Continuum Dislocation Theory Size

8/18/2019 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



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Indentation test

H = F

A

(Hardness)


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.



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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



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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

|.


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),



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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



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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.



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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.


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.


Evolution of plastic distortion


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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



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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


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


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



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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


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|



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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



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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



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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.



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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




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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


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


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




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β (y ) = γ r β 1(y ), where γ r = γ − γ cr

y

_

1

Graphs of β 1(ȳ )


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.


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


Comparison


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y/h

h/d=2.3h/d=80energy minimizationwith dissipation

Γ

Γ

Γ

Γ

u ,y

Shear strain profile


Loading program


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t

Γ

Γ

Γ

O

A

B

C

Loading path


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◦


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


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 )



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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


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


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.


Continuum model of deformation twinning


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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


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


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.


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


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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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


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


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


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 ,



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Energy minimization

In the equilibrium state the true plastic distortion minimizes the total

energy


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◦.


l


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2 4 6 8 10

0.02

0.04

0.06

0.08

x

Function l (x ).


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 ).


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.


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


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


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


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


Documents

Continuum Dislocation Theory Size