teorija elasticnost

8/16/2019 teorija elasticnost

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Friday, May 29, 2015

Teorija elasti!nosti

ideje

da bi utvrdio normalni i tangencijalni napon, ispitas spil karata i od toga materijala kocku,primenis tangencijalni napon, i kad pocnu isto da se ponasaju, to predstavlja granicu tognapona

inkrementalno merenje napon dilatacija, svaki sledeci inkrement napona restartovanjedilatacija

platicnost, faza ali da se ne povecava napon, kako se ponasa, znaci zaustaviti povecanjenapona i posmatrati deformacije, plasticno tecenje

definisanje tangencijalnog napona kao trenje, normalni napon puta m, ne vredi ti tribometar jer

zavisi to od toga kako ces iseci, nego sameljes i ugao unutrasnjeg gtrenja

kad bi posmatrali gradu ne kao kontinualnu nego spil zalepljenih karata, lepak + trenje

sve vise razmisljaj o trenju, i mehanizmima prostiranja napona

axis zatezanje grede, povecanje smanjenje fleksione krutosti?

liniska grada i shell modeluj im uporedi, i onda resetka integral diferencija mata definicij, suma limesa, limesi

nekako utvrditi vektor veze medjumolekularnih sila, na taj nacin modelirati na tom malom nivou

1.1 uvod

Cartesian tensor, zbog jednostavnosti samo kori!"en

book of Marsden and Hughes3 is recommended, za klasicna i tenzorska analiza

theory of elasticity, problems and solutions.

Chapter 1 treats the theory of stress. Euler-Cauchy's principle

Analiza unutra!nje sile deformabilnog tela, mo#e biti $vrsto ili fluid-(ne prima smicanje)

We define a solid as a deformable body that possesses shear strength,

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In the analysis of stress we are concerned with the study of inner forces and couples that

hold a solid body in the shape that it has in the undeformed and deformed states. On the

atomic level, those forces are ionic, metallic, and van der Waals forces that act between

individual atoms.

The notion of stress is introduced in continuum mechanics to analyze integrated effects

of atomic forces. This is the so-called macroscopic view of the solid body. In it we ignore

the discrete nature of the body and assume that the mass of the body is continuously

distributed in part of three-dimensional Euclidean space.

stress free samo sile na nivou atoma koje grade veze da bi materijal egzistirao

telo se nalazi u naponskom stanju kada sile deluju i menjaju ravnote#u atoma, to rezultira

deformacijeom tela

sledi da analiza napona koncentrisana na promeni atomskih sila, usled deformacije

spolja!nje sile mogu biti povr!inske i body forces (deluju na sve ta$ke tela, gravitacija, inercijalne

sile)

The materials in which prescribed couples are acting are called the polar

materials., napravili parove spregova, tako posmatrali momenat

axiom of Euler and Cauchy (see Truesdell and Toupin (1960)) is introduced.

Upon any surface (real or imaginary) that divides the body, the action of one part of the body on

the other is equivalent (equipollent) to the system of distributed forces and couples on the surface

dividing the body.

posmatrano kao polarno telo mn=0

Cauchy's lemma: the stress vectors acting upon

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opposite sides of the same surface are equal in magnitude and opposite in direction.

As we

stated,

the infinitely many vectors Pn determine the stress state at the point. We show next that all

those vectors are not independent and that all of them can be determined if the stress

vectors on coordinate planes of a rectangular Cartesian coordinate system Xi are known.

TENZORSKI GLEDAS USLOVE RAVNOTE%E PREKO NAPONA

7 There is disagreement on the origin of the name. Veblen states that "The convenient though historically

not well justified name, tenS07' was introduced by A. Einstein (0. Ve- blen: Inva7'iants of Quadratic

Forms, Cambridge University Press, Cambridge 1962). On the other hand Brillouin elaims that the namewas given by German physicist Voight in his Lehrbuch del' K7'istalphysik (L. Brillouin: COU7'S de

Physique The07'ique-Les tenseu7'S en mechanique et en elasticite, Masson et Cie., Paris 1960).

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Therefore, with T symmetric we can state the following theorem: there exist always three

mutually orthogonal principal directions independently of any multiplicity of the

eigenvalues.

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By multiplying (29) with m1, m2, and m3 and (30) with n1, n2, and n3 and by subtracting

the results, we get

From (31) it follows that n-.lm; that is: the principal directions corresponding to distinct

principal values of a are mutually orthogonal.

