continuum dislocation theory size
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Continuum dislocation theory: size effects and
formation of microstructure
K.C. Le
Lehrstuhl für Mechanik - MaterialtheorieRuhr-Universität Bochum, 44780 Bochum, Germany
Workshop on ”Multiscale Material Modeling”, Bad Herrenalb 2012
Collaborators: V. Berdichevsky, D. Kochmann, Q.S. Nguyen,P. Sembiring, B.D. Nguyen
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 1 / 78
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Outline of my lecture
Size effects in crystal plasticity and formation of microstructure
Why continuum dislocation theory?
Thermodynamic framework of CDT
Antiplane constrained shear
Plane constrained shear
Double slip systems
Mechanism of twin formation
Continuum model of deformation twinning
Bending
Polygonization
Conclusion
Further works
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Size effects in crystal plasticity and formation of
microstructures
Hall-Petch relation
Indentation
Shear, torsion, bending
Deformation twinning
Polygonization
Recrystallization
Grain growth
Texturing
ect.
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Hall-Petch relation
σY ∝ σY 0 + λ√ d
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 4 / 78
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Indentation test
H = F
A
(Hardness)
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 5 / 78
T i f i
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Twin formation
TEM micrograph of the [1̄11] oriented single crystal under tension. Strain
30%, Karaman et al., 2000
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Polygonization
Polygonized state of a bent single crystal beam
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View along intersections of slip planes with polygon boundaries in a
polygonized zinc crystal, Gilman, 1955
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Why continuum dislocation theory?
Plastic deformations are due to nucleation, multiplication and motion
of dislocationsDislocations appear to reduce energy of crystals
Motion of dislocations produces energy dissipation which causes theresistance to their motion
Under favorable condition the rearrangement of dislocations bydislocation climbing and gliding may reduce further energy of crystals
Deformation twinning and polygonization are low energy dislocationstructures
Continuum description is dictated by high dislocation densities(108-1014m−2). To compare: dislocation density 4×1011m−2 meansthat the total length of dislocation loops in one cubic meter of crystalequals the distance from the earth to the moon
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Related literature
Continuum dislocation theory: Kondo,1952; Nye,1953; Bilby et
al.,1955; Kröner,1955,1958; Berdichevsky & Sedov,1967; Le &Stumpf,1996a,b,c; Ortiz & Repetto,1999; Ortiz et al.,2000;Acharya,2001; Svendsen,2002; Gurtin,2002,2004; Berdichevsky,2006;Berdichevsky & Le,2007; Le & Sembiring,2008a,b,2009; Kochmann &Le,2008,2009a,b; Le &Nguyen,2010,2011; Le & Nguyen,2012.
Strain gradient plasticity: Fleck et al.,1994; Shu & Fleck,1999; Gao et al.,1999; Acharya & Bassani,2000; Huang et al.,2000,2004; Fleck &Hutchinson,2001; Han et al.,2005; Aifantis & Willis,2005.
