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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Direct Breakthrough Curve Prediction From Statistics of Heterogeneous Conductivity Fields

In this paper, we present a methodology to predict the shape of solute breakthrough curves in heterogeneous aquifers at early times and/or under high degrees of heterogeneity, both cases in which the classical macrodispersion theory may not be applicable. The methodology relies on the observation that breakthrough curves in heterogeneous media are generally well described by lognormal distributions, and mean breakthrough times can be predicted analytically. The log-variance of solute arrival is thus sufficient to completely specify the breakthrough curves, and this is calibrated as a function of aquifer heterogeneity and dimensionless distance from a source plane by means of Monte Carlo analysis and statistical regression. Using the ensemble of simulated groundwater flow and solute transport realizations employed to calibrate the predictive regression, reliability estimates for the prediction are also developed. Additional theoretical contributions include heuristics for the time until an effective macrodispersion coefficient becomes applicable, and also an expression for its magnitude that applies in highly heterogeneous systems. Finally, it is seen that the results here represent a way to derive continuous time random walk transition distributions from physical considerations rather than from empirical field calibration.

58 GEOSCIENCES↗
DOE OSTI2018

Bacterial diversity is positively correlated with soil heterogeneity

Plant and animal diversity generally increases with increasing environmental heterogeneity. Here, we test whether this relationship also holds for bacterial communities in soil. Specifically, we investi-gate whether invasive annual grasslands have reduced soil heterogeneity and, thereby, decreased bacterial alpha- and beta-diversity. Soils were sampled at nine sites within a 5-km stretch of Southern California, at five depths in three habitats, including non-native invasive annual grassland, native oak woodland, and native coastal sage scrub. We characterized soil heterogeneity as well as bacterial alpha- and beta-diversity by 16S ribosomal RNA sequencing. We found that invasive annual grasslands harbored less soil heterogeneity and reduced bacterial alpha-diversity relative to the two native woody habitats. Further, across all habi-tats and depths, bacterial alpha- and beta-diversity was positively related to soil heterogeneity. These results suggest that plant invasions associated with soil homogenization may lead to reduced microbial diversity.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Leveraging the digital thread for physics-based prediction of microstructure heterogeneity in additively manufactured parts

A major limitation of additive manufacturing (AM) processes is that local conditions of material deposition frequently lead to unintentional heterogeneities in microstructure and properties within a single component, despite nominally uniform process conditions. Up to now, there has been no way to a priori determine the distribution of these heterogeneities, requiring expensive trial-and-error approaches to fabrication, testing, and characterization. Here, a physics-based framework for creating a digital representation of the laser powder bed fusion (PBF) process is proposed to predict the variation in solidification behavior that leads to heterogeneous microstructures in an as-built part. By leveraging in situ process data stored in the part’s digital thread, the scan path and process parameters were input into a heat transfer model which predicted solidification data at the melt pool scale. A two-step unsupervised clustering algorithm was used to first cluster the local solidification conditions (12.5µm 3 voxels) and then to cluster the regional behavior on the scale of multiple scan passes and print layers (250µm 3 super-voxels). This process was used to identify regions with similar solidification characteristics for multiple locations in a Stainless Steel 316-L component. The corresponding as-built part was sectioned and characterized using electron backscatter diffraction (EBSD). Quantitative analysis of the pole figures confirmed that the predicted regions of heterogeneity in the solidification conditions corresponded with differences in the observed microstructure. In conclusion, this work shows a viable path for estimating the microstructural heterogeneity for additively manufactured parts to either limit microstructural variation throughout a part or to enable functionality-based variation of the microstructure.

