Prediction of thermal loads on nuclear reactor vessel lower plenum after core melting and relocation during a severe accident requires knowledge about the core melt behavior, especially the circulation pattern. To analyze the heat transfer dynamics on the lower plenum walls, two-dimensional numerical simulations of a fluid flow with internal heat generation were performed for Rayleigh numbers 10 6 , 10 7 , 10 8 , 10 9 , 10 11 and 10 13 at Prandtl number 0.8. For subgrid motion modeling, a Large-Eddy Simulation Smagorinsky model was implemented. The minimum, time-average and maximum Nusselt numbers on the boundaries were calculated. The dynamics of fluid structures were analyzed to reveal the instability mechanisms and transition to turbulence. Results disclose Rayleigh-Taylor instabilities as a dominant mechanism for turbulence appearance, which occurs when the Rayleigh number is increased over 10 8 . The structure dependence of fluid motion at high Rayleigh numbers makes the time-average of heat transfer hard to assess. The time-average values should be supplemented with probability distributions of related variables.

Aluminum alloys reinforced with B 4 C particle of average particle size of 10.5 μm and 20.9 μm were exposed to 535°C for various duration. The material degradation was evaluated by both mechanical and acoustic emission monitoring during tensile test. The micromechanisms of fracture changed from particle fracture and debonding, for as received specimens, to particle/matrix debonding after exposure to high temperature. The cumulative acoustic emission (AE) events during post heat exposure reduced with the increase of heal damage on the specimen. A Weibull probability distribution function that incorporates heat damage and relates to the cumulative events is developed. This model could be used to evaluate the extent of heat damage to a composite specimen.

Goodman Diagram method or similar methods are used to estimate safety of a mechanical structure under high cycle fatigue loading for any combination of alternating and mean stresses. Magnitude of the factor of safety (FS) indicates margin from nominal design capable of desired performance. The value of larger than one of FS is desired to account for uncertainty and variability in loads and material properties. This FS based on stress does not provide any direct knowledge about the life of the mechanical structure. A FS based on life can be derived and used in conjunction with Goodman concept. This method yields an estimate of FS based on life (FN) for a given stress based FS for any combination of alternating and mean stresses. A procedure is described in this paper that helps in estimating reliability of a mechanical structure. Reliability depends on the magnitude of stresses and material properties. Usually variability in load and in material properties can be quantified by a statistical distribution. Methods of probabilistic theories can be used to determine the influence of these variations on the reliability. The procedure utilizes established methods and theories to yield practical evaluation of reliability. First, the modified Goodman equation of factor of safety is combined with the life equation proposed by Jo Dean Morrow (Dowling, 1999). This provides a relationship between calculated factor of safeties based on stress and life. Finally, the developed equations are utilized in a probabilistic approach that incorporates statistical distribution of uncertainties. This procedure yields reliability assessment of a mechanical structure to perform an expected task.

In this study, a technique to predict the equivalent initial crack size (EICS) in a particulate composite material, containing hard particles embedded in a rubber matrix, was developed using constant strain rate crack propagation test data. The accuracy of the developed technique was determined. In addition, the statistical distribution of the equivalent initial crack size follows the second asymptotic distribution of maximum value.

Evolution of distributed damage in heterogeneous solids is modeled using the Transformation Field Analysis method [ Proc. R. Soc. Lond. A (1992) 437, 311–327] and selected models of interface debonding in fibrous or particulate composites, as described in detail in the forthcoming paper [ J. Mech. Phys. Solids Boehler Memorial Volume, 2001]. In this approach, stress changes caused by local debonding under increasing overall loads are represented by residual stresses generated by damage-equivalent eigenstrains that act together with the applied mechanical loading program and physically based local transformation strains on an undamaged elastic aggregate. Damage rates are derived from a prescribed probability distribution of interface strength and local energy released by debonding. Numerical simulations of damage evolution in a glass/elastomer composite indicate which of these two conditions controls the process at different reinforcement densities and overall stress states. In general, the energy released by a single particle at given overall stress decreases with increasing reinforcement density, and in proportion to particle size. Therefore, dense reinforcement by smaller-diameter particles or fibers should enhance damage resistance of composite systems.

A novel algorithm was developed to facilitate segmentation of the articular cartilage of the knee from magnetic resonance (MR) image data. The new approach uses a model of the bone and cartilage elements of the knee as a template. The operator initializes the template by aligning certain bony anatomic landmarks. The inner cartilage surface (ICS) is determined through a standard seed growing segmentation approach; the bone elements of the template are then fit to the ICS through an elastic registration technique. The deformation fields thus obtained are subsequently applied to the appropriate cartilage templates. The deformed templates and ICS normals, in conjunction with a gaussian probability distribution function and the subject image data, are used to produce the final estimate of the cartilage surfaces.