Statistical mechanics of dislocations: Le & Berdichevsky,2001,2002;
Groma et al.,2003,2005; Berdichevsky,2005,2006.Discrete dislocation simulations: Needleman & Van der Giessen,2001;Shu et al.,2001; Yefimov et al.,2004.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 10 / 78
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Thermodynamic framework of CDT
Kinematics
s
m
p
e
= +e p
Small strain theory: unknown functions u i (x ), β ij (x )Plastic distortion: β ij =
na=1 β a(x )s
a
i ma
j
Total (compatible) strain: εij = 12 (u i , j + u j ,i )
Plastic (incompatible) strain: εp ij = 12 (β ij + β ji )
Elastic (incompatible) strain: εe ij = εij −
εp ij
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 11 / 78
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Dislocation density
da
Bi=αijn jda
1 2 3 4 5
6
7
89101112
13
1 2 3 4 5
6
7
891011
12
13
14 14
b
a) b)
Dislocation density: αij = jkl β il ,k
Geometrical interpretation: Stokes theorem
S αij n j da =
L β ij τ j ds = B i
B i is the resultant Burgers vector of all GND (excess dislocations), whosedislocation lines cuts the area S . For single slip the scalar dislocationdensity is ρ = 1
b
| jkl β ,k ml n j
|.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 12 / 78
E ti
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Energetics
State variables: elastic strain εe ij and dislocation density αij (plastic strainand its gradient are history dependent and are not the state variables)
Free energy density (Kröner)
ψ(εij − εp ij , αij ) = ψ0(εij − εp ij ) + ψm(αij )Elastic energy of the crystal lattice
ψ0(εe ij ) = 12C ijkl εe ij ε
e kl
Energy of microstructure (single slip system, Berdichevsky, 2006)
ψm(αij ) = µk ln 1
1 − ρ/ρs Small up to moderate dislocation density
ψm(αij ) µk ρ
ρs +
1
2
ρ2
ρ2s K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 13 / 78
V i ti l i i l f CDT
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Variational principles of CDT
Negligible resistance to dislocation motion: the energy minimization
I (u i , β ij ) =
Ω ψ(εij − εp
ij , αij )dx → minu ,βFinite resistance to dislocation motion: Variational equation (Sedov,Berdichesky, 1967)
δ I +
Ω∂ D ∂ β̇ ij
δβ ij dx = 0
implying the evolution equation (of Biot type) for the plastic distortion
∂ D
∂ β̇ ij = −
δ εψ
δβ ij ≡ −∂ψ
∂β ij + ∂
∂ x k
∂ψ
∂β ij ,k
Note that, if the dissipation potential D = D ( β̇ ij ) is a homogeneousfunction of first order (rate-independent theory), the variational equation
reduces to the minimization of “relaxed” energy.K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 14 / 78
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Antiplane constrained shear
x
y
a
z
h L
A single crystal strip is placed in a “hard” device with the prescribed
displacements at the boundary
u z = γ y
γ overall shear strain (control parameter)
Assumption: a h, a LK.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 15 / 78
Single slip system
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Single slip system
Plastic distortion
β zy = β (x )
Since dislocation cannot reach the boundary x = 0, a, the plastic distortionβ (y ) satisfies the constraints:
β (0) = β (a) = 0
The plastic strains are given by
εp yz = εp zy =
1
2β (x ).
The only non-zero component of tensor of dislocation density is
αzz = β ,x .
The free energy density of the crystal with dislocations takes a simple form
ψ = 1
2µ(γ − β )2 + µk ( |β ,x |
ρs b +
β 2,x ρ2s b
2),
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 16 / 78
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Energy minimization
If the resistance to dislocation motion is negligible, the true plastic
distortion minimizes the total energy
I [β (x )] = hL
a0
1
2µ(γ − β )2 + µk ( |β ,x |
ρs b +
β 2,x ρ2s b
2)
dx
Introducing the dimensionless quantities
x̄ = xb ρs , ā = ab ρs , E = b ρs µhL
I
to rewrite the energy functional in the form
E [β (x )] =
a0
1
2(γ − β )2 + k (|β | + 1
2β 2)
dx
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 17 / 78
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Energetic threshold
x
Β
Ε aa-Ε
Β
E = 1
2(γ − β 0)2a + 2k |β 0|
Minimization of this function shows that there exists the threshold value
γ en = 2k ab ρs
Size effect (typical to all gradient theories): the threshold value is inverselyproportional to the size a of the specimen.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 18 / 78
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Minimizer
Ansatz (based on the feature of dislocation pile-up)
β (x ) =
β 1(x ) for x ∈ (0, l ),β m for x ∈ (l , a − l ),β 1(a − x ) for x ∈ (a − l , a),
Functional
E = 2
l 0
1
2(γ − β 1)2 + k
β 1 +
1
2β 21
dx +
1
2(γ − β m)2(a − 2l ).
