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Analysis of Flexure Strength Data of Ceramics
Abstract
A method to obtain the two and three Weibull parameters from the statistical strength distribution of ceramics, when either surface flaws or volumetric flaws govern fracture, is outlined. The advantages of obtaining confidence in the parameter estimates are given realizing the flaw severity variations within a test population. The inadequacy of testing a very limited number of specimens to gather reliability data to assess service performance is discussed.
Pulsed laser annealing and ion beam mixing are currently being investigated as surface modification techniques in an attempt to enhance the mechanical properties of poly crystalline α-SiC. Thin Ni overlayers (20nm -100nm) are evaporated onto the SiC surface. The specimens are subsequently irradiated with pulses of a ruby or excimer (KrF or XeCl) laser or bombarded with high energy Si+ or Xe+ ions. Following laser irradiation, the fracture strength of the SiC is increased by as much as ≃50%, but after ion beam mixing, no strength increase is observed. Weibull statistics are used to characterize the SiC fracture strength data before and after surface modification. Cross-sectional transmission electron microscopy (X-TEM), scanning electron microscopy (SEM), secondary ion mass spectroscopy (SIMS), and Rutherford backs cattering (RBS) techniques are used to characterize the modified SiC in order to determine the cause for the increase in fracture strength and the extent of mixing as a result of the surface modification.
Testing Metrology is a generic name used to describe the combination of measurement techniques of sufficient accuracy necessary to determine materials property data. The appropriate accuracy is to some extent dependent upon the purpose for which the data is required, since the level of precision sufficient for product release may be inadequate for determining reliable design data for high technology plant operating in a hostile high temperature environment. Relatively simple product release tests, such as hardness, impact strength, tensile testing (RT and HT) or even the simple one hundred hour stress rupture test, do not provide the design engineer or the plant operator with the necessary data to safely design or predict remnant life of components. Sophisticated testing techniques, such as tests to provide multiaxial creep data, low cycle fatigue, with or without combined creep dwell cycles, or creep crack growth measurements are becoming more important. Despite an improved understanding of the behaviour of materials, in some circumstances it is still regarded as necessary to resort to model component testings (1) or to test full size samples for which testing nuclear containment pressure vessels has provided the impetus to develop 100 MN testing facilities - see Fig. 1 (2). This latter machine will be capable of testing specimens up to 2.5 m in width, under uni-axial or bi-axial stressing conditions, up to approximately 300°C. The specific subject of high temperature testing metrology has been promoted in the UK during the last few years by the activities of the High Temperature Mechanical Testing Committee, resulting in a number of publications covering a broad range of testing techniques (1, 3 to 7).
In this study, the dependency of the variation in the shape parameter m, which was determined in the two-parameter Weibull distribution function fitted to a strength distribution, on the number of samples, n, was investigated by using data in numerically simulated distributions. By random sampling, five groups, i. e. (a) forty sets of n=5, (b) twenty sets of n=10, (c) eight sets of n=25, (d) five sets of n=40, and (e) two sets of n=100, were prepared for the present analysis.It was clarified that the variation of m-value was enlarged with decreasing the number of samples, and its variation range as an absolute value became larger as the shape parameter determined for the original population is increased. Values of the shape parameter, which were obtained for respective numbers of samples, were normalized by the shape parameter for the original population. By such normalization, it was revealed that a relative m-variation with respect to the number of samples was hardly dependent on the shape parameter for the original population. The simulated behavior of m-variation was also found to coincide with that observed in four-point bending test using a pressureless sintered alpha silicon carbide.Finally, it should be noted that the number of samples to be used in a strength test, which is recommended in JIS, is not always enough for a better understanding of the statistical strength characteristics in ceramics.
Four-point bending tests of a gas pressure sintered silicon nitride were conducted using notched specimens with different notch shapes as well as smooth specimens of distinct sizes, and the effect of the specimen geometry on the bending strength was experimentally clarified. The mean strength in smooth specimens could be almost correlated with the effective volume, though the mean strength in notched specimens shifted toward the lower strength side compared with the relation for smooth specimens. From observation of fracture surface, no unique correlation of the strength with the effective volume was suggested to be ascribed to a difference in flaw morphology between smooth and notched specimens. To discuss the efficiency of the effective volume, a Monte Carlo simulation was also carried out by assuming the same distribution characteristics of cracks in a material. The simulated result revealed that the effect of the specimen geometry on the strength was explained by using the effective volume. The effective volume, however, was found to be inefficient because of a failure of the primitive assumption in the Weibull theory when the flaw density was extremely low in a material.
In structural ceramics, a remarkable scatter of strength is observed due to brittle behavior in deformation and flaws unavoidably formed during material processing. Therefore, information on strength characteristics of ceramics is required for the mechanical design in their applications to high-performance components. In the present paper, the strength properties of typical ceramics are reviewed from statistical aspects, especially. As for the static strength, general features of ceramic strength are first outlined, and then several factors affecting strength, such as environments, specimen geometry, stress state and machining, are described. The fatigue strength is reviewed in two categories of the static and cyclic fatigue. As one of screening procedures to exclude components with a lower strength, the proof testing in ceramics is described and the problems in the testing are also mentioned.
For ceramic materials, the statistical distribution of defects is closely connected to the fracture stresses. This relationship has been used to derive the distribution of flaw sizes and their positions from different fracture experiments. Reversely, fracture stresses were predicted from the measurement of flaw populations, which would offer a non-destructive tool to measure mechanical properties. There are two obstacles for precise predictions, on the one hand the small size of the defects and on the other hand their complicated geometry. In the literature, considerable effort was undertaken to find an answer to this question and to develop suited tools for determining the size distribution of defects. For the description of the mechanical behaviour, the most frequently used approach is the Weibull theory. The Weibull statistics naturally arises if the number of the defects shows an inverse power law with respect to their size. Because of the technical importance for the prediction of strength and life-time of ceramic components, the limits and the restrictions by the statistical description were of particular interest. From theoretical considerations by analytical models as well as from computer simulations, the reliability of the Weibull theory is predicted, i.e. the variation coefficient of the Weibull modulus and the scale parameter are calculated for a certain material and a certain number of experimental tests. This allows to calculate the number of specimens required to predict the strength at a certain given accuracy.
This paper not only analyzed the Weibull modulus through reliability engineering but also obtained the optimal parameters with multiple performance characteristics using the fuzzy-based Taguchi method for polishing ceramic blocks. Optimization with the multiple performance characteristics is found to be the highest speed and greatest load, and diamond particle of 3 μm and 30% concentration. The analysis of variance shows that the most significant polishing parameters are load (57.75%) and speed (24.43%), followed by concentration (12.52%) and diamond particle size (5.30%). Experimental results have shown that the multiple performance characteristics can be improved effectively through this approach.
Surface cracks on sintered SiCMOR bars are ultrasonically measured by a dB drop technique, using surface acoustic waves. These values of crack size are compared with those obtained by applying linear elastic fracture mechanics to destructive measurements of bar strength. The magnitudes of two error components in the ultrasonic measurement are evaluated: a random component, due primarily to the finite beam width, and a consistent error component, due to closure effects at the crack tip.
Statistical fatigue tests in rotating bending were carried out by using round bar specimens of alumina ceramics, in which fatigue life distributions were examined at five stress levels by assigning 20 specimens to each level. Based on these results, the probabilistic fatigue property of this material was given as the quantitative P-S-N characteristics in wide range of fatigue life 1 < N < 108. Furthermore, SEM observations were made on the fracture surfaces, and the effect of material defects on the fatigue life was also discussed.
The properties of nuclear grade graphites exhibit anisotropy and could vary considerably within a manufactured block. Graphite strength is affected by the direction of alignment of the constituent coke particles, which is dictated by the forming method, coke particle size, and the size, shape, and orientation distribution of pores in the structure. In this paper, a Weibull failure probability analysis for components is presented using the American Society of Testing Materials strength specification for nuclear grade graphites for core components in advanced high-temperature gas-cooled reactors. The risk of rupture (probability of fracture) and survival probability (reliability) of large graphite blocks are calculated for varying and discrete values of service tensile stresses. The limitations in these calculations are discussed from considerations of actual reactor environmental conditions that could potentially degrade the specification properties because of damage due to complex interactions between irradiation, temperature, stress, and variability in reactor operation.
An approach to optimisation of the polishing of ceramic blocks with multiple performance characteristics, based on reliability analysis, Taguchi and grey methods, has been studied. Speed, load, diamond size and concentration are optimised polishing parameters when the performance characteristics, which include Weibull modulus and removal rate, are taken into consideration. Optimisation with multiple performance characteristics is found to be the highest speed and greatest load, and diamond particles of 3 μm size and 45% concentration. Analysis of variance shows that the most significant polishing parameters are load (52.46%) and speed (33.09%), followed by diamond particle size (8.21%) and concentration (6.24%). Experimental results have shown that the multiple performance characteristics of polishing of ceramic blocks are greatly improved through this approach.
Four-point bending tests of a pressureless sintered alumina and a gas pressure sintered silicon nitride were conducted using notched specimens with different notch shapes as well as smooth specimens of distinct sizes, and the effect of the specimen geometry on the bending strength was experimentally clarified. The mean strength in the alumina was correlated with the effective volume independently of specimen geometry. In the silicon nitride, however, the mean strength could not be correlated with the effective volume; the mean strength in notched specimens shifted toward the lower strength side compared with the relation for smooth specimens. The poor correlation in the silicon nitride was suggested to be caused by the fact that the flaw distribution in smooth and notched specimens was different. To discuss the efficiency of the effective volume, a Monte Carlo simulation in the framework of fracture mechanics was also carried out by assuming the same characteristics of crack distribution in a material. The simulated result revealed that the effect of the specimen geometry on the strength was explained by using the effective volume. The effective volume, however, was found to be inefficient because of failure of the basic assumptions in the Weibull theory when the Weibull modulus was not consistent or the flaw density was extremely low in a material.
This paper discusses the optimal parameters using the Taguchi method, but also analyses the Weibull modulus through reliability engineering for polishing ceramic gauge blocks. The optimal polishing parameters for removal rate thus obtained include highest speed, greatest load and a diamond particle size of 3m and 30% concentration. The analysis of variance shows that the most significant polishing parameters for removal rate are speed (39.8%) and load (39.2%), followed by concentration (17.9%) and diamond size (3.1%). The optimal polishing parameters for the Weibull modulus are found to be low speed and medium or high load, regardless of diamond size and concentration.
AbstractCeramic cutting tools have been previously tested according to their wear. During the present research turning tests have been carried out with failure as tool life criteria. Relationships between statistical distribution of cutting time until failure during machining tests and flexural strength during bending tests, have been established, according to a same probability of presence of critical micro-structural flaws into the stressed volume of ceramic material. A new concept of efficiency of ceramic tools is proposed: the safe tool life which means tool life with low probability of catastrophic failure of tool.
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