28 research outputs found
On the variational limits of lattice energies on prestrained elastic bodies
We study the asymptotic behaviour of the discrete elastic energies in
presence of the prestrain metric , assigned on the continuum reference
configuration . When the mesh size of the discrete lattice in
goes to zero, we obtain the variational bounds on the limiting (in the sense of
-limit) energy. In case of the nearest-neighbour and
next-to-nearest-neibghour interactions, we derive a precise asymptotic formula,
and compare it with the non-Euclidean model energy relative to
On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime
We consider a two-dimensional atomic mass spring system and show that in the
small displacement regime the corresponding discrete energies can be related to
a continuum Griffith energy functional in the sense of Gamma-convergence. We
also analyze the continuum problem for a rectangular bar under tensile boundary
conditions and find that depending on the boundary loading the minimizers are
either homogeneous elastic deformations or configurations that are completely
cracked generically along a crystallographic line. As applications we discuss
cleavage properties of strained crystals and an effective continuum fracture
energy for magnets
On the commutability of homogenization and linearization in finite elasticity
We study non-convex elastic energy functionals associated to (spatially)
periodic, frame indifferent energy densities with a single non-degenerate
energy well at SO(n). Under the assumption that the energy density admits a
quadratic Taylor expansion at identity, we prove that the Gamma-limits
associated to homogenization and linearization commute. Moreover, we show that
the homogenized energy density, which is determined by a multi-cell
homogenization formula, has a quadratic Taylor expansion with a quadratic term
that is given by the homogenization of the quadratic term associated to the
linearization of the initial energy density
On the effect of interactions beyond nearest neighbours on non-convex lattice systems
We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation
Elastic limit of square lattices with three point interactions
26 pagesInternational audienceWe derive the equivalent energy of a square lattice that either deforms into the three-dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy-Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomic models
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
Linearized plasticity is the evolutionary \Gamma-limit of finite plasticity
We provide a rigorous justification of the classical linearization approach
in plasticity. By taking the small-deformations limit, we prove via
\Gamma-convergence for rate-independent processes that energetic solutions of
the quasi-static finite-strain elastoplasticity system converge to the unique
strong solution of linearized elastoplasticity.Comment: To appear on J. Eur. Math. Soc. (JEMS
A bending-torsion theory for thin and ultrathin rods as a Γ-limit of atomistic models
The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as Γ-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness h and interatomic distance ε. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibres (ε∼h), which incorporates surface energy and new discrete terms in the limiting functional. This can be thought of as a contribution to the mechanical modelling of nanowires. Second, we treat the case where ε≪h and recover a nonlinear rod model − the modern version of Kirchhoff's rod theory