A Linear Theory of Porous Elastic Solids

Linear Wave Motions in Continua with Nano-Pores

“…In addition to balances (6)(7)(8) and (10)(11)(12), we need the balance equations at a surface of discontinuity, namely a propagating wave Σ. As it is customary, we assume that the smooth movable surface Σ, that traverses the body B, is oriented and we denote by n the unit normal vector to Σ in the reference placement B * and by υ n the corresponding non-zero normal speed of displacement of Σ at point (x * , τ) in the reference placement.…”

“…where the second-order referential microinertia tensor field J * ∈ Sym + is constant (see [3,12,28]). As a consequence of this definition, the left-hand side of Eq.…”

“…A direct way to proceed is to consider the thermoelastic solid with nano-pores as a continuum with an ellipsoidal microstructure (see [4,12]) which describes media whose each material element contains a large cavity, that does not diffuse through the skeleton, filled by an elastic inclusion, or an inviscid fluid, both of negligible mass (e.g., composite materials reinforced with chopped elastic fibers, porous media with elastic granular inclusions, real ceramics, etc. ): this cavity is able to have a microstretch different from, and independent of, the local affine deformation deriving from the macromotion and so can allow distinct microstrains along the principal axes of microdeformation, in absence of microrotations .…”

“…The quoted model [4] is surely complementary to the use of the Cosserat theory [9], when microrotations are of interest in the analysis, but no the microdeformations: merely, we wish to observe that our theory contains naturally the voids theory by constraining the microstrain to be spherical. In particular, in [12] it has been observed that, during quasi-static homogeneous motion, the porous solid behaves like an isotropic simple material with fading memory in the linear range and it reduces to a viscoelastic medium when the microstructural variable remains spherical as in [2]. More generally, for complete microdeformations, the framework of media with affine structure better depicts macro-and micro-motion (see [3,17]).…”

“…In this chapter, we extend the linear theory [12] of elastic solids with nano-pores to the thermoelastic case and include a rate effect in the holes' response, which results in internal dissipation from experimental evidence [21]; after we make a complete study of the propagation of linear waves. In particular, in §2 we apply the general theory of continua with microstructure to the ellipsoidal case and furnish balance equations and jump conditions; in §3 we present constitutive equations for kinetic energy and co-energy density and for dependent constitutive fields and, after, we use thermodinamic restrictions; in §4 we define small thermoelastic deformations from the reference placement and obtain the linear field balance equations; in §5 we study linear micro-vibrations for which we obtain three admissible modes; in §6 we analyze the propagation of harmonic plane waves and comment the secular equations governing the eight solutions: two shear optical micro-elastic modes, two coupled transverse elastic waves and four coupled longitudinal thermo-elastic waves; in §7 we get the propagation conditions of the macro-acceleration waves for either a heat-conducting or non-conducting isotropic thermoelastic material with nano-pores (corresponding, respectively, to homothermal and homentropic waves), as well as for generalized transverse waves; in §8 we gain the growth equations which govern the propagation of the macro-acceleration waves and discuss the couplings between the higher order discontinuities.…”