**7. Macro-acceleration waves**

We are now in a position to study acceleration wave propagation. We shall consider the surfaces Σ that are *weak singularities*, defined as those carrying only jumps of the derivatives of order 2 of the macro- and micro-displacement vectors and of order 1 of the thermal variables; these singular surfaces of order 2 are called *macro-acceleration waves* and all external forces and supplies, *i.e.* **f**, **B** and *λ*, are supposed continuous across them with all the derivatives (see, also, [1, 37]).

Therefore, these peculiar discontinuity surfaces suffer 2nd-order derivative jumps of the displacement **u** and 1st-order derivative jumps of the temperature variation *ϑ* and of the microstrain tensor **V**, which is, in our context, directly related to the left Cauchy-Green tensor **U** of the micro-deformation.

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

$$\mathbb{E}\left[\nabla\mathbf{Y}\right] = \nabla\left[\left.\mathbf{Y}\right\| - \upsilon\_n^{-1}\left[\left.\dot{\mathbf{Y}}\right]\right] \otimes \mathbf{n}.\tag{70}$$

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

$$
\upsilon\_n \llbracket \nabla \mathbf{Y} \rrbracket = - \llbracket \dot{\mathbf{Y}} \rrbracket \otimes \mathbf{n}.\tag{71}
$$

#### **7.1. Homothermal case**

14 Will-be-set-by-IN-TECH

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.

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

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*

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

*<sup>d</sup>* <sup>=</sup> <sup>2</sup>*v*<sup>2</sup>

*<sup>v</sup>* = <sup>5</sup> 3 *v*2 *sm* <sup>−</sup> <sup>2</sup> 3 *v*2

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

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

We are now in a position to study acceleration wave propagation. We shall consider the surfaces Σ that are *weak singularities*, defined as those carrying only jumps of the derivatives of order 2 of the macro- and micro-displacement vectors and of order 1 of the thermal variables; these singular surfaces of order 2 are called *macro-acceleration waves* and all external forces and supplies, *i.e.* **f**, **B** and *λ*, are supposed continuous across them with all the derivatives (see,

Therefore, these peculiar discontinuity surfaces suffer 2nd-order derivative jumps of the displacement **u** and 1st-order derivative jumps of the temperature variation *ϑ* and of the

acceleration waves in the same material, undeformed and at rest, obtained in [26].

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

*sm* + *λ*<sup>1</sup> > 0 by inspection,

*tm* + 3*λ*<sup>2</sup> > 0 for (30)4,5, are

*th* is similar in character to a thermal wave;

of pure dilatation, the last one corresponding to *δ*<sup>2</sup>

are zero, we can observe the phenomenon of the resonance.

If frequencies are high, the *δ<sup>d</sup>* micro-wave, of speed *v*<sup>2</sup>

and the *δ<sup>v</sup>* one, which travels with velocity *v*<sup>2</sup>

solutions of voids theory [36].

quite small constant.

dilatation micro-elastic modes.

also, [1, 37]).

**7. Macro-acceleration waves**

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

$$-\gamma \|\nabla \theta\| \cdot \mathbf{n} = \gamma v\_n^{-1} \|\dot{\theta}\| = 0,\tag{72}$$

where Eq. (71) were used. Thus, *in linear porous thermoelasticity*, the first derivatives of *ϑ* are continuous and *every macro-acceleration wave is homothermal*.

The last jump condition (44) gives the following relation

$$\upsilon\_{n}\llbracket\mathop{\mathbf{V}}\rrbracket + \upsilon\_{\text{sm}}^{2}\llbracket\nabla\mathop{\mathbf{V}}\rrbracket \,\mathbf{n} + 2(\upsilon\_{tm}^{2} - \upsilon\_{\text{sm}}^{2})\mathop{\text{sym}}\,(\lbrack\mathop{\text{Div}}\mathbf{V}\rrbracket\rangle \otimes \mathbf{n}) + $$

$$+ \lambda\_{1}\left[\left(\mathbf{n} \cdot \llbracket\mathop{\text{div}}\mathbf{V}\rrbracket\right)\mathbf{I} + \text{sym}\left(\lbrack\mathop{\nabla}\boldsymbol{\nu}\rrbracket\otimes\mathbf{n}\right)\right] + \lambda\_{2}\left(\mathbf{n} \cdot \llbracket\mathop{\nabla}\boldsymbol{\nu}\rrbracket\right)\mathbf{I} = \mathbf{O},\tag{73}$$

while the jumps of the balance laws (39) and (41) and of the derivative of Eq. (38) furnish these other ones, when Eq. (72) is also used:

$$\mathbb{E}\left[\left.\ddot{\mathbf{u}}\right\| = v\_t^2 \left\|\Delta \mathbf{u}\right\| + \left(v\_l^2 - v\_l^2\right) \left\|\nabla \left(\text{Div}\,\mathbf{u}\right)\right\| + \kappa\_\* \left(\lambda\_5 \left\|\left.\nabla \boldsymbol{\nu}\right\| + \lambda\_6 \left\|\left.\text{Div}\,\mathbf{V}\right\|\right)\right),\tag{74}$$

$$
\gamma \left[ \| \Delta \boldsymbol{\theta} \| + \gamma\_2 \| \operatorname{Div} \dot{\mathbf{u}} \| + \kappa\_\* \gamma\_3 \| \dot{\boldsymbol{\nu}} \| \right] = 0, \quad \left[ \dot{\boldsymbol{\rho}} \right] + \rho\_\* \left[ \operatorname{Div} \dot{\mathbf{u}} \right] = 0. \tag{75}
$$

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, [26, 39]):

$$\mathcal{U}(\mathbf{n}\otimes\mathbf{n}) := \left[v\_t^2 \mathbf{I} + (v\_l^2 - v\_t^2)\mathbf{n}\otimes\mathbf{n}\right]\_{\text{'}}\tag{76}$$

but no discontinuities along Σ in the mass and temperature fields, for equations (78) and

[[ **<sup>V</sup>**˙ ]]*em* <sup>=</sup> *<sup>δ</sup>*(*<sup>ζ</sup>* **<sup>n</sup>** <sup>⊗</sup> **<sup>n</sup>** <sup>+</sup> **<sup>e</sup>** <sup>⊗</sup> **<sup>e</sup>** <sup>+</sup> **<sup>f</sup>** <sup>⊗</sup> **<sup>f</sup>**);

<sup>8</sup> <sup>+</sup> <sup>4</sup>(*v*<sup>2</sup>

inspection for inequalities (30)4,5. This wave propagates at a constant speed *υ<sup>n</sup>* = *vem*,

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

and the discontinuity amplitudes in the mass and temperature variations are, respectively:

*em*[*κ*∗*γ*3(2 + *ζ*) − *γ*2*�*].

**Remark.** The *longitudinal micro-waves* of compaction or distention, usually predicted in the voids theory [1, 38], is here excluded, in general, unless we impose the additional condition

i) *one purely transverse micro-wave,* of vector amplitude [[ **<sup>V</sup>**˙ ]]*pm* <sup>=</sup> *�*(**<sup>I</sup>** <sup>−</sup> **<sup>n</sup>** <sup>⊗</sup> **<sup>n</sup>**) and speed

ii) *a purely longitudinal micro-wave,* which recovers the quoted prediction of voids theories

*<sup>δ</sup> �*<sup>ˆ</sup> *vlm* **<sup>n</sup>**, with *�*<sup>ˆ</sup> <sup>=</sup> *<sup>κ</sup>*∗(*λ*<sup>5</sup> <sup>+</sup> *<sup>λ</sup>*6)

*<sup>δ</sup> �*ˆ, [[ *<sup>ϑ</sup>*¨]]*lm* = *<sup>ϑ</sup>*<sup>ˆ</sup> <sup>ˆ</sup>

*sm* − 2*λ*1, which corresponds to another *longitudinal macro-, mass, temperature wave*

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

*lm* :<sup>=</sup> <sup>2</sup>*v*<sup>2</sup>

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

*<sup>δ</sup>*, with *<sup>ϑ</sup>*<sup>ˆ</sup> :<sup>=</sup> *<sup>v</sup>*<sup>2</sup>

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

*sm*)(*v*<sup>2</sup>

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

[[ *<sup>ρ</sup>*˙]]*em* <sup>=</sup> *<sup>ρ</sup>*∗*δ �* and [[ *<sup>ϑ</sup>*¨]]*em* <sup>=</sup> *<sup>δ</sup> <sup>ϑ</sup>*ˇ, (84)

