**2. Balance laws and jump conditions**

We identify the continuous material body with nano-pores B with a fixed homogeneous and free of residual stresses region of the three dimensional Euclidean space G, called the "natural" reference placement B∗ (see, *e.g.*, §83 of [22]). We suppose that each material element of the continuum contains a nano-pore which is capable to have a microstretch different from, and independent of, the local affine deformation ensuing from the macromotion. Therefore, if we denote the generic material element of B∗ by **x**∗, the thermomechanical behaviour of B is described by three smooth mappings on B∗ × � (� is the set of real numbers): the spatial position **x** ∈ G, at time *τ*, of the material point which occupied the position **x**<sup>∗</sup> in the reference placement B∗, the left Cauchy–Green tensor of the micro-deformation **<sup>U</sup>** <sup>∈</sup> Sym+, at time *<sup>τ</sup>*, of the associated nano-pore (Sym<sup>+</sup> being the collection of second-order symmetric and positive definite tensor fields) and the absolute positive temperature *θ* > 0.

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 B∗ of the body B.

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

$$\mathcal{A}(\mathbf{U}) := \left. \frac{d\mathbf{U}\_s}{d\mathbf{s}} \right|\_{\mathbf{s}=\mathbf{0}} \,. \tag{1}$$

that is, in components:

2 Will-be-set-by-IN-TECH

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

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.

We identify the continuous material body with nano-pores B with a fixed homogeneous and free of residual stresses region of the three dimensional Euclidean space G, called the "natural" reference placement B∗ (see, *e.g.*, §83 of [22]). We suppose that each material element of the continuum contains a nano-pore which is capable to have a microstretch different from, and independent of, the local affine deformation ensuing from the macromotion. Therefore, if we denote the generic material element of B∗ by **x**∗, the thermomechanical behaviour of B is described by three smooth mappings on B∗ × � (� is the set of real numbers): the spatial position **x** ∈ G, at time *τ*, of the material point which occupied the position **x**<sup>∗</sup> in the reference placement B∗, the left Cauchy–Green tensor of the micro-deformation **<sup>U</sup>** <sup>∈</sup> Sym+, at time *<sup>τ</sup>*, of the associated nano-pore (Sym<sup>+</sup> being the collection of second-order symmetric and positive

with, and without, microstructure [19, 20].

**2. Balance laws and jump conditions**

definite tensor fields) and the absolute positive temperature *θ* > 0.

$$\mathcal{A}\_{ijk} = \mathbf{U}\_{il}\mathcal{E}\_{ljk} + \mathcal{E}\_{ikl}\mathbf{U}\_{lj}.\tag{2}$$

A is a third-order tensor, symmetric and positive definite in the first two indices, that is A**c** ∈ Sym<sup>+</sup> for all vectors **c**.

The expression of the kinetic energy density per unit mass of microstructured bodies is the sum of two terms, the classical one <sup>1</sup> <sup>2</sup> **<sup>x</sup>**˙ <sup>2</sup> due to the translational inertia and the microstructured 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 non-negative scalar function, homogeneous in **U**, such that *κ*(**U**, **0**) = 0 and *<sup>∂</sup>*2*<sup>κ</sup> <sup>∂</sup>***U**˙ <sup>2</sup> �= **<sup>0</sup>**, and it is related to the kinetic co-energy density *χ*(**U**, **U**˙ ) by the Legendre transform

$$
\frac{\partial \chi}{\partial \dot{\mathbf{U}}} \cdot \dot{\mathbf{U}} - \chi = \kappa \tag{3}
$$

(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

$$\mathcal{A}^\* \frac{\partial \chi}{\partial \dot{\mathbf{U}}} = -\mathcal{A}^\* \frac{\partial \chi}{\partial \mathbf{U}'} \tag{4}$$

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

$$\text{skw}\left[\dot{\mathbf{U}}\frac{\partial\chi}{\partial\mathbf{U}} + \mathbf{U}\frac{\partial\chi}{\partial\mathbf{U}}\right] = \mathbf{0},\tag{5}$$

where 'skw' denotes the skew part of a second–order tensor: skw (·) :<sup>=</sup> <sup>1</sup> 2 (·) <sup>−</sup> (·)*T* the symmetric one being sym (·) :<sup>=</sup> <sup>1</sup> 2 (·)+(·)*T* .

#### 4 Will-be-set-by-IN-TECH 64 Wave Processes in Classical and New Solids Linear Wave Motions in Continua with Nano-Pores <sup>5</sup>

All the admissible thermo-kinetic processes for porous solids with large irregular voids are governed by the following general system of balance equations proposed in [4]; they are the mass conservation, the Cauchy equation, the micromomentum and moment of momentum balances, the Neumann energy equation and the entropy inequality in the Lagrangian description, respectively:

$$
\rho\_\* = \rho \det \mathbf{F}\_\* \tag{6}
$$

field related to the motion of B (excepting **x**, **U** and *θ*) suffers jump discontinuity across Σ and

where *f* <sup>+</sup> or *f* <sup>−</sup> refers to the limit of *f* as the wave is approached from the right or left,

Therefore, we can write classical Kotchine's equations, as modified in order to take into account microstructural effects, and a relation that restricts the jump of micromomentum (see

**<sup>h</sup>** <sup>−</sup> **<sup>P</sup>***T***x**˙ <sup>−</sup> **<sup>Λ</sup>**∗**U**˙

The form of the jumps across a propagating wave of higher order time derivatives of principal

The first proposal is of kinematic character and leads to an expression for the microstructural kinetic co-energy *χ* for B. We are interested in the linear theory from the next section, so the

where the second-order referential microinertia tensor field **<sup>J</sup>**<sup>∗</sup> <sup>∈</sup> Sym<sup>+</sup> is constant (see [3, 12, 28]). As a consequence of this definition, the left-hand side of Eq. (8) reduces to (*ρ*∗**UJ**¨ <sup>∗</sup>) and we meet also the requirements of positivity and quadratic form for *κ*: in fact from (3) we have

2

In order to study thermodynamic restrictions for thermoelastic materials with nano-pores we must define the Helmholtz free energy density per unit mass *ψ* := *�* − *θη* and insert it in the axiom of dissipation (11); after, by using Eq. (10), we obtain the reduced version of the

Here we generalize constitutive prescriptions presented in [12, 26] by considering both conduction of heat and inelastic surface effects associated to changes in the deformation of nano-pores in the vicinity of the hole boundaries. Therefore, let us call the array S := {**F**, **U**, ∇**U**, *θ*} of independent variables the elastic state of the material with nano–pores and, assuming the equipresence principle, let us postulate the following constitutive relations of

*ψ*˜, *η*˜, **P˜**, **Y˜** , **Λ˜** , **h˜**

**3. Thermodynamic restrictions on the constitutive assumptions**

*<sup>χ</sup>* <sup>=</sup> <sup>1</sup> 2

*<sup>κ</sup>* <sup>=</sup> *<sup>χ</sup>* <sup>=</sup> <sup>1</sup>

 *ρ*∗*υ<sup>n</sup> ∂κ*

> · **n**

[[ *<sup>f</sup>* ]] = *<sup>f</sup>* <sup>+</sup> <sup>−</sup> *<sup>f</sup>* <sup>−</sup>, (14)

Linear Wave Motions in Continua with Nano-Pores 65

*<sup>∂</sup>***U**˙ (**U**, **<sup>U</sup>**˙ ) + **<sup>Λ</sup> <sup>n</sup>**

(**UJ**˙ <sup>∗</sup>) · **<sup>U</sup>**˙ , (17)

(**UJ**˙ <sup>∗</sup>) · **<sup>U</sup>**˙ . (18)

(S, **<sup>U</sup>**˙ , <sup>∇</sup> *<sup>θ</sup>*). (20)

<sup>≤</sup> **<sup>P</sup>** · **<sup>F</sup>**˙ <sup>+</sup> **<sup>Y</sup>** · **<sup>U</sup>**˙ <sup>+</sup> **<sup>Λ</sup>** · ∇ **<sup>U</sup>**˙ <sup>−</sup> *<sup>θ</sup>*−1**<sup>h</sup>** · ∇ *<sup>θ</sup>*. (19)

, [[ *ρ*∗*υnθη*]] ≥ [[ **h** · **n**]]. (16)

= **0**, (15)

so we employ the usual notation [[ ·]] for jumps, so that

[[ *ρ*∗(*υ<sup>n</sup>* − **x**˙ · **n**)]] = 0, [[ *ρ*∗*υn***x**˙ + **P n**]] = **0**,

fields can be obtained from the balance equations (6-8) and (10).

simplest assumption is to assume the function *χ* to be quadratic in **U**˙

{*ψ*, *<sup>η</sup>*, **<sup>P</sup>**, **<sup>Y</sup>**, **<sup>Λ</sup>**, **<sup>h</sup>**} <sup>=</sup>

 =

<sup>2</sup> + *κ*(**U**, **U**˙ )

respectively.

 *ρ*∗*υ<sup>n</sup> �* + 1 2 **x**˙

entropy inequality:

thermoelastic kind:

*ρ*∗ *ψ*˙ + ˙ *θη*

[25]-[27]) as it follows:

$$
\rho\_\* \ddot{\mathbf{x}} = \rho\_\* \mathbf{f} + \text{Div } \mathbf{P}\_\* \tag{7}
$$

$$
\rho\_\* \left[ \frac{d}{d\tau} \left( \frac{\partial \chi}{\partial \dot{\mathbf{U}}} \right) - \frac{\partial \chi}{\partial \mathbf{U}} \right] = \rho\_\* \mathbf{B} - \mathbf{Y} + \text{Div } \mathbf{A}\_\prime \tag{8}
$$

$$\mathcal{E}\left(\mathbf{P}\mathbf{F}^T\right) = \mathcal{A}^\*\mathbf{Y} + \left(\nabla\mathcal{A}^\*\right)\mathbf{A},\tag{9}$$

$$\rho\_\* \,\varepsilon = \mathbf{P} \cdot \dot{\mathbf{F}} + \mathbf{Y} \cdot \dot{\mathbf{U}} + \mathbf{A} \cdot \nabla \,\dot{\mathbf{U}} + \rho\_\* \lambda - \text{Div } \mathbf{h}\_\* \tag{10}$$

$$
\rho\_\* \rho\_i \theta \dot{\eta} \ge \rho\_\* \lambda - \text{Div} \, \mathbf{h} + \theta^{-1} \mathbf{h} \cdot \nabla \theta \,\tag{11}
$$

where *ρ* is the mass density and *ρ*<sup>∗</sup> its referential value; Div means the trace of the nabla: Div (·) := tr(∇ (·)); **f** is the vector body force, **P** the Piola-Kirchhoff stress tensor, *ε* the specific internal energy density per unit mass, *λ* the rate of heat generation due to irradiation or heating supply, **h** the referential heating flux, *η* the density of entropy.

Moreover, on the left hand side of the balance of micro-momentum (8) the Lagrangian derivative of the kinetic co-energy *χ* appears, while, on the right hand side, *ρ*∗**B** and −**Y** are the resultant second-order symmetric tensor densities of external and internal microactions, respectively: the first one is interpreted as a controlled pore pressure and the other one includes interactive forces between the gross and fine structures as well as internal dissipative contributions due to the stir of the pores' surface. Finally, **Λ** is the referential microstress third-order tensor, symmetric in the first two indices, which is related to the capability of recognizing boundary microtractions, even if, in some cases, it expresses weakly non-local internal effects due to the impossibility of defining a physically significant connection on the manifold of the microstructural kinetic parameter **U** (see [24]).

The balance of moment of momentum (9) assumes a more significant expression when we use the representation (2) for A; in fact we have

$$\text{skw}\left(\mathbf{PF}^T\right) = 2\,\text{skw}\left[\mathbf{UY} + \nabla\,\mathbf{U} \odot \mathbf{A}\right] \,\text{\textdegree}\tag{12}$$

where the tensor product � between third-order tensors is so defined:

$$(\nabla \mathbf{U} \odot \Lambda)\_{ij} := \mathbf{U}\_{i\hbar,L} \Lambda\_{j\hbar L}.\tag{13}$$

**Remark.** We observe that the voids theories [1, 5] are immediately recovered when the microstretch **U** is constrained to be spherical (see, also, §5 of [4]).

In addition to balances (6-8) and (10-12), 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<sup>n</sup>* the corresponding non-zero normal speed of displacement of Σ at point (**x**∗, *τ*) in the reference placement. We further assume that some field related to the motion of B (excepting **x**, **U** and *θ*) suffers jump discontinuity across Σ and so we employ the usual notation [[ ·]] for jumps, so that

4 Will-be-set-by-IN-TECH

All the admissible thermo-kinetic processes for porous solids with large irregular voids are governed by the following general system of balance equations proposed in [4]; they are the mass conservation, the Cauchy equation, the micromomentum and moment of momentum balances, the Neumann energy equation and the entropy inequality in the Lagrangian

where *ρ* is the mass density and *ρ*<sup>∗</sup> its referential value; Div means the trace of the nabla: Div (·) := tr(∇ (·)); **f** is the vector body force, **P** the Piola-Kirchhoff stress tensor, *ε* the specific internal energy density per unit mass, *λ* the rate of heat generation due to irradiation

Moreover, on the left hand side of the balance of micro-momentum (8) the Lagrangian derivative of the kinetic co-energy *χ* appears, while, on the right hand side, *ρ*∗**B** and −**Y** are the resultant second-order symmetric tensor densities of external and internal microactions, respectively: the first one is interpreted as a controlled pore pressure and the other one includes interactive forces between the gross and fine structures as well as internal dissipative contributions due to the stir of the pores' surface. Finally, **Λ** is the referential microstress third-order tensor, symmetric in the first two indices, which is related to the capability of recognizing boundary microtractions, even if, in some cases, it expresses weakly non-local internal effects due to the impossibility of defining a physically significant connection on the

The balance of moment of momentum (9) assumes a more significant expression when we use

**Remark.** We observe that the voids theories [1, 5] are immediately recovered when the

In addition to balances (6-8) and (10-12), 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<sup>n</sup>* the corresponding non-zero normal speed of displacement of Σ at point (**x**∗, *τ*) in the reference placement. We further assume that some

*ρ*<sup>∗</sup> = *ρ* det **F**, (6) *ρ*∗**x**¨ = *ρ*∗**f** + Div **P**, (7)

<sup>E</sup> (**PF***T*) = <sup>A</sup>∗**<sup>Y</sup>** + (∇ A∗) **<sup>Λ</sup>**, (9) *<sup>ρ</sup>*<sup>∗</sup> *<sup>ε</sup>*˙ <sup>=</sup> **<sup>P</sup>** · **<sup>F</sup>**˙ <sup>+</sup> **<sup>Y</sup>** · **<sup>U</sup>**˙ <sup>+</sup> **<sup>Λ</sup>** · ∇ **<sup>U</sup>**˙ <sup>+</sup> *<sup>ρ</sup>*∗*<sup>λ</sup>* <sup>−</sup> Div **<sup>h</sup>**, (10) *<sup>ρ</sup>*∗*θη*˙ <sup>≥</sup> *<sup>ρ</sup>*∗*<sup>λ</sup>* <sup>−</sup> Div **<sup>h</sup>** <sup>+</sup> *<sup>θ</sup>*−1**<sup>h</sup>** · ∇ *<sup>θ</sup>*, (11)

skw (**PF***T*) = 2 skw [**UY** <sup>+</sup> <sup>∇</sup> **<sup>U</sup>** � **<sup>Λ</sup>**] , (12)

(∇ **U** � Λ)*ij* := U*ih*,*L*Λ*jhL*. (13)

= *ρ*∗**B** − **Y** + Div **Λ**, (8)

description, respectively:

*ρ*∗ *d dτ*  *∂χ ∂***U**˙ <sup>−</sup> *∂χ ∂***U** 

or heating supply, **h** the referential heating flux, *η* the density of entropy.

manifold of the microstructural kinetic parameter **U** (see [24]).

where the tensor product � between third-order tensors is so defined:

microstretch **U** is constrained to be spherical (see, also, §5 of [4]).

the representation (2) for A; in fact we have

$$\mathbb{E}\left[f\right] = f^+ - f^-,\tag{14}$$

where *f* <sup>+</sup> or *f* <sup>−</sup> refers to the limit of *f* as the wave is approached from the right or left, respectively.

