Finite element analysis of fracture behavior in ceramics: Competition between artificial notch and internal defects under three-point bending

Evaluation of the nonlinear relationship between the surface defect size and fracture strength of ceramics is important for engineering applications. In this study, we aim to predict the apparent nonlinear relationship between the defect size and fracture strength of single-edge notched beams (SENBs) using the finite element method. Specifically, we applied the methodology for predicting fracture strength from microstructure distribution data (relative density, pore size, aspect ratio, and grain size) to a finite element analysis (FEA) model in which the shape and size of the initial defects are defined at notch locations. By reproducing the apparent nonlinearity caused by the competition between the surface and internal defects within the framework of linear elastic fracture mechanics, the effectiveness of the FEA methodology for the evaluation of strength scatter and allowable crack size in ceramics was demonstrated.     

Finite element analysis of fracture behavior in ceramics: Prediction of strength distribution using microstructural features

The estimation of the strength scatter caused by internal defects is necessary in analyzing a reliable design of advanced ceramic components. In this study, we proposed a finite element analysis methodology to predict the stochastic fracture behavior of ceramics based on the microstructural features obtained by the scanning electron microscopy and X-ray computed tomography. Here, the two- and three-dimensional distribution microstructural data are approximated by various probability density functions and are reflected in the dispersion of parameters of the damage model via a fracture mechanics model. We then applied the proposed method to alumina fine ceramics sintered at three different temperatures, and performed the three-point bending test. Furthermore, the numerically created Weibull distributions were compared with those obtained experimentally. Our analysis results confirm that the proposed method can reasonably predict the strength scatter in ceramics.     

Statistical size scaling of ceramic strength

This work is based on the belief that the uniform spatial distribution of flaws underlying Weibull statistics is not necessarily always true for all ceramics. The weakest-link statistics for a power-law spatial distribution of flaws is adopted to synchronize the size dependence and random variation of ceramic strength. Three sets of published strength data of ceramics from different sized specimens are used for this purpose. The assumption of power-law spatial distribution of flaws in ceramics is validated, with both uniform and nonuniform spatial distributions being showcased possible. For all the three case studies, there exists a master curve to correlate a compound parameter encompassing cumulative failure probability and size effect with the nominal fracture strength.     

Novel model for the sintering of ceramics with bimodal pore size distributions: Application to the sintering of lime

A mathematical model for the sintering of ceramics with bimodal pore size distributions at intermediate and final stages is developed. It considers the simultaneous effects of coarsening by surface diffusion, and densification by grain boundary diffusion and lattice diffusion. This model involves population balances for the pores in different zones determined by each porosimetry peak, and is able to predict the evolution of pore size distribution function, surface area, and porosity over time. The model is experimentally validated for the sintering of lime and it is reliable in predicting the so called “initial induction period” in sintering, which is due to a decrease in intra‐aggregate porosity offset by an increase inter‐aggregate porosity. In addition, a novel methodology for determination of mechanisms based on the analysis of the pore size distribution function is proposed, and with this, it was demonstrated that lattice diffusion is the controlling mechanism in the CaO sintering. © 2016 American Institute of Chemical Engineers AIChE J, 63: 893–902, 2017     

Fracture Statistics Based on Pore/Grain-Size Interaction

The statistics of fracture in ceramics are discussed based on a model that describes crack instability that occurs at a configuration of a microcrack positioned in the stress-concentrating effect of a large pore. The interaction of pore size and grain-size distribution is considered, and the effect of a locally varying stress field is included. Results are presented as predictions of the critical pore size and microcrack size that cause fracture for the two assumed average grain sizes of 1 and 5 µm.     

From nature geometry to material design: Advanced fractal nature analysis for predicting experimental elastic properties

In this work, combined experimental and fractal nature analysis procedures are proposed in order to both model and design mechanical properties of porous ceramics. Several porous ceramics samples have been considered both from an in-situ experimental campaign and from the literature. Microstructural information concerning pore size distribution has been approximated by the Intermingled Fractal units (IFU) approach and effective mechanical properties are derived by a simple discrete model. The capability of the proposed methodology to reproduce high-scattered mechanical properties is fully shown and a comparison with classical bounds and estimates is also reported. Finally, the combined experimental, fractal nature analysis and homogenisation scheme is implemented as a design procedure for the technological production of advanced porous ceramics.     

