On comparing the shape parameters of two Weibull distributions

Probability estimators developed previously by the authors have been used to compare unbiased estimates of the shape parameters (Weibull modulus) from two distributions by the linear regression method for thirteen sample sizes ranging from 10 to 100. Percentiles of the ratio of two estimated Weibull moduli for various confidence levels have been developed. The use of these percentiles for hypothesis testing, i.e., comparison of the two Weibull moduli is demonstrated by using datasets from the literature.     

Lattice modelling of size effect in concrete strength

This paper uses a recently improved lattice network model to study the size effect in the strength of plain concrete structures. The several improvements made to the lattice network model are: (i) tension softening of the matrix phase is included in the material modelling; (ii) the structural response is modelled by incrementing the deformation rather than the load. This eliminates the need for introducing arbitrary scaling parameters in the beam element failure criteria and; (iii) a square rather than a triangular lattice beam network is found to be adequate for modelling concrete, thus greatly reducing the computational time.The improved square lattice network has been used to simulate the complete load-deformation response of notched three-point bend beams of different sizes with a view to checking the validity of several size effect models available in the literature. Lattice simulation was found to identify microcracking, crack branching, crack tortuosity and bridging, thus allowing the fracture process to be followed until complete failure. The improved lattice model predicted smooth structural response curves in excellent agreement with test results.The simulated nominal strengths also correlated very well with the test results, apart from that for the smallest beams (depth 38.1 mm). However, even in the relatively broad range of sizes (1:8) of the test beams, there was no clear evidence that one size effect model is superior to the other. In fact, rather surprisingly the test data would appear to be equally well described by all the available size effect models. The lattice simulations however indicated a trend which is better predicted by the multifractal scaling model.     

Fragment size distributions from the dynamic fragmentation of brittle solids

Experimental and theoretical size distributions resulting from dynamic fragmentation are briefly surveyed. The power-law character unique to brittle solids is contrasted to fragment distributions of other materials. The catastrophic fracture of competent brittle solids is shown to have close parallels to hydrodynamic turbulence in fluids. Ideas that emerge from the physical similarities suggest methods for extending an earlier energy-based theory of dynamic fragmentation. Features of this theory are compared with limited fragmentation data for brittle solids.     

On the validity of the Weibull failure model for brittle particles

The Weibull theory of material strength and fracture assumes that the Weibull modulus m is a material parameter, which does not depend on shape and size of the loaded object. Based on large data sets from single-particle fracture experiments with brittle materials (glass, clinker cement, limestone), the authors show that the Weibull modulus of nearly spherical particles seems to decrease with increasing particle diameter. A possible explanation is that the inner structure of the particles depends on their size so that small particles are much stronger than large ones. Received: 9 December 1999     

Probability distribution of energetic-statistical size effect in quasibrittle fracture

The physical sources of randomness in quasibrittle fracture described by the cohesive crack model are discussed and theoretical arguments for the basic form of the probability distribution are presented. The probability distribution of the size effect on the nominal strength of structures made of heterogeneous quasibrittle materials is derived, under certain simplifying assumptions, from the nonlocal generalization of Weibull theory. Attention is limited to structures of positive geometry failing at the initiation of macroscopic crack growth from a zone of distributed cracking. It is shown that, for small structures, which do not dwarf the fracture process zone (FPZ), the mean size effect is deterministic, agreeing with the energetic size effect theory, which describes the size effect due to stress redistribution and the associated energy release caused by finite size of the FPZ formed before failure. Material randomness governs the statistical distribution of the nominal strength of structure and, for very large structure sizes, also the mean. The large-size and small-size asymptotic properties of size effect are determined, and the reasons for the existence of intermediate asymptotics are pointed out. Asymptotic matching is then used to obtain an approximate closed-form analytical expression for the probability distribution of failure load for any structure size. For large sizes, the probability distribution converges to the Weibull distribution for the weakest link model, and for small sizes, it converges to the Gaussian distribution justified by Daniels' fiber bundle model. Comparisons with experimental data on the size-dependence of the modulus of rupture of concrete and laminates are shown. Monte Carlo simulations with finite elements are the subject of ongoing studies by Pang at Northwestern University to be reported later.     