Therefore, we can write classical Kotchine's equations, as modified in order to take into account microstructural effects, and a relation that restricts the jump of micromomentum (see [25]-[27]) as it follows:

$$\left[\rho\_\*(\upsilon\_{\textit{n}} - \dot{\mathbf{x}} \cdot \mathbf{n})\right] = 0, \quad \left[\rho\_\* \upsilon\_{\textit{n}} \dot{\mathbf{x}} + \mathbf{P} \mathbf{n}\right] = \mathbf{0}, \quad \left[\rho\_\* \upsilon\_{\textit{n}} \frac{\partial \kappa}{\partial \dot{\mathbf{U}}}(\mathbf{U}, \mathbf{U}) + \mathbf{A} \,\mathbf{n}\right] = \mathbf{0}, \tag{15}$$

$$\left\| \left[ \rho\_\* \upsilon\_{\hbar} \left( \boldsymbol{\varepsilon} + \frac{1}{2} \dot{\mathbf{x}}^2 + \kappa (\mathbf{U}, \dot{\mathbf{U}}) \right) \right] \right\| = \left[ \left( \mathbf{h} - \mathbf{P}^T \dot{\mathbf{x}} - \boldsymbol{\Lambda}^\* \dot{\mathbf{U}} \right) \cdot \mathbf{n} \right], \left[ \rho\_\* \upsilon\_{\hbar} \theta \eta \right] \geq \left[ \mathbf{h} \cdot \mathbf{n} \right]. \tag{16}$$

The form of the jumps across a propagating wave of higher order time derivatives of principal fields can be obtained from the balance equations (6-8) and (10).

#### **3. Thermodynamic restrictions on the constitutive assumptions**

The first proposal is of kinematic character and leads to an expression for the microstructural kinetic co-energy *χ* for B. We are interested in the linear theory from the next section, so the simplest assumption is to assume the function *χ* to be quadratic in **U**˙

$$\chi = \frac{1}{2} (\dot{\mathbf{U}} \mathbf{J}\_\*) \cdot \dot{\mathbf{U}} \, \tag{17}$$

where the second-order referential microinertia tensor field **<sup>J</sup>**<sup>∗</sup> <sup>∈</sup> Sym<sup>+</sup> is constant (see [3, 12, 28]). As a consequence of this definition, the left-hand side of Eq. (8) reduces to (*ρ*∗**UJ**¨ <sup>∗</sup>) and we meet also the requirements of positivity and quadratic form for *κ*: in fact from (3) we have

$$
\boldsymbol{\kappa} = \boldsymbol{\chi} = \frac{1}{2} (\dot{\mathbf{U}} \mathbf{J}\_\*) \cdot \dot{\mathbf{U}}.\tag{18}
$$

In order to study thermodynamic restrictions for thermoelastic materials with nano-pores we must define the Helmholtz free energy density per unit mass *ψ* := *�* − *θη* and insert it in the axiom of dissipation (11); after, by using Eq. (10), we obtain the reduced version of the entropy inequality:

$$\rho\_\*\left(\dot{\psi} + \dot{\theta}\eta\right) \le \mathbf{P} \cdot \dot{\mathbf{F}} + \mathbf{Y} \cdot \dot{\mathbf{U}} + \mathbf{A} \cdot \nabla \, \dot{\mathbf{U}} - \theta^{-1} \mathbf{h} \cdot \nabla \theta. \tag{19}$$

Here we generalize constitutive prescriptions presented in [12, 26] by considering both conduction of heat and inelastic surface effects associated to changes in the deformation of nano-pores in the vicinity of the hole boundaries. Therefore, let us call the array S := {**F**, **U**, ∇**U**, *θ*} of independent variables the elastic state of the material with nano–pores and, assuming the equipresence principle, let us postulate the following constitutive relations of thermoelastic kind:

$$\{\psi,\eta,\mathbf{P},\mathbf{Y},\mathbf{A},\mathbf{h}\} = \{\tilde{\psi},\tilde{\eta},\tilde{\mathbf{P}},\tilde{\mathbf{Y}},\tilde{\mathbf{A}},\tilde{\mathbf{h}}\} \left(\mathcal{S},\dot{\mathbf{U}},\nabla\theta\right). \tag{20}$$

#### 6 Will-be-set-by-IN-TECH 66 Wave Processes in Classical and New Solids Linear Wave Motions in Continua with Nano-Pores <sup>7</sup>

Now we have to check the compatibility of these prescriptions with the Clausius-Duhem inequality in its reduced version (19) that must be valid for any choice of the fields in the set (S, **<sup>U</sup>**˙ , <sup>∇</sup> *<sup>θ</sup>*). Consequently, by using the chain rule of differentiation, when the terms are appropriately ordered, the inequality reads:

$$\begin{split} & \left(\rho\_{\*}\frac{\partial\boldsymbol{\psi}}{\partial\mathbf{F}} - \mathbf{P}\right) \cdot \mathbf{F} + \left(\rho\_{\*}\frac{\partial\boldsymbol{\psi}}{\partial\mathbf{U}} - \mathbf{Y}\right) \cdot \mathbf{U} + \left(\rho\_{\*}\frac{\partial\boldsymbol{\psi}}{\partial(\nabla\mathbf{U})} - \mathbf{A}\right) \cdot \nabla\cdot\mathbf{U} + \\ & + \rho\_{\*}\left(\eta + \frac{\partial\boldsymbol{\psi}}{\partial\boldsymbol{\theta}}\right)\dot{\boldsymbol{\theta}} + \rho\_{\*}\frac{\partial\boldsymbol{\psi}}{\partial\dot{\mathbf{U}}} \cdot \ddot{\mathbf{U}} + \rho\_{\*}\frac{\partial\boldsymbol{\psi}}{\partial(\nabla\boldsymbol{\theta})} \cdot \nabla\,\dot{\boldsymbol{\theta}} + \frac{1}{\theta}\mathbf{h} \cdot \nabla\,\theta \le 0. \end{split} \tag{21}$$

The left-hand member of 21 is linear with respect to **<sup>F</sup>**˙ , <sup>∇</sup> **<sup>U</sup>**˙ , ˙ *<sup>θ</sup>*, **<sup>U</sup>**¨ and <sup>∇</sup> ˙ *θ*, quantities that take up arbitrary values; thus the respective coefficients in the linear expression must vanish, and hence:

$$\psi = \tilde{\psi}(\mathcal{S}), \quad \mathbf{P} = \rho\_\* \frac{\partial \psi}{\partial \mathbf{F}}, \quad \Lambda = \rho\_\* \frac{\partial \psi}{\partial (\nabla \mathbf{U})}, \quad \eta = -\frac{\partial \psi}{\partial \theta}. \tag{22}$$

These relations mean that the Helmholtz free energy *ψ*, the Piola-Kirchhoff stress tensor **P**, the microstress **Λ** and the entropy *η* depend upon the elastic state of the material only; moreover, **P**, **Λ** and *η* are determined as soon as the constitutive equation for *ψ* is known.

The residual inequality defines the dissipation D of the thermo-kinetic process

$$\mathcal{D} := \mathbf{H} \cdot \dot{\mathbf{U}} + \frac{1}{\theta} \mathbf{h} \cdot \nabla \theta \le 0,\tag{23}$$

**J**<sup>∗</sup> = *κ*∗**I**, where *κ*<sup>∗</sup> ≥ 0 is the non-negative microinertia coefficient depending on the reference

Besides, we may take the displacement field **u**, the infinitesimal strain tensor **E**, the microstrain tensor **V**, with its reference gradient ∇**V**, and the temperature *ϑ* and mass *�* variations from the reference placement as measures of "small" thermoelastic deformations from the reference

**E** := sym (∇**u**), **V** := **U** − **I**, ∇ **V** = ∇ **U**, *ϑ* := *θ* − *θ*<sup>∗</sup> and *�* := *ρ* − *ρ*∗. (28)

Hence, in the linear theory we can change the choice of variables of the elastic state S, the new ones being the following <sup>S</sup>˜ :<sup>=</sup> {**E**, **<sup>V</sup>**, <sup>∇</sup>**V**, *<sup>ϑ</sup>*}; then, in the natural reference placement B∗, it is

As observed in the previous section, the free energy *ψ* determines much of the behaviour of the nano-porous material, thus we suppose that the reference placement B∗ of the body is also a placement of minimum for the free energy and, therefore, we choose the most general homogeneous, quadratic and positive definite form for the free energy *ψ* valid for a centrosymmetric isotropic linear thermoelastic solids with nano-pores (see [12, 30, 31]):

> *κ*∗*λ*<sup>3</sup> <sup>2</sup> (tr **<sup>V</sup>**)

*sm*) Div **V** · Div **V** +

<sup>2</sup> <sup>∇</sup> (tr **<sup>V</sup>**) · ∇ (tr **<sup>V</sup>**) <sup>+</sup> *<sup>γ</sup>*<sup>1</sup>

The positiveness of the expression in (29) assures us that the thirteen constant thermoelastic coefficients *vl*, *vt*, *λ<sup>i</sup>* (*i* = 1, . . . , 6), *γ<sup>j</sup>* (*j* = 1, 2, 3), *vtm* and *vsm* must resolve the following

> *tm* + *<sup>v</sup>*<sup>2</sup> *sm*

*<sup>l</sup>* <sup>−</sup> <sup>4</sup>*v*<sup>2</sup>

*<sup>t</sup>*) > <sup>3</sup>*γ*<sup>2</sup>

*κ*∗ 2 *v*2

*<sup>t</sup>*)(<sup>3</sup> *<sup>λ</sup>*<sup>3</sup> <sup>+</sup> <sup>2</sup> *<sup>λ</sup>*4) <sup>&</sup>gt; *<sup>κ</sup>*∗(<sup>3</sup> *<sup>λ</sup>*<sup>5</sup> <sup>+</sup> *<sup>λ</sup>*6)2, 4 *<sup>λ</sup>*4*v*<sup>2</sup>

> 4(*v*<sup>2</sup>

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>t</sup>*), *<sup>γ</sup>*1(3*λ*<sup>3</sup> <sup>+</sup> <sup>2</sup>*λ*4) <sup>&</sup>gt; <sup>3</sup>*κ*∗*γ*<sup>2</sup>

+ 6*γ*2*γ*3(3 *λ*<sup>5</sup> + *λ*6) >

<sup>2</sup> <sup>+</sup> *<sup>κ</sup>*∗*λ*<sup>4</sup> **<sup>V</sup>** · **<sup>V</sup>** <sup>+</sup> *<sup>κ</sup>*∗*λ*<sup>5</sup> (tr **<sup>E</sup>**) (tr **<sup>V</sup>**) <sup>+</sup>

Linear Wave Motions in Continua with Nano-Pores 67

*sm* ∇**V** · ∇**V** + (29)

<sup>2</sup> *<sup>ϑ</sup>*<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*<sup>2</sup> *<sup>ϑ</sup>* tr **<sup>E</sup>** <sup>+</sup> *<sup>κ</sup>*∗*γ*<sup>3</sup> *<sup>ϑ</sup>* tr **<sup>V</sup>**,

*<sup>t</sup>* <sup>&</sup>gt; *<sup>κ</sup>*∗*λ*<sup>2</sup> 6,

3;

(30)

*sm*)2, *γ*<sup>1</sup> > 0,

<sup>2</sup>. (31)

and **<sup>h</sup>** <sup>=</sup> <sup>−</sup> *<sup>ξ</sup>*∗∇ *<sup>θ</sup>*, (32)

geometric features of the pores.

S˜

<sup>∗</sup> = {**0**, **0**, **O**, 0}.

*<sup>ψ</sup>*(S˜) = (*v*<sup>2</sup>

system of inequalities:

3*v*<sup>2</sup> *<sup>l</sup>* <sup>&</sup>gt; <sup>4</sup>*v*<sup>2</sup>

*v*2 *tm* > *<sup>v</sup>*<sup>2</sup>

*γ*1  *<sup>l</sup>* <sup>−</sup> <sup>2</sup>*v*<sup>2</sup> *t*) <sup>2</sup> (tr **<sup>E</sup>**)

<sup>+</sup> *<sup>κ</sup>*∗*λ*<sup>6</sup> **<sup>E</sup>** · **<sup>V</sup>** <sup>+</sup> *<sup>κ</sup>*∗(*v*<sup>2</sup>

+ *κ*∗*λ*1∇ (tr **V**) · Div **V** +

*<sup>t</sup>* > 0, (3*v*<sup>2</sup>

*<sup>l</sup>* <sup>−</sup> <sup>4</sup>*v*<sup>2</sup>

**H** = −*ρ*∗*κ*<sup>∗</sup>

*sm* > 0, *v*<sup>2</sup>

*<sup>ρ</sup>*∗*κ*−<sup>1</sup> <sup>∗</sup> (3*v*<sup>2</sup>

> 3*γ*<sup>2</sup> <sup>2</sup>*κ*<sup>∗</sup>

placement B∗; they are defined by Eq. (26) and as it follows:

<sup>2</sup> + *v*<sup>2</sup>

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>l</sup>* <sup>−</sup> <sup>4</sup>*v*<sup>2</sup>

<sup>−</sup>1(3 *λ*<sup>3</sup> + 2 *λ*4) + 3*γ*<sup>2</sup>

where tr(·) denotes the trace of a tensor, *i.e.*, tr **E** := **E** · **I**.

