Chapter IV: Verification Examples
4.2 Infinite Straight Screw Dislocation
Consider the segment of screw dislocation line with positive direction along the x3−axis as shown in Figure 4.1. The segment has length2Land is discretized using 2M+1uniformly distributed monopoles so that the element of line length associated with each monopole is l = 2L
2M + 1. In light of the method of monopoles, we approximate the nonzero component of the dislocation density tensor for this segment as
αh33 =
a=M
X
a=−M
blδ(x−xa) =
a=M
X
a=−M
2L
2M+ 1bδ(x1)δ(x2)δ(x3−xa3), (4.1) where b is the magnitude of the Burgers vector of the dislocation and
xa3 =al = 2aL
2M + 1, a=−M, . . . , M . Self-stresses
Using the definition of the dislocation density tensor (see equation (3.2)), the self stress of a continuous distribution of dislocations in an infinite solid Ω can be written as [62]
σij = ˆ
Ω
CijklCpqmnelnhGkp,q(x−x0)αmh(x0)dx0. (4.2)
l
e<latexit sha1_base64="+h5Gv+bP5kCcL6p/iEmNWnVm7dQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdY8/rVWtu3Z2BLBOvIDUo0OxVv7r9hGUxSsME1brjuakJcqoMZwInlW6mMaVsRAfYsVTSGHWQz06dkBOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HQqNgRv8eVl4p/Vr+re3XmtcV2kUYYjOIZT8OACGnALTfCBwQCe4RXeHOG8OO/Ox7y15BQzh/AHzucPXJaNYQ==</latexit><latexit sha1_base64="+h5Gv+bP5kCcL6p/iEmNWnVm7dQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdY8/rVWtu3Z2BLBOvIDUo0OxVv7r9hGUxSsME1brjuakJcqoMZwInlW6mMaVsRAfYsVTSGHWQz06dkBOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HQqNgRv8eVl4p/Vr+re3XmtcV2kUYYjOIZT8OACGnALTfCBwQCe4RXeHOG8OO/Ox7y15BQzh/AHzucPXJaNYQ==</latexit><latexit sha1_base64="+h5Gv+bP5kCcL6p/iEmNWnVm7dQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdY8/rVWtu3Z2BLBOvIDUo0OxVv7r9hGUxSsME1brjuakJcqoMZwInlW6mMaVsRAfYsVTSGHWQz06dkBOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HQqNgRv8eVl4p/Vr+re3XmtcV2kUYYjOIZT8OACGnALTfCBwQCe4RXeHOG8OO/Ox7y15BQzh/AHzucPXJaNYQ==</latexit><latexit sha1_base64="+h5Gv+bP5kCcL6p/iEmNWnVm7dQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdY8/rVWtu3Z2BLBOvIDUo0OxVv7r9hGUxSsME1brjuakJcqoMZwInlW6mMaVsRAfYsVTSGHWQz06dkBOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HQqNgRv8eVl4p/Vr+re3XmtcV2kUYYjOIZT8OACGnALTfCBwQCe4RXeHOG8OO/Ox7y15BQzh/AHzucPXJaNYQ==</latexit> 1 e<latexit sha1_base64="3O6YLVvrmffK6RJaJaHBAAZiLcg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvRi8eKxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTP3WEyrNE/lgxikGMR1IHnFGjZXusVfvVapuzZ2BLBOvIFUo0OxVvrr9hGUxSsME1brjuakJcqoMZwIn5W6mMaVsRAfYsVTSGHWQz06dkFOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HTKNgRv8eVl4tdrVzXv7rzauC7SKMExnMAZeHABDbiFJvjAYADP8ApvjnBenHfnY9664hQzR/AHzucPXhmNYg==</latexit><latexit