*pm*

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

*δ* **n** ⊗ **n**: thus it is a wave of compaction, if the scalar amplitude

*δ* > 0. In this last case the coupled wave is again *longitudinal*

*lm*

*tm*−*v*<sup>2</sup>

Linear Wave Motions in Continua with Nano-Pores 77

<sup>2</sup>*λ*<sup>2</sup> > 0 for the same inequalities and so

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

*sm*+*λ*1)−*λ*2+

<sup>√</sup>*<sup>D</sup>* <sup>2</sup>(*λ*1+*λ*2) , *<sup>D</sup>* the

*sm* + 2*λ*<sup>1</sup> − *λ*2) > 0 by

[2*λ*<sup>5</sup> + (*λ*<sup>5</sup> + *λ*6)*ζ*], (83)

, (85)

*<sup>γ</sup>* (2*κ*∗*γ*<sup>3</sup> <sup>−</sup> *<sup>γ</sup>*2*ς*); (86)

*sm* + *λ*1; the amplitude of the

, (87)

*<sup>γ</sup>* (*κ*∗*γ*<sup>3</sup> <sup>−</sup> *<sup>γ</sup>*2*�*ˆ). (88)

(81), as in the classical case, in fact [[ *ρ*˙]]*tm* = 0 and [[ *ϑ*¨]]*tm* = 0.

here *<sup>δ</sup>* is its scalar amplitude, *<sup>ζ</sup>* the constant so defined: *<sup>ζ</sup>* :<sup>=</sup> <sup>2</sup>(*v*<sup>2</sup>

[[ **<sup>u</sup>**¨]]*em* <sup>=</sup> *δ � vem* **<sup>n</sup>**, with the constant *�* :<sup>=</sup> *<sup>κ</sup>*<sup>∗</sup>

of the subsequent point ii) on the constitutive thermoelastic constants.

[[ **<sup>u</sup>**¨]]*pm* <sup>=</sup> *� ς vpm* **<sup>n</sup>**, with *<sup>ς</sup>* <sup>=</sup> <sup>2</sup>*κ*<sup>∗</sup> *<sup>λ</sup>*<sup>5</sup>

[[ *<sup>ρ</sup>*˙]]*pm* <sup>=</sup> *<sup>ρ</sup>*∗*� ς*, [[ *<sup>ϑ</sup>*¨]]*pm* <sup>=</sup> *ϑ�*˜ , with *<sup>ϑ</sup>*˜ :<sup>=</sup> *<sup>v</sup>*<sup>2</sup>

If (*λ*<sup>1</sup> + *λ*2) = 0, we have other two significant subcases:

and which propagates at constant speed *v*<sup>2</sup>

discontinuity is [[ **<sup>V</sup>**˙ ]]*lm* = <sup>ˆ</sup>

*δ* < 0, and of distention, if ˆ

and the connected amplitudes are

[[ **<sup>u</sup>**¨]]*lm* = <sup>ˆ</sup>

[[ *<sup>ρ</sup>*˙]]*lm* <sup>=</sup> *<sup>ρ</sup>*<sup>∗</sup> <sup>ˆ</sup>

<sup>2</sup>*λ*<sup>2</sup> <sup>+</sup> <sup>1</sup> 2 <sup>√</sup>*D*: *<sup>v</sup>*<sup>2</sup>

The coupled *macro-wave* is now *longitudinal*, that is,

*vem* > *vtm*; obviously this result holds only if (*λ*<sup>1</sup> + *λ*2) �= 0.

◦) *One extensional micro-wave,* whose vector amplitude is

discriminant *D* := 12*λ*1(*λ*<sup>1</sup> + *λ*2) + 9*λ*<sup>2</sup>

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

with *v*<sup>2</sup>

*v*2 *pm* := *v*<sup>2</sup>

ˆ

of amplitudes

*em* := *v*<sup>2</sup>

with the constant *ϑ*ˇ := *γ*−1*v*<sup>2</sup>

$$\mathcal{L}\left(\mathbf{n}\otimes\mathbf{n}\right) := \left[v\_{\rm sm}^2 \mathcal{I} + (v\_{\rm tm}^2 - v\_{\rm sm}^2)(\Phi\otimes\mathbf{n} + \mathbf{n}\otimes\mathbf{I}\otimes\mathbf{n}) + \\ \mathbf{ }\right.\tag{77}$$

$$+ \lambda\_2 \mathbf{I}\otimes\mathbf{I} + \lambda\_1 \left(\mathbf{I}\otimes\mathbf{n}\otimes\mathbf{n} + \mathbf{n}\otimes\mathbf{n}\otimes\mathbf{I}\right)\!\!/\!/ \tag{77}$$

where we introduced the fourth-order tensor I and third-order tensor **Φ** of components, respectively: I*ijkl* := *δikδjl* and Φ*ijk* := *δiknj*.

Therefore, from Eqs. (73)-(75), we are able to write the system of equations for the unknown amplitudes in the following manner (see, also, [41]):

$$\left[\boldsymbol{\upsilon}\_{\boldsymbol{\pi}}\left[\boldsymbol{\rho}\right]\right] = \rho\_{\*}\left[\left[\boldsymbol{\bar{u}}\right]\right] \cdot \mathbf{n}\_{\prime} \tag{78}$$

$$
\mathbb{E}\left[\upsilon\_n^2 \mathbf{I} - \mathcal{U}(\mathbf{n}\otimes \mathbf{n})\right] \left[\|\mathbf{\ddot{u}}\|\right] = -\kappa\_\* \upsilon\_\mathbb{H} (\lambda\_5 \mathbf{n}\otimes \mathbf{I} + \lambda\_6 \mathbf{I}\otimes \mathbf{n}) \|\,\mathsf{V}\|\,\mathsf{V} \tag{79}
$$

$$
\left[\upsilon\_n^2 \mathcal{L} - \mathcal{C}(\mathbf{n} \otimes \mathbf{n})\right] \left[\mathbf{\dot{V}}\right] = \mathbf{0},\tag{80}
$$

$$
\gamma \left\| \vec{\theta} \right\| = \kappa\_\* \gamma\_3 \upsilon\_n^2 \mathbf{I} \cdot \left\| \dot{\mathbf{V}} \right\| - \gamma\_2 \upsilon\_n \left\| \vec{\mathbf{u}} \right\| \cdot \mathbf{n}.\tag{81}
$$

In conclusion, the jump of macro-acceleration [[ **u**¨]], ruled by Eq. (79), is in general coupled to the discontinuities in the microstructural variable [[ **V**˙ ]], unlike the purely acceleration waves studied in [26], as observed in the previous remark.

Hence to classify possible macro-acceleration waves Σ we must analyze in advance Eq. (80) that gives three possible speed of displacement *υ<sup>n</sup>* for the surface Σ related to:

◦) *Two shear optical micro-waves,* whose speed propagation is *υ<sup>n</sup>* = *vsm*; the amplitude of the discontinuity is [[ **<sup>V</sup>**˙ ]]*sm* <sup>=</sup> *<sup>α</sup>*(**<sup>e</sup>** <sup>⊗</sup> **<sup>e</sup>** <sup>−</sup> **<sup>f</sup>** <sup>⊗</sup> **<sup>f</sup>**) + *<sup>β</sup>*(**<sup>e</sup>** <sup>⊗</sup> **<sup>f</sup>** <sup>+</sup> **<sup>f</sup>** <sup>⊗</sup> **<sup>e</sup>**), where *<sup>α</sup>* and *<sup>β</sup>* are the scalar components of the wave amplitude and, as before in the plane waves study, **e** and **f** are the unit vectors in the plane orthogonal to **n** such that **e** · **f** = 0.

By inserting this solution in the other three Eqs. (78), (79) and (81), we obtain, in general, that [[ **u**¨]]*sm* = **0**, [[ *ρ*˙]]*sm* = 0 and [[ *ϑ*¨]]*sm* = 0, that is, in essence, the waves carry predominantly a change in the nano-pore structure without altering the thermoelastic features of the matrix material.

**Observation.** Instead, in the particular case in which one eigenvalue of U coincides with that of C, *i.e.*, *vt* = *vsm*, (or *vl* = *vsm*), there could be also an associated transverse (or longitudinal, respectively) macro-wave of free amplitude which not alters (or which alters, respectively) mass and temperature fields; in this last case we have [[ *<sup>ρ</sup>*˙]]*sm* <sup>=</sup> *<sup>ρ</sup>*∗*v*−<sup>1</sup> *sm* [[ *<sup>u</sup>*¨*n*]]*sm* and [[ *<sup>ϑ</sup>*¨]]*sm* <sup>=</sup> <sup>−</sup>*γ*−1*γ*2*vsm*[[ *<sup>u</sup>*¨*n*]]*sm*.

