Contents 4
From the translation editor 10
Preface to the third edition 13
Preface to the second edition 15
Preface to the first edition 16
Designations 20
Chapter 1 Introduction 22
§ 1. Elasticity 22
§ 2. Stresses 23
§ 3. Notation for forces and stresses 24
§ 4. Stress components 25
§ 5. Deformation components 26
§ 6. Hooke's Law 28
§ 7. Index notations 32
Tasks 34
Chapter 2 Plane Stress and Plane Strain 35
§ 8. Plane stress consisted 35
§ 9. Plane deformation 35
§ 10. Stresses at point 37
§ 11. Deformations at point 42
§ 12. Measurement of surface deformations 44
§ 13. Construction of the circle of Mohr deformations for rosette 46
§ 14. Differential equilibrium equations 46
§ 15. Boundary conditions 47
§ 16. Compatibility equations 48
§ 17. Stress function 50
Tasks 52
Chapter 3. 2D Cartesian Problems 54
§ 18. Solution in polynomials 54
§ 19. End effects. Principle of Saint-Venant 58
§ 20. Definition of displacements 59
§ 21. Bending of the console loaded at the end 60
§ 22. Bending of a beam with a uniform load 64
§ 23. Other cases of beams with continuous load distribution 69
§ 24. Solution of a two-dimensional problem using Fourier series 71
§ 25. Other applications of Fourier series. Self-weight load 77
§ 26. Influence of condos. Native Functions 78
Tasks 80
Chapter 4. 2D Problems in Polar Coordinates 83
§ 27. General equations in polar coordinates 83
§ 28. Polar-symmetric stress distribution 86
§ 29. Pure bending of curved bars 89
§ 30. Deformation components in polar coordinates 93
§ 31. Displacements with symmetrical stress zeros 94
§ 32. Rotating discs 97
§ 33. Bending of a curved beam with a force applied at the end 100
§ 34. Edge dislocations 105
§ 35. Influence of a round hole on the distribution of stresses in a plate 106
§ 36. Concentrated force applied at some point of a rectilinear boundary 113
§ 37. Arbitrary vertical load on a rectilinear boundary 119
§ 38. The force acting on the tip of the wedge 125
§ 39. Bending moment acting on the tip of the wedge 127
§ 40. Action on a beam of a concentrated force 128
§ 41. Stresses in a round disk 137
§ 42. Force acting at a point of an infinite plate 141
§ 43. Generalized solution of a two-dimensional problem in polar coordinates 146
§ 44. Applications of the generalized solution in polar coordinates 150
§ 45. Wedge loaded along the faces 153
§ 46. Own solutions for wedges and cutouts 155
Tasks 158
Chapter 5. Experimental methods. Photoelasticity and moiré method 163
§ 47. Experimental methods and verification of theoretical solutions 163
§ 48. Measurement of stresses by photoelastic method 163
§ 49. Circular polariscope 169
§ 50. Examples of determining stresses by the photoelastic method 171
§ 51. Determination of principal stresses 174
§ 52. Methods of photoelasticity in the three-dimensional case 175
§ 53. Moire method 177
Chapter 6
§ 54. Functions of a complex variable 180
§ 55. Analytic functions and the Laplace equation 182
§ 56. Stress functions expressed in terms of harmonic and complex functions 184
§ 57. Displacements corresponding to a given stress function 186
§ 58. Expression of stresses and displacements through complex potentials 188
§ 59. Resultant stress acting on a certain curve. Boundary conditions 190
§ 60. Curvilinear coordinates 193
§ 61. Stress components in curvilinear coordinates 196
Tasks 198
§ 62. Solutions in elliptic coordinates. Elliptical Hole in a Plate with a Uniform Stress State 198
§ 63. Elliptical hole in a plate subjected to uniaxial tension 202
§ 64. Hyperbolic boundaries. Cutouts 206
§ 65. Bipolar coordinates 208
§ 66. Solutions in bipolar coordinates 209
§ 67. Determination of complex potentials by given boundary conditions. Methods of N. I. Muskhelishvili 214
§ 68 Formulas for complex potentials 217
§ 69. Properties of stresses and strains corresponding to complex potentials, analytical in the area of ​​the material located around the hole 219
§ 70. Theorems for boundary integrals 221
§ 71. Mapping function ω(ξ) for an elliptic hole. Second boundary integral 224
§ 72. Elliptical hole. Formula for ψ(ζ) 225
§ 73. Elliptical hole. Particular tasks 226
Tasks 229
Chapter 7 Stress and Strain Analysis in the Spatial Case 230
§ 74 Introduction 230
§ 75. Principal stresses 232
§ 76. Ellipsoid of stresses and directing surface of stresses 233
§ 77. Determination of principal stresses 234
§ 78. Stress invariants 235
§ 79. Determination of the maximum shear stress 236
§ 80. Homogeneous deformation 238
§ 81. Deformations at a point of a body 239
§ 82. Principal strain axes 242
§ 83. Rotation 243
Tasks 245
Chapter 8 General Theorems 246
§ 84. Differential equilibrium equations 246
§ 85. Conditions of compatibility 247
§ 86. Definition of displacements 250
§ 87. Equilibrium equations in displacements 251
§ 88. General solution for displacements 252
§ 89. Superposition principle 253
§ 90. Deformation energy 254
§ 91. Strain energy for an edge dislocation 259
