Dislocation density tensor
Samuel Forest
Mines ParisTech / CNRS
Centre des Matériaux/UMR 7633
BP 87, 91003 Evry, France
Samuel.Forest@mines-paristech.fr
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Resulting Burgers vector
[Kröner, 1969]
Resulting Burgers B s for slip system s for a closed
circuit limiting the surface S
„Z
«
ξ (x ).n dS b s
Bs =
Z S
α
.n dS
=
∼
S
where
α
(x ) = b s ⊗ ξ (x )
∼
Burgers vector
b s (x )
Consider contributions of all systems and ensemble
average it
Z
α
.n dS
B =
∼
S
α
=
∼
Dislocation line vector
ξ (x )
X
< bs ⊗ ξ >
s
Ergodic hypothesis: compute the dislocation density
tensor by means of a volume average in DDD simulations
Statistical theory of dislocations
5/33
Dislocation density tensor for edge dislocations
Resulting Burgers vector
B
α
∼
=
=
=
nb e 1
=
α
.e 3 S
∼
n
b ⊗ξ
S
−ρG b e 1 ⊗ e 3
G
n edge dislocations piercing the
surface S
convention:
z =
b
×ξ
b
Statistical theory of dislocations
ρ = n/S is the density of geometrically necessary dislocations according to
(Ashby, 1970).
2
3
0 0 α13
4 0 0
0 5
0 0
0
out of diagonal component of α
∼
diagonal component α33 for screw dislocations with b = b e 3
6/33
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Fourth order and scalar dislocation densities
[Kröner, 1969]
Two–point correlation tensor
α
(x , x 0 ) =< b (x ) ⊗ ξ (x ) ⊗ b (x 0 ) ⊗ ξ (x 0 ) >
≈
The invariant quantity
Z
Z
1
b2
L
αijij (x , x ) dV =
χ(x ) dV = b 2 = b 2 ρ
V V
V V
V
where L is the total length of dislocation lines inside V and χ(x ) equals 1 when
there is a dislocation at x , 0 otherwise.
ρ is the scalar dislocation density traditionally used in physical metallurgy
Statistical theory of dislocations
8/33
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Reminder on tensor analysis (1)
The Euclidean space is endowed with an arbitrary coordinate
system characterizing the points M(q i ). The basis vectors are
defined as
∂M
ei =
∂q i
The reciprocal basis (e i )i=1,3 of (e i )i=1,3 is the unique triad of
vectors such that
e i · e j = δji
If a Cartesian orthonormal coordinate system is chosen, then both
bases coincide.
Continuum crystal plasticity approach
10/33
Reminder on tensor analysis (2)
The gradient operator for a tensor field T (X ) of arbitrary rank is
then defined as
∂T
⊗ ei
grad T = T ⊗ ∇ :=
∂q i
The gradient operation therefore increases the tensor rank by one.
Continuum crystal plasticity approach
11/33
Reminder on tensor analysis (2)
The gradient operator for a tensor field T (X ) of arbitrary rank is
then defined as
∂T
⊗ ei
grad T = T ⊗ ∇ :=
∂q i
The gradient operation therefore increases the tensor rank by one.
The divergence operator for a tensor field T (X ) of arbitrary rank is
then defined as
∂T
· ei
div T = T · ∇ :=
∂q i
The divergence operation therefore decreases the tensor rank by one.
Continuum crystal plasticity approach
12/33
Reminder on tensor analysis (2)
The gradient operator for a tensor field T (X ) of arbitrary rank is then
defined as
grad T = T ⊗ ∇ :=
∂T
⊗ ei
∂q i
The gradient operation therefore increases the tensor rank by one.
The divergence operator for a tensor field T (X ) of arbitrary rank is then
defined as
div T = T · ∇ :=
∂T
· ei
∂q i
The divergence operation therefore decreases the tensor rank by one.
The curl operator (or rotational operator) for a tensor field T (X ) of
arbitrary rank is then defined as
∂T
∧ ei
∂q i
a ∧ b = ijk aj bk e i = ∼
: (a ⊗ b )
curl T = T ∧ ∇ :=
where the vector product is ∧
The component ijk of the third rank permutation tensor is the signature of the
permutation of (1, 2, 3).
