Some uniqueness results for thermoelastic materials with double porosity structure
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O R I G I NA L A RT I C L E
Anamaria N. Emin · Olivia A. Florea · Eduard M. Cr˘aciun
Some uniqueness results for thermoelastic materials with double porosity structure
Received: 10 October 2020 / Accepted: 2 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The main goal of the present paper was to obtain some new uniqueness results for the anisotropic thermoelastic bodies with double porosity structure. There are obtained some auxiliary results based on the Betti reciprocity relation that involve some thermoelastic processes. Keywords Double porosity bodies · Thermoelasticity · Unicity · Betti type
1 Introduction The applications of double porosity materials spread accross a wide range of domains, e.g., civil engineering, geotechniques [1,2], biomechanics [3]. First mathematical model of a thermoelastic material with double porosity structure was studied by Barenblatt [4,5]. Straughan [6] expressed the connection between the two porosities of a material referring to the pores of the body and the cracks of the skeleton. Based on the results obtained by Biot [7] for the materials with single porosity and by Barenblatt et al. [4] for the bodies with double porosity, Wilson and Aifantis [8] have continued the research in the field of deformable bodies with double porosity. The uniqueness, reciprocity and variational theorems for the basic equations that govern the elastic materials with voids were proved by Ie¸san [9] and in [10] Ie¸san included the thermal effect for such media. Based on the Nunziato–Cowin theory for materials with voids [11], Ie¸san and Quintanilla [12] developed a nonlinear theory for thermoelastic bodies with double porosity. The dynamical problems of the theory od elasticity, viscoelasticity, and thermoelasticity for bodies with double porosity were studied in many papers by Svanadze [13–19]. In [20], Svanadze obtained some theorems for the isotropic materials with double porosity structure. Straughan studied the stability and uniqueness for the materials with double porosity by introducing a novel functional in order to obtain Holder stability estimates [21]. Kansal [22] obtained uniqueness and reciprocity theorems for anisotropic materials with double porosity Communicated by Andreas Öchsner. A. N. Emin · E. M. Cr˘aciun (B) Faculty of Mechanical, Industrial and Maritime Engineering, Ovidius University of Constanta, Bd. Mamaia 124, 900527 Constan¸ta, Romania E-mail: [email protected]; [email protected] A. N. Emin E-mail: [email protected] O. A. Florea Faculty of Mathematics and Computer Science, Transilvania University of Bra¸sov, Constanta, Romania E-mail: [email protected]
A. N. Emin et al.
based on the Lord–Shuman model [23]. Mixed problem with initial and boundary conditions, in the context of thermoelasticity of dipolar bodies were recently studied in [24,25]. Uniqueness results for a mixed initial boundary problem for dipolar thermoelastic bodies were studied by Marin and Craciun [26] and for the microstretch thermoelastic mate
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