◦) *Two transverse micro-waves,* with propagation velocity *υ<sup>n</sup>* = *vtm*. Their amplitude is of the form [[ **<sup>V</sup>**˙ ]]*tm* <sup>=</sup> *<sup>χ</sup>e*(**<sup>n</sup>** <sup>⊗</sup> **<sup>e</sup>** <sup>+</sup> **<sup>e</sup>** <sup>⊗</sup> **<sup>n</sup>**) + *<sup>χ</sup>f*(**<sup>n</sup>** <sup>⊗</sup> **<sup>f</sup>** <sup>+</sup> **<sup>f</sup>** <sup>⊗</sup> **<sup>n</sup>**), with *<sup>χ</sup><sup>e</sup>* and *<sup>χ</sup><sup>f</sup>* the components of the amplitude itself. Now, there is a coupled *transverse macro-wave*, obtained by the study of equation (79), of amplitude:

$$\mathbb{I}\left[\ddot{\mathbf{u}}\right]\_{tm} = \frac{\kappa\_\* v\_{tm} \lambda\_6}{v\_t^2 - v\_{tm}^2} (\chi\_\varepsilon \mathbf{e} + \chi\_f \mathbf{f})\_\prime \tag{82}$$

but no discontinuities along Σ in the mass and temperature fields, for equations (78) and (81), as in the classical case, in fact [[ *ρ*˙]]*tm* = 0 and [[ *ϑ*¨]]*tm* = 0.

◦) *One extensional micro-wave,* whose vector amplitude is

16 Will-be-set-by-IN-TECH

*<sup>t</sup>*)**n** ⊗ **n** 

where we introduced the fourth-order tensor I and third-order tensor **Φ** of components,

Therefore, from Eqs. (73)-(75), we are able to write the system of equations for the unknown

In conclusion, the jump of macro-acceleration [[ **u**¨]], ruled by Eq. (79), is in general coupled to the discontinuities in the microstructural variable [[ **V**˙ ]], unlike the purely acceleration waves

Hence to classify possible macro-acceleration waves Σ we must analyze in advance Eq. (80)

◦) *Two shear optical micro-waves,* whose speed propagation is *υ<sup>n</sup>* = *vsm*; the amplitude of the discontinuity is [[ **<sup>V</sup>**˙ ]]*sm* <sup>=</sup> *<sup>α</sup>*(**<sup>e</sup>** <sup>⊗</sup> **<sup>e</sup>** <sup>−</sup> **<sup>f</sup>** <sup>⊗</sup> **<sup>f</sup>**) + *<sup>β</sup>*(**<sup>e</sup>** <sup>⊗</sup> **<sup>f</sup>** <sup>+</sup> **<sup>f</sup>** <sup>⊗</sup> **<sup>e</sup>**), where *<sup>α</sup>* and *<sup>β</sup>* are the scalar components of the wave amplitude and, as before in the plane waves study, **e** and **f** are the

By inserting this solution in the other three Eqs. (78), (79) and (81), we obtain, in general, that [[ **u**¨]]*sm* = **0**, [[ *ρ*˙]]*sm* = 0 and [[ *ϑ*¨]]*sm* = 0, that is, in essence, the waves carry predominantly a change in the nano-pore structure without altering the thermoelastic

**Observation.** Instead, in the particular case in which one eigenvalue of U coincides with that of C, *i.e.*, *vt* = *vsm*, (or *vl* = *vsm*), there could be also an associated transverse (or longitudinal, respectively) macro-wave of free amplitude which not alters (or which alters, respectively) mass and temperature fields; in this last case we have [[ *<sup>ρ</sup>*˙]]*sm* <sup>=</sup> *<sup>ρ</sup>*∗*v*−<sup>1</sup> *sm* [[ *<sup>u</sup>*¨*n*]]*sm*

◦) *Two transverse micro-waves,* with propagation velocity *υ<sup>n</sup>* = *vtm*. Their amplitude is of the form [[ **<sup>V</sup>**˙ ]]*tm* <sup>=</sup> *<sup>χ</sup>e*(**<sup>n</sup>** <sup>⊗</sup> **<sup>e</sup>** <sup>+</sup> **<sup>e</sup>** <sup>⊗</sup> **<sup>n</sup>**) + *<sup>χ</sup>f*(**<sup>n</sup>** <sup>⊗</sup> **<sup>f</sup>** <sup>+</sup> **<sup>f</sup>** <sup>⊗</sup> **<sup>n</sup>**), with *<sup>χ</sup><sup>e</sup>* and *<sup>χ</sup><sup>f</sup>* the components of the amplitude itself. Now, there is a coupled *transverse macro-wave*, obtained by the study

> [[ **<sup>u</sup>**¨]]*tm* <sup>=</sup> *<sup>κ</sup>*∗*vtmλ*<sup>6</sup> *v*2 *<sup>t</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *tm*

*sm*)(**Φ** ⊗ **n** + **n** ⊗ **I** ⊗ **n**) +

*υn*[[ *ρ*˙]] = *ρ*<sup>∗</sup> [[ **u**¨]] · **n**, (78)

, (76)

+*λ*<sup>2</sup> **I** ⊗ **I** + *λ*<sup>1</sup> (**I** ⊗ **n** ⊗ **n** + **n** ⊗ **n** ⊗ **I**)], (77)

[[ **<sup>u</sup>**¨]] = <sup>−</sup>*κ*∗*υn*(*λ*<sup>5</sup> **<sup>n</sup>** <sup>⊗</sup> **<sup>I</sup>** <sup>+</sup> *<sup>λ</sup>*<sup>6</sup> **<sup>I</sup>** <sup>⊗</sup> **<sup>n</sup>**)[[ **<sup>V</sup>**˙ ]], (79)

[[ **V**˙ ]] = **0**, (80)

(*χ<sup>e</sup>* **e** + *χ<sup>f</sup>* **f**), (82)

*<sup>n</sup>* **<sup>I</sup>** · [[ **<sup>V</sup>**˙ ]] <sup>−</sup> *<sup>γ</sup>*2*υn*[[ **<sup>u</sup>**¨]] · **<sup>n</sup>**. (81)

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

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

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

that gives three possible speed of displacement *υ<sup>n</sup>* for the surface Σ related to:

unit vectors in the plane orthogonal to **n** such that **e** · **f** = 0.

U(**n** ⊗ **n**) :=

respectively: I*ijkl* := *δikδjl* and Φ*ijk* := *δiknj*.

 *υ*2

 *υ*2

features of the matrix material.

and [[ *<sup>ϑ</sup>*¨]]*sm* <sup>=</sup> <sup>−</sup>*γ*−1*γ*2*vsm*[[ *<sup>u</sup>*¨*n*]]*sm*.

of equation (79), of amplitude:

amplitudes in the following manner (see, also, [41]):

*<sup>n</sup>***I** − U(**n** ⊗ **n**)

*<sup>n</sup>* I−C(**n** ⊗ **n**)

*<sup>γ</sup>* [[ *<sup>ϑ</sup>*¨]] = *<sup>κ</sup>*∗*γ*3*υ*<sup>2</sup>

studied in [26], as observed in the previous remark.

<sup>C</sup>(**<sup>n</sup>** <sup>⊗</sup> **<sup>n</sup>**) := [*v*<sup>2</sup>

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

$$\|\dot{\mathbf{V}}\|\_{em} = \delta(\zeta \mathbf{n} \otimes \mathbf{n} + \mathbf{e} \otimes \mathbf{e} + \mathbf{f} \otimes \mathbf{f});$$

here *<sup>δ</sup>* is its scalar amplitude, *<sup>ζ</sup>* the constant so defined: *<sup>ζ</sup>* :<sup>=</sup> <sup>2</sup>(*v*<sup>2</sup> *tm*−*v*<sup>2</sup> *sm*+*λ*1)−*λ*2+ <sup>√</sup>*<sup>D</sup>* <sup>2</sup>(*λ*1+*λ*2) , *<sup>D</sup>* the discriminant *D* := 12*λ*1(*λ*<sup>1</sup> + *λ*2) + 9*λ*<sup>2</sup> <sup>8</sup> <sup>+</sup> <sup>4</sup>(*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm*)(*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm* + 2*λ*<sup>1</sup> − *λ*2) > 0 by inspection for inequalities (30)4,5. This wave propagates at a constant speed *υ<sup>n</sup>* = *vem*, with *v*<sup>2</sup> *em* := *v*<sup>2</sup> *tm* <sup>+</sup> *<sup>λ</sup>*<sup>1</sup> <sup>+</sup> <sup>3</sup> <sup>2</sup>*λ*<sup>2</sup> <sup>+</sup> <sup>1</sup> 2 <sup>√</sup>*D*: *<sup>v</sup>*<sup>2</sup> *tm* <sup>+</sup> *<sup>λ</sup>*<sup>1</sup> <sup>+</sup> <sup>3</sup> <sup>2</sup>*λ*<sup>2</sup> > 0 for the same inequalities and so *vem* > *vtm*; obviously this result holds only if (*λ*<sup>1</sup> + *λ*2) �= 0.