36 MATERIALS SCIENCE↗
DOE OSTI2017

Combining eddy-covariance and chamber measurements to determine the methane budget from a small, heterogeneous urban floodplain wetland park

Methane (CH 4 ) emissions and carbon uptake in temperate freshwater wetlands act in opposing directions in the context of global radiative forcing. Large uncertainties exist for the rates of CH 4 emissions making it difficult to determine the extent that CH 4 emissions counteract the carbon sequestration of wetlands. Urban temperate wetlands are typically small and feature highly heterogeneous land cover, posing an additional challenge to determining their CH 4 budget. The data analysis approach we introduce here combines two different CH 4 flux measurement techniques to overcome scale and heterogeneity problems and determine the overall CH 4 budget of a small, heterogeneous, urban wetland landscape. Temporally intermittent point measurements from non-steady-state chambers provided information about patch-level heterogeneity of fluxes, while continuous, high temporal resolution flux measurements using the eddy-covariance (EC) technique provided information about the temporal dynamics of the fluxes. Patch-level scaling parameterization was developed from the chamber data to scale eddy covariance data to a ‘fixed-frame’, which corrects for variability in the spatial coverage of the eddy covariance observation footprint at any single point in time. Finally, by combining two measurement techniques at different scales, we addressed shortcomings of both techniques with respect to heterogeneous wetland sites.

03 NATURAL GAS↗
DOE OSTI2019

On the importance of phase saturation heterogeneity in the analysis of laboratory studies of hydrate dissociation

Methane hydrates (MHs) have been considered as the future source of energy because of its vast resource volume and high energy density. Energy recovery from MH-bearing sediments has attracted intensifying research activities. Fundamentally, heat transfer, fluid flow through porous media, and the kinetics of hydrate reaction are the three key processes controlling the behavior of MH dissociation and the associated fluid production. Earlier studies have suggested that heterogeneous spatial distribution of SH is inevitable in MH-bearing samples synthesized in laboratory. In this paper, we extend our study to analyze numerically the simulation results from the two realizations of the samples (homogeneous and heterogeneous) to identify differences in the fluid production and to determine if they are sufficiently different. Additionally, we conduct a sensitivity analysis and a statistical analysis on the key transport and kinetic rate parameters that could affect hydrate dissociation and fluid production in the context of a heterogeneous hydrate-bearing sample, in an effort to provide insights that could lead to improved designs for laboratory experiments and (possibly) field applications. Our results suggest that the approximation of an artificial hydrate-bearing core with heterogeneous phase saturations by an assumption of uniform phase saturation distributions results in practically similar fluid production profile except for the very early stage with maximum 20.0% deviation in the water production. From the sensitivity and statistical analysis, we determine that gas production depends strongly on the kinetic rate constant, K d 0 and the composite thermal conductivity of the hydrate-bearing sediments, λ θ ; while, water production is very sensitive to K d 0 and the absolute permeability of the sandy medium, k. Understanding the effect of phase heterogeneity and the relative importance of key parameters on the production behavior of hydrate-bearing sediments could provide basis for novel production technologies that lead to enhanced gas production and energy efficiency in the energy recovery process.

42 ENGINEERING↗
DOE OSTI2023

Color as a tool for quantitative analysis of heterogeneous polymer degradation

Polymers are widely used industry, but degrade due to environmental conditions, such as elevated temperature or radiation. The degradation of polymers is dominated oxidation in oxygenated environments. Such oxidation is commonly referred to as thermal-oxidation or photo-oxidation and in many situations is heterogeneous through thickness of the polymer. While heterogeneous oxidation of polymers is a common phenomenon, the effect on the lifetime of polymers is not well understood as methods to evaluate such oxidation are time consuming and challenging to implement. Here we show a novel method for quantitative evaluation of heterogeneous oxidation in polymers using a fast and accurate color approach. The proposed color approach was observed to improve the efficiency of obtaining heterogenous oxidation profiles in half the time of common approaches, such as indention or Fourier transform infrared spectroscopy. Furthermore, the method was found to produce trends similar to that observed using common approaches and the applicability of the method was verified by mapping oxygen concentration through thickness using energy dispersive X-ray spectroscopy. Lastly, the utility of the proposed method was demonstrated through the generation of an oxidation design space, which is envisioned to assist with designing experiments towards estimation of polymer lifetime. Finally, we anticipate this work to be a starting point for further evaluation of heterogeneous oxidation in polymers using optical properties.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Surface Functional Groups Affect Iron (Hydr)oxide Heterogeneous Nucleation: Implications for Membrane Scaling

