Per C. Hemmer

Burst avalanches in bundles of fibers: Local versus global load-sharing

Abstract Bursts in bundles of many parallel fibers with stochastically distributed failure thresholds are studied. The distribution of the sizes Δ of burst avalanches has a power-law behavior, ∞Δ-ξ. When the load is shared equally among surviving fibers, the power-law exponent is ξ=2.5. When, however, the increased stresses that result from a fiber failure are concentrated to the nearest-neighbor fibers, the exponent is considerably higher, for randomly distributed thresholds ξ=4.5.

Crossover Behavior in Burst Avalanches: Signature of Imminent Failure

The statistics of damage avalanches during a failure process typically follows a power law. When these avalanches are recorded only near the point at which the system fails catastrophically, one finds that the power law has an exponent which is different from that one finds if the recording of events starts away from the vicinity of catastrophic failure. We demonstrate this analytically for bundles of many fibers, with statistically distributed breakdown thresholds for the individual fibers and where the load is uniformly distributed among the surviving fibers. In this case the distribution D(Delta) of the avalanches (Delta) follows the power law Delta-xi with xi=3/2 near catastrophic failure and xi=5/2 away from it. We also study numerically square networks of electrical fuses and find xi=2.0 near catastrophic failure and xi=3.0 away from it. We propose that this crossover in xi may be used as a signal of imminent failure.

Solid-solid transitions induced by repulsive interactions

We show that a rich variety of crystalline structures, and a corresponding diversity of the associated phase diagrams, result from the presence in the pair potential of a soft repulsion in addition to a hard core. We use different forms for the soft repulsion, and show that the results are sensitive to the details of the potentials (in particular, their convexity) even if the range of the soft repulsion is limited to a small fraction of the hard-core diameter. Our demonstration combines exact ground-state analysis with first-order perturbation theory at finite temperatures. The relevance of our work to certain features found in real systems is also discussed.

The fiber bundle model : modeling failure in materials

Description: The book contains different applications of the fiber bundle model. As example, the authors have included fatigue and creep in their discussion. The understanding of materials and their behavior under different conditions is undergoing a revolution due to the coming–of–age of computational atomistic modeling. Even though this technique allows for the precise knowledge of what every atom is doing in such materials, it is still necessary to understand the results in terms of models such as the fiber bundle model. From the contents: The Fiber Bundle Model Average Properties Fluctuation Effects Local and Intermediate Load Sharing Recursive Breaking Dynamics Predicting Failure Fiber Bundle Model in Material Science Snow Avalanches and Landslides

The random parking problem

A new approach to the random parking problem is given.

Crossover behavior in failure avalanches.

Composite materials, with statistically distributed thresholds for breakdown of individual elements, are considered. During the failure process of such materials under external stress (load or voltage), avalanches consisting of simultaneous rupture of several elements occur, with a distribution D(Delta) of the magnitude Delta of such avalanches. The distribution is typically a power law D(Delta) proportional to Delta (-xi). For the systems we study here, a crossover behavior is seen between two power laws, with a small exponent xi in the vicinity of complete breakdown and a larger exponent xi for failures away from the breakdown point. We demonstrate this analytically for bundles of many fibers where the load is uniformly distributed among the surviving fibers. In this case xi=3/2 near the breakdown point and xi=5/2 away from it. The latter is known to be the generic behavior. This crossover is a signal of imminent catastrophic failure of the material. Near the breakdown point, avalanche statistics show nontrivial finite size scaling. We observe similar crossover behavior in a network of electric fuses, and find xi=2 near the catastrophic failure and xi=3 away from it. For this fuse model power dissipation avalanches show a similar crossover near breakdown.

Phase transitions in a solution of rodlike macromolecules of two different sizes

Phase equilibria for a solution of long rigid rods of diameters Di and lengths Li (i = 1,2) are studied analytically using Gaussian trial functions. The conditions on the molecular size parameters for a nematic—nematic phase transition to be possible are determined, and found to be satisfied everywhere except in a closed domain in the diameter ratio—length ratio plane. Information on the relative ordering of the two components is also provided.

Rupture Processes in Fibre Bundle Models

Fibre bundles with statistically distributed thresholds for breakdown of individual fibres are interesting models of the statics and dynamics of failures in materials under stress. They can be analyzed to an extent that is not possible for more complex materials. During the rupture process in a fibre bundle avalanches, in which several fibres fail simultaneously, occur. We study by analytic and numerical methods the statistics of such avalanches, and the breakdown process for several models of fibre bundles. The models differ primarily in the way the extra stress caused by a fibre failure is redistributed among the surviving fibres. When a rupture occurs somewhere in an elastic medium, the stress elsewhere is increased. This may in turn trigger further ruptures, which can cascade to a final complete breakdown of the material. To describe or model such breakdown processes in detail for a real material is difficult, due to the complex interplay of failures and stress redistributions. Few analytic results are available, so computer simulations is the main tool (see [1, 2, 3] for reviews). Fibre bundle models, on the other hand, are characterized by simple geometry and clear-cut rules for how the stress caused by a failed element is redistributed on the intact fibres. The attraction and interest of these models lies in the possibility of obtaining exact results, thereby providing inspiration and reference systems for studies of more complicated materials. In this review we survey theoretical and numerical results for several models of bundles of N elastic and parallel fibres, clamped at both ends, with statistically distributed thresholds for breakdown of individual fibres (Fig. 1). The individual thresholds xi are assumed to be independent random variables with the same cumulative distribution function P (x) and a corresponding density function p(x):

Energy bursts in fiber bundle models of composite materials.

A bundle of many fibers with stochastically distributed breaking thresholds for the individual fibers is considered as a model of composite materials. The bundle is loaded until complete failure, to capture the failure scenario of composite materials under external load. The fibers are assumed to share the load equally, and to obey Hookean elasticity right up to the breaking point. We determine the distribution of bursts in which an amount of energy E is released. The energy distribution follows asymptotically a universal power law E(-5/2) , for any statistical distribution of fiber strengths. A similar power law dependence is found in some experimental acoustic emission studies of loaded composite materials.

Exact treatment of mode locking for a piecewise linear map

A piecewise linear map with one discontinuity is studied by analytic means in the two-dimensional parameter space. When the slope of the map is less than unity, periodic orbits are present, and we give the precise symbolic dynamic classification of these. The localization of the periodic domains in parameter space is given by closed expressions. The winding number forms a devils terrace, a two-dimensional function whose cross sections are complete devilss staircases. In such a cross section the complementary set to the periodic intervals is a Cantor set with dimensionD=0.