The coupled *macro-wave* is now *longitudinal*, that is,

$$\mathbb{E}\left[\left.\hat{\mathbf{u}}\right\|\_{\mathcal{C}\mathbf{m}} = \delta \oslash \boldsymbol{v}\_{\mathcal{C}\mathbf{m}} \,\mathbf{n}\_{\prime} \quad \text{with the constant } \boldsymbol{\mathcal{O}} := \frac{\kappa\_{\ast}}{v\_{l}^{2} - v\_{\mathcal{C}\mathbf{m}}^{2}} \big[2\lambda\_{5} + (\lambda\_{5} + \lambda\_{6})\zeta\right],\tag{83}$$

and the discontinuity amplitudes in the mass and temperature variations are, respectively:

$$\|\left[\dot{\rho}\right]\_{em} = \rho\_\* \delta \,\boldsymbol{\sigma} \quad \text{and} \quad \|\left[\ddot{\theta}\right]\_{em} = \delta \,\check{\theta} \,\tag{84}$$

with the constant *ϑ*ˇ := *γ*−1*v*<sup>2</sup> *em*[*κ*∗*γ*3(2 + *ζ*) − *γ*2*�*].

**Remark.** The *longitudinal micro-waves* of compaction or distention, usually predicted in the voids theory [1, 38], is here excluded, in general, unless we impose the additional condition of the subsequent point ii) on the constitutive thermoelastic constants.


$$\left[\left[\bar{\mathbf{u}}\right]\_{pm} = \varrho \odot v\_{pm}\mathbf{n}\right.\quad\text{with}\quad\boldsymbol{\varsigma} = \frac{2\boldsymbol{\kappa}\_{\*}\,\boldsymbol{\lambda}\_{5}}{v\_{l}^{2} - v\_{pm}^{2}}.\tag{85}$$

$$\mathbb{E}\left[\dot{\rho}\right]\_{pm} = \rho\_\* \varrho \,\,\varsigma\_{\prime} \quad \mathbb{E}\left[\ddot{\theta}\right]\_{pm} = \tilde{\theta}\varrho\_{\prime} \quad \text{with} \quad \tilde{\theta} := \frac{v\_{pm}^2}{\gamma} (2\kappa\_\* \gamma\_3 - \gamma\_2 \varsigma);\tag{86}$$

ii) *a purely longitudinal micro-wave,* which recovers the quoted prediction of voids theories and which propagates at constant speed *v*<sup>2</sup> *lm* :<sup>=</sup> <sup>2</sup>*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm* + *λ*1; the amplitude of the discontinuity is [[ **<sup>V</sup>**˙ ]]*lm* = <sup>ˆ</sup> *δ* **n** ⊗ **n**: thus it is a wave of compaction, if the scalar amplitude ˆ *δ* < 0, and of distention, if ˆ *δ* > 0. In this last case the coupled wave is again *longitudinal* and the connected amplitudes are

$$\mathbb{E}\left[\vec{\mathbf{u}}\right]\_{lm} = \hat{\delta}\,\hat{\omega}\,v\_{lm}\mathbf{n}, \quad \text{with} \quad \hat{\omega} = \frac{\kappa\_\*(\lambda\_5 + \lambda\_6)}{v\_l^2 - v\_{lm}^2},\tag{87}$$

$$\mathbb{E}\left[\widehat{\rho}\right]\_{lm} = \rho\_\* \widehat{\delta} \, \widehat{\mathfrak{d}}, \quad \left[\widehat{\mathfrak{d}}\right]\_{lm} = \widehat{\mathfrak{d}} \widehat{\delta}, \quad \text{with} \quad \widehat{\mathfrak{d}} := \frac{v\_{lm}^2}{\gamma} (\kappa\_\* \gamma\_3 - \gamma\_2 \widehat{\mathfrak{d}}).\tag{88}$$

#### **7.2. Homentropic modes**

In the particular case when the solid with nano-pores does not conduct heat, *i.e.*, **h** ≡ 0 whatever ∇ *ϑ* we choose, then the energy balance (25) may be written, in the linear theory, in the following form *θ*∗*η*˙ = *λ* and the jump across a macro-acceleration wave Σ shows that

$$\left[\left[\dot{\eta}\right]\right] = 0.\tag{89}$$

Both conditions [[ *ϑ*˙]] = 0 and *γ* = 0 apply; thus, from Eq. (91) (or from (81)), we obtain the

Also now we have shear optical and transverse micro-waves as in the previous instances; instead, in general, extensional macro-acceleration waves do not occur, unless the condition

Let us study now the growth or the decay of the macro-acceleration waves Σ which travel through the thermo-elastic material with nano-pores, thus we restrict ourselves to plane waves which are of uniform scalar amplitude with assigned initial value, uniform in the sense

For this purpose, we differentiate twice with respect to time each term of Eq. (38) and once those of Eqs. (39) and (41), take into account the balance of micromomentum (40) and form

*<sup>λ</sup>*1[[ Div (Div **<sup>V</sup>**)]] + *<sup>λ</sup>*2[[ <sup>Δ</sup>(tr **<sup>V</sup>**)]] <sup>−</sup> *<sup>ω</sup>* **<sup>I</sup>** · [[ **<sup>V</sup>**˙ ]]

where **d** (≡ *dn***n** + *de***e** + *df***f**) represents the jump in the third time-derivative of the

Algebraic computations, very similar to those carried out in [26, 42], with the use of the Hugoniot-Hadamard compatibility condition (70) and of definitions (76) and (77) of the homothermal acoustic tensors, we obtain the following evolution equations for the

*<sup>λ</sup>*5<sup>∇</sup> (tr[[ **<sup>V</sup>**˙ ]]) + *<sup>λ</sup>*6Div [[ **<sup>V</sup>**˙ ]]

*tm*) sym

<sup>−</sup><sup>2</sup> *<sup>υ</sup><sup>n</sup> <sup>λ</sup>*2[**<sup>n</sup>** · ∇ (tr[[ **<sup>V</sup>**˙ ]])]**<sup>I</sup>** <sup>−</sup> *<sup>υ</sup>*<sup>2</sup>

with Θ that indicates the jump in the third time-derivative of the temperature change field *ϑ*.

*<sup>n</sup> dn* − Div [[ **u**¨]]), (97)

sym [<sup>∇</sup> (tr[[ **<sup>V</sup>**˙ ]]) <sup>⊗</sup> **<sup>n</sup>**]+(**<sup>n</sup>** · Div [[ **<sup>V</sup>**˙ ]])**<sup>I</sup>**

*<sup>γ</sup>*Δ[[ *<sup>ϑ</sup>*˙]] + *<sup>γ</sup>*1[[ *<sup>ϑ</sup>*¨]] + *<sup>γ</sup>*2Div [[ **<sup>u</sup>**¨]] + *<sup>κ</sup>*∗*γ*3tr[[ **<sup>V</sup>**¨ ]]

*<sup>λ</sup>*5[[ <sup>∇</sup> (tr **<sup>V</sup>**˙ )]] + *<sup>λ</sup>*6[[ Div **<sup>V</sup>**˙ ]]

*sm*) sym [[ <sup>∇</sup> (Div **<sup>V</sup>**)]] + *<sup>λ</sup>*1[[ <sup>∇</sup> <sup>2</sup>(tr **<sup>V</sup>**)]] + (96)

−

*<sup>t</sup>*(<sup>∇</sup> [[ **<sup>u</sup>**¨]])**<sup>n</sup>** <sup>−</sup> *<sup>γ</sup>*2[[ *<sup>ϑ</sup>*¨]]**<sup>n</sup>**

(Div [[ **<sup>V</sup>**˙ ]]) <sup>⊗</sup> **<sup>n</sup>** <sup>+</sup> <sup>∇</sup> ([[ **<sup>V</sup>**˙ ]]**n**)

<sup>+</sup> *<sup>γ</sup>*2[[ <sup>∇</sup> *<sup>ϑ</sup>*˙]],

**<sup>I</sup>** <sup>−</sup> <sup>2</sup>*σ*[[ **<sup>V</sup>**˙ ]],

*λ*5(tr[[ **V**¨ ]])**n** + *λ*6[[ **V**¨ ]]**n**

+ *γ*2*υ*<sup>2</sup>

 −

*<sup>n</sup>*[*<sup>ω</sup>* tr[[ **<sup>V</sup>**˙ ]]**<sup>I</sup>** + <sup>2</sup> *<sup>σ</sup>* [[ **<sup>V</sup>**˙ ]]],

+

*<sup>n</sup>*<sup>∇</sup> [[ *<sup>ϑ</sup>*˙]], (98)

− (99)

, (100)

*em*) = *γ*2[2*λ*<sup>5</sup> + (*λ*<sup>5</sup> + *λ*6)*ζ*].

