Characterization of Nonlinear Material Response in the Presence of Large Uncertainties – A Bayesian Approach

The aim of the current work is to develop a Bayesian approach to model and simulate the behavior of materials with nonlinear mechanical response in the presence of significant uncertainties in the experimental data as well as the applicability of models. The core idea of this approach is to combine deterministic approaches by the use of physics based models, with ideas from Bayesian inference to account for such uncertainties. Traditionally, parameters of models in mechanics have been identified through deterministic approaches to obtain single point estimates. Such methods perform very well for linear models and are the preferred approach in identifying model parameters, especially for precisely engineered systems such as structures and machinery. But in the presence of large variations such as in the response of biological materials, such deterministic approaches do not sufficiently capture the uncertainty in the response. We propose that the model parameters need to encode the spread that is observed in the data in addition to modeling the physics of the system. To this end, we propose the idea of probability distributions for model parameters in order to incorporate the uncertainty in the data. We demonstrate this probabilistic approach to identifying model parameters with the example of two problems: the characterization of sheep arteries using data from inflation experiments and the problem of detecting an inhomogeneity in a cantilever beam. The parameters in the artery characterization problem are the model parameters in the constitutive models and in the cantilever problem the parameters are the stiffnesses of the inhomogeneity and the material of the beam. For each of these problems, we compute the probability distribution of the parameters using Bayesian inference. We show that the probability distributions of parameters can be used towards two kinds of diagnostics: assigning probability to a hypothesis (inhomogeneity detection problem) and using the probability distribution for classifying newly obtained data (characterization of artery data). For the inhomogeneity detection problem, the hypothesis is a statement on the ratio of the stiffnesses and it is observed that the probability of the hypothesis matches well with the data. In the case of the artery characterization problem, new data was successfully classified using the probability distributions computed with training data

Discrete Preisach Model for the Superelastic Response of Shape Memory Alloys

The aim of this work is to present a model for the superelastic response of Shape Memory Alloys (SMAs) by developing a Preisach Model with thermodynamics basis. The special features of SMA superelastic response is useful in a variety of applications (eg. seismic dampers and arterial stents). For example, under seismic loads the SMA dampers undergo rapid loading{unloading cycles, thus going through a number of internal hysteresis loops, which are responsible for dissipating the vibration energy. Therefore the design for such applications requires the ability to predict the response, particularly internal loops. It is thus intended to develop a model for the superelastic response which is simple, computationally fast and can predict internal loops. The key idea here is to separate the elastic response of SMAs from the dissipative response and apply a Preisach Model to the dissipative response as opposed to the popular notion of applying the Preisach Model to the stress{strain response directly. Such a separation allows for the better prediction of internal hysteresis, avoids issues due to at/negative slopes in the stress{strain plot, and shows good match with experimental data, even when minimal input is given to the model. The model is developed from a Gibbs Potential, which allows us to compute a driving force for the underlying phase transformation in the superelastic response. The hysteresis between the driving force for transformation and the extent of transformation (volume fraction of martensite) is then used with a Preisach model. The Preisach model parameters are identi ed using a least squares approach. ASTM Standards for the testing of NiTi wires (F2516-07^sigma 2), are used for the identi cation of the parameters in the Gibbs Potential. The simulations are run using MATLAB R . Results under di erent input conditions are discussed. It is shown that the predicted response shows good agreement with the experimental data. A couple of attempts at extending the model to bending and more complex response of SMAs is also discussed