Fractional Calculus for Space-Fractional Thermoelasticity

Classical continuum theories cannot accurately describe thermoelastic behavior at the micro- and nano-scales due to scale-dependent effects. At these scales, long-range interactions significantly influence the thermomechanical response of structural systems. Fractional calculus has emerged as an effective framework for representing nonlocal interactions through space-fractional formulations within continuum mechanics. Although space-fractional structural models are capable of capturing scale-dependent mechanical behavior, thermoelastic effects in such formulations have not yet been adequately addressed. This study develops a space-fractional thermoelastic framework to analyse thermal effects in micro- and nano-scale truss structures. Thermal influences are incorporated through temperature-dependent material properties, allowing the effect of temperature variation on structural response to be examined within a continuum-based formulation. A parametric investigation is carried out to study the influence of Young’s modulus variation, material grain size, and material order, represented by the fractional order, under different temperature levels and boundary conditions. The analysis is performed within a thermal continuum-based space-fractional formulation, in which nonlocal interactions are described through a fractional-order representation embedded in the governing equations. The proposed framework advances thermomechanical modelling at small scales, and it supports the design and optimisation of micro- and nano-scale devices that account for the thermal effects, including MEMS and NEMS structural components.