Analytical results are discussed for the chain-of-bundles probability model for the strength of fibrous materials. Two load sharing rules are considered for failed and nonfailed fibers in a bundle. The first is the equal load sharing rule of classical analysis, and the second is a local load sharing rule which is more realistic for composite materials. A rather detailed discussion of past statistical analysis is given. From a careful study of previous results, several conjectures and key questions about the behavior of the strength are generated. Also, an exact analysis of failure is per formed so that the properties of the strength distribution can be studied. Difficulties of a general analysis are discussed in detail. The sequel will contain a thorough numerical investigation of the model with emphasis on studying the convergence of certain transformed distributions and on answering key questions raised in this study.

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Fiber bundle models, where fibers have random lifetimes depending on their load histories, are useful tools in explaining time-dependent failure in heterogeneous materials. Such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings. Various asymptotic and approximate theories have been developed for bundles with various geometries and fiber load-sharing mechanisms, but numerical verification has been hampered by severe computational demands in larger bundles. To gain insight at large size scales, interest has returned to idealized fiber bundle models in 1D. Such simplified models typically assume either equal load sharing (ELS) among survivors, or local load sharing (LLS) where a failed fiber redistributes its load onto its two nearest flanking survivors. Such models can often be solved exactly or asymptotically in increasing bundle size, N, yet still capture the essence of failure in real materials. The present work focuses on 1D bundles under LLS. As in previous works, a fiber has failure rate following a power law in its load level with breakdown exponent rho. Surviving fibers under fixed loads have remaining lifetimes that are independent and exponentially distributed. We develop both new asymptotic theories and new computational algorithms that greatly increase the bundle sizes that can be treated in large replications (e.g., one million fibers in thousands of realizations). In particular we develop an algorithm that adapts several concepts and methods that are well-known among computer scientists, but relatively unknown among physicists, to dramatically increase the computational speed with no attendant loss of accuracy. We consider various regimes of rho that yield drastically different behavior as N increases. For 1/2< or =rho< or =1, ELS and LLS have remarkably similar behavior (they have identical lifetime distributions at rho=1) with approximate Gaussian bundle lifetime statistics and a finite limiting mean. For rho>1 this Gaussian behavior also applies to ELS, whereas LLS behavior diverges sharply showing brittle, weakest volume behavior in terms of characteristic elements derived from critical cluster formation. For 0

A thorough numerical investigation is conducted for the chain-of- bundles probability model. The emphasis is on studying the convergence of certain transformed distributions. Strong numerical evidence of con vergence is presented in support of a weakest link transformation for scal ing to large fibrous materials. Also, exact results are graphed for small bundles. The numerical results are discussed in light of the key questions and conjectures raised in the preceding paper of this series. For practical purposes the Weibull distribution may be used to represent the strength distribution of fibrous materials. The associated shape and scale parameter values for the fibrous material agree extremely well with those that are experimentally observed, as is shown in a specific example. The variability in strength is substantially reduced in passing from a single fiber to the fibrous material, but for customary specimen sizes, the median strength of the fibrous material is somewhat less than that of the fiber.

The existence of anomalous scaling exponents very near the surface should not influence the present discussion: see, for example: de Gennes, P.-G.; Pincus, P. J. Phys. Lett. 1984,45, L 953 and also Binder, K.; Kremer, K. NATO Adu. Study Znst.ABSTRACT: Craze fibril stability of polymer glasses can be characterized by measuring the median strain (craze fibril stability) at the onset of craze fibril breakdown in thin films under a constant low strain rate.Monodisperse polystyrenes (PS's) of molecular weight M = 37000-20000000 were used. A strong increase in craze fibril stability was found as M increased from 50000 to 200000. The increase occurred over the same range as the increase in macroscopic fracture toughness of PS. Voids were observed to always nucleate at the bulk-craze interface and never in the craze midrib. Higher strain rates reduced the craze fibril stability. Foreign particle inclusions, e.g., dust, in specimens can significantly decrease the fibril stability from its intrinsic (clean) value. Even in the presence of such particlea, however, the M dependence of fibril stability is qualitatively similar to that of crazes in "dust-free" films. The fibril breakdown statistica can be investigated by examining simultaneously the failure events in a large number of independent film specimens. The statistics of craze fibril breakdown are found to follow a Weibull distribution. Two parameters may be extracted by fitting this distribution to the breakdown data: (1) a Weibull scale parameter CW, which is a measure of craze fibril stability, and (2) a Weibull modulus p, which is a measure of variability, that is, the breadth of the distribution of fibril stability (high p's produce narrow distributions and vice versa). The Weibull distribution in a weakest-link setting can be used to predict the effect of sample size on the probability of craze fibril breakdown somewhere in the sample. A microscopic statistical model in which craze fibrils fail by random disentanglement of molecular strands at the craze-bulk interface is developed. The experimental observations are in good agreement with the predictions of the model.