Defects Elasticity Theory

8/8/2019 Defects Elasticity Theory

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3.2 Elasticity theory of dislocations

Basics of linear elasticity theory

Stress field of a straight dislocation

Strain energy

Forces on dislocations

Hartmut S. Leipner: Defects in crystals

(Dislocation core structure)

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hsl 2008 Defects in crystals 3.2 Elasticity theory

Displacement vectoru = (ux, uy, uz)

Nine components of the strain tensor

ij


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Considering a small cubic volume element in a solid, the total stress state

can be described by the forces perpendicular and parallel to the faces ofthe cube.

On each face, three stresses: 1 normal ii, 2 shearij(ij; i,j =x,y,z) All together nine components of the stress

Stress in the solid

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Stress tensor

Stress tensor is symmetrical, ij

= ji

(rotational equilibrium).

Magnitude of the individual components depends on the orientation of

the coordinate system.

A special coordinate system can always be found, where there are only

normal stresses,

Positive normal stress as tension

Hydrostatic pressure is the average normal stress,

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Strain

Generally, the elastic deviation of the shape of the solid can beexpressed as astrain tensor,

ii elongations, ij shear (ij)

Strain tensor also symmetrical

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Stressstrain relationships

Stress as force per unit area of surface; consider orientationof the surface and direction of the force Uniaxial tension =, shear = G Special cases ofHookes law

Relation between stress and strain tensors

Expression of 9 equation like

Chas 34 = 81 components Cijkl(4th rank tensor)

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Elastic constants C

In praxi, number of constants is reduced due to symmetry. For isotropic solids only two parameters (e.g.G and Lam constant )

In cubic crystals, three constants are needed.

~

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Elastic moduls

Other constants to be used: Youngs modulus, Poissons constant ,and bulk modulusK,

Poissons constant

Elongation inx-direction connected with reduction of cross section

yy = zz=xx

~

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hsl 2008 Defects in crystals 3.2 Elasticity theory Strain field screw

Volterra screw dislocation [Hull, Bacon 1992]

Representation as a cylinderof elastic material

Slit LMNO ||zaxis, surfacedisplaced by b

Displacements:

ux = uy = 0

Strain field of a straight screw dislocation

bb

Cylinder with radius r0

not taken into account:

assumptions oflinearelasticity theory not valid

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Simpler form in cylindrical coordinates:

Using rz=xzcos+ yzsin z= xzsin+ yzcos

(the only non-zero

components)

Straight screw dislocation

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Strain and stress 1/r, diverge withr 0

Linear elasticity approach not valid at the center of the dislocation

Dislocation core with atomistic model Theoretical stress limit reached atrb

Reasonable core radius 1 nm

Discussion of the strain and stress fields

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hsl 2008 Defects in crystals 3.2 Elasticity theory Strain field edge

Volterra edge dislocation[Hull, Bacon 1992]

bb

Stress field of a straight edge dislocation

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hsl 2008 Defects in crystals 3.2 Elasticity theory Stress field edge dislocation

Contours of equal stress

about an edge dislocation[Hirth, Lothe 1992]

Stress field contours of an edge dislocation

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Deformation is basically a plane strain. Both dilatational and shear components exist.

Largest normal stress xx || Burgers vector

Max. compressive stress immediately abovey = 0 (slip plane)

max. tensile stress immediately belowy = 0

Pressure on a volume element

Stress field of an edge dislocation

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Elastic strain energy in theory of elasticity:

Two parts of the total strain energy of a body containing a dislocation:E=Ecore +Eel

Strain energy of a dislocation

(Total elastic energy per unit length)

Elastic energyper unit length of a screw:

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Strain energy of an edge more complicated to calculate(lower symmetry)

Elastic energy of an edge dislocation higher by about 3/2 than that of a screw

Eeldepends onr0andR(core radius and cut-off radius). Example:G = 41010 Nm2, r0 = 1 nm,R = 1 mm, b = 0.25 nm

Eel 6 eV per unit length of a dislocation

R corresponds to crystal dimensions.

For many dislocations in a crystal,

superposition of the long-range strain fields

Discussion of the strain energy

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hsl 2008 Defects in crystals 3.2 Elasticity theory Elastic energy screw

Elastic energy of dislocations

r

Elastic energy in a ring cylinder of the thickness dr

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Superposition of edge and screw parts

EelGb2, with 0.51.0

Shortest lattice translation vectors

preferred asEelis min.

Splitting of a dislocation withb1 = 2ainto two dislocations,E1 4a

2,E2 2a2

[Bohm 1992]

Dislocation splitting

Energy of mixed dislocations

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+ =

b1

b2

b3

b1+ b

2= b

3

Energy criterion for dislocation reaction

Reaction favorable

Condition with angle:! /2 < Reaction preferred0 < /2 Dissociation preferred

Franks rule

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Elementary process of plastic deformation

The motion of dislocations is the elementary process of the plastic deformationof crystals. is here the shear stress.

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External forces on dislocations

Dislocation separates slipped regionfrom unslipped one

Deformation work may be done by

external force shift of the dislocation

Plastic deformation by motion of a dislocation[Kelly:2000]

Screw motion

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Dislocation of lengthL, swept distance on the slip planex

Applied shear force on the crystal (per lengthL): F= x

Work done by the crystal (per lengthL): W= x b

L

x

b

Definition of a force (per lengthL) to move the dislocation in

x-direction: workW= force on dislocation x

Fd = b

Pure motion in the glide plane

Fd = b (per unit lengthL)

Definition of force

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PeachKoehler force

General case of a shear force on a crystal: F= n

(n unit vector of the swept-out surface, n =u, with line vector, direction of motion u)

Displacement by Burgers vectorb if dislocation moves

Work done by the crystal:

W= (n)b =(b)n

W= (b)(u) = (b)u

PeachKoehlerforce Fd = (b)

Force acting on a plane perpendicular to b || dislocation: Fd always normal to the dislocation

Force to move the dislocation in the direction u

(All eq. are given per unit length of dislocationL.)

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Strain energy proportional to the length,

increase in length causes increase in energyExistence of a line tension trying to reduce the line

lengthTo be defined via the increase in energy per unit

length T= Gb2

[Bohm 1992]

Line tension

Line tension

Calculation of the shear stress 0 to maintaina certain radius of curvatureR

Inward force:Ks = 2Tsin(/2),

for small angles: dKs = Td

Outward force on segment ds due to appliedstress is 0 bds

Equilibrium: Td= 0 bds (ds =Rd)

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Edge dislocations with the same slipplanerepulsive force ifbhas the same

sign, attractive if opposite sign More complicated, if slip planes

different Displacement in dislocation I is

Burgers vectorb of dislocation II

Interaction between edge dislocations

Forces between dislocations

y

x

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Calculation of the force

Components of the force on dislocation II per unit length(bx = b, by = bz= 0):

y

Fx

Fy

x

I

II

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F b ll l d di l i


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hsl 2008 Defects in crystals 3.2 Elasticity theory Forces between edge dislocations

Force between parallel edge dislocations. Unit of forceFx is Gb2y/[2(1 )].

Curve A for like dislocations, curve B for unlike dislocations.[Hull, Bacon 1993/Cottrell 1953]

Force between parallel edge dislocations

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I t ti f b t d di l ti


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hsl 2008 Defects in crystals 3.2 Elasticity theory Forces between edge dislocations

Dislocations motion only in the slip plane

most importantFx

Forx > 0,Fx < 0 (attractive) ifx 0 (attractive) ifx

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Defects Elasticity Theory

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