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Dielectric elastomers are employed for a wide variety of adaptive structures. Many of these soft elastomers exhibit significant rate-dependencies in their response. Accurately quantifying this viscoelastic behavior is non-trivial and in many cases a nonlinear modeling framework is required. Fractional-order operators have been applied to modeling viscoelastic behavior for many years, and recent research has shown fractional-order methods to be effective for nonlinear frameworks. This implementation can become computationally expensive to achieve an accurate approximation of the fractional-order derivative. Accurate estimation of the elastomer’s viscoelastic behavior to quantify parameter uncertainty motivates the use of Markov Chain Monte Carlo (MCMC) methods. Since MCMC is a sampling based method, requiring many model evaluations, efficient estimation of the fractional derivative operator is crucial. In this paper, we demonstrate the effectiveness of using quadrature techniques to approximate the Riemann–Liouville definition for fractional derivatives in the context of estimating the uncertainty of a nonlinear viscoelastic model. We also demonstrate the use of parameter subset selection techniques to isolate parameters that are identifiable in the sense that they are uniquely determined by measured data. For those identifiable parameters, we employ Bayesian inference to compute posterior distributions for parameters. Finally, we propagate parameter uncertainties through the models to compute prediction intervals for quantities of interest. 

Auxetic foams exhibit novel mechanical properties due to their unique microstructure for improved energy-absorption and cavity expansion applications that have fascinated the scientific community since their inception. Given the advancements in material processing and performance of polymer open cell auxetic foams, there is a strong desire to fully understand the nonlinear rate-dependent deformation of these materials. The influence of nonlinear compressibility is introduced here along with relaxation effects to improve model predictions for different stretch rates and finite deformation regimes. The viscoelastic behavior of the material is analyzed by comparing fractional order and integer order calculus models. All results are statistically validated using maximum entropy methods to obtain Bayesian posterior densities for the hyperelastic, auxetic, and viscoelastic parameters. It is shown that fractional order viscoelasticity provides [Formula: see text]–[Formula: see text] improvement in prediction over integer order viscoelastic models when the model is calibrated at higher stretch rates where viscoelasticity is more significant.

 

We perform parameter subset selection and uncertainty analysis for phase-field models that are applied to the ferroelectric material lead titanate. A motivating objective is to determine which parameters are influential in the sense that their uncertainties directly affect the uncertainty in the model response, and fix noninfluential parameters at nominal values for subsequent uncertainty propagation. We employ Bayesian inference to quantify the uncertainties of gradient exchange parameters governing 180° and 90° tetragonal phase domain wall energies. The uncertainties of influential parameters determined by parameter subset selection are then propagated through the models to obtain credible intervals when estimating energy densities quantifying polarization and strain across domain walls. The results illustrate various properties of Landau and electromechanical coupling parameters and their influence on domain wall interactions. We employ energy statistics, which quantify distances between statistical observations, to compare credible intervals constructed using a complete set of parameters against an influential subset of parameters. These intervals are obtained from the uncertainty propagation of the model input parameters on the domain wall energy densities. The investigation provides critical insight into the development of parameter subset selection, uncertainty quantification, and propagation methodologies for material modeling domain wall structure evolution, informed by density functional theory simulations. 

The quantification of uncertainty in intelligent material systems and structures requires methods to objectively compare complex models to measurements, where the majority of cases include multiple model outputs and quantities of interests given multiphysics coupling. This creates questions about constructing appropriate measures of uncertainty during fusion of data and comparisons between data and models. Novel materials with complex or poorly understood coupling can benefit from advanced statistical analysis to judge models in light of multiphysics data. Here, we apply the Maximum Entropy (ME) method to more complicated ferroelectric single crystals containing domain structures and soft electrostrictive membranes under both mechanical and electrical loading. Multiple quantities of interest are considered, which requires fusing heterogeneous information together when quantifying the uncertainty of lower fidelity models. We find that parameters, which were initially unidentifiable using a single quantity of interest, become identifiable using multiple quantities of interest. We also show that posterior densities may broaden or narrow when multiple data sets are fused together. This is likely due to conflict or agreement, respectively, between the different quantities of interest and the multiple model outputs. Such information is important to advance our predictions of intelligent materials and structures from multi-model inputs and heterogeneous data.