Elasticity (eBook)

Theory, Applications, and Numerics

Martin H. Sadd (Autor)

eBook Download: PDF

2010 | 1. Auflage
480 Seiten
Elsevier Science (Verlag)
978-0-08-047747-3 (ISBN)

Infos

Although there are several books in print dealing with elasticity, many focus on specialized topics such as mathematical foundations, anisotropic materials, two-dimensional problems, thermoelasticity, non-linear theory, etc. As such they are not appropriate candidates for a general textbook. This book provides a concise and organized presentation and development of general theory of elasticity. Complemented by a Solutions Manual and including MatLab codes and coding, this text is an excellent book teaching guide.

- Contains exercises for student engagement as well as the integration and use of MATLAB Software

- Provides development of common solution methodologies and a systematic review of analytical solutions useful in applications of engineering interest

- Presents applications of contemporary interest

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Although there are several books in print dealing with elasticity, many focus on specialized topics such as mathematical foundations, anisotropic materials, two-dimensional problems, thermoelasticity, non-linear theory, etc. As such they are not appropriate candidates for a general textbook. This book provides a concise and organized presentation and development of general theory of elasticity. This text is an excellent book teaching guide. - Contains exercises for student engagement as well as the integration and use of MATLAB Software- Provides development of common solution methodologies and a systematic review of analytical solutions useful in applications of

Inhalt

Cover 1
Frontmatter 2
Half Title Page 2
Title Page 4
Copyright 5
Preface 6
Contents 10
About the Author 13
Part I: Foundations and Elementary Applications 14
1. Mathematical Preliminaries 16
1.1 Scalar, Vector, Matrix, and Tensor Definitions 16
1.2 Index Notation 17
1.3 Kronecker Delta and Alternating Symbol 19
1.4 Coordinate Transformations 20
1.5 Cartesian Tensors 22
1.6 Principal Values and Directions for Symmetric Second-Order Tensors 25
1.7 Vector, Matrix, and Tensor Algebra 28
1.8 Calculus of Cartesian Tensors 29
1.9 Orthogonal Curvilinear Coordinates 32
References 37
Exercises 38
2. Deformation: Displacements and Strains 40
2.1 General Deformations 40
2.2 Geometric Construction of Small Deformation Theory 43
2.3 Strain Transformation 47
2.4 Principal Strains 48
2.5 Spherical and Deviatoric Strains 49
2.6 Strain Compatibility 50
2.7 Curvilinear Cylindrical and Spherical Coordinates 54
References 56
Exercises 56
3. Stress and Equilibrium 62
3.1 Body and Surface Forces 62
3.2 Traction Vector and Stress Tensor 64
3.3 Stress Transformation 67
3.4 Principal Stresses 68
3.5 Spherical and Deviatoric Stresses 71
3.6 Equilibrium Equations 72
3.7 Relations in Curvilinear Cylindrical and Spherical Coordinates 74
References 77
Exercises 77
4. Material Behavior—Linear Elastic Solids 82
4.1 Material Characterization 82
4.2 Linear Elastic Materials—Hooke’s Law 84
4.3 Physical Meaning of Elastic Moduli 87
4.4 Thermoelastic Constitutive Relations 90
References 92
Exercises 92
5. Formulation and Solution Strategies 96
5.1 Review of Field Equations 96
5.2 Boundary Conditions and Fundamental Problem Clasifications 97
5.3 Stress Formulation 101
5.4 Displacement Formulation 102
5.5 Principle of Superposition 104
5.6 Saint-Venant’s Principle 105
5.7 General Solution Strategies 106
References 111
Exercises 112
6. Strain Energy and Related Principles 116
6.1 Strain Energy 116
6.2 Uniqueness of the Elasticity Boundary-Value Problem 121
6.3 Bounds on the Elastic Constants 122
6.4 Related Integral Theorems 123
6.5 Principle of Virtual Work 125
6.6 Principles of Minimum Potential and Complementary Energy 127
6.7 Rayleigh-Ritz Method 131
References 133
Exercises 133
7. Two-Dimensional Formulation 136
7.1 Plane Strain 136
7.2 Plane Stress 139
7.3 Generalized Plane Stress 142
7.4 Antiplane Strain 144
7.5 Airy Stress Function 145
7.6 Polar Coordinate Formulation 146
References 148
Exercises 149
8. Two-Dimensional Problem Solution 152
8.1 Cartesian Coordinate Solutions Using Polynomials 152
8.2 Cartesian Coordinate Solutions Using Fourier Methods 162
8.3 General Solutions in Polar Coordinates 170
8.4 Polar Coordinate Solutions 173
References 204
Exercises 205
9. Extension, Torsion, and Flexure of Elastic Cylinders 214
9.1 General Formulation 214
9.2 Extension Formulation 215
9.3 Torsion Formulation 216
9.4 Torsion Solutions Derived from Boundary Equation 226
9.5 Torsion Solutions Using Fourier Methods 232
9.6 Torsion of Cylinders With Hollow Sections 236
9.7 Torsion of Circular Shafts of Variable Diameter 240
9.8 Flexure Formulation 242
9.9 Flexure Problems Without Twist 246
References 250
Exercises 251
Part II: Advanced Applications 256
10. Complex Variable Methods 258
10.1 Review of Complex Variable Theory 258
10.2 Complex Formulation of the Plane Elasticity Problem 265
10.3 Resultant Boundary Conditions 269
10.4 General Structure of the Complex Potentials 270
10.5 Circular Domain Examples 272
10.6 Plane and Half-Plane Problems 277
10.7 Applications Using the Method of Conformal Mapping 282
10.8 Applications to Fracture Mechanics 287
10.9 Westergaard Method for Crack Analysis 290
References 291
Exercises 292
11. Anisotropic Elasticity 296
11.1 Basic Concepts 296
11.2 Material Symmetry 298
11.3 Restrictions on Elastic Moduli 304
11.4 Torsion of a Solid Possessing a Plane of Material Symmetry 305
11.5 Plane Deformation Problems 312
11.6 Applications to Fracture Mechanics 325
References 328
Exercises 329
12. Thermoelasticity 332
12.1 Heat Conduction and the Energy Equation 332
12.2 General Uncoupled Formulation 334
12.3 Two-Dimensional Formulation 335
12.4 Displacement Potential Solution 338
12.5 Stress Function Formulation 339
12.6 Polar Coordinate Formulation 342
12.7 Radially Symmetric Problems 343
12.8 Complex Variable Methods for Plane Problems 347
References 355
Exercises 355
13. Displacement Potentials and Stress Functions 360
13.1 Helmholtz Displacement Vector Representation 360
13.2 Lamé’s Strain Potential 361
13.3 Galerkin Vector Representation 362
13.4 Papkovich-Neuber Representation 367
13.5 Spherical Coordinate Formulations 371
13.6 Stress Functions 376
References 378
Exercises 378
14. Micromechanics Applications 384
14.1 Dislocation Modeling 385
14.2 Singular Stress States 389
14.3 Elasticity Theory with Distributed Cracks 398
14.4 Micropolar/Couple-Stress Elasticity 401
14.5 Elasticity Theory with Voids 410
14.6 Doublet Mechanics 416
References 421
Exercises 422
15. Numerical Finite and Boundary Element Methods 426
15.1 Basics of the Finite Element Method 427
15.2 Approximating Functions for Two-Dimensional Linear Triangular Elements 429
15.3 Virtual Work Formulation for Plane Elasticity 431
15.4 FEM Problem Application 435
15.5 FEM Code Applications 437
15.6 Boundary Element Formulation 442
References 448
Exercises 448
Appendix A: Basic Field Equations in Cartesian, Cylindrical, and Spherical Coordinates 450
Strain-Displacement Relations 451
Equilibrium Equations 451
Hooke’s Law 452
Equilibrium Equations in Terms of Displacements (Navier’s Equations) 453
Appendix B: Transformation of Field Variables Between Cartesian, Cylindrical, and Spherical Components 455
Cylindrical Components from Cartesian 456
Spherical Components from Cylindrical 456
Spherical Components From Cartesian 457
Appendix C: MATLAB Primer 458
C.1 Getting Started 458
C.2 Examples 460
Reference 469
Index 470

