There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Elements Of X Ray Diffraction

A dislocation-based constitutive law for pure Zr including temperature effects

Disorder and long-range interactions are two of the key components that make material failure an interesting playfield for the application of statistical mechanics. The cornerstone in this respect has been lattice models of the fracture in which a network of elastic beams, bonds, or electrical fuses with random failure thresholds are subject to an increasing external load. These models describe on a qualitative level the failure processes of real, brittle, or quasi-brittle materials. This has been particularly important in solving the classical engineering problems of material strength: the size dependence of maximum stress and its sample-to-sample statistical fluctuations. At the same time, lattice models pose many new fundamental questions in statistical physics, such as the relation between fracture and phase transitions. Experimental results point out to the existence of an intriguing crackling noise in the acoustic emission and of self-affine fractals in the crack surface morphology. Recent advances ...

Statistical models of fracture

Defect-interface interactions

Monte Carlo simulation and theoretical modeling are used to study the statistical failure modes in unidirectional composites consisting of elastic fibers in an elastic matrix. Both linear and hexagonal fiber arrays are considered, forming 2D and 3D composites, respectively. Failure is idealized using the chain-of-bundles model in terms of δ-bundles of length δ, which is the length-scale of fiber load transfer. Within each δ-bundle, fiber load redistribution is determined by local load-sharing models that approximate the in-plane fiber load redistribution from planar break clusters, as predicted from 2D and 3D shear-lag models. As a result the δ-bundle failure models are 1D and 2D, respectively. Fiber elements have random strengths following either a Weibull or a power-law distribution with shape and scale parameters ρ and σδ, respectively. Under Weibull fiber strength, failure simulations for 2D δ-bundles, reveal two regimes: When fiber strength variability is low (roughly ρ>2) the dominant failure mode is by growing clusters of fiber breaks, one of which becomes catastrophic. When this variability is high (roughly 0<ρ<2) cluster formation is suppressed by a dispersed failure mode due to the blocking effects of a few strong fibers. For 1D δ-bundles or for 2D δ-bundles under power-law fiber strength, the transitional value of ρ drops to 1 or lower, and overall, it may slowly decrease with increasing bundle size. For the two regimes, closed-form approximations to the distribution of δ-bundle strength are developed under the local load-sharing model and an equal load-sharing model of Daniels, respectively. The results compare favorably with simulations on δ-bundles with up to 1500 fibers.

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Strength distributions and size effects for 2D and 3D composites with Weibull fibers in an elastic matrix

Monte-Carlo simulations and theoretical modeling are used to study the statistical failure modes and associated lifetime distribution of unidirectional 2D and 3D fiber-matrix composites under constant load. Within the composite the fibers weaken over time and break randomly, and the matrix undergoes linear viscoelastic creep in shear. The statistics of fiber failure are governed by the breakdown model of Coleman (1958a), which embodies a Weibull hazard functional of fiber load history imparting power-law sensitivity to fiber load with exponent ρ, and Weibull lifetime characteristics with shape parameter β. The matrix has a power-law creep compliance in shear with exponent α. Fiber load redistribution at breaks is calculated using a shear-lag mechanics model, which is much more realistic than idealized rules based on equal, global or local load-sharing. The present study is concerned only with the `avalanche' failure regime discussed by Curtin and Scher (1997) which occurs for sufficiently large ρ, and whereby the composite lifetime distribution follows weakest-link scaling. The present Monte-Carlo failure simulations reveal two distinct failure modes within the avalanche regime: For larger ρ, where fiber failure is very sensitive to load level, the weakest link volume fails in a `brittle' manner by the gradual growth of a cluster of mostly contiguous fiber breaks, which then abruptly transitions into a catastrophic crack. For smaller ρ, where this load sensitivity is much less, the weakest link volume shows `tough' behavior, i.e., distributed damage in terms of random fiber failures until the failure of a critical volume and its catastrophic extension. The transition from brittle to tough failure mode for each ρ within the avalanche regime is gradual and depends on the matrix creep exponent α and Weibull exponent β. Also, as α increases above zero the sensitivity of median composite lifetime to load level increasingly deviates from power-law scaling known to occur in the elastic matrix case, α=0. By probabilistic modeling of the dominant failure modes in each regime we obtain distribution forms and various scalings for damage growth, and for carefully chosen sets of parameter values we analytically extend simulation results on small composites (limited by current computer power) to more realistic sizes. Our analytical strength distributions are applicable for ρ>2 in 2D, and ρ≳4 in 3D. The 2D bound coincides with the avalanche-percolation threshold derived by Curtin and Scher (1997) using entirely different arguments.

Lifetime distributions for unidirectional fibrous composites under creep-rupture loading

In addition to texture, plastic anisotropy of a polycrystalline fcc metal stems from the directional nature of the dislocation substructure within individual grains. This produces the marked work hardening or softening observed immediately following load path changes. Following the framework of Peeters et al., in bcc steel, we develop a dislocation substructure evolution-based stage III hardening model for copper, capable of capturing the constitutive response under load path changes. The present model accounts for the more complicated substructure geometry in fcc metals than in bcc. Using an optimization algorithm, the parameters governing substructure evolution in the model are fit to experimental stress-strain curves obtained during compression along the three orthogonal directions in samples previously rolled to various reductions. These experiments approximate monotonic, reverse, and cross-load paths. With parameters suitably chosen, the substructure model, embedded into a self-consistent polycrystal plasticity model, is able to reproduce the measured flow stress response of copper during load path change experiments. The sensitivity of the parameters to the assumed substructure geometry and their uniqueness are also discussed.

Sivasambu Mahesh

Papers

Strength distributions and size effects for 2D and 3D composites with Weibull fibers in an elastic matrix

Lifetime distributions for unidirectional fibrous composites under creep-rupture loading

Application of a Substructure-Based Hardening Model to Copper under Loading Path Changes

Size and heterogeneity effects on the strength of fibrous composites

On the influence of fiber shape in bone-shaped short-fiber composites