§ 92. The principle of virtual work 261
§ 93. Castigliano's theorem 266
§ 94. Applications of the principle of minimum work. Rectangular plates 270
§ 95. Effective width of wide flanges of beams 273
Tasks 279
§ 96. Uniqueness of the solution 280
§ 97. Reciprocity theorem 282
§ 98. Approximate nature of solutions for a plane stress state 285
Tasks 287
Chapter 9. Elementary three-dimensional problems of the theory of elasticity 289
§ 99. Uniform stress state 289
§ 100. Tension of a prismatic rod under its own weight 290
§ 101. Torsion of round shafts of constant cross section 293
§ 102. Pure bending of prismatic rods 294
§ 103. Pure bending of plates 298
Chapter 10
§ 104. Torsion of rectilinear rods 300
§ 105. Elliptical cross section 305
§ 106. Other elementary solutions 307
§ 107. Membrane analogy 310
§ 108. Torsion of a rod of narrow rectangular cross section 314
§ 109. Torsion of rectangular rods 317
§ 110. Additional results 320
§ 111. Solution of torsion problems by the energy method 323
§ 112. Torsion of rods of rolled profiles 329
§ 113. Experimental analogies 331
§ 114. Hydrodynamic analogies 332
§ 115. Torsion of hollow shafts 335
§ 116. Torsion of thin-walled pipes 339
§ 117. Screw dislocations 343
§ 118. Torsion of a rod, one of the cross sections of which remains flat 345
§ 119. Torsion of round shafts of variable diameter 347
Tasks 355
Chapter 11
§ 120. Bending of the console 359
§ 121. Stress function 361
§ 122. Round cross-section 363
§ 123. Elliptical cross section 364
§ 124. Rectangular cross section 365
§ 125. Additional results 371
§ 126. Asymmetric cross-sections 373
§ 127. Bend center 375
§ 128. Solving bending problems using the soap film method 378
Section 129 Transfers 381
§ 130. Further studies of the bending of the bars 382
Chapter 12
§ 131. General equations 384
§ 132. Solution in polynomials 387
§ 133. Bending of a round plate 388
§ 134. The three-dimensional problem of a rotating disk 391
§ 135. Force applied at some point of an infinite body 393
§ 136. Spherical vessel under the influence of internal or external uniform pressure 396
§ 137. Local stresses around a spherical cavity 399
§ 138. Force applied on the boundary of a semi-infinite body 401
§ 139. Load distributed over part of the boundary of a semi-infinite body 405
§ 140. Pressure between two contacting spherical bodies 412
§ 141. Pressure between two bodies in contact. More general case 417
§ 142. Collision of balls 422
§ 143. Symmetrical deformation of a round cylinder 424
§ 144. Round cylinder under the action of girdle pressure 428
§ 145. Solution of Boussinesq in the form of two harmonic functions 430
§ 146. Stretching of a helical spring (screw dislocations in a ring) 431
§ 147. Pure bending of a part of a round ring 434
Chapter 13 Thermal Stresses 436
§ 148. The simplest cases of distribution of thermal stresses. Deformation Method 436
Tasks 442
§ 149. Longitudinal temperature change in the strip 442
§ 150. Thin round disk: temperature distribution symmetrical about the center 445
§ 151. Long round cylinder 447
Tasks 455
§ 152. Scope 455
§ 153. General equations 459
§ 154. The reciprocity theorem in thermoelasticity 463
§ 155. Total thermoelastic deformations. Arbitrary temperature distribution 464
§ 156. Thermoelastic displacements. Integral solution of V. M. Maizel 466
Tasks 469
§ 157. Initial stresses 469
§ 158. General change in volume associated with initial stresses 472
§ 159. Plane deformation and plane stress state. Deformation Method 472
§ 160. Two-dimensional problems with stationary heat flow 474
§ 161. Plane thermally stressed state caused by perturbation of a homogeneous heat flow by an insulated hole 480
§ 162. Solutions of general equations. Thermoelastic displacement potential 481
§ 163. General two-dimensional problem for circular regions 485
§ 164. General two-dimensional problem. Solution in complex potentials 487
Chapter 14. Propagation of waves in an elastic continuous medium 490
§ 165 Introduction 490
§ 166. Expansion waves and distortion waves in an isotropic elastic medium 491
§ 167. Plane waves 492
§ 168. Longitudinal waves in rods of constant section. Elementary Theory 497
§ 169. Longitudinal impact of rods 502
§ 170. Rayleigh surface waves 510
§ 171. Waves with spherical symmetry in an infinite medium 513
§ 172. Explosive pressure in a spherical cavity 514
Application. Application of finite difference equations in the theory of elasticity 518
§ 1. Derivation of finite difference equations 518
§ 2. Methods of successive approximations 522
§ 3. Relaxation method 525
§ 4. Triangular and hexagonal grids 530
§ 5. Block and group relaxation 535
§ 6. Torsion of rods with multiply connected cross sections 536
§ 7. Points located near the boundary 538
§ 8. Biharmonic equation 540
§ 9. Torsion of circular shafts of variable diameter 548
§ 10. Solving problems with the help of a computer 551
Name Index 553
Index 558

