Thermodynamics of fractional-order nonlocal continua and its application to the thermoelastic response of beams

Abstract: This study presents a comprehensive framework for constitutive modeling of a frame-invariant fractional-order approach to nonlocal thermoelasticity in solids. For this purpose, thermodynamic and mechanical balance laws are derived for nonlocal solids modeled using the fractional-order continuum theory. This includes revisiting the Cauchy's hypothesis for surface traction vector in order to account for long-range interactions across the domain of nonlocal solid. Remarkably, it is shown that the fractional-order model allows the rigorous localized application of thermodynamic balance principles unlike existing integral approaches to nonlocal elasticity. Further, the mechanical governing equations of motion for the fractional-order solids obtained here are consistent with existing results from variational principles. These fractional-order governing equations involve self-adjoint operators and admit unique solutions, in contrast to analogous studies following the local Cauchy's hypothesis. To illustrate the efficacy of this framework, case-studies for the linear and the geometrically nonlinear responses of nonlocal beams subject to combined thermomechanical loads are considered here. Comparisons with existing integer-order integral nonlocal approaches highlight a consistent softening response of nonlocal structures predicted by the fractional-order framework, irrespective of the boundary and thermomechanical loading conditions. This latter aspect addresses an important incongruence often observed in strain-based integral approaches to nonlocal elasticity.