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failure evaluation of the component is made possible by appropriate scaling of the fracture strength distribution from test specimens under laboratory conditions to complex shaped components experiencing multi axial stress state. In addition, strength of the ceramic component is affected by its surface area and volume. The Weibull parameters for surface flaws were obtained from four-point bend strength data and parameters for volume flaws were obtained from uni-axial tension tests as a function of temperature using standard specimen geometries in laboratory testing. Fractography of failed specimens was performed to ascertain the failure source from a volume or surface flaw. The sub-critical crack growth parameters were determined from dynamic fatigue or stress rupture tests. From these lab-scale data, the reliability of a component with complex geometry and loading is then predicted [5]. The process is captured in Fig 1. The reliability prediction process described for structural ceramics described above is traditionally practiced in industry. Here the failure prediction is probabilistic since the strength of material is input as a probabilistic (Weibull) distribution. But, the predicted state of stress from finite element modeling in the material is deterministic. The predicted stress state is also probabilistic as uncertainty in modeling exists which induces uncertainty in reliability prediction. Incorporating uncertainty in Reliability Prediction In designing with ceramics (as with other materials), designers are confronted with numerous uncertainties in modeling and material property variability. The designer must consider manufacturing tolerances (geometry), variability in loads (engine operating conditions), material properties (thermal conductivity, specific heat, elastic modulus, Poisson’s ratio, thermal expansion, etc.), and boundary conditions (constraints or load/displacement dependent boundary conditions, etc.) during design to ensure that products are reliable and safe. To address these random variations in traditional (deterministic) design, safety factors have traditionally been used. Safety factors have been established for mature applications and materials, like bridges, aircraft parts etc. Choosing a factor of safety for new and novel application and materials can be subjective, and either may lead to a very conservative design or compromise reliability. The reliability-based design approach uses probabilistic methods to account for uncertainties and expected variations in consistent objective fashion. Each random variable is assigned a probability distribution. In general, the distribution can be defined by a mean, a standard deviation, and a distribution type. Some distributions will require bounds or additional shape parameters. If the relevant data exists, the distribution may be found from data by standard statistical methods. However, frequently the structural analyst or the design engineer only has a rough notion of the mean and perhaps the standard deviation. Even though these statistics must be estimated, this approach still provides more accurate results than using safety or knockdown factors [6], which can in some cases stack up in the analysis process to yield over-conservative and consequently uneconomical designs. Although the predicted reliability or probability of failure is necessarily approximate, the probabilistic approach will still provide very useful information in the form of sensitivity factors. The sensitivity factors, which are by-products of the probabilistic analysis, tell the analyst which sources of uncertainty are contributing most to the uncertainty in the predicted performance. Thus, it is possible to determine which variables should be better controlled to attain the best improvement in product reliability. Alternatively, one can determine which tolerances could be relaxed without substantially affecting product reliability. Component geometry, loading and material properties Build Finite Element Model Heat Transfer Analysis Temperature at each node Elastic Stress Analysis Principal Stress at each element Element volume and surface area CARES Figure 1. Flow chart illustrating the current reliability evaluation methodology using CARES [5] Fracture strength data Crack growth parameters Risk of rupture for each element Volume Surface Overall component reliability 126 Copyright © 2007 by ASMEPDF Image | ADVANCED MICROTURBINE SYSTEMS Final Report for Tasks 1 Through 4 and Task 6
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