Textauszug

1

Mathematical Preliminaries

Similar to other field theories such as fluid mechanics, heat conduction, and electromagnetics, the study and application of elasticity theory requires knowledge of several areas of applied mathematics. The theory is formulated in terms of a variety of variables including scalar, vector, and tensor fields, and this calls for the use of tensor notation along with tensor algebra and calculus. Through the use of particular principles from continuum mechanics, the theory is developed as a system of partial differential field equations that are to be solved in a region of space coinciding with the body under study. Solution techniques used on these field equations commonly employ Fourier methods, variational techniques, integral transforms, complex variables, potential theory, finite differences, and finite and boundary elements. Therefore, to develop proper formulation methods and solution techniques for elasticity problems, it is necessary to have an appropriate mathematical background. The purpose of this initial chapter is to provide a background primarily for the formulation part of our study. Additional review of other mathematical topics related to problem solution technique is provided in later chapters where they are to be applied.

1.1 Scalar, Vector, Matrix, and Tensor Definitions

Elasticity theory is formulated in terms of many different types of variables that are either specified or sought at spatial points in the body under study. Some of these variables are scalar quantities, representing a single magnitude at each point in space. Common examples include the material density ρ and material moduli such as Young’s modulus E, Poisson’s ratio v, or the shear modulus μ. Other variables of interest are vector quantities that are expressible in terms of components in a two- or three-dimensional coordinate system. Examples of vector variables are the displacement and rotation of material points in the elastic continuum. Formulations within the theory also require the need for matrix variables, which commonly require more than three components to quantify. Examples of such variables include stress and strain. As shown in subsequent chapters, a three-dimensional formulation requires nine components (only six are independent) to quantify the stress or strain at a point. For this case, the variable is normally expressed in a matrix format with three rows and three columns. To summarize this discussion, in a three-dimensional Cartesian coordinate system, scalar, vector, and matrix variables can thus be written as follows:

where e1, e2, e3 are the usual unit basis vectors in the coordinate directions. Thus, scalars, vectors, and matrices are specified by one, three, and nine components, respectively.