The curl operation therefore leaves the tensor rank unchanged.
Continuum crystal plasticity approach
13/33
Reminder on tensor analysis (3)
With respect to a Cartesian orthonormal basis, the previous formula
simplify. We give the expressions for a second rank tensor T
∼
grad T
= Tij,k e i ⊗ e j ⊗ e k
∼
div T
= Tij,j e i
∼
We consider then successively the curl of a vector field and of a
second rank vector field, in a Cartesian orthonormal coordinate
frame
∂u
∧ e j = ui,j e i ∧ e j = kij ui,j e k
curl u =
∂Xj
curl A
=
∼
∂A
∼
∧ e k = Aij,k e i ⊗ e j ∧ e k = mjk Aij,k e i ⊗ e m
∂xk
Continuum crystal plasticity approach
14/33
Reminder on tensor analysis (4)
We also recall the Stokes formula for a vector field for a surface S
with unit normal vector n and oriented closed border line L:
I
Z
I
Z
u · dl = − (curl u ) · n ds,
ui dli = −kij
ui,j nk ds
L
S
L
S
Applying the previous formula to uj = Aij at fixed i leads to the
Stokes formula for a tensor field of rank 2:
Z
I
Z
I
) · n ds,
Aij dlj = −mjk
Aij,k nm ds
A
· dl = − (curl A
∼
∼
L
S
Continuum crystal plasticity approach
L
S
15/33
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Incompatibility of elastic and plastic deformations
F
=E
.P
∼
∼ ∼
F
P
E
In continuum mechanics, the
previous differential operators
are used with respect to the
initial coordinates X or with
respect to the current coordinates x of the material points.
In the latter case, the notation
∇, grad , div and curl are used
but in the former case we adopt
∇X , Grad, Div and Curl.
F
=∼
1 + Grad u =⇒ Curl F
=0
∼
∼
The deformation gradient is a compatible field which derives from the
displacement vector field. This is generally not the case for elastic and plastic
deformation:
Curl E
6= 0, Curl P
6= 0
∼
∼
It may happen incidentally that elastic deformation be compatible for instance
when plastic or elastic deformation is homogeneous.
Continuum crystal plasticity approach
17/33
Incompatibility of elastic and plastic deformations
F
=E
.P
∼
∼ ∼
E
relates the infinitesimal vec∼
tors dζ and dx , where dζ results from the cutting and releasing operations from the infinitesimal current lattice vector dx
F
P
E
−1
dζ = E
· dx
∼
If S is a smooth surface containing x in the current configuration and bounded
by the closed line c, the true Burgers vector is defined as
I
−1
B =
E
.dx
∼
c
Continuum crystal plasticity approach
18/33
Dislocation density tensor in continuum crystal
plasticity
F
=E
.P
∼
∼ ∼
E
relates the infinitesimal vec∼
tors dζ and dx , where dζ results from the cutting and releasing operations from the infinitesimal current lattice vector dx
F
E
P
−1
dζ = E
· dx
∼
If S is a smooth surface containing x in the current configuration and bounded
by the closed line c, the true Burgers vector is defined as
I
Z
Z
−1
−1
B =
E
.dx = − (curl E
).n ds =
α
· n ds
∼
∼
∼
c
S
S
according to Stokes formula which gives the definition of the true dislocation
density tensor
−1
−1
α
= −curl E
= −jkl Eik,l
ei ⊗ ej
∼
∼
Continuum crystal plasticity approach
19/33
Dislocation density tensor in continuum crystal
plasticity
The Burgers vector can also be computed by means of a closed
circuit c0 ⊂ Ω0 convected from c ⊂ Ω:
I
I
I
−1
−1
B =
E
· dx =
E
·F
· dX =
P
· dX
∼
∼
∼
∼
c
c0
c0
Z
Z
T ds
= −
(Curl P
) · dS = − (Curl P
)·F
·
∼
∼
∼
J
S0
S
−T
Nanson’s formula ds = JF
