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This study presents the application of variable-order (VO) fractionalcalculus to the modeling of nonlocal solids. The reformulation of nonlocalfractional-order continuum mechanic framework, by means of VO kinematics,enables a unique approach to the modeling of solids exhibitingposition-dependent nonlocal behavior. The frame-invariance of the strain tensoris leveraged to identify constraints on the definition of the VO. The VOnonlocal continuum formulation is then applied to model the response ofEuler-Bernoulli type beams whose governing equations are derived in strong formby means of variational principles. The VO formulation is shown to beself-adjoint and positive-definite, which ensure that the governing equationsare well-posed and free from boundary effects. These characteristics stand incontrast to classical integral approaches to nonlocal elasticity, where it isnot always possible to obtain positive-definite and self-adjoint systems. A keystep in promoting the use of VO approaches is to identify methodologies todetermine the VO describing a given physical system. This study presents a deeplearning based framework capable of solving the inverse problem consisting inthe identification of the VO describing the behavior of the nonlocal beam onthe basis of its response. It is established that the internal architecture ofbidirectional recurrent neural networks makes them suitable for nonlocalboundary value problems, similar to the one treated in this study. Results showthat the network accurately solves the inverse problem even for nonlocal beamswith VO patterns inconsistent with the network training data set. Althoughpresented in the context of a 1D Euler-Bernoulli beam, both the VO nonlocalformulation and the deep learning techniques are very general and could beextended to the solution of any general higher-dimensional VO boundary valueproblem.

Variable-Order Approach to Nonlocal Elasticity: Theoretical Formulation and Order Identification via Deep Learning Techniques