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The octahedral stresses are important in fracture mechanics. Namely, in the octahedral

shear stress theory of elastic failure it is assumed that the failure in elastic material takes

place when the octahedral shear stress (4) reaches the critical value. This theory of elastic

failure is known as the Maxwell-von Mises distortion energy theory. The octahedral stress(J oct is sometimes called the hydrostatic or volumetric stress.

plo$e primer zanemaren sigma 3 komponente

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Chapter 2

analysis of strain

The engineering strain is the most common definition applied to materials

used in mechanical and structural engineering, which are subjected to very

small deformations. On the other hand, for some materials, e.g. elastomers

and polymers, subjected to large deformations, the engineering definition of

strain is not applicable, e.g. typical engineering strains greater than 1%, [6] thus other more complex definitions of strain are required, such as stretch ,

logarithmic strain , Green strain , and Almansi strain .

The engineering shear strain is defined as the tangent of that angle, and is

equal to the length of deformation at its maximum divided by the

perpendicular length in the plane of force application which sometimes

makes it easier to calculate.

The engineering shear strain is defined as ( ) the change in angle

between lines and .

Chapter 3

Hookes law

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The relation between the stress and deformation, in continuum mechanics, is called the

constitutive equation. G.W. Leibniz was the first to realize that the constitutive equations

must be determined experimentally.

Anisotropic, orthotropic, and isotropic elastic body

Anisotropy /!æna"#s$tr %pi/ is the property of being directionallydependent, as opposed to isotropy, which implies identicalproperties in all directions.

ortotropan dva pravca razlika

In elasticity theory, instead of Lame constants A and Il two other constants are often used:

modulus of elasticity E and Poisson's ratio v. We now derive the connections between

Lame constants and "engineering" constants E and v.

The controversy about the number of independent constants needed to describe an

isotropic elastic body was, actually, centered about the value of Poisson ratio. As we see

from (6) Poisson and Cauchy molecular theory predict that only one constant is sufficient

to describe isotropic elastic material. If E is taken as this constant, then the ratio oflongitudinal and

lateral strain (and v is just this) is equal to v = 1/4. Experiments that were performed,especially by G. Wertheim in 1848, clearly indicated that v "# 1/4 for some isotropic

elastic materials so that two constants are needed in the relation connecting the stress and

strain tensors.

The coefficient of thermal expansion describes how the size of an object

changes with a change in temperature. Specifically, it measures the

fractional change in size per degree change in temperature at a constant

pressure. Several types of coefficients have been developed: volumetric,

area, and linear. Which is used depends on the particular application and

which dimensions are considered important. For solids, one might only be

concerned with the change along a length, or over some area.

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Temperature is a monotonic function of the average molecular kinetic

energy of a substance. When a substance is heated, the kinetic energy of

its molecules increases.

Thermal radiation is electromagnetic radiation generated by the thermal

motion of charged particles in matter. All matter with a temperature greater

than absolute zero emits thermal radiation. When the temperature of the

body is greater than absolute zero, interatomic collisions cause the kinetic

energy of the atoms or molecules to change. This results in charge-

acceleration and/or dipole oscillation which produces electromagnetic

radiation, and the wide spectrum of radiation reflects the wide spectrum ofenergies and accelerations that occur even at a single temperature.

The four laws of thermodynamics define fundamental physical quantities

(temperature, energy, and entropy) that characterize thermodynamic

systems. The laws describe how these quantities behave under variouscircumstances, and forbid certain phenomena (such as perpetual motion).

The four laws of thermodynamics are:[1][2][3][4][5][6]

• Zeroth law of thermodynamics: If two systems are in thermal

equilibrium respectively with a third system, they must be in thermal

equilibrium with each other. This law helps define the notion of

temperature.

• First law of thermodynamics: When energy passes, as work, as heat,

or with matter, into or out from a system, its internal energy changes

in accord with the law of conservation of energy. Equivalently,perpetual motion machines of the first kind are impossible.

• Second law of thermodynamics: In a natural thermodynamic process,

the sum of the entropies of the interacting thermodynamic systems

increases. Equivalently, perpetual motion machines of the second

kind are impossible.

