1,358 research outputs found
A Griffith-Euler-Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics
We study a planar thin brittle beam subject to elastic deformations and
cracks described in terms of a nonlinear Griffith energy functional acting on
deformations of the beam. In particular we consider the case in which
elastic bulk contributions due to finite bending of the beam are comparable to
the surface energy which is necessary to completely break the beam into several
large pieces. In the limit of vanishing aspect ratio we rigorously derive an
effective Griffith-Euler-Bernoulli functional which acts on piecewise
regular curves representing the midline of the beam. The elastic part of this
functional is the classical Euler-Bernoulli functional for thin beams in the
bending dominated regime in terms of the curve's curvature. In addition there
also emerges a fracture term proportional to the number of discontinuities of
the curve and its first derivative
Global Energy Matching Method for Atomistic-to-Continuum Modeling of Self-Assembling Biopolymer Aggregates
This paper studies mathematical models of biopolymer supramolecular aggregates that are formed by the self-assembly of single monomers. We develop a new multiscale numerical approach to model the structural properties of such aggregates. This theoretical approach establishes micro-macro relations between the geometrical and mechanical properties of the monomers and supramolecular aggregates. Most atomistic-to-continuum methods are constrained by a crystalline order or a periodic setting and therefore cannot be directly applied to modeling of soft matter. By contrast, the energy matching method developed in this paper does not require crystalline order and, therefore, can be applied to general microstructures with strongly variable spatial correlations. In this paper we use this method to compute the shape and the bending stiffness of their supramolecular aggregates from known chiral and amphiphilic properties of the short chain peptide monomers. Numerical implementation of our approach demonstrates consistency with results obtained by molecular dynamics simulations
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
A bending-torsion theory for thin and ultrathin rods as a Ī-limit of atomistic models
The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as Ī-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness h and interatomic distance Īµ. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibres (Īµā¼h), 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 ĪµāŖh and recover a nonlinear rod model ā the modern version of Kirchhoff's rod theory
Concurrently coupled solid shell-based adaptive multiscale method for fracture
ArtĆculo Open Access en el sitio web del editor. Pago por publicar en abierto.A solid shell-based adaptive atomisticācontinuum numerical method is herein proposed to simulate complex crack growth patterns in thin-walled structures. A hybrid solid shell formulation relying on the combined use of the enhanced assumed strain (EAS) and the assumed natural strain (ANS) methods has been considered to efficiently model the material in thin structures at the continuum level. The phantom node method (PNM) is employed to model the discontinuities in the bulk. The discontinuous solid shell element is then concurrently coupled with a molecular statics model placed around the crack tip. The coupling between the coarse scale and the fine scale is realized through the use of ghost atoms, whose positions are interpolated from the coarse scale solution and enforced as boundary conditions to the fine scale model. In the proposed numerical scheme, the fine scale region is adaptively enlarged as the crack propagates and the region behind the crack tip is adaptively coarsened in order to reduce the computation costs. An energy criterion is used to detect the crack tip location. All the atomistic simulations are carried out using the LAMMPS software. A computational framework has been developed in MATLAB to trigger LAMMPS through system command. This allows a two way interaction between the coarse and fine scales in MATLAB platform, where the boundary conditions to the fine region are extracted from the coarse scale, and the crack tip location from the atomistic model is transferred back to the continuum scale. The developed framework has been applied to study crack growth in the energy minimization problems. Inspired by the influence of fracture on currentāvoltage characteristics of thin Silicon photovoltaic cells, the cubic diamond lattice structure of Silicon is used to model the material in the fine scale region, whilst the Tersoff potential function is employed to model the atomāatom interactions. The versatility and robustness of the proposed methodology is demonstrated by means of several fracture applications.UniĆ³n Europea ERC 306622Ministerio de EconomĆa y Competitividad DPI2012-37187, MAT2015-71036-P y MAT2015-71309-PJunta de AndalucĆa P11-TEP-7093 y P12-TEP -105
Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure
Emphasis in modern day efforts in mechanics of materials is increasingly directed towards integration with computational materials science, which itself rests on solid physical and mathematical foundations in thermodynamics and kinetics of processes. Practical applications demand attention to length and time scales which are sufficiently large to preclude direct application of quantum mechanics approaches; accordingly, there are numerous pathways to mathematical modelling of the complexity of material structure during processing and in service. The conventional mathematical machinery of energy minimization provides guidance but has limited direct applicability to material systems evolving away from equilibrium. Material response depends on driving forces, whether arising from mechanical, electromagnetic, or thermal fields. When microstructures evolve, as during plastic deformation, progressive damage and fracture, corrosion, stress-assisted diffusion, migration or chemical/thermal aging, the associated classical mathematical frameworks are often ad hoc and heuristic. Advancing new and improved methods is a major focus of 21st century mechanics of materials of interfaces and evolving microstructure
Theory and computation of higher gradient elasticity theories based on action principles
In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packagesāby using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect
A continuum model for brittle nanowires derived from an atomistic description by Ī-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 Ī-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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