The formulation of elasticity problems not only involves these types of variables, but also incorporates additional quantities that require even more components to characterize. Because of this, most field theories such as elasticity make use of a tensor formalism using index notation. This enables efficient representation of all variables and governing equations using a single standardized scheme. The tensor concept is defined more precisely in a later section, but for now we can simply say that scalars, vectors, matrices, and other higher-order variables can all be represented by tensors of various orders. We now proceed to a discussion on the notational rules of order for the tensor formalism. Additional information on tensors and index notation can be found in many texts such as Goodbody (1982) or Chandrasekharaiah and Debnath (1994).

1.2 Index Notation

Index notation is a shorthand scheme whereby a whole set of numbers (elements or components) is represented by a single symbol with subscripts. For example, the three numbers a1, a2, a3 are denoted by the symbol ai, where index i will normally have the range 1, 2, 3. In a similar fashion, aij represents the nine numbers a11, a12, a13, a21, a22, a23, a31, a32, a33. Although these representations can be written in any manner, it is common to use a scheme related to vector and matrix formats such that

(1.2.1)

In the matrix format, a1j represents the first row, while ai1 indicates the first column. Other columns and rows are indicated in similar fashion, and thus the first index represents the row, while the second index denotes the column.

In general a symbol aij … k with N distinct indices represents 3N distinct numbers. It should be apparent that ai and aj represent the same three numbers, and likewise aij and amn signify the same matrix. Addition, subtraction, multiplication, and equality of index symbols are defined in the normal fashion. For example, addition and subtraction are given by

(1.2.2)

and scalar multiplication is specified as

(1.2.3)

The multiplication of two symbols with different indices is called outer multiplication, and a simple example is given by

(1.2.4)

The previous operations obey usual commutative, associative, and distributive laws, for example:

(1.2.5)

Note that the simple relations ai = bi and aij = bij imply that a1 = b1, a2 = b2, … and a11 = b11, a12 = b12, … However, relations of the form ai = bj or aij = bkl have ambiguous meaning because the distinct indices on each term are not the same, and these types of expressions are to be avoided in this notational scheme. In general, the distinct subscripts on all individual terms in an equation should match.

It is convenient to adopt the convention that if a subscript appears twice in the same term, then summation over that subscript from one to three is implied; for example:

(1.2.6)

It should be apparent that aii = ajj = akk = …, and therefore the repeated subscripts or indices are sometimes called dummy subscripts. Unspecified indices that are not repeated are called free or distinct subscripts. The summation convention may be suspended by underlining one of the repeated indices or by writing no sum. The use of three or more repeated indices in the same term (e.g., aiii or aiijbij) has ambiguous meaning and is to be avoided. On a given symbol, the process of setting two free indices equal is called contraction. For example, aii is obtained from aij by contraction on i and j. The operation of outer multiplication of two indexed symbols followed by contraction with respect to one index from each symbol generates an inner multiplication; for example, aijbjk is an inner product obtained from the outer product aijbmk by contraction on indices j and m.

A symbol aij … m … n … k is said to be symmetric with respect to index pair mn if

(1.2.7)

while it is antisymmetric or skewsymmetric if

(1.2.8)

Note that if aij … m … n … k is symmetric in mn while bpq … m … n … r is antisymmetric in mn, then the product is zero:

(1.2.9)

A useful identity may be written as

(1.2.10)

The first term a(ij) = 1/2(aij + aji) is symmetric, while the second term a[ij] = 1/2(aij − aji) is antisymmetric, and thus an arbitrary symbol aij can be expressed as the sum of symmetric and antisymmetric pieces. Note that if aij is symmetric, it has only six independent components. On the other hand, if aij is antisymmetric, its diagonal terms aii (no sum on i) must be zero, and it has only three independent components. Note that since a[ij] has only three independent components, it can be related to a quantity with a single index, for example, ai (see Exercise 1-14).

1.3 Kronecker Delta and Alternating Symbol

A useful special symbol commonly used in index notational schemes is the Kronecker delta defined by

(1.3.1)

Within usual matrix theory, it is observed that this symbol is simply the unit matrix. Note that the Kronecker delta is a symmetric symbol. Particular useful properties of the Kronecker delta include the following:

(1.3.2)

Another useful special symbol is the alternating or permutation symbol defined by

(1.3.3)

Consequently, ε123 = ε231 = ε312 = 1, ε321 = ε132 = ε213 = −1, ε112 = ε131 = ε222 = … = 0. Therefore, of the 27 possible terms for the alternating symbol, 3 are equal to +1, three are equal to −1, and all...

Produkt-Details

• Erscheint lt. Verlag: 4.8.2010
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Luft- / Raumfahrttechnik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-047747-X / 008047747X
• ISBN-13: 978-0-08-047747-3 / 9780080477473