Weibull Theory Research Articles

Reliability prediction of I&C cable insulation materials by DSC and Weibull theory for probabilistic safety assessment of NPPs

Instrumentation and control (I&C) cables used in nuclear power plants (NPPs) are exposed to various deteriorative environmental effects during their operational lifetime. The factors consisting of long-term irradiation and enhanced temperature eventually result in insulation degradation. Monitoring of the actual state of the cable insulation and the prediction of their residual service life consist of the measurement of the properties that are directly proportional to the functionality of the cables (usually, elongation at break is used as the critical parameter). Although, several condition monitoring (CM) and life estimation techniques are available, currently there is no any standard methodology or an approach towards incorporating the cable ageing effects into probabilistic safety assessment (PSA) of NPPs. In view of this, accelerated thermal and radiation ageing of I&C cable insulation materials have been carried out and the degradation due to thermal and radiation ageing has been assessed using oxidation induction time (OIT) and oxidation induction temperature (OITp) measurements by differential scanning calorimetry (DSC). As elongation at break (EAB) is considered to be a benchmark characterization technique for polymeric materials, tensile tests have also been carried out on these cable materials for correlating with DSC findings. The Fourier transform infrared (FTIR) spectroscopy has also been carried out on fresh and irradiated samples to monitor the structural changes in the insulation materials. The correlations obtained from DSC and FTIR studies are relatively in good agreement with the tensile testing results. The images of scanning electron microscopy (SEM) on fresh and aged samples support the degradation predicted from infrared spectroscopy, thermal and mechanical measurements. The time dependent reliability predicted from the OIT based on Weibull theory will be useful in incorporating the cable ageing into PSA of NPPs.

Fatigue performance of rubber-modified recycled aggregate concrete (RRAC) for pavement

Based on the performance requirements for airport pavement concrete, this study analyzes the mechanical properties, fatigue properties and damage characteristics of rubber-modified recycled aggregate concrete (RRAC) with waste rubber replacement rates of 10%, 20% and 30%. The test results for compressive strength, four-point flexural bending, elasticity modulus and fatigue life on RRAC showed that the compressive strength of recycled aggregate concrete (RAC) increased by 10.1% from that of normal concrete (NC) (rubber-free and recycled-aggregate-free concrete). The compressive strength, flexural strength and elasticity modulus of the RRAC decreased with the increase in rubber particle content. However, the peak deflection, peak strain and ultimate strain increased with the increase of rubber particle content with a certain range. When rubber particle content reached 20% of the sand, the ultimate strain of RRAC was 3.45times that of rubber-free RAC. Based on the Weibull theory, statistical analysis was performed on the test results, and double logarithm fatigue equations for RRAC under different failure probabilities were provided, which could be used to predict the ultimate fatigue strength of RRAC. An analysis of the damage characteristics showed that both recycled aggregate and rubber particle could enhance the concrete’s fatigue life. This enhancement was most significant when the rubber particle content reached 20%.

Flexural properties of fully biodegradable alpha-grass fibers reinforced starch-based thermoplastics

This work has the aim of study the flexural properties of alpha-grass reinforced starch-based composites. The composite materials contain alpha-fibers in the range from 5 to 35 wt%. The reinforcing fibers were submitted to an alkali treatment to create a good interphase between the fibers and the matrix. It was observed that a mild 2.5 h cooking process was enough to create a good interphase, while longer periods rendered lesser improvements. The surface charges of the fibers and the matrix were determined by polyelectrolyte titration, and it was found that after the alkaline treatment both were similar. The composite materials were injection molded and tested under flexural conditions. All the flexural properties of the studies composites increased linearly with the reinforcement contents. The micromechanics of the flexural modulus and strength were studied and compared with that of tensile modulus and strength. It was established that the efficiency factors for the tensile and flexural properties were statistically similar. Three different methods were used to compute the intrinsic flexural strength from the available data. Finally the Weibull theory was used to study the best prediction of the standard deviation of the intrinsic flexural modulus.