<sup>2</sup> *<sup>υ</sup><sup>n</sup>* **<sup>I</sup>** · [[ **<sup>V</sup>**˙ ]]. (95)

Linear Wave Motions in Continua with Nano-Pores 79

[[ **<sup>u</sup>**¨]] · **<sup>n</sup>** <sup>=</sup> *<sup>κ</sup>*∗*γ*3*γ*−<sup>1</sup>

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

*<sup>t</sup>*)[[ ∇ (Div **u**˙ )]] + *κ*<sup>∗</sup>

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

*<sup>γ</sup>*[[ <sup>Δ</sup>*ϑ*˙]] + *<sup>γ</sup>*1[[ *<sup>ϑ</sup>*¨]] + *<sup>γ</sup>*2[[ Div **<sup>u</sup>**¨]] + *<sup>γ</sup>*<sup>3</sup> **<sup>I</sup>** · [[ **<sup>V</sup>**¨ ]] = 0,

following compatibility condition for the generalized transverse wave:

(95) is satisfied, that is: *γ*3(2 + *ζ*)(*v*<sup>2</sup>

[[ *ρ*¨]] = −*ρ*∗[[ Div **u**¨]],

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

**d** = *v*<sup>2</sup>

[[ **V**¨ ]] = *v*<sup>2</sup>

displacement field **u**.

propagating wave Σ:

*<sup>n</sup>***I** − U(**n** ⊗ **n**)

*<sup>n</sup>* I−C(**n** ⊗ **n**)

−2 *υ<sup>n</sup> v*2

**d** = *κ*∗*υ<sup>n</sup>*

 *υn* 

[[ **V**¨ ]] = 2*υn*(*v*<sup>2</sup>

*sm*(<sup>∇</sup> [[ **<sup>V</sup>**˙ ]])**<sup>n</sup>** <sup>+</sup> *<sup>λ</sup>*<sup>1</sup>

<sup>−</sup>*υ*<sup>2</sup> *n* 

<sup>2</sup>*γ*(**<sup>n</sup>** · ∇ [[ *<sup>ϑ</sup>*¨]]) + *<sup>γ</sup>*2*dn*

*<sup>l</sup>* )[(Div [[ **<sup>u</sup>**¨]])**<sup>n</sup>** <sup>+</sup> <sup>∇</sup> ([[ **<sup>u</sup>**¨]] · **<sup>n</sup>**)] <sup>−</sup> <sup>2</sup>*v*<sup>2</sup>

 −

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

[[ *<sup>ρ</sup>*¨]] = *<sup>ρ</sup>*∗(*υ*−<sup>1</sup>

+*υ<sup>n</sup>* (*v*<sup>2</sup> *<sup>t</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*γ*Θ = *υ<sup>n</sup>*

 *υ*2

 *υ*2

**8. Evolution equations for wave amplitudes**

that the scalar amplitude does not vary with position on Σ.

jumps of all equations across the wave Σ to have:

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

+

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

Thus, we have established the following result: *In a non-conducting thermoelastic body with nano-pores, every macro-acceleration wave is homentropic*.

The jump Eqs. (78) and (80) remain unchanged, while, since [[ *<sup>ϑ</sup>*˙]] �<sup>=</sup> 0, Eqs. (79) and (81) are replaced by the following ones:

$$\mathbb{E}\left[\upsilon\_n^2 \mathbf{I} - \mathcal{U}(\mathbf{n}\otimes \mathbf{n})\right] \left[\mathbf{\bar{u}}\right] = -\kappa\_\* \upsilon\_n (\lambda\_5 \mathbf{n}\otimes \mathbf{I} + \lambda\_6 \mathbf{I}\otimes \mathbf{n}) \left[\mathbf{V}\right] - \gamma\_2 \upsilon\_n \|\left[\boldsymbol{\vartheta}\right] \mathbf{n},\tag{90}$$

$$
\gamma\_1 \upsilon\_n \lbrack \dot{\theta} \rbrack = \gamma\_2 \lbrack \ddot{\mathbf{u}} \rbrack \cdot \mathbf{n} - \kappa\_\* \gamma\_3 \upsilon\_n \mathbf{I} \cdot \llbracket \dot{\mathbf{V}} \rbrack,\tag{91}
$$

where the relation (35)4 and the condition (71) were used in the jump of Eq. (39) and in Eq. (89); by substituting Eq. (91) into Eq. (90) we obtain the following jump, similar to (79),

$$
\mathbb{E}\left[v\_n^2 \mathbf{I} - \tilde{\mathcal{U}}(\mathbf{n} \otimes \mathbf{n})\right] \|\,\mathbf{\ddot{u}}\| = -\kappa\_\* \upsilon\_{\mathbb{H}}(\tilde{\lambda}\_5 \mathbf{n} \otimes \mathbf{I} + \lambda\_6 \mathbf{I} \otimes \mathbf{n}) \|\,\mathbf{\ddot{V}}\|,\tag{92}
$$

but where we introduced the instantaneous *homentropic acoustic macro-tensor*

$$\mathcal{U}(\mathbf{n}\otimes\mathbf{n}) := \left[v\_t^2 \mathbf{I} + \left(\tilde{v}\_l^2 - v\_t^2\right) \mathbf{n}\otimes\mathbf{n}\right],\tag{93}$$

with *v*˜<sup>2</sup> *<sup>l</sup>* :<sup>=</sup> *<sup>v</sup>*<sup>2</sup> *<sup>l</sup>* <sup>−</sup> *<sup>γ</sup>*<sup>2</sup> 2*γ*−<sup>1</sup> <sup>1</sup> , and the constant *<sup>λ</sup>*˜ <sup>5</sup> :<sup>=</sup> *<sup>λ</sup>*<sup>5</sup> <sup>−</sup> *<sup>γ</sup>*2*γ*3*γ*−<sup>1</sup> <sup>1</sup> .

The linear algebraic system of Eqs. (78), (92) and (80) for the amplitudes of the macroacceleration waves Σ in the homentropic case has the same five solutions with the same propagation speeds, as the homothermal one: two shear optical micro-waves, two transverse micro-waves and one extensional micro-wave. The only variation is in the extensional one and consists in the change of constants *vl* and *<sup>λ</sup>*<sup>5</sup> with *<sup>v</sup>*˜*<sup>l</sup>* and *<sup>λ</sup>*˜ 5, respectively.

Finally, the temperature jump (91) is absent in all shear optical and transverse micro-waves, while in the extensional one we have:

$$\left[\dot{\theta}\right]\_{\rm cm} = \gamma\_1^{-1} \left[\gamma\_2 \varsigma - \kappa\_\* \gamma\_3 (2 + \zeta)\right] \delta\_\prime \tag{94}$$

with the constant *�*˜ := *<sup>κ</sup>*<sup>∗</sup> *v*˜2 *<sup>l</sup>* −*v*<sup>2</sup> *em* [2*λ*˜ <sup>5</sup> + (*λ*˜ <sup>5</sup> + *λ*6)*ζ*].

#### **7.3. Generalized transverse case**

This last subsection concerns the behaviour of macro-acceleration waves that are both homothermal and homentropic and which are usually called *generalized transverse waves*. In physical terms these waves are uninfluenced by thermo-mechanical coupling effects in the transmitting matrix material of the porous solid.

Both conditions [[ *ϑ*˙]] = 0 and *γ* = 0 apply; thus, from Eq. (91) (or from (81)), we obtain the following compatibility condition for the generalized transverse wave:

$$\mathbb{E}\left[\ddot{\mathbf{u}}\right] \cdot \mathbf{n} = \kappa\_\* \gamma\_3 \gamma\_2^{-1} \upsilon\_n \mathbf{I} \cdot \left[\dot{\mathbf{V}}\right].\tag{95}$$

Also now we have shear optical and transverse micro-waves as in the previous instances; instead, in general, extensional macro-acceleration waves do not occur, unless the condition (95) is satisfied, that is: *γ*3(2 + *ζ*)(*v*<sup>2</sup> *<sup>l</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *em*) = *γ*2[2*λ*<sup>5</sup> + (*λ*<sup>5</sup> + *λ*6)*ζ*].

