| M. Arndt and M. Griebel (Bonn): Derivation of higher order gradient continuum models from atomistic models for crystalline solids | |||||||||
|
Many materials exhibit a complex behaviour which needs to be resolved
on different length scales. For example microscopical effects can
often be well described on an atomistic level using methods such as
Molecular Dynamics (MD), whereas macroscopic effects can be described
by models on the continuum mechanical level. The computation of the
behaviour on coarser length scales usually cannot be done on finer
length scales due to computational limits. This shows the need of
advanced analytical and numerical techniques to bridge the gap between
the different scales. \newcommand{\NN}{{\mathbb N}}
\newcommand{\RR}{{\mathbb R}} \newcommand{\CC}{{\mathbb C}}
Usually the link between the two scales is established by the
Cauchy-Born hypothesis: the continuum energy density $\Phi$ is assumed
to be a function only of the deformation gradient $\nabla y(x)$. Here
$y:\Omega\to\RR^3$ describes the deformation of the specimen, and
$\Omega\subset\RR^3$ is the reference configuration. This leads to an
energy of the form
\begin{equation} \label{EqOrderOne}
E(y) = \int_\Omega \Phi(\nabla y(x)) \, \textrm{d}x.
\end{equation}
Such models can be derived from atomistic models by means of scaling
techniques, see e.g.~\cite{BlancLeBrisLions:2002a}. They can describe
the behaviour of several solids quite well. But many solids exhibit
effects that cannot be captured within this formulation, such as a
microstructure of a determined length scale in shape memory alloys
\cite{ArndtGriebelRoubicek:2003}. This especially holds for materials
with many-body potentials on the atomistic scale, which are necessary
to describe the behaviour of complex solids adequately.
To remedy this deficiency, higher order gradient models have been
proposed, see e.g.~\cite{TriantafyllidisBardenhagen:1993}. Here the
energy density $\Phi$ is assumed to depend on the derivatives of $y$
up to some order $K\in\NN$:
\begin{equation} \label{EqOrderK}
E(y) = \int_\Omega \Phi(x, \nabla y(x), \nabla^2 y(x),
\ldots, \nabla^K y(x)) \, \textrm{d}x
\end{equation}
We propose an upscaling scheme, based on a power series expansion, to
derive continuum models of type \eqref{EqOrderK} for any $K$ from
atomistic systems, see \cite{ArndtGriebel:2003}. It is shown that the
resulting models capture essential material responses to nonlinear
deformations such as bending etc., which are lost by standard scaling
techniques. Furthermore boundary effects are taken into account
correctly. The difference between the two continuum models and the
original atomistic system is examined by numerical calculations, both
for simple model problems and complex atomistic potentials for real
crystals.
\bibliographystyle{abbrv}
\bibliography{arndt.bib}
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| A. Braides (Rom,
Italy): Gamma-limits of discrete systems | |||||||||
| We apply the techniques of Gamma-convergence to describe the limit of variational problems for lattice systems as the lattice spacing tends to zero. In particular we address the questions: the general form of limit energies, formulas and bounds for continuum limits, multi-scale analysis. | |||||||||
| A. Cherkaev, E. Cherkaev, L. Slepyan (Berlin): Wave of transition in chains and lattices from bistable elements | |||||||||
|
The paper investigates structural resistance to a dynamic impact
(collision). Structures are destroyed due to material instabilities that
cause stress concentration and localized failure; after the whole
structure fails, most of the material in a failed structure remains in a
fairly good shape. The model of breakable structures necessarily involves
an atomistic elements since the breakage is a concentrated event that
that involves a finite energy release and structural change. In the same
time,
consideration of multiple breakages calls for a continuum description of
the
dynamics or failure. We are developing atomistic model of breakable
structures,
derive the equation of their dynamics, and homogenize them.
To increase the structural resistivity, we suggest a
nonlinear composite structure that dissipates energy by distributing a
"partial damage" over a large area and by transforming the energy of the
impact into high-frequency modes that quickly dissipate.
The cellular element (link) of the structure contains two roughly
parallel rods of different length; the stronger and longer rod is
initially inactive and starts to resist when the strain is large enough
and the shorter rod is broken. The tensile force in such a link is a
nonmonotonic function of the elongation and the stored energy is a
nonconvex function. Chains and lattices of such elements experience a
phase transition when the waves of partial damage propagate along them.
The wave of partial damage consists of a sequence of breakages of the weak
elements; the wave absorbs the energy of a collision and transforms it to
high-frequency vibration mode. In the same time, the partial damage
does not lead to failure of the whole structure.
Our model of cellular lattices with discontinuous piecewise linear
properties allows for explicit calculation of damage waves and multiple
equilibria in partially damaged chain and lattice. The ``house of cards
problem'' is addressed: Under what conditions the local damage will
propagate through the structure of damage demonstrates
the controllability of the damage process; in
particular, the waves of partial damage can be directed in a desirable
direction. An adequate "dynamic homogenization" allows to derive the
continuous model of the process and to take into account the energy of the
high-frequency modes and the found from the discrete model speed of the
phase transition wave.
This project is conducted in collaboration with Elena Cherkaev,
Department of Mathematics, University of Utah and
Leonid Slepyan, Tel Aviv University; the project is supported by ARO and
by NSF.
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| J.
Giannoulis (Stuttgart): Macroscopic pulse evolution for a nonlinear oscillator chain | |||||||||
|
The talk addresses the question of the macroscopic evolution of modulated
microscopic patterns of an oscillator chain. In order to capture dispersive
effects we use a near-linear modulation ansatz with an appropriate macroscopic
time scale. This formal multiscale ansatz yields a nonlinear Schr{\"o}dinger
equation describing the macroscopic evolution of the pulse. The talk is
focused on the mathematical justification of this multiscale approximation
procedure. We generalize previous work on cubic nonlinearities to general ones,
using a normal form transformation.