Some Effects of Cavities on the Fracture of Ceramics: II, Spherical Cavities

Fracture probabilities associated with spherical cavities were analyzed by combining principles of fracture mechanics and fracture statistics. The analysis is based on the theory that fracture occurs from a distribution of cracks located at the cavity surfaces. It predicts effects on strength of cavity size and cavity volume content (porosity) that depend sensitively on the flaw population in relation to the cavity size distribution. The theory has limit solutions that coincide with several models of fracture derived for porous ceramics. The predicted effects of pore size on strength are compared with some available data.     

Evolution of defect size and strength of porous alumina during sintering

The evolution of fracture strength with increasing density in ceramics, using alumina as a model system, is discussed in terms of the interplay between a defect serving as stress concentrator, a crack lying in its enhanced stress field and the fracture toughness of the porous ceramic. Introduction of crack-free fracture-causing artificial pores of various sizes allows detailed measurement of their shrinkage with ongoing densification, while fractography describes the location and type of fracture initiation. A fracture mechanics model, describing growth of a semicircular crack emanating from the pore until instability, yields good agreement with experiment. In particular, the result that the radius of the artificial, spherical defect in a size regime between 25 and 120 μm has only a small influence on fracture strength for samples with an average grain size smaller than 1μm, can be explained.     

A multi-scale model for predicting the thermal shock resistance of porous ceramics with temperature-dependent material properties

This paper develops a novel multi-scale thermal/mechanical analysis model which not only can efficiently measure the thermal shock response but also highly reflects the effects of diversiform micro-structures of porous ceramics. Knowledge of the temperature distribution and time-varied thermal stress intensity factors (SIF) is derived by finite element/finite difference method and the weight function method in the macro continuum model. The finite element analysis employs a micro-mechanical model in conjunction with the macro model for the purpose of relating the SIF to the thermal stress in the struts of the porous ceramics. The micro model around the crack tip was established by using Voronoi lattices to accurately explore the micro-architectural features of porous ceramics. Hot shock induced center crack and cold shock induced edge crack are both considered. Effects of relative density and pore size on the thermal shock resistance are investigated and the results are well coincident with the experimental tests. The influence of cell regularity and cross section shape of the cell struts is discussed and the corresponding explanations are provided. The importance of incorporating temperature-dependent material properties on the thermal shock resistance prediction is quantitatively represented. These multi-faceted models and results provide a significant guide to the design and selection of porous ceramics against the thermal shock fracture failure for the future thermal protection system of space shuttle.     

Effects of poling state and pores on fracture toughness of Pb(Zr0·95Ti0·05)O3 ferroelectric ceramics

Abstract

Abstract

The effects of poling state and pores on the fracture toughness of Pb(Zr_0·95Ti_0·05)O₃ (PZT 95/5) ferroelectric ceramics were investigated. X-ray diffraction analysis and piezoelectric constant measurements reveal that the phase structures of PZT 95/5 ceramics change with the poling state, which significantly affects the fracture toughness. The poled PZT 95/5 ceramics demonstrate higher fracture toughness than the unpoled ceramics, and their fracture toughness significantly increases after the pressure depoling. As the porosity of ceramics increases with addition of poreformer during preparation, their fracture toughnesses all decrease accordingly either in poled state or unpoled state. The effect of pore size on the fracture toughness is subtle for the poled ceramics, but for the hydrostatic pressure depoled porous PZT 95/5 ceramics, their fracture toughness increases with the increase in pore size. A new stress model is proposed to explain the pore size effect on the fracture toughness of hydrostatic pressure depoled PZT 95/5 ceramics.     

