Introduction of the Model
Main Results
Conclusion
Study of a Model for Reference-free Plasticity
J. Wohlgemuth1
1 Max
S. Luckhaus2
Planck Institute for Mathematics in Sciences
2 University
of Leipzig
10.9.2014
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Outline
1
Introduction of the Model
2
Main Results
3
Conclusion
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Setting of continuum elasticity theory
φ
ΩR
Ω=φ(Ω R)
Stress-free reference configuration: ΩR
Differentiable map: φ
R
Energy: H = ΩR F̃ (∇φ(z)) dz
Energy landscape determines the properties of the model.
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Limits of elasticity theory
Global minimizer of the elastic energy functional
Local minimizer of the physical energy
Local order fixed by the reference configuration
Plastic deformation leads to lower energy states
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Replacing the reference configuration
Transformation to Eulerian coordinates:
Z
−1
dx
H=
F̃ ∇φ φ−1 (x) det ∇φ φ−1 (x)
Ω
Consider for zi ∈ Zd
xi := φ(zi ) ≈ φ(z) + ∇φ(z)(zi − z)
Main variable: Atom positions χ ∈ ΩN ⊂ R3N
Fit Bravais lattice A−1 (x) Zd − τ (x) + x locally to the
atom positions
Matrix A(x) → Argument of F (A) = F̃ A−1 det(A)
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
The energy functional
Ω
ϕ
λ
L
R
P
−1
A
s
Hλ (χ) = hλ (x, χ) dx + i,j V |xi − xj |
Energy density: hλ (x) = minA {Jλ + ϑ |det A − ρλ | + F }
Jλ (A, χ, x) measures the squared distance between the
atoms and the lattice.
Particle density: ρλ (χ, x)
Elastic energy functional: F (A)
Hard-core potential: V
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Definition of the density ρλ and Jλ
ρλ (χ, x) :=
Jλ (A, χ, x) :=
1 X −1
ϕ
λ
|x
−
x|
,
i
Cϕ λd χ
−1 2
A X
−1
W
(A
(x
−
x)
+
τ
)ϕ
λ
|x
−
x|
i
i
Cϕ λd χ
ϕ ∈ C ∞ (R+ )is a cut-off function
W ∈ C ∞ Rd has the properties for all zn ∈ Zd , z ∈ Rd and
x ∈ BΘW (0)
W (z) = W (z + zn )
(Periodicity)
0 y 2 ≤ y ∇2 W (x)y ≤ c 1 y 2
cΘ
Θ
C0W dist2 (z, Zd ) ≤ W (z) ≤ C1W dist2 (z, Zd )
J.Wohlgemuth, S.Luckhaus
(Local convexity)
(Coercivity)
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
The elastic energy
The elastic energy F ∈ C2 (Gld (R)) has the following properties
F (A) = F (AR), ∀A ∈ Gld (R), ∀R ∈ SOd (R)
(Frame indifference)
∃E ∈ Gld (R) with F (E) = 0
(Existence of minimizer)
F (A) ≥ C1El (det(E) − det(A))2 + C2El dist2 (A, E SOd )
(Coercivity)
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Upper bound for the energy barrier of plastic
relaxation: Basic Idea
There exists a continuous path of configurations with the
following properties
1) Equal to the elastically deformed configuration at the
boundary
2) Starting point: Elastically deformed configuration
3) Ending point: Configuration of lower energy
4) Energy barrier: At most O(λ2 )
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Upper bound: Theorem
There exists λ̂, C, Cmax ∈ R and ˆ > 0 such that for all λ > λ̂,
2
3
AR ∈ Gl2 (R), b ∈ A−1
R Z , x0 ∈ Ω and L > Cλ satisfying the
assumptions
1) F (AR ) ≤ ˆ,
⊥ ≤ 0,
2) (∇F )AV A−1
b
⊗
b
R
3) B|A−1 |L (x0 ) ⊂ Ω,
R
then there exists a continuous path χ(s) : [0, 1] → R2
the properties
2
1) χ(s) = A−1
R Z in B4λ (Ω) /Ω for all s ∈ [0, 1],
2
2) χ(0) = A−1
R Z ∩ B4λ (Ω),
3) H(χ(1)) < H(χ(0)),
4) maxs∈[0,1] H(χ(s)) ≤ H(0) + Cmax λ2 .
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
N
with
Introduction of the Model
Main Results
Conclusion
Idea of the Proof:
Main difficulties
The fitted A(x) does not have to be continuous.
Get an upper bound that is independent on the system
size.
Strategy
Compare of the configuration in each point to a Bravais
lattice.
Compare this lattice in each point to the corresponding
point in the original lattice.
Estimate the energy density of a configuration φ(Zd ) with:
ĥλ (χ, x) ≤ Fλ (∇φ(φ−1 (x))) + Cλ4 |∇2 φ(φ−1 (x))|
Construct an explicit path consisting of two dislocation
lines and estimate its energy.