· dS has been used. We obtain the
∼
alternative definition of the dislocation density tensor
1
−1
T
α
= −curl E
= − (Curl P
)·F
∼
∼
∼
∼
J
Continuum crystal plasticity approach
20/33
Dislocation density tensor in continuum crystal
plasticity
The Burgers vector can also be computed by means of a closed
circuit c0 ⊂ Ω0 convected from c ⊂ Ω:
I
I
I
−1
−1
B =
E
· dx =
E
·F
· dX =
P
· dX
∼
∼
∼
∼
c
c0
c0
Z
Z
T ds
= −
(Curl P
) · dS = − (Curl P
)·F
·
∼
∼
∼
J
S0
S
−T
Nanson’s formula ds = JF
· dS has been used. We obtain the
∼
alternative definition of the dislocation density tensor
1
−1
T
α
= curl E
= − (Curl P
)·F
∼
∼
∼
∼
J
or equivalently
−1
−T
T
J(curl E
)·E
= (Curl P
)·P
∼
∼
∼
∼
which is a consequence of curl F
= curl (E
·P
)=0
∼
∼
∼
Continuum crystal plasticity approach
21/33
Dislocation density tensor at small deformation
We introduce the notations
e
p
H
= Grad u = H
+H
,
∼
∼
∼
e
e
e
with H
=ε
+ω
,
∼
∼
∼
p
p
p
H
=ε
+ω
∼
∼
∼
Within the small perturbation framework
e
p
p
e
p
F
=1
+H
=1
+εe +ω
+ε
+ω
' (1
+εe +ω
).(1
+εp +ω
)'E
.P
∼
∼
∼
∼ ∼
∼
∼
∼
∼ ∼
∼
∼ ∼
∼
∼ ∼
We have
e
E
'1
+H
,
∼
∼
∼
p
P
'1
+H
∼
∼
∼
−1
e
E
'1
−H
∼
∼
∼
so that the dislocation density tensor can be computed as
e
p
α
' Curl H
= −Curl H
∼
∼
∼
since Curl H
= 0 due to the compatibility of the deformation
∼
gradient.
Continuum crystal plasticity approach
22/33
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Lattice rotation full field measurement
initial orientations
lattice orientation field after deformation
Continuum crystal plasticity approach
24/33
Dislocation density vs. lattice curvature
Experimental techniques like EBSD provide the field of lattice orientation and,
e
consequently, of lattice rotation R
during deformation. Since
∼
−1
e−1
eT
α
= −curl E
= −curl (U
·R
)
∼
∼
∼
∼
the hypothesis of small elastic strain (and in fact of small elastic strain
gradient) implies
eT
α
' −curl R
∼
∼
If, in addition, elastic rotations are small, we have
e
e
α
' −curl (1
−ω
) = curl ω
∼
∼
∼
∼
The small rotation axial vector is defined as
×
1
e
ωe = − ∼
:ω
,
∼
2
Continuum crystal plasticity approach
×
e
ω
= −
·ωe
∼
∼
25/33
Dislocation density vs. lattice curvature
or, in matrix form,
2
0
6
6
e
e
[ω
]
=
6 −ω12
∼
4
e
ω31
e
ω12
e
−ω31
0
e
ω23
e
−ω23
0
3
2
0
7 6
7 6 ×e
7 = 4 ω3
5
×
−ω e2
×
−ω e3
0
×
ω e1
×
ω e2
3
7
×
−ω e1 7
5
0
The gradient of the lattice rotation field delivers the lattice curvature tensor. In
the small deformation context, the gradient of the rotation tensor is
represented by the gradient of the axial vector:
×
κ
:= grad ω e
∼
e
One can establish a direct link between curl ω
and the gradient of the axial
∼
vector associated with ω
.
∼
Continuum crystal plasticity approach
26/33
Dislocation density vs. lattice curvature
e
[curl ω
]
∼
=
=
2
e
e
ω12,3
+ ω31,2
e
−ω31,1
e
−ω12,1
3
6
6
6
4
e
−ω23,2
e
e
ω12,3
+ ω23,1
e
−ω12,2
e
−ω23,3
e
−ω31,3
e
e
ω23,1
+ ω31,2
7
7
7
5
2
−ω e3,3 − ω e2,2
6
6
6
4
ω e1,2
×
×
×
ω e2,1
×
−ω e3,3 − ω e1,1
×
ω e2,3
ω e1,3
×
×
3
×e
7
7
7
5
ω 3,1
×
×
ω 3,2
×
×
−ω e1,1 − ω e2,2
from which it becomes apparent that
T
α
=κ
− (trace κ
)1,
∼
∼
∼ ∼
T
κ
=α
−
∼
∼
1
(trace α
)1
∼ ∼
2
This is a remarkable relation linking, with the context of small elastic strains1
and rotations, the dislocation density tensor to lattice curvature. It is known as
Nye’s formula [Nye, 1953].
1
and in fact of small gradient of elastic strain.
Continuum crystal plasticity approach
27/33
Lattice curvature due to edge dislocations
α
= −ρG b e 1 ⊗ e 3
∼
so that
κ
= −ρG b e 3 ⊗ e 1
∼
the only non–vanishing component is
κ31 = Φe3,1
which corresponds to bending with
respect to axis 3
Continuum crystal plasticity approach
28/33
Tilt boundary
(Read, 1958)
Continuum crystal plasticity approach
29/33
Lattice torsion due to screw dislocations
screw dislocations parallel to e 3
α
= ρG b e 3 ⊗ e 3
∼
T
κ
=α
−
∼
∼

ρG b 
[κ
]=
∼
2
1
(trace α
)1
∼ ∼
2

−1 0 0
0 −1 0 
0
0 1
torsion with respect to all axes!!!
relaxed Volterra cylinder
[Friedel, 1964]
Continuum crystal plasticity approach
30/33
Twist boundary
two families of orthogonal screw dislocations
α
= −ρG b(e 1 ⊗e 1 +e 2 ⊗e 2 )
∼

0 0 0
[κ
] = ρG b  0 0 0 
∼
0 0 1

torsion with respect
to axis 3
κ33 = Φe3,3
(Read, 1958)
Continuum crystal plasticity approach
31/33
Plan
1
Statistical theory of dislocations
The dislocation density tensor
Scalar dislocation densities
2
Continuum crystal plasticity approach
Incompatibility and dislocation density tensor
Lattice curvature tensor
3
Need for generalized continuum crystal plasticity
Towards generalized single crystal plasticity
A continuum crystal plasticity model should at least include
• the effect of scalar dislocation density ρS ; this is the case of classical
crystal plasticity according to Mandel, Teodosiu, Sidoroff, Asaro which
incorporate hardening rules from physical metallurgy
• the effect of dislocation density tensor; it is the main ingredient of the
continuum theory of dislocations (closure problem)
geometrically necessary dislocation density ρG
Combine both! But acknowledge then the fact that the presence of α
in the
∼
constitutive equations leads to higher order partial differential equations when
inserted in the equilibrium equations. Additional boundary conditions are
necessary. Several possibilities:
• since α
is implicitly related to P
⊗ ∇ and F
⊗ ∇, consider a strain
∼
∼
∼
gradient model or strain gradient plasticity model;
[Mindlin and Eshel, 1968] [Fleck and Hutchinson, 1997]
• since α
is related to the lattice curvature tensor, raise the lattice rotation
∼
to internal degrees of freedom and consider a Cosserat theory.
[Günther, 1958] [Kröner, 1963] [Mura, 1963]
Need for generalized continuum crystal plasticity
33/33
Fleck N.A. and Hutchinson J.W. (1997).
Strain gradient plasticity.
Adv. Appl. Mech., vol. 33, pp 295–361.
Friedel J. (1964).
Dislocations.
Pergamon.
Günther W. (1958).
Zur Statik und Kinematik des Cosseratschen Kontinuums.
Abhandlungen der Braunschweig. Wiss. Ges., vol. 10, pp
195–213.
Kröner E. (1963).
On the physical reality of torque stresses in continuum
mechanics.
Int. J. Engng. Sci., vol. 1, pp 261–278.
Mindlin R.D. and Eshel N.N. (1968).
On first strain gradient theories in linear elasticity.
Int. J. Solids Structures, vol. 4, pp 109–124.
Need for generalized continuum crystal plasticity
33/33
Mura T. (1963).
On dynamic problems of continuous distribution of
dislocations.
Int. J. Engng. Sci., vol. 1, pp 371–381.
Nye J.F. (1953).
Some geometrical relations in dislocated crystals.
Acta Metall., vol. 1, pp 153–162.
Need for generalized continuum crystal plasticity
33/33