summer school

Generalized Continua

and Dislocation Theory

Theoretical Concepts,

Computational Methods

And Experimental Verification

July 9-13, 2007

International Centre for Mechanical Science

Udine, Italy

Lectures on:

Introduction to and fundamentals of

discrete dislocations and dislocation

dynamics. Theoretical concepts and

computational methods

Hussein M. ZbibSchool of Mechanical and Materials Engineering

Washington State University

Pullman, WA

zbib@wsu.edu

Contents

Lecture 1: The Theory of Straight Dislocations – Zbib

Lecture 2: The Theory of Curved Dislocations –Zbib

Lecture 3: Dislocation-Dislocation & Dislocation-Defect Interactions -Zbib

Lecture 4: Dislocations in Crystal Structures - Zbib

Lecture 5: Dislocation Dynamics - I: Equation of Motion, effective mass - Zbib

Lecture 6: Dislocation Dynamics - II: Computational Methods - Zbib

Lecture 7 : Dislocation Dynamics - Classes of Problems – Zbib

Text books:

• D. Hull and D. J. Bacon, D. J., Introduction to Dislocations (Butterworth-Heinemann, Oxford,

1984).

• J.P. Hirth, and J. Lothe, 1982. Theory of dislocations. New York, Wiley.

• Elastic Strain and Dislocation Mobility, eds. V. L. Indenbom and J. Lothe (Elsevier Science

Publishers, 1992)

Manuscripts: • Zbib, H.M. and Khraishi, T. Size Effects and Dislocation-Wave Interaction in Dislocation

Dynamics Chapter in the Book Series entitled: Dislocations in Solids, edited by F.R.N. Nabarro

and John P. Hirth. Elsevier, to be published in 2007

• Zbib, H.M., and Khraishi, T.A., Dislocation Dynamics. In: Handbook of Materials Modeling. Ed.

Sidney Yip, pp. 1097-1114, Springer, 2005.

• J.P. Hirth, H.M. Zbib and J. Lothe, Modeling & Simulations in Maters. Sci. & Enger., 6 (1998)165.

• I. Demir, J.P. Hirth and H.H. Zbib, The Somigliana Ring Dislocation, J. Elasticity, 28, 223-246,

1992.

• Khraishi, T.A., Zbib, H.M., Hirth, J.P. and de La Rubia, T.D., “The stress Field of a General

Volterra Dislocation Loop: Analytical and Numerical Approaches”, Philosophical Magazine, 80,

95-105, 2000.

• Khraishi, T. and Zbib, H.M., The Displacement Field of a Rectangular Volterra Dfislocation Loop,

Phil Mag,82, 265-277, 2002.

• Zbib and Diaz de la Rubia, A Multiscale Model of Plasticity, Int. J. Plasticity, 18, 1133-1163-2002.

Recommended Reading

Lecture 1: The Theory of Straight Dislocations

Defects in Crystalline materials:

vacancies,

interstitials and impurity atoms (point defects),

dislocations (line defects),

grain boundaries,

heterogeneous interfaces and microcracks (planar defects),

chemically heterogeneous precipitates,

twins and

other strain-inducing phase transformations (volume defects).

These defects determine to a large extent the strength and mechanical

behavior of the crystal.

Most often, dislocations define plastic yield and flow behavior, either as

the dominant plasticity carriers or through their interactions with the

other strain-producing defects.

Macroscopic experiment

“Macroscopic Scale”

representative

“homogeneous” element

Continuum Plasticity

“Mesoscopic Scale”

Polycrystalline

plasticity

“Microscopic Scale”

dislocations in single

crystal

,

1m

Dislocation structure in a high

purity copper single crystal

deformed in tension (Hughes)

Dislocation Dynamics

Cu

Nb

75 nm

Dislocation – Fundamentals

Dislocations: Continuum concept

•Volterra, V., 1907. Sur l’equilibre des cirps elastiques

multiplement connexes. Ann. Ecole Norm. Super. 24, 401-

517.

•Somigliana, C., 1914. Sulla teoria delle distorsioni elastiche.

Atti Acad. Lincii, Rend. CI. Sci. Fis. Mat. Natur 23, 463-472.

They considered the elastic properties of a cut in a continuum,

corresponding to slip, disclinations, and/or dislocations.

But associating these geometric cuts to dislocations in crystalline materials was not

made until the year 1934.

1926 Frenkel estimated the theoretical shear strength using a periodic force law

b

x

b

xthth

22 sin

when the shear strain (x/d) is small,

d

x

Equating the two equations yields:

110

2

d

bth

d; interplanar spacing,

J. Frenkel, Z. Phys., p574, (1962)

But the experimentally observed shear stress was much smaller thanthat

410y

Dislocations in Crystalline materials

In order to explain the less than ideal strength of crystalline materials,

Orowan (1934), Polanyi (1934) and Taylor (1934) simultaneously hypothesized the existence of dislocation as a crystal defect.

Later in the late 50.s, the existence of dislocations was experimentally confirmed by

Hirsch, et al. (1956) and Dash (1957).

Presently these crystal defects are routinely observed by various means of electron microscopy

Pure edge dislocation

Pure Screw dislocation

b

b

RH Burgers circuit

Burgers vector b

Line sense

Dislocations & Slip in Crystalline materials

RH Burgers circuit

Burgers vector b

Axiom:

reversing the direction of the line sense causes the Burgers vector to

reverse its direction

b must be conserved over the entire dislocation length (Volterra

dislocation)

Dislocations can never end in a crystal. It either: Forms a closed loops,

intersect with a surface or boundary, or branch into other dislocations

known as dislocation reaction.