Function β 1(x ) is subject to the boundary conditions
β 1(0) = 0, β 1(l ) = β m.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 19 / 78
V i hi f i l i h β ( ) b i h E l
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Varying this energy functional with respect to β 1(x ) we obtain the Eulerequation for β 1(x ) on the interval (0, l )
γ
−β 1 + k β
1 = 0.
The variation of energy functional with respect to β m and l yields the twoadditional boundary conditions at x = l
β 1(l ) = 0, 2k = (γ
−β m)(a
−2l ).
Solution
β 1(x ) = γ − γ (cosh x √ k − tanh l √
k sinh
x √ k
), 0 ≤ x ≤ l .
Transcendental equation to determine l
f (l ) ≡ 2l + 2 k γ
cosh l √
k = a.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 20 / 78
Evolution of plastic distortion
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Evolution of plastic distortion
0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.002
0.004
0.006
0.008
x
β
a
b
c
Evolution of β : a)γ = 0.001, b)γ = 0.005, c)γ = 0.01
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 21 / 78
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Stress strain curve
0.001 0.002 0.003 0.004 0.005
0.0002
0.0004
0.0006
0.0008
σ/μ
γO
A
B
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Non-zero dissipation
For the case with dissipation ( ˙β > 0) we have to solve the evolutionequation
γ − β + k β ,xx = γ c , β (0) = β (a) = 0,
with γ c
= K /µ. Introduce the deviation of γ (t ) from the critical shear, γ c
,γ r = γ − γ c > 0
γ r − β + k β = 0.
Thus, β = γ r β 1. Similarly, for β̇
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g p g
t
c
Loading path
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 24 / 78
Hysteresis and Bauschinger effect
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g
-0.0005 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030
-0.0004
-0.0002
0.0002
0.0004
O
A
B
C
D
σ/μ
γ
Average shear stress versus shear strain curve
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 25 / 78
Plane strain constrained shear
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Plane strain constrained shear
0
y
x
h
sm
a
L
z
γh
A single crystal strip is placed in a “hard” device with the prescribed
displacements at the boundary
u = γ y , v = 0 at y = 0, h
γ overall shear strain (control parameter)
Assumption: h a LK.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 26 / 78
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Single slip system
The plastic distortion tensor
β ij = β (y )s i m j
where
s = (cos ϕ, sin ϕ, 0), m = (− sin ϕ, cos ϕ, 0)
Since dislocation cannot reach the boundary y = 0, h, the plasticdistortion β (y ) satisfies the constraints:
β (0) = β (h) = 0
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 27 / 78
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Strain measures
Total strain
εxx = 0, εxy = 1
2u ,y , εyy = v ,y
Plastic strains
εp xx = −12 β sin 2ϕ, εp xy = 12 β cos 2ϕ, εp yy = 12 β sin 2ϕ
Elastic strain
εe xx = 1
2β sin 2ϕ, εe xy =
1
2(u ,y
−β cos 2ϕ),
εe yy = v ,y − 1
2β sin 2ϕ
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 28 / 78
Nye’s dislocation density
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da
Bi=αijn jda
Non-zero components of dislocation density tensor
αxz = β ,y sin ϕ cos ϕ, αyz = β ,y sin2 ϕ
Scalar dislocation density
ρ = 1
b α2xz + α
2yz =
1
b |β ,y || sin ϕ|
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 29 / 78
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Energy density
Energy per unit volume of dislocated crystal
ψ(εe ij , αij ) = 1
2 λ (εe ii )2 + µεe ij εe ij + µk ln 1
1 − ρρs
λ, µ - Lamé constants, b - magnitude of Burgers’ vector, ρs - saturateddislocation density, k - material constant
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 30 / 78
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Energy functional