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| M. Herrmann (Berlin): Atomic chain with temperature | |||||||||
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| G. James
(Toulouse, france): Centre manifold reduction for time-periodic oscillations in infinite lattices | |||||||||
| Time-periodic oscillations in infinite one-dimensional lattices can be expressed in many cases as solutions of an ill-posed "spatial" recurrence relation on a loop space. We give simple spectral conditions under which all small amplitude solutions lie on an invariant finite-dimensional centre manifold. This result reduces the problem locally to the study of a finite-dimensional mapping. In the case of hardening FPU chains, this map is reversible and admits homoclinic orbits corresponding to ``discrete breather'' solutions. | |||||||||
| T.
Kriecherbauer (Bochum): Travelling waves in nonlinear lattices | |||||||||
|
We will present results on the existence of families of travelling
wave solutions (periodic in time, (quasi-) periodic in space)
for infinite lattices of particles with non-linear nearest-neighbor
interactions.
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| C. Le Bris
(Marne la Vallee, France): A variational approach for the definition of mechanical energies | |||||||||
|
We will review some joint work with X. Blanc (Paris 6), I .
Catto (Paris 9) and PL Lions (College de France) devoted to the
definition of energy densities at the continuum level starting from the
discrete level. The case under consideration is the case of crystals at
zero temperature.
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| F. Legoll (Marne la Vallee, France): Mathematical analysis of a simple 1D micro-macro method for materials simulation | |||||||||
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Materials can be described at different scales with different models.
The two models we consider in this work are the atomistic model
(describing the material at a fine scale by using interatomic
potentials) and the continuum mechanics model (describing the material
at a macroscopic scale by using elastic energy density functions).
When one wants to describe fine scale localized phenomena arising in a
material (like nanoindentation or fracture), the macroscopic model is
not precise enough, whereas the atomistic model is too expensive to be
used in the whole domain. Multiscale methods have been proposed to deal
with such situations.
In this talk, we will present a joint work with X. Blanc (Paris 6) and
Claude Le Bris (CERMICS): we propose a mathematical analysis of a simple
1D micro-macro method, which combines both scales, the atomistic one and
the continuum mechanics one, into a single model.
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| K. Matthies (Berlin): Solitary waves in 2D Hamiltonian Lattices | |||||||||
|
We discus the existence of travelling waves in various 2D Hamiltonian
lattcies.
For example, the existence of longitudinal solitary waves is shown for the
Hamiltonian dynamics of a 2D elastic lattice of particles interacting
via harmonic springs between nearest and next nearest
neighbours. A contrasting nonexistence result for transversal
solitary waves is given.
The presence of the longitudinal waves is related to the
two-dimensional geometry of the lattice which creates a universal
overall anharmonicity.
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| I. Müller, A. Sisman (Berlin): Casimir-like Size Effects in Ideal Gases | |||||||||
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The wave character of atoms can produce Casimir-like size effects
in ideal gases confined in a narrow box. Thus the pressure tensor is not
isotropic anymore and the size effect becomes a driving force for isothermal
diffusion through a permeable wall. Such size effects give rise to
"thermosize effects" not unlike thermoelectric effects.
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| M. O. Rieger (Pisa, Italy): Young measure solutions for shape memory alloys | |||||||||
| We prove global existence for a modification of one-dimensional thermoelasticity with nonconvex energy by means of a vanishing capillarity regularization. The limiting system respects balance laws of momentum and a modified energy balance. A special feature is that the free energy is nonconvex as a function of the deformation gradient for temperatures below a threshold temperature. This allows for modeling of structural phase transitions in solids. We prove the existence of Young measure valued solutions, since in general the existence of weak solutions cannot be expected. This is joint work with Johannes Zimmer (MPI Leipzig). | |||||||||
| C. Schütte (Berlin): Stochastic Modelling of Nonadiabatic Processes: Some Surprising Insights | |||||||||
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| H. Spohn
(München): The phonon Boltzmann equation for weakly disordered wave equations | |||||||||
|
We report on work in progress jointly with L. Lukarrinen. Our goal is
to prove the validity of the Boltzmann equation for energy transport
in harmonic lattices with random masses. The natural object is the
Wigner function for wave equations and the tool is the Erd{\"o}s-Yau
graphical expansion.
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| G. Stoltz (Paris, France): To model shock propagations of explosives in 1D | |||||||||
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| F. Theil (Warwick, UK): Surface energies in two-dimensional mass-spring models for crystals | |||||||||
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We study an atomistic pair potential-model that describes the elastic
behavior of crystals in two dimensions. The main focus is the computation
of the ground state energy as a function of the number of particles
that are involved in the minimization.
A popular method for the extraction of continuum concepts such as bulk or
surface energy density is to evaluate the energy on affine
arrangements (clamped particles). We prove for a two-dimensional model
that the surface energy is proportional to the square root of the number
of particle and can be written as a surface integral. Moreover we
show that the clamped-particle approach leads to an overestimation of the
surface energy.
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| L. Truskinovsky (Paris,
France): From NN to NNN | |||||||||
|
We consider a one dimensional chain where nearest neighbors (NN) are
connected by bi-stable springs and discuss the regularizing nature of
the addition of next to nearest neighbor (NNN) interaction. Such a
regularization leads to the nontrivial macroscopic effects in both
statics and dynamics.
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| H. Zapolsky (Rouen,
France): Discrete atomistic model of coarsening of ordered intermetallic precipitates with coherency stress | |||||||||
| J. Zimmer (Leipzig): On relaxation and Young measures | |||||||||
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