*sm*

*<sup>t</sup>* **E** · **E** +

*κ*∗*λ*<sup>2</sup>

6*λ*<sup>1</sup> + 9*λ*<sup>2</sup> + 2*v*<sup>2</sup>

alternatively to the last inequality of relation (30), the following one holds:

balance equations, thus, since **H** must vanish whenever **V**˙ = **0**, we take

*γ*1(3*v*<sup>2</sup>

*ω*(tr **V**˙ )**I** + 2 *σ* **V**˙

*<sup>t</sup>*)(<sup>3</sup> *<sup>λ</sup>*<sup>3</sup> <sup>+</sup> <sup>2</sup> *<sup>λ</sup>*4) <sup>−</sup> (<sup>3</sup> *<sup>λ</sup>*<sup>5</sup> <sup>+</sup> *<sup>λ</sup>*6)<sup>2</sup>

3(3*v*<sup>2</sup>

*<sup>l</sup>* <sup>−</sup> <sup>4</sup>*v*<sup>2</sup>

Finally, we need to express the dissipative part **H** of internal microactions **Y** and the referential heating flux **h** within the same linear approximation as the other constitutive terms in the

where **H** := *ρ*<sup>∗</sup> *∂ψ*˜ *<sup>∂</sup>***<sup>U</sup>** (S) <sup>−</sup> **<sup>Y</sup>**˜(S, **<sup>U</sup>**˙ , <sup>∇</sup> *<sup>θ</sup>*) is the dissipative part of internal microactions, a symmetric second-order tensor.

For a thermally isotropic porous material which is a *definite heat conductor* [29], Fourier's law gives

$$\mathbf{h} = -\boldsymbol{\xi}(\mathcal{S})\,\nabla\theta,\quad\text{with}\quad\boldsymbol{\xi} \ge 0;\tag{24}$$

thus, by using in the energy Eq. (10) relations (22), (24) and *�* = *ψ* + *θη*, we have that, for a thermoelastic medium with nano–pores, it becomes

$$
\rho\_\* \theta \frac{d\eta}{d\tau} + \mathbf{H} \cdot \dot{\mathbf{U}} = \rho\_\* \lambda + \text{Div}\left(\xi^\tau \nabla \theta\right). \tag{25}
$$

#### **4. Linear field equations**

Now we need to introduce the displacement field **u** of a material element of the body: it is

$$\mathbf{u}(\mathbf{x}\_{\*\prime}\tau) := \mathbf{x}(\mathbf{x}\_{\*\prime}\tau) - \mathbf{x}\_{\*\prime} \tag{26}$$

therefore, the deformation gradient **F** is expressed by:

$$\mathbf{F}(\mathbf{x}\_{\*},\tau) = \mathbf{I} + \nabla \mathbf{u} \tag{27}$$

The natural reference placement B∗ is homogeneous, free of residual macro- and micro-stresses, so S∗ = {**I**,**I**, **O**, *θ*∗} and **P**∗, **Y**∗, **Λ**∗, *ψ*<sup>∗</sup> all vanish in B∗ (**I** := (*δik*) is the identity tensor); moreover, the reference microinertia tensor field **J**<sup>∗</sup> has spherical value: **J**<sup>∗</sup> = *κ*∗**I**, where *κ*<sup>∗</sup> ≥ 0 is the non-negative microinertia coefficient depending on the reference geometric features of the pores.

Besides, we may take the displacement field **u**, the infinitesimal strain tensor **E**, the microstrain tensor **V**, with its reference gradient ∇**V**, and the temperature *ϑ* and mass *�* variations from the reference placement as measures of "small" thermoelastic deformations from the reference placement B∗; they are defined by Eq. (26) and as it follows:

$$\mathbf{E} := \text{sym} \left( \nabla \mathbf{u} \right), \quad \mathbf{V} := \mathbf{U} - \mathbf{I}, \quad \nabla \cdot \mathbf{V} = \nabla \cdot \mathbf{U}, \quad \theta := \theta - \theta\_\* \quad \text{and} \quad \boldsymbol{\varrho} := \boldsymbol{\rho} - \boldsymbol{\rho}\_\*. \tag{28}$$

Hence, in the linear theory we can change the choice of variables of the elastic state S, the new ones being the following <sup>S</sup>˜ :<sup>=</sup> {**E**, **<sup>V</sup>**, <sup>∇</sup>**V**, *<sup>ϑ</sup>*}; then, in the natural reference placement B∗, it is S˜ <sup>∗</sup> = {**0**, **0**, **O**, 0}.

As observed in the previous section, the free energy *ψ* determines much of the behaviour of the nano-porous material, thus we suppose that the reference placement B∗ of the body is also a placement of minimum for the free energy and, therefore, we choose the most general homogeneous, quadratic and positive definite form for the free energy *ψ* valid for a centrosymmetric isotropic linear thermoelastic solids with nano-pores (see [12, 30, 31]):

$$\psi(\mathcal{S}) = \frac{(v\_l^2 - 2v\_l^2)}{2} \left( \text{tr} \, \mathbf{E} \right)^2 + v\_l^2 \, \mathbf{E} \cdot \mathbf{E} + \frac{\kappa\_\* \lambda\_3}{2} \left( \text{tr} \, \mathbf{V} \right)^2 + \kappa\_\* \lambda\_4 \, \mathbf{V} \cdot \mathbf{V} + \kappa\_\* \lambda\_5 \left( \text{tr} \, \mathbf{E} \right) \left( \text{tr} \, \mathbf{V} \right) + \frac{\kappa\_\* \lambda\_4}{2} \left( \text{tr} \, \mathbf{V} \right) \left( \text{tr} \, \mathbf{V} \right) + \frac{\kappa\_\*}{2} \lambda\_4 \left( \text{tr} \, \mathbf{V} \right) \left( \text{tr} \, \mathbf{V} \right)$$

$$+ \kappa\_\* \lambda\_6 \, \mathbf{E} \cdot \mathbf{V} + \kappa\_\* (v\_{lm}^2 - v\_{sm}^2) \, \text{Div} \, \mathbf{V} \cdot \text{Div} \, \mathbf{V} + \frac{\kappa\_\*}{2} v\_{sm}^2 \, \nabla \mathbf{V} \cdot \nabla \mathbf{V} + \tag{29}$$

$$\pm \kappa\_\* \lambda\_7 \nabla \left( \text{tr} \, \mathbf{V} \right) \cdot \text{Div} \, \mathbf{V} \mid \quad \nabla \left( \text{tr} \, \mathbf{V} \right) \cdot \nabla \left( \text{tr} \, \mathbf{V} \right) \downarrow \quad \gamma\_1 \, \omega\_1^2 \, \text{ker} \, \theta \, \text{tr} \, \mathbf{E} \mid \, \mathbf{x} \approx \theta \, \text{tr} \, \mathbf{V}$$

$$+\,\kappa\_{\ast}\lambda\_{1}\nabla\left(\text{tr}\,\mathbf{V}\right)\cdot\text{Div}\,\mathbf{V} + \frac{\kappa\_{\ast}\lambda\_{2}}{2}\nabla\left(\text{tr}\,\mathbf{V}\right)\cdot\nabla\left(\text{tr}\,\mathbf{V}\right) + \frac{\gamma\_{1}}{2}\,\theta^{2} + \gamma\_{2}\,\theta\,\text{tr}\,\mathbf{E} + \kappa\_{\ast}\gamma\_{3}\,\theta\,\text{tr}\,\mathbf{V}\,\theta$$

where tr(·) denotes the trace of a tensor, *i.e.*, tr **E** := **E** · **I**.

6 Will-be-set-by-IN-TECH

Now we have to check the compatibility of these prescriptions with the Clausius-Duhem inequality in its reduced version (19) that must be valid for any choice of the fields in the set (S, **<sup>U</sup>**˙ , <sup>∇</sup> *<sup>θ</sup>*). Consequently, by using the chain rule of differentiation, when the terms are

> · **<sup>U</sup>**˙ <sup>+</sup> *ρ*∗

up arbitrary values; thus the respective coefficients in the linear expression must vanish, and

*<sup>∂</sup>***<sup>F</sup>** , **<sup>Λ</sup>** <sup>=</sup> *<sup>ρ</sup>*<sup>∗</sup>

These relations mean that the Helmholtz free energy *ψ*, the Piola-Kirchhoff stress tensor **P**, the microstress **Λ** and the entropy *η* depend upon the elastic state of the material only; moreover,

1

For a thermally isotropic porous material which is a *definite heat conductor* [29], Fourier's law

thus, by using in the energy Eq. (10) relations (22), (24) and *�* = *ψ* + *θη*, we have that, for a

Now we need to introduce the displacement field **u** of a material element of the body: it is

The natural reference placement B∗ is homogeneous, free of residual macro- and micro-stresses, so S∗ = {**I**,**I**, **O**, *θ*∗} and **P**∗, **Y**∗, **Λ**∗, *ψ*<sup>∗</sup> all vanish in B∗ (**I** := (*δik*) is the identity tensor); moreover, the reference microinertia tensor field **J**<sup>∗</sup> has spherical value:

*∂ψ <sup>∂</sup>*(<sup>∇</sup> *<sup>θ</sup>*) · ∇ ˙

> *∂ψ ∂*(∇**U**)

*<sup>∂</sup>***<sup>U</sup>** (S) <sup>−</sup> **<sup>Y</sup>**˜(S, **<sup>U</sup>**˙ , <sup>∇</sup> *<sup>θ</sup>*) is the dissipative part of internal microactions, a

**h** = −*ξ*(S) ∇ *θ*, with *ξ* ≥ 0; (24)

<sup>+</sup> **<sup>H</sup>** · **<sup>U</sup>**˙ <sup>=</sup> *<sup>ρ</sup>*∗*<sup>λ</sup>* <sup>+</sup> Div (*<sup>ξ</sup>* <sup>∇</sup> *<sup>θ</sup>*). (25)

**u**(**x**∗, *τ*) := **x**(**x**∗, *τ*) − **x**∗; (26)

**F**(**x**∗, *τ*) = **I** + ∇**u** (27)

*∂ψ*

*<sup>∂</sup>***U**˙ · **<sup>U</sup>**¨ <sup>+</sup> *<sup>ρ</sup>*<sup>∗</sup>

*∂ψ*

**P**, **Λ** and *η* are determined as soon as the constitutive equation for *ψ* is known. The residual inequality defines the dissipation D of the thermo-kinetic process

<sup>D</sup> :<sup>=</sup> **<sup>H</sup>** · **<sup>U</sup>**˙ <sup>+</sup>

*∂ψ <sup>∂</sup>*(∇**U**) <sup>−</sup> **<sup>Λ</sup>**

> *θ* + 1

*<sup>θ</sup>*, **<sup>U</sup>**¨ and <sup>∇</sup> ˙

, *<sup>η</sup>* <sup>=</sup> <sup>−</sup>*∂ψ*

*<sup>θ</sup>* **<sup>h</sup>** · ∇ *<sup>θ</sup>* <sup>≤</sup> 0, (23)

· ∇ **<sup>U</sup>**˙ <sup>+</sup>

*<sup>θ</sup>* **<sup>h</sup>** · ∇ *<sup>θ</sup>* <sup>≤</sup> 0. (21)

*θ*, quantities that take

*∂θ* . (22)

appropriately ordered, the inequality reads:

 · **<sup>F</sup>**˙ <sup>+</sup> *ρ*∗ *∂ψ <sup>∂</sup>***<sup>U</sup>** <sup>−</sup> **<sup>Y</sup>**

The left-hand member of 21 is linear with respect to **<sup>F</sup>**˙ , <sup>∇</sup> **<sup>U</sup>**˙ , ˙

*<sup>ψ</sup>* <sup>=</sup> *<sup>ψ</sup>*˜(S), **<sup>P</sup>** <sup>=</sup> *<sup>ρ</sup>*<sup>∗</sup>

 *ρ*∗ *∂ψ <sup>∂</sup>***<sup>F</sup>** <sup>−</sup> **<sup>P</sup>**

+*ρ*<sup>∗</sup> *<sup>η</sup>* <sup>+</sup> *∂ψ ∂θ* ˙ *θ* + *ρ*<sup>∗</sup>

hence:

where **H** := *ρ*<sup>∗</sup>

gives

*∂ψ*˜

thermoelastic medium with nano–pores, it becomes

therefore, the deformation gradient **F** is expressed by:

*ρ*∗*θ dη dτ*

symmetric second-order tensor.