E = aL h
0 1
2λv 2,y +
1
2µ(u ,y − β cos 2ϕ)2 + 1
4µβ 2 sin2 2ϕ
+ µ(v ,y − 12
β sin 2ϕ)2 + µk ln 1
1 − |β,y || sinϕ|b ρs
dy
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 31 / 78
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Reduced energy functional
Minimization with respect to u and v leads to
E (β ) =aL h
0
µ12
(1
−κ)β 2 sin2 2ϕ +
1
2
κ
β
2 sin2 2ϕ
+ 1
2(γ − β cos2ϕ)2 + k ln 1
1 − |β,y || sinϕ|b ρs
dy
where β = 1
h h
0 β (y ) dy
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Energy minimization
If the resistance to dislocation motion is negligible, the true plastic
distortion minimizes the total energy
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Energetic threshold
γ en = 2k
hb ρs
| sin ϕ|| cos2ϕ|
Size effect: the threshold value is inversely proportional to the size h.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 34 / 78
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Material parameter
Material µ (GPa) ν b (Å) ρs (µm−2) k
Aluminum 26.3 0.33 2.5 1.834103 0.000156
Table: Material characteristics
In all simulations h = 1µm
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 35 / 78
Evolution of plastic distortion
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0.1 0.2 0.3 0.4
-0.0075
-0.005
-0.0025
0.0025
0.005
0.0075
a
b
c
d
e
f
y-
=30°
=60°
Β
Evolution of β : a,d)γ = 0.0068, b,e)γ = 0.0118, c,f)γ = 0.0168
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 36 / 78
Stress strain curve
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0.002 0.004 0.006 0.008 0.01
0.002
0.004
0.006
0.008
=30°
=60°
O
A
B
A´
B´
Γ
ΤΜ
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 37 / 78
N di i i
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Non-zero dissipation
Non-zero resistance to dislocation motion: energy minimization is replaced
by the flow rule
∂ D
∂ β̇ = −δ γ ψ
δβ
for β̇ = 0, whereD = K |β̇ |
κ
≡ −δ γ ψ
δβ
=
−∂ψ
∂β
+ ∂
∂ y
∂ψ
∂β ,y Plastic distortion may evolve only if the yield condition |κ | = K is fulfilled.If |κ |
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Yield condition
Differential equation
|k β ,yy sin2 ϕ
b 2ρ2s − (1 − κ)β sin2 2ϕ
− (cos2 2ϕ + κ sin2 2ϕ)β + γ cos 2ϕ| = K /µ = γ cr cos 2ϕBoundary conditions
β (0) = β (h) = 0
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 39 / 78
Evolution of plastic distortion
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β (y ) = γ r β 1(y ), where γ r = γ − γ cr
y
_
1
Graphs of β 1(ȳ )
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 40 / 78
Dislocation density
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α(y ) = γ r α1(y )
y _
1
Graphs of α1(ȳ )
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Size effect for hardening rate
It is interesting to calculate the shear stress τ which is a measurablequantity. During the loading, we have for the normalized shear stress
τ
µ = γ cr + γ r
1 −1 − 2tanh ηh2
ηh
β 1p cos 2ϕ
where β 1p is calculated from the solution. The second term of thisequation causes the hardening due to the dislocations pile-up anddescribes the size effect in this model.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 42 / 78
Comparison
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ΤΤ
h/d=1.15
h/d=2.3h/d=80with dissipationenergy minimization
Γ
Shear stress vs. shear strain curve
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 43 / 78
Comparison
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y/h
h/d=2.3h/d=80energy minimizationwith dissipation
Γ
Γ
Γ
Γ
u ,y
Shear strain profile
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Loading program
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t
Γ
Γ
Γ
O
A
B
C
Loading path
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 45 / 78
Bauschinger effect
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-0.002 0.002 0.004 0.006 0.008 0.01-0.002