In this paper, a Bayesian Belief Network (BBN) approach to the modeling and diagnosis of xerographic printing systems is proposed. First, a continuous BBN model based on physics of the printing process and field data is developed. The model captures the causal relationships between the various physical variables in the system using conditional probability distributions. Next, the continuous BBN is discretized based on the principle of maximum entropy so that it can be implemented on commercially available software, Hugin. The resulting BBN can be used for the prediction of print quality behaviors, as well as for inference and fault diagnosis. Examples of network deduction and inference are presented to illustrate the usefulness of the BBN model.

We propose a framework to develop turbulence models based on single-point closure through a probability distribution function. The evolution equation for the characteristic function is formed and the possible way of constructing specific models is addressed.

The existing constitutive models of composite were usually built on the relation between the phase-average stress and strain, as treated in monolithic materials. In the present work, however, it was found that the averaged characteristic of the phase material, in which the plastic damage occurs, is quite different from that of the monolithic material. And the morphology of the non-uniform local properties remarkably gives effects on the global property of the composite. In order to model the local phenomenon in composite, a statistical distribution function was proposed to describe the local stress and the local plasticity. Furthermore by applying Localized Rigidity Method (LRM), which was proposed by the authors, a mixture law to estimate the percentage of the local plastic damage regions is developed. A numerical calculation was performed to investigate the nonlinear phenomena of the composite with the meso-localized plasticity, and the high performance of the present model was verified.

The development of stochastic models for representing the uncertainties present in a problem is a very important component of probabilistic reliability analysis. Stochastic modeling of engineering parameters require the determination of appropriate probability distribution functions and their associated statistical parameters. In cases that of practical interest, there are usually two or more random variables or processes involved, and hence it is further necessary to establish the correlations between the uncertain parameters or functions. The choice of stochastic models often influences the reliability values as well as the probabilistic sensitivity factors computed from analyses. This is especially so for probabilistic fracture mechanics and fatigue reliability calculations which constitute the cornerstone for durability and damage tolerance assessments. The present paper reports the development of a methodology for probabilistic fatigue data characterization. The characterization utilizes raw fatigue damage (crack initiation or crack growth) data to determine the most appropriate probability distributions, statistical parameters and confidence bounds for the modeling parameters that are involved in fatigue damage initiation and propagation relations used in durability and damage tolerance prediction. Correlations between different parameters are also computed on the basis of well established deterministic models and standards (such as the ASTM E647-93 standard) that have been widely accepted in the industry. Statistical tests are implemented on the basis of Kolmogorov-Smirnov measures of model acceptability. The probabilistic models are determined using both the method of moments (MOM) and maximum likelihood estimators (MLE) and are packaged within the framework of a user-friendly computational tool (PRADAC). Illustrative practical examples are presented to demonstrate the utility of this probabilistic pre-processing computational tool and the important effects on fatigue reliability prediction of aerospace structural components.

This paper presents the results of a combined experimental and analytical study of the probabilistic nature of fatigue crack growth in Ti-6Al-4V. A simple experimental fracture mechanics is presented for obtaining statistical fatigue crack growth parameters from one or two fatigue tests. The experimental studies of long fatigue crack growth show that the variabilities in the long fatigue crack growth rate and the Paris coefficient are well described by the log-normal distribution. The variabilities in the Paris exponent are also shown to be well characterized by a Weibull distribution. The measured statistical distributions are incorporated into a probabilistic fracture mechanics framework for the estimation of material reliability. The implications of the results are also discussed for the probabilistic analysis of fatigue crack growth in engineering components and structures.

Composite durability under conditions where the material survives the initial application of load but then deteriorates with time is a major issue in most engineering applications of composites. The degradation can be caused by the propagation of many damage modes: delamination, matrix cracking, interface degradation, creep, and fiber degradation, among others. In many situations, ultimate and total failure is associated with failure of the fiber bundle supporting the dominant portion of the applied load. Here, the lifetime of a composite due to fiber degradation is investigated for two important modes of fiber degradation: fatigue crack growth under cyclic loading and slow crack growth under stress rupture conditions of constant load. In both cases, individual fiber failure is caused by the growth of pre-existing cracks to a critical size via fatigue or rupture, which is modeled here by a Paris law. A simulation model is used to determine the time-dependent stresses on a bundle of uniaxial brittle fibers, each containing an initial distribution of cracks corresponding to a Weibull strength distribution. As individual fibers fail, their stresses are transferred to nearby fibers, which increases the rate of degradation of the nearby fibers. The damage evolution thus accelerates locally, culminating in the very rapid growth of damage across the entire specimen at the failure time. The average failure time for a set of nominally identical specimens is primarily a function of the fiber Weibull modulus, the Paris law exponent in the fatigue or rupture model, the initial fiber strength, and the rate coefficient of the Paris law. Guidance for the major dependencies of the failure time is obtained by considering an analogous “Global Load Sharing” (GLS) model in which broken fibers transfer load equally to all remaining fibers in the same cross-section. The statistical distribution of failure times at fixed composite size and the size scaling of the mean failure time with increasing size, neither of which can be obtained from the GLS model, are presented and discussed.