Because of its favorable thermodynamics and fast kinetics, heterogeneous solid nucleation on membranes triggers early-stage mineral scaling. Iron (hydr)oxide, a typical membrane scale, initially forms as nanoparticles that interact with surface functional groups on membranes, but these nanoscale phenomena are difficult to observe in real time. In this study, we utilized in situ grazing incidence small angle X-ray scattering and ex situ atomic force microscopy to examine the heterogeneous nucleation of iron (hydr)oxide on surface functional groups commonly used in membranes, including hydroxyl (OH), carboxyl (COOH), and fluoro (F) groups. We found that, compared to nucleation on hydrophilic OH- and COOH-surfaces, the high hydrophobicity of an F-modified surface significantly reduced the extents of both heterogeneously and homogeneously formed iron (hydr)oxide nucleation. Moreover, on the OH-surface, the high functional group density of 0.76 nmol/cm 2 caused faster heterogeneous nucleation than that on a COOH-surface, with a density of 0.28 ± 0.04 nmol/cm 2 . The F-surface also had the highest heterogeneous nucleation energy barrier (26 ± 0.6 kJ/mol), followed by COOH- (23 ±0.8 kJ/mol) and OH- (20 ± 0.9 kJ/mol) surfaces. The kinetic and thermodynamic information provided here will help us better predict the rates and extents of early-stage scaling of iron (hydr)oxide nanoparticles in membrane processes. Finally, this work provides both kinetic and thermodynamic information about iron (hydr)oxide nucleation controlled by membrane-related functional groups (OH, COOH, and F).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Effect of Formation Heterogeneity on CO 2 Dissolution in Subsurface Porous Media

Dissolution trapping is one of the most dominant mechanisms for the secure storage of CO 2 injected in porous subsurface formations saturated with brine. This trapping mechanism is enhanced by convective mixing, which occurs due to the gravitational instability between the different fluid layers in the aquifer. The reservoir permeability also plays a crucial role in the dissolution rate and overall fluid flow dynamics during the density-driven convection in porous media. Herein this study investigates the role of complex heterogeneity, i.e., irregular permeability distribution in CO 2 dissolution, using a novel experimental approach to create medium permeability heterogeneity in Hele–Shaw cells. Complex subsurface transport phenomena such as a preferential dissolution path, CO 2 sweep efficiency, changes in finger morphology, and CO 2 concentration distribution are visualized by creating heterogeneous media. Experimental results showed that reservoir permeability heterogeneity causes significant channeling effects and poor sweep efficiency. A scaling relationship between average finger growth rate (Gr) and permeability (k) was obtained as Gr [m s –1 ] = 266.8k [m 2 ] + 1.20 × 10 –6 . Furthermore, the mass of CO 2 dissolved is calculated using the spectrophotometric method to characterize the convective instability. The convective flux was analyzed by comparing the experimental dissolution flux with the theoretical diffusion flux, calculating a maximum Sherwood number of 6.8. The study’s findings improve the current understanding of the CO 2 convection morphology in heterogeneous media, allowing better assessment of long-term CO 2 storage.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Impacts of Permeability Heterogeneity and Background Flow on Supercritical CO 2 Dissolution in the Deep Subsurface

Motivated by CO 2 capture and sequestration (CCS) design considerations, we consider the coupled effects of permeability heterogeneity and background flow on the dissolution of a supercritical CO 2 lens into an underlying deep, confined aquifer. We present the results of a large-scale Monte Carlo simulation study examining the interaction of background flow rate and three parameters describing multi-Gaussian log-permeability fields: mean, variance, and correlation length. Hundreds of high-resolution simulations were performed using the PFLOTRAN finite volume software to model CO 2 dissolution in a kilometer-scale aquifer over 1,000 years. Predictive dimensionless scaling relationships relating CO 2 dissolution rate to heterogeneity statistics, Rayleigh (Ra) and Péclet (Pe) numbers were developed for both gravitationally dominated free convection to background flow-dominated forced convection regimes. An empirical criterion, Pe = Ra 3/4 , was discovered for regime transition. All simulations converged quickly to a quasi-steady, approximately linear dissolution rate. However, this rate displayed profound variability between permeability field realizations sharing the same heterogeneity statistics, even under mild permeability heterogeneity. In general, increased heterogeneity was associated with a lower mean and higher variance of dissolution rate, undesirable from a CCS design perspective. The relationship between dissolution rate and background flow was found to be complex and nonlinear. Dimensionless scaling relationships were uncovered for a number of special cases. Results call into question the validity of the Boussinesq approximation in the context of modest-to-high background flow rates and the general applicability of numerical simulations without background flow.