• Third law of thermodynamics: The entropy of a system approaches a

constant value as the temperature approaches absolute zero.[2] With

the exception of glasses the entropy of a system at absolute zero is

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http://en.wikipedia.org/wiki/Glasshttp://en.wikipedia.org/wiki/Absolute_zerohttp://en.wikipedia.org/wiki/Third_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Perpetual_motion_machinehttp://en.wikipedia.org/wiki/Thermodynamic_systemhttp://en.wikipedia.org/wiki/Thermodynamic_processhttp://en.wikipedia.org/wiki/Second_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Perpetual_motion_machinehttp://en.wikipedia.org/wiki/Conservation_of_energyhttp://en.wikipedia.org/wiki/Internal_energyhttp://en.wikipedia.org/wiki/Heathttp://en.wikipedia.org/wiki/Work_(thermodynamics)http://en.wikipedia.org/wiki/First_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Thermal_equilibriumhttp://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Perpetual_motionhttp://en.wikipedia.org/wiki/Thermodynamic_systemhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Thermodynamicshttp://en.wikipedia.org/wiki/Electromagnetic_radiationhttp://en.wikipedia.org/wiki/Kinetic_energyhttp://en.wikipedia.org/wiki/Absolute_zerohttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Matterhttp://en.wikipedia.org/wiki/Charged_particleshttp://en.wikipedia.org/wiki/Thermal_motionhttp://en.wikipedia.org/wiki/Electromagnetic_radiation


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typically close to zero, and is equal to the log of the multiplicity of the

quantum ground state.

Solutions for Some Problems of

Elasticity Theory

elementary strength of materials and rod theories

important problems of extension, torsion, and bending formulated for an elastic rod

These problems (determination of the stress strain and deformation state of a prismatic

rod loaded by a system of forces and couples applied at the ends of the rod) are called the

Saint-Venant problems.

rod

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torzija

From (69) we conclude that by torsion of a rod with circular cross-section there is no

warping (deplanation) of the cross-section.

In the elementary approach (strength of materials) this is taken as the hypothesis (the

Bernoulli hypothesis). From (19) it follows that

Serrin (1971) showed that: the only cross-section for which in torsion the

shear stress is constant on the boundary C is circular cross-section.

BENDING PURE

Poisson ratio equal to v = 1/4 (one constant elasticity)

chapter 6

Plane State of Strain and Plane State

of Stress

Generally speaking a solution of three-dimensional problems of elasticity theory is hard toobtain.

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It is therefore important to analyze a simpler case in which the quantities involved

(stresses, strains, etc.) depend on two spatial coordinates only.

However both components of the displacement vector and the stress tensor components,

in general, depend on Xl, x2, and X3. That is why the plane stress problem is a three-

dimensional elasticity problem.

In physics, Hamilton's principle is William Rowan Hamilton's formulation

of the principle of stationary action (see that article for historical

formulations). It states that the dynamics of a physical system is determined

by a variational problem for a functional based on a single function, the

Lagrangian, which contains all physical information concerning the system

and the forces acting on it. The variational problem is equivalent to and

allows for the derivation of the differential equations of motion of the

physical system. Although formulated originally for classical mechanics,

Hamilton's principle also applies to classical fields such as the

electromagnetic and gravitational fields, and has even been extended to

quantum mechanics, quantum field theory and criticality theories.

In mathematics, and particularly in functional analysis and the Calculus of

variations, a functional is a function from a vector space into its underlyingscalar field, or a set of functions of the real numbers. In other words, it is a

function that takes a vector as its input argument, and returns a scalar.

Calculus of variations is a field of mathematical analysis that deals with

maximizing or minimizing functionals, which are mappings from a set of

functions to the real numbers. Functionals are often expressed as definite

integrals involving functions and their derivatives. The interest is in extremal

functions that make the functional attain a maximum or minimum value – or

stationary functions – those where the rate of change of the functional iszero.

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http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Definite_integralhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Map_(mathematics)http://en.wikipedia.org/wiki/Functional_(mathematics)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Scalar_fieldhttp://en.wikipedia.org/wiki/Vector_spacehttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Calculus_of_variationshttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Quantum_field_theoryhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Field_(physics)http://en.wikipedia.org/wiki/Gravityhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Field_(physics)http://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Equations_of_motionhttp://en.wikipedia.org/wiki/Differential_equationhttp://en.wikipedia.org/wiki/Lagrangianhttp://en.wikipedia.org/wiki/Functional_(mathematics)http://en.wikipedia.org/wiki/Calculus_of_variationshttp://en.wikipedia.org/wiki/Dynamics_(mechanics)http://en.wikipedia.org/wiki/Principle_of_stationary_actionhttp://en.wikipedia.org/wiki/William_Rowan_Hamiltonhttp://en.wikipedia.org/wiki/Physics


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A simple example of such a problem is to find the curve of shortest length

connecting two points. If there are no constraints, the solution is obviously a

straight line between the points. However, if the curve is constrained to lie

on a surface in space, then the solution is less obvious, and possibly many

solutions may exist. Such solutions are known as geodesics. A related

problem is posed by Fermat's principle: light follows the path of shortest

optical length connecting two points, where the optical length depends upon

the material of the medium. One corresponding concept in mechanics is the

principle of least action.

