Lattice deformations and elastic energy functionals in d dimensions a variational and functional analytic approach
Stefan Kahler - Supervisor: Prof. Dr Gero Friesecke
Introduction
Sketch of proof
My thesis [4] makes a contribution to the fundamental and widely open question how, on which length- and timescales, and with which accuracy detailed, fully
atomistic models of crystalline solids can be approximated by (classical or noclassi-
The proof relies on a mixture of classical calculus, functional analysis and
geometry in
R
d
.
The problem can be simplied by switching over to a linear approximation via
cal) continuum models of linear elasticity, nite elasticity, dislocations, plasticity, or
Taylor's theorem, which is developed in [3] for a similar setting. The quadratic
fracture.
error terms can be estimated by a long, yet elementary calculation.
Specically, in [4] I am interested in putting the standard atomistic energy functionals for crystalline solids into a rigorous mathematical framework. My main
The proof of (1) is a consequence of this approximation.
The proof of (2) and (3), however, is more complicated and consists - after a
further application of an idea presented in [3] - of the following two steps:
Step 1:
result is a precise mathematical way in which to isolate the `elastic' part of the
Dene for each k
total energy from the `binding' part of the total energy. The former consists of
the total interaction energy of the atoms minus their total interaction energy in
2N
an auxiliary function wk
P

 x 2 L \ Bk V 0(jx
x
0
( ) :=
wk x
an (undeformed, crystalline, equilibrium) reference conguration. To dene it mathematically for general localized deformations requires performing a careful limit

0;
x
0
6= x
0j)
0
x
x
x
x0
j
:L!R
j ;if x
2
d
,
Bk
else:
process, since in an innite crystal both contributions are innite even though
The Banach-Steinhaus theorem establishes a connection between the
their dierence, the elastic energy of interest, is nite.
assertions and the
Step 2:
(w ) 2N
k
k
k k2
:
-boundedness of
(w ) 2N.
k
k
Reduce the problem of proving or disproving the
k k2
:
to a simple geometric fact: Assume that in the gure below, the blue
atoms are those on @Bk and the red ones are those on @Bk
rst coordinate equal to k
Consider
L := Z
the unbounded cubic lattice
d
(d
2N
1.
1,
with only the
Then the number of the red atoms is invariant
with respect to k if and only if d
Setting and problem
-boundedness of
= 1.
)
a smooth and suciently rapidly decreasing interatomic pair potential function
V
: R+ ! R
describing an interaction of the atoms (the following gure shows
a typical potential function)
Since in detail the proof of (3) makes use of special potential functions
p
[ 2;1) = 0
satisfying V
an outer force f
:L!R
the energy functional EB
R
1
(
y ) :=
R
2
the atoms are exposed to
: fy : L ! R
X
EB
x; x
x
d
0
V
2 L \ BR
6= x
j
:
: fy : L ! R
R
(y ) := E R (y )
d
R
EB
B
is the identity.
lim !1 (y )?
,
X
f
(x )y (x );
Open challenges
2 L \ BR
-ball with radius R >
Which conditions on injective displacements y
R
0
y (x )j)
x
R
of the limit
g!R
y injective
(jy (x )
k k1
the associated energy functional
0:L!R
d
0
in which BR denotes the open
in which y
(short-range potentials), (4) is quite evident.
:
d
j
0
The next step will be to study conditions under which minimizers exist. Specical-
g!R
y injective
ly, I will pursue the following program.
,
R (y0);
L!R
I would like to clarify the conditions under which atomistic minimizers of elastic
energy exist, and study their basic properties.
In principle, the set-up I am studying allows the physically important possibility
that the reference deformation y
d
0
is not the identity, but corresponds to the
presence of defects. The simplest examples would be a single edge- or
guarantee the existence
screw-dislocation. I would like to extend my results to this case, establishing in
This question is the main problem examined in [4]. It
particular the existence of atomistically energy minimizing congurations with
is inspired by [2] and [1].
defects.
In the longer run I would also begin to address the question of rigorous passage
to continuum limits in various interesting scaling regimes, via
-convergence.
Main results
: L ! R is injective, if y y0 is summable and if f is bounded, then
lim !1 (y ) does exist.
If d = 1, if y : L ! R is injective and if y
y0 and f are square summable,
then lim !1 (y ) does exist.
If d > 1 and if f is square summable, then V can be chosen such that there
exists an injective y : L ! R such that y
y0 is square summable and
lim !1 (y ) does not exist.
d
(1) If y
R
(2)
d
R
(3)
References
R
[1] Conti, S., Dolzmann, G., Kirchheim, B., Müller, S. (2006): Sucient conditions for the
validity of the Cauchy-Born rule close to SO(n), J. Eur. Math. Soc. 8, 515-530.
[2] Friesecke, G., Theil, F. (2002): Validity and failure of the Cauchy-Born hypothesis in a
two-dimensional mass-spring lattice, J. Nonl. Sci. 12 No. 5, 445-478.
[3] Friesecke, G., Theil, F. (2005): Periodic crystals as local minimizers of pair potential
energies, unpublished notes.
[4] Kahler, S. (2008): Lattice deformations and elastic energy functionals in dimensions - a
variational and functional analytic approach, Bachelor's Thesis.
R
d
R
R
(4) The unexpected dierence between (2) and (3), i.e., the surprising dependence
on the dimension, is solely of geometric origin and cannot be overcome by
d
strengthening the assumptions on the decay of V .
Chair of Analysis (M7)
j
Boltzmannstr. 3
j
85748 Garching
j
www-m7.ma.tum.de