  Constitutive modeling based on continuum mechanics theory has been a
classical approach for modeling the mechanical responses of materials. However,
when constitutive laws are unknown or when defects and/or high degrees of
heterogeneity are present, these classical models may become inaccurate. In
this work, we propose to use data-driven modeling, which directly utilizes
high-fidelity simulation and/or experimental measurements to predict a
material's response without using conventional constitutive models.
Specifically, the material response is modeled by learning the implicit
mappings between loading conditions and the resultant displacement and/or
damage fields, with the neural network serving as a surrogate for a solution
operator. To model the complex responses due to material heterogeneity and
defects, we develop a novel deep neural operator architecture, which we coin as
the Implicit Fourier Neural Operator (IFNO). In the IFNO, the increment between
layers is modeled as an integral operator to capture the long-range
dependencies in the feature space. As the network gets deeper, the limit of
IFNO becomes a fixed point equation that yields an implicit neural operator and
naturally mimics the displacement/damage fields solving procedure in material
modeling problems. We demonstrate the performance of our proposed method for a
number of examples, including hyperelastic, anisotropic and brittle materials.
As an application, we further employ the proposed approach to learn the
material models directly from digital image correlation (DIC) tracking
measurements, and show that the learned solution operators substantially
outperform the conventional constitutive models in predicting displacement
fields.