#### **8. Evolution equations for wave amplitudes**

18 Will-be-set-by-IN-TECH

In the particular case when the solid with nano-pores does not conduct heat, *i.e.*, **h** ≡ 0 whatever ∇ *ϑ* we choose, then the energy balance (25) may be written, in the linear theory, in the following form *θ*∗*η*˙ = *λ* and the jump across a macro-acceleration wave Σ shows that

Thus, we have established the following result: *In a non-conducting thermoelastic body with*

The jump Eqs. (78) and (80) remain unchanged, while, since [[ *<sup>ϑ</sup>*˙]] �<sup>=</sup> 0, Eqs. (79) and (81) are

where the relation (35)4 and the condition (71) were used in the jump of Eq. (39) and in Eq. (89); by substituting Eq. (91) into Eq. (90) we obtain the following jump, similar to (79),

The linear algebraic system of Eqs. (78), (92) and (80) for the amplitudes of the macroacceleration waves Σ in the homentropic case has the same five solutions with the same propagation speeds, as the homothermal one: two shear optical micro-waves, two transverse micro-waves and one extensional micro-wave. The only variation is in the extensional one

Finally, the temperature jump (91) is absent in all shear optical and transverse micro-waves,

This last subsection concerns the behaviour of macro-acceleration waves that are both homothermal and homentropic and which are usually called *generalized transverse waves*. In physical terms these waves are uninfluenced by thermo-mechanical coupling effects in the

*<sup>γ</sup>*<sup>1</sup> *<sup>υ</sup>n*[[ *<sup>ϑ</sup>*˙]] = *<sup>γ</sup>*2[[ **<sup>u</sup>**¨]] · **<sup>n</sup>** <sup>−</sup> *<sup>κ</sup>*∗*γ*3*υ<sup>n</sup>* **<sup>I</sup>** · [[ **<sup>V</sup>**˙ ]], (91)

[[ *η*˙]] = 0. (89)

[[ **<sup>u</sup>**¨]] = <sup>−</sup>*κ*∗*υn*(*λ*<sup>5</sup> **<sup>n</sup>** <sup>⊗</sup> **<sup>I</sup>** <sup>+</sup> *<sup>λ</sup>*<sup>6</sup> **<sup>I</sup>** <sup>⊗</sup> **<sup>n</sup>**)[[ **<sup>V</sup>**˙ ]] <sup>−</sup> *<sup>γ</sup>*2*υn*[[ *<sup>ϑ</sup>*˙]]**n**, (90)

[[ **<sup>u</sup>**¨]] = <sup>−</sup>*κ*∗*υn*(*λ*˜ <sup>5</sup> **<sup>n</sup>** <sup>⊗</sup> **<sup>I</sup>** <sup>+</sup> *<sup>λ</sup>*<sup>6</sup> **<sup>I</sup>** <sup>⊗</sup> **<sup>n</sup>**)[[ **<sup>V</sup>**˙ ]], (92)

<sup>1</sup> [*γ*2*ς* − *κ*∗*γ*3(2 + *ζ*)] *δ*, (94)

<sup>1</sup> .

, (93)

**7.2. Homentropic modes**

replaced by the following ones:

*<sup>n</sup>***I** − U(**n** ⊗ **n**)

 *υ*2

 *υ*2

with *v*˜<sup>2</sup>

*<sup>l</sup>* :<sup>=</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>l</sup>* <sup>−</sup> *<sup>γ</sup>*<sup>2</sup> 2*γ*−<sup>1</sup>

while in the extensional one we have:

**7.3. Generalized transverse case**

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

transmitting matrix material of the porous solid.

with the constant *�*˜ := *<sup>κ</sup>*<sup>∗</sup>

*nano-pores, every macro-acceleration wave is homentropic*.

*<sup>n</sup>***<sup>I</sup>** <sup>−</sup> <sup>U</sup>˜(**<sup>n</sup>** <sup>⊗</sup> **<sup>n</sup>**)

<sup>U</sup>˜(**<sup>n</sup>** <sup>⊗</sup> **<sup>n</sup>**) :<sup>=</sup>

but where we introduced the instantaneous *homentropic acoustic macro-tensor*

 *v*2 *<sup>t</sup>* **I** + *v*˜ 2 *<sup>l</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *t* **n** ⊗ **n** 

<sup>1</sup> , and the constant *<sup>λ</sup>*˜ <sup>5</sup> :<sup>=</sup> *<sup>λ</sup>*<sup>5</sup> <sup>−</sup> *<sup>γ</sup>*2*γ*3*γ*−<sup>1</sup>

and consists in the change of constants *vl* and *<sup>λ</sup>*<sup>5</sup> with *<sup>v</sup>*˜*<sup>l</sup>* and *<sup>λ</sup>*˜ 5, respectively.

[2*λ*˜ <sup>5</sup> + (*λ*˜ <sup>5</sup> + *λ*6)*ζ*].

[[ *ϑ*˙]]*em* = *γ*−<sup>1</sup>

Let us study now the growth or the decay of the macro-acceleration waves Σ which travel through the thermo-elastic material with nano-pores, thus we restrict ourselves to plane waves which are of uniform scalar amplitude with assigned initial value, uniform in the sense that the scalar amplitude does not vary with position on Σ.

For this purpose, we differentiate twice with respect to time each term of Eq. (38) and once those of Eqs. (39) and (41), take into account the balance of micromomentum (40) and form jumps of all equations across the wave Σ to have:

[[ *ρ*¨]] = −*ρ*∗[[ Div **u**¨]], **d** = *v*<sup>2</sup> *<sup>t</sup>* [[ <sup>Δ</sup>**u**˙ ]] + (*v*<sup>2</sup> *<sup>l</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *<sup>t</sup>*)[[ ∇ (Div **u**˙ )]] + *κ*<sup>∗</sup> *<sup>λ</sup>*5[[ <sup>∇</sup> (tr **<sup>V</sup>**˙ )]] + *<sup>λ</sup>*6[[ Div **<sup>V</sup>**˙ ]] <sup>+</sup> *<sup>γ</sup>*2[[ <sup>∇</sup> *<sup>ϑ</sup>*˙]], [[ **V**¨ ]] = *v*<sup>2</sup> *sm*[[ <sup>Δ</sup>**V**]] + <sup>2</sup>(*v*<sup>2</sup> *tm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *sm*) sym [[ <sup>∇</sup> (Div **<sup>V</sup>**)]] + *<sup>λ</sup>*1[[ <sup>∇</sup> <sup>2</sup>(tr **<sup>V</sup>**)]] + (96) + *<sup>λ</sup>*1[[ Div (Div **<sup>V</sup>**)]] + *<sup>λ</sup>*2[[ <sup>Δ</sup>(tr **<sup>V</sup>**)]] <sup>−</sup> *<sup>ω</sup>* **<sup>I</sup>** · [[ **<sup>V</sup>**˙ ]] **<sup>I</sup>** <sup>−</sup> <sup>2</sup>*σ*[[ **<sup>V</sup>**˙ ]], *<sup>γ</sup>*[[ <sup>Δ</sup>*ϑ*˙]] + *<sup>γ</sup>*1[[ *<sup>ϑ</sup>*¨]] + *<sup>γ</sup>*2[[ Div **<sup>u</sup>**¨]] + *<sup>γ</sup>*<sup>3</sup> **<sup>I</sup>** · [[ **<sup>V</sup>**¨ ]] = 0,

where **d** (≡ *dn***n** + *de***e** + *df***f**) represents the jump in the third time-derivative of the displacement field **u**.