In chapters 4-6, the basic equations of the theory of elasticity were derived, which establish the laws of change in stresses and strains in the vicinity of an arbitrary point of the body, as well as relations relating stresses to strains and strains to displacements. We present the complete system of elasticity equations in Cartesian coordinates.

Navier equilibrium equations:

Cauchy relations:

Hooke's law (in direct and inverse forms):

Recall that here e = e x + e y + ez- relative volumetric deformation, and according to the law of pairing of shear stresses Xj. = Tj; and accordingly y~ = ^ 7 . The Lame constants included in (16.3, a) are determined by formulas (6.13).

It can be seen from the above system that it includes 15 differential and algebraic equations containing 15 unknown functions (6 components of the stress tensor, 6 components of the strain tensor, and 3 components of the displacement vector).

Due to the complexity of the complete system of equations, it is impossible to find a general solution that would be valid for all problems of the theory of elasticity encountered in practice.

There are various ways to reduce the number of equations if, for example, only stresses or displacements are taken as unknown functions.

If, when solving the problem of the theory of elasticity, we exclude displacements from consideration, then instead of the Cauchy relations (16.2), we can obtain equations that relate the components of the strain tensor to each other. Differentiate the deformation r x, defined by the first equality (16.2), two times in y, deformation g y - twice for x and add the resulting expressions. As a result, we get

The expression in brackets, according to (16.2), determines the angular deformation y. Thus, the last equality can be written as

Similarly, two more equalities can be obtained, which, together with the last relation, form the first group compatibility equations of Saint-Venant deformations:

Each of the equalities (16.4) establishes a connection between deformations in one plane. The Cauchy relations can also be used to obtain compatibility conditions relating deformations in different planes. We differentiate expressions (16.2) for angular deformations as follows: y - by z y - by X;

By y; Add the first two equalities and subtract the third. As a result, we get

Differentiating this equality with respect to y and considering that,

we arrive at the following relation:

With the help of a circular substitution, we obtain two more equalities, which, together with the last relation, constitute the second group of Saint-Venant deformation compatibility equations:

The deformation compatibility equations are also called the conditions continuity or continuity. These terms characterize the fact that the body remains solid during deformation. If we represent the body as consisting of individual elements and accept the deformations e x, y as arbitrary functions, then in the deformed state it will not be possible to add up a solid body from these elements. When conditions (16.4), (16.5) are met, the displacements of the boundaries of individual elements will be such that the body will remain solid even in the deformed state.

Thus, one of the ways to reduce the number of unknowns in solving problems of the theory of elasticity is to exclude displacements from consideration. Then, instead of the Cauchy relations, the complete system of equations will include the compatibility equations for Saint-Venant deformations.

Considering the complete system of equations of the theory of elasticity, one should pay attention to the fact that it practically does not contain factors that determine the stress-strain state of the body. Such factors include the shape and dimensions of the body, methods of fixing it, loads acting on the body, with the exception of body forces. X, Y, Z.

Thus, the complete system of equations of the theory of elasticity establishes only general patterns of changes in stresses, strains, and displacements in elastic bodies. The solution of a specific problem can be obtained if the loading conditions of the body are given. This is given in the boundary conditions, which distinguish one problem of the theory of elasticity from another.

From a mathematical point of view, it is also clear that the general solution of a system of differential equations includes arbitrary functions and constants, which must be determined from the boundary conditions.

4. STRUCTURE OF THE EARTH ACCORDING TO SEISMOLOGY DATA

Fundamentals of the theory of elasticity: strain tensor, stress tensor, Hooke's law, elastic moduli, uniform deformations, elastic waves in an isotropic medium, Fermat's, Huygens', Snell's laws. seismic waves. Development of seismometric observations: seismic stations and their networks, hodographs, wave trajectories inside the Earth. Determination of the propagation velocity of seismic waves using the Gertlotz-Wiechert equation. P- and S-wave velocities as a function of the Earth's radius. The state of the Earth's matter according to seismology. Earth's crust. Lithosphere and asthenosphere. Seismology and global tectonics.

Fundamentals of the theory of elasticity[Landau, Lifshitz, 2003, p. 9-25, 130-144]

Strain tensor

The mechanics of solids, considered as continuous media, is the content elasticity theory. The basic equations of the theory of elasticity were established by O.L. Cauchy and S.D. Poisson in the 1920s (see Chapter 15 for details).

Under the influence of applied forces, solid bodies are deformed to one degree or another, i.e. change their shape and volume. For a mathematical description of the deformation of the body proceed as follows. The position of each point of the body is determined by its radius vector r (with components x 1 = x, x 2 = y, x 3 = z) in some coordinate system. When a body is deformed, all its points, generally speaking, are displaced. Let us consider some definite point of the body; if its radius vector before deformation was r , then in the deformed body it will have some other

value r / (with x i / components). The displacement of the point of the body during deformation will then be represented by the vector r / - r, which we denote by the letter u:

u = x/ − x .

Vector u name strain vector(or displacement vector). Knowledge of the vector u

as a function of x i completely determines the deformation of the body.

When a body is deformed, the distances between its points change. If the radius vector between them before deformation was dx i , then in the deformed body the radius

the vector between the same two points will be dx i / = dx i + du i . The very distance between the points before deformation was equal to:

dl = dx1 2 + dx2 2 + dx3 2 ,

and after deformation:

dl / = dx 1 / 2 + dx 2 / 2 + dx 3 / 2 .