Transmittance predictions for transparent alumina ceramics based on the complete grain size distribution or a single mean grain size replacing the whole distribution

A simple method, based on the equivalent composite model or the dense polycrystalline model, is proposed for predicting the in-line transmittance in ceramics with a grain size distribution of birefringent crystallites. It is shown that the transmittance predictions based on the Rayleigh-Gans approximation are indistinguishable from those based on Mie theory, so that the latter can be replaced by the former. Based on normal distributions of different position and width it is shown that size distributions with smaller mean sizes lead to higher transmittance, whereas the effect of the distribution width on the transmittance is comparatively weak (with the narrowest distribution being the most favorable for high transmittance). The key result of this paper is the fact that for the distributions investigated (normal and log-normal) the transmittance prediction based the whole grain size distribution can be replaced by a transmittance prediction based on the geometric mean of the intensity-weighted distribution.     

Notch radius effect on fracture toughness of ceramics pertinent to grain size

Unlike fracture toughness, the notch fracture toughness of a ceramic is not a constant; rather, it increases with the notch-root radius ρ in a notched specimen. In this study, by analyzing the fracture measurements of eight different notched ceramics with an average grain size G of 3–40 μm, a simple model describing the relation between the notch fracture toughness and fracture toughness is proposed as a function of the relative notch-root radius ρ/G. The normal distribution is incorporated to consider the inevitable scatter in measurements where fracture mechanisms and errors are present. The results demonstrate that the model can effectively predict the quasi-brittle fracture variation trend for ceramics, including the upper and lower bounds, with 96% reliability, from a normal distribution; thus, it can address virtually all of the experimental data. We also determined that the notch fracture toughness approximates the fracture toughness if ρ ≤ G.     

混凝土孔结构及其分形维数与氯离子扩散性能的关系

氯离子扩散系数是研究海洋环境下混凝土结构耐久性的重要参数之一。通过开展不同水胶比混凝土的压汞试验和盐雾扩散试验,研究了混凝土内部孔隙率、孔径分布及临界孔径对氯离子扩散系数的影响规律。结合Menger海绵体模型,建立孔体积分形维数与氯离子扩散系数的关系。结果表明:孔隙率和临界孔径与无量纲化氯离子扩散系数的相关性很高,可作为反映混凝土氯离子扩散性能的重要参数;通过数学分析计算得到的孔表面分形维数分布在2.56~3.86之间,孔体积分形维数分布在2.85~2.98之间;基于压汞法和分形理论计算得到的孔体积分形维数可以作为评价氯离子扩散系数的指标,在孔径小于10 nm、10～100 nm、100～1 000 nm以及大于1 000 nm四类区间,氯离子扩散系数随孔体积分形维数的增加而下降。     

Capillary Pressure Estimation Using Statistical Pore Size Functions

Capillary pressure curves, which have been employed for a long period of time by researchers interested in pore size distribution, are commonly obtained from experimental measurements. The dynamic capillary pressure that influences the flow is affected by many factors including the pore size characteristics and pore scale dynamics. Hence, it is important to investigate the variation of the estimated pore size distribution with capillary number. In this study, a glass type micromodel is considered as the porous media sample. A parametric probability density function is proposed to express the pore size distribution of the porous model, which is also measured using an image analysis technique. The capillary pressure saturation mathematical model is developed by integrating the pore size distribution function. Model parameters with a physical significance are estimated by fitting the model to the measured capillary pressure data at different capillary numbers. The results of capillary pressure obtained are well matched to the measured values. The results show that the trends of the extracted pore size distribution curves have similar trends, but they are not exactly the same. Therefore, the dynamic capillary pressure data alone are not sufficient for estimation of the pore size distribution. As a related development, the prediction of the capillary pressure curves based on measured pore size distributions is also presented. The proposed probability distribution function has the flexibility of representing a wide variety of pore size distributions.     

A new analysis method for the determination of the pore size distribution of porous carbons from nitrogen adsorption measurements

A new analysis method has been developed for the determination of the pore size distribution of porous carbons from nitrogen adsorption measurements. The method is based on a molecular model for the adsorption of nitrogen in porous carbon. It allows, for the first time, the distribution of pore sizes to be determined over both the micropore and mesopore size ranges using a single analysis method. In addition to carbons, this method is also applicable to a range of adsorbents, such as silicas and aluminas.     

Pore‐size evaluation and gas transport behaviors of microporous membranes: An experimental and theoretical study

A modified gas‐translation (GT) model based on a GT mechanism was successfully applied to the pore‐size evaluation and gas transport behavior analysis of microporous membranes with different pore‐size distributions. Based on the gas permeation results of three microporous membranes derived from different alkoxides, the effects of activation energy and the selection of a standard gas on the pore‐size evaluation were discussed in a comparative study. The presence of nano‐sized defects had an important influence on the gas permeation performance of microporous membranes, depending largely on the original pore size of the membrane in question. Moreover, the gas‐separation effect of the pore‐size distribution in a silica membrane was theoretically studied and revealed a significant increase in gas permeance for relatively large gas species but not for small ones. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2268–2279, 2015     

Optimal design on the mechanical and thermal properties of porous alumina ceramics based on fractal dimension analysis

In order to investigate the relationship between pore structure and thermal conductivity as well as mechanical strength, porous alumina ceramics (PAC) with various pore structures were fabricated, using starch as the pore‐forming agent. Fractal theory was employed to characterize the pore size distribution more accurately than ever used parameters. The results show that the increase in starch content in PAC leads to an enhanced porosity, a higher mean pore size, and reduced fracture dimension, thermal conductivity and strength. The fractal analysis indicated that the fractal dimension decreases gradually and reaches its minimum value with increasing the starch content up to 25 wt%, but the further incorporation results in an opposite trend. It is suggested from micro‐pore fractographic analysis that the optimization of both thermal insulation performance and mechanical strength are positively correlated with the increase in the mean pore size and proportion of 2‐14 μm pores but negatively corrected with the porosity. These results provide a new perspective and a deeper understanding for fabrication of PAC with both excellent thermal insulation and mechanical performance.     

确定多孔介质孔结构参数的渗流网络分析法

According to the simulation of nitrogen sorption process in porous media with three-dimensional network model, and the analysis for such a process with percolation theory, a new method is proposed to determine a pore structure parameter--mean coordination number of pore network, which represents the connectivity among a great number of pores. Here the “chamber-throat“ model and the Weibull distribution are used to describe the pore geometry and the pore size distribution respectively. This method is based on the scaling law of percolation theory after both effects of sorption thermodynamics and pore size on the sorption hysteresis loops are considered. The results show that it is an effective procedure to calculate the mean coordination number for micro- and meso-porous media.     

A study on the nonuniform deformation of PTFE membrane during its tentering transverse stretching

To understand the deformation of biaxially stretched polytetrafluoroethylene (PTFE) membrane during the tenter frame transverse stretching process, a finite element analysis (FEA) model was established to study the stress and displacement distribution during the transverse stretching. The morphology, pore size, and mechanical properties of the membrane were also characterized. It has been found from the experimental and FEA simulation results that the tentering transverse stretching of PTFE membrane is a nonuniform stretching, the stress and displacement distribution of the PTFE membrane during tentering is nonuniform because of the nonuniform deformation and the ease of yield and plastic deformation originated from the specific structure of the virgin PTFE particles. The nonuniform thickness and pore size distribution across the membrane width resulted from this nonuniform deformation was also characterized and discussed. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010     

Prediction of convective transport within unsaturated concrete utilizing pore size distribution data

The durability of concrete structures is often compromised by physical and chemical interaction with the external environment that leads to ongoing maintenance and, in the worst cases, can lead to reduced structural integrity and consequent asset replacement. Concrete is a porous material and most field-exposed concrete is partially saturated with water. Where the concrete is unsaturated and there is no external water pressure acting on a concrete surface, the primary mechanisms of transport into concrete are convective-diffusion ingress (i.e. uptake of water and water-borne agents due to capillary attraction). This paper assesses capillarity and outlines a predictive model of the uptake of water by concrete based on analysis of the pore size distribution. It is acknowledged that concrete has a multitude of internal pores with a broad range of lengths and cross-sectional shapes, surface roughness, tortuosities, random meeting and divergence with adjacent pores, microcracks and fractures, and variable pore-water chemical composition, however the prediction model shows reasonable agreement with water sorptivity test data.