sha1_base64="3O6YLVvrmffK6RJaJaHBAAZiLcg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvRi8eKxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTP3WEyrNE/lgxikGMR1IHnFGjZXusVfvVapuzZ2BLBOvIFUo0OxVvrr9hGUxSsME1brjuakJcqoMZwIn5W6mMaVsRAfYsVTSGHWQz06dkFOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HTKNgRv8eVl4tdrVzXv7rzauC7SKMExnMAZeHABDbiFJvjAYADP8ApvjnBenHfnY9664hQzR/AHzucPXhmNYg==</latexit><latexit sha1_base64="3O6YLVvrmffK6RJaJaHBAAZiLcg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvRi8eKxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTP3WEyrNE/lgxikGMR1IHnFGjZXusVfvVapuzZ2BLBOvIFUo0OxVvrr9hGUxSsME1brjuakJcqoMZwIn5W6mMaVsRAfYsVTSGHWQz06dkFOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HTKNgRv8eVl4tdrVzXv7rzauC7SKMExnMAZeHABDbiFJvjAYADP8ApvjnBenHfnY9664hQzR/AHzucPXhmNYg==</latexit><latexit sha1_base64="3O6YLVvrmffK6RJaJaHBAAZiLcg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvRi8eKxhbaUDbbSbt0swm7G6GE/gQvHlS8+o+8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6iRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTP3WEyrNE/lgxikGMR1IHnFGjZXusVfvVapuzZ2BLBOvIFUo0OxVvrr9hGUxSsME1brjuakJcqoMZwIn5W6mMaVsRAfYsVTSGHWQz06dkFOr9EmUKFvSkJn6eyKnsdbjOLSdMTVDvehNxf+8TmaiyyDnMs0MSjZfFGWCmIRM/yZ9rpAZMbaEMsXtrYQNqaLM2HTKNgRv8eVl4tdrVzXv7rzauC7SKMExnMAZeHABDbiFJvjAYADP8ApvjnBenHfnY9664hQzR/AHzucPXhmNYg==</latexit> 2
2L
e3
<latexit sha1_base64="PusC0/YE5zU+QTH7YuXNFlwmavc=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lUUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+ag1QcDj/dmmJkXpoJr47pfTmlpeWV1rbxe2djc2t6p7u496CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxxdT/3WIyrNE3lvxikGMR1IHnFGjZXusHfaq9bcujsD+Uu8gtSgQLNX/ez2E5bFKA0TVOuO56YmyKkynAmcVLqZxpSyER1gx1JJY9RBPjt1Qo6s0idRomxJQ2bqz4mcxlqP49B2xtQM9aI3Ff/zOpmJLoKcyzQzKNl8UZQJYhIy/Zv0uUJmxNgSyhS3txI2pIoyY9Op2BC8xZf/Ev+kfln3bs9qjasijTIcwCEcgwfn0IAbaIIPDAbwBC/w6gjn2Xlz3uetJaeY2YdfcD6+AV+cjWM=</latexit><latexit sha1_base64="PusC0/YE5zU+QTH7YuXNFlwmavc=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lUUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+ag1QcDj/dmmJkXpoJr47pfTmlpeWV1rbxe2djc2t6p7u496CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxxdT/3WIyrNE3lvxikGMR1IHnFGjZXusHfaq9bcujsD+Uu8gtSgQLNX/ez2E5bFKA0TVOuO56YmyKkynAmcVLqZxpSyER1gx1JJY9RBPjt1Qo6s0idRomxJQ2bqz4mcxlqP49B2xtQM9aI3Ff/zOpmJLoKcyzQzKNl8UZQJYhIy/Zv0uUJmxNgSyhS3txI2pIoyY9Op2BC8xZf/Ev+kfln3bs9qjasijTIcwCEcgwfn0IAbaIIPDAbwBC/w6gjn2Xlz3uetJaeY2YdfcD6+AV+cjWM=</latexit><latexit