Betti's theorem

The relation (8) is Betti's theorem: the work of outer forces I on the displacement field

corresponding to the system of forces II is equal to the work of the system of forces II on

the displacement field corresponding to the system of forces 1.

Principle of virtual workThe Principle of virtual work belongs to the class of differential variational principles and

is the basis of all variational principles in elasticity theory. It was introduced in mechanicsby Bernoulli and now is taken as an axiom that holds for all mechanical systems. Its clear

statement has been given by J. R. D'Alembert (see Lindsay (1975)). and reads: in the

equilibrium state of a mechanical system the work of all outer forces on virtual

displacements (infinitesimal displacements that satisfy kinematical constraints) is equal to

zero.

The principle of virtual work is important for the following reasons (Marsden and Hughes1983).

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1. it is useful numerically.

2. it is believed that it remains valid under the conditions for which

differential equations do not necessarily make sense.

3. Equations expressing a mathematical statement of the principle of virtual work coincide

with the weak form of differential equations for which there are many relevant

mathematical theorems.

The principle of virtual work is used in two different forms: the principle of virtual

displacement and the principle of virtual forces. In what follows we analyze both

principles.

Next we define the concept of infinitesimal change of configuration for elastic bodies.

Consider an elastic body that is in equilibrium under the action of body forces f andsurface forces Pn (see Fig. 6).

polje virtualnog rada

kastiljanova teorema

TANJIRI, teorija ploca

As we stated earlier the plane problems of elasticity theory describe, ap- proximately, the

behavior of a thin elastic body (see Chapter 6.1). The plate theory is also an

approximation to the three-dimensional problems of elasticity theory. The plate represents

approximation of an elastic body when one dimension of the body is much smaller thanother two. A plane dividing the thickness of the plate in half is called the middle plane.

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The stress tensor and displacement vector in plate theory are expressed as functions of

points in the middle plane. As a matter of fact, this can be taken as a definition: a plate is

an elastic body (no matter how thick) for which the stress tensor and displacement vector

are functions of points on a single plane (called the middle plane) of a body.

Plates have wide application in engineering. We mention concrete and reinforced concrete

plates in civil engineering, plates in naval architecture, in the automotive industry, and so

on.

In this Chapter we present first a nonlinear theory of plates formulated by von Karman

(1910). Then we also present a generalization of classical linear plate theory that includes

the influence of shear stresses on the deformation of a plate. As far as loading is

concerned we assume that the plate is loaded by an arbitrary system of forces having

nonzero components in the direction normal to the middle plane as well as in the direction

of the middle plane.

kirkohovepredpostavke

the Kirchhoff-Love hypothesis

1. The line element normal to the middle plane before deformation transforms into a line

element normal to the middle surface to which the middle plane is deformed. Also the

length of the element before and after deformation remains the same.bernuli mariot hipoteza kod stapa, geometrisko znacenje

Reissner-Mindlin theory kaze suprotno

2. Equidistant points from the middle plane do not influence each other; that is, there are

no stress components in the direction normal to the middle plane. (}33 = 0 o3 sigma

kod tankih ploca

It is obvious that the condition (3) is not satisfied exactly if the plate is loaded with thedistributed forces normal to the middle plane.

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chapter 9

Pressure Between Two Bodies inContact

To solve the contact problem basic assumptions of the classical theory of contact due to

Hertz (1892) are introduced. In presenting them we follow Johnson (1987)

1. Bodies in contact are linearly elastic homogeneous, with possibly different Lame

constants.

2. In the unloaded state the bodies are in contact at the single point 0 that lies on asmooth part of the outer surface of both bodies. The point 0 is a regular point of

both surfaces and there is a well-defined tangent plane at O.

3. The bodies are compressed along the normal to the tangent plane at O.

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chapter 10

elastic stability

338 strana, kul pocinje

Having defined three concepts of stability analysis we may ask: are all definitions of

stability compatible? Is, the critical value of the load pa- rameter the same when

determined on the basis of the Euler, energy, and dynamic methods? If the critical valuesare the same, is it then possible to keep one definition of stability and derive the other two

theorems? In

Euler, energy, and dynamic methods

such attempts the dynamic method has a good chances to be taken as the definition ofstability and the value of other two methods judged accord- ing to restrictions that have to

be introduced in order to derive those two methods from the dynamic method. We discuss

those questions now.

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teorija elasticnost

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