490 research outputs found
Existence and Convergence of Solutions of the Boundary Value Problem in Atomistic and Continuum Nonlinear Elasticity Theory
We show existence of solutions for the equations of static atomistic
nonlinear elasticity theory on a bounded domain with prescribed boundary
values. We also show their convergence to the solutions of continuum nonlinear
elasticity theory, with energy density given by the Cauchy-Born rule, as the
interatomic distances tend to zero. These results hold for small data close to
a stable lattice for general finite range interaction potentials. We also
discuss the notion of stability in detail.Comment: new version with only minor change
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Atomistic Models of Materials: Mathematical Challenges
[no abstract available
On the passage from atomistic systems to nonlinear elasticity theory
We derive continuum limits of atomistic models in the realm of nonlinear
elasticity theory rigorously as the interatomic distances tend to zero. In
particular we obtain an integral functional acting on the deformation gradient
in the continuum theory which depends on the underlying atomistic interaction
potentials and the lattice geometry. The interaction potentials to which our
theory applies are general finite range models on multilattices which in
particular can also account for multi-pole interactions and bond-angle
dependent contributions. Furthermore, we discuss the applicability of the
Cauchy-Born rule. Our class of limiting energy densities consists of general
quasiconvex functions and the class of linearized limiting energies consistent
with the Cauchy-Born rule consists of general quadratic forms not restricted by
the Cauchy relations
A continuum model for brittle nanowires derived from an atomistic description by Î-convergence
Starting from a particle system with short-range interactions, we derive a continuum model for the bending, torsion, and brittle fracture of inextensible rods moving in three-dimensional space. As the number of particles tends to infinity, it is assumed that the rod's thickness is of the same order as the interatomic distance. Fracture energy in the Î-limit is expressed by an implicit cell formula, which covers different modes of fracture, including (complete) cracks, folds and torsional cracks. In special cases, the cell formula can be significantly simplified. Our approach applies e.g. to atomistic systems with Lennard-Jones-type potentials and is motivated by the research of ceramic nanowires
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Microstructures in Solids: From Quantum Models to Continua
The mathematical theory of solids was studied from the modern perspective of materials with microcstructures. The discussed topics ranged from experimental findings, via numerical simulations and mathematical modeling to the analysis of models with microstructures. A special emphasis was given to theories providing rigorous insight into and justification of the limit passage between different scales
An analysis of crystal cleavage in the passage from atomistic models to continuum theory
We study the behavior of atomistic models in general dimensions under
uniaxial tension and investigate the system for critical fracture loads. We
rigorously prove that in the discrete-to-continuum limit the minimal energy
satisfies a particular cleavage law with quadratic response to small boundary
displacements followed by a sharp constant cut-off beyond some critical value.
Moreover, we show that the minimal energy is attained by homogeneous elastic
configurations in the subcritical case and that beyond critical loading
cleavage along specific crystallographic hyperplanes is energetically
favorable. In particular, our results apply to mass spring models with full
nearest and next-to-nearest pair interactions and provide the limiting minimal
energy and minimal configurations.Comment: The final publication is available at springerlink.co
Finite element methods for the non-linear mechanics of crystalline sheets and nanotubes
The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a twoâdimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyperâelastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thinâshell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multiâmillion systems illustrate the computational savings which can be achieve
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Material Theories
Material theories is a series of workshops concerned with a broad range of topics related to the mechanics and mathematics of materials. As such, this edition brought together researchers from diverse fields converging toward the interaction between mathematics, mechanics, and material science
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Variational Methods for the Modelling of Inelastic Solids
This workshop brought together two communities working on the same topic from different perspectives. It strengthened the exchange of ideas between experts from both mathematics and mechanics working on a wide range of questions related to the understanding and the prediction of processes in solids. Common tools in the analysis include the development of models within the broad framework of continuum mechanics, calculus of variations, nonlinear partial differential equations, nonlinear functional analysis, Gamma convergence, dimension reduction, homogenization, discretization methods and numerical simulations. The applications of these theories include but are not limited to nonlinear models in plasticity, microscopic theories at different scales, the role of pattern forming processes, effective theories, and effects in singular structures like blisters or folding patterns in thin sheets, passage from atomistic or discrete models to continuum models, interaction of scales and passage from the consideration of one specific time step to the continuous evolution of the system, including the evolution of appropriate measures of the internal structure of the system
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