The ratio of flexural strength to uniaxial tensile strength in bulk epoxy resin polymeric materials

The flexural behavior of epoxies was investigated by performing mechanical tests and applying statistical Weibull theory and analytical methods to the results. The effects of loading systems and environmental conditions were also considered. Three kinds of epoxies were studied: Epon E863, PRI 2002, and PR520. In total, 53 three-point-bending (3PB) Epon E863 samples and 26 3PB PR520 were tested immediately after curing, together with 26 four-point-bending (4PB) PRI2002 samples stored at 60°C and 90% Rh for 48 weeks. The Weibull parameters were estimated using both linear regression and the moments method. The statistical character of the Weibull model leads to uncertainty in the evaluated parameters, even for a large number of experiments. This study analyzed the ratio of flexural strength to tensile strength in bulk epoxy resin polymers. An analytical method previously developed by the authors to study the relationship between uniaxial tension/compression stress-strain curves and flexural load-deflection response was used to obtain the ratio. The results show that the Weibull model overpredicted the aforementioned ratio in different load arrangements.

Thickness-dependence of the breakdown strength: Analysis of the dielectric and mechanical failure

Abstract The breakdown strength as well as the mechanical strength of ceramic materials decreases with increasing volume. The volume-effect of the mechanical strength can be explained by the Weibull theory. For the breakdown strength the same explanation has been often assumed. In order to validate this assumption breakdown strength and mechanical strength of alumina samples with defined porosities were compared. Differences in the Weibull moduli of breakdown and mechanical strength distributions indicate that the volume-effect cannot explain the thickness-dependence of the breakdown strength. In particular, the thickness-dependence of the breakdown strength always leads to a Weibull modulus of two which is not in agreement with the measured Weibull moduli for samples with constant thickness. It can be concluded that the thickness-dependence of the breakdown strength cannot be explained by the Weibull concept. A recently developed breakdown model which is based on space charge injection is able to explain the experimental results.

Impact cratering experiments into quartzite, sandstone and tuff: The effects of projectile size and target properties on spallation

Impact cratering experiments were performed on quartzite, tuff, and dry and water-saturated sandstones in the framework of the MEMIN research unit. 2.5–12mm diameter projectiles were accelerated to ∼5km/s. Evaluation of the resulting craters shows that crater volumes and crater efficiencies of large-scale experiments are greater than predicted by strength scaling laws. A method to approximate the transient crater volume shows that this effect is largely due to an increase in spallation. Strength scaling laws are used to determine the reduction of tensile strength in large-scale experiments and show a decrease by a factor of 1.8–3.6. This strength reduction can be correlated with a decrease in strain rate for larger projectiles, and with the Weibull theory of strength reduction for larger rock sample sizes. Further variations in spallation are observed between different target materials; a decrease in spall is suggested to be controlled by increased porosity.

Improvement of the shape of ceramic piston head by optimization and FEM

In the present study, the shape of ceramic piston head was improved by optimization methods. The objective function used was failure probability. The polyaxiality of the stress field was in principle taken into account by the Principle of Independent Action, which is based on Weibull's theory. However because of the large differences in principal stresses, the expression could be simplified. According to the results obtained the probability of failure decreased to 1/10 of the original the volume was reduced by 20 percent principal stress decreased.

An experimental investigation into size effects in quasi-isotropic carbon/epoxy laminates with sharp and blunt notches

An experimental investigation into size effects in notched [45/90/-45/0] 4s carbon/epoxy laminates is carried out. The in-plane dimensions of the quasi-isotropic laminates are scaled by up to a factor of 8. Larger Scale 16 specimens with only their width and notch length being doubled were also tested as a further comparison. Interrupted tests and X-ray Computed Tomography (CT) scanning are carried out to study the damage at the crack tips. Sharp centre-notched tensile tests are compared to open-hole tests of the same notch length (hole diameter), material and stacking sequence. A similar strength reduction scaling trend is found for both configurations at the small sizes, except with higher tensile strength for the centre-notched specimens than the open-hole specimens. However, there is a cross-over point when the sizes increase, with the sharp notched results approaching an asymptote based on Linear Elastic Fracture Mechanics (LEFM), and the open hole results approaching an asymptote based on Weibull theory.

