Stochastic Modeling of Physical Parameters on Complex Domains, with Applications to 3D Printed Materials

The proper modeling of uncertainties in constitutive models is a central concern in mechanics of materials and uncertainty quantification. Within the framework of probability theory, this entails the construction of suitable probabilistic models amenable to forward simulations and inverse identification based on limited data. The development of new manufacturing technologies, such as additive manufacturing, and the availability of data at unprecedented levels of resolution raise new challenges related to the integration of geometrical complexity and material inhomogeneity — both aspects being intertwined through processing.

In this dissertation, we address the construction, identification, and validation of stochastic models for spatially-dependent material parameters on nonregular (i.e., nonconvex) domains. We focus on metal additive additive manufacturing, with the aim of closely integrating experimental measurements obtained by collaborators, and consider the randomization of anisotropic linear elastic and plasticity constitutive models. We first present a stochastic modeling framework enabling the definition and sampling of non-Gaussian models on complex domains. The proposed methodology combines a stochastic partial differential approach, which is used to account for geometrical features on the fly, with an information-theoretic construction, which ensures well-posedness in the associated stochastic boundary value problems through the derivation of ad hoc transport maps.

We then present three case studies where the framework is deployed to model uncertainties in location-dependent anisotropic elasticity tensors and reduced Hill’s plasticity coefficients (for 3D printed stainless steel 316L). Experimental observations at various scales are integrated for calibration (either through direct estimators or by solving statistical inverse problems by means of the maximum likelihood method) and validation (whenever possible), including structural responses and multiscale predictions based on microstructure samples. The role of material symmetries is specifically investigated, and it is shown that preserving symmetries is, indeed, key to appropriately capturing statistical fluctuations. Results pertaining to the correlation structure indicate strong anisotropy for both types of behaviors, in accordance with fine-scale observations.