The mechanics of flexible bodies has a wide range of applications in engineering. Especially in structural and mechanical engineering, the dimensioning of flexible solid carrier elements requires knowledge of engineering mechanics calculation methods. In our book Elasticity Theory, the aim is to present the basic concepts and calculation methods of flexible bodies and carrier elements in a modern way. Care has been taken to ensure that the narrative language of our book Elasticity is in Pure Turkish. In this context, the “Turkish Dictionary” (2005), “Mathematics Terms Dictionary” (1983), “Physics Terms Dictionary” (1983) of the Turkish Language Association founded and led by Great Atatürk were selected as guides and Turkish equivalents were derived for French terms that have not yet been translated into Turkish. For example, instead of “integral”, “integral”, instead of “differential equation”, “derivative equation”, instead of “singular integral equation”, “singular integral equation”, instead of “elastic stability”, “flexible durability”, instead of “plastic”, “yoğurumcul”, instead of “snap-through buckling”, “transition buckling”, etc. The equivalents of the important Turkish terms used in English, the most common French language, are written in parentheses, and since English writing is not phonetic, their pronunciations are also given in square brackets. In writing the English pronunciations, Fahir İZ’s “English-Turkish Dictionary” (1971) (TDK Publication) book was used. The book consists of a total of 10 chapters. The first five of the chapters explain the Fundamental Topics of Flexible Science, and the following five chapters explain the Application Topics used in engineering. Unlike books with similar content, the first of the Basic Topics, Chapter 1, and the first of the Application Topics, Chapter 6, are presented with titles that will facilitate understanding of the topics. Thus, the necessary mathematical information for easy understanding of the book is summarized. Our Flexible Science book aims to be presented to a wide range of students at undergraduate, graduate and doctoral levels. The topic explanations are enriched with numerous examples and figures, and the solutions to the Problems to be Solved at the end of each chapter are also provided, thus making efforts to ensure that the topics are easily understood by a wide range of readers. The subject of Mechanical Vibrations and Wave Equations was added to the book (Chapter 8), with the aim of providing necessary information to engineers who will perform mechanical vibration dimensioning and to engineers who will perform earthquake engineering dimensioning in our country, which is located on the earthquake zone. Since the simplified dimensioning of the load-bearing structural elements such as rods, thin plates and thin shells is very important in structural engineering, it is explained in a separate section (Chapter 9). In the dimensioning of flexible structural elements, the stability of rods, plates, shells and their systems is very important and is explained under the heading of Flexible Stability in Chapter 10.
Since the number of pages of our Flexible Science book is more than 1350, we have reluctantly accepted that it be divided into two volumes: Volume 1 Basic Topics and Volume 2 Application Topics. Despite the care we have shown, there may be spelling and drawing errors and mistakes in our Flexible Science book. I hope that our esteemed readers will contribute to the next editions by forwarding the deficiencies they detect to the author's e-mail address.
I would like to express my gratitude to my teachers Prof. Dr. Sacit TAMEROĞLU, Prof. Dr. Erdoğan S. ŞUHUBİ and the good man, the late Prof. Dr. Vural CİNEMRE, who made great contributions to my development from my university student years to my academic career.
I would like to thank Ali Rıza HALIS, who did the challenging computer typing and drawing of the figures of my two-volume and comprehensive ESNEYBİLİM book.