Finally we get:

dl / 2 = dl 2 + 2u

∂u i

∂uk

∂u l

∂u l

∂xk

∂xk

∂x i

∂x i

These expressions determine the change in the length element when the body is deformed. The tensor u ik is called strain tensor; by definition, it is symmetrical:

u ik = u ki .

Like any symmetric tensor, the tensor u ik at each point can be reduced to

main axes and make sure that in each volume element of the body, the deformation can be considered as a set of three independent deformations in three perpendicular directions - the main axes of the strain tensor. Almost in almost all cases of deformation of bodies, the deformations turn out to be small. This means that the change in any distance in the body is small compared to the distance itself. In other words, the relative elongations are small compared to unity.

Except for some special cases, which we will not touch on, if the body is subjected to a small deformation, then all components of the strain tensor are also small. Therefore, in expression (4.3), the last term can be neglected as a small quantity of the second order. Thus, in the case of small strains, the strain tensor is determined by the expression:

u = 1			∂u i	+ ∂u k ) .
			∂xk	∂x i

So, forces are the cause of movements (displacements) arising in the body, and deformations are the result of movements [Khaikin, 1963, p. 176].

The main assumption of the classical theory of elasticity

In an undeformed body, the arrangement of molecules corresponds to the state of its thermal equilibrium. At the same time, all its parts are with each other and in mechanical equilibrium. This means that if any volume is singled out inside the body, then the resultant of all forces acting on this volume from other parts is equal to zero.

When deformed, the arrangement of molecules changes, and the body is removed from the state of equilibrium in which it was originally. As a result, forces will arise in it, striving to return the body to a state of equilibrium. These internal forces arising from deformation are called internal stresses. If the body is not deformed, then there are no internal stresses in it.

Internal stresses are caused by molecular bonds, i.e. forces of interaction of body molecules with each other. Very important for the theory of elasticity is the fact that molecular forces have a very small radius of action. Their influence extends around the particle that creates them only at a distance of the order of intermolecular. But in the theory of elasticity, as in the macroscopic theory, only distances that are large compared to intermolecular ones are considered. Therefore, the "range of action" of molecular forces in the theory of elasticity must be considered equal to zero. It can be said that the forces that cause internal stresses are "short-range" forces in the theory of elasticity, which are transmitted from each point only to the points closest to it.

Thus, in the classical theory of elasticity, forces acting on any part of the body from the parts surrounding it exhibit this action. only directly through the surface this part of the body.

In fact, the author of the fundamental work [Khaikin, 1963, p. 484].

Stress tensor

The conclusion that all forces exert their action only through the surface is the key to the classical theory of elasticity. It allows for any body volume each of the three components of the resultant of all internal stress forces

∫ F i dV (where F i is the force acting on a unit volume dV ) is converted to an integral over the surface of this volume. In this case, as follows from the vector analysis, the vector F i must be the divergence of some tensor of the second rank, i.e. look like:

F i = ∂ σ ik . (4.6)

∂xk

Then the force acting on a certain volume can be written as an integral over a closed surface covering this volume:

∫ Fi dV = ∫ ∂ ∂ σ x ik			= / σ ik df k ,
where vector d f = df 2			Df 2	directed	along the outer normal to the surface,

covering the volume dV .

The tensor σ ik is called stress tensor. As can be seen from (4.7), σ ik df k is the i -th

component of the force acting on the surface element d f . Choosing the surface elements in the planes xy, уz, xz, we find that the component σ ik of the stress tensor

is the ith component of the force acting on a unit surface perpendicular to the x k axis. So, on a unit area perpendicular to the x axis, normal to

her (directed along the x axis) force σ xx and tangential (directed along the y and z axes)

forces σ yx and σ zx .

Note that the force acting from the side of internal stresses on the entire surface of the body, in contrast to (4.7), is:

− ∫ σ ik df k .

Writing the moment of forces M ik acting on a certain volume of the body in the form:

M ik = ∫ (F i x k − F k x i ) dV

and requiring that it be expressed as an integral over the surface only, we get that the stress tensor is symmetrical:

σ ik = σ ki .

A similar conclusion can be reached in a simpler way [Sivukhin, 1974, p. 383]. Namely. The moment dM ik is directly proportional to the moment of inertia of the elementary

volume dM ik ≈ I ≈ (dV )5 / 3 and, therefore, we obtain (F i x k − F k x i )dV = dM ik ≈ (dV )5 / 3 ≈ 0 , whence relation (4.8) follows automatically.

The symmetry of the stress tensor allows it to be brought to the principal axes at each point, i.e. at each point, the stress tensor can be represented as:

σ ik = σ xx + σ yy + σ zz .

In equilibrium, the forces of internal stresses must be mutually compensated in each element of the volume of the body, i.e. should be F i = 0 . So the equations

equilibrium of a deformed body have the form:

∂σ ik = 0 .

∂xk

If the body is in the field of gravity, then the sum F + ρ g of the forces of internal stresses F and the force of gravity ρ g acting on a unit volume, ρ -

the density of the body, g is the free fall acceleration vector. The equilibrium equations in this case have the form:

∂ σ ik + ρ g i = 0 .
∂xk

Deformation energy

Consider some deformed body and assume that its deformation changes so that the deformation vector u i changes by a small amount δ u i .

Let us determine the work done in this case by the forces of internal stresses. Multiplying the force (4.6) by the displacement δ u i and integrating over the entire volume of the body, we obtain:

		∫ ∂ xk
	δRdV =		∂σik	δ ui dV .

The symbol δ R denotes the work of the forces of internal stresses in a unit volume of the body. Integrating by parts, considering an unbounded medium not deformed at infinity, letting the integration surface go to infinity, then σ ik = 0 on it, we obtain:

∫ δ RdV = − ∫ σ ik δ uik dV .

Thus, we find:

δ R = − σ ikδ u ik .

The resulting formula determines the work to change the strain tensor, which determines the change in the internal energy of the body.

Russian State University

oil and gas them. I.M. Gubkina

Department of Technical Mechanics

ABSTRACT

"Theory of elasticity"

Completed by: Polyakov A. A.

Checked by: Evdokimov A.P.

Moscow 2011

elasticity theory equation

1. Introduction

Theory of stress-strain state at a point of a body

2.1 Stress theory

2 Deformation theory

3 Relation between stressed and deformed state for elastic bodies

Basic equations of the theory of elasticity. Types of problems in the theory of elasticity

1 Basic equations of elasticity theory

2 Types of problems in the theory of elasticity

4 Equations of the theory of elasticity in displacements (Lame equations)

Variational principles of elasticity theory

1 Principle of possible displacements (Lagrange principle)

2 Principle of possible states (Castillano principle)

3 Relationship between the exact solution and solutions obtained based on the Lagrange and Castigliano principles

List of used literature

1. Introduction

Theories of stresses and deformations were created by O. Cauchy. They are set forth in a work submitted to the Paris Academy of Sciences in 1822, a summary of which was published in 1823 and a number of subsequent articles. O. Koshi derived three equations of equilibrium of an elementary tetrahedron, proved the law of pairing of tangential stresses, introduced the concepts of principal axes and principal stresses, and derived differential equilibrium equations (usually they are not derived in the course of the strength of materials). He also introduced the surface of normal stresses (the Cauchy quadric), on which the ends of the radius vectors are located, the directions of which coincide with the direction of the normals to the areas, and the value is inversely proportional to the square root of the absolute value of the normal stress in this area, and it was proved that this surface is surface of the second order centered at the origin. The possibility of transforming the surface of normal stresses to principal axes indicates the existence of three mutually principal perpendicular areas at each point.

A similar shear stress surface was introduced by the Russian mechanic G.V. Kolosov in 1933

The geometric interpretation of the stress state in space in the form of an ellipsoid of stresses was given by G. Lame and B. Clapeyron in their memoirs submitted to the Paris Academy of Sciences in 1828 and published in 1833.

The geometric representation of the stress state on a plane for one series of platforms passing through the main axis, in the form of a circle of stresses, was proposed by K. Kuhlman in his book in 1866.

For the general case of the stress state, a very clear geometric interpretation of it on the plane was given by O. Mohr (the so-called circular Mohr diagram) in 1882. From it, a number of important conclusions can be drawn about the extremity of the principal stresses, the position of the areas in which the tangential stresses are maximum, and about the values ​​of these maximum shear stresses.

O. Cauchy gave a definition of strains, derived their dependence on displacements in the particular case of small strains (these dependencies, as a rule, are not derived in the course of the resistance of materials), defined the concepts of principal stresses and principal strains, and obtained the dependences of stress components on strain components, as for isotropic, and for an anisotropic elastic body. In the resistance of materials, the dependences of the strain components on the stress components for an isotropic body are usually established. They are called the generalized Hooke's law, although, of course, this name is arbitrary, since R. Hooke did not know the concept of stress.

In these dependencies, Cauchy first introduced two constants and wrote down the dependences of stresses on strains in the form

m, ,

However, later O. Koshi adopted the concept of L. Navier. According to it, elastic bodies consist of molecules, between which, during deformation, forces arise that act in the directions of straight lines connecting the molecules and are proportional to the change in the distances between the molecules. Then the number of elastic constants for the general case of an anisotropic body is 15, and for an isotropic body we obtain one elastic constant. S. Poisson adhered to this hypothesis, and at the beginning - G. Lame and B. Clapeyron. Based on it, Poisson found that the coefficient of transverse strain is 1/4.

D. Green in 1839 derived the relationship between strains and stresses without using the hypothesis of the molecular structure of elastic bodies. He obtained them based on the principle of conservation of energy, introducing the concept of elastic potential, and showed that when using the linear dependences of six strain components on six stress components, 21 out of 36 coefficients are independent, i.e. in the general case of an anisotropic body, the number of elastic constants is 21 For an isotropic body, the number of elastic constants is reduced to two. A theory in which the number of elastic constants for an anisotropic body is 15, and for an isotropic body 1, was sometimes called "raric constant" or "uniconstant", and a theory in which the number of elastic constants for an anisotropic body is 21, and for an isotropic 2 - "multi-constant" .

The dispute between the supporters of these theories prompted physicists to experimental research.

G. Wertheim, based on measurements of the internal volumes of glass and metal pipes in axial tension, established in 1848 that the coefficient of transverse deformation is not equal to 1/4. He considered it different for various materials, but for many materials it was close to 1/3.