Efficient prediction of deterministic size effects using the scaled boundary finite element method

This paper develops an efficient numerical approach to predict deterministic size effects in structures made of quasi-brittle materials using the scaled boundary finite element method (SBFEM). Depending on the structure’s size, two different SBFEM-based crack propagation modelling methodologies are used for fracture analyses. When the length of the fracture process zone (FPZ) in a structure is of the order of its characteristic dimension, nonlinear fracture analyses are carried out using the finite element-SBFEM coupled method. In large-sized structures, a linear elastic fracture mechanics (LEFM)-based SBFEM is used to reduce computing time due to small crack propagation length required to represent the FPZ in an equivalent nonlinear analysis. Remeshing is used in both methods to model crack propagation with crack paths unknown a priori. The resulting peak loads are used to establish the size effect laws. Three concrete structures were modelled to validate the approach. The predicted size effect is in good agreement with experimental data. The developed approach was found more efficient than the finite element method, at least in modelling LEFM problems and is thus an attractive tool for predicting size effect.     

Experimental characterization of the bending fatigue strength of threaded fasteners

The fatigue strength in bending of pre-stressed steel bolts is investigated and compared to the fatigue strength in axial tension. The strength is measured in terms of maximum engineering stress amplitude, neglecting any stress concentration in the threads. The experimental results reveal that the fatigue limit is 76% higher in bending than in axial tension. A finite element model is used to compute the stress state in the threaded region for both axial tension and bending. It allows fitting a volume based weakest link model to the experimentally observed failure probabilities. Based on the good fit of the weakest link model it is argued that randomly distributed defects in the highly stressed thread root determine the fatigue strength.     

Experimental evaluation of the strength distribution of fibers under high strain rates by bimodal Weibull distribution

This paper proposes a bimodal Weibull distribution model for strain- rate- and temperature-dependent fiber strength. The relationships of the mechanical quantities between fiber and fiber bundles at different strain rates and temperatures under tensile impact are established. A method for determining mechanical parameters of fibers by tensile impact tests of fiber bundles is established. Experiments on E-glass bundles have been performed at six strain rates (90, 300, 800, 1100, 1300 and 1700 s⁻¹) at three different temperatures (−70, 14, 80°C). According to the statistical analysis and models, the mechanical parameters for the fiber and their relationships with strain rate and temperature are obtained from the tensile impact experimental results. The emulated stress/strain curves from the model are in good agreement with the test data. The theoretical model and test results show that the shape parameters, β^d₁ and β^d₂, are not only strain rate independent but also temperature independent. The scale parameters σ^d₀₁ and σ^d₀₂, which change with strain rate and temperature, are not constant.     

A modified size effect model for brittle nonmetallic materials

Formulae of both existing size effect models for brittle materials, `effective volume' and `effective surface' were theoretically derived for bar specimens with rectangular and circular cross-sections, for three- and four-point bending load configuration, respectively. Additionally, a modified model called `effective shell model' is introduced. Exemplarily three- and four-point bending tests were performed on bar specimens of different sizes of a leucite reinforced glass ceramic material. The results were analyzed based on the three size effect models. It was found that the defect population of the specimens could be characterized better with the effective surface than with the effective volume model. The new effective shell model improves the statistical reliability even more.     

An investigation of the loading rate dependence of the Weibull stress parameters

This paper examines the dependence of the Weibull stress parameters on loading rate for a 22NiMoCr37 pressure vessel steel. Extensive fracture tests, including both quasi-static and dynamic tests, are conducted using deep- and shallow-cracked SE(B) specimens. The fracture specimens are carefully prepared to ensure the crack fronts are placed at the location where the material is homogeneous. Three dynamic loading rates (in terms of the stress intensity factor rate, in the low-to-moderate range are considered. The load-line velocities for the dynamic tests are chosen so that the resulted values for the deep- and shallow-cracked specimens are the same. Independent calibrations performed at each loading rate (quasi-static and the three dynamic loading rates) using deep- and shallow-cracked fracture toughness data show that the Weibull modulus, m, is invariant of loading rate. The calibrated m-value is 7.1 for this material. Rate dependencies of the scale parameter (σ_u) and the threshold parameter (σ_w-min) are computed using the calibrated m and the results indicate that σ_u decreases and σ_w-min increases with higher loading rates. The demonstrated loading rate invariant of m, when combined with the master curve for dynamic loading, can provide a practical approach which simplifies the process to estimate σ_u as a function of loading rate.     