İbrahim BAKIRTAŞ
Contents
1. GENERAL INFORMATION ABOUT NUMBERS IMPRINT AND COMPLEX NUMBERS
1.1. History of Complex Numbers
1.2. Definitions of Imaginary Numbers and Complex Numbers
1.3. Algebraic Operations Defined on the Set of Complex Numbers
1.4. Geometric Description of Complex Numbers
1.5. Polar Form of Complex Numbers
1.6. Euler's Formula, De Moivre's Identity
1.7. Roots of Complex Numbers of Various Orders
1.8. Calculation of Roots of Polynomials (Polynomials)
1.8.1. Fundamental Theorem of Algebra
1.9. Generalization of Complex Variables to Three-Dimensional Space: Quaternion
2. FUNCTIONS OF COMPLEX VARIABLES
2.1. Analysis of Various Complex Functions
2.1. 1. Polynomial Functions
2.1. 2. Rational Algebraic Functions
2.1. 3. Exponential Functions
2.1. 4. Logarithmic Functions
2.1. 5. Trigonometric Functions
2.1. 5. 1. Description of the cosz and sinz Functions
2.1. 5. 2. Description of the tanz Function
2.1. 6. Complex Hyperbolic Functions
2.1. 6. 1. Description of the coshz and sinhz Functions
2.1. 7. Inverse Trigonometric Functions
2.1. 8. Inverse Hyperbolic Functions
2.1. 9. Multi-valued Functions of the Form za and az
2.1. 9. 1. Multi-valued Functions of the Form (z – c)a
2.1. 9. 2. less Formed Multi-Valued Functions
2.2. Algebraic and Transcendental Function DefinitionsSimple Functions
2.3. Branch Points and Branch Sections Collective Information on Riemann Surfaces
2.3.1. Branch Points and Branch Sections of the Function w = z
2.3.2. Branch Points and Branch Sections of the Function w = z–a
2.3.3. Branch Points of the Function w = n (z–a)m
2.3.4. Branch Points of the Function w = (z–a)(z–b)
2.3.5. Branch Points of the Function w = (z–a)(z–b)(z–c)
2.3.6. Branch Points and Branch Sections of the Root of Degree n of the Polynomial P(z) with Simple and Multiple Roots
3. DERIVATIVES OF COMPLEX FUNCTIONS AND CAUCHY-RIEMANN EQUATIONS DERIVATIVE OPERATORS
3.1. Derivatives of Complex Functions and Cauchy-Riemann Conditions
3.2. Harmonic Functions and Laplace Equation
3.3. Derivative Rules
3.4. Formal Similarity of Derivatives of Multivalued Functions
3.5. Derivatives of Some Functions Important in Application
3.6. Entire Function and Meromorrque Function
3.7. Derivative Operators of Complex Functions
3.7.1. Gradient Operators ∇(del) and –∇(del line)
3.7.2. Divergence of a Vector
3.7.3. Curl of a Vector
3.7.4. Laplace Operator
4. INTEGRALS OF COMPLEX FUNCTIONS
4.1. Integrals of Complex Functions Along Lines
4.1. 1. Curve Forms in the Plane
4.1. 2. Regions
4.1. 3. Complex Integration
4.1. 4. General Properties of Complex Integrals
4.1. 5. Real Components of Complex Integrals
4.1. 6. Complex Integrals Depending on Parameters
4.1. 7. Antiderivative, Antiderivative Curve of Many-Valued Functions
4.1. 7.1. Calculation of Antiderivatives for C Consisting of Finite Number of Regular Ck(k = 1, 2, ..., n–1) Curves Piece by Piece
4.1. 7.2. Green's Theorem, Cauchy's Theorem and Cauchy-Goursat Theorem in the Plane . 162
4.1. 7.3. Cauchy-Goursat Theorem in n-fold Dependent Regions
4.1. 7.4. Deformation and Shifting of Integral Lines
4.1. 7.5. Morera's Theorem . 168
4.2. Cauchy-Integral Formula and Related Formulas . 168
4.2. 1. Cuchy Integral Formula for Regions Containing More Than One Pole Point
4.2. 2. Results of Cauchy's Formula . 172
4.2. 2.1. Locating Singularity Points on the Circumference Curve
4.2. 2. Cauchy Integral for n-fold Dependent Regions