0.002
0.004
0.006
0.008
O
A
B
C
D
Γ
ΤΜ
Normalized shear stress versus shear strain curve for ϕ = 60◦
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 46 / 78
Double slip system
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r l
y
h
0
z
x a
L
h
sl
m l
m r
sr
Tensor of plastic distortion
β ij (y ) = β l (y )s l i ml j + β r (y )s r i mr j
with β l (y ) and β r (y ) satisfying
β l (0) = β l (h) = β r (0) = β r (h) = 0
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 47 / 78
Strain measures
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Total strain
εxx = 0, εxy = 1
2
u ,y , εyy = v ,y
Plastic strains
εp xx = −1
2(β l sin 2ϕl + β r sin 2ϕr ),
εp xy = 1
2
(β l cos 2ϕl + β r cos 2ϕr ),
εp yy = 1
2(β l sin 2ϕl + β r sin 2ϕr )
Elastic strain
ε
e
xx =
1
2 (β l sin 2ϕl + β r sin 2ϕr ),
εe xy = 1
2(u ,y − β l cos 2ϕl − β r cos 2ϕr ),
εe yy = v ,y − 1
2(β l sin 2ϕl + β r sin 2ϕr )
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 48 / 78
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Nye’s dislocation density
Non-zero components of Nye’s dislocation density tensor
αxz = β l ,y sin ϕl cos ϕl + β r ,y sin ϕr cos ϕr
αyz = β l ,y sin2 ϕl + β r ,y sin
2 ϕr
Scalar dislocation density
ρ = ρl + ρr
= 1
b |β l ,y sin ϕl
|+
1
b |β r ,y sin ϕr
|
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Energetic threshold
γ en = 2k
hb ρs
| sin ϕ|| cos2ϕ|
Size effect: the threshold value is inversely proportional to the size h.Mention that the energetic threshold value for the symmetric double slip isequal to that of the single slip found in (Le & Sembiring, 2007).
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Comparison
h/d=80
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/energy minimizationwith dissipation
y _
h
u,y
The total shear strain profiles
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Mechanism of twins formation
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T T T
T T T
x
y
s
m
h
a x
y t
x
y
hs
h (1-s )
u l
rotation
shear
T T
T T
T T
T T
The twin phase is formed by a twinning shear produced by the movement
of pre-existing dislocations to the boundary followed by a rigid rotation.The described mechanism of twin formation is closely related to that of Bullough (1957). The difference is that dislocations need not to glidethrough each lattice plane.
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Continuum model of deformation twinning
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Tensor of plastic distortion
β ij (y ) =
β s l i m
l j for 0 < y
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The in-plane components of the plastic strain tensor εp ij = 12 (β ij + β ji ) read
εp xx = −1
2β sin 2ϕ− 1
2β T sin 2ϕl ,
εp xy = 1
2β cos 2ϕ +
1
2β T cos 2ϕl ,
εp yy = 1
2 β sin 2ϕ + 1
2 β T sin 2ϕl ,
with the following quantities defined in the upper and lower part of thecrystal:
[β (y ), ϕ, β T ] =
[β u (y ), ϕu , β t ] h(1 − s ) < y
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da
Bi=αijn jda
Non-zero components of dislocation density tensor
αxz = β ,y sin ϕ cos ϕ, αyz = β ,y sin2 ϕ
Scalar dislocation density
ρ = 1
b
α2xz + α
2yz =
1
b |β ,y || sin ϕ|
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Energy density
Free energy per unit volume of dislocated crystal
ψ(ε
e
ij , αij ) =
1
2 λ (ε
e
ii )
2
+ µε
e
ij ε
e
ij + µk ln
1
1 − ρρs λ, µ - Lamé constants, b - magnitude of Burgers’ vector, ρs - saturateddislocation density, k - material constant
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 57 / 78
E f i l
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Energy functional
E (u , v , β, s ) = aL
h0
12 λv
2,y +
14 µ(β sin 2ϕ + β T sin 2ϕl )
2
+ 12 µ(u ,y − β cos 2ϕ − β T cos 2ϕl )2
+ µ(v ,y − 1
2 β sin 2ϕ − 1
2 β T sin 2ϕl )2
+ µk ln 1
1 − |β,y sinϕ|b ρs
dy .