On the significance and description of the size effect in multimodal fracture behavior. Experimental assessment on E-glass fibers

The existence of a size effect is an integral part of the physical phenomenon of brittle fracture. The literature dealing with the influence of fiber size on its ultimate strength has mainly focused on specimens whose failure is governed by a single category of flaw. In this work, we are discussing the size effect from a statistical point of view for fibers exhibiting multiple failure modes.     

A microstructural approach to flaw size dependence of strength in engineering ceramics

In this work, microstructural effects on the flaw size dependence of ceramic strength were investigated from aspects of stress analysis in the grain just ahead of the crack tip and also R-curve behaviour. In the analysis, it was assumed that the stress averaged in one grain just ahead of the crack tip, in ceramics, might control the fracture from a flaw. A microstructurally modified fracture criterion using the averaged stress was established by introducing the R-curve due to the grain bridging effect for longer cracks. A new R-curve of an exponential type was proposed for the fracture criterion. The criterion could adequately express the central trend in the dispersal of experimental results in the strength versus flaw size relation. To explain the scatter of results, the size distribution and the crystallographic anisotropy of the grain ahead of the crack tip were examined as dominant factors. The lower bound of strength scatter was estimated from the largest grain size, and the strength dispersion was reduced by decreasing the range of grain size variation. In FEM simulations, each element was regarded as one grain with a different crystallographic orientation, which was randomly selected by using a series of quasi-uniform random numbers. It was revealed that the scatter of strength due to crystallographic variations was smaller than the strength dispersion caused by a distributed grain size.     

Development of an LEFM dynamic crack criterion for correlated size and rate effects in concrete beams

The strength of concrete under severe dynamic loading depends on the specimen size and the loading rate. Although the size effect, under quasi-static loading, has been explained by the size-dependent strain energy rate, the main causes of the size and rate effects for dynamic loading cases have not been clarified. In this study, a linear elastic fracture mechanics (LEFM) dynamic crack criterion for a notched three-point bend specimen is developed to explain the size and rate effects, and the possible correlation of these effects. This was achieved by using energy balance, force equilibrium, and Griffith's crack model. From the proposed LEFM dynamic crack criterion, it was shown that (1) the kinetic energy rate seems to be the main cause of the rate effect, (2) the size and rate effects are not independent phenomena.     

Temperature dependence of Weibull stress parameters: Studies using the Euro-material

This work demonstrates the temperature invariance of the Weibull stress modulus, m, for a 22Ni-MoCr37 pressure vessel steel through calibrations at two extreme temperatures of the ductile-to-brittle transition. This temperature invariance reflects the characterization of microcrack size distribution in the material described by the modulus. The calibrations performed here also demonstrate the clear dependence of the Weibull scale parameter, σ_u, on temperature. The increase of σ_u with temperature reflects the increase in microscale toughness of ferritic steels. The calibration procedure employs a three-parameter Weibull stress model which includes the effects of a minimum (threshold) toughness, K_min. The calibrations suggest that K_min increases gradually with temperature. Finally, an engineering procedure is presented to enable practical applications of the Weibull stress model for defect assessments. This procedure combines the demonstrated temperature invariance of m, a recently developed method for predicting the variation of σ_u with temperature using the Master Curve, and calibration of the Weibull stress parameters at one temperature. The (calibrated) temperature invariant m and the estimated σ_u as a function of temperature are used to predict the cumulative probability of fracture for several large datasets without direct calibration.     