4.2. 2.3. Successive Derivatives of Cauchy's Integral Formula
4.2. 2.4. Cauchy's Inequality
4.2.2.5. Gauss's Mean Value Theorem
4.2.2.6. Liouville's Theorem
4.2.2.7. Maximum Module Theorem
4.2.2.8. Minimum Module Theorem
4.2.2.9. Angular Variation TheoremLogarithmic Residue
4.2.2.10. Rouché's Theorem
4.2.2.11. Poisson's Integral Formulas
5. COMPLEX NUMBER SEQUENCES AND SERIES Absolutes and singularities of complex functions
5.1. Complex number sequences
5.2. Complex number series
5.2.1. Convergence of complex number series
5.2.2. Criteria for convergence of complex number series
5.3. Taylor series
5.3.1. Maclaurin series
5.3.2. Taylor series of rational functions
5.3.3. Addition, conjugation, real and imaginary parts of series
5.4. Cauchy product of series
5.4.1. Multiplicative inversion of power series
5.4.2. Rearrangement of power series
5.4.3. Nesting one series into another
5.4.4. Weierstrass double series theorem
5.5. Laurent Series
5.6. Zero Points and Singular Points of Complex Functions
5.6.1. Zero Points and Zero Point Orders of Complex Functions
5.6.2. Singularity and Discrete Singular Points of Complex Functions
6. RESIDUE THEOREM AND ITS APPLICATIONS
6.1. Cauchy Residue Theorem
6.2. Methods Used in Calculating Residues
6.2.1. Calculating Residues by Integration
6.2.2. Calculating Residues in the Case of a Function Having a Simple Pole at Point z0
6.2.3. Calculating Residues of Functions Having a Higher Order Pole
6.2.4. Other Methods in Calculating Residues
6.2.4.1. Calculation of Residue of Function of Form f(z) = ϕ(z)/ψ(z) in Case ψ(z0) = 0 and ψ′(z0) ≠ 0
6.2.4.2. Calculation of Residue of Higher Functions in Complex Structure
6.2.4.3. Definite Integrals of Trigonometric Functions in [0, 2π)
6.3. Improper Integrals
6.3.1. Infinite Region Integrals
6.3.2. Infinite Region Integrals
6.4. Auxiliary Theorems Used in Calculation of Infinite Integrals
6.4.1. First Jordan Auxiliary Theorems
6.4.2. Second Jordan Auxiliary Theorem
6.4.2.1. Second Jordan Lemma for Functions with Exponential Term Factors
6.5. Calculation of Residue at Infinity
6.5.1. Two Finite Regions and Calculation of Residue at Infinity
6.6. Examples of Integrals of Meromorc Functions
6.7. Plemelj-Sokhostki Prime Value Integral and Prime Value Integral Involving a Semicircular Arc
6.7.1. Plemelj-Sokhostki Prime Value Integral
6.7.2. Prime Value Formulas Involving a Semicircular Arc
6.8. Integral of Functions with Branch Points
6.9. Integral of Functions with More Than One Branch Point in a Finite Region, Other Indirect Integrals
6.10. Addition of Series Using the Residue Theorem
6.11. Mittag-Lefsher Expansion Theorem
7. ANALİTİK SÜRDÜRME, SONSUZ ÇARPIM SERİLERİ
7.1. Analitik Sürdürme
7.1.1. Tanımlar ve Temel Kavramlar
7.1.2. Taylor Serilerine Açma Yöntemiyle Analitik Sürdürme
7.1.3. Analitik Sürdürmenin Doğal Sınırı ya da Engeli
7.1.4. Analitik Sürdürmeye İlişkin Riemann
Teoremi ve Schwartz Yansıma İlkesi
8. AÇIKORUR BETİMLEMELER
8.1. Açıkorur Betimlemenin Temel Özellikleri
8.2. Küçük Şekillerin Açıkorur Dönüşümleri
8.3. Betimleme Fonksiyonunun Türevinin Yüksek Mertebeden Sıfır Olması Durumunda Dönüşüm
8.4. Riemann Betimleme Teoremi .428
8.5. Bazı Temel Dönüşümler .430
8.5.1. Öteleme, Dönme,
Büyültme ve Evritim Dönüşümleri .430
8.5.2. Çiftdoğrusal Dönüşüm .432
8.5.3. Jaukovsky Dönüşümü .435
8.5.4. Schwarz-Christoffel Dönüşümü
8.5.4a.