**4. Linear field equations**

The positiveness of the expression in (29) assures us that the thirteen constant thermoelastic coefficients *vl*, *vt*, *λ<sup>i</sup>* (*i* = 1, . . . , 6), *γ<sup>j</sup>* (*j* = 1, 2, 3), *vtm* and *vsm* must resolve the following system of inequalities:

$$\begin{split} &3v\_{l}^{2} > 4v\_{l}^{2} > 0, \quad (3v\_{l}^{2} - 4v\_{l}^{2})(3\,\lambda\_{3} + 2\,\lambda\_{4}) > \kappa\_{\*}(3\,\lambda\_{5} + \lambda\_{6})^{2}, \quad 4\,\lambda\_{4}v\_{l}^{2} > \kappa\_{\*}\lambda\_{6}^{2}, \\ &v\_{lm}^{2} > v\_{sm}^{2} > 0, \quad v\_{sm}^{2} \left(6\lambda\_{1} + 9\lambda\_{2} + 2v\_{lm}^{2} + v\_{sm}^{2}\right) > 4(v\_{lm}^{2} - v\_{sm}^{2})^{2}, \quad \gamma\_{1} > 0, \\ &\gamma\_{1}\left[\rho\_{\*}\kappa\_{\*}^{-1}(3v\_{l}^{2} - 4v\_{l}^{2})(3\,\lambda\_{3} + 2\,\lambda\_{4}) - (3\,\lambda\_{5} + \lambda\_{6})^{2}\right] + 6\gamma\_{2}\gamma\_{3}(3\,\lambda\_{5} + \lambda\_{6}) > \\ & > 3\gamma\_{2}^{2}\kappa\_{\*}^{-1}(3\,\lambda\_{3} + 2\,\lambda\_{4}) + 3\gamma\_{3}^{2}(3v\_{l}^{2} - 4v\_{l}^{2}), \quad \gamma\_{1}(3\lambda\_{3} + 2\lambda\_{4}) > 3\kappa\_{\*}\gamma\_{3}^{2}. \end{split} \tag{30}$$

alternatively to the last inequality of relation (30), the following one holds:

$$
\gamma\_1(3v\_l^2 - 4v\_l^2) > 3\gamma\_2^2. \tag{31}
$$

Finally, we need to express the dissipative part **H** of internal microactions **Y** and the referential heating flux **h** within the same linear approximation as the other constitutive terms in the balance equations, thus, since **H** must vanish whenever **V**˙ = **0**, we take

$$\mathbf{H} = -\rho\_\* \kappa\_\* \left[ \omega(\text{tr}\,\dot{\mathbf{V}}) \mathbf{I} + 2\,\sigma \,\dot{\mathbf{V}} \right] \quad \text{and} \quad \mathbf{h} = -\sf{\mathcal{J}}\_\* \nabla \theta \,\tag{32}$$

with *<sup>ω</sup>* and *<sup>σ</sup>* inelastic constants and *<sup>ξ</sup>*<sup>∗</sup> :<sup>=</sup> *<sup>ξ</sup>*(S˜ <sup>∗</sup>). By inserting relations (32) in the dissipation imbalance (23), we obtain in the linear approximation

$$
\rho\_\* \kappa\_\* \left[ \omega (\text{tr} \, \mathbf{V})^2 + 2 \, \sigma \, \mathbf{V}^2 \right] + \frac{\tilde{\xi}\_\*}{\theta} (\nabla \theta)^2 \ge 0,\tag{33}
$$

Moreover, we observe that the balance of moment of momentum (12) in the linear approximation assumes the following more simple expression: skw (**P** − 2**Y**) = **0**, which is

At the end we can write the linear equations of jumps by inserting constitutive relations (35) in Eqs.(15) and (16) and by ignoring terms of higher order: therefore they are the following

*<sup>t</sup>*) Div **u** + *κ*∗*λ*<sup>5</sup> tr **V** + *γ*2*ϑ*

*<sup>t</sup>* [Div (**<sup>u</sup>** <sup>⊗</sup> **<sup>n</sup>**) + <sup>∇</sup> (**<sup>u</sup>** · **<sup>n</sup>**)] <sup>+</sup> *<sup>κ</sup>*∗*λ*6**Vn**

*sm*) sym (Div **<sup>V</sup>** <sup>⊗</sup> **<sup>n</sup>**) <sup>+</sup> *<sup>v</sup>*<sup>2</sup>

 Δ*ν* + 2(*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

[[ *ρ*∗(*υ<sup>n</sup>* − **x**˙ · **n**)]] = 0, (42)

 **n** +

*sm* (∇ **V**) **n**

+ [[*λ*<sup>1</sup> {(**n** · Div **V**)**I** + sym [∇ (tr **V**) ⊗ **n**]} + *λ*<sup>2</sup> [**n** · ∇ (tr **V**)]**I**]] = **O**, (44)

[[*υ<sup>n</sup>* (*γ*1*ϑ* + *γ*<sup>2</sup> Div **u** + *κ*∗*γ*<sup>3</sup> tr **V**) − *γ* ∇ *ϑ* · **n**]] = 0, (45)

−(3*λ*<sup>3</sup> + 2*λ*4) *ν* − (3*ω* + 2*σ*)*ν*˙ − (3*λ*<sup>5</sup> + *λ*6)Div **u** − 3*γ*3*ϑ* + *ι*, (46)

<sup>∇</sup> (Div **<sup>V</sup>***D*)

(<sup>∇</sup> <sup>2</sup>*ν*)*<sup>D</sup>* <sup>−</sup> *<sup>λ</sup>*<sup>6</sup> [sym (<sup>∇</sup> **<sup>u</sup>**)]*<sup>D</sup>* <sup>+</sup> *<sup>κ</sup>*−<sup>1</sup>

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

<sup>3</sup> ∇ *ν*

<sup>⊗</sup> **<sup>n</sup>**

Now we can uncouple the spherical and deviatoric components of the linear balance of

where *<sup>ν</sup>* and *<sup>ι</sup>* are the traces of **<sup>V</sup>** and *<sup>κ</sup>*−<sup>1</sup> <sup>∗</sup> **<sup>B</sup>**, respectively, while the deviatoric part is defined

With the same procedure, the respective jump (44) is splitted in the following spherical and

Micro-vibrations, produced during various operations from railway and/or roads to foot traffic and propagated from one medium to another, are one of the main factor for fatigue

<sup>∇</sup> *<sup>ν</sup>* · **<sup>n</sup>** + [<sup>2</sup> (*v*<sup>2</sup>

Div **V***<sup>D</sup>* + <sup>1</sup>

 +

*sm*) + 3*λ*<sup>1</sup>

*<sup>D</sup>*

= **0**, (43)

Linear Wave Motions in Continua with Nano-Pores 69

Div (Div **<sup>V</sup>***D*) <sup>−</sup>

<sup>∗</sup> **<sup>B</sup>***D*, (47)

 = 0,

(48)

<sup>−</sup> <sup>2</sup>*λ*4**V***<sup>D</sup>* <sup>−</sup> <sup>2</sup>*σ***V**˙ *<sup>D</sup>* <sup>+</sup>

*sm*) + <sup>3</sup> *<sup>λ</sup>*1]Div **<sup>V</sup>***<sup>D</sup>* · **<sup>n</sup>**

*D*

= **O**.

satisfied by the symmetric tensors (35)1,2 identically.

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

last imbalance being satisfied identically by Eq. (45).

*tm* + 2*λ*<sup>1</sup> + 3*λ*<sup>2</sup>

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm*) sym

by: **<sup>A</sup>***<sup>D</sup>* :<sup>=</sup> **<sup>A</sup>** <sup>−</sup> <sup>3</sup>−1(tr **<sup>A</sup>**)**I**, for each symmetric second-order tensor **<sup>A</sup>**.

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm*) 

*sm*) + *λ*<sup>1</sup>

*tm*) + 2 *λ*<sup>1</sup> + 3 *λ*<sup>2</sup>

micromomentum (39) to obtain, respectively:

*sm* <sup>Δ</sup>**V***<sup>D</sup>* + <sup>2</sup>(*v*<sup>2</sup>

ones:

 *υn***u**˙ + (*v*<sup>2</sup> *<sup>l</sup>* <sup>−</sup> <sup>2</sup>*v*<sup>2</sup>

*ν*¨ = 1 3 *v*2 *sm* + 2 3 *v*2

**V**¨ *<sup>D</sup>* = *v*<sup>2</sup>

+ 2 3 (*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

deviatoric relations, respectively:

*sm*

*sm* + 2*v*<sup>2</sup>

<sup>∇</sup> **<sup>V</sup>***D*

**n**+

**5. Micro-vibrations in solids with nano-pores**

sym *λ*1<sup>∇</sup> *<sup>ν</sup>* <sup>+</sup> <sup>2</sup> (*v*<sup>2</sup>

 *υnν*˙ + 1 <sup>3</sup> (*v*<sup>2</sup>

*υn***V**˙ *<sup>D</sup>* + *v*<sup>2</sup>

+  + *v*2

*υn***V**˙ + 2 (*v*<sup>2</sup>

which is verified if and only if

$$
\Im 3\omega + 2\sigma \ge 0, \quad \sigma \ge 0, \quad \text{and} \quad \mathfrak{f}\_\* \ge 0. \tag{34}
$$

Therefore, we derive from constitutive equations (22) and (32) and the definition of **H** (see, also, [12]) the following linear constitutive expressions for the dependent fields:

$$\begin{aligned} \mathbf{P} &= \rho\_\* \left\{ \left[ (v\_l^2 - 2v\_l^2) \operatorname{tr} \mathbf{E} + \kappa\_\* \lambda\_5 \operatorname{tr} \mathbf{V} + \gamma\_2 \theta \right] \mathbf{I} + 2\, v\_l^2 \operatorname{E} + \kappa\_\* \lambda\_6 \mathbf{V} \right\}, \\ \mathbf{Y} &= \rho\_\* \kappa\_\* \left\{ \left[ \operatorname{tr} \left( \lambda\_3 \mathbf{V} + \lambda\_5 \mathbf{E} + \omega \dot{\mathbf{V}} \right) + \gamma\_3 \theta \right] \mathbf{I} + 2\lambda\_4 \mathbf{V} + \lambda\_6 \mathbf{E} + 2\sigma \dot{\mathbf{V}} \right\}, \\ \mathbf{A} &= \rho\_\* \kappa\_\* \left\{ 2 \left( v\_{\rm fm}^2 - v\_{\rm sm}^2 \right) \operatorname{syml} \left( \operatorname{Div} \mathbf{V} \otimes \mathbf{I} \right) + v\_{\rm sm}^2 \nabla \mathbf{V} + \\ & \quad + \lambda\_1 \left[ \mathbf{I} \otimes \operatorname{Div} \mathbf{V} + \operatorname{syml} \left( \nabla \left( \operatorname{tr} \mathbf{V} \right) \otimes \mathbf{I} \right) \right] + \lambda\_2 \mathbf{I} \otimes \nabla \left( \operatorname{tr} \mathbf{V} \right) \right\}, \\ \eta &= -\gamma\_1 \theta - \gamma\_2 \operatorname{tr} \mathbf{E} - \kappa\_\* \gamma\_3 \operatorname{tr} \mathbf{V}, \end{aligned} \tag{35}$$

where the left-symmetric part "syml " of a third-order tensor **Ω** means:

$$(\text{sym}\,\mathbf{1}\,\Omega)\_{ijl} := \frac{1}{2} \left(\Omega\_{ijl} + \Omega\_{jil}\right), \quad \forall i, j, k = 1, 2, 3. \tag{36}$$

In addition we need the expression of the determinant of the deformation gradient **F** in the linear theory:

$$\det \mathbf{F} = \det(\mathbf{I} + \nabla \mathbf{u}) \simeq \mathbf{1} + \text{Div } \mathbf{u} \quad \left[ \Rightarrow (\det \mathbf{F})^{-1} \simeq \mathbf{1} - \text{Div } \mathbf{u} \right]. \tag{37}$$

**Remark.** When the pores are absent, *vl* and *vt* reduce to be the usual propagation speeds of dilatational and distortional waves in the linear isothermal elasticity, respectively.

The balance equations for a thermoelastic solid with nano-pores, governing the mass density *ρ*, the displacement field **u**, the microstrain tensor **V** and the temperature change *ϑ*, are obtained by substituting constitutive relations (17) and (35) and Eq. (37)2 into the Eqs. (6)-(8) and after by using (28)1 and the fact that **J**<sup>∗</sup> = *κ*∗**I**; besides, last equation is get by Eq. (25) when linearized relations (28)4, (32) and (35)4 are applied:

$$
\rho = -\rho\_\* \text{Div}\,\mathbf{u}\_\* \tag{38}
$$

$$\ddot{\mathbf{u}} = v\_t^2 \Delta \mathbf{u} + \nabla \left[ (v\_l^2 - v\_t^2) \text{Div } \mathbf{u} + \kappa\_\* \lambda\_5 \text{ tr } \mathbf{V} + \gamma\_2 \theta \right] + \kappa\_\* \lambda\_6 \text{Div } \mathbf{V} + \mathbf{f},\tag{39}$$

$$\begin{aligned} \ddot{\mathbf{V}} &= v\_{\text{sm}}^2 \Delta \mathbf{V} + 2(v\_{\text{lm}}^2 - v\_{\text{sm}}^2) \text{sym} \left[ \nabla \left( \text{Div} \, \mathbf{V} \right) \right] + \lambda\_1 \nabla^2 \left( \text{tr } \mathbf{V} \right) + \\ &+ \left[ \lambda\_1 \text{Div} \left( \text{Div} \, \mathbf{V} \right) + \lambda\_2 \Delta \left( \text{tr } \mathbf{V} \right) - \text{tr} \left( \lambda\_3 \mathbf{V} + \omega \dot{\mathbf{V}} \right) - \lambda\_5 \text{Div} \, \mathbf{u} - \gamma\_3 \theta \right] \text{I} - \end{aligned} \tag{40}$$

$$-\lambda\_6 \operatorname{sym} \left( \nabla \mathbf{u} \right) - 2\lambda\_4 \mathbf{V} - 2\sigma \dot{\mathbf{V}} + \kappa\_\*^{-1} \mathbf{B} \qquad \text{and}$$

$$0 = \gamma \Delta \theta + \gamma\_1 \dot{\theta} + \gamma\_2 \text{Div } \dot{\mathbf{u}} + \kappa\_\* \gamma\_3 \text{tr } \dot{\mathbf{V}} + \theta\_\*^{-1} \lambda\_\prime \tag{41}$$

with *<sup>γ</sup>* :<sup>=</sup> *<sup>ξ</sup>*∗(*ρ*∗*θ*∗)−<sup>1</sup> <sup>≥</sup> 0.

Moreover, we observe that the balance of moment of momentum (12) in the linear approximation assumes the following more simple expression: skw (**P** − 2**Y**) = **0**, which is satisfied by the symmetric tensors (35)1,2 identically.