A micron-resolution particle image velocimetry (PIV) system has been developed to spatially and temporally resolve electroosmotic flow fields within microfluidic bioanalytical devices. A second diagnostic technique, particle tracking velocimetry (PTV) has been used to determine the distribution of electrophoretic mobilities of seed particles and thereby make the PIV measurements quantitative. This second particle tracking technique has been used to determine probability distribution functions of the seed particles. Results from simulations of electric fields yield local electric field strengths in the geometries of interest. The measured mean mobility of the seed particles (obtained from PTV measurements) is then multiplied by the local electric field vector to obtain the electrophoretic velocity. The variance on the particle mobility measurement influences the errors introduced in the electroosmotic flow measurements. After total particle velocities are measured within a microfluidic system of interest, the seed particle electrophoretic velocities are subtracted from the PIV total velocity data to obtain electroosmotic flow field velocities. Ensemble-averaged velocity field measurements for electroosmotic flow at the intersection of a cross-channel are presented.

Material parameters related to deterministic models can have different values due to variation of experiments outcome. From a mathematical point of view, probabilistic modeling can improve this problem. It means that material parameters of constitutive models can be characterized as random variables with a probability distribution. To this end, we propose a constitutive models of rubber-like materials based on uncertainty quantification (UQ) approach. UQ reduces uncertainties in both computational and real-world applications. Constitutive models in elastomers play a crucial role in both science and industry due to their unique hyper-elastic behavior under different loading conditions (uni-axial extension, biaxial, or pure shear). Here our goal is to model the uncertainty in constitutive models of elastomers, and accordingly, identify sensitive parameters that we highly contribute to model uncertainty and error. Modern UQ models can be implemented to use the physics of the problem compared to black-box machine learning approaches that uses data only. In this research, we propagate uncertainty through the model, characterize sensitivity of material behavior to show the importance of each parameter for uncertainty reduction. To this end, we utilized Bayesian rules to develop a model considering uncertainty in the mechanical response of elastomers. As an important assumption, we believe that our measurements are around the model prediction, but it is contaminated by Gaussian noise. We can make the noise by maximizing the posterior. The uni-axial extension experimental data set is used to calibrate the model and propagate uncertainty in this research.

Fatigue is one of the most frequently encountered failure modes of rotary shouldered connections (RSC) used in drill strings. Once initiated, a fatigue crack tends to grow and ultimately lead to a twist-off, which is catastrophic and often results in lengthy non-producing time and expensive fishing operations. The complexity of the fatigue mechanism, the variabilities of material properties, and the nonlinear contact interactions of the pin and the box elements of an RSC pose a substantial challenge to accurately predicting the fatigue life of the RSC. This would require considerable conservatism to be exercised to prevent a twist-off, which causes premature retirement of drilling assets. Using a statistical approach to predict the risk of twist-off (ROTO) of each RSC on the drill string could be a more economically viable solution as it would enable quantified risk assessment and scientifically calculated tradeoffs between performance, cost, and risk of failures. In this study, a methodology for statistical prediction of the ROTO of rotary shouldered threaded connections was developed. First, static material properties, including yield strength, tensile strength, elongation, and reduction in area, were extracted from a wealth of available material certificates. Feature engineering was carried out to arrive at two independent properties, tensile strength and reduction in area. Fatigue properties were then generated with the retrieved static material data and earlier established correlations between static and fatigue properties. Afterwards, elasto-plastic finite element analyses were performed on RSCs made of the same material but with different properties to determine critical fatigue indicators, stress and strain states as respective functions of the tensile strength. Finally, Monte-Carlo simulations were conducted with respect to statistical distributions of the two independent material variables to predict the ROTO as a function of fatigue life. The predictions were found to be favorable agreement with the available full-scale fatigue test data of an API connection type.