Schwarz-Christoffel Dönüşümünün Korunması İçin bilgiler
8.5.4b. Schwarz-Christoffel Dönüşümü
5. COMPLEX NUMBER SEQUENCES AND SERIES Absolutes and singularities of complex functions
5.1. Complex number sequences
5.2. Complex number series
5.2.1. Convergence of complex number series
5.2.2. Criteria for convergence of complex number series
5.3. Taylor series
5.3.1. Maclaurin series
5.3.2. Taylor series of rational functions
5.3.3. Addition, conjugation, real and imaginary parts of series
5.4. Cauchy product of series
5.4.1. Multiplicative inversion of power series
5.4.2. Rearrangement of power series
5.4.3. Nesting one series into another
5.4.4. Weierstrass double series theorem
5.5. Laurent Series
5.6. Zero Points and Singular Points of Complex Functions
5.6.1. Zero Points and Zero Point Orders of Complex Functions
5.6.2. Singularity and Discrete Singular Points of Complex Functions
6. RESIDUE THEOREM AND ITS APPLICATIONS
6.1. Cauchy Residue Theorem
6.2. Methods Used in Calculating Residues
6.2.1. Calculating Residues by Integration
6.2.2. Calculating Residues in the Case of a Function Having a Simple Pole at Point z0
6.2.3. Calculating Residues of Functions Having a Higher Order Pole
6.2.4. Other Methods in Calculating Residues
6.2.4.1. Calculation of Residue of Function of Form f(z) = ϕ(z)/ψ(z) in Case ψ(z0) = 0 and ψ′(z0) ≠ 0
6.2.4.2. Calculation of Residue of Higher Functions in Complex Structure
6.2.4.3. Definite Integrals of Trigonometric Functions in [0, 2π)
6.3. Improper Integrals
6.3.1. Infinite Region Integrals
6.3.2. Infinite Region Integrals
6.4. Auxiliary Theorems Used in Calculation of Infinite Integrals
6.4.1. First Jordan Auxiliary Theorems
6.4.2. Second Jordan Auxiliary Theorem
6.4.2.1. Second Jordan Lemma for Functions with Exponential Term Factors
6.5. Calculation of Residue at Infinity
6.5.1. Two Finite Regions and Calculation of Residue at Infinity
6.6. Examples of Equivalent Integrals of Meromorc Functions
6.7. Plemelj-Sokhostki Prime Value Integral and Prime Value Integral Involving a Semicircular Arc
6.7.1. Plemelj-Sokhostki Prime Value Integral
6.7.2. Prime Value Formulas Involving a Semicircular Arc
6.8. Integral of Functions with Branch Points
6.9. Integral of Functions with More than One Branch Point in a Finite Region, Other Indirect Integrals
6.10. Addition of Series by Using the Residue Theorem
6.11. Mittag-Lefsher Expansion Theorem
7. ANALYTICAL CONTINUATION, INFINITE PRODUCTION SERIES
7.1. Analytical Continuation
7.1.1. Definitions and Basic Concepts
7.1.2. Analytical Continuation by Expansion of Taylor Series
7.1.3. Natural Limit or Barrier of Analytical Continuation
7.1.4. Riemann Theorem and Schwartz Reflection Principle on Analytical Continuation
8. OPEN CONFORMAL DESCRIPTIONS
8.1. Basic Properties of Open Conformal Descriptives
8.2. Open Conformal Transformations of Small Shapes
8.3. Transformation in the Case Where the Derivative of the Describing Function is Zero to Higher Order
8.4. Riemann Descriptive Theorem .428
8.5. Some Basic Transformations .430
8.5.1. Translation, Rotation, Magnification and Convolution Transformations .430
8.5.2. Bilinear Transformation .432
8.5.3. Jaukovsky Transform .435
8.5.4. Schwarz-Christoffel Transform
8.5.4a. Information for Preservation of Schwarz-Christoffel Transform
8.5.4b. Schwarz-Christoffel Transform
9. INTRODUCTION TO SINGULAR INTEGRATED EQUATIONS
9.1. Classification of Integral Equations
9.1.1. Fredholm Integral Equations
9.1.2. Volterra Integral Equations
9.1.3. Singular Integral Equations
9.2. Fundamentals of Singular Integral Equations
9.2.1. Integral Lines of Singular Integral Equations
9.2.2. Topics on Functions Entering Singular Integral Equations Hölder-Lipschitz (H-L) (Hα) Conditions
9.2.3. Integrals of Cauchy Type .462
9.2.3a. Cauchy Formulas Defined on Simple, Piecewise, Regular, Continuous and Closed Lines (Addendum to Section 4.2)
9.2.3b. Cauchy Formulas for Special Density Functions
9.2.3c. Cauchy Type Integral Evaluated on Non-Self-Intersecting (Simple), Piecewise, Regular, Open or Closed Integration Curves