Algebraic computations, very similar to those carried out in [26, 42], with the use of the Hugoniot-Hadamard compatibility condition (70) and of definitions (76) and (77) of the homothermal acoustic tensors, we obtain the following evolution equations for the propagating wave Σ:

[[ *<sup>ρ</sup>*¨]] = *<sup>ρ</sup>*∗(*υ*−<sup>1</sup> *<sup>n</sup> dn* − Div [[ **u**¨]]), (97) *υ*2 *<sup>n</sup>***I** − U(**n** ⊗ **n**) **d** = *κ*∗*υ<sup>n</sup> υn <sup>λ</sup>*5<sup>∇</sup> (tr[[ **<sup>V</sup>**˙ ]]) + *<sup>λ</sup>*6Div [[ **<sup>V</sup>**˙ ]] − *λ*5(tr[[ **V**¨ ]])**n** + *λ*6[[ **V**¨ ]]**n** + +*υ<sup>n</sup>* (*v*<sup>2</sup> *<sup>t</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *<sup>l</sup>* )[(Div [[ **<sup>u</sup>**¨]])**<sup>n</sup>** <sup>+</sup> <sup>∇</sup> ([[ **<sup>u</sup>**¨]] · **<sup>n</sup>**)] <sup>−</sup> <sup>2</sup>*v*<sup>2</sup> *<sup>t</sup>*(<sup>∇</sup> [[ **<sup>u</sup>**¨]])**<sup>n</sup>** <sup>−</sup> *<sup>γ</sup>*2[[ *<sup>ϑ</sup>*¨]]**<sup>n</sup>** + *γ*2*υ*<sup>2</sup> *<sup>n</sup>*<sup>∇</sup> [[ *<sup>ϑ</sup>*˙]], (98) *υ*2 *<sup>n</sup>* I−C(**n** ⊗ **n**) [[ **V**¨ ]] = 2*υn*(*v*<sup>2</sup> *sm* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *tm*) sym (Div [[ **<sup>V</sup>**˙ ]]) <sup>⊗</sup> **<sup>n</sup>** <sup>+</sup> <sup>∇</sup> ([[ **<sup>V</sup>**˙ ]]**n**) − −2 *υ<sup>n</sup> v*2 *sm*(<sup>∇</sup> [[ **<sup>V</sup>**˙ ]])**<sup>n</sup>** <sup>+</sup> *<sup>λ</sup>*<sup>1</sup> sym [<sup>∇</sup> (tr[[ **<sup>V</sup>**˙ ]]) <sup>⊗</sup> **<sup>n</sup>**]+(**<sup>n</sup>** · Div [[ **<sup>V</sup>**˙ ]])**<sup>I</sup>** − (99) <sup>−</sup><sup>2</sup> *<sup>υ</sup><sup>n</sup> <sup>λ</sup>*2[**<sup>n</sup>** · ∇ (tr[[ **<sup>V</sup>**˙ ]])]**<sup>I</sup>** <sup>−</sup> *<sup>υ</sup>*<sup>2</sup> *<sup>n</sup>*[*<sup>ω</sup>* tr[[ **<sup>V</sup>**˙ ]]**<sup>I</sup>** + <sup>2</sup> *<sup>σ</sup>* [[ **<sup>V</sup>**˙ ]]], *γ*Θ = *υ<sup>n</sup>* <sup>2</sup>*γ*(**<sup>n</sup>** · ∇ [[ *<sup>ϑ</sup>*¨]]) + *<sup>γ</sup>*2*dn* − <sup>−</sup>*υ*<sup>2</sup> *n <sup>γ</sup>*Δ[[ *<sup>ϑ</sup>*˙]] + *<sup>γ</sup>*1[[ *<sup>ϑ</sup>*¨]] + *<sup>γ</sup>*2Div [[ **<sup>u</sup>**¨]] + *<sup>κ</sup>*∗*γ*3tr[[ **<sup>V</sup>**¨ ]] , (100)

with Θ that indicates the jump in the third time-derivative of the temperature change field *ϑ*.

Therefore, the transport equations in the linearized case will give standard evolution laws of the type *<sup>f</sup>* � <sup>=</sup> <sup>−</sup>*<sup>μ</sup> <sup>f</sup>* and hence *<sup>f</sup>* <sup>=</sup> *<sup>f</sup>*<sup>0</sup> <sup>e</sup>−*μφ*, where *<sup>μ</sup>* is a constant, *<sup>f</sup>*<sup>0</sup> is the strength of the wave at *τ* = *τ*<sup>0</sup> and *φ* is the increasing distance, measured along the normal to the wave, from the wave front at the same time.

Therefore, we have from Eq. (106) that all the components of [[ **V**¨ ]]*tm* are equal to zero except

− *σ vtm φ* 

*tm*

Also in this case the scalar amplitudes decay to zero as the time interval (*τ* − *τ*0) increases indefinitely, but, unlike shear optical waves, we have here a macro-acceleration jump with a third order discontinuity related to the elastic properties of nano-pores and to a part that

In this case the solutions of §7.1 for the amplitude and the speed are [[ **V**˙ ]]*em* and *vem*,

*em***I** − U(**n** ⊗ **n**)

*dδ dn* **<sup>n</sup>** <sup>−</sup>

*dn* <sup>+</sup> *vemσδ*

*γ*1*ϑδ*ˇ + *γ*2*vem�*

 *v*2

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

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

*v*2 *sm dδ*

2 *dδ*

*tm*)*ζ* − *λ*1(2 + *ζ*)

The solutions of the system (111) are [[ *<sup>V</sup>*¨ ]]*ij* <sup>=</sup> 0, if *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*, and [[ *<sup>V</sup>*¨ ]]<sup>33</sup> = [[ *<sup>V</sup>*¨ ]]22, while, to determine *δ*, it is necessary to know either of [[ *V*¨ ]]<sup>11</sup> or [[ *V*¨ ]]<sup>22</sup> previously, otherwise it remains undetermined and we cannot say anything about the growth or the decay of this wave; vice versa, if we are able to assign the behaviour of the amplitude *δ*, we can resolve the remaining two jumps. Moreover, from Eqs. (110) and (112), we obtain that (*de*)*em* = (*df*)*em* = 0, while also [[ *ρ*¨]]*em*, (*dn*)*em* and Θ*em* suffer of the same undeterminacies already spoken about. The micro-wave is then accompanied by second and third order discontinuities

[[ *V*¨ ]]1*<sup>i</sup>* +

*σ*(*v*<sup>2</sup>

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

*dn* [*λ*1*<sup>ζ</sup>* <sup>+</sup> *<sup>λ</sup>*2(<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>*)] <sup>+</sup> *vemω*(<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>*)*<sup>δ</sup>*

<sup>−</sup> *vemσζδ*

*dδ*

(**e** ⊗ **e** + **f** ⊗ **f**),

*dn* <sup>+</sup> *<sup>κ</sup>*∗*γ*<sup>3</sup> tr[[ **<sup>V</sup>**¨ ]]

*tm* + *<sup>v</sup>*<sup>2</sup> *t*)

, for *i* = *e*, *f* , (108)

Linear Wave Motions in Continua with Nano-Pores 81

**d** = (110)

 ,

 **I** +

(**n** ⊗ **n**) (111)

. (112)

*λ*5(tr[[ **V**¨ ]])**n** + *λ*6[[ **V**¨ ]]**n**

, for *i* = *e*, *f* . (109)

for the undetermined [[ *V*¨ ]]<sup>12</sup> and [[ *V*¨ ]]13, while the scalar amplitudes are given by

with *χi*<sup>0</sup> the values at *τ* = *τ*0. Instead, Eqs. (105), (107) and (108) establish that

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

*<sup>χ</sup>i*(*τ*) = *<sup>χ</sup>i*<sup>0</sup> exp

(*dn*)*tm* = [[ *<sup>ρ</sup>*¨]]*tm* <sup>=</sup> <sup>Θ</sup>*tm* <sup>=</sup> 0, (*di*)*tm* <sup>=</sup> *<sup>κ</sup>*∗*vtm <sup>λ</sup>*<sup>6</sup>

respectively, and thus, by applying Eqs (83) and (84), we have:

*dδ dn* ,

[[ **<sup>V</sup>**¨ ]] = <sup>−</sup>*vem*

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

<sup>−</sup>2*vem*

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

*em* − *ρ*∗*vem�*

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

<sup>2</sup> *<sup>γ</sup> <sup>ϑ</sup>*<sup>ˇ</sup> *<sup>d</sup><sup>δ</sup>*

in macro-mechanical, mass and thermal fields.

In this peculiar subcase we observed in §7.1:

*vem* [2*λ*<sup>5</sup> + (*λ*<sup>5</sup> + *λ*6)*ζ*]

*dn* <sup>+</sup> *<sup>γ</sup>*2*dn*

decays to zero.