TWIP-alloys have rather low stacking fault energies, so the contribution of surface energy to this functional can be neglected.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 58 / 78
Condensed energy
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0.05 0.1 0.15 0.2 0.25
0.0001
0.0002
0.0003
0.0004
0.0005
0.002 0.004 0.006 0.008 0.01
0.00001
0.000012
s = 0
s = 1
s = 0.9
s = 0.7
s = 0.6
s = 0.5
s = 0.8
s = 0.4
s = 0.3
s = 0.2
s = 0.1
cond
E
8-6
·10
6-6
·10
4-6
·10
2-6
·10
cond
E
s = 0
s
Condensed energy E cond(s , γ ) = minβ E (β, s , γ ) versus overall shear strain
γ for various volume fractions s (left) with a magnification (right) forsmall values of s and small strains (clearly indicating that s = 0 minimizesthe energy in that region). The actual energy with evolving s follows thepath of least energy
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Stress-strain curve
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0.05 0.1 0.15 0.2 0.25 0.3
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
/
A
B
C
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Bending
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x
y
r
a
hs m
A single crystal beam is bent along a rigid cylinder. The displacements of
the lower face are prescribed
u x (x , 0) = r sin(x /r ) − x , u y (x , 0) = r cos(x /r ) − r
Assumption: h aK.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 62 / 78
Single slip system
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Tensor of plastic distortion
β ij = β (x , y )s i m j , s = (cos ϕ, sin ϕ), m = (− sin ϕ, cos ϕ)Plastic strains
εp xx = −1
2β sin 2ϕ, εp yy =
1
2β sin 2ϕ, εp xy =
1
2β cos 2ϕ
Dislocation density
αxz = β ,x cos2 ϕ + β ,y cos ϕ sin ϕ,
αyz = β ,x cos ϕ sin ϕ + β ,y sin2 ϕ
Scalar dislocation density
ρ = 1
b
(αxz )2 + (αyz )2 =
1
b |β ,x cos ϕ + β ,y sin ϕ|
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Energy density
Free energy per unit volume of dislocated crystal
ψ(ε
e
ij , αij ) =
1
2 λ (ε
e
ii )
2
+ µε
e
ij ε
e
ij + µk
ln
1
1 − ρρs λ, µ - Lamé constants, b - magnitude of Burgers’ vector, ρs - saturateddislocation density, k - material constant
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 64 / 78
Energy functional of the bent beam
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I = a
0
h0 [
1
2 λ(u x ,x + u y ,y )2
+ µ(u x ,x +
1
2 β sin 2ϕ)2
+ µ(u y ,y − 12
β sin 2ϕ)2 + 1
2µ(u x ,y + u y ,x − β cos 2ϕ)2
+ µk ln 1
1 − |β
,x cos ϕ+β,y sinϕ|b ρs
] dxdy
Because of the prescribed displacements at y = 0 dislocations cannotreach the lower face of the beam which is in contact with the bending jigin the deformed state, therefore
β (x , 0) = 0
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 65 / 78
Reduced energy functional
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Variational-asymptotic analysis reduces the energy functional containingthe small parameter h/a to
E 1 = a
0 h
0[κ(cos
x
r − 1 + 1
2β sin2ϕ)2 + k |β ,y sin ϕ|
+ 12
k (β ,y sin ϕ)2] dxdy
where κ is given by
κ = 11 − ν ,
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 66 / 78
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Energy minimization
In the equilibrium state the true plastic distortion minimizes the total
energy
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 67 / 78
Smooth minimizer
The plastic distortion
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β
(x , y ) =β 1(x , y ) for y ∈
(0, l (x )),
β 0(x ) for y ∈ (l (x ), h)where
β 1(x , y ) = β 1p (1
−cosh χy + tanh χl (x )sinh χy )
β 1p = − 2sin2ϕ
(cos x
r − 1), χ =
2κ
k cos ϕ
Transcendental equation to determine l (x )
k sin ϕ + κ[(cos x r − 1)
+1
2β 1p (1 − 1
cosh χl (x ))]sin2ϕ)sin2ϕ(h − l (x )) = 0.
