Conditional Probability of Failure and Accept/Reject Criteria

A discussion is given of a general probabilistic approach to the derivation of the failure probability conditioned by nondestructive (ND) measurements and of an optimal accept/reject procedure. This approach involves the use of explicit stochastic models of both the ND measurement process and the failure process (including a postulated stress environment). The overall decision logic involves a number of online and off-line inputs and outputs which will be described in detail with some indications of the kinds that are of interest to various categories of users. Particular emphasis will be placed upon the operating characteristic curve (i.e., the false-rejection probability vs. the false-acceptance probability representing a broad spectrum of optimal decision procedures) and its significance as a measure of the performance and cost-effectiveness of NDE systems. Explicit results will be given for the case o ceramic NDE with acoustical scattering measurements and two alternative failure models. The first is one in which the fracture process originates at a void surrounded by peripheral microcracks and the second involves fracture originating in a subcritical inclusion. Particular attention will be devoted to limiting situations in which the unconditional failure probability is small and/or in which the ND measurements are accurate and sufficiently diverse. INTRODUCTION The purpose of thi~ paper is to present a description of progress made since the last review of quantitative NDE on the subject of probabilistic failure prediction and optimization of accept/ reject criteria. This work goes beyond other work on reliability theory by making use of explicit physical models of both the failure and measurement processes. The resultant formalism enables one to bridge the gap between the ND measurements arid .the concerns of the ultimate user. Although our methodology applies in principle to any maerial, our explicit applications will be made to structural cerami·cs. We will make the following simplifying assumptions: 1. The ND measurement, or set of such measurements, will be performed at a single time and a single accept/reject decision will be made on the basis of the result. 2. Only the most significant (e.g., the largest defect in a first-piece causes failure, if any, and the less significant defects in aggregate have a negligible probability of causing failure. 3. The applied stress is spatially uniform. The probability of failure depends only on the maximum stress and is independent of the stress history up to that time. In the following sections we discuss first the general theory of failure prediction and accept/ reject decisions. In later sections, we discuss the applications to failure in ceramics due to voids and subcritical inclusions, respectively. Various output properties are considered with particular emphasis on plots of false-reject vs false-accept probabilities. Although the failure models have been validated with real data on fracture and the associated causative defect in each 646 of a number of test pieces, the assessments of the overall decision formalisms have been carried out only with artif1cial theoretical data based upon the relevant models of the measurement and failure processes. GENERAL PROBABILISTIC THEORY OF FAILURE PREDICTION AND ACCEPT/REJECT DECISIONS In this section we discuss the formalism that is required to derive an optimal accept/reject decision procedure using the results of nondestrutive measurements as input data. The major part of the formalism deals with the calculation of the probability of failure (or survival) of the test piece (under an assumed stress environment) based upon the nondestructive measurements. The decision to be considered involves a single inspection before the component is put into service. In the case of a brittle ceramic, failure is defined as the inability to avoid catastrophic fracture under a specified time-invariant uniform applied stress (1,2). The procedure outlined here applies to any material subject to a single inspection. As shown in Fig. 1 the input information to the decision process is composed of nondestructive measurements on the test piece. The output is a