AND I. Kupffer, in 1853, testing metal rods for tension and torsion, also found that the ratio of moduli in shear and tension does not correspond to a transverse strain equal to 1/4.

In 1855, F. Neumann tested samples of rectangular cross-section for bending and measured the angles of rotation of two faces of the beam (the cross section takes on a trapezoidal shape). As a result, he showed that the coefficient of transverse deformation is not equal to 1/4. G. Kirchhoff, a student of F. Neumann, came to the same conclusion on the basis of tests carried out in 1859 for joint bending and torsion of round brass rods, sealed at one end and loaded at the other with a concentrated force, with the measurement of the angle of twist of the rod and the angle of rotation of the section .

A large experimental study of the coefficients of transverse deformation for various grades of steel was carried out by one of the students of G. Kirchhoff, M.F. Okatov in 1865 - 1866 The results are given in his doctoral dissertation. Torsion and bending tests of thin prisms cut from single crystals, as well as tests of the compressibility of crystals under uniform equal compression, were carried out by V. Voigt and described in his numerous articles, later combined in a book published in 1910 They confirmed the correctness of the multi-constant theory.

A deep study of the mathematical structure of Hooke's law for anisotropic bodies was carried out by the mechanic and engineer Jan Rychlevsky in 1984 on the basis of the concept of an elastic eigenstate he introduced. In particular, he showed that 21 elastic constants represent six true stiffness moduli, 12 stiffness distributors, and three angles.

2. Theory of stress-strain state at a point of a body

1 Stress theory

Internal force factors that arise when an elastic body is loaded characterize the state of a particular section of the body, but do not answer the question of which point of the cross section is the most loaded, or, as they say, the dangerous point. Therefore, it is necessary to introduce into consideration some additional quantity characterizing the state of the body at a given point.

If a body to which external forces are applied is in equilibrium, then internal resistance forces arise in any of its sections. Denote by the internal force acting on the elementary area , and the normal to this area through then the value

called full voltage.

In the general case, the total stress does not coincide in direction with the normal to the elementary area, so it is more convenient to operate with its components along the coordinate axes -

If the external normal coincides with any coordinate axis, for example, with the X axis, then the stress components will take the form, while the component turns out to be perpendicular to the section and is called normal stress, and the components will lie in the section plane and are called shear stresses.

To easily distinguish between normal and shear stresses, other designations are usually used: - normal stress, - shear.

Let us single out from the body under the action of external forces an infinitely small parallelepiped, whose faces are parallel to the coordinate planes, and whose edges have length . On each face of such an elementary parallelepiped, there are three stress components that are parallel to the coordinate axes. In total, we get 18 stress components on six faces.

Normal stresses are denoted as , where the index denotes the normal to the corresponding face (that is, it can take the values ​​). Shear stresses have the form ; here the first index corresponds to the normal to the site on which the given shear stress acts, and the second indicates the axis parallel to which this stress is directed (Fig. 1).

Fig.1. Normal and shear stresses

For these voltages, the following sign rule is adopted. The normal stress is considered positive in tension, or equivalently, when it coincides with the direction of the outer normal to the site on which it acts. The tangential stress is considered positive if on the site, the normal to which coincides with the direction of the coordinate axis parallel to it, it is directed towards the positive coordinate axis corresponding to this voltage.

Stress components are functions of three coordinates. For example, the normal stress at a point with coordinates can be denoted

At a point that is at an infinitesimal distance from the one under consideration, the voltage, up to infinitesimals of the first order, can be expanded in a Taylor series:

For platforms that are parallel to the plane, only the x-coordinate changes, and the increments Therefore, on the face of the parallelepiped, which coincides with the plane, the normal stress will be Therefore, out of 18 voltage components, only nine are unknown.

In the theory of elasticity, the law of pairing of shear stresses is proved, according to which, along two mutually perpendicular areas, the components of shear stresses, perpendicular to the lines of intersection of these areas, are equal to each other:

Equalities (2) lead to the fact that out of nine stress components characterizing the stress state at a point of the body, only six remain:

It can be shown that stresses (3) not only characterize the stress state of the body at a given point, but determine it uniquely. The combination of these stresses forms a symmetrical matrix, which is called the stress tensor:

(4)

When multiplying a tensor by a scalar value, a new tensor is obtained, all components of which are times larger than the components of the original tensor.

2 Deformation theory

Under the influence of external loads, the elastic body changes its shape and deforms. In this case, the points of the body take some new position. To determine the deformation of an elastic body, we compare the positions of the points of the body before and after the application of the load.

Consider a point of an unloaded body and its new position after the application of the load. The vector is called the point displacement vector (Fig. 2).

Fig.2. Point moving vector

Two types of displacements are possible: displacement of the whole body as a whole without deformation - such displacements are studied by theoretical mechanics as displacements of an absolutely rigid body, and displacement associated with the deformation of the body - such displacements are studied by the theory of elasticity.

Let us designate the projections of the point displacement vector on the coordinate axes as respectively. They are equal to the difference between the corresponding coordinates of the points and :

and are functions of coordinates:

The deformation of the body is caused by the difference in the displacements of its various points. An infinitely small parallelepiped with edges cut out of an elastic body near an arbitrary point , due to various displacements of its points, is deformed in such a way that the length of its edges changes and the initially right angles between the faces are distorted.