sha1_base64="PusC0/YE5zU+QTH7YuXNFlwmavc=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lUUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+ag1QcDj/dmmJkXpoJr47pfTmlpeWV1rbxe2djc2t6p7u496CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxxdT/3WIyrNE3lvxikGMR1IHnFGjZXusHfaq9bcujsD+Uu8gtSgQLNX/ez2E5bFKA0TVOuO56YmyKkynAmcVLqZxpSyER1gx1JJY9RBPjt1Qo6s0idRomxJQ2bqz4mcxlqP49B2xtQM9aI3Ff/zOpmJLoKcyzQzKNl8UZQJYhIy/Zv0uUJmxNgSyhS3txI2pIoyY9Op2BC8xZf/Ev+kfln3bs9qjasijTIcwCEcgwfn0IAbaIIPDAbwBC/w6gjn2Xlz3uetJaeY2YdfcD6+AV+cjWM=</latexit><latexit sha1_base64="PusC0/YE5zU+QTH7YuXNFlwmavc=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lUUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+ag1QcDj/dmmJkXpoJr47pfTmlpeWV1rbxe2djc2t6p7u496CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxxdT/3WIyrNE3lvxikGMR1IHnFGjZXusHfaq9bcujsD+Uu8gtSgQLNX/ez2E5bFKA0TVOuO56YmyKkynAmcVLqZxpSyER1gx1JJY9RBPjt1Qo6s0idRomxJQ2bqz4mcxlqP49B2xtQM9aI3Ff/zOpmJLoKcyzQzKNl8UZQJYhIy/Zv0uUJmxNgSyhS3txI2pIoyY9Op2BC8xZf/Ev+kfln3bs9qjasijTIcwCEcgwfn0IAbaIIPDAbwBC/w6gjn2Xlz3uetJaeY2YdfcD6+AV+cjWM=</latexit>
Figure 4.1: Discretization of the screw dislocation segment in the method of monopoles.
Inserting approximation (4.1) into Equation (4.2), the resulting stress field is σijh(x) =
ˆ
Ω
CijklCpq3nln3Gkp,q(x−x0)
a=M
X
a=−M
2L
2M + 1bδ(x01)δ(x02)δ(x03−xa3)dx0
=CijklCpq3nln3b
a=M
X
a=−M
2L
2M + 1Gkp,q(x1, x2, x3 −xa3)
= (Cijk1Cpq32−Cijk2Cpq31)b
a=M
X
a=−M
Gkp,q(x1, x2, x3−xa3) 2L 2M + 1,
(4.3) where we have used the sifting property of the Dirac distribution—see Equation (2.59)—to write the second equality.
For a fixed L, the last expression in (4.3) is a Riemann sum [124–126] for
´L
−LGkp,q(x1, x2, x3−x03)dx03. Therefore, taking the limit as M → ∞, we get σijh(x)→(Cijk1Cpq32−Cijk2Cpq31)b
ˆ L
−L
Gkp,q(x1, x2, x3−x03)dx03. (4.4) Furthermore, in the limit L→ ∞, (4.4) becomes
σijh(x)→(Cijk1Cpq32−Cijk2Cpq31)b ˆ ∞
−∞
Gkp,q(x1, x2, x3−x03)dx03. (4.5) For an isotropic medium with Lam´e constant λ and shear modulus µ, we have, recalling Equation (2.10),
Cijkl=λδijδkl+µ(δilδjk+δikδjl).
so that (4.5) simplifies to
σijh →µbCijk1[Fk32(x) +Fk23(x)]−µbCijk2[Fk31(x) +Fk13(x)], (4.6) where
Fkpq(x) = ˆ ∞
−∞
Gkp,q(x1, x2, x3−x03)dx03.
But recall from Equation (2.70) that the Green’s tensor for an infinite isotropic medium with shear modulus µand Poisson ratio ν is given by
Gij(x−y) = 1 8πµ|x−y|
2δij − δij −TiTj
2(1−ν)
so that
Gij,k(x−y) = −1 8πµ|x−y|2
2δijTk− δijTk+δikTj +δjkTi−3TiTjTk 2(1−ν)
, (4.7) where
T = x−y
|x−y|.
