A comparative method for improving the reliability of brittle components
Introduction
Often, the absolute reliability of a product cannot be revealed. This is because of
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the complexity of the physical processes and physical mechanisms underlying the failure modes, most of which remain unknown or are associated with large uncertainties;
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the complex influence and uncertainty associated with the environment, the operational loads and duty cycles;
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the variability associated with reliability-critical design parameters (e.g. the state of manufactured surfaces, components tolerances, unbalanced forces, internal environment, duty cycles, etc.);
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the non-robustness of the reliability prediction models.
Key reliability-controlling parameters are associated with uncertainty which does not allow to reveal the absolute reliability level. Major sources of this type of uncertainty are associated with the natural variation of the material properties and the uncertainty associated with their measurement, the uncertainty in determining the times to failure, estimated load magnitudes, etc. Furthermore, even if this information were available, for common, widely used reliability models, even a relatively small amount of uncertainty in the reliability parameters leads to large errors in the model predictions which renders such predictions of questionable value.
Here are some examples illustrating the problem. Reliability predictions during multiple loading are often based on the load–strength interference (Freudenthal, 1954, Carter, 1986) model which involves two basic random variables ‘Load’ and ‘Strength’, characterised by distinct distributions. The reliability in this case is determined by the probability of a relative configuration in which the load is smaller than strength.
Suppose that a load with cumulative distribution function F(x) has been applied n times. The probability density function of the strength is given by s(x). The reliability R (probability of no failure) during multiple loading is then given by the integral (Carter, 1986):where s min and s max are the lower and the upper limit of the strength. The application of expression (1) for reliability predictions however is associated with large errors if n is relatively large. For a large ‘n’, Fn(x) is very sensitive to variations ΔF in the load distribution F(x). Small inaccuracies in the parameters of F(x) will then cause large variations of Fn(x). Indeed, if, for the sake of simplicity, the strength has been taken to be constant c (the variance of s(x) is zero), the integral (1) becomes R = Fn(c).
An error in the parameters of the load distribution F(x) which leads to a relative error ΔF/F in F(c) (for x = c) will result in a large relative error ΔR/R = n(ΔF/F) in the predicted reliability value. In other words, the model (1) is not robust for large n and predicts the reliability R with large errors. Inevitable errors in the parameters of the load and strength distributions will cause very large errors in the calculated reliability value. Since the load distributions are always associated with uncertainty in the parameters, using Eq. (1) to make reliability predictions for a large number of load applications is of questionable value.
Very similar is the case where reliability of a system with a large number of components logically arranged in series is considered. With the continual increase in the complexity of the existing engineering equipment, such complex systems are now very common. The reliability of a system with components logically arranged in series, working independently from one another is estimated fromFor the special case where the system is built on a single component with reliability R, the reliability of the system becomes Rsys = RM. Estimating the reliability R of the component however is always associated with uncertainty. Indeed, let us assume that the reliability of the component is estimated from testing n components of the same type. At the beginning, the reliability is unknown and the initial distribution (prior) is uniform in the interval 0,1:
If x denotes the number of components (out of n components) which survived a single test, the probability of a given sequence of x components surviving the test is:( is not necessary because the surviving sequence is given). The prior distribution f(R) regarding the reliability R can then be revised by using the Bayes’ theorem (Ang and Tang, 1975):and f(R|x) = 0, otherwise. The posterior distribution f(R|x) is the Beta probability distribution.
Increasing the number of tests n will reduce the uncertainty associated with the unknown reliability R but uncertainty will still remain. In other words, uncertainty and errors associated with the reliability parameters in the reliability models are inherent and cannot be eliminated. An error ΔR in the reliability of a single component will lead to a large error ΔRsys/Rsys = n(ΔR/R) in the predicted reliability value for the system.
Predicting the probability of failure locally initiated by flaws is also associated with uncertainty related to the type of existing flaws initiating fracture, the size distributions of the flaws, the locations and the orientations of the flaws, the microstructure around the flaws (crystallographic orientation, chemical and structural inhomogeneity, local fracture toughness, etc.). Some of these random variables are not statistically independent. Capturing this uncertainty, necessary for a correct prediction of the reliability of components is a formidable task.
An efficient way of resolving this predicament, when the focus is on improving the reliability of designs, is not to attempt prediction of the absolute reliability. Competing designs are simply compared on the basis of their reliability which is calculated with the same predefined set of input parameters. In this respect, despite that the absolute reliability level remains unknown, the relative reliability level can be increased irrespective of the absolute reliability level.
Section snippets
A relative reliability measure for selecting designs with improved resistance to failure initiated by flaws
In order to isolate and assess the impact of the design shape or type of loading on the probability of failure of brittle components, the same notional material properties, number density of flaws, fracture criterion and distribution of the flaw size, can be assumed for the competing designs. Given these common assumed properties, the probabilities of overstress failure characterizing the competing designs are compared and the design characterized by the smallest probability of overstress
Distribution of the minimum fracture stress and a mathematical formulation of the weakest-link concept
Consider now a bar made of brittle material containing random flaws, loaded in tension (Fig. 2). Since the loading stress σ is below the minimum fracture stress σM of the homogeneous matrix, in this case, failure can only be initiated by a flaw residing in the stressed volume. A flaw that will initiate failure with certainty, if it is present in the volume of the loaded bar will be referred to as critical flaw (Batdorf and Crose, 1974). A critical flaw for example can be a flaw whose size
Conclusions
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A new comparative method has been developed for selecting design shapes and loading with improved resistance to failure initiated by flaws. The method is very precise because statistical information is collected from all parts of the stressed component.
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The method is also very efficient, because it does not rely on a Monte Carlo simulation. The algorithm is of linear complexity O(n), where n is the number of finite elements into which the component has been divided.
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The Weibull distribution is
Acknowledgements
The author thanks the anonymous reviewers for the useful comments and suggestions.
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