A Survey of Techniques to Estimate the Uncertainty in Material Parameters Todd Simmermacher Sandia National Laboratories P. O. Box 5800 Albuquerque, NM 87185-0557 ABSTRACT When estimating parameters for a material model from experimental data collected during a separate effects physics experiment, the quality of fit is only a part of the required data. Also necessary is the uncertainty in the estimated parameters so that uncertainty quantification and model validation can be performed at the full system level. The uncertainty and quality of fit of the data are many times not available and should be considered when fitting the data to a specified model. There are many techniques available to fit data to a material model and a few of them are presented in this work using a simple acoustical emission dataset. The estimated parameters and the affiliated uncertainty will be estimated using a variety of techniques and compared. 1. INTRODUCTION In model validation and uncertainty quantification studies, uncertainty in model parameters such as Young’s modulus and Poisson’s ratio need to be characterized. Many times the uncertainty in these parameters is assumed to be uniformly distributed between two values that are chosen based upon expert opinion. A more quantitative and defendable estimate of the uncertainty in material parameters can be found through the analysis of data from material tests. Material tests are used to identify parameters for constitutive models used in structural models. These material tests exercise the material over the load range, in terms of strain typically, of the environments that the model will be used to predict. A constitutive model is then fit to the data by calibrating the parameters of the model in either an ad hoc or formal optimization framework. It is these fitted parameters that are used in the structural model. A typical set of parameters are usually considered deterministic and are assumed to represent the mean of the data. A probabilistic model can be used to characterize the uncertainty in calculated responses from the full structural model by forward propagating the parametric uncertainty through the structural model. Using the collection of estimated parameters such as Young’s modulus, a Kernel Density Estimator (KDE,[1]) is used to identify a probability density function (PDF) . The PDF is then used to generate samples of the parameter in a Monte Carlo ([2]), Latin Hypercube sampling (LHS) or other sampling techniques. In this work, a Bayesian technique [3] will be used to estimate the probabilistic model of the constitutive model parameters. In this work, probabilistic models obtained using six techniques will be compared. The first four techniques use a traditional optimization procedure and the KDE to develop the probability model of the elastic parameters Young’s modulus and Poisson’s ratio. The fifth and sixth methods use a Bayesian technique to develop the probabilistic model of the elastic parameters. The data being used for this analysis is described in [4]. The material was a syntactic carbon foam. The model required for the foam was a linear elastic model which has three parameters: density, Young’s modulus and Poisson’s ratio. The density can be estimated for a sample by measuring and weighing the samples. Estimating the Young’s modulus and Poisson’s ratio is the main objective of this paper. The data to be fitted is acoustic data obtained from impact tests on the foam samples. The geometry of the test samples is given in Figure 1. The test was performed by dropping small steel spheres onto the sample and T. Proulx (ed.), Linking Models and Experiments, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series 5, 219 2. DATA SET DESCRIPTION DOI 10.1007/978-1-4419-9305-2_15, © The Society for Experimental Mechanics, Inc. 2011
RkJQdWJsaXNoZXIy MTMzNzEzMQ==