Mechanics Research Communications 48 (2013) 52–58 Contents lists available at SciVerse ScienceDirect Mechanics Research Communications jo ur nal homep age : www.elsevier.com/locate/mechrescom On a strain gradient theory of thermoviscoelasticity D. Ies ¸ an a , R. Quintanilla b,∗ a Octav Mayer Institute of Mathematics (Romanian Academy), Bd. Carol I, nr. 8, 700508 Ias ¸ i, Romania b Matemática Aplicada 2, UPC, Colón 11, 08222 Terrassa, Barcelona, Spain a r t i c l e i n f o Article history: Received 6 July 2012 Received in revised form 28 November 2012 Available online 17 December 2012 Keywords: Thermoviscoelasticity Strain gradient theory Existence and uniqueness results Chiral materials a b s t r a c t This paper is concerned with a strain gradient theory of thermoviscoelasticity in which the time deriva- tives of the strain tensors are included in the set of independent constitutive variables. The theory is motivated by the recent interest in the study of gradient theories. First, we establish the basic equations of the linear theory and present two uniqueness results. Then, we use a semigroup approach to derive an existence result. Finally, we establish the constitutive equations for an isotropic chiral material and derive a solution of the field equations. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction The origin of the theories of nonsimple elastic solids goes back to works of Toupin (1962, 1964), Mindlin (1964), and Mindlin and Eshel (1968). Aifantis and coworkers formulated gradient elastic- ity theories for finite deformations (Triantafyllidis and Aifantis, 1986) and infinitesimal deformations (Aifantis, 1992; Altan and Aifantis, 1992; Ru and Aifantis, 1993). These theories contain fewer higher-order terms. Many authors have investigated the Aifantis theory and have published a large number of papers (see Askes and Aifantis, 2011 and references therein). The experimental observa- tions have shown that the classical continuum theories cannot be used to describe satisfactorily some phenomena. The interest in the gradient theory of elasticity is stimulated by the fact that this theory is adequate to investigate important problems related to size effects and nanotechnology (Aifantis, 1999, 2000, 2009; Askes and Aifantis, 2011). The strain gradient theory has been also used to study the behaviour of chiral materials (see, for example, Auffray et al., 2009; Papanicolopulos, 2011). The gradient theories of thermomechan- ics have been studied in various papers (Ahmadi and Firoozbaksh, 1975; Ies ¸ an, 1983, 2004; Forest et al., 2000, 2002; Polizzotto, 2003; Forest and Amestoy, 2008; Forest and Aifantis, 2010). Forest and Aifantis (2010) have introduced higher order gradients of temper- ature and concentration to investigate the transport theories. The gradient theory of viscoelasticity has been studied by Valanis (1997) and Aifantis (2011). In the first part of this ∗ Corresponding author. Tel.: +34 937398162; fax: +34 937398101. E-mail address: ramon.quintanilla@upc.edu (R. Quintanilla). paper we derive a gradient theory of thermoviscoelasticity of the Kelvin-Voigt type. We have studied the implications of the Clausius–Duhem inequality by using the method presented by Eringen (1999). In the theory of nonsimple thermoelastic mate- rials some qualitative results have been obtained. The existence and asymptotic stability of solutions in the grade-consistent the- ory of Cosserat thermoelasticity have been obtained by Ies ¸ an and Quintanilla (1992). The exponential stability of solutions to the boundary-initial-value problems for nonsimple thermoelastic bars was investigated by Fernández-Sare et al. (2010). Existence and decay of solutions in the isothermal theories of nonsimple materi- als have been derived by Pata and Quintanilla (2010). In the present paper we establish two uniqueness results in the gradient theory of thermoviscoelasticity. Moreover, we use a semigroup approach to derive an existence theorem. In the case of isotropic chiral mate- rials we present a general solution of the field equations that is analogous to the Cauchy–Kowalewski–Somigliana solution in the isothermal theory of classical elasticity. 2. Basic equations We consider a continuum which at time t 0 occupies the region B of Euclidean three-dimensional space and is bounded by the surface ∂B. The configuration of the body at time t 0 is taken as reference configuration. We refer the motion of the continuum to a fixed system of rectangular Cartesian axes Ox i (i = 1, 2, 3). We shall employ the usual summation and differentiation conventions: Latin subscripts (unless otherwise specified) are understood to range over the integers (1, 2, 3), summation over repeated sub- scripts is implied and subscripts preceded by a comma denote 0093-6413/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2012.12.003