24 research outputs found
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
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
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
An integral-representation result for continuum limits of discrete energies with multi-body interactions
We prove a compactness and integral-representation theorem for sequences of
families of lattice energies describing atomistic interactions defined on
lattices with vanishing lattice spacing. The densities of these energies may
depend on interactions between all points of the corresponding lattice
contained in a reference set. We give conditions that ensure that the limit is
an integral defined on a Sobolev space. A homogenization theorem is also
proved. The result is applied to multibody interactions corresponding to
discrete Jacobian determinants and to linearizations of Lennard-Jones energies
with mixtures of convex and concave quadratic pair-potentials
Overall properties of a discrete membrane with randomly distributed defects
A prototype for variational percolation problems with surface energies is considered: the description of the macroscopic properties of a (two-dimensional) discrete membrane with randomly distributed defects in the spirit of the weak membrane model of Blake and Zisserman (quadratic springs that may break at a critical length of the elongation). After introducing energies depending on suitable independent identically distributed random variables, this is done by exhibiting an equivalent continuum energy computed using Delta-convergence, geometric measure theory, and percolation arguments. We show that below a percolation threshold the effect of the defects is negligible and the continuum description is given by the Dirichlet integral, while above that threshold an additional (Griffith) fracture term appears in the energy, which depends only on the defect probability through the chemical distance on the "weak cluster of defects"
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