Workshop on

Discrete Atomistic Models and their Continuum Limits

List of Abstracts

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}

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.

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.

M. Herrmann (Berlin):
Atomic chain with temperature

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.

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.

F. Legoll (Marne la Vallee, France):
Mathematical analysis of a simple 1D micro-macro method for materials simulation
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.

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.

I. Müller, A. Sisman (Berlin):
Casimir-like Size Effects in Ideal Gases
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.

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

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.

G. Stoltz (Paris, France):
To model shock propagations of explosives in 1D

F. Theil (Warwick, UK):
Surface energies in two-dimensional mass-spring models for crystals
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.

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.

H. Zapolsky (Rouen, France):
Discrete atomistic model of coarsening of ordered intermetallic precipitates with coherency stress
J. Zimmer (Leipzig):
On relaxation and Young measures