42 ENGINEERING↗
DOE OSTI2015

Coarse-Grained Clustering Dynamics of Heterogeneously Coupled Neurons

The formation of oscillating phase clusters in a network of identical Hodgkin–Huxley neurons is studied, along with their dynamic behavior. The neurons are synaptically coupled in an all-to-all manner, yet the synaptic coupling characteristic time is heterogeneous across the connections. In a network of N neurons where this heterogeneity is characterized by a prescribed random variable, the oscillatory single-cluster state can transition—through (possibly perturbed) period-doubling and subsequent bifurcations—to a variety of multiple-cluster states. The clustering dynamic behavior is computationally studied both at the detailed and the coarse-grained levels, and a numerical approach that can enable studying the coarse-grained dynamics in a network of arbitrarily large size is suggested. Among a number of cluster states formed, double clusters, composed of nearly equal sub-network sizes are seen to be stable; interestingly, the heterogeneity parameter in each of the double-cluster components tends to be consistent with the random variable over the entire network: Given a double-cluster state, permuting the dynamical variables of the neurons can lead to a combinatorially large number of different, yet similar “fine” states that appear practically identical at the coarse-grained level. For weak heterogeneity we find that correlations rapidly develop, within each cluster, between the neuron’s “identity” (its own value of the heterogeneity parameter) and its dynamical state. For single- and double-cluster states we demonstrate an effective coarse-graining approach that uses the Polynomial Chaos expansion to succinctly describe the dynamics by these quickly established “identity-state” correlations. This coarse-graining approach is utilized, within the equation-free framework, to perform efficient computations of the neuron ensemble dynamics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Experimental Investigation of Buoyant Flow in Realistic Bedforms With Heterogeneous Wettability

Submeter-scale geologic heterogeneity greatly affects CO 2 plume migration and retention. In this work, we present meter-scale laboratory experiments that can capture the impact of realistic submeter-scale geologic heterogeneity on multiphase flow and trapping. We produce realistic sedimentary formations consisting of ripple deposits with varying grain size contrast and wettability in a meter-scale slab chamber. Then, we conduct multiphase flow experiments with analog fluids through these structures and measure the saturation patterns, capillary heterogeneity trapping (CHT), and overall trapping performance. When we alter the ripple bedform architecture, variations in trapped saturation and CHT (10–20%) increment are exhibited. Similar growth in trapping performance is also observed when grain size contrast increases. Finally, wettability changes (water- to oil-wet) can increase nonwetting saturation and CHT up to 5% and 10–20%, respectively. These results emphasize the importance of correctly characterizing the impact of small-scale heterogeneities and wettability changes. We believe this is the first time that multiphase flow experiments were conducted in meter-scale domains with realistic ripple bedforms and heterogeneous wettability to investigate plume migration and trapping.

58 GEOSCIENCES↗
DOE OSTI2012

Homogenization and Upscaling for Diffusion, Heat Conduction, and Wave Propagation in Heterogeneous Materials

We present a general homogenization method for diffusion, heat conduction, and wave propagation in a periodic heterogeneous material with piecewise constants. The method is relevant to the frequently encountered upscaling issues for heterogeneous materials. The dispersion relation for each problem is first expressed in the general form where the frequency w (or wavenumber k) is expanded in terms of the wavenumber k (or frequency w). A general homogenization model can be directly obtained with any given dispersion relation. Next step we study the unit cell of the heterogeneous material and derive the exact dispersion relation. The final homogenized equations include both leading order terms (effective properties) and high order contributions that represent the effect of the microscopic heterogeneity on the macroscopic behavior. That effect can be lumped into a single dimensionless heterogeneity parameter, which is bounded between -1/12 less than or equal to less than or equal to 0 and has a universal expression for all three problems. Numerical examples validate the proposed method and demonstrate a significant computational saving.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2014

Heterogeneity‐enhanced gas phase formation in shallow aquifers during leakage of CO 2 ‐saturated water from geologic sequestration sites