Determination of concrete fracture parameters based on two-parameter and size effect models using split-tension cubes

The notched beam specimens have been commonly used in concrete fracture. In this study, the splitting-cube specimens, which have some advantages - compactness and lightness - compared to the beams, were analyzed for the effective crack models: two-parameter model and size effect model. The linear elastic fracture mechanics formulas of the cube specimens namely the stress intensity factor, the crack mouth opening displacement, and the crack opening displacement profile were first determined for different load-distributed widths using the finite element method. Subsequently, four series of experimental studies on cubic, cylindrical, and beam specimens were performed. The statistical investigations indicated that the results of the split-cube tests look viable and very promising.     

Identification of the effective grain size responsible for the ductile to brittle transition temperature for steel with an ultrafine grain size ferrite/cementite microstructure with a bimodal ferrite grain size distribution

This research investigated the mechanism responsible for the ductile to brittle transition temperature for the newly developed steels with a bimodal, ultrafine grain size, ferrite/cementite microstructure (UGF/C), which are produced by caliber warm rolling followed by annealing. The microstructure of the steel was characterised. Charpy impact tests were carried out in the temperature range from 373 K to 4.2 K and the fracture surfaces were analysed. The effective grain size responsible for the ductile-to-brittle transition temperature corresponded to the grain size of the large grain size regions. The mechanism of this phenomenon was attributed to the characteristics of the grain boundaries, as high angle grain boundaries are more effective in impeding cleavage crack propagation. The grain size of the large grain size regions was important in determining the DBTT because these grain boundaries were high angle grain boundaries, whereas the small gain size regions were dominated by the low angle grain boundaries.     

Fatigue life prediction of ZE41A magnesium alloy using Weibull distribution

In this investigation, the fatigue life prediction of ZE41A magnesium alloy has been statistically analyzed by Weibull distribution. The mechanical fatigue tests are conducted under R = 0.1 axial tension condition on specimen machined at as cast and welded materials. The micro structural investigations performed shows strong influence of precipitation on the fatigue failure of material. The curve for maximum stress and cycles to failure has been constructed for above stated materials. Using Weibull, the probability distribution according to which the material will fail is obtained. The fracture surface of the specimens is studied using scanning electron microscope.     

Micromechanical modelling of size effects in failure of porous elastic solids using first order plane strain gradient elasticity

In this paper a first order porous strain gradient elasticity model is presented. The constitutive equations have been obtained by higher order homogenization and the model is used with a failure criterion in order to discuss size effects in failure of porous elastic solids. The model contains two microstructural parameters namely the void volume fraction and the half void spacing. After an extended numerical validation of the porous strain gradient elasticity model, the boundary value problem of a plate with a hole under bi- and uniaxial remote tension is investigated. The numerical simulations have been performed varying both microstructural parameters in order to study the influence of different microstructural dimensions on the onset of macroscopic failure. The numerical results show that the presented model is able to predict size effects and that size effects in failure do not only depend on the microstructural properties but also on the macroscopic geometry, loading conditions, and the failure mechanism.     

An analysis of the Brazilian disk fracture test using the Weibull probabilistic treatment of brittle strength

It is pointed out that the Weibull multiaxial treatment of brittle strength contains limitations which are not present in the more familiar uniaxial formulation. Provided these limitations are satisfied, it is possible to use tension or bending data to predict multiaxial behavior when at least one principal stress is tensile. This is illustrated for the Brazilian disk test (diametral compression of a disk). Predictions based on bending tests agree well with observed strength values in disk tests on two types of rocks.     

Robust Regression using Probability Plots for Estimating the Weibull Shape Parameter

The Weibull shape parameter is important in reliability estimation as it characterizes the ageing property of the system. Hence, this parameter has to be estimated accurately. This paper presents a study of the efficiency of using robust regression methods over the ordinary least‐squares regression method based on a Weibull probability plot. The emphasis is on the estimation of the shape parameter of the two‐parameter Weibull distribution. Both the case of small data sets with outliers and the case of data sets with multiple‐censoring are considered. Maximum‐likelihood estimation is also compared with linear regression methods. Simulation results show that robust regression is an effective method in reducing bias and it performs well in most cases. Copyright © 2006 John Wiley & Sons, Ltd.