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Definition: Reparametrisations
Reparametrisations: B = (B, t) ∈ Gld (Z) × Zd
Applied BA := (BAR , BτR + t)
χA = χBA
Product of reparametrisations B = B0,1 ...BK −1,K


K
K
kY
−1
Y
X

B = B0,1 ....BK −1,K =
Bj−1,j , t =
Bj−1,j  tk −1,k .
j=1
J.Wohlgemuth, S.Luckhaus
k =1
j=1
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Definition: Regular points
x is called (ρ , J , CA )-regular with A = (A, τ ) ∈ Gld (R) × Rd , if
it holds
1) Jλ (A, χ, x) < J ρλ (χ, x),
2) |ρλ (χ, x) − det A| < ρ det A,
3) kA−1 k < CA ,
4) |xi − xj | > so for i 6= j .
Heuristics: x is regular with A means the configuration looks
like the lattice χA in B2λ (x).
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Basic Idea: Connecting regular points
yj is regular with Aj means χ ≈ χAj + x in B2λ (yj )
If |yi − yj | < 1, 5λ, the configuration looks like both lattices
in B2λ (y1 ) ∩ B2λ (y2 )
Hence, A1 has to be close to a reparametrisation of A2 .
kδAk2λ = λ2 kδAk2 + kδτ k2 = O (J )
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Theorem: Connecting regular points
For all CA > so there exists λ̂ ∈ R and ˆJ > 0 such that for all
λ > λ̂ and all J ≤ ˆJ it holds: If yj is (2−3−2d , J , CA )-regular
with Aj for j = 1, 2 and |y1 − y2 | ≤ 32 λ, then there exists an
uniquely defined reparametrisation
B1,2 = B1,2 , t1,2 ∈ Gld (Z) × Zd such that
√ J
kid −
<O
,
λ
√ B1,2 τ2 + t1,2 − τ1 − B1,2 A2 + A1 (y2 − y1 ) <O
J
2
A−1
1 B1,2 A2 k
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Applications
Sequence of regular points → sequence of
reparamertisation
The reparametrisation product is a topological quantity
The t part of the reparametrisation product is the Burgers
vector.
The core energy of a dislocation scales like λ2 in d = 2
The core energy of a topological defect in A scales like λ3
in d = 2
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Basic Idea: Lower bounds on the energy density
The global minimizer  of hλ (·, χ, x) is discontinuous
The jumps are nearly given by reparametrisations.
Jλ (·, χ, x) has differentiable local minimizers ÃB .
Lower bound for the energy density
ĥλ (χ, y ) ≥ FC (∇τ̃B (y )) + Cλ4 k∇2 τ̃B (y )k2 .
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Theorem: Lower bounds on the energy density
For all CA > 0 there exists λ̂, ˆ > 0 such that for λ > λ̂ the
following holds. If y (s) : [0, 1] → Ω is differentiable curve with
ĥλ (χ, y (s)) ≤ ˆ for all s, Â ∈ Gld (R) × Rd is the global
minimizer of hλ (·, χ, y (0)), the reparametrisation
B = (B, t) ∈ Gld (Zd ) × Zd fulfills kÂ−1 B −1 k ≤ CA /2 then there
exists a differentiable function ÃB (y (s)) such that ÃB (y (s)) is a
local minimizer of Jλ (·, χ, y (s)) for every s and the energy
density fulfills
ĥλ (χ, y ) ≥FC (∇τ̃B (y )) + Cλ4 k∇2 τ̃B (y )k2
,
where
n
o
FC (A) = min F (BA)|B ∈ Gld (Zd ) + O(λ−2 ) .
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Ideas of the proof
Low energy states have a regular minimizing Â
Reparametrisation
ofa regular
A is regular and
Jλ B Â, χ, x ≤ CJλ Â, χ, x
Local convexity of Jλ (·, χ, x) for regular points.
Use the local convexity to prove that in the neighborhood
of a regular point there is a unique local minimizer of
Jλ (·, χ, x).
2
Jλ (B Â, χ, x) ≥ Jλ (ÃB , χ, x) + C B Â − ÃB λ
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Ideas of the proof
Use implicit function theorem and local convexity to prove
that this local minimizer is a differentiable function of x and
fulfills
Jλ ÃB , χ, x ≥Cλ2 λ2 k∇ÃB k2 + k∇τ̃B − ÃB k2
Jλ ÃB , χ, x ≥Cλ4 λ2 k∇2 ÃB k2 + k∇2 τ̃B − ∇ÃB k2
Use these estimates to rewrite ĥλ (χ, x) in terms of ∇τ̃B .
Minimization over  and ÃB only decreases the energy.
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Summary
We have
an upper bound for the energy barrier of plastic relaxation
a method to identify topological defects in the model
a lower bound for the core energy of dislocations
a lower bound for the energy density of low energy states
What we do not have
lower bound for the energy barrier of plastic relaxation.
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity
Introduction of the Model
Main Results
Conclusion
Open questions
How to calculate the lower bound? When is it even
possible?
In most theorems λ is assumed to be large. That leads to a
large area arround the dislocation core that is not
described well. How large does λ has to be?
J.Wohlgemuth, S.Luckhaus
Defence: Reference-free Plasticity