mention .. The Chapters and Sections of the original text have been called Parts and Chapters respectively, where the latter have been numbered consecutively. The subject index was not contained in the Russian original and the authors' index represents an extension of the original list of references. In this way the reader should be able to find quickly the pages on which anyone reference is discussed. The transliteration problem has been overcome by printing the names of Russian authors and journals also in Russian type. While preparing this translation in the first place for my own informa tion, the knowledge that it would also become accessible to a large circle of readers has made the effort doubly worthwhile. I feel sure that the reader will share with me in my admiration for the simplicity and lucidity of presentation.
- Sales Rank: #3513408 in Books
- Brand: Brand: Springer
- Published on: 1977-04-30
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.63" w x 6.14" l, 2.78 pounds
- Binding: Hardcover
- 732 pages
- Used Book in Good Condition
Most helpful customer reviews
4 of 4 people found the following review helpful.
Classic work from a Master
By docn
I.N. Muskhelishvili (with Kolosov and numerous colleagues, many of them in Tbilisi) developed and set the subject of complex variable methods applied to planar elastic problems on a firm mathematical footing in the early half of the 20th century. This excellent, authoritative and comprehensive work is organized as follows-
(1) Part I, chapters 1-3, pp 1-84 : Introduces stress, strain and the concept of the generalized Hooke's law for elasticity. This is fairly standard material, except that the more familiar Einstein summation convention is not used.
(2) Part II, chapters 4-7, pp 85-189: In chapter 4, planar approximations (plane stress and plane strain) are introduced. Chapter 5 introduces the concept of complex stress functions and the way they can be used to solve the bi-harmonic equation of planar elasticity. The strength of INM's work is its generality: multiply connected and infinite domains are also considered. The latter part of the chapter deals with questions of uniqueness and regularity of solutions. Chapter 6 is a brief but excellent introduction to the meaning of multi-valued displacements and elastic dislocations. Chapter 7 introduces the conformal transform for the Kolosov-Muskhelishvili equations.
(3) Part III, chapters 8-11, pp 190-256: Chapter 8 introduces the complex fourier series. Chapter 9 uses this technique to solve the first and second boundary value problems (specified traction and specified displacement respectively) for a disk and its complementary region (infinite plane with a hole). After solving the problems in generality, special cases where the potential functions have only a finite number of terms in expansion are considered. Chapter 10 tackles the ring in a similar way. Chapter 11 demonstrates elementary applications of conformal mapping.
(4) Part IV, chapters 12-13, pp 257-306: These two chapters provide a quick introduction to Cauchy integrals and boundary values of complex functions. Entire texts have been written on the subject, but the material contained here is adequate for the rest of this work.
(5) Part V, chapters 14-17, pp. 307-434: In this part, the reader will begin to appreciate the power of the complex variable approach. The organization is similar to Part III, i.e. problems are solved in great generality first and specific examples of interest considered subsequently. In the examples one may find such classic problems as the stretching of an infinite plate with an elliptic hole in addition to less common ones such as an arbitrary number of concentrated forces acting at interior points of a disk. The problem of an elliptic inclusion is also considered. Chapter 16 tackles half-plane problems and also the (rarer) case of a general semi-infinite region. Chapter 17 briefly considers numerous generalizations (multiply connected regions, mixed boundary value problems)
(6) Part VI, chapters 18-20, pp. 435-563: Chapter 18 introduces the Riemann Boundary Value Problem (called the 'Hilbert Problem' by INM) for finding a sectionally holomorphic function that satisfies a given 'jump' at the boundary. This elegant and powerful theory is then applied to solve numerous important problems for the half-plane (chapter 19). In the examples, the reader will encounter classic mixed boundary-value problems e.g. contact of a punch and a half-plane and crack-problems (infinite plane with straight cuts). Chapter 20 applies the BVP technique to regions with a circular boundary. Chapter 21 extends this to the case where the region of interest maps conformally to a circle.
(7) Part VII, chapters 22-25, pp. 564-661: This is a quite distinct part of the book and it considers torsion and bending of bars, both homogeneous and compound. The classic St. Venant problem is considered and the examples are at least mathematically fascinating (torsion of a Booth's lemniscate sectioned rod?)
(8) 3 useful appendices and numerous references. Unfortunately, many of the historic works referred here (by Magnaradze, Savin, Sherman, Shtaerman, Vekua) are in 30s-50s era Soviet journals and not available in translation.
The book was translated by Prof. J.R.M Radok (himself an elastician of repute) and the translation is generally excellent with very few errors.
It is unfortunate is that this beautiful work is being sold at an exorbitant price (> $200). This makes it prohibitively expensive for graduate students. One hopes that Kluwer academic will consider the value of this text and the importance of its wide availability and lower the price.
12 of 12 people found the following review helpful.
A masterpiece
By Sot P. Filopoulos
For those who are involved in any way with the mathematical theory of 2-D elasticity, this book is really a must have. Though the first edition goes back to 1949 (English edition by Noordhoff Ltd in 1953), it still contains everything that is known in this field. It actually employs the theory of holomorphic functions, Cauchy integrals and conformal mapping in order to solve the various boundary value problems met in plane elasticity. It is almost self contained as far as it concerns its mathematical background. Its results can be used to treat numerous applications of practical interst, such as cracks, inclusions, holes, contact problems, etc. Its a pitty it is out of print (or in print-on-demand only). It can be used along with the same author's "Singular Integral Equations" (another masterpiece on the subject) and Gakhov's:"Boundary Value Problems". Basic knowledge of complex analysis is a prerequisite. It covers 700 pages, contents are: 1. Analysis of stress 2. Analysis of strain 3. Basic Equations 4. Basic eqns of the plane theory of elasticity 5. Stress function-Complex representation 6.Multi-valued displacements. Thermal stresses 7. Transformation of the basic formulae for conformal mapping 8.On Fourier series 9. Solutions for regions bounded by a circe 10.The circular ring 11. Application of conformal mapping 12.Fundamental properties of Cauchy integrals 13.Boundary values of holomorphic functions 14.General solution for the fundamental problems for regions bounded by one contour 15.solution of the fundamental problems for regions mapped onto a circle by rational functions 16.solution for the half-plane and for semi-infinite regions 17.some general methods for solution of bouundary value problems 18.The problem of linear relationship 19.Solution for the half-plane and the plane with straight cuts 20.solution of boundary problems for regions bounded by circle and the infinite plane cut along circular arcs 21.solution of the boundary problems for regions mapped onto a circle by rational functions 22.Torsion and bending of homogenous bars 23.Torsion of bars consisting of different materials 24.Extension and bending of bars consisting of different materials with uniform Poisson's ratio 25.Extension and bending for different Poisson's ratio. Appendix 1. On the concept of a tensor Appendix 2.On the determination of functions from their differentials in multiply connected regions. Appendix 3. Determination of a function of a complex variable from its real part. Indefinite integrals of holomorphic functions
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