It follows that Fk32, Fk23, Fk31,and Fk13 evaluate to zero for k = 1,2. Likewise, F323 =F313 = 0. However,F332 and F331 evaluate respectively to
− 1 2πµ
x2 x21+x22 and
− 1 2πµ
x1 x21+x22. Therefore, we obtain
σhij(x)→µbCij31F332(x)−µbCij32F331(x)
→µ2b(δi3δj1+δi1δj3)F332(x)−µ2b(δi3δj2+δi2δj3)F331(x). (4.8) It is immediate that σ11h , σ22h , σh12,and σ33h →0everywhere in the solid, whereas
σh13(x)→ −µb 2π
x2 x21+x22 , σ23h (x)→ µb
2π x1 x21+x22 .
In other words, we recover the stress expressions of Equation (2.28) asL, M → ∞.
Transport Equation
Recall the dislocation transport equation derived in Chapter 3:
˙
αij −ejlkemnk(vmαin),l = 0, (4.9) where v, the dislocation velocity, is a function of the force on the dislocation.
We assume that the dislocation velocity is zero for driving forces below the Peierls barrier and that the Peierls barrier is small enough to be neglected [127]. This is an especially reasonable assumption in FCC metals where the Peierls stress is very low [128]. In this so-called "viscous drag" regime, the appropriate kinetic potential is given by Equation (3.36) and the dislocation velocity, which becomes a linear function of stress, is usually limited by the viscosity due to dislocation interaction with lattice vibrations (i.e. sound waves) [40]. Thus, in order to determine the dislocation evolution over time, it is necessary to obtain the driving forces along the dislocation. In doing so, we use the Peach-Koehler formula derived in Chapter 2 and repeated below for convenience:
f = (σ·b)×t, (4.10)
where f is the force per unit length, σ is the total stress experienced by the dislocation,b is the Burgers vector, andt is the tangent to the dislocation line.
For a screw dislocation along the x3−axis, we have t= (0,0,1) andb = (0,0, b) with b >0so that
fi =ij3σj3b =i13σ13b+i23σ23b . (4.11) It has already been established that within the linear elasticity theory of dislocations, the stress field diverges logarithmically along the dislocation line. As a result, a regularization scheme must be employed to evaluate the stresses along the dislocation. Using the core regularization described in Chapter 3, the regularized force along the dislocation is obtained as
fi =φ∗φ∗fi
=i13σ13b+i23σ23b ,
(4.12) where
σ =φ∗φ∗σ, (4.13)
with
φ(x) = 1
4π2|x|e|x|/
as given in Chapter 3.
In light of this regularization, we make use of Equation (2.72) to write σαβ (x) =− µ
4π
˛
L
bmimα∂S
∂x0i dx0β− µb 4π
˛
L
bmimβ∂S
∂x0i dx0α
− µ
4π(1−ν)
˛
L
bmimk ∂3R
∂x0i∂x0α∂x0β −2δαβ∂S
∂x0i
! dx0k,
(4.14)
where
R =φ∗φ∗R =R−e−R/+42
R 1−e−R/a
(4.15) and
S =φ∗φ∗ 1 R = 1
R 1−e−R/a
− 1
2e−R/ (4.16)
with
R=x0−x and R =|R|. (4.17)
Therefore, we have σ13 =− µ
4π
˛
L
bi31∂S
∂x0i dx03− µb 4π
˛
L
bi33∂S
∂x0i dx01
− µ
4π(1−ν)
˛
L
bi3k
∂3R
∂x0i∂x01x03 −2δ13∂S
∂x0i
dx0k. Upon further simplification, we arrive at
σ13=−µb 4π
ˆ ∞
−∞
R2 R
dS
dR dx03. (4.18)
Similar calculations lead to
σ23=−µb 4π
ˆ ∞
−∞
R1 R
dS
dR dx03. (4.19)
Along the dislocation line, i.e. when x1 = x2 = 0, we have R1 = R2 = 0, R3 =x03−x3 so that R =|x03−x3|. It follows that,
σ13=σ23= 0. (4.20)
Therefore, by Equation (4.12), the self-force on the dislocation vanishes and as a result, the velocity is identically zero along the dislocation. Owing to Equation (4.9), this implies that α˙ij = 0, which means that the dislocation density tensor—and thus the dislocation configuration—is constant in time. This is to be expected since it is well known that an infinite straight dislocation is an equilibrium configuration [68].