Weakest link and localisation WL2: a method to conciliate probabilistic and energetic scale effects in numerical models

The random nature of geomaterial tensile strength, combined with a softening behaviour, leads to a strong and systematic dependency of the first crack stress value on the loaded volume. This problem is so important that the predictive capability of efficient damage models can be occulted by the phenomenon. The method proposed here allows this aspect to be efficiently considered in a non-linear finite element context thanks to a variant of the Weibull theory, in which the classical weakest link theory is rearranged in a non-local form permitting a second gradient implementation.

Weibull statistics effective area and volume in the ball-on-ring testing method

The ball-on-ring method is together with other biaxial bending methods often used for measuring the strength of plates of brittle materials, because machining defects are remote from the high stresses causing the failure of the specimens. In order to scale the measured Weibull strength to geometries relevant for the application of the material, the effective area or volume for the test specimen must be evaluated. In this work analytical expressions for the effective area and volume of the ball-on-ring test specimen is derived. In the derivation the multiaxial stress field has been accounted for by use of the Weibull theory, and the multinomial theorem has been used to handle the integration of multiple terms raised to the power of the Weibull modulus. The analytical solution is verified with a high number of finite element models for various geometric parameters. The finite element model was also used to study the difference from other multiaxial failure theories, i.e. the Batdorf theories.

Aleatory quantile surfaces in damage mechanics

Abstract In statistical damage mechanics, a deterministic failure limit surface is replaced with a scale-dependent family of quantile surfaces. An idealized homogeneous isotropic matrix material containing cracks of random size and orientation is used to elucidate expected mathematical character of aleatory uncertainty and scale effects for initiation of damage in a brittle material. Scope is limited to statistics and scale dependence for the ONSET (not subsequent progression) of shear-driven failure. Exact analytical solutions for probability of such failure (with an interesting pole-point visualization) are derived for axisymmetric extension or compression of a single-crack sample. A semi-analytical bound on the failure CDF is found for a multi-crack specimen by integrating the single-crack probability over an exponential crack size distribution for which the majority of flaws are small enough to be safe from failure at any orientation. Resulting tails of the predicted failure distribution differ from Weibull theory, especially in the third invariant.

Study of Tower Surface Crack Size Effect Based on Weibull Theory

This paper presents the evaluation of safety state of for tower surface with Weibull and Fractal theory which simulate stress situation of the different scales on the cable-stayed tower surface with PFC. The combination of the Weibull theory and the Fractal theory evaluated the crack condition on the surface of the cable-stayed tower. Calculation results of three specimens showed that the number of the cracks increased when the load step and surface size increased. At the same time, the largest size of specimen had the biggest fractal dimension. The discrete element numerical simulation shows that Weibull modulus was successfully used to evaluate the crack extension of the cable bent tower.

Weibull analysis of stiffness and strength in bulk epoxy adhesives reinforced with particles

The mechanical and statistical behaviour of aerospace structural adhesives has been investigated by performing tensile tests on moulded samples and applying Weibull theory to results. Effects of loading rate, sample size and process parameters have been studied. The experimental design method has been employed in an attempt to extend the experimental results to other combinations of parameters. Three kinds of structural adhesive have been characterised, viz, Hysol® EA9321, Hysol® EA9394 and Hysol® EA9395, which are particle reinforced epoxy adhesives, all of them containing voids with statistically distributed sizes (partially dependent on bonding technique). These highly heterogeneous, but natural, microstructures explain the dispersion of measured strength and Young’s modulus of the materials. These two quantities are demonstrated not to be correlated. With these results, simple fibre bundle theory is used to evaluate the mean value and variance of the failure stress of a bonded joint loaded in tension.