Abstract A primary concern for geologic carbon storage is the potential for leakage of stored carbon dioxide (CO 2 ) into the shallow subsurface where it could degrade the quality of groundwater and surface water. In order to predict and mitigate the potentially negative impacts of CO 2 leakage, it is important to understand the physical processes that CO 2 will undergo as it moves through naturally heterogeneous porous media formations. Previous studies have shown that heterogeneity can enhance the evolution of gas phase CO 2 in some cases, but the conditions under which this occurs have not yet been quantitatively defined, nor tested through laboratory experiments. This study quantitatively investigates the effects of geologic heterogeneity on the process of gas phase CO 2 evolution in shallow aquifers through an extensive set of experiments conducted in a column that was packed with layers of various test sands. Soil moisture sensors were utilized to observe the formation of gas phase near the porous media interfaces. Results indicate that the conditions under which heterogeneity controls gas phase evolution can be successfully predicted through analysis of simple parameters, including the dissolved CO 2 concentration in the flowing water, the distance between the heterogeneity and the leakage location, and some fundamental properties of the porous media. Results also show that interfaces where a less permeable material overlies a more permeable material affect gas phase evolution more significantly than interfaces with the opposite layering.

Plampin, Michelle R.↗
DOE OSTI2015

Experimental study on effects of geologic heterogeneity in enhancing dissolution trapping of supercritical CO 2

Abstract Dissolution trapping is one of the primary mechanisms that enhance the storage security of supercritical carbon dioxide (scCO 2 ) in saline geologic formations. When scCO 2 dissolves in formation brine produces an aqueous solution that is denser than formation brine, which leads to convective mixing driven by gravitational instabilities. Convective mixing can enhance the dissolution of CO 2 and thus it can contribute to stable trapping of dissolved CO 2 . However, in the presence of geologic heterogeneities, diffusive mixing may also contribute to dissolution trapping. The effects of heterogeneity on mixing and its contribution to stable trapping are not well understood. The goal of this experimental study is to investigate the effects of geologic heterogeneity on mixing and stable trapping of dissolved CO 2 . Homogeneous and heterogeneous media experiments were conducted in a two‐dimensional test tank with various packing configurations using surrogates for scCO 2 (water) and brine (propylene glycol) under ambient pressure and temperature conditions. The results show that the density‐driven flow in heterogeneous formations may not always cause significant convective mixing especially in layered systems containing low‐permeability zones. In homogeneous formations, density‐driven fingering enhances both storage in the deeper parts of the formation and contact between the host rock and dissolved CO 2 for the potential mineralization. On the other hand, for layered systems, dissolved CO 2 becomes immobilized in low‐permeability zones with low‐diffusion rates, which reduces the risk of leakage through any fault or fracture. Both cases contribute to the permanence of the dissolved plume in the formation.

Agartan, Elif↗
DOE OSTI2019

Modelling the Fluctuations of Reactive Shock Waves in Heterogeneous Solid Explosives as Stochastic Processes

Although current phenomenological burn models are useful for describing the average or bulk reactive flow behaviour of heterogeneous explosives, one fundamental weakness inherent to these models is the loss of detailed microstructural information at the scale of the calculation. In order to include the effects of the microstructure, and in particular the underlying material heterogeneities that influence the build-up to detonation, a new paradigm is put forth for modelling sub-grid, reaction-induced fluctuations (i.e. “hot spots”) at the continuum level. This modelling approach assumes that the reaction rate is stochastic, rather than deterministic, and it uses Langevin-type equations with a mathematical framework built upon Itô calculus and Lambourn's CIM model for shocked heterogeneous explosives. This approach follows directly from our previous letter, Ref. [1], and is inspired by the probability density function (pdf) methods used in turbulent reactive flows. Here, the stochastic burn model is derived, implemented, and exercised far beyond what has been shown in previous work. New hydrocode simulation results demonstrate the role of stochastic fluctuations during shock initiation; these fluctuations are approximated by collections of discrete particles, that evolve with drift (i.e. deterministic) and diffusion (i.e. stochastic) coefficients. Furthermore, the particle values are propagated and averaged to calculate the heat release, yield strength, and material impedance in each computational cell. Hydrocode simulation results further show how the fluctuating hot spot energy may or may not be transmitted to the wave front, and result in a detonation wave-like structure. The fundamental stochastic nature of this model permits simulations to have varying outcomes with the same initial conditions; this allows for go/no-go estimation (e.g., marginal or failed detonations), which might possibly be calibrated using the statistical distributions from real materials. Mesoscale calculations of shocked heterogeneous explosives also show that these fluctuations are physically justified (i.e., Ref. [2]), and it is hypothesized that the pdf functions provide a link between the meso(grain) and continuum scales for practical engineering calculations. Hence, our approach is a paradigm shift for building efficient, reduced order continuum burn models, that represent the sub-grid stochastic behaviour of shocked heterogeneous solid explosives.