In reliability-based mechanical design, reliability replaces the traditional factor of safety as the measurement index of the safety of mechanical components. More than 90% of metal components under cyclic fatigue loadings in industries fail because of fatigue. The P-S-N curve fatigue theory (Probability - Stress level - Number of cycles) is one of the current important fatigue theories. It is very important to know how to determine the reliability of components under different loading-induced cyclic stresses for reliability-based mechanical design. The Monte Carlo method is a powerful numerical simulation in almost every field such as optimization, numerical integration, and generating draws from a probability distribution. Literature reviews show the Monte Carlo method is successfully implemented to estimate the reliability of components under single loading-induced cyclic stress. However, there is little literature about implementing the Monte Carlo method to estimate the reliability of components under multiple loading-induced cyclic stress by using the P-S-N curve fatigue theory. The purpose of this paper is to develop a new Monte Carlo computational algorithm to calculate the reliability of components under several cyclic loadings using the P-S-N curve fatigue theory. Two key concepts in the widely-accepted Miner rule in fatigue theory are that fatigue damage is linear cumulative and the fatigue damage because of different cyclic stress is independent. Based on these two key concepts, this paper has successfully developed a new Monte Carlo computational algorithm to calculate the reliability of components under multiple loading-induced cyclic stresses using the P-S-N curve fatigue theory. The results obtained by the developed computational algorithm is validated by results obtained from two published methods. The results by the developed computational algorithm is again validated by the K-D probabilistic model. Based on validation studies, the relative differences in the results between the proposed method and the published methods are in the range of 0.66% to 2.98%. Therefore, the developed Monte Carlo computational algorithm is validated and can provide an acceptable estimation of the reliability of components under several cyclic fatigue loadings using the P-S-N curve fatigue theory.

For the development of design rules for nuclear piping using the Load and Resistance Factor Design (LRFD) method, the probabilistic properties of steel, namely, the mean value, bias, coefficient of variation, and probability distribution are needed. The paper presents background information for the existing material tables in the ASME Boiler and Pressure Vessel Code, Section II. Then it investigates the probabilistic properties for the most representative materials used for nuclear piping such as a carbon, stainless austenitic, and low alloy steels. Properties up to temperature 700°F are examined through a review of studies for the mechanical behavior of these materials. The paper discusses approaches for grouping materials in broader categories than the consideration of each type of steel separately. The impact of the steel probabilistic properties on the development of LRFD equations and the associated target reliability index is provided.

Ceramic matrix composites (CMCs) are a promising subclass of composite materials suitable for high temperature applications. CMCs exhibit multiple damage mechanisms such as matrix cracking, interphase debonding, fiber sliding, fiber pullout, delaminations etc. Additionally, process induced defects such as matrix porosity exists at multiple length scales and has a considerable influence on the mechanical and failure behavior of CMCs. In the current work, the effect of intra-tow porosity, which exist at the micro-scale, on the mechanical behavior of CMCs has been investigated by numerical homogenization. Micro-scale response of 3 phase CMCs with intra tow pores has been obtained by finite element analysis based homogenization. Pores have been modeled as non-intersecting ellipsoids in a square unit cell representative of matrix material. The effective mechanical properties of porous matrix at the micro scale has been obtained from numerical homogenization, which are in good agreement with Mori-Tanaka mean field theory. The obtained matrix elastic properties have then been included in a three phase unit cell consisting of fiber, interphase and matrix representative of CMC microstructure. The effect of porosity volume fraction and aspect ratio on the effective elastic properties of the composite have been reported. Homogenization approach to model statistical distribution of pore size obtained from X-ray computed tomography of CMC minicomposite has been proposed.

To accurately simulate fracture, it is necessary to account for small-scale randomness in the properties of a material. Apparent properties of Statistical Volume Elements (SVE), can be characterized below the scale of a Representative Volume Element (RVE). Apparent properties cannot be defined uniquely for an SVE, in the manner that unique effective properties can be defined for an RVE. Both constitutive behavior and material strength properties in SVE must be statistically characterized. The geometrical partitioning method can be critically important in affecting the probability distributions of mesoscale material property parameters. Here, a Voronoi tessellation based partitioning scheme is applied to generate SVE. Resulting material property distributions are compared with those from SVE generated by square partitioning. The proportional limit stress of the SVE is used to approximate SVE strength. Superposition of elastic results is used to obtain failure strength distributions from boundary conditions at variable angles of loading.

The suspension spring compression introduce local deformation in the suspension shock tower, and it brings the imprecision or uncertainty of the computer aided tolerancing (CAT) results of the suspension assembly. This work presents an integrated tolerancing and motion error analysis method of suspension assembly to consider the local deformation in the suspension shock tower. Tolerancing model of the suspension assembly is constructed in CAT software. FEA model provides the local deformation in the suspension shock tower introduced by the suspension spring compression under full loaded condition. The local deformation is represented with the modified probability distribution function (PDF) of the joining point between the suspension and the suspension shock tower. The modified PDFs of the joining points, together with the manufacturing deviations of the suspension parts are input to the constructed tolerancing model. The results are obtained that the final assembly variations of the suspension assembly under empty and full loaded conditions. Based on the tolerancing results, motion errors of the suspension from empty condition to full loaded condition are analyzed with the ADAMS model of the suspension. The results have shown that tolerancing and motion error analysis results considering the local deformation in suspension shock tower are more accurate than the initial normal distribution. The tolerancing and motion error analysis work presented in the paper will enhance the understanding of the suspension assembling, and help systematically improving the precision control efficiency in automobile industry.