9.2.4. Original Integrals Principal Value of Cauchy Type Integrals
9.2.5. Sokhotski-Plemelj Formulas
9.2.5.1. Sokhotski-Plemelj Formulas for Angular Integration Lines7
9.2.6. Cauchy Type Integral Evaluated on Infinite Straight Lines
9.2.7. Calculation of Cauchy Type Integral Given on Infinite Straight Lines by Residue Theorem
9.2.8. Conditions for the Function ϕ(t) Given on a Non-Self-Intersecting Regular and Closed L Line to be the Boundary Value of the Analytic Function in the Interior Region and to be the Boundary Value of the Analytic Function in the Exterior Region
9.2.9. Changing the Order of Successive Integrations Poincaré-Bertrand Formula
9.2.9.1. Poincaré-Bertrand Formula with a Single Variable Density Function and Calculation of Cauchy Singular Integral Using This Formula
9.2.10. Harnack's Theorem
9.2.11. Extension of the Integration Line to (n + 1)-Fold Connected Regions
9.2.12. Piecewise Analytic Function for a Set of Closed Loop Curves L
9.2.13. Relation Between the Cauchy Kernel and Other Strong Singular Kernels
9.2.13.1. Relation Between Hilbert Kernel and Cauchy Kernel
9.2.14. Behavior of Cauchy Type Integration at Ends of Integration Line and Discontinuity Points of Density Function
9.2.14.1. Asymptotic Expansions of Singularity at Ends a and b of Open L Integration Line
9.2.14.2. Case of Density Function Showing Finite Jump Discontinuity
9.2.14.3. Case of Density Function Having Power Singularity at Ends and on Integration Arc
9.2.15. Conversion of Singular Integration with Logarithmic Kernel to Singular Integration with Cauchy Kernel
9.3. Riemann-Hilbert Boundary Value Problem (RHP) .507
9.3.1. Definition of Riemann-Hilbert Boundary Value Problem (RHP)
9.3.2. Explanations on Calculating the Extent of the Extent Function G(t)
9.3.3. Solution of Riemann-Hilbert Boundary Value Problem (RHP) in a Singly Connected Region
9.3.3.1. Solution of RHP where the Exponential Function G(t) is a Rational Function
9.3.3.2. Transformation of the Exponential Function G(t) to a Rational Function by Padé Approximation
9.3.4. Solution of Riemann-Hilbert Boundary Value Problem (RHP) in a Semi-Infinite Plane
9.3.5. Solution of Riemann-Hilbert Boundary Value Problem (RHP) in a Multiply Connected Region
9.3.5.1. Solution of Homogeneous RHP in a Multiply Connected Region
9.3.5.2. Solution of Heterogeneous RHP in a Multiply Connected Region
9.3.6. Solution of Hilbert Boundary Value Problem (HP)
9.3.6.1. Conjugates of Differentiable (Analytic) Functions for Semi-Infinite Plane and Unit Radius Loops
9.3.6.2. Hilbert Problem in Semi-Plane Region
9.3.6.3. Hilbert Problem in Unit Radius Loop Region
9.3.7. Exceptional (Discrete) Cases of RHP
9.3.7.1. Solution of RHP when the Exponential Function G(t) Goes from Integer Order to Zero and Infinity at Some Points on the Closed Boundary Curve
9.3.7.2. Case of RHP when the Closed Boundary Line L is Angular
9.4. Solution of Singular Integral Equations with Cauchy and Hilbert Kernels in the Case of Closed Boundary Lines by Transforming them into RHP
9.4.1. Solution of Singular Integral Equations with Cauchy Kernels
9.4.2. Solution of Singular Integral Equations with Hilbert Kernels
9.4.3. Approximate Solution of Singular Integral Equations with Closed Integral Lines
9.4.4. Solution of Singular Integral Equations in the Case of Angular Closed Integral Lines
9.4.5. Solution of Singular Integral Equations by Reducing them to Regular Integral Equations (Fredholm type) by the Arrangement Method
9.4.5.1. Some Properties of Singular Operators
9.4.5.2. Arrangement Operations, Conjugate Operations
9.4.5.3. Solution of Singular Integral Equations by the Carlemann–Vekua Method
9.5. Riemann-Hilbert Boundary Value Problems with Open Boundary Lines and Open Ends Converging at Nodal Points
9.5.1. Behavior of Cauchy Type Integral at Ends of Integral Lines and Density Function Discontinuities
9.5.2. Density Function with First Kind Discontinuity Cauchy Type Integral
