introduction to and fundamentals of discrete dislocations ... · dislocations in crystalline...
TRANSCRIPT
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
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)