Elastic distortions β(d) created by the presence of a dislocation field will give rise to an associated stress field π(d) through the stiffness relationship
π(d) = C : β(d) (3.2.1)
where C is the fourth rank stiffness tensor of the material.
It is desirable then that a direct relation between the dislocation density tensor and the dislo- cation stress be determined. This can be achieved using Mura’s (1963) formula [69] for elastic distortion at position X arising from the contributions of points X0 along dislocation line L, this is often represented in literature as:
βji(X)[ei ej] =
I
L
where G is a Green’s function for isotropic elasticity. The notation Gip,q refers to the gradient of
the Green’s function, taken such that ∇ G ≡ Gip,q [ei⊗ ep] ⊗ [eq]. Making use of the dot-cross
operator ˙×; defined such that (a b) ˙×(c d) = (a · c)(b × d); it is possible to write eqn (3.2.2) in vector notation:
β(X) = I
L
∇ G(X − X0) : C ˙× (b t) dl(X0) The full form of G is given by
G(X − X0) = 1 8πµ h I ∇2R − 1 2(1 − ν) ∇ ∇R i (3.2.3) where µ is the shear modulus, I is identity, ν is Poisson’s ratio and R =k X − X0 k. The points X and X0 exist within the domain Ω.
In order to homogenise the Mura formula it is beneficial to consider a dislocation distribution within a small volume element ω ⊂ Ω centred at X, with volume δVω(X). This volume surrounds
a dislocation segment of length δl. A dislocation distribution function D(X) may be introduced across Ω such that the dislocation content at point X is described by
D(X) = b δ(X − X0)
if a single dislocation is present. The Dirac delta function equals 1 whenever the point X falls on a dislocation line, and 0 otherwise. For a number of parallel dislocations (with the same Burgers vector) at the same point, the distribution function becomes a sum across these dislocations
D(X) =
N
X
k=1
b δ(X − X0 k) (3.2.4)
where N is the number of dislocations and k is the index for the different dislocation lines.
The volume average of the distribution function over small element ω becomes
hD(X)iω = b δVω(X) N X k=1 Z Z Z ω δ(X− X0 k) d3X (3.2.5)
For straight parallel dislocation segments on the same slip plane the coordinates along a dislocation segment will vary in only one dimension, collapsing the volume integral into a line integral along the segment length δl, i.e:
Z Z Z ω δ(X− X0 k) dX1 dX2 dX3 = Z δl δ(X− X0 k) dX3 = δl (3.2.6)
Using this equivalence eqn (3.2.5) becomes
hD(X)iω = b N (X) δl δVω(X)
= b ρ(X) (3.2.7)
where the scalar dislocation density ρ(X) is introduced as the total length of dislocation contained in element ω.
If Mura’s formula is now considered for a number of identical dislocations N , following line L β(X) =
I
L
∇ G(X − X0) : C ˙× (N b t) dl(X0) then the distortion at X contributed by a line segment at point X0 is
δβ(X − X0) = ∇ G(X − X0) : C ˙× (N (X0) b t(X0)) δl(X0) (3.2.8) Using the definition in eqn (3.2.7) the distribution function may now be introduced, allowing the distortion from volume element ω to be calculated
δβ(X − X0) = ∇ G(X − X0) : C ˙× (hD(X0)iω t(X0)) δVω(X0)
and the expression for dislocation density to be introduced
δβ(X − X0) = ∇ G(X − X0) : C ˙× α(X0) δVω(X0) (3.2.9)
α(X0) = hD(X)iω t(X0) (3.2.10)
The total dislocation stress at X relates to the sum of the distortions from all other points in the Ω domain, indexed as X0 m π(d)(X) = C : X m∈Ω δβ(X − X0 m) π(d)(X) = X m∈Ω C : ∇ G(X − X0 m)) : C ˙× α(X0 m)) δVω(X0 m)) (3.2.11)
If all volumes δVω are the same, then taking the limit δVω → 0 forms the integral π(d)(X) = Z Z Z Ω C : ∇ G(X − X0) : C ˙× α(X0)) dV j(X − X0) = C : ∇ G(X − X0) : C (3.2.12) where the third rank tensor j(X − X0) is introduced to collect material specific terms.
The dislocation stress field may now be directly defined at any point in a domain by a function of the dislocation density at all other points
π(d)(X) = Z Z Z
Ω
Chapter 4
Field Dislocation Mechanics II:
Application to Nickel-based
Superalloys
This chapter will apply the general FDM formulation to the specific case of a nickel-based super- alloy material deforming under plane strain conditions. Plane strain simplifications for the SSD density α, dislocation velocity and stress fields, and interaction with appropriate microstructural features will all be considered. Finally the integration within a crystal plasticity formulation is described.
