Heterogeneous peridynamic neural operators: Discover biotissue constitutive law and microstructure from digital image correlation measurements

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Abstract
Human tissues are highly organized structures with collagen fiber arrangements varying from point to point. Anisotropy of the tissue arises from the natural orientation of the fibers, resulting in location-dependent anisotropy. Heterogeneity also plays an important role in tissue function. It is therefore critical to discover and understand the distribution of fiber orientations from experimental mechanical measurements such as digital image correlation (DIC) data. To this end, we introduce the Heterogeneous Peridynamic Neural Operator (HeteroPNO) approach for data-driven constitutive modeling of heterogeneous anisotropic materials. Our goal is to learn a nonlocal constitutive law together with the material microstructure, in the form of a heterogeneous fiber orientation field, from load-displacement field measurements. We propose a two-phase learning approach. First, we learn a homogeneous constitutive law in the form of a neural network-based kernel function and a nonlocal bond force. This captures the complex homogeneous material responses from data. Second, we reinitialize the learnt bond force and train it together with the kernel and the fiber orientation field for each material point. Due to the inherent properties of the state-based peridynamic framework that we use, our HeteroPNO-learned material models are guaranteed to be objective and to satisfy the balances of linear and angular momenta. The effects of heterogeneity and the nonlinearity of the constitutive relationship are captured by the nonlocal kernel function, enabling physical interpretability. These features enable our HeteroPNO architecture to learn a constitutive model for biological tissue with anisotropic, heterogeneous response undergoing large deformation. To demonstrate the applicability of our approach, we apply the HeteroPNO in learning a material model and fiber orientation field from DIC measurements in biaxial testing. We evaluate the learnt fiber architecture, which is found to be consistent with observations from polarized spatial frequency domain imaging. The framework is capable of providing displacement and stress field predictions for new and unseen loading instances.

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