Revisiting the Modeling of Porous Structures using Variable-Order Calculus*

Several theoretical and experimental studies have highlighted the prominent role of size-dependent (or nonlocal) effects in porous structures. The spatial distribution of porosity, in terms of the pore location and size, leads to a position-dependent nonlocal phenomenon that results in a position-dependent softening of the structure. This study presents the application of variable-order fractional calculus to the modeling of position-dependent nonlocal effects in porous structures. Variable-order (VO) operators can evolve seamlessly, guided by a VO law, to describe dissimilar physics without the need to modify the underlying structure of the governing equations. The reformulation of the classical continuum framework, by means of VO kinematics, enables a unique approach to model structures exhibiting position-dependent nonlocal behavior. Further, this study presents a deep learning based framework capable of solving the inverse problem consisting in the identification of the VO describing the behavior of the structure on the basis of its response. The combined capabilities of the VO theory and the deep learning framework are illustrated by simulating the response of porous core beams. The VO model shows excellent accuracy when compared to traditional homogenization techniques that are widely used in the modeling of porous structures. The reference solution for the aforementioned comparison was obtained from a 3D finite element model (in a commercial finite element software) that fully resolves the porous beam geometry. The latter analysis also highlighted the significant computational efficiency of the VO model when compared to 3D finite element analysis. Although presented in the context of a slender beam, both the VO nonlocal model and the deep learning techniques are very general and can be extended to the simulation of any general higher-dimensional porous structure.