Analysis on the mechanical strength of WC-Co cemented carbides under uniaxial and biaxial bending

The mechanical strength of three WC-Co grades was determined and compared under uniaxial and biaxial bending. Uniaxial four-point bending was conducted on bar-shaped specimens; biaxial testing was performed on discs using the ball-on-three-balls (B3B) method. Strength results were analysed within the frame of the Weibull theory. Differences in characteristic strength between uniaxial and biaxial bending were explained as an effect of the effective surface tested in each case. A higher Weibull modulus was obtained in one grade, independent of the testing method, which was related to its higher fracture toughness. The use and validity of the B3B biaxial test to determine the strength distribution of cemented carbides is discussed.

Dynamic cratering of graphite: Experimental results and simulations

The cratering process in brittle materials under hypervelocity impact (HVI) is of major relevance for debris shielding in spacecraft or high-power laser applications. Amongst other materials, carbon is of particular interest since it is widely used as elementary component in composite materials. In this paper we study a porous polycrystalline graphite under HVI and laser impact, both leading to strong debris ejection and cratering. First, we report new experimental data for normal impacts at 4100 and 4200 m s−1 of a 500-μm-diameter steel sphere on a thick sample of graphite. In a second step, dynamic loadings have been performed with a high-power nanosecond laser facility. High-resolution X-ray tomographies and observations with a scanning electron microscope have been performed in order to visualize the crater shape and the subsurface cracks. These two post-mortem diagnostics also provide evidence that, in the case of HVI tests, the fragmented steel sphere was buried into the graphite target below the crater surface. The current study aims to propose an interpretation of the results, including projectile trapping. In spite of their efficiency to capture overall trends in crater size and shape, semi-empirical scaling laws do not usually predict these phenomena. Hence, to offer better insight into the processes leading to this observation, the need for a computational damage model is argued. After discussing energy partitioning in order to identify the dominant physical mechanisms occurring in our experiments, we propose a simple damage model for porous and brittle materials. Compaction and fracture phenomena are included in the model. A failure criterion relying on Weibull theory is used to relate material tensile strength to deformation rate and damage. These constitutive relations have been implemented in an Eulerian hydrocode in order to compute numerical simulations and confront them with experiments. In this paper, we propose a simple fitting procedure of the unknown Weibull parameters based on HVI results. Good agreement is found with experimental observations of crater shapes and dimensions, as well as debris velocity. The projectile inclusion below the crater is also reproduced by the model and a mechanism is proposed for the trapping process. At least two sets of Weibull parameters can be used to match the results. Finally, we show that laser experiment simulations may discriminate in favor of one set of parameters.

A heuristic model of stone comminution in shock wave lithotripsy

A heuristic model is presented to describe the overall progression of stone comminution in shock wave lithotripsy (SWL), accounting for the effects of shock wave dose and the average peak pressure, P+(avg), incident on the stone during the treatment. The model is developed through adaptation of the Weibull theory for brittle fracture, incorporating threshold values in dose and P+(avg) that are required to initiate fragmentation. The model is validated against experimental data of stone comminution from two stone types (hard and soft BegoStone) obtained at various positions in lithotripter fields produced by two shock wave sources of different beam width and pulse profile both in water and in 1,3-butanediol (which suppresses cavitation). Subsequently, the model is used to assess the performance of a newly developed acoustic lens for electromagnetic lithotripters in comparison with its original counterpart both under static and simulated respiratory motion. The results have demonstrated the predictive value of this heuristic model in elucidating the physical basis for improved performance of the new lens. The model also provides a rationale for the selection of SWL treatment protocols to achieve effective stone comminution without elevating the risk of tissue injury.

