Scaling of Fracture Strength in Disordered Quasi-Brittle Materials

This paper presents two main results. The first result indicates that in materials with broadly distributed microscopic heterogeneities, the fracture strength distribution corresponding to the peak load of the material response does not follow the commonly used Weibull and (modified) Gumbel distributions. Instead, a {\it lognormal} distribution describes more adequately the fracture strengths corresponding to the peak load of the response. Lognormal distribution arises naturally as a consequence of multiplicative nature of large number of random distributions representing the stress scale factors necessary to break the subsequent "primary" bond (by definition, an increase in applied stress is required to break a "primary" bond) leading up to the peak load. Numerical simulations based on two-dimensional triangular and diamond lattice topologies with increasing system sizes substantiate that a {\it lognormal} distribution represents an excellent fit for the fracture strength distribution at the peak load. The second significant result of the present study is that, in materials with broadly distributed microscopic heterogeneities, the mean fracture strength of the lattice system behaves as $\mu_f = \frac{\mu_f^\star}{(Log L)^\psi} + \frac{c}{L}$, and scales as $\mu_f \approx \frac{1}{(Log L)^\psi}$ as the lattice system size, $L$, approaches infinity.Comment: 24 pages including 11 figure

Statistical properties of fracture in a random spring model

Using large scale numerical simulations we analyze the statistical properties of fracture in the two dimensional random spring model and compare it with its scalar counterpart: the random fuse model. We first consider the process of crack localization measuring the evolution of damage as the external load is raised. We find that, as in the fuse model, damage is initially uniform and localizes at peak load. Scaling laws for the damage density, fracture strength and avalanche distributions follow with slight variations the behavior observed in the random fuse model. We thus conclude that scalar models provide a faithful representation of the fracture properties of disordered systems.Comment: 12 pages, 17 figures, 1 gif figur

Crack roughness and avalanche precursors in the random fuse model

We analyze the scaling of the crack roughness and of avalanche precursors in the two dimensional random fuse model by numerical simulations, employing large system sizes and extensive sample averaging. We find that the crack roughness exhibits anomalous scaling, as recently observed in experiments. The roughness exponents ($\zeta$, $\zeta_{loc}$) and the global width distributions are found to be universal with respect to the lattice geometry. Failure is preceded by avalanche precursors whose distribution follows a power law up to a cutoff size. While the characteristic avalanche size scales as $s_0 \sim L^D$, with a universal fractal dimension $D$, the distribution exponent $\tau$ differs slightly for triangular and diamond lattices and, in both cases, it is larger than the mean-field (fiber bundle) value $\tau=5/2$

An Efficient Block Circulant Preconditioner For Simulating Fracture Using Large Fuse Networks

{\it Critical slowing down} associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block circlant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present alorithm per iteration is $O(rs log s) + delops$, where the stiffness matrix ${\bf A}$ is partioned into $r$-by-$r$ blocks such that each block is an $s$-by-$s$ matrix, and $delops$ represents the operational count associated with solving a block-diagonal matrix with $r$-by-$r$ dense matrix blocks. This algorithm using the block circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) algorithm, and alleviates the {\it critical slowing down} that is especially severe close to the critical point. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.Comment: 16 pages including 2 figure

Three-dimensional second-order inelastic analysis of steel frames

This research addresses accurate, efficient and rigorous finite element modeling of the three-dimensional second-order inelastic response of beam-column members and their interaction in general three-dimensional framing systems. To this end, two beam-column finite elements based on displacement-type and mixed approaches, that can represent the inelastic three-dimensional stability behavior of a member of open-walled cross-section, arbitrarily curved in space, are formulated. The beam-column kinematics of deformation include finite rotation, torsional warping of cross-sections and flexural-torsional coupling. Shear stresses due to uniform torsion are considered in addition to normal stresses due to axial force, biaxial bending and bimoments. A generalized variational principle is used to formulate the mixed finite element. Variationally consistent force recovery algorithms that are suitable for elastoplastic analysis are derived for two- and three-parameter mixed variational formulations. The finite elements are formulated both in Total Lagrangian (TL) and Total Lagrangian Co-Rotational (TL-CR) descriptions. A consistent transformation between TL and TL-CR formulations is derived. Along with the above aspects of formulation, this research also addresses return mapping algorithms and consistent tangent operators for multi-dimensional J\sb {2} flow theory in a given constrained configuration subspace. The classical radial return mapping algorithm is adopted for strain constrained subspaces, and a plane stress return mapping algorithm developed by Simo is generalized for any stress constrained subspace. The constitutive model is implemented in an object-oriented environment such that the return mapping algorithms and the consistent tangent operators are obtained for any constrained configuration subspace. The above displacement-based and mixed bean-column finite elements are used in studying the inelastic two- and three-dimensional response of frames. The studies involve comparison with several well established, and recent insightful experimental investigations. The numerical studies indicate that the mixed element has better coarse mesh accuracy than the displacement element especially in the inelastic analysis. The developed finite elements are capable of representing the inelastic stability behavior of beam-column members that are arbitrarily deformed in space, and their interaction in general three-dimensional framing systems. The proposed research provides the necessary capabilities to perform a more realistic and rational three-dimensional second-order inelastic analysis of steel frames