9.5.3. Asymptotic Values of Cauchy Type Integral in the Case of Exponential Branching Singularity of Density Function
9.5.4. Asymptotic Behavior of Cauchy Type Integral and Principal Value Integral at Nodal Points Where Ends of Open Boundary Lines Meet
9.5.5. Solution of Riemann-Hilbert Problem (RHP) Consisting of Regular Open Boundary Lines (Arcs)
9.5.5.1. Solution of RHP in the Case of Single Open Boundary Line
9.5.5.2. Solution of RHP in the Case of n Open and Non-Intersecting Regular Boundary Lines
9.5.5.3. Solution of RHP with Open Boundary Lines Meet at Nodal Points
9.5.5.4. Closed Solution of Dominant Singular Integral Cycle
9.5.6. Introduction to Numerical Solution Methods of Singular Integral Equations with Open Boundary Lines
9.5.6.1. Jacobi Polynomial Information to be Used in the Solution
9.5.6.2. Solution of Integral Equations with Cauchy Kernels of the First Kind with Jacobi Polynomials
9.5.6.3. Solution of Integral Equation with Cauchy Kernel of the Second Kind
Prof. Dr. İbrahim BAKIRTAŞ
He was born in 1945 in the village of Turnalı (former name: Os) in the Araklı district of Trabzon province. Since there was no primary school in his village, he went to the Quran course in his village between the ages of 6 and 11. After becoming a hafiz at the age of 11, he started the primary school that opened in his village in 1956 after his age was reduced by a court decision to 1948. He finished primary school in 1960 in four years, ranking first in his class. He studied middle school and high school as a free boarding student at Trabzon High School and graduated from this school in 1966 with high honors and ranking first. He entered the most popular university of his time, the Istanbul Technical University Faculty of Civil Engineering, which he entered in 1966, and graduated with high honors in 1971 as a civil engineer after 5 years.
He worked in the construction of the Bosphorus 1st Suspension Bridge and in the Kocaeli YSE Provincial Directorate. He made building, bridge and factory projects in his private structural engineering office. During this time, he worked as an assistant at the higher education institutions affiliated with Yıldız Technical University and had students complete reinforced concrete and steel structure projects.
In July 1973, he was appointed as an assistant at the Faculty of Engineering and Architecture of ITU. In 1977, he received the title of Dr. with his thesis titled SOLUTION OF SOME ELASTICITY PROBLEMS FOR VARIABLE MEDIA OF ELASTICITY MODULUS. In 1980, he won a TÜBİTAK scholarship and conducted scientific studies as a postdoctoral scientific researcher at the Université Pierre et Marie Curie in Paris. The article he prepared here titled “Ondes de surface SH pures en élasticité in homogène” was published in the Journal de Mécanique théorique et appliquée and later in the book “Continuum Mechanics through the Ages - From the Renaissance to the Twentieth Century : From Hydraulics to Plasticity” by the famous French scientist Gerard A. Maugin. In this article, the closed solution of the very destructive Love waves formed in the earth's crust, in the case of the ground elasticity coefficients changing with depth, was given for the first time. İbrahim Bakırtaş published articles in international journals included in the science zikir index, and received the title of associate professor in 1981 and professor in 1988. He participated as a speaker in various science workshops and presented papers. In 2012, he was deemed worthy of the “Mustafa İnan Award” by the National Mechanics National Committee.
Prof. Dr. İbrahim BAKIRTAŞ, during his academic years at ITU Faculty of Civil Engineering, gave lectures on Statics, Dynamics, Strength of Materials, Theory of Elasticity, Theory of Plasticity, Solution Methods in Theory of Elasticity, Engineering Mathematics, Theory of Complex Functions to undergraduate, graduate and doctoral students and also took part in many projects in the field of Structural and Earthquake Engineering.