Effect of tantalum content on the microstructure and mechanical behavior of cermets based on (TixTa1−x)(C0.5N0.5) solid solutions

Titanium–tantalum carbonitride, (Ti, Ta)(C, N), based cermets with different Ti and Ta contents were prepared using a mechanically induced self-sustaining reaction and then densified using a pressureless sintering process. Complete microstructural and mechanical characterizations were performed on the materials, which revealed that the size of the carbonitride ceramic particle was significantly reduced when the Ta content was increased. The flexural strength and fracture toughness were measured using the ball on three balls test and the indentation microfracture test, respectively. The strength profile was analyzed under the framework of Weibull theory. The change in the mechanical properties as a function of the Ta content was correlated with the normalized microstructural parameters, such as the binder mean free path. The decrease in toughness and flexural strength was attributed to the presence of intermetallic compounds in the binder phase, which was also corroborated by the nanoindentation tests.

On the application of Weibull analysis to experimentally determined single fibre strength distributions

The application of Weibull theory to the analysis of experimental data obtained from the tensile testing of reinforcement fibres is widespread in composites research and development. One basic assumption implicit in the use of Weibull analysis is that all values of fibre strength described by any set of unimodal or multimodal Weibull parameters are accessible experimentally. However, this is not the case, as a minimum level of fibre strength is necessary in order to be able to isolate, prepare and test any fibre. In this paper the consequences of this experimental limitation are explored in terms of the commonly applied Weibull graphical analysis method. It is demonstrated that this can result in significant curvature in a standard Weibull plot at the low strength end of the data. Furthermore, at low sampling numbers this effect can be misinterpreted as evidence of multiple defect populations. The phenomenon significantly affects the values of the Weibull parameters obtained from the graphical analysis and also from the average strength versus gauge length analysis. The presence of this lower limit presents a serious challenge to those wishing to support conclusions on the physics and mechanics of fibre fracture from Weibull analysis of single fibre tensile data.

Dynamic Behavior of a Porous Brittle Material: Experiments and Modeling

The purpose of our research is to study porous polycrystalline graphite under various loading conditions. New experimental data are provided. Some of them concern impacts of a 500μm diameter steel sphere at velocities above 4000 m/s on thick carbon targets, leading to strong debris ejection and cratering. A high speed frame camera showed the debris velocity distribution to lie in the 10-200 m/s range. Post-mortem tomographies have also been performed. They reveal some subsurface cracks, but also provide some evidence that the fragmented sphere lies below the target surface. Dynamic loadings involving similar energy densities (above 2000J/cm2) can also be reached through the interaction of a nanosecond intense laser focalized on a carbon target. An experimental result obtained on a laser facility is presented. Numerical simulations have been performed in order to explain the observed results. An Eulerian hydrocode has been chosen because of the large deformation occurring under considered experiments. We have used a classical model to describe the porous behavior, including equation of state, elasticity, shear strength and densification. In this paper, we focus on the effect of the addition to this porous model of a failure criterion relying on the Weibull theory. The selection of the parameters is based on one set of data and the fit is demonstrated against another test. The overall agreement with the experimental data is good.

A two-scale Weibull approach to the failure of porous ceramic structures made by robocasting: Possibilities and limits

This paper introduces our approach to modeling the mechanical behavior of cellular ceramics, through the example of calcium phosphate scaffolds made by robocasting for bone-tissue engineering. The Weibull theory is used to deal with the scaffolds' constitutive rods statistical failure, and the Sanchez-Palencia theory of periodic homogenization is used to link the rod- and scaffold-scales. Uniaxial compression of scaffolds and three-point bending of rods were performed to calibrate and validate the model. If calibration based on rod-scale data leads to over-conservative predictions of scaffold's properties (as rods' successive failures are not taken into account), we show that, for a given rod diameter, calibration based on scaffold-scale data leads to very satisfactory predictions for a wide range of rod spacing, i.e. of scaffold porosity, as well as for different loading conditions. This work establishes the proposed model as a reliable tool for understanding and optimizing cellular ceramics' mechanical properties.