Sequential calibration of material constitutive model using mixed-effects calibration

Abstract

Identifying model parameters is nowadays intrinsically linked with quantifying the associated uncertainties. While classical methods allow to handle some types of uncertainties such as experimental noise, they are not designed to take into account the variability between the different test specimens, significant in particular for composites materials. The estimation of the impact of this intrinsic variability on the material properties can be achieved using population approaches where this variability is modeled by a probability distribution (e.g., a multivariate Gaussian distribution). The objective is to calibrate this distribution (or equivalently its parameters for a parametric distribution). Among the estimation methods can be found mixed-effects models where the parameters that characterize each replication are decomposed between the population averaged behavior (called fixed-effects) and the impact of material variability (called random-effects). Yet, when the number of model parameters or the computational time of a single run of the simulations increases (for multiaxial models for instance), the simultaneous, global identification of all the material parameters is difficult because of the number of unknown quantities to estimate and because of the required model evaluations. Furthermore, the parameters do not have the same influence on the material constitutive model depending for instance on the nature of the load (e.g., tension, compression). The method proposed in this paper enables to calibrate the model on multiple experiments. It decomposes the overall calibration problem into a sequence of calibrations, each subproblem allowing to calibrate the joint distribution of a subset of the model parameters. The calibration process is eased as the number as the number of unknown parameters is reduced compared to the full problem. The proposed calibration process is applied to an orthotropic elastic model with non linear longitudinal behavior, for a unidirectional composite ply made of carbon fibers and epoxy resin. The ability of the method to sequentially estimate the model parameters distribution is investigated. Its capability to ensure consistency throughout the calibration process is also discussed. Results show that the methodology allows to handle the calibration of complex material constitutive models in the mixed-effects framework.