Exact solutions of isotropic viscoelastic functionally graded Kirchhoff plates Raffaele Barretta a , Raimondo Luciano b,⇑ a Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy b Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, via G. Di Biasio 43, 03043 Cassino, FR, Italy article info Article history: Available online 5 August 2014 Keywords: Kirchhoff plate Viscoelasticity Functionally graded material Analytical modelling Elliptic domain Fiber-reinforced composites abstract Viscoelastic equilibrium problems of KIRCHHOFF plates can be solved in a closed form only under special geometric assumptions, loading conditions and kinematic constraints on the boundary. A new solution procedure, based on a correspondence principle between a linearly elastic, homogeneous and orthotropic SAINT-VENANT beam under torsion and an isotropic linearly viscoelastic and functionally graded KIRCHHOFF plate with no kinematic constraints on the boundary, is proposed. The methodology is adopted to eval- uate displacement, bending–twisting curvature and moment fields of an elliptic plate, with viscoelastic constitutive behavior and loading conditions described by convolution integrals, assessing thus new benchmarks for computational mechanics. The analysis is specialized to periodic fiber-reinforced com- posites with polymeric matrix described by a four-parameter viscoelastic model. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction The analysis of structures composed of functionally graded materials is a thematic of major interest in literature both from the theoretical and computational point of view, due to the inter- esting potentialities exhibited in engineering applications [1]. Numerous theories and skillful computational methods have been proposed by different authors to analyze composite plates, see e.g. [2,3] and reference cited therein. A recent review on the analysis of thermo-elastic static, vibration and stability problems of function- ally graded plates has been contributed in [4]. The elastostatic equilibrium problem of a plate, possibly inhomogeneous, can how- ever be solved in a closed form only for special geometries, loading distributions and boundary kinematic constraints [5]. Noteworthy solutions of simply supported laminated plates were established in [6–9]. A novel Hamiltonian system-based symplectic superposition approach has been recently contributed in [10] to detect new benchmark solutions of rectangular plates with a corner point- supported. This skillful methodology was previously adopted in [11] to solve free rectangular plates. Further benchmarks in linear elasticity were provided in [12,13] for elliptic domains and equilateral triangles by resorting to an analogy involving bend- ing–twisting moments of KIRCHHOFF plates and warping fields of SAINT-VENANT beams under torsion and flexure. The equivalence method there exposed is further resorted to in the present paper to assess new exact solutions of isotropic plates, functionally graded in the mid-plane and kinematically free on the boundary, but with linearly viscoelastic constitutive behavior. The treatment makes explicit recourse to the correspondence principle in linear viscoelasticity introduced by CHRISTENSEN [14], based on the theory of LAPLACE transforms (see e.g. [15]), and generalized to inhomoge- neous materials by SHAPERY in [16]. It is know that the correspon- dence principle may not hold for functionally graded materials [17]. We consider the class of isotropic functionally graded visco- elastic materials with relaxation moduli in separable form in space and time. In this context, as shown by PAULINO,JIN and MUKHERJEE [18–20], the correspondence principle remains valid. The plan of the present paper is the following. Basic assumptions and equa- tions governing linearly viscoelastic functionally graded KIRCHHOFF plates are collected in Section 2 in a coordinate-free form. A new solution procedure is proposed in Section 4 to assess the exact expressions of displacements, bending–twisting curvatures and moments of inhomogeneous isotropic viscoelastic thin plates, with no boundary kinematic constraints. The treatment is based on the elastic–viscoelastic correspondence principle [21] and on the equivalence method contributed in [12] between the elastic equi- librium problems of isotropic KIRCHHOFF plates and orthotropic SAINT-VENANT beams, as illustrated in Section 3. The proposed meth- odology is exploited in Section 5 to detect new closed-form solu- tions of functionally graded elliptic plates with homogenized http://dx.doi.org/10.1016/j.compstruct.2014.07.044 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: rabarret@unina.it (R. Barretta), luciano@unicas.it (R. Luciano). Composite Structures 118 (2014) 448–454 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct