Scaling behaviour of damage in the fracture of two-dimensional elastic beam lattices (original) (raw)
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Fracture roughness in three-dimensional beam lattice systems
Physical Review E, 2010
We study the scaling of three-dimensional crack roughness using large-scale beam lattice systems. Our results for prenotched samples indicate that the crack surface is statistically isotropic, with the implication that experimental findings of anisotropy of fracture surface roughness in directions parallel and perpendicular to crack propagation is not due to the scalar or vectorial elasticity of the model. In contrast to scalar fuse lattices, beam lattice systems do not exhibit anomalous scaling or an extra dependence of roughness on system size. The local and global roughness exponents ͑ loc and , respectively͒ are equal to each other, and the threedimensional crack roughness exponent is estimated to be loc = = 0.48Ϯ 0.03. This closely matches the roughness exponent observed outside the fracture process zone. The probability density distribution p͓⌬h͑ᐉ͔͒ of the height differences ⌬h͑ᐉ͒ = ͓h͑x + ᐉ͒ − h͑x͔͒ of the crack profile follows a Gaussian distribution, in agreement with experimental results.
Brittle Crack Roughness in Three-Dimensional Beam Lattices
Eprint Arxiv Cond Mat 0505633, 2005
The roughness exponent is reported in numerical simulations with a three-dimensional elastic beam lattice. Two different types of disorder have been used to generate the breaking thresholds, i.e., distributions with a tail towards either strong or weak beams. Beyond the weak disorder regime a universal exponent of 0.59(1) is obtained. This is within the range 0.4-0.6 reported experimentally for small scale quasi-static fracture, as would be expected for media with a characteristic length scale.
Physical Review B, 1989
We studied numerically the fracture of three types of disordered media: a scalar, a centralforce, and a beam model. We discovered the following novel, universal laws: in an initial regime, force and displacement both scale as L with the system size L; the number of bonds that break scales during the whole process as L ', and the distribution of local forces is multifractal just before the system breaks, whereas it has constant-gap scaling when catastrophic breaking sets in.
Physical Review B, 2006
We analyze statistical and scaling properties of the fracture of two-dimensional ͑2D͒ central-force spring lattices with strong disorder by means of computer simulation. We run fracture simulations for two types of boundary conditions and compare the results both with the simulation of random damage percolation on the same lattices and with the analytical scaling relations of percolation theory. We investigate the scaling behavior of the macroscopic failure thresholds, the main features of the developing microscopic cluster statistics and damage pattern, and the roughness scaling of the final crack. Our observations show that simulated fracture has three clearly distinguished regimes. The initial phase displays short-range localization of damage, but it is soon replaced by a regime where damage develops in a uniform manner, qualitatively as in random percolation. Already before the maximum-stress point macroscopic localization and anisotropy come into play, resulting in final crack formation. The data of the second, uniform-damage regime can be fitted consistent with the scaling laws of random percolation. Beyond this regime a clear difference is observed with percolation theory and with earlier results from fuse-network models. Nevertheless, the final-crack roughness is found to scale accurately over at least three decades, with a roughness exponent consistent with limited available data for 2D systems and marginally consistent with the value for 2D percolation in a gradient.
Author ' s personal copy Crack in a 2 D beam lattice : Analytical solutions for two bending modes
2010
We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the ‘macrolevel’ state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass–spring lattices. Considering the static prob...
Discrete element modeling of brittle crack roughness in three dimensions
Frontiers in Physics, 2014
Crack morphology obtained in the fracture of materials with a disordered micro-structure is studied using numerical simulations. Physical properties are embedded on a regular three dimensional lattice as discrete stochastic elements which conform to the laws of linear elasticity. In this model, also known as the beam lattice, these elements are analogous to beams in that relative displacements between neighboring nodes induce axial, bending, and shearing forces, as in a real elastic solid. The stochastic nature enters via the introduction of random breaking thresholds on the individual elements. Using this model, the exponent characterizing the scaling with system size of the crack roughness perpendicular to the fracture plane is reported. Two different types of disorder have been used to generate the thresholds, i.e., distributions with a tail toward strong elements or with a tail toward weak elements. At weak disorders the self-affine regime seems to lie beyond the system sizes presently included. At stronger disorders a self-affine regime appears, for which we obtain exponents consistent with ζ 0.6 for both types of disorder. The latter result is in fair agreement with the experimental value reported for large length scales, ζ 0.50.
The damage tolerance of elastic–brittle, two-dimensional isotropic lattices
Journal of the Mechanics and Physics of Solids, 2007
The fracture toughness of elastic-brittle 2D lattices is determined by the finite element method for three isotropic periodic topologies: the regular hexagonal honeycomb, the Kagome lattice and the regular triangular honeycomb. The dependence of mode I and mode II fracture toughness upon relative density is determined for each lattice, and the fracture envelope is obtained in combined mode I-mode II stress intensity factor space. Analytical estimates are also made for the dependence of mode I and mode II toughness upon relative density. The high nodal connectivity of the triangular grid ensures that it deforms predominantly by stretching of the constituent bars, while the hexagonal honeycomb deforms by bar bending. The Kagome microstructure deforms by bar stretching remote from the crack tip, and by a combination of bar bending and bar stretching within a characteristic elastic deformation zone near the crack tip. This elastic zone reduces the stress concentration at the crack tip in the Kagome lattice and leads to an elevated macroscopic toughness. Predictions are given for the tensile and shear strengths of a centre-cracked panel with microstructure given explicitly by each of the three topologies. The hexagonal and triangular honeycombs are flaw-sensitive, with a strength adequately predicted by linear elastic fracture mechanics (LEFM) for cracks spanning more than a few cells. In contrast, the Kagome microstructure is damage tolerant, and for cracks shorter than a transition length its tensile strength and shear strength are independent of crack length but are somewhat below the unnotched strength. At crack lengths exceeding the transition value, the strength decreases with increasing crack length in accordance with the LEFM estimate. This transition crack length scales with the parameter of bar length divided by relative density of the Kagome grid, and can be an order of magnitude greater than the cell size at low relative densities. Finally, the presence of a boundary layer is noted at the free edge of a crack-free Kagome grid loaded in tension and in shear. Deformation within this boundary layer is by a combination of bar bending and stretching whereas remote from the free edge the Kagome
Author's personal copy Crack in a 2D beam lattice: Analytical solutions for two bending modes
2020
a b s t r a c t We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the 'macrolevel' state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass-spring lattices. C...
Fractures in heterogeneous two-dimensional systems
Physical Review E, 2001
A two-dimensional triangular lattice with bond disorder is used as a testing ground for fracture behavior in heterogeneous materials in strain-controlled conditions. Simulations are performed with two interaction potentials ͑harmonic and Lennard-Jones types͒ and different breaking thresholds. We study the strain range where the fracture progressively develops from the first to the last breakdown. Scaling properties with the lattice size are investigated: no qualitative difference is found between the two interaction potentials. Clustering properties of the broken bonds are also studied by grouping them into disjoint sets of connected bonds. Finally, the role of kinetic energy is analyzed by comparing overdamped with dissipationless dynamics.