Figure 3.3 shows two edges of this parallelepiped: and the length of the edge is equal to and the edge is

After deformation, the points take a position. In this case, the point will receive a displacement, the components of which in the plane of the drawing are equal, and the point separated from the point at an infinitely small distance will receive a displacement, the components of which will differ from the components of the point displacement by an infinitesimal value due to a change in the coordinate

Fig.3. Linear and angular deformations

The components of the point displacement will differ from the components of the point displacement by an infinitesimal value due to a change in the coordinate

The length of the projection of the rib on the axis after deformation:

Projection of the absolute elongation of the rib on the axis

Relative elongation along the axis

(6)

is called linear deformation in the direction of the axis.

Similarly, linear deformations along the directions of the axes and

(7)

Consider the change in the angles between the edges of the parallelepiped (Fig. 3). The tangent of the angle of rotation of the rib in the plane

Due to the smallness of the deformations a, the linear deformation can be neglected due to its smallness compared to unity, and then

Similarly, you can determine the angle of rotation of the rib in the same plane:

The distortion of a right angle is called angular deformation and is defined as the sum of the angles of rotation of the ribs and:

(8)

In the same way, angular deformations in two other coordinate planes are determined:

(9)

Formulas (6)-(9) give six basic dependencies for linear and angular deformations on the displacement components. These dependencies are called the Cauchy equations:

(10)

In the limit when the lengths of the edges of the parallelepiped tend to zero, the Cauchy relations determine the linear and angular deformations in the vicinity of the point

Positive linear deformations correspond to elongations, and negative ones to shortenings. The shift angle is considered positive when the angle between the positive directions of the corresponding coordinate axes decreases and negative - otherwise.

Similarly to the stress tensor, the deformed state of the body at a given point is described by the strain tensor

(11)

Like the stress tensor, the strain tensor is a symmetrical matrix that contains nine components, six of which are different.

2.3 Relationship between stress and strain for elastic bodies

The relationships between stresses and strains are of a physical nature. Restricted to small strains, the relationship between stresses and strains can be considered linear.

When testing a rod in tension (mechanical testing of materials will be discussed in detail in the next section), a proportional relationship is established between normal stress and linear deformation in one direction, which is called Hooke's law:

where the elastic constant is called the modulus of longitudinal elasticity.

In the same experimental way, a relationship was established between linear deformations in the longitudinal and transverse directions:

where - linear deformation in the transverse direction, - the second elastic constant, called the Poisson's ratio.

In mechanical tests for pure shear, a directly proportional relationship was established between shear stress and angular deformation in the plane of action of this stress, which was called Hooke's law in shear:

where the value is the third elastic constant and is called the shear modulus. However, this elastic constant is not independent, because associated with the first two

To establish the relationship between strains and stresses, we select an infinitely small parallelepiped from the body (Fig. 1) and consider the action of only normal stresses. it leads to deformations of a higher order of smallness.

Let us determine the elongation of the rib parallel to the stress Under the action of this stress, according to Hooke's law (3.12), the relative elongation of the rib will occur

Stress causes a similar elongation in the direction perpendicular to the rib

and in the direction of the rib - shortening, which according to (13) is

or, taking into account the deformation expression

Similarly, the relative shortening of the rib under the action of stress is determined

Based on the principle of independence of the action of forces, the total relative elongation of the rib can be defined as the sum of the elongations from the action of each stress:

Similarly, one can define linear deformations along the directions of the other two axes:

In accordance with Hooke's law in shear (14), the relationship between angular deformations and shear stresses can be represented independently for each of the three planes parallel to the coordinate planes:

Thus, six formulas have been obtained that express a linear relationship between the strain and stress components in an isotropic elastic body and are called the generalized Hooke's law:

(16)

3. Basic equations of the theory of elasticity. Types of problems in the theory of elasticity

The main task of the theory of elasticity is the determination of the stress-strain state according to the given conditions of loading and fixing the body.

The stress-strain state is determined if the components of the stress tensor (s) and the displacement vector , nine functions are found.

3.1 Basic equations of elasticity theory

In order to find these nine functions, one must write down the basic equations of the theory of elasticity, or:

Differential Cauchies

(17)

where are the components of the tensor of the linear part of the Cauchy deformations;

components of the tensor of the derivative of displacement along the radius.

Differential equilibrium equations

where are the stress tensor components; is the projection of the body force on the j axis.

Hooke's law for a linearly elastic isotropic body

where are the Lame constants; for an isotropic body. Here are normal and shear stresses; strain and shear angles, respectively.

The above equations must satisfy the Saint-Venant dependencies

In the theory of elasticity, the problem is solved if all the basic equations are satisfied.

2 Types of problems in the theory of elasticity

Boundary conditions on the surface of the body must be satisfied and, depending on the type of boundary conditions, there are three types of problems in the theory of elasticity.

First type. Forces are given on the surface of the body. Border conditions

Second type. Problems in which displacement is specified on the body surface. Border conditions

Third type. Mixed problems of the theory of elasticity. Forces are given on a part of the body surface, displacement is given on a part of the body surface. Border conditions

Problems in which forces or displacements are specified on the surface of the body, but it is required to find the stress-strain state inside the body and what is not specified on the surface, are called direct problems. If, however, stresses, deformations, displacements, etc. are specified inside the body, and it is required to determine what is not specified inside the body, as well as displacements and stresses on the surface of the body (that is, to find the causes that caused such a stress-strain state)), then such problems are called inverse.

