PAMM · Proc. Appl. Math. Mech. 17, 475 – 476 (2017) / DOI 10.1002/pamm.201710207
Numerical solution of plane constrained shear problem for single crystals
within continuum dislocation theory
Tuan Minh Tran
1, ∗
, Matthias Baitsch
2
, and Khanh Chau Le
1
1
Lehrstuhl für Mechanik-Materialtheorie, Ruhr-Universität Bochum
2
Bauinformatik und Numerische Verfahren, Hochschule Bochum
Within the recently proposed Continuum Dislocation Theory (CDT), numerical solutions of simple shear test for single
crystals strip using finite elements are performed. Under the assumption of plane constrained deformation of crystal having
only one active slip system, the plastic slip, the dislocation density as well as the stress-strain curve are computed. The results
of numerical simulations are compared with those obtained from analytical solutions which show good agreement.
© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Finite element implementation
We apply the free energy proposed by [1] (zero dissipation) to the model with λ,μ is elastic Lamé moduli, elastic strains
ε
e
= ε − ε
p
, material constant k, dislocation density ρ, and saturated dislocation density ρ
s
:
I (ε
e
,ρ)=
V
1
2
λ(trε
e
)
2
+ με
e
:ε
e
+ μk ln
1
1 −
ρ
ρs
dv·
In the above formula, the first two terms describe the elastic energy whereas the last term is nothing else but the energy of
dislocation network. Let us depict plastic distortion β = β(x,y) s ⊗ m so that ε
p
=
1
2
(β + β
T
). With Nye’s dislocation
density tensor α = −β×∇, the scalar dislocation density for plane strain can be computed by ρ =
1
b
α
2
xz
+ α
2
yz
=
1
b
|∇β · s|.
It is more convenient to reformulate the above total energy using Voigt’s notation with generalized displacement vector w =
(u,v,β)
T
:
I [w]=
Ω
1
2
(D
ε
w)
T
E(D
ε
w)+
˜
ψ
m
(D
ρ
w)
dxdy,
where E is plane strain elasticity tensor, differential operator D
ε
=
∂
x
0
1
2
sin 2ϕ
0 ∂
y
−
1
2
sin 2ϕ
∂
y
∂
x
− cos 2ϕ
, D
ρ
= (0, 0, cos ϕ∂
x
+sin ϕ∂
y
)
and
˜
ψ
m
(∇β · s) is generalized energy of dislocation network.
After basic steps in linearization and discretization using isoparametric bilinear quadrilateral elements, the vector of internal
forces which is the basis for Newton-Raphson solution procedure is obtained:
δI (ˆ u
e
,δ ˆ u
e
)=
ne
e=1
(δ ˆ u
e
)
T
ˆ
Ω
e
B
T
EBˆ u
e
+
˜
ψ
′
m
(Qˆ u
e
)Q
T
J
e
dξdη =
ne
e=1
(δ ˆ u
e
)
T
f
e
int
(ˆ u
e
)
where B = D
ε
N, Q = D
ρ
N, with vector valued shape functions N
i
(ξ,η)= N
j
(ξ,η)e
k
,i = 3(j −1)+k,j =1, ··· , 4,k =
1, ··· , 3. Similar procedure is also used to find the tangent stiffness matrix. In addition, the boundary conditions are realized
by putting the strip into a “hard” device (see Figure 1) with prescribed displacements at its upper and lower sides: u(x, 0) =
v(x, 0) = 0; u(x,h)= γh,v(x,h)=0, left and right sides: u(x, 0) = v(x, 0) = 0; u(x,h)= γh,v(x,h)=0.
In order to mitigate numerical problems associated with the non-smooth absolute function in the formulation of dislocation
density, we approximate sign function by smooth sigmoid function sig
ν
(x)=
2
1+e
-νx
− 1, with ν ∈ R
+
and has to be chosen
sufficiently large enough. It is also interesting to introduce an alternative energy density (see [2]), which helps to overcome
the potentially dangerous in numerical setting when ρ>ρ
s
: Ψ
m
(ρ)= kμ
∑
n
i=1
1
i
ρ
ρs
i
+ e
c(ρ/χρs−1)
− e
−c
, where
constants n,c,χ are selected according to desired precision.
The absolute value introduced by the energy density of the dislocation network causes the integrands related with geometri-
cally necessary dislocations ρ a jump across the line ∇β · s =0 such that standard Gauss-Legendre quadrature scheme cannot
be applied efficiently. It is therefore a piecewise integration algorithm using Duffy transformation and Delaunay triangulation
is introduced (see [4]).
*
Corresponding author: e-mail tuan.tran-z6c@rub.de
© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim