IFAC PapersOnLine 51-2 (2018) 601–606 ScienceDirect ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2018.03.102 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: fractional order strain, dipolar thermoelasticity, reciprocity, microstructure. 1. INTRODUCTION The history of the theory of generalized thermoelasticity dates back, on the one hand, to Lord and Shulman, who introduced this theory in 1967 with one relaxation time for an isotropic body, and to Dhaliwal and Sherief, who extended this theory in 1980 for the anisotropic case, and on the other hand, to Green and Lindsay, who developed in 1972 this theory with two relaxation times, based on a generalized inequality of thermodynamics. As it is shown in Sherief et al. (2010), fractional calculus, represented by the application of fractional derivatives, was used for the first time by Abel in approaching the generalized version of the tautochrone problem. The sec- ond half of the nineteenth century was representative for the development of the theory of fractional derivatives and integrals, Caputo being one of those who used, in 1974, fractional derivatives for the description of viscoelastic materials and to establish the relation between fractional derivatives and the linear theory of viscoelasticity. Frac- tional calculus has brought a successful change to many existing models in physical processes such as those of polymers. The theory of multipolar structures, which includes dipo- lar elasticity, leads to the publication of the first results on dipolar theory by Mindlin in 1963. Then he formulated, in Mindlin (1964), a linear theory of a three-dimensional, elastic continuum, having some features of a crystal lattice, as a consequence of considering the unit cell as a molecule of a polymer, a crystallite of a pollycristal or a grain ⋆ corresponding author: Lavinia F. Codarcea-Munteanu, e-mail: codarcealavinia@unitbv.ro of a granular material. Moreover, Green and Rivlin are considered the promoters of the theory of dipolar bodies as considered in Green et al. (1964), since they dedicated their research to the study of these new structures. A study of thermoelastic materials which presents theories of deformable solids with particles that have more than three degrees of freedom is presented in Ie¸ san (1980). In the theory of dipolar continua, each material point is constrained to deform homogeneously, in this case having twelve degrees of freedom, i.e. three for translations and nine for micro-deformations. The theory of dipolar bodies has been the focus of at- tention for many authors. For example, in Marin (1997), Marin, Agarwal, Codarcea (2017), Marin (2010), and Marin, Codarcea, Chiril˘ a (2017), the authors developed and generalized the previously obtained results for some particular dipolar materials, precisely because there has always been a concern to find a mathematical model for describing the materials that surround us and the natural phenomena. For instance, some specific dipolar materials (with voids or double porosity) are applied in the de- scription of rocks, soil or in the manufacturing process of porous materials. Moreover, a simplified theory of dipolar gradient elasticity can be applied to textile materials, see Kordolemis et al. (2015), or to crack problems from fracture mechanics in which this theory is more adequate than the classical one, see Georgiadis (2003). A new theory of thermoelasticity was derived in El- Karamany et al. (2011), and more recently in Youssef (2015), based on fractional order of strain, which is consid- ered as a new modification of Duhamel-Neumann’s stress- * Department of Mathematics and Computer Science, Transilvania University of Bra¸ sov, Bra¸ sov, Romania (e-mail: codarcealavinia@unitbv.ro). ** Department of Mathematics and Computer Science, Transilvania University of Bra¸ sov, Bra¸ sov, Romania (e-mail: adina.chirila@unitbv.ro) *** Department of Mathematics and Computer Science, Transilvania University of Bra¸ sov, Bra¸ sov, Romania (e-mail: m.marin@unitbv.ro) Abstract: The theory of dipolar thermoelasticity or, equivalently, thermoelasticity with microstructure arises in the field of mechanics of generalized continua. This article proposes a new mathematical model by considering the strain with fractional order in this theory. Introducing the Caputo fractional derivative in the classical case leads to new constitutive equations and to a reciprocity relation. The results presented in this article could be used to better model materials with microstructure for which classical mechanics fails to give the same output as expected by experiments. Lavinia F. Codarcea-Munteanu * Adina N. Chiril˘ a ** Marin I. Marin *** Modeling Fractional Order Strain in Dipolar Thermoelasticity