A dislocation can be easily understoodby considering that a crystal can deform irreversibly by slip, i.e. shifting or sliding along one of its atomic

planes. If the slip displacement is equal to a lattice vector, the material across the slip plane will preserve its

lattice structure and the change of shape will become permanent. However, rather than simultaneous

sliding of two half-crystals, slip displacement proceeds sequentially, starting from one crystal surface and

propagating along the slip plane until it reaches the other surface. The boundary between the slipped and

still unslipped crystal is a dislocation and its motion is equivalent to slip propagation.

In this picture, crystal plasticity by slip is a net result of the motion of a large number of dislocation lines,

in response to applied stress. It is interesting to note that this picture of deformation by slip in crystalline

materials was first observed in the nineteenth century by Mügge (1883) and Ewing and Rosenhain

(1899). They observed that deformation of metals proceeded by the formation of slip bands on the surface

of the specimen. Their interpretation of these results was obscure since metals were not viewed as

crystalline at that time.

Mixed Dislocation

Linear theory of elasticity

)( zyx u,u,uuThe displacement of a material point in a strained body from its position in the unstrained state can

be represented by the vector form:

i

j

j x

u

x

uiij

1

1Strain tensor

Hooke’s law

)elasticity anistopic (General constantselasticijkl

klijklij

C

C

Equilibrium equation

0 ijij f,body forceStress

tensor

Basic filed equation: Combining the above equations yields

0 iljkijkl fuC ,

)/(:, jijjij xNote and repeated index (e.g. j) means summation over the

index; j=1,2,3

ratio sPoisson' is and

modulusshearis

21

2

klijjkiljlikijkl

klijklij

C

C

)(

Linear isotropic elasticity

The stress field of a straight dislocations

• Screw dislocation

y

xbbu

u

u

z

y

x

1tan22

0

0

The displacement of a material point in a strained body from its position in the unstrained state can

be represented by the vector form: )( zyx u,u,uu

Strain tensor

i

j

j x

u

x

uiij

1

1

Strain in Cartesian coordinates - screw dislocation

22

22

yx

x

π2

b

yx

y

π2

b

yzyz

xzxz

xyyyyyxx

2

2

0

ratio sPoisson' is and

modulusshearis

21

2

klijjkiljlikijkl

klijklij

C

C

)(

Linear isotropic elasticity

Stress - Screw dislocation

22

22

2

2

0

yx

xb

yx

yb

yz

xz

xyyyyyxx

Because normal stress are all null, the screw dislocation has a strainfield which has no dilation – it results in pure distortion (only change in shape not in volume)

In cylindrical coordinate

r

bz

zzrrrrz

2

Screw dislocations will interact strongly with a defect which has a large shear strain associated with it.

Example: Screw dislocation with an interstitial atom in a BCC metal (interstitial atom produces shear strain approx equal to = 0.5

Thus, only shear strain around a screw dislocation exists >> No dilation stain

Note:

1) The stress is proportional to 1/r ….Long-range

2) as

ty.singulari.....,0 r

The assumed linear elasticity behavior breaks down near the dislocation line….The dislocation Core…

The dislocation Core…

As the center of the dislocation is approached the linear elasticity theory ceases to be valid and non-linear, atomistic model must be used. The region where linear elasticity breaks down is called the core of the dislocation or radius

0r

0r

The stress reached the theoretical limit and the strain exceeds about 10% when br Typically br0 2

Edge Dislocation

0,0

z

StrainPlane

iz

uu

Airy stress

function04

yx

x

y

xy

yy

xx

2

2

2

2

2

Solution leads to:

0u

))(1(4)ln(

)1(4

21

2π

bu

)()1(2

1tan

2π

bu

z

22

2222

y

22

1

x

yx

yxyx

yx

xy

x

y

and the non-zero stress components are:

)y(x

yb

)y(x

)yx(xb

)y(x

)yy(xb

)y(x

)yy(3xb

22

222

22

222

22

222

22

)1(

)1(2

)1(2

)1(2

zz

xy

yy

xx

Since edge dislocations have both shear and normal stress they will interact with defects that produce both

shear and normal strains.

Edge dislocation interacts with another edge

dislocation

Edge dislocation does not interact with pure

screw dislocation.

Strain Energy

Consider the energy stored per unit length in the elastic filed of the infinite screw dislocation, in a region bounded by cylinders of radius and R

0r

0

s

r

R

4

bdrr

L

Wln2

2

22

0

R

r

z

The energy diverges as Rand as

surface)freethetondislocatiofromdistance(ShortestlisR

Thus R can’t be infinite, an approximate choice for

00 r

Similar expression can be obtained for the edge dislocation:

0

e

r

R

4

b

L

Wln

)1(

2

and br0 2For bR 310

2ln 0r

R

and for 3/1

2

2

1b

L

W

L

W se

Observations

W is proportional to 2b

Therefore, we want b to be as short as possible to minimize the energy

-- close packed directions are chosen are the preferred ones.

W/L is a force that acts along the dislocation line (line tension)If a stress is applied the dislocation will bend until force balance is reached between the applied stress and the line tension

ij

1)

2)

Dislocation problems are solved by either:

a) Energy balance –

work done on a dislocation by a stress field =energy increase of the dislocation due to its increase in line length

or

b) Force balanceThe force on a dislocation due to a stress field = resisting force on a dislocation due to its line tension.

3)

Strain energy is actually made up of

elastic energy + core energy

2

2

1b

atomeV /8

atomeV /1

Therefore, elastic strain >> core energy

4)

In addition the 8 eV/atom energy is a large energy compared to formation of a vacancy ~ 1 eV/vacancy

Therefore,

0 STHGdisl even at high temp.

Thus, dislocations are thermodynamically unstable, and hence the number of dislocations which might be preset due to thermal activation is small.

5)