A reparameterization-invariant Bayesian framework for uncertainty estimation and calibration of simple materials

Abstract In this work, we attempt to formalize the many concepts involved in the calibration of constitutive models to experimental data, restricting ourselves to the class of simple materials and spatially-homogeneous experiments. To begin with, we revisit the widely used method of least-squares and discuss its ambiguities and shortcomings. Here, Bayesian inference presents an alternative and closely-related approach to parameter identification, which we introduce with the help of a simple mechanical example using Student’s t -distribution. We then derive a reparameterization-invariant posterior for the probabilistic calibration of a simple material given a general collection of spatially-homogeneous experiments, which constitutes the main result of the work. The proposed Bayesian framework is subsequently applied to a simple but illustrative example: parameter identification in a three-term Ogden model to the classic Treloar data on rubber. Finally, the general properties of the posterior and the results of its application invite a discussion about the subtleties and consequences of model calibration in continuum mechanics, such as the information content of different experimental setups or the interaction between number of experimental data versus number of material parameters.