At the end we can write the linear equations of jumps by inserting constitutive relations (35) in Eqs.(15) and (16) and by ignoring terms of higher order: therefore they are the following ones:

$$\mathbb{E}\left[\rho\_\* (\upsilon\_n - \dot{\mathbf{x}} \cdot \mathbf{n})\right] = 0,\tag{42}$$

$$\begin{aligned} \left[\boldsymbol{\upsilon}\_{n}\dot{\mathbf{u}} + \left[\left(\boldsymbol{v}\_{l}^{2} - 2\boldsymbol{v}\_{l}^{2}\right)\text{Div }\mathbf{u} + \kappa\_{\*}\lambda\_{5}\operatorname{tr}\mathbf{V} + \gamma\_{2}\theta\right]\mathbf{n}\right] + \\ + \left[\boldsymbol{v}\_{l}^{2}\left[\operatorname{Div}\left(\mathbf{u}\otimes\mathbf{n}\right) + \nabla\left(\mathbf{u}\cdot\mathbf{n}\right)\right] + \kappa\_{\*}\lambda\_{6}\mathbf{V}\mathbf{n}\right] = \mathbf{0}, \end{aligned} \tag{43}$$

$$\begin{aligned} \left[\upsilon\_n \dot{\mathbf{V}} + 2 \left(\upsilon\_{tm}^2 - \upsilon\_{sn}^2\right) \text{sym } \left(\text{Div } \mathbf{V} \otimes \mathbf{n}\right) + \upsilon\_{sn}^2 \left(\nabla \mathbf{V}\right) \mathbf{n}\right] + \\ + \left[\lambda\_1 \left\{\left(\mathbf{n} \cdot \text{Div } \mathbf{V}\right) \mathbf{I} + \text{sym } \left[\nabla \left(\text{tr}\,\mathbf{V}\right) \otimes \mathbf{n}\right]\right\} + \lambda\_2 \left[\mathbf{n} \cdot \nabla \left(\text{tr}\,\mathbf{V}\right)\right] \mathbf{I}\right] = \mathbf{O}, \end{aligned} \tag{44}$$

$$\left[\upsilon\_{\boldsymbol{\theta}}\left(\gamma\_{1}\boldsymbol{\theta}+\gamma\_{2}\operatorname{Div}\mathbf{u}+\kappa\_{\*}\gamma\_{3}\operatorname{tr}\mathbf{V}\right)-\gamma\operatorname{\nabla}\boldsymbol{\theta}\cdot\mathbf{n}\right]=\mathbf{0},\tag{45}$$

last imbalance being satisfied identically by Eq. (45).

8 Will-be-set-by-IN-TECH

Therefore, we derive from constitutive equations (22) and (32) and the definition of **H** (see,

 <sup>+</sup> *<sup>ξ</sup>*<sup>∗</sup>

3*ω* + 2*σ* ≥ 0, *σ* ≥ 0, and *ξ*<sup>∗</sup> ≥ 0. (34)

 **I** + 2 *v*<sup>2</sup>

+*λ*<sup>1</sup> [**I** ⊗ Div **V** + syml (∇ (tr **V**) ⊗ **I**)] + *λ*<sup>2</sup> **I** ⊗ ∇ (tr **V**)},

*�* = −*ρ*∗Div **u**, (38)

*sm*) sym [<sup>∇</sup> (Div **<sup>V</sup>**)] <sup>+</sup> *<sup>λ</sup>*1<sup>∇</sup> <sup>2</sup>(tr **<sup>V</sup>**) +

*ω*(tr **V**˙ )<sup>2</sup> + 2 *σ* **V**˙ <sup>2</sup>

also, [12]) the following linear constitutive expressions for the dependent fields:

*λ*3**V** + *λ*5**E** + *ω***V**˙

where the left-symmetric part "syml " of a third-order tensor **Ω** means:

2 

*<sup>t</sup>*)tr **E** + *κ*∗*λ*<sup>5</sup> tr **V** + *γ*2*ϑ*

 + *γ*<sup>3</sup> *ϑ* 

*sm*) syml (Div **<sup>V</sup>** <sup>⊗</sup> **<sup>I</sup>**) <sup>+</sup> *<sup>v</sup>*<sup>2</sup>

Ω*ijl* + Ω*jil*

In addition we need the expression of the determinant of the deformation gradient **F** in the

**Remark.** When the pores are absent, *vl* and *vt* reduce to be the usual propagation speeds of

The balance equations for a thermoelastic solid with nano-pores, governing the mass density *ρ*, the displacement field **u**, the microstrain tensor **V** and the temperature change *ϑ*, are obtained by substituting constitutive relations (17) and (35) and Eq. (37)2 into the Eqs. (6)-(8) and after by using (28)1 and the fact that **J**<sup>∗</sup> = *κ*∗**I**; besides, last equation is get by Eq. (25)

*<sup>t</sup>*)Div **u** + *κ*∗*λ*<sup>5</sup> tr **V** + *γ*2*ϑ*

*<sup>λ</sup>*1Div (Div **<sup>V</sup>**) + *<sup>λ</sup>*2Δ(tr **<sup>V</sup>**) <sup>−</sup> tr(*λ*3**<sup>V</sup>** <sup>+</sup> *<sup>ω</sup>***V**˙ ) <sup>−</sup> *<sup>λ</sup>*5Div **<sup>u</sup>** <sup>−</sup> *<sup>γ</sup>*3*<sup>ϑ</sup>*

<sup>−</sup>*λ*<sup>6</sup> sym (<sup>∇</sup> **<sup>u</sup>**) <sup>−</sup> <sup>2</sup>*λ*4**<sup>V</sup>** <sup>−</sup> <sup>2</sup>*σ***V**˙ <sup>+</sup> *<sup>κ</sup>*−<sup>1</sup>

<sup>0</sup> <sup>=</sup> *<sup>γ</sup>*Δ*<sup>ϑ</sup>* <sup>+</sup> *<sup>γ</sup>*1*ϑ*˙ <sup>+</sup> *<sup>γ</sup>*2Div **<sup>u</sup>**˙ <sup>+</sup> *<sup>κ</sup>*∗*γ*3tr **<sup>V</sup>**˙ <sup>+</sup> *<sup>θ</sup>*−<sup>1</sup> <sup>∗</sup> *<sup>λ</sup>*, (41)

dilatational and distortional waves in the linear isothermal elasticity, respectively.

<sup>∗</sup>). By inserting relations (32) in the dissipation

*<sup>t</sup>* **E** + *κ*∗*λ*6**V**

**I** + 2 *λ*<sup>4</sup> **V** + *λ*<sup>6</sup> **E** + 2 *σ* **V**˙

<sup>⇒</sup> (det **<sup>F</sup>**)−<sup>1</sup> � <sup>1</sup> <sup>−</sup> Div **<sup>u</sup>**

*<sup>θ</sup>* (<sup>∇</sup> *<sup>θ</sup>*)<sup>2</sup> <sup>≥</sup> 0, (33)

 ,

*sm*∇ **V** + (35)

, ∀ *i*, *j*, *k* = 1, 2, 3. (36)

+ *κ*∗*λ*<sup>6</sup> Div **V** + **f**, (39)

<sup>∗</sup> **<sup>B</sup>** and

**I** − (40)

. (37)

 ,

with *<sup>ω</sup>* and *<sup>σ</sup>* inelastic constants and *<sup>ξ</sup>*<sup>∗</sup> :<sup>=</sup> *<sup>ξ</sup>*(S˜

which is verified if and only if

**P** = *ρ*<sup>∗</sup>

**u**¨ = *v*<sup>2</sup>

**V**¨ = *v*<sup>2</sup>

+

with *<sup>γ</sup>* :<sup>=</sup> *<sup>ξ</sup>*∗(*ρ*∗*θ*∗)−<sup>1</sup> <sup>≥</sup> 0.

*<sup>t</sup>* Δ**u** + ∇

*sm* <sup>Δ</sup>**<sup>V</sup>** + <sup>2</sup>(*v*<sup>2</sup>

linear theory:

**Y** = *ρ*∗*κ*<sup>∗</sup>

 (*v*<sup>2</sup> *<sup>l</sup>* <sup>−</sup> <sup>2</sup>*v*<sup>2</sup>

**<sup>Λ</sup>** <sup>=</sup> *<sup>ρ</sup>*∗*κ*∗{<sup>2</sup> (*v*<sup>2</sup>

tr

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

(syml **<sup>Ω</sup>**)*ijl* :<sup>=</sup> <sup>1</sup>

det **F** = det(**I** + ∇**u**) � 1 + Div **u**

when linearized relations (28)4, (32) and (35)4 are applied:

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

 (*v*<sup>2</sup> *<sup>l</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*η* = −*γ*1*ϑ* − *γ*<sup>2</sup> tr **E** − *κ*∗*γ*<sup>3</sup> tr **V**,

imbalance (23), we obtain in the linear approximation

*ρ*∗*κ*<sup>∗</sup> 

> Now we can uncouple the spherical and deviatoric components of the linear balance of micromomentum (39) to obtain, respectively:

$$\dot{\boldsymbol{\nu}} = \left(\frac{1}{3}\boldsymbol{v}\_{\rm sm}^{2} + \frac{2}{3}\boldsymbol{v}\_{\rm lm}^{2} + 2\lambda\_{1} + 3\lambda\_{2}\right)\Delta\boldsymbol{\nu} + \left[2(\boldsymbol{v}\_{\rm lm}^{2} - \boldsymbol{v}\_{\rm sm}^{2}) + 3\lambda\_{1}\right] \text{Div}\left(\text{Div}\,\mathbf{V}^{D}\right) - $$
 
$$ -(3\lambda\_{3} + 2\lambda\_{4})\boldsymbol{\nu} - (3\boldsymbol{\omega} + 2\boldsymbol{\sigma})\boldsymbol{\dot{\nu}} - (3\lambda\_{5} + \lambda\_{6})\text{Div}\,\mathbf{u} - 3\gamma\_{3}\boldsymbol{\theta} + \boldsymbol{\iota}\_{\rm s} \tag{46}$$
 
$$\boldsymbol{\omega} \quad \boldsymbol{\kappa} \quad \boldsymbol{\nu} \ge 0$$

$$
\ddot{\mathbf{V}}^D = v\_{\rm sm}^2 \Delta \mathbf{V}^D + 2(v\_{\rm tm}^2 - v\_{\rm sm}^2) \left\{ \text{sym} \left[ \nabla \left( \text{Div} \, \mathbf{V}^D \right) \right] \right\}^D - 2\lambda\_4 \mathbf{V}^D - 2\sigma \dot{\mathbf{V}}^D + 
$$

$$
+ \left[ \frac{2}{3} (v\_{\rm tm}^2 - v\_{\rm sm}^2) + \lambda\_1 \right] (\nabla^2 \nu)^D - \lambda\_6 \left[ \text{sym} \left( \nabla \, \mathbf{u} \right) \right]^D + \kappa\_\*^{-1} \mathbf{B}^D \tag{47}
$$

where *<sup>ν</sup>* and *<sup>ι</sup>* are the traces of **<sup>V</sup>** and *<sup>κ</sup>*−<sup>1</sup> <sup>∗</sup> **<sup>B</sup>**, respectively, while the deviatoric part is defined by: **<sup>A</sup>***<sup>D</sup>* :<sup>=</sup> **<sup>A</sup>** <sup>−</sup> <sup>3</sup>−1(tr **<sup>A</sup>**)**I**, for each symmetric second-order tensor **<sup>A</sup>**.

With the same procedure, the respective jump (44) is splitted in the following spherical and deviatoric relations, respectively:

$$\begin{aligned} \left[\upsilon\_{n}\dot{\upsilon} + \left[\frac{1}{3}(\upsilon\_{\rm sm}^{2} + 2\upsilon\_{\rm tm}^{2}) + 2\,\lambda\_{1} + 3\,\lambda\_{2}\right] \nabla\,\upsilon\cdot\mathbf{n} + \left[2\left(\upsilon\_{\rm tm}^{2} - \upsilon\_{\rm sm}^{2}\right) + 3\,\lambda\_{1}\right] \text{Div}\,\mathbf{V}^{D}\cdot\mathbf{n}\right] &= 0, \\ \left[\upsilon\_{\rm n}\dot{\mathbf{V}}^{D} + \upsilon\_{\rm sm}^{2}\left(\nabla\,\mathbf{V}^{D}\right)\mathbf{n} + \\ &+ \left\{\text{sym}\left\{\left[\lambda\_{1}\nabla\upsilon + 2\left(\upsilon\_{\rm tm}^{2} - \upsilon\_{\rm sm}^{2}\right)\left(\text{Div}\,\mathbf{V}^{D} + \frac{1}{3}\nabla\upsilon\right)\right] \otimes \mathbf{n}\right\}\right\}^{D}\right] = \mathbf{O}. \end{aligned} \tag{48}$$

### **5. Micro-vibrations in solids with nano-pores**

Micro-vibrations, produced during various operations from railway and/or roads to foot traffic and propagated from one medium to another, are one of the main factor for fatigue in structures; moreover, they could also cause serious damages in producing micro and nano scale equipments, other than errors during experiments in high-precision laboratories equipped with lasers, sensors or microscopes.

