General theorems and fundamental solutions in the dynamical theory of mixtures

Abstract

This paper is concerned with the dynamic theory of a binary mixture consisting of a gas and an elastic solid. First, we derive general theorems (reciprocal theorem, uniqueness theorem, minimum principle). Then, we establish a Galerkin representation and the fundamental solutions in the case of steady vibrations.

Representation theorems and fundamental solutions for micropolar solid–fluid mixtures under steady state vibrations

“…The equations of the theory developed by Eringen (2003) combine the usual linearized system of equations for isotropic micropolar elastic solids with those for incompressible micropolar viscous fluid and become a complex system of partial differential equations. We outline that the fundamental solutions of the static problems and of the steady oscillations problem for other types of mixtures are constructed in (Svanadze, 1993(Svanadze, , 1996Ciarletta, 1995;Ieşan, 1996;de Boer & Svanadze, 2004;Galeş, 2004 a,b;Svanadze & de Boer, 2005). We also note that in the theories of pure materials with microstructure the proposed problems have been studied by Ieşan & Pompei (1995); Ieşan & Nappa (2001), Svanadze et al (2007), Nappa (2008) and Cialetta et al (2009).…”

On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation

“…Representations of Galerkin-type solutions in the theory of micropolar thermoelasticity without energy dissipation, in the theory of thermoelastic materials with voids, and in the dynamical theory of binary mixture consisting of a gas and an elastic solid were established by Ciarletta [6,10,11]. In the theories of binary mixtures of elastic solids and fluid-saturated porous media, representations of general solutions were presented by Basheleishvili [12], Svanadze [13], and Svanadze and de Boer [14].…”

On the representations of solutions in the theory of generalized thermoelastic diffusion

“…In the theory of thermoelasticity with micro-temperatures, the representation of general solutions was presented by Scalia and Svanadze [20]. The representation of Galerkin-type solutions in the theory of micropolar thermoelasticity without energy dissipation and the dynamical theory of binary mixture consisting of gas and an elastic solid were established by Ciarletta [31,32]. Mukhopadhyay et al [33] established the representation theorem of Galerkin type in the theory of generalized thermoelasticity with three phase lags.…”