The conformal mapping of the certain simple-connected area of the plane

FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY

Note 3.7. The conformal mapping of the certain simple-connected area of the plane

i

] [ K into the analogical area of the plane ^{z x iy} is realized by the analytical function z z ] . The word conformal is used here because this kind of mapping preserves the angles between two curves at the point of their intersection.

The simplest example of using conformal mapping. The exterior of the circle of radius ^ro is mapped into the interior of the unit circle. The mapping function has the form

^o

z ] r ] . (3.63)

Then for the case, when the infinite plate with the circular hole of the radius ^ro is stretched in direction ^Ox by the constant stress ^Vo, a solution in the complex potentials is as follows

^{o 4}

^{2 o 2}

z z r z

M V ª«¬ º»¼ ^{, \ z}

^{Vo 2}

^ª«¬^{z ro 2}

^{1 z ro 2 z3}

^º»¼. (3.64) The considered problem is, in fact, the Kirsch’s problem. Therefore, the corresponding stresses are shown by formulas (3.59)-(3.61).

Section 3. One-dimensional problems

The analysis of these problems is caused by their simplicity and great technical importance.

So, if the simplicity of the class of one-dimensional problems is mentioned, then first of all the one more class of problems should be considered here. They are also simple and one-dimensional by the mathema- tical description. At that, they have a general character and unusual practical applications.

Thus, the class of universal deformations should be briefly considered. The universal deforma- tions (uniform deformations, universal states) occupy a special place in the theory of elasticity just owing to their universality. It consists in that the theoretically and experimentally determined elastic constants of material in samples, in which the universal deformation is created purposely, are valid also for all other deformed states both samples and any different products made of this material.

It is considered therefore that the particular importance of universal deformation (their fundamentality) consists of a possibility to use them in the determination of properties of materials from tests. To realize the universal deformation, two conditions have to be fulfilled:

1. Uniformity of deformation must not depend on the choice of material.

2. The deformation of material has to occur by using only the surface loads.

In the theory of infinitesimal deformations, the next kinds of universal deformations are studied in detail:

simple shear,

simple (uniaxial) tension-compression,

uniform volume (omniaxial) tension-compression.

In the linear theory of elasticity, the experiment with a sample, in which the simple shear is realized, allows determining the elastic shear modulus ^P. The experiment with a sample, in which the uniaxial tension is realized, allows determining Young’s elastic modulus ^E and Poisson’s ratio ^Q. The experiment with a sample, in which the uniform compression is realized, allows determining the elastic bulk modulus k.

While being passed from the linear model of very small deformations to the models of nonsmall (moderate or large) ones, that is, from the linear mechanics of materials to nonlinear mechanics of materials, the universal states permit to describe theoretically and experimentally many nonlinear phenomena. The history of mechanics testifies the experimental observation in the XIX century of the nonlinear effects that arose under the simple shear and were named later by names of Poynting and Kelvin. After about a hundred years in the XX century, these effects were described theoretically within the framework of the nonlinear Mooney-Rivlin model.

The mechanics of composite materials is one more area of application of universal deformations. The simplest and most used model, in this case, is the model of averaged (effective, reduced) moduli. In the theory of effective moduli, the composite materials of the complex internal structure with internal links are treated usually as the homogeneous

FOUNDATIONS OF MECHANICS OF MATERIALS PART 1

SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY

It was found that it is sufficient for isotropic composites to study the energy stored in the elementary volumes of composites under only two kinds of universal deformations: simple shear and omniaxial compression.

Universal deformation of simple shear. The experiments on simple shear are realized on the sufficiently long beam of quadratic cross-section, in which the uniform deformation is created on some distance from the ends. The lower side of the beam is fixed rigidly and the surface tangential constant load T2 is app- lied to the upper side. The deformation of the beam can be described by one component of the deforma-tion gradient u1,2 w u x1 w 2. The component u1,2 and the shear angle ^J are linked as follows

1,2 tan 0.

u J W ! (3.65)

In the linear theory, the shear angle is assumed to be small and then ^{J |tanJ W}. The Cauchy-Green strain tensor is characterized by only three nonzero components

²

11 1 2 u1,1 u1,1 u u1,_k 1,_k 1 2 u u1,2 1,2 u u1,3 1,3

H W ;

12 21 1 2 u1,2 u2,1 u u1,_k 2,_k 1 2 .

H H W

The principal extensions are written through the shear angle by formulas O1 1, O₂ O W₃ .

Universal deformation of uniaxial tension. A rod in the form of a straight long cylinder (of circular or quadratic cross-section)with the axis in direction of axis Ox1 is considered when the lateral surface of the rod is free of stresses. The rod is stretched in the axial direction.

Then the uniform stress-strain state is formed in the rod except for the area near the ends.

It is characterized by only one nonzero component V11 of the stress tensor and two nonzero components ^{H H11, 22 H33} of the strain tensor (or two principal extensions O O1, 2 O3).

This kind of deformations is used for the introduction of the Young modulus and Poisson’s ratio instead of two classical Lame elastic constants. Perhaps, the oldest and exhausting procedu-res are shown in classical Love’s book. Let us use the adopted at that time notations and write the standard representation of the Hooke law through the Lame moduli ^{O P,}

2 ; 2 ; 2 ,

y xy x zx z yz

X PH Z PH Y PH

2 ; 2 ; 2 ;

x xx y yy z zz

X ' O PH Y ' O PH Z ' O PH (3.66)

where the notation of dilatation is used ' HxxHyyHzz.

The classical procedure of introducing the Young modulus and the Poisson’s ratio is shown below.

Toward this end, the universal deformation of uniaxial tension is considered, when the axis is chosen in direction Ox and the prism is stretched at the ends by a uniform tension ^T. The stress state of a prism is uniform and is characterized by only the stress Xx T (other stresses are zero ones). In this case, the Hooke law becomes simpler

2 _xx,

T ' O PH ₀ ' O ₂PH_yy, 0 ' O 2PH_zz.

The expression for dilatation is obtained by adding all three equalities above

(3 2 ) (3 2 ).

T O P ' o ' T O P

The substitution of the last expression for dilatation into the first equality (3.66) gives relations 3 2 2 _xx 3 2 _xx

T ª¬O O P º¼T PH o T ª¬P O P O P H º¼ .

The last expression represents the elementary law T EHxx of link between tension and strain of prism, in which the Young modulus ^E is used. Comparison of this law with relation (3.66) gives the classical expression for the Young modulus through the Lame moduli

3 2 .

E ª¬P O P O P º¼ (3.67)

The substitution of expression for dilatation into the second and third equalities (2) gives relations

2

yy zz xx

H H O O P H

ª¬ º¼ ,

which express the classical Poisson’s law on the transverse compression under the longitudinal extension and permit to introduce the Poisson’s ratio

_{yy xx}

_{zz xx} 2

V H H H H O O P . (3.68)

Thus, the Poisson’s ratio is one of the characteristics of linear deformation of elastic material and is considered as the basic notion of linear elasticity. But the ratio of transverse strain to the longitudinal one can be used in any model of nonlinear elasticity (and not only elasticity).

FOUNDATIONS OF MECHANICS OF MATERIALS PART 1

SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY

In this case, this ratio will have its representation in each model and possibly will not be constant quantity for any level of strain.

Universal deformation of uniform (omniaxial) compression-tension. A sample has the shape of a cube, to sides of which the uniform surface load (hydrostatic compression) is applied. Then the uniform stress state is formed in the cube. The normal stresses are equal with each other

11 22 33

V V V

Definition 3.2.

Definition 3.3.

, and the shear stresses V_iki kz

Definition 3.2.

Definition 3.3.

are absent. This type of universal deformation is defined by the following components of displacement gradients

1,1 2,2 3,3 0;

u u u !H u_1,1u_2,2u_3,3 3H e; u_{k m}, w u_k wx_m 0 k mz .

Definition 3.2.

Definition 3.3.

(3.69) The Cauchy-Green strain tensor is as follows

2

11 22 33 1 2 ; _ik 0 i k ,

H H H H H H z

Definition 3.2.

Definition 3.3.

(3.70) and the algebraic invariants are written in the form

1 11 22 33 ;

I H H H e I2 H11 ² H22 ² H33 ²; I3 H11 ³ H22 ³ H33 ³.

Definition 3.2.

(3.71) The principal extensions are equal to each other O O1 2 O3.

Definition 3.2.