To study their propagation, let us assume that external volume contributions are null, *i.e.* **f** = **0**, **B** = **O** and *λ* = 0; moreover, there are no macro-displacements in the system, *i.e.* **u** = **0** and *�* = 0, then let us consider solutions of Eqs. (39), (46), (47) and (41) of the form of thermal micro-vibrations in absence of dissipation as the following ones:

$$\boldsymbol{\nu} = \boldsymbol{\Psi}\,\boldsymbol{e}^{i\boldsymbol{b}\boldsymbol{\tau}},\ \mathbf{V}^{D} = \mathbf{\mathcal{V}}\,\boldsymbol{e}^{i\boldsymbol{b}\boldsymbol{\tau}},\ \boldsymbol{\theta} = \boldsymbol{\theta}\,\boldsymbol{e}^{i\boldsymbol{b}\boldsymbol{\tau}},\ \boldsymbol{\omega} = \boldsymbol{\sigma} = \mathbf{0},\tag{49}$$

where **w**, *μ*, **S** *ϑ*¯ and *�*¯ are constants which represent the wave amplitude of, respectively, the macro-displacement vector, the micro-strain trace, the deviatoric part of the micro-strain tensor and the temperature and mass fluctuation; besides, the wave function *φ* can be

where *δ* is the wave number; *b* > 0 is the frequency; *a*(*b*) > 0 and *c*(*b*) are the wave attenuation and the wave speed, respectively; **n** is the unit vector representing the direction of wave propagation, while the unit vector **w**ˆ defines the direction of motion. The specific loss *l* is

Again we suppose that all external sources are zero, *i.e.* **f** = **0**, **B** = **O** and *λ* = 0, hence, by substituting Eqs. (54) and (55) in the linear system (38), (39), (46), (47) and (41), we obtain the

(*iγδ*<sup>2</sup> <sup>−</sup> *<sup>b</sup>γ*1)*ϑ*¯ <sup>+</sup> *<sup>γ</sup>*2*bδ***<sup>w</sup>** · **<sup>n</sup>** <sup>−</sup> *<sup>κ</sup>*∗*γ*3*b<sup>ν</sup>* <sup>=</sup> 0 and *�*¯ <sup>=</sup> *<sup>ρ</sup>*∗*ψhn*. (59)

This algebraic system of eleven equations may be combined into five independent systems through linear combinations of those equations: two uncoupled relations, two coupled systems of two equations each and one coupled system of five equations. The study of all five systems needs the introduction of two unit vectors, **e** and **f**, in the plane orthogonal to the direction of propagation **n** and such that **e** · **f** = 0. Therefore, we have the following particular

From the deviatoric Eq. (58), we obtain two independent dispersion relations relating

frequencies *b* and wave numbers *δ*; they are two different shear optical micro-waves :

*smδ*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>2</sup>*iσ<sup>b</sup>*

*smδ*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>2</sup>*iσ<sup>b</sup>*

<sup>2</sup> + 4 *σ*2*b*<sup>2</sup>

These shear optical micro-modes propagate with attenuation *as* = *<sup>σ</sup><sup>c</sup>*

(2*λ*<sup>4</sup> − *<sup>b</sup>*2)

*λ*6**Sn** +

 *λ*<sup>6</sup> <sup>3</sup> <sup>+</sup> *<sup>λ</sup>*<sup>5</sup>

*sm*)*δ*<sup>2</sup> [sym (**Sn** <sup>⊗</sup> **<sup>n</sup>**)]*<sup>D</sup>* <sup>+</sup>

*sm*) + *λ*<sup>1</sup>

*<sup>δ</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>3</sup>*λ*<sup>3</sup> <sup>−</sup> *ib*(2*<sup>σ</sup>* <sup>+</sup> <sup>3</sup>*ω*)

 *μ* **n** 

*<sup>δ</sup>*2(**Sn**) · **<sup>n</sup>** <sup>+</sup> (*λ*<sup>6</sup> <sup>+</sup> <sup>3</sup>*λ*5) *<sup>δ</sup>*(**<sup>w</sup>** · **<sup>n</sup>**) <sup>−</sup> <sup>3</sup>*γ*3*ϑ*¯ <sup>=</sup> 0, (57)

 *μ* +

Linear Wave Motions in Continua with Nano-Pores 71

*Se f* = 0 and (60)

(*See* − *Sf f*) = 0, (61)

and velocity given

*v*2 *sm*

without modifying the thermo-elastic

*<sup>δ</sup>*2*μ*(**<sup>n</sup>** <sup>⊗</sup> **<sup>n</sup>**)*<sup>D</sup>* <sup>=</sup> **<sup>O</sup>**, (58)

<sup>−</sup> *<sup>γ</sup>*2*δϑ*¯**<sup>n</sup>** <sup>=</sup> **<sup>0</sup>**, (56)

*φ*(**x**∗, *τ*) = exp(*ibτ* − *δ* **n** · **x**∗) with *δ* := *a* + *ib*/*c*, (55)

generally represented by the real (or the imaginary) part of a complex function

*<sup>t</sup>*)*δ*2(**<sup>w</sup>** · **<sup>n</sup>**)**<sup>n</sup>** <sup>−</sup> *<sup>κ</sup>*∗*<sup>δ</sup>*

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

defined by *l* := 4*π*

following relations:

(*b*<sup>2</sup> + *v*<sup>2</sup>

 *b*<sup>2</sup> + 1 3 *v*2 *sm* + 2 3 *v*2

 *b*<sup>2</sup> + *v*<sup>2</sup>

occurrences.

by *c*<sup>2</sup>

*<sup>s</sup>* <sup>=</sup> *<sup>v</sup>*<sup>2</sup> *sm* 2*σ*<sup>2</sup> 

**6.1. Shear optical waves**

 *ac b* .

*<sup>t</sup> <sup>δ</sup>*2)**<sup>w</sup>** + (*v*<sup>2</sup>

+ 2(*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*smδ*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>2</sup>*iσ<sup>b</sup>*

*<sup>l</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

+*λ*6*δ* [sym (**w** ⊗ **n**)]

 *b*<sup>2</sup> + *v*<sup>2</sup>

 *b*<sup>2</sup> + *v*<sup>2</sup>

where the subscripts indicates tensor components.

<sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> <sup>+</sup>

*tm* + 2*λ*<sup>1</sup> + 3*λ*<sup>2</sup>

*sm*) + 3*λ*<sup>1</sup>

**S** + 2(*v*<sup>2</sup>

*<sup>D</sup>* + 2 3 (*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

where *ν*ˆ, **V**ˆ and *ϑ*ˆ are constant amplitudes, *b* is the frequency and *i* is the imaginary unit. Then Eq. (39) is satisfied identically, while the other equations become

$$\left(b^{2} - 3\lambda\_{3} - 2\lambda\_{4}\right)\hat{\boldsymbol{\upsilon}} = 3\gamma\_{3}\hat{\boldsymbol{\theta}}, \quad \left(b^{2} - \lambda\_{4}\right)\hat{\mathbf{V}}^{D} = \mathbf{O}, \quad \gamma\_{1}\hat{\boldsymbol{\theta}} + \kappa\_{\*}\gamma\_{3}\hat{\boldsymbol{\upsilon}} = 0. \tag{50}$$

Therefore we have the following admissible results:

◦) *dilatational mode:*

$$b\_d = \sqrt{3\lambda\_3 + 2\lambda\_4 - 3\kappa\_\*\gamma\_1^{-1}\gamma\_{3'}^2} \tag{51}$$

$$\hat{\mathbf{V}}\_{11} = \hat{\mathbf{V}}\_{22} = \hat{\mathbf{V}}\_{33} \ (\Rightarrow \hat{\mathbf{v}} = 3\,\hat{\mathbf{V}}\_{11}), \quad \hat{\mathbf{V}}\_{ij} = \mathbf{0}, \ \forall \ i \neq j, \quad \hat{\boldsymbol{\theta}} = -\kappa\_\*\gamma\_1^{-1}\gamma\_3\hat{\mathbf{v}}\_{i'} $$

we observe that the frequency *bd* of this spatio-thermal oscillation is real for the restriction (30)8 of the free energy density *ψ* to be a positive definite form;

◦) *extensional modes with a constant volume:*

$$b\_{\varepsilon} = \sqrt{\lambda}\_{4\prime} \quad \text{ $\hat{\imath} = \hat{\theta} = 0$ } \left( \Rightarrow \hat{\mathbf{V}}\_{33} = -\hat{\mathbf{V}}\_{11} - \hat{\mathbf{V}}\_{22} \right) , \quad \hat{\mathbf{V}}\_{i\bar{j}} = 0 , \forall \ \mathbf{i} \neq \mathbf{j}; \tag{52}$$

also in these modes the frequency *be* of the micro-oscillations is real for the restriction (30)3, while no thermal vibrations are present;

◦) *shear modes:*

$$b\_{\rm s} = \sqrt{\lambda}\_{4\prime} \quad \hat{\mathbf{V}}\_{\rm ij} \neq 0, \forall \ i \neq j, \quad \hat{\mathbf{V}}\_{\rm ii} = 0, \forall \ i, \Rightarrow \vartheta = \hat{\boldsymbol{\theta}} = 0; \tag{53}$$

their frequency *bs* coincides with the real frequency *be* of the extensional modes and neither here there are thermal vibrations.

**Remark.** When we neglect thermic phenomena, our oscillating solutions recover three of the mechanical micro-vibrations obtained for general microstructure in [17].

#### **6. Dispersion relations for plane waves**

Now we draw here some results on the propagation of plane wave motion in a linear thermoelastic solids with big pores. We seek solutions of the system of linear balance Eqs. (38), (39), (46), (47) and (41) in the form of traveling harmonic waves (see, also, [32]):

$$\mathbf{u} = \boldsymbol{\phi}(\mathbf{x}\_{\ast}, \boldsymbol{\tau}) \text{ w. } \boldsymbol{\nu} = \mu \, \boldsymbol{\phi}(\mathbf{x}\_{\ast}, \boldsymbol{\tau}), \text{ } \mathbf{V}^{D} = \boldsymbol{\phi}(\mathbf{x}\_{\ast}, \boldsymbol{\tau}) \text{ } \mathbf{S}, \text{ } \boldsymbol{\theta} = \boldsymbol{\bar{\theta}} \, \boldsymbol{\phi}(\mathbf{x}\_{\ast}, \boldsymbol{\tau}), \text{ } \boldsymbol{\varrho} = \boldsymbol{\bar{\varrho}} \, \boldsymbol{\phi}(\mathbf{x}\_{\ast}, \boldsymbol{\tau})\tag{54}$$

where **w**, *μ*, **S** *ϑ*¯ and *�*¯ are constants which represent the wave amplitude of, respectively, the macro-displacement vector, the micro-strain trace, the deviatoric part of the micro-strain tensor and the temperature and mass fluctuation; besides, the wave function *φ* can be generally represented by the real (or the imaginary) part of a complex function

$$\phi(\mathbf{x}\_{\*},\tau) = \exp(ib\tau - \delta \,\mathbf{n} \cdot \mathbf{x}\_{\*}) \quad \text{with} \quad \delta := a + ib/c,\tag{55}$$

where *δ* is the wave number; *b* > 0 is the frequency; *a*(*b*) > 0 and *c*(*b*) are the wave attenuation and the wave speed, respectively; **n** is the unit vector representing the direction of wave propagation, while the unit vector **w**ˆ defines the direction of motion. The specific loss *l* is defined by *l* := 4*π ac b* .

Again we suppose that all external sources are zero, *i.e.* **f** = **0**, **B** = **O** and *λ* = 0, hence, by substituting Eqs. (54) and (55) in the linear system (38), (39), (46), (47) and (41), we obtain the following relations:

$$(b^2 + v\_l^2 \delta^2) \mathbf{w} + (v\_l^2 - v\_t^2) \delta^2 (\mathbf{w} \cdot \mathbf{n}) \mathbf{n} - \kappa\_\* \delta \left[ \lambda\_6 \mathbf{S} \mathbf{n} + \left( \frac{\lambda\_6}{3} + \lambda\_5 \right) \mu \ \mathbf{n} \right] - \gamma\_2 \delta \bar{\theta} \mathbf{n} = \mathbf{0}, \text{ (56)}$$

$$\begin{split} \left[ b^2 + \left( \frac{1}{3} v\_{\text{sm}}^2 + \frac{2}{3} v\_{\text{lm}}^2 + 2 \lambda\_1 + 3 \lambda\_2 \right) \delta^2 - 2 \lambda\_4 - 3 \lambda\_3 - ib (2\sigma + 3\omega) \right] \mu + \\ &+ \left[ 2 (v\_{\text{lm}}^2 - v\_{\text{sm}}^2) + 3 \lambda\_1 \right] \delta^2 (\mathbf{S} \mathbf{n}) \cdot \mathbf{n} + (\lambda\_6 + 3 \lambda\_5) \delta (\mathbf{w} \cdot \mathbf{n}) - \mathcal{Y} \gamma\_3 \bar{\theta} = 0, \quad \text{(57)} \end{split}$$

$$\begin{aligned} \left(b^2 + v\_{\rm sm}^2 \delta^2 - 2\lambda\_4 - 2i\sigma b\right) \mathbf{S} + 2(v\_{\rm tm}^2 - v\_{\rm sm}^2) \delta^2 \left[\text{sym } (\mathbf{S} \mathbf{n} \otimes \mathbf{n})\right]^D + \\ + \lambda\_6 \delta \left[\text{sym } (\mathbf{w} \otimes \mathbf{n})\right]^D + \left[\frac{2}{3}(v\_{\rm tm}^2 - v\_{\rm sm}^2) + \lambda\_1\right] \delta^2 \mu (\mathbf{n} \otimes \mathbf{n})^D &= \mathbf{O}, \end{aligned} \tag{58}$$

$$
\rho\_1(i\gamma\delta^2 - b\gamma\gamma\_1)\ddot{\theta} + \gamma\_2 b \delta \mathbf{w} \cdot \mathbf{n} - \kappa\_\* \gamma\_3 b \nu = 0 \quad \text{and} \quad \ddot{\varrho} = \rho\_\* \psi h\_n. \tag{59}
$$

This algebraic system of eleven equations may be combined into five independent systems through linear combinations of those equations: two uncoupled relations, two coupled systems of two equations each and one coupled system of five equations. The study of all five systems needs the introduction of two unit vectors, **e** and **f**, in the plane orthogonal to the direction of propagation **n** and such that **e** · **f** = 0. Therefore, we have the following particular occurrences.

#### **6.1. Shear optical waves**

10 Will-be-set-by-IN-TECH

in structures; moreover, they could also cause serious damages in producing micro and nano scale equipments, other than errors during experiments in high-precision laboratories

To study their propagation, let us assume that external volume contributions are null, *i.e.* **f** = **0**, **B** = **O** and *λ* = 0; moreover, there are no macro-displacements in the system, *i.e.* **u** = **0** and *�* = 0, then let us consider solutions of Eqs. (39), (46), (47) and (41) of the form of thermal

*ibτ*, *ϑ* = *ϑ*ˆ *e*

<sup>3</sup>*λ*<sup>3</sup> <sup>+</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>3</sup>*κ*∗*γ*−<sup>1</sup>

**<sup>V</sup>**<sup>ˆ</sup> <sup>11</sup> <sup>=</sup> **<sup>V</sup>**<sup>ˆ</sup> <sup>22</sup> <sup>=</sup> **<sup>V</sup>**<sup>ˆ</sup> <sup>33</sup> (<sup>⇒</sup> *<sup>ν</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>3</sup> **<sup>V</sup>**<sup>ˆ</sup> <sup>11</sup>), **<sup>V</sup>**<sup>ˆ</sup> *ij* <sup>=</sup> 0, <sup>∀</sup> *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*, *<sup>ϑ</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup>*κ*∗*γ*−<sup>1</sup>

we observe that the frequency *bd* of this spatio-thermal oscillation is real for the restriction

also in these modes the frequency *be* of the micro-oscillations is real for the restriction (30)3,

their frequency *bs* coincides with the real frequency *be* of the extensional modes and neither

**Remark.** When we neglect thermic phenomena, our oscillating solutions recover three of the

Now we draw here some results on the propagation of plane wave motion in a linear thermoelastic solids with big pores. We seek solutions of the system of linear balance Eqs.

