HETEROGENEOUS PERIDYNAMIC NEURAL OPERATORS: DISCOVER BIOTISSUE CONSTITUTIVE LAW AND MICROSTRUCTURE FROM DIGITAL IMAGE CORRELATION MEASUREMENTS.

Human tissues are highly organized structures with specific collagen fiber arrangements varying from point to point. The effects of such heterogeneity play an important role for tissue function, and hence it is of critical to discover and understand the distribution of such fiber orientations from experimental measurements, such as the digital image correlation data. To this end, we introduce the heterogeneous peridynamic neural operator (HeteroPNO) approach, for data-driven constitutive modeling of heterogeneous anisotropic materials. The goal is to learn both a nonlocal constitutive law together with the material microstructure, in the form of a heterogeneous fiber orientation field, from loading field-displacement field measurements. To this end, we propose a two-phase learning approach. Firstly, we learn a homogeneous constitutive law in the form of a neural network-based kernel function and a nonlocal bond force, to capture complex homogeneous material responses from data. Then, in the second phase we reinitialize the learnt bond force and the kernel function, and training them together with a fiber orientation field for each material point. Owing to the state-based peridynamic skeleton, our HeteroPNO-learned material models are objective and have the balance of linear and angular momentum guaranteed. Moreover, the effects from heterogeneity and nonlinear constitutive relationship are captured by the kernel function and the bond force respectively, enabling physical interpretability. As a result, our HeteroPNO architecture can learn a constitutive model for a biological tissue with anisotropic heterogeneous response undergoing large deformation regime. The anisotropy and heterogeneity of this tissue stems from collagen fibers with unknown natural orientation, resulting in a location-dependent anisotropy. To demonstrate the applicability of our approach, we apply the heterogeneous PNO in learning the material model and fiber orientation field from digital image correction (DIC) data containing the planar displacement field on the tissue and the reaction forces in a biaxial testing. We find the learnt fiber architecture consistent with observations from polarized spatial frequency domain imaging. Moreover, the framework is capable to provide displacement and stress field predictions for new and unseen loading instances.