  This study presents the application of variable-order (VO) fractional
calculus to the modeling of nonlocal solids. The reformulation of nonlocal
fractional-order continuum mechanic framework, by means of VO kinematics,
enables a unique approach to the modeling of solids exhibiting
position-dependent nonlocal behavior. The frame-invariance of the strain tensor
is leveraged to identify constraints on the definition of the VO. The VO
nonlocal continuum formulation is then applied to model the response of
Euler-Bernoulli type beams whose governing equations are derived in strong form
by means of variational principles. The VO formulation is shown to be
self-adjoint and positive-definite, which ensure that the governing equations
are well-posed and free from boundary effects. These characteristics stand in
contrast to classical integral approaches to nonlocal elasticity, where it is
not always possible to obtain positive-definite and self-adjoint systems. A key
step in promoting the use of VO approaches is to identify methodologies to
determine the VO describing a given physical system. 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 nonlocal beam on
the basis of its response. It is established that the internal architecture of
bidirectional recurrent neural networks makes them suitable for nonlocal
boundary value problems, similar to the one treated in this study. Results show
that the network accurately solves the inverse problem even for nonlocal beams
with VO patterns inconsistent with the network training data set. Although
presented in the context of a 1D Euler-Bernoulli beam, both the VO nonlocal
formulation and the deep learning techniques are very general and could be
extended to the solution of any general higher-dimensional VO boundary value
problem.