**<sup>u</sup>** <sup>=</sup> *<sup>φ</sup>*(**x**∗, *<sup>τ</sup>*) **<sup>w</sup>**, *<sup>ν</sup>* <sup>=</sup> *μ φ*(**x**∗, *<sup>τ</sup>*), **<sup>V</sup>***<sup>D</sup>* <sup>=</sup> *<sup>φ</sup>*(**x**∗, *<sup>τ</sup>*) **<sup>S</sup>**, *<sup>ϑ</sup>* <sup>=</sup> *ϑ φ*¯ (**x**∗, *<sup>τ</sup>*), *�* <sup>=</sup> *� φ*¯ (**x**∗, *<sup>τ</sup>*) (54)

(38), (39), (46), (47) and (41) in the form of traveling harmonic waves (see, also, [32]):

<sup>1</sup> *<sup>γ</sup>*<sup>2</sup> 3,

*<sup>λ</sup>*4, *<sup>ν</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>ϑ</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>0</sup> (<sup>⇒</sup> **<sup>V</sup>**<sup>ˆ</sup> <sup>33</sup> <sup>=</sup> <sup>−</sup>**V**<sup>ˆ</sup> <sup>11</sup> <sup>−</sup> **<sup>V</sup>**<sup>ˆ</sup> <sup>22</sup>), **<sup>V</sup>**<sup>ˆ</sup> *ij* <sup>=</sup> 0, <sup>∀</sup> *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*; (52)

*<sup>λ</sup>*4, **<sup>V</sup>**<sup>ˆ</sup> *ij* �<sup>=</sup> 0, <sup>∀</sup> *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*, **<sup>V</sup>**<sup>ˆ</sup> *ii* <sup>=</sup> 0, <sup>∀</sup> *<sup>i</sup>*, <sup>⇒</sup> *<sup>ν</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>ϑ</sup>*<sup>ˆ</sup> <sup>=</sup> 0; (53)

where *ν*ˆ, **V**ˆ and *ϑ*ˆ are constant amplitudes, *b* is the frequency and *i* is the imaginary unit. Then

 *<sup>b</sup>*<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*<sup>4</sup>  *ibτ*, *ω* = *σ* = 0, (49)

**<sup>V</sup>**<sup>ˆ</sup> *<sup>D</sup>* <sup>=</sup> **<sup>O</sup>**, *<sup>γ</sup>*1*ϑ*<sup>ˆ</sup> <sup>+</sup> *<sup>κ</sup>*∗*γ*3*ν*<sup>ˆ</sup> <sup>=</sup> 0. (50)

<sup>1</sup> *γ*3*ν*ˆ;

(51)

equipped with lasers, sensors or microscopes.

micro-vibrations in absence of dissipation as the following ones:

Eq. (39) is satisfied identically, while the other equations become

*ν*ˆ = 3*γ*3*ϑ*ˆ,

*bd* = 

(30)8 of the free energy density *ψ* to be a positive definite form;

mechanical micro-vibrations obtained for general microstructure in [17].

Therefore we have the following admissible results:

*ibτ*, **V***<sup>D</sup>* = **V**ˆ *e*

*ν* = *ν*ˆ *e*

◦) *dilatational mode:*

◦) *shear modes:*

*<sup>b</sup>*<sup>2</sup> <sup>−</sup> <sup>3</sup>*λ*<sup>3</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup>

◦) *extensional modes with a constant volume: be* <sup>=</sup> <sup>√</sup>

while no thermal vibrations are present;

*bs* <sup>=</sup> <sup>√</sup>

**6. Dispersion relations for plane waves**

here there are thermal vibrations.

From the deviatoric Eq. (58), we obtain two independent dispersion relations relating frequencies *b* and wave numbers *δ*; they are two different shear optical micro-waves :

$$\left(b^2 + v\_{\rm sm}^2 \delta^2 - 2\lambda\_4 - 2i\sigma b\right) S\_{\varepsilon f} = 0 \quad \text{and} \tag{60}$$

$$\left(b^2 + v\_{\rm sm}^2 \delta^2 - 2\lambda\_4 - 2i\sigma b\right) \left(\mathcal{S}\_{\rm ce} - \mathcal{S}\_{ff}\right) = 0,\tag{61}$$

where the subscripts indicates tensor components.

The these shear optical micro-modes propagate with attenuation  $a\_s = \frac{\sigma \varepsilon}{v\_{su}^2}$  and velocity given by  $c\_s^2 = \frac{v\_{su}^2}{2\sigma^2} \left[2\lambda\_4 - b^2 + \sqrt{\left(2\lambda\_4 - b^2\right)^2 + 4\sigma^2 b^2}\right]$  without modifying the thermo-elastic

#### 12 Will-be-set-by-IN-TECH 72 Wave Processes in Classical and New Solids Linear Wave Motions in Continua with Nano-Pores <sup>13</sup>

features of the matrix material of the porous medium; then the specific loss is *ls* = 4*π* <sup>2</sup>*λ*4−*b*<sup>2</sup> <sup>2</sup>*b<sup>σ</sup>* + 1 + <sup>2</sup>*λ*4−*b*<sup>2</sup> 2*bσ* 2 .

For high frequencies all quantities grow with *b*, while for low frequencies, the speed and the attenuation approach *vsm*√2*λ*4/*<sup>σ</sup>* and <sup>√</sup>2*λ*4/*vsm*, respectively, while *ls* is big. Moreover, it is also possible a static solution with attenuation *as* = <sup>√</sup>2*λ*<sup>4</sup> *vsm* .

#### **6.2. Transverse waves**

We get also two different systems of transverse waves, for *j* = *e*, *f* , from Eqs. (56) and (58):

$$\left(\left(b^{2} + v\_{t}^{2}\delta^{2}\right)w\_{j} - \kappa\_{\*}\lambda\_{6}\delta\right.\left.S\_{nj}\right] = 0,\tag{62}$$

The amplitude ratio *R* of the micro-wave to the macro-wave is obtained from (62):

large frequencies gives a ratio *Rtm* very big, while at low frequencies it is constant.

*<sup>t</sup> <sup>a</sup> Snj*, for *<sup>j</sup>* <sup>=</sup> *<sup>e</sup>*, *<sup>f</sup>* .

At the end, we can also have a static solution with attenuation *atm* =

*v*2

*λ*6*Snn* +

*tm* + 2*λ*<sup>1</sup> + 3*λ*<sup>2</sup>

 2 3 (*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

waves, the only ones which present thermal effects:

*<sup>l</sup> <sup>δ</sup>*2)*wn* <sup>−</sup> *<sup>κ</sup>*∗*<sup>δ</sup>*

+ (*λ*<sup>6</sup> + 3*λ*5) *δwn* +

 *b*<sup>2</sup> + 1 3 4*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm*

equivoluminal microelastic wave, the third one *δ*<sup>2</sup>

amplitudes are related by *wj* = *<sup>κ</sup>*∗*λ*<sup>6</sup>

**6.3. Longitudinal waves**

(*b*<sup>2</sup> + *v*<sup>2</sup>

2 3

(69)2.

and *δ*<sup>2</sup>

one *δ*<sup>2</sup>

*λ*6*δ wn* +

 *b*<sup>2</sup> + 1 3 *v*2 *sm* + 2 3 *v*2 <sup>=</sup> (*b*<sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup>

For the solution predominantly elastic *δ<sup>t</sup>* at large frequencies, the ratio *Rt* is a constant; at small frequencies it approaches zero with the frequency itself. Instead, the micro-mode *δtm* at

The remaining equations of the system (56)-(59) furnish the solutions for the longitudinal

+ 2 3 2 3 (*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

Last equation rules the propagation of mass wave and it relates the mass amplitude *�*¯ directly with that of normal displacement *wn*, so when this is calculated, the first one is get by Eq.

For the residual amplitudes *wn*, *μ*, *Sn* and *ϑ*¯, we must pose the determinant of their coefficients equal to zero in order to have a nontrivial solution of the system (66)-(69)1; therefore, we get a 4th-order equation in *δ*2, which can be resolved with the Ferrari-Cardano derivation of the quartic formula, after the application of the Tchirnhaus transformation by mean of numerical techniques (see [32] and [35]). By the way, an exact analytical solution of the dispersion relation for longitudinal waves is very complicated and without interest to be reported here explicitly: instead we are concerned to summarize the behaviour of all wave numbers *δ*<sup>2</sup>

*th*, which are dominated by displacement, deviatoric and spherical parts of microstrain and thermal fields, respectively. Hence, there exist four coupled longitudinal waves: the first

*<sup>e</sup>* is predominantly an elastic wave of dilatation, the second one *δ*<sup>2</sup>

 *μ* 

*sm*) + *λ*<sup>1</sup>

*<sup>δ</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>2</sup>*iσ<sup>b</sup>*

*<sup>γ</sup>*2*b<sup>δ</sup> wn* <sup>−</sup> *<sup>κ</sup>*∗*γ*3*b<sup>μ</sup>* + (*iγδ*<sup>2</sup> <sup>−</sup> *<sup>b</sup>γ*1)*ϑ*¯ <sup>=</sup> 0 and *�*¯ <sup>=</sup> *<sup>ρ</sup>*∗*ψwn*. (69)

*<sup>δ</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>3</sup>*λ*<sup>3</sup> <sup>−</sup> *ib*(2*<sup>σ</sup>* <sup>+</sup> <sup>3</sup>*ω*)

*sm*) + *λ*<sup>1</sup>

 *λ*<sup>6</sup> <sup>3</sup> <sup>+</sup> *<sup>λ</sup>*<sup>5</sup> *<sup>t</sup> <sup>δ</sup>*2)

*<sup>κ</sup>*∗*λ*6*<sup>δ</sup>* . (65)

Linear Wave Motions in Continua with Nano-Pores 73

<sup>√</sup>2*λ*4(1−*ζ*)

<sup>−</sup> *<sup>γ</sup>*2*δϑ*¯ <sup>=</sup> 0, (66)

*<sup>δ</sup>*2*Snn* <sup>−</sup> <sup>3</sup>*γ*3*ϑ*¯ <sup>=</sup> 0, (67)

*δ*2*μ* = 0,

 *μ* +

*Snn* + (68)

*<sup>e</sup>* , *δ*<sup>2</sup> *<sup>d</sup>*, *<sup>δ</sup>*<sup>2</sup> *v*

*<sup>d</sup>* is associated with an

*<sup>v</sup>* is predominantly a volume fraction wave

*vtm* in which the

*<sup>R</sup>* <sup>=</sup> *Snj wj*

*λ*6*δ wj* + 2 *b*<sup>2</sup> + *v*<sup>2</sup> *tmδ*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>2</sup>*iσ<sup>b</sup> Snj* = 0. (63)

This homogeneous system has a nontrivial solution for the amplitudes *wj* and *Snj* if and only if the following dispersion relation is satisfied by *δ*:

$$2\left(v\_t^2\delta^2 + b^2\right)\left(v\_{lm}^2\delta^2 + b^2 - 2\lambda\_4 - 2i\sigma b\right) + \kappa\_\*\lambda\_6^2\delta^2 = 0.\tag{64}$$

Eq. (64) is similar to the dispersion relation for plane thermoelastic waves studied in [33] and our analysis will use results there obtained. The first transverse solution of (64) is associated predominantly with the elastic properties of the material (*vt*) and denoted by *δt*; the second one, *δtm*, with the properties governing elastic and dissipative changes in porosity (*vtm*, *λ*4, *λ*<sup>6</sup> and *σ*).

The analytical solutions of the dispersion relation (64) are quite cryptic and they are summarized in Table 1 of [34] (*modulo* some innocuous identification in notations); in this work we only report their physical interpretation without big difficulties. The coupling of motion Eqs. (56) and (58) of linear macro- and micro-momentum does the wave of dispersive kind, while the presence of big voids adds a dissipative mechanism associated with nano-pores which yields both waves to attenuate. If the dissipation coefficient *σ* is null, then we recover the presence of the resonance.

When frequencies are low, the elastic wave propagates with speed *vt* <sup>√</sup><sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*, where 0 <sup>≤</sup> *ζ* :<sup>=</sup> *<sup>κ</sup>*∗*λ*<sup>2</sup> 6 4*λ*4*v*<sup>2</sup> *t* < 1 for the inequality (30)3, while the attenuation coefficient and the specific loss remain very small and approach zero with the frequency itself. The predominantly micro-transverse wave propagates with constant speed *vtm σ* 2*λ*4(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*) and with constant attenuation <sup>√</sup>2*λ*4(1−*ζ*) *vtm* ; nevertheless, its specific loss *ltm* <sup>=</sup> <sup>2</sup>*λ*4(1−*ζ*) *<sup>σ</sup><sup>b</sup>* is large and in inverse proportion to the frequency *b*.

At high frequencies, the predominantly elastic transverse wave propagates with the classical speed *vt* and, as the frequency approaches infinity, the attenuation coefficient *ate* and the specific loss *lte* are very small and approach zero, as for low frequencies. Instead the predominantly micro-transverse wave propagates with attenuation *σ*/*vtm* and with constant speed *vtm*, but with a small specific loss which approach zero when the frequency approaches infinity.

The amplitude ratio *R* of the micro-wave to the macro-wave is obtained from (62):

$$R = \frac{S\_{nj}}{w\_{\!\!\!/}} = \frac{(b^2 + v\_t^2 \delta^2)}{\kappa\_\* \lambda\_6 \delta}. \tag{65}$$

For the solution predominantly elastic *δ<sup>t</sup>* at large frequencies, the ratio *Rt* is a constant; at small frequencies it approaches zero with the frequency itself. Instead, the micro-mode *δtm* at large frequencies gives a ratio *Rtm* very big, while at low frequencies it is constant.

At the end, we can also have a static solution with attenuation *atm* = <sup>√</sup>2*λ*4(1−*ζ*) *vtm* in which the amplitudes are related by *wj* = *<sup>κ</sup>*∗*λ*<sup>6</sup> *v*2 *<sup>t</sup> <sup>a</sup> Snj*, for *<sup>j</sup>* <sup>=</sup> *<sup>e</sup>*, *<sup>f</sup>* .

