A General Approach for the Statistical Analysis of Multiaxial Fracture

A general approach for the statistical analysis of fracture under multiaxial states of stress was developed. The approach invokes a critical coplanar strain-energy release-rate fracture criterion and considers distributions of penny cracks in random and preferred orientations. The analysis predicts strength ratios that depend on the strength dispersion and the proximity to the lower bound. For example, the uniaxial and equibiaxial strength ratios well above the lower bound are 1.25 to 1.09 for a typical range of dispersions. Uniaxial and equibiaxial fracture data for alumina were analyzed and compared with the theory. A good correlation was obtained.     

Fracture of Brittle Solids in the Presence of Thermoelastic Stresses

A micromechanical model for strength behavior as a function of grain size in two-phase materials with thermal expansion mismatch and in single-phase materials with thermal expansion anisotropy is presented. The strength vs grain-size plot is interpreted in terms of internal stresses and the ratio of grain to flaw size. The strength of thermally isotropic material is predicted to exhibit a weak grain-size dependence with a negative grain-size exponent normally different from 0.5. In thermally anisotropic poly crystalline solids, before the critical grain size for spontaneous cracking is reached, there will be a region of decreasing strength with increasing grain size due to an increase in the grain-to-flaw size ratio. When a critical grain size is reached, the ratio of grain to flaw size will decrease instantaneously and the strength will decrease in the same fashion. Very good agreement was obtained between predicted and experimentally observed strength behavior for TiO₂ and MgO ceramics.     

Equivalence of Physically Based Statistical Fracture Theories for Reliability Analysis of Ceramics in Multiaxial Loading

Two weakest-link fracture statistics formulations for multiaxial loading, Batdorf's flaw density and orientation distribution approach and Evans' elemental strength approach, are compared for identical fracture criteria and distribution functions. It is demonstrated that despite some fundamental differences in the methodology used in calculating fracture probabilities for multiaxial loading the two approaches give identical predictions. A recent contradictory conclusion reported in the literature is shown to be incorrect.     

Analysis of Brittle Fracture Stress Statistics

Methods for determining the parameters of the Weibull distribution function from the results of brittle fracture tests are considered. Linearization, direct (least-squares) curve fitting, and maximum likelihood techniques are applied to a set of data for the fracture of glass specimens in cross-bending and the results compared. It is suggested that the maximum likelihood method is to be preferred since it is completely objective and involves no arbitrary assumptions. However, none of the methods yields reliable results for functions involving three parameters for small sets of data.     

Statistics of Brittle Failure in Multiaxial Stress States

Two methods for relating failure statistics for different stress states in brittle materials are compared. Despite the fact that the two methods are based on the same fundamental assumptions on crack distributions, they give apparently very different formulations. In this paper it is shown that the two methods are equivalent, and the apparent ambiguity resulting from having two seemingly different results from the same set of basic assumptions is removed.     

A New Analytical Model for Estimation of Scratch-Induced Damage in Brittle Solids

Scratch tests are of fundamental interest both for understanding machining-induced damage and for evaluating the scratch resistance of brittle materials. An improved blister field model for the scratch process is proposed where the blister field strength is explicitly determined in terms of the material properties, loading conditions, and geometry of the scratch tool. Additionally, one new expanding cylindrical cavity model is implemented to estimate the plastic zone size surrounding the scratch groove. A quantitative evaluation of the damage zone size is conducted by combining the above two models. The predicted damage zone sizes are in good agreement with the results available elsewhere in literature.     

Localized Surface Modification of Low-Thermal-Conductivity Brittle Solids

A new method utilizing temporal and spatial control of a CO₂ laser beam to generate temperatures sufficient to melt and fuse a vitreous material without concomitant development of crack-inducing thermal gradients has been developed for the localized surface modification and repair of small manufacturing defects in brittle and low-thermal-conductivity solids. Heat transfer and mechanical modeling of the process have allowed development of specialized heating algorithms. Low-positive-expansion frits with low glass transition temperature also have been developed that enable rapid treatment of small areas without the generation of parasitic critical stresses.     

Ceramics Reliability: Statistical Analysis of Multiaxial Failure Using the Weibull Approach and the Multiaxial Elemental Strength Model

Methodology for designing reliable ceramic components requires a precise evaluation and correlation of strengths in different stress states. The present paper compares the merits of the Weibull approach and the multiaxial elemental strength model on an experimental case involving mixed-mode failure in the presence of bimodal flaw populations (surface and volume flaws). The experimental data were obtained using flexure specimens of Si₃N₄ tested at various spans, with the purpose of enhancing shearing effects. The analysis of data was refined by developing an advanced postprocessor program to finite-element codes for failure probability determination based upon the Barnett-Freudenthal approximation of the Weibull approach and the multiaxial elemental strength model. In a second step, the strengths of the specimens exhibiting failures from the two concurrent populations of flaws (intermediate span) were predicted using both approaches from data obtained with different span lengths (long and short spans). Comparison with experimental data showed that the multiaxial elemental strength model is an improvement over the Weibull approach. It also allowed the short-span bending test to be assessed. Finally, important implications for structural design with ceramics are discussed.     

Generalized Orowan-Petch Plot for Brittle Fracture

The fracture of brittle solids is considered to be based on a defect configuration consisting of a stress concentrator, in our case a hemispherical pore, and a starter crack, in our case either a semicircular crack, a circular crack, or a circumferential crack. Computation of the stress field, application of the relevant weight functions, and an averaging scheme based on the assumption of a virtual crack increment provides stress intensity factors for all defect configurations considered. A normalization scheme yields a generalized version of the Orowan-Petch plot in the form of a normalized strength to normalized defect size relationship.     

Quantitative Analysis of Brittle Fracture Surfaces Using Fractal Geometry

Fractal geometry is a non-Euclidean geometry which has been developed to analyze irregular or fractional shapes. In this paper, fracture in ceramic materials is analyzed as a fractal process. This means that fracture is viewed as a self-similar process. We have examined the fracture surfaces of six different alumina materials and five glass-ceramics, with different microstructures, to test for fractal behavior. Slit island analysis and Fourier transform methods were used to determine the fractal dimension, D , of successively sectioned fracture surfaces. We found a correlation between increasing the fractional part of the fractal dimension and increasing toughness. In other words, as the toughness increasing the fracture surface increases in roughness. However, more than just a measure of roughness, the applicability of fractal geometry to fracture implies a mechanism for generation of the fracture surface. The results presented here imply that brittle fracture is a fractal process; this means that we should be able to determine processes on the atomic scale by observing the macroscopic scale by finding the generator shape and the scheme for generation inherent in the fractal process.     

Analysis of Fracture Probabilities in Nonuniformly Stressed Brittle Materials

The results of theoretical studies on the effect of nonuniform stress fields encountered in prismatic beams under bending on the fracture of brittle materials are described. Derivations were carried out to determine the risk of rupture of bending specimens subjected to a symmetrical four-point load of arbitrary spacing, the symmetric three-point loading and pure bending forming limiting cases of this more general loading. The analysis was based on materials obeying the Weibull distribution function with assumptions for either volumetric or surface flaw dispersion conditions. The predicted strengths of bending and tensile specimens are compared. An analytical method for the determination of the three Weibull parameters from a pure bending test is proposed. This method, based on the best fit of a theoretical curve to the experimental data, was applied successfully to experimental results on Columbia Resin, a brittle amorphous polymer.     

Effects of Oxidation on the Flaw-Limited Strengths of Brittle Materials

The weakest-link statistical model is used to derive an approximate relation describing the influence of oxidation or corrosion on the fracture strengths of brittle materials. Differences in strength under comparable short-term oxidizing and nonoxidizing exposures are found to be proportional to δ m/m² where m is the Weibull modulus.     

Brittle Fracture versus Quasi Plasticity in Ceramics: A Simple Predictive Index

Simple relations for the onset of competing brittle and quasi-plastic damage modes in Hertzian contact are presented. The formulations are expressed in terms of well-documented material parameters, elastic modulus, toughness, and hardness, enabling a priori predictions for given ceramics and indenter radii. Data from a range of selected ceramic (and other) materials are used to demonstrate the applicability of the critical load relations, and to evaluate coefficients in these relations. The results confirm that quasi plasticity is highly competitive with fracture in ceramics, over a sphere radius range 1–10 mm. Implications concerning the brittleness of ceramics in the context of indentation size effects are discussed.     

Median Cracking of Brittle Solids Due to Scribing with Sharp Indenters

The process of moving a loaded hard indenter across the surface of a brittle solid to produce a median crack is referred to as "scribing." As a first step in predicting the depth of median cracking due to sharp indenters, an approximate plastic plane strain analysis is presented for the region under the indenter. This allows the horizontal force transverse to the scribing direction to be predicted. Taking this force and an approximate solution for the stress intensity factor for the crack geometry in scribing leads to a prediction for crack depth as a function of load. The residual stresses which arise on unloading are shown to be a strong function of indenter geometry and may act to extend or partially close the original median crack formed during loading. Reasonable agreement is found between prediction and experiment for six indenter configurations.     

裂解炉炉管弯曲及断裂原因分析

分析计算了裂解炉炉管受力情况，指出裂解炉自身结构的相互约束力和炉管渗碳是导致炉管弯曲、断裂的主要原因，并提出了相应的措施。     

Evaluation of a Fundamental Approach for the Statistical Analysis of Fracture

Generalized relations between the strength distribution function and derivatives of the fracture probability are developed for five widely used testing methods for measuring the strengths of ceramic materials. The derived relations for three of these methods are used to analyze room-temperature fracture strength data for hot-pressed silicon nitride. Fracture of this material is shown to be controlled by internal flaws at strength levels up to and including the range where the tensile and flexural distributions overlap, but it is primarily surface-flaw-controlled toward the upper end of the flexural distribution.     

Fracture Toughness Predictions for Crack Bowing in Brittle Particulate Composites

The bowing of cracks between obstacles has been proposed as a mechanism for increasing the .fracture toughness of brittle materials. An analytical expression is presented based on previous theoretical studies that can be used to predict the increase in fracture toughness. The approach incorporates the effect of obstacle failure and the assumptions needed to compare the predictions with experimental data.