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Energetic threshold
Size effect: the threshold value depends on a and h
r cr = a
arccos(1− k 2κh cosϕ)
Thus, if the radius of the bending jig r > r cr , then l (x ) = 0 and β = 0everywhere yielding the purely elastic deformation without dislocations.For r x ∗ forming there the boundary layer.
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Material parameter
Material µ (GPa) ν b (Å) ρs (µm−2) k
Zinc 43 0.25 2.68 145.4 0.000156
Table: Material characteristics
In all simulations a = 10mm, h = 1.3mm, ϕ = 35◦.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 70 / 78
l
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2 4 6 8 10
0.02
0.04
0.06
0.08
x
Function l (x ).
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β0
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2 4 6 8 10
0.5
1.0
1.5
2.0
2.5
3.0
x
Function β 0(x ).
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Polygonization
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During annealing dislocations may climb in the transversal direction andthen glide along the slip direction and be rearranged as shown in thisFigure.K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 73 / 78
After annealing
In the final polygonized relaxed state the dislocations form low angle tilt
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In the final polygonized relaxed state the dislocations form low angle tiltboundaries between polygons which are perpendicular to the slip direction,
while inside the polygons there are no dislocations. We want to show thatthis rearrangement of dislocations correspond to a sequence of piecewiseconstant β̌ (x , y ) reducing energy of the beam. Here and below check isused to denote the polygonized relaxed state after annealing. The jump of β̌ means the dislocations concentrated at the surface, therefore we ascribe
to each jump point the normalized Read-Shockley surface energy density
γ ([[β̌ ]]) = γ ∗|[[β̌ ]]| ln e β ∗|[[β̌ ]]| ,
with [[β̌ ]](x i ) = β̌ (x i + 0) − β̌ (x i − 0) denoting the jump of β̌ (x ),γ ∗ =
b 4π(1−ν ) , and β ∗ the saturated misorientation angle.
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 74 / 78
Energy reducing sequence
β0
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2 4 6 8 10
0.5
1.0
1.5
2.0
2.5
3.0
x
β0
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 75 / 78
Number of polygonsThe number of polygons can be estimated from above by requiring that theincrease of the surface energy is less than the reduction of the bulk energy
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in gradient terms. For the material parameters of zinc, the estimatedaverage polygon distance (taken as the length of the beam divided by thenumber of polygons) is equal to around 2.7 × 10−7 m which is in excellentagreement with the experimental result obtained by Gilman in 1955.
0 2 4 6 8 10
0
20
40
60
80
100
r
ln N
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 76 / 78
Conclusions
CDT enables ones to model dislocation pile-up and size effects
Th i h h ld l f h di l i l i
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There exists a threshold value for the dislocation nucleationdepending on the grain size
Work hardening and Bauschinger effect can properly be described
Deformation twins exhibit another type of non-convexity
Existence of distinct thresholds for the onset of deformation twinning
The stress-strain response exhibits a sharp load drop (followed by astress plateau) upon the onset of twinning
Polygonization occurs due to the smallness of the Read-Shockleysurface energy as compared with the bulk energy of distributeddislocations
The rearrangement of dislocations is realized by the dislocation climbwith the subsequent dislocation glide.
High-temperature dislocation climb during annealing is crucial for thekinetics of polygonization
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 77 / 78
Further works
Indentation
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Bending and torsion
Particle strengtheningMultiple slip
CDT for polycrystals
Hall-Petch relation
Plastic zone near the crack tip and ductile fracture
Finite twinning shear and finite rotation
Parameter identification and applications to TWIP-alloys
3-D model
Dislocation climb taking into account the interaction with vacancies
Kinetics of polygonization
Formation of dislocation cell structure
K.C. Le (Mechanik - Materialtheorie) CDT: size effects, microstructure Bad Herrenalb workshop 2012 78 / 78
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