4 Equations of the theory of elasticity in displacements (Lame equations)

To determine the equations of the theory of elasticity in displacements, we write: differential equilibrium equations (18) Hooke's law for a linearly elastic isotropic body (19)

If we take into account that deformations are expressed in terms of displacements (17), we write:

It should also be recalled that the shear angle is related to displacements by the following relation (17):

(23)

Substituting expression (22) into the first equation of equalities (19), we obtain that the normal stresses

(24)

Note that the notation u in this case does not imply summation over i.

Substituting expression (23) into the second equation of equalities (19), we obtain that shear stresses

(25)

Let us write the equilibrium equations (18) in expanded form for j = 1

(26)

Substituting into equation (26) expressions for normal (24) and tangential (25) stresses, we obtain

where λ is the Lame constant, which is determined by the expression:

We substitute expression (28) into equation (27) and write,

where is determined by expression (22), or in expanded form

We divide expression (29) by G and add similar terms and obtain the first Lame equation:

(30)

where is the Laplace operator (harmonic operator), which is defined as

(31)

Similarly, you can get:

(32)

Equations (30) and (32) can be written as follows:

(33)

Equations (33) or (30) and (32) are the Lame equations. If the body forces are zero or constant, then

(34)

moreover, the notation in this case does not imply summation over i. Here

It can be shown that such a representation of displacements in terms of a harmonic function turns the Lame equations (33) into an identity. Often they are called the Popkovich-Grodsky conditions. The four harmonic functions are not necessary, because φ0 can be equated to zero.

4. Variational principles of the Theory of elasticity.

1 Principle of possible displacements (Lagrange principle)

Lagrange principle. For a body in equilibrium, the work of external and internal forces on any possible infinitesimal displacement increments is zero.

Using the Clapeyron theorem, that for an elastically deformed body by varying the displacement, we obtain the Lagrange principle

Possible in the mechanics of deformable bodies are such displacements that satisfy the external and internal constraints imposed on the body.

External connections are conditions for fixing, internal connections are a condition of continuity.

In order to satisfy the internal constraints, it is necessary that the displacement increments be continuous single-valued functions of the coordinates.

In this form, the Lagrange principle is valid for any deformable bodies.

For elastic bodies, it was obtained that

(41)

Then (40), taking into account (41), can be written as

(42)

where W is the specific strain, and

Here U is a variation of the entire potential energy of the body.

We substitute expression (43) into (42), and, since the forces do not vary, we write that

(44)

Equation (44) is a variational Lagrange equation.

If the forces are conservative, then the first two integrals represent the change in the potential of external forces during the transition from the undeformed state to the deformed one.

Potential of external forces

(45)

where - the possible work of external forces during the transition from the undeformed to the deformed state is calculated under the assumption that the external forces remain unchanged. Total energy of the system

Then, taking into account expressions (44) - (46), the Lagrange principle will be written:

that is, the variation of the total energy of the system in the equilibrium position on possible displacements is equal to zero. Expression (47) is the variational Lagrange equation in the case of the action of only conservative forces.

In the position of stable equilibrium, the total energy P is minimal,

The Lagrange principle is the principle of minimum energy.

2 Principle of possible states (Castillano principle)

We will call possible states those that are in accordance with external and internal forces, that is, satisfying the equilibrium equations.

Equation (57) writes the Castigliano Principle. With possible changes in the stress state of the body, the variation is equal to the integral over that part of the body surface on which displacements are given from the products of possible surface forces and displacements.

3 Relationship between the exact solution and solutions obtained based on the Lagrange and Castigliano principles

Based on the Lagrange principle, choosing some functions, or a set of them, and since the set of functions is limited, we get a smaller number of degrees of freedom of the system, thus reducing the degrees of freedom of the structure. That is, in the energy sense, the solution turns out to be more rigid than the exact one.

If we take the integral characteristics, then the approximate solution is more rigidly integral.

When solving the problem of loading a hinged beam with a transverse force in the middle of the span (Fig. 1), the approximate solution will give a smaller displacement under the force than with the exact solution.

exact solution

When solving the same problem using the Castigliano variational principle, since the continuity condition is not satisfied, the system gets more freedom than in reality.

The exact solution lies between these two approximate methods (Lagrange and Castigliano). Sometimes the difference between the obtained solutions is small.

5. List of used literature

1. Aleksandrov A.V., Potapov V.D. Fundamentals of the theory of elasticity and plasticity. 400 pages. Higher school. 1990.

2. Veretimus D.K. Fundamentals of the theory of elasticity. Part I. Stress theory. Methodological guide for the course "Fundamentals of the theory of elasticity and plasticity." 2005.-37s.

Veretimus D.K. Fundamentals of the theory of elasticity. Part II. Theory of deformations. Relationship between the stressed and deformed state. Methodological guide for the course "Fundamentals of the theory of elasticity and plasticity", 2005.-53p.

Veretimus D.K. Fundamentals of the theory of elasticity. Part III. Basic equations of the theory of elasticity. Types of problems in the theory of elasticity. Methodological guide for the course "Fundamentals of the theory of elasticity and plasticity", 2005.-45p.