**1. Introduction**

The first attempts to describe the behaviour of porous materials with the use of an additional degree of kinematical freedom, in order to refine the Cauchy's theory, are due principally to Nunziato and Cowin [1, 2] and co-workers. Nevertheless, their voids theory can be considered as a particular case of a general theory of continua with microstructure [3] and so, when we have to consider more complex media with nano-pores, we need to use suggestions of this last theory [4]. In fact, a nano-pore in a thermoelastic solid is roughly ellipsoidal, unlike small lacunae finely dispersed in the solid matrix that can be supposed all spherical and for which the volume fraction suffices to describe the microdeformation (see, also, [5, 6]). Cowin itself remarked the importance of the shape of the holes in the description of lacunae containing osteocytes or of bone canaliculi [7, 8]: in the human bone, *e.g.*, the lacunae are almost ellipsoidal with mean values along the axes of about 4 *μ*m, 9 *μ*m and 22 *μ*m. And, as a matter of fact, the voids theory does not predict size effects in torsion of bars in an isotropic material, while they occur both in torsion and in bending, as observed for bones and polymer foam materials in [9]. Even if some problem of physical concreteness or of mathematical hardness could arise [10, 11], a better improvement of the voids theory, within a microstructured scheme, is necessary in order to characterize the more complex structure.

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 "tortuosity" matrix, a macroscopic geometrical symmetric tensor that expresses the effects of the geometry of the microscopic pores' surface, was previously presented in [13], but in [14] the model of a microstretched medium has been firstly used to study materials with distributions of aligned ellipsoidal vacuous pores and explicit computations have been carried

©2012 Giovine, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 62 Wave Processes in Classical and New Solids Linear Wave Motions in Continua with Nano-Pores <sup>3</sup>

out for matrix materials subjected to axisymmetric and plane strain loading conditions; then the model has been used to formulate more general multiphase theories of microstructured media [15]. Furthermore, some tension tests and numerical results have been presented in [16] for a similar model of a microstretched medium. 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]). Finally, our model [4] was used to analyze nonlinear wave propagation in constrained porous media [18] and to examine adsorption and diffusion of polluttants in soils, viewed as an immiscible mixture of materials with, and without, microstructure [19, 20].

The spatial position **x**(**x**∗, *τ*) is a one-to-one correspondence, for each *τ*, between the reference placement B∗ and the current placement B*<sup>τ</sup>* = **x**(B∗, *τ*) of the body B and, so, the deformation gradient **F** := ∇**x**(**x**∗, *τ*) (= *∂***x**/*∂***x**∗) is a second order tensor with positive determinant. Through the inverse mapping **x**∗(**x**, *τ*) of **x**, we can consider all the relevant fields in the theory as defined over the current placement B*<sup>τ</sup>* = **x**(B∗, *τ*) as well as over the reference placement

Hence, a body with nano-pores is like a medium with ellipsoidal microstructure [4] and a rotation **<sup>Q</sup>** <sup>=</sup> <sup>e</sup>−E**<sup>s</sup>** of the observer of characteristic vector **<sup>s</sup>**, where <sup>E</sup> is Ricci's permutation tensor and e the basis of natural logarithms, causes the symmetric tensor **U** to change into **<sup>U</sup>***<sup>s</sup>* <sup>=</sup> **QUQ***T*; moreover, the infinitesimal generator <sup>A</sup> of the group of rotations on the microstructure in Sym+, *i.e.*, the operator describing the effect of a rotation of the observer

on the value **U***<sup>s</sup>* of the microstructure to the first order in **s** (see §3 of [3]), is given by

<sup>A</sup>(**U**) :<sup>=</sup> *<sup>d</sup>***U***<sup>s</sup>*

*d***s s**=**0**

A is a third-order tensor, symmetric and positive definite in the first two indices, that is A**c** ∈

The expression of the kinetic energy density per unit mass of microstructured bodies is the

one *κ*(**U**, **U**˙ ) due to the inertia related to the admissible expansional micromotions of the pores' boundaries (the superposed dot denotes material time derivative). This additional term is a

(see, also, [23]). The kinetic co-energy *χ*, as *κ*, must have the same value for all observers at rest, *i.e.*, it must be invariant under the Galilean group and hence satisfy the condition

*<sup>∂</sup>***U**˙ <sup>=</sup> −A<sup>∗</sup> *∂χ*

where the third-order tensor A<sup>∗</sup> is defined through the relation (A∗**C**) · **c** := **C** · (A**c**), for all second-order tensors **C** and all vectors **c**. The use of Eq. (2) into Eq. (4) and the multiplication of both sides by the Ricci's tensor E gives the following kinematic compatibility relation

> *<sup>∂</sup>***U**˙ <sup>+</sup> **<sup>U</sup>** *∂χ ∂***U**

(·)+(·)*T*

.

non-negative scalar function, homogeneous in **U**, such that *κ*(**U**, **0**) = 0 and *<sup>∂</sup>*2*<sup>κ</sup>*

*∂χ*

<sup>A</sup>˙ <sup>∗</sup> *∂χ*

skw

**<sup>U</sup>**˙ *∂χ*

where 'skw' denotes the skew part of a second–order tensor: skw (·) :<sup>=</sup> <sup>1</sup>

2 

related to the kinetic co-energy density *χ*(**U**, **U**˙ ) by the Legendre transform

, (1)

Linear Wave Motions in Continua with Nano-Pores 63

*<sup>∂</sup>***U**˙ <sup>2</sup> �= **<sup>0</sup>**, and it is

A*ijk* = U*il*E*ljk* + E*ikl*U*lj*. (2)

<sup>2</sup> **<sup>x</sup>**˙ <sup>2</sup> due to the translational inertia and the microstructured

*<sup>∂</sup>***U**˙ · **<sup>U</sup>**˙ <sup>−</sup> *<sup>χ</sup>* <sup>=</sup> *<sup>κ</sup>* (3)

*<sup>∂</sup>***U**, (4)

= **0**, (5)

2  (·) <sup>−</sup> (·)*T*

B∗ of the body B.

that is, in components:

Sym<sup>+</sup> for all vectors **c**.

sum of two terms, the classical one <sup>1</sup>

the symmetric one being sym (·) :<sup>=</sup> <sup>1</sup>

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.
