1,447 research outputs found
Quasicontinuum simulation of fracture at the atomic scale
We study the problem of atomic scale fracture using the recently developed quasicontinuum method in which there is a systematic thinning of the atomic-level degrees of freedom in regions where they are not needed. Fracture is considered in two distinct settings. First, a study is made of cracks in single crystals, and second, we consider a crack advancing towards a grain boundary (GB) in its path. In the investigation of single crystal fracture, we evaluate the competition between simple cleavage and crack-tip dislocation emission. In addition, we examine the ability of analytic models to correctly predict fracture behaviour, and find that the existing analytical treatments are too restrictive in their treatment of nonlinearity near the crack tip. In the study of GB-crack interactions, we have found a number of interesting deformation mechanisms which attend the advance of the crack. These include the migration of the GB, the emission of dislocations from the GB, and deflection of the crack front along the GB itself. In each case, these mechanisms are rationalized on the basis of continuum mechanics arguments
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
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
Coarse-Graining and Renormalization of Atomistic Binding Relations and Universal Macroscopic Cohesive Behavior
We present two approaches for coarse-graining interplanar potentials and
determining the corresponding macroscopic cohesive laws based on energy
relaxation and the renormalization group. We analyze the cohesive behavior of a
large---but finite---number of interatomic planes and find that the macroscopic
cohesive law adopts a universal asymptotic form. The universal form of the
macroscopic cohesive law is an attractive fixed point of a suitably-defined
renormalization-group transformation.Comment: 15 pages, 6 figures, submitted to the Journal of the Mechanics and
Physics of Solid
Continuum Surface Energy from a Lattice Model
We investigate some connections between the continuum and atomistic descriptions of de- formable crystals, using some interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. When the crystal shape is a lattice polygon, we show that the energy equals the bulk elastic energy, plus the boundary integral of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for infinite-range potentials it is asymptotically valid as the lattice parameter zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. For more general convex domains with possibly curved boundary, the surface energy density depends on the unit normal in a striking way. It is continuous at irrational directions, discontinuous at rational ones and nowhere differ- entiable. This pathology is alarming since it renders the surface energy minimization problem (under domain variations) ill-posed. An alternative approach of defining the continuum region is introduced, that restores continuity of the surface energy density function
An Overview of the State of the Art in Atomistic and Multiscale Simulation of Fracture
The emerging field of nanomechanics is providing a new focus in the study of the mechanics of materials, particularly in simulating fundamental atomic mechanisms involved in the initiation and evolution of damage. Simulating fundamental material processes using first principles in physics strongly motivates the formulation of computational multiscale methods to link macroscopic failure to the underlying atomic processes from which all material behavior originates. This report gives an overview of the state of the art in applying concurrent and sequential multiscale methods to analyze damage and failure mechanisms across length scales
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