Note: The contributions from geometrically necessary dislocations (GNDs) are omitted from this work, but will provide grounds for future extension of this model. It is assumed that for the single crystal, single-slip arrangement in this work the GND effect would be minimal as the deformation gradients on a slip plane are not severe. This assumption is later discussed in Section 9.3.
4.1
Adapting the Transport Equation to Plane Strain
This section will see the FDM transport equation simplified to 2D plane strain conditions. The plane of interest is the X1 - X2 plane. The out-of-plane strain components are all removed apart
It has been stated earlier in eqn (3.1.4) that the dislocation density tensor can be defined as
α = ρ b ⊗ ξ αip = ρ bi ξp
in the general case. However if only edge type dislocations are considered with tangent vector parallel to the X3 axis then the equations can be reduced to a 2D form. All Burgers vectors are set
parallel to the X1 axis, meaning the system contains density that evolves along a single slip plane
(aligned to this axis). This gives the dislocation density the parameters: b = [b 0 0] ξ = [0 0 1] α = ρ 0 0 b 0 0 0 0 0 0 α13= ρ b (4.1.1)
Here it can be seen that the α tensor is reduced to a single component α13. As the Burgers
magnitude will be constant upon a single slip system then the dislocation density is now described by a scalar field ρ(X). A graphical representation of the dislocation alignment is shown in Fig. (4.1.1), with the slip velocity vector ˙q1 indicated.
If the density tensor in the general form eqn (3.1.24) is replaced with α13 then a scalar continuity
equation can be developed
∂α13
∂t − α13 L
(p)
33 − 3nm ∂m L(p)1n = 0
As there is no velocity in the X3 direction within the plane strain setup then L(p)33 = 0
∂α13
∂t − 3nm ∂m L
(p)
1n = 0 (4.1.2)
It now remains to evaluate the flux tensor in plane strain. This involves first calculating the value of the curl of the dislocation flux in general space, then reducing to the scalar setup.
Expanding the Dislocation Flux Term
Expanding the dislocation flux term fully using the definition L(p)ik = kmnαimq˙nfrom eqn (3.1.19)
gives the following components:
L(p)11 = α12 q˙3 − α13 q˙2 L(p)12 = α13 q˙1 − α11 q˙3 L(p)13 = α11 q˙2 − α12 q˙1 L(p)21 = α22 q˙3 − α23 q˙2 L(p)22 = α23 q˙1 − α21 q˙3 L(p)23 = α21 q˙2 − α22 q˙1 L(p)31 = α32 q˙3 − α33 q˙2 L(p)32 = α33 q˙1 − α31 q˙3 L(p)33 = α31 q˙2 − α32 q˙1
Meaning that for the plane strain setup in eqn (4.1.1), where ˙q3 = 0, only two terms survive
L(p)11 = − α13 q˙2
Expanding the curl of the flux tensor using placeholder tensor Aip= pnm∂mL(p)in gives the following components: A11 = ∂3 L (p) 12 − ∂2 L (p) 13 A12 = ∂1 L (p) 13 − ∂3 L (p) 11 A13 = ∂2 L (p) 11 − ∂1 L (p) 12 A21 = ∂3 L(p)22 − ∂2 L(p)23 A22 = ∂1 L(p)23 − ∂3 L(p)21 A23 = ∂2 L(p)21 − ∂1 L(p)22 A31 = ∂3 L(p)32 − ∂2 L(p)33 A32 = ∂1 L(p)33 − ∂3 L(p)31 A33 = ∂2 L(p)31 − ∂1 L(p)32
As only L(p)11 and L(p)12 exist, and because the spatial differential in the third direction has no value (i.e ∂3 = 0), this leaves only
A13 = ∂2 L(p)11 − ∂1 L(p)12
3nm ∂m L(p)1n = A13 = − ∂1 (α13 q˙1) − ∂2 (α13 q˙2) (4.1.3)
The Plane Strain Scalar Continuity Equation
Now the expanded flux term in eqn (4.1.3) can be inserted into eqn (4.1.2) ∂α13
∂t + ∂1 (α13 q˙1) + ∂2 (α13 q˙2) = 0
substituting for the scalar density introduces the Burgers magnitude to every term. As this is a constant for a single dislocation variety then it has no effect on the continuity.
b ∂ρ
∂t + b ∂1 (ρ ˙q1) + b ∂2 (ρ ˙q2) = 0 ∂ρ
∂t + ∂1 (ρ ˙q1) + ∂2 (ρ ˙q2) = 0 Using the chain rule and collecting terms now gives a more concise form:
∂ρ
∂t + ∇ · ( ˙q ρ) = 0
∂ρ
This equation now describes the transport of a field of parallel dislocations within a 2D plastic state. It requires only the input of the dislocation velocity field within the domain and the boundary conditions associated with the environment in order to function as a model for dislocation behaviour.