42 ENGINEERING↗
DOE OSTI2022

OpenMP Target Task: Tasking and Target Offloading on Heterogeneous Systems

This work evaluated the use of OpenMP tasking with target GPU offloading as a potential solution for programming productivity and performance on heterogeneous systems. Also, it is proposed a new OpenMP specification to make the implementation of heterogeneous codes simpler by using OpenMP target task, which integrates both OpenMP tasking and target GPU offloading in a single OpenMP pragma. As a test case, the authors used one of the most popular and widely used Basic Linear Algebra Subprogram Level-3 routines: triangular solver (TRSM). To benefit from the heterogeneity of the current high-performance computing systems, the authors propose a different parallelization of the algorithm by using a nonuniform decomposition of the problem. This work used target GPU offloading inside OpenMP tasks to address the heterogeneity found in the hardware. This new approach can outperform the state-of-the-art algorithms, which use a uniform decomposition of the data, on both the CPU-only and hybrid CPU-GPU systems, reaching speedups of up to one order of magnitude. The performance that this approach achieves is faster than the IBM ESSL math library on CPU and competitive relative to a highly optimized heterogeneous CUDA version. One node of Oak Ridge National Laboratory’s supercomputer, Summit, was used for performance analysis.

Valero Lara, Pedro↗
DOE OSTI2024

MatRIS: Addressing the Challenges for Portability and Heterogeneity Using Tasking for Matrix Decomposition (Cholesky)

The ubiquitous in-node heterogeneity of HPC and cloud computing platforms makes software portability and performance optimization extremely challenging. Described here, the MatRIS multilevel math library abstraction framework employs tasking to alleviate these difficulties. MatRIS includes the IRIS task-based runtime on the bottom level and exposes different layers of abstraction to render algorithms architecturally agnostic. MatRIS ensures the decomposition and creation of tasks that represent the necessary encapsulation of the optimized kernels from both vendor and open-source math libraries. Once built, MatRIS can select different combinations of accelerators at runtime, making it portable even on diverse heterogeneous architectures. By leveraging the IRIS runtime’s features for managing heterogeneity, MatRIS deploys algorithms that remove the need to specify orchestration and data transfer. This study describes how the serial task abstraction of a tiled Cholesky factorization is made portable and scalable in the case of multi-device and multi-vendor heterogeneity on a node with NVIDIA and AMD GPUs by using MatRIS. First, we demonstrate that Cholesky in MatRIS provides multi-GPU scalability that offers competitive performance versus cuSolverMG. Then, we present the challenges and opportunities for heterogeneous execution.

Monil, M. A. H.↗
DOE OSTI2018

Fiber Bundle Model Under Heterogeneous Loading

The present work deals with the behavior of fiber bundle model under heterogeneous loading condition. The model is explored both in the mean-field limit as well as with local stress concentration. In the mean field limit, the failure abruptness decreases with increasing order k of heterogeneous loading. In this limit, a brittle to quasi-brittle transition is observed at a particular strength of disorder which changes with k. On the other hand, the model is hardly affected by such heterogeneity in the limit where local stress concentration plays a crucial role. The continuous limit of the heterogeneous loading is also studied and discussed in this paper. Some of the important results related to fiber bundle model are reviewed and their responses to our new scheme of heterogeneous loading are studied in details. Our findings are universal with respect to the nature of the threshold distribution adopted to assign strength to an individual fiber.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