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

• Published in: European journal of mechanics, A, Solids Vol. 88; p. 104238
• Main Authors: Sidhardh, Sai, Patnaik, Sansit, Semperlotti, Fabio
• Format: Journal Article
• Language: English
• Published: Berlin Elsevier Masson SAS 01.07.2021; Elsevier BV
• Subjects: Constitutive modeling; Equations of motion; Fractional calculus; Geometric nonlinearity; Hypotheses; Integrals; Mathematical models; Nonlinear response; Nonlocal elasticity; Operators (mathematics); Strain; Thermoelasticity; Variational principles; Thermoelasticity; Geometric nonlinearity; Nonlocal elasticity; Fractional calculus; Constitutive modeling
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2021.104238

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 following the strain-based integral approaches to nonlocal elasticity. •Frame-invariant model for nonlocal thermoelasticity developed using fractional calculus.•Localized imposition of thermodynamic laws enabled by fractional continuum mechanics.•Mechanical balance laws employed to develop governing equations for fractional-order solids.•Constitutive models for fractional-order thermoelasticity derived following thermodynamic principles.•Thermomechanical response of nonlocal solid presents consistent softening response.