#### **6.3. Longitudinal waves**

12 Will-be-set-by-IN-TECH

features of the matrix material of the porous medium; then the specific loss is *ls* =

For high frequencies all quantities grow with *b*, while for low frequencies, the speed and the

We get also two different systems of transverse waves, for *j* = *e*, *f* , from Eqs. (56) and (58):

This homogeneous system has a nontrivial solution for the amplitudes *wj* and *Snj* if and only

*tmδ*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>2</sup>*iσ<sup>b</sup>*

Eq. (64) is similar to the dispersion relation for plane thermoelastic waves studied in [33] and our analysis will use results there obtained. The first transverse solution of (64) is associated predominantly with the elastic properties of the material (*vt*) and denoted by *δt*; the second one, *δtm*, with the properties governing elastic and dissipative changes in porosity (*vtm*, *λ*4, *λ*<sup>6</sup>

The analytical solutions of the dispersion relation (64) are quite cryptic and they are summarized in Table 1 of [34] (*modulo* some innocuous identification in notations); in this work we only report their physical interpretation without big difficulties. The coupling of motion Eqs. (56) and (58) of linear macro- and micro-momentum does the wave of dispersive kind, while the presence of big voids adds a dissipative mechanism associated with nano-pores which yields both waves to attenuate. If the dissipation coefficient *σ* is null, then we recover

loss remain very small and approach zero with the frequency itself. The predominantly

At high frequencies, the predominantly elastic transverse wave propagates with the classical speed *vt* and, as the frequency approaches infinity, the attenuation coefficient *ate* and the specific loss *lte* are very small and approach zero, as for low frequencies. Instead the predominantly micro-transverse wave propagates with attenuation *σ*/*vtm* and with constant speed *vtm*, but with a small specific loss which approach zero when the frequency approaches

*vtm* ; nevertheless, its specific loss *ltm* <sup>=</sup> <sup>2</sup>*λ*4(1−*ζ*)

< 1 for the inequality (30)3, while the attenuation coefficient and the specific

*σ*

<sup>√</sup>2*λ*4/*<sup>σ</sup>* and <sup>√</sup>2*λ*4/*vsm*, respectively, while *ls* is big. Moreover, it is

*<sup>t</sup> <sup>δ</sup>*2) *wj* <sup>−</sup> *<sup>κ</sup>*∗*λ*6*<sup>δ</sup> Snj* <sup>=</sup> 0, (62)

<sup>+</sup> *<sup>κ</sup>*∗*λ*<sup>2</sup>

*Snj* = 0. (63)

<sup>6</sup>*δ*<sup>2</sup> <sup>=</sup> 0. (64)

<sup>√</sup><sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*, where 0 <sup>≤</sup>

2*λ*4(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*) and with constant

*<sup>σ</sup><sup>b</sup>* is large and in inverse

<sup>√</sup>2*λ*<sup>4</sup> *vsm* .

*tmδ*<sup>2</sup> <sup>−</sup> <sup>2</sup>*λ*<sup>4</sup> <sup>−</sup> <sup>2</sup>*iσ<sup>b</sup>*

4*π* 

and *σ*).

*ζ* :<sup>=</sup> *<sup>κ</sup>*∗*λ*<sup>2</sup> 6 4*λ*4*v*<sup>2</sup> *t* 

attenuation

infinity.

<sup>2</sup>*λ*4−*b*<sup>2</sup> <sup>2</sup>*b<sup>σ</sup>* +  1 +

attenuation approach *vsm*

**6.2. Transverse waves**

<sup>2</sup>*λ*4−*b*<sup>2</sup> 2*bσ*

also possible a static solution with attenuation *as* =

(*b*<sup>2</sup> + *v*<sup>2</sup>

if the following dispersion relation is satisfied by *δ*:

2 *v*2 *<sup>t</sup> <sup>δ</sup>*<sup>2</sup> + *<sup>b</sup>*<sup>2</sup>

the presence of the resonance.

<sup>√</sup>2*λ*4(1−*ζ*)

proportion to the frequency *b*.

*λ*6*δ wj* + 2

 *v*2

 *b*<sup>2</sup> + *v*<sup>2</sup>

When frequencies are low, the elastic wave propagates with speed *vt*

micro-transverse wave propagates with constant speed *vtm*

2 .

> The remaining equations of the system (56)-(59) furnish the solutions for the longitudinal waves, the only ones which present thermal effects:

$$(b^2 + v\_l^2 \delta^2) w\_n - \kappa\_\* \delta \left[\lambda\_6 \text{S}\_{nn} + \left(\frac{\lambda\_6}{3} + \lambda\_5\right) \mu\right] - \gamma\_2 \delta \bar{\theta} = 0,\tag{66}$$

$$\left[b^2 + \left(\frac{1}{3} v\_{sm}^2 + \frac{2}{3} v\_{tm}^2 + 2\lambda\_1 + 3\lambda\_2\right) \delta^2 - 2\lambda\_4 - 3\lambda\_3 - ib(2\sigma + 3\omega)\right] \mu + $$

$$+ \left(\lambda\_6 + 3\lambda\_5\right) \delta w\_n + \left[\frac{2}{3} (v\_{tm}^2 - v\_{sm}^2) + \lambda\_1\right] \delta^2 S\_{nn} - 3\gamma\_3 \bar{\theta} = 0,\tag{67}$$

$$\frac{2}{3}\lambda\_6 \delta \, w\_n + \left[b^2 + \frac{1}{3}\left(4v\_{tm}^2 - v\_{sm}^2\right)\delta^2 - 2\lambda\_4 - 2irb\right]S\_{mn} + \tag{68}$$

$$\begin{aligned} +\frac{2}{3}\left[\frac{2}{3}(v\_{tm}^2 - v\_{sm}^2) + \lambda\_1\right]\delta^2\mu &= 0, \\ \gamma\_2 b\delta\,w\_n - \kappa\_\*\gamma\_3 b\mu + (i\gamma\delta^2 - b\gamma\_1)\bar{\theta} &= 0 \quad \text{and} \quad \bar{\varrho} = \rho\_\*\psi w\_n. \end{aligned} \tag{69}$$

Last equation rules the propagation of mass wave and it relates the mass amplitude *�*¯ directly with that of normal displacement *wn*, so when this is calculated, the first one is get by Eq. (69)2.

For the residual amplitudes *wn*, *μ*, *Sn* and *ϑ*¯, we must pose the determinant of their coefficients equal to zero in order to have a nontrivial solution of the system (66)-(69)1; therefore, we get a 4th-order equation in *δ*2, which can be resolved with the Ferrari-Cardano derivation of the quartic formula, after the application of the Tchirnhaus transformation by mean of numerical techniques (see [32] and [35]). By the way, an exact analytical solution of the dispersion relation for longitudinal waves is very complicated and without interest to be reported here explicitly: instead we are concerned to summarize the behaviour of all wave numbers *δ*<sup>2</sup> *<sup>e</sup>* , *δ*<sup>2</sup> *<sup>d</sup>*, *<sup>δ</sup>*<sup>2</sup> *v* and *δ*<sup>2</sup> *th*, which are dominated by displacement, deviatoric and spherical parts of microstrain and thermal fields, respectively. Hence, there exist four coupled longitudinal waves: the first one *δ*<sup>2</sup> *<sup>e</sup>* is predominantly an elastic wave of dilatation, the second one *δ*<sup>2</sup> *<sup>d</sup>* is associated with an equivoluminal microelastic wave, the third one *δ*<sup>2</sup> *<sup>v</sup>* is predominantly a volume fraction wave

#### 14 Will-be-set-by-IN-TECH 74 Wave Processes in Classical and New Solids Linear Wave Motions in Continua with Nano-Pores <sup>15</sup>

of pure dilatation, the last one corresponding to *δ*<sup>2</sup> *th* is similar in character to a thermal wave; from numerical outcomes, the two micro-waves result to be slower than elastic and thermal waves, the elastic one being the fastest: this is in accordance with experimental evidence.

microstrain tensor **V**, which is, in our context, directly related to the left Cauchy-Green tensor

**Remark:** It is noteworthy that our definition of macro-acceleration wave differs from usual definitions of acceleration waves in, *e.g.*, [27, 38, 39] because they examine singularities of order 2 for both macro- and micro-structural kinematic variables **u** and **V**. Moreover, our study of standard acceleration waves in [26] shows that, in the linear theory, these jumps are in general uncoupled unless the instantaneous acoustic macro- and micro-tensor have some eigenvalue coincident (see, also, the comments about the non-linear theory in §8 of [39]).

The normal velocity of displacement *υ<sup>n</sup>* of the macro-acceleration wave is continuous everywhere in the body and, hence, the following Hugoniot-Hadamard compatibility condition for the jump across Σ of the derivatives of an arbitrary field **Ψ** in B∗ holds (see,

In the linear approximation, jump Eqs. (42) and (43) along a macro-acceleration wave Σ are identically satisfied; instead the jump balance of energy (45) reduces to the Fourier condition

where Eq. (71) were used. Thus, *in linear porous thermoelasticity*, the first derivatives of *ϑ* are

while the jumps of the balance laws (39) and (41) and of the derivative of Eq. (38) furnish

Now we can use the Hugoniot-Hadamard condition (71) to get a system of algebraic equations where the amplitudes of the discontinuities [[ *ρ*˙]], [[ **u**¨]], [[ **V**˙ ]] and [[ *ϑ*¨]] are the unknown quantities. With this end in view, we employ the following definitions of instantaneous *homothermal acoustic macro-tensor* U(**n** ⊗ **n**) and *micro-tensor* C(**n** ⊗ **n**) (see, for analogy,

*sm*) sym ([[ Div **V**]] ⊗ **n**) +

*<sup>t</sup>*)[[ ∇ (Div **u**)]] + *κ*<sup>∗</sup> (*λ*5[[ ∇ *ν*]] + *λ*6[[ Div **V**]]), (74)

+*λ*<sup>1</sup> [(**n** · [[ div **V**]])**I** + sym ([[ ∇ *ν*]] ⊗ **n**)] + *λ*<sup>2</sup> (**n** · [[ ∇ *ν*]])**I** = **O**, (73)

*γ* [[ Δ*ϑ*]] + *γ*2[[ Div **u**˙ ]] + *κ*∗*γ*3[[ *ν*˙]] = 0, [[ *ρ*˙]] + *ρ*∗[[ Div **u**˙ ]] = 0. (75)

<sup>−</sup> *<sup>γ</sup>*[[ <sup>∇</sup> *<sup>ϑ</sup>*]] · **<sup>n</sup>** <sup>=</sup> *γυ*−<sup>1</sup>

*tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>n</sup>* [[ **<sup>Ψ</sup>**˙ ]] <sup>⊗</sup> **<sup>n</sup>**. (70)

Linear Wave Motions in Continua with Nano-Pores 75

*<sup>n</sup>* [[ *<sup>ϑ</sup>*˙]] = 0, (72)

*<sup>υ</sup>n*[[ <sup>∇</sup> **<sup>Ψ</sup>**]] = <sup>−</sup>[[ **<sup>Ψ</sup>**˙ ]] <sup>⊗</sup> **<sup>n</sup>**. (71)

[[ <sup>∇</sup> **<sup>Ψ</sup>**]] = <sup>∇</sup> [[ **<sup>Ψ</sup>**]] <sup>−</sup> *<sup>υ</sup>*−<sup>1</sup>

In particular, if the field **Ψ** is continuous in B∗, Eq. (70) reduces to

continuous and *every macro-acceleration wave is homothermal*. The last jump condition (44) gives the following relation

*sm*[[ <sup>∇</sup> **<sup>V</sup>**]] **<sup>n</sup>** <sup>+</sup> <sup>2</sup>(*v*<sup>2</sup>

*<sup>l</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

**U** of the micro-deformation.

*e.g.*, (2.7)2 of [40]):

**7.1. Homothermal case**

*υn*[[ **V**˙ ]] + *v*<sup>2</sup>

[[ **u**¨]] = *v*<sup>2</sup>

[26, 39]):

these other ones, when Eq. (72) is also used:

*<sup>t</sup>* [[ <sup>Δ</sup>**u**]] + (*v*<sup>2</sup>

By disregarding micro-rotation and thermal effects, we are able to observe that the micro-wave solution above can be acknowledged in some developments of §8 of [17] for elastic plates and, peculiarly, the velocity of the elastic wave is less than that which would be calculated for classical elasticity *vl* due to the phenomenon of the compliance of pores. In addition, if we neglect non-spherical contributions to the microstrain in constitutive equations (35), we find solutions of voids theory [36].

The longitudinal waves are all dispersive in character, because of the coupling of Eqs. (66)-(69)1, and suffer attenuation (the thermal mode with a large coefficient) due to the thermal coupling and to the presence of voids. Furthermore, if the two dissipation coefficient *σ* and *ω* are zero, we can observe the phenomenon of the resonance.

For low frequencies, there is no damping effect in either of the four modes. The only significant wave is the *δe*-one, because the other three modes almost do not exist and their attenuation coefficients remain very small and approach zero with the frequency itself; instead the velocity *vl* of the *δe*-wave is increased by a small amount due to the thermomechanical coupling, but decreased significantly because of nano-porosity effects; the attenuation is a quite small constant.

If frequencies are high, the *δ<sup>d</sup>* micro-wave, of speed *v*<sup>2</sup> *<sup>d</sup>* <sup>=</sup> <sup>2</sup>*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm* + *λ*<sup>1</sup> > 0 by inspection, and the *δ<sup>v</sup>* one, which travels with velocity *v*<sup>2</sup> *<sup>v</sup>* = <sup>5</sup> 3 *v*2 *sm* <sup>−</sup> <sup>2</sup> 3 *v*2 *tm* + 3*λ*<sup>2</sup> > 0 for (30)4,5, are not accompanied by elastic or thermal modes, which are instead coupled, and vice versa; attenuation coefficients for micro-modes remains small but constant. Elastic mode propagates with the classical speed *vl* and attenuation coefficient which approaches zero slowly with the frequency itself; instead the propagation velocity and the attenuation coefficient of the thermal wave *δth* sharply increase with the frequency itself, being diffusive in nature. We notice that the high frequency limits of the two micro-elastic waves correspond to the velocities of acceleration waves in the same material, undeformed and at rest, obtained in [26].

Finally, it can be observed numerically that the specific loss *l* is significantly large when the wave velocity has quite small value in some regions of frequency. The loss due to energy dissipation is comparatively high in case of *δ<sup>e</sup>* and *δth*-modes and moderate for predominantly dilatation micro-elastic modes.