*8.1.3. Extensional mode*

[[ *<sup>ρ</sup>*¨]] = *<sup>ρ</sup>*∗*dnv*−<sup>1</sup>

*em* **I** − C(**n** ⊗ **n**)

<sup>+</sup>2*vem <sup>d</sup><sup>δ</sup>*

<sup>=</sup> *<sup>κ</sup>*∗*vem*

*<sup>γ</sup>*<sup>Θ</sup> <sup>=</sup> *vem*

*8.1.4. λ*<sup>1</sup> + *λ*<sup>2</sup> = 0 *case.*

 *v*2

### **8.1. Evolution of homothermal waves**

Now, since for a plane homothermal macro-acceleration wave entering the natural reference placement B∗ the jump of *<sup>ϑ</sup>*˙ vanishes for the Fourier condition (72), by developing the analysis of equations (97)-(100) we get the following consequences:

#### *8.1.1. Shear optical case*

By inserting the micro-wave solution of §7.1 of amplitude [[ **V**˙ ]]*sm* and speed *vsm*, which have fields *u*¨, *ρ*˙ and *ϑ*¨ continuous through the wave, we obtain that:

$$\begin{aligned} \left[ \left[ \vec{\rho} \right] \right] &= \rho\_\* \, d\_n v\_{\text{sm}}^{-1}, \quad \left[ v\_{\text{sm}}^2 \mathbf{I} - \mathcal{U}(\mathbf{n} \otimes \mathbf{n}) \right] \mathbf{d} = -\kappa\_\* v\_{\text{sm}} \left[ \lambda\_5 (\text{tr} \left[ \ddot{\mathbf{V}} \right]) \mathbf{n} + \lambda\_6 \llbracket \ddot{\mathbf{V}} \rceil \mathbf{n} \right], \quad \text{(101)}\\ \left[ v\_{\text{sm}}^2 \mathcal{Z} - \mathcal{C}(\mathbf{n} \otimes \mathbf{n}) \right] \left[ \ddot{\mathbf{V}} \right] &= \end{aligned}$$

$$\mathbf{r}\_{s} = 2v\_{sm}^{3} \left\{ \left[ \frac{d\mathbf{a}}{dn} + \frac{\sigma a}{v\_{sm}} \right] (\mathbf{f} \otimes \mathbf{f} - \mathbf{e} \otimes \mathbf{e}) - \left[ \frac{d\boldsymbol{\beta}}{dn} + \frac{\sigma \boldsymbol{\beta}}{v\_{sm}} \right] (\mathbf{e} \otimes \mathbf{f} + \mathbf{f} \otimes \mathbf{e}) \right\}, \tag{102}$$

$$
\gamma \Theta = \gamma\_2 v\_{sm} d\_{\rm tr} - \kappa\_\* \gamma\_3 v\_{sm}^2 \text{tr} \left[ \ddot{\mathbf{V}} \right]; \tag{103}
$$

hence, from Eq. (102), it must be [[ *<sup>V</sup>*¨ ]]<sup>11</sup> = [[ *<sup>V</sup>*¨ ]]<sup>12</sup> = [[ *<sup>V</sup>*¨ ]]<sup>13</sup> <sup>=</sup> 0 and [[ *<sup>V</sup>*¨ ]]<sup>33</sup> <sup>=</sup> <sup>−</sup>[[ *<sup>V</sup>*¨ ]]22, while [[ *V*¨ ]]<sup>22</sup> and [[ *V*¨ ]]<sup>23</sup> remain undefined,

$$\mathfrak{a}(\tau) = \mathfrak{a}\_0 \exp\left(-\frac{\sigma}{\upsilon\_{sm}}\phi\right) \text{ and } \mathfrak{f}(\tau) = \mathfrak{f}\_0 \exp\left(-\frac{\sigma}{\upsilon\_{sm}}\phi\right),\tag{104}$$

with *α*<sup>0</sup> and *β*<sup>0</sup> the values at *τ* = *τ*0. Instead, by analysing Eq. (101)2, we have that **d***sm* = **0** and thus [[ *ρ*¨]]*sm* = 0 and Θ*sm* = 0.

This kind of micro-wave does not cause any disturbance in the mechanical and thermal fields and the scalar amplitudes *α* and *β* decay to zero as the time interval (*τ* − *τ*0) increases indefinitely (because *φ* behaves so).

#### *8.1.2. Transverse micro-wave*

For the transverse solutions of §7.1, whose amplitude is [[ **V**˙ ]]*tm*, speed *vtm* and jump [[ **u**¨]]*tm* given by equation (82) (while *ρ*˙ and *ϑ*¨ are continuous), the algebraic system of evolution Eqs. (97)-(100) reduces to the following one:

$$\begin{split} \left[\mathbb{\hat{p}}\right] = \rho\_\* v\_{tm}^{-1} d\_{\mathcal{U}} \quad \left[v\_{tm}^2 \mathbf{I} - \mathcal{U}(\mathbf{n} \otimes \mathbf{n})\right] \mathbf{d} = \\ = \kappa\_\* v\_{tm}^2 \lambda\_{\mathcal{H}} \frac{v\_{tm}^2 + v\_t^2}{v\_{tm}^2 - v\_t^2} \left(\frac{d\chi\_{\varepsilon}}{dn} \mathbf{e} + \frac{d\chi\_f}{dn} \mathbf{f}\right) - \kappa\_\* v\_{tm} \left[\lambda\_5 (\text{tr}\left[\dot{\mathbf{V}}\right]) \mathbf{n} + \lambda\_6 \mathbb{I}\left[\dot{\mathbf{V}}\right] \mathbf{n}\right] \right. \\ \left. \left[v\_{tm}^2 \mathcal{L} - \mathcal{L}(\mathbf{n} \otimes \mathbf{n})\right] \left[\dot{\mathbf{V}}\right] = -4v\_{tm}^3 \text{sym}\left[\left(\frac{d\chi\_{\varepsilon}}{dn} + \frac{\sigma\chi\_{\varepsilon}}{v\_{tm}}\right) \mathbf{e} + \left(\frac{d\chi\_f}{dn} + \frac{\sigma\chi\_f}{v\_{tm}}\right) \mathbf{f}\right] \otimes \mathbf{n}, \tag{106} \\ \gamma \boldsymbol{\Theta} = \gamma\_2 v\_{\mathrm{Im}} d\_{\mathrm{I}} - \kappa\_5 \gamma\_3 v\_{\mathrm{Im}}^2 \text{tr}\left[\dot{\mathbf{V}}\right]. \end{split} \tag{107}$$

Therefore, we have from Eq. (106) that all the components of [[ **V**¨ ]]*tm* are equal to zero except for the undetermined [[ *V*¨ ]]<sup>12</sup> and [[ *V*¨ ]]13, while the scalar amplitudes are given by

$$\chi\_{i}(\tau) = \chi\_{i0} \exp\left(-\frac{\sigma}{v\_{tm}}\phi\right), \text{ for } i = e, f\_{\prime} \tag{108}$$

with *χi*<sup>0</sup> the values at *τ* = *τ*0. Instead, Eqs. (105), (107) and (108) establish that

$$(d\_h)\_{lm} = \|\ddot{\rho}\|\_{lm} = \Theta\_{lm} = 0,\\
(d\_i)\_{lm} = \frac{\kappa\_\* v\_{lm} \lambda\_6}{v\_t^2 - v\_{lm}^2} \left(\|\ddot{V}\|\_{1i} + \frac{\sigma(v\_{lm}^2 + v\_t^2)}{v\_{tm}^2 - v\_t^2} \chi\_i\right), \text{ for } i = e, f. \tag{109}$$

Also in this case the scalar amplitudes decay to zero as the time interval (*τ* − *τ*0) increases indefinitely, but, unlike shear optical waves, we have here a macro-acceleration jump with a third order discontinuity related to the elastic properties of nano-pores and to a part that decays to zero.

#### *8.1.3. Extensional mode*

20 Will-be-set-by-IN-TECH

Therefore, the transport equations in the linearized case will give standard evolution laws of the type *<sup>f</sup>* � <sup>=</sup> <sup>−</sup>*<sup>μ</sup> <sup>f</sup>* and hence *<sup>f</sup>* <sup>=</sup> *<sup>f</sup>*<sup>0</sup> <sup>e</sup>−*μφ*, where *<sup>μ</sup>* is a constant, *<sup>f</sup>*<sup>0</sup> is the strength of the wave at *τ* = *τ*<sup>0</sup> and *φ* is the increasing distance, measured along the normal to the wave, from the

Now, since for a plane homothermal macro-acceleration wave entering the natural reference placement B∗ the jump of *<sup>ϑ</sup>*˙ vanishes for the Fourier condition (72), by developing the analysis

By inserting the micro-wave solution of §7.1 of amplitude [[ **V**˙ ]]*sm* and speed *vsm*, which have

(**f** ⊗ **f** − **e** ⊗ **e**) −

hence, from Eq. (102), it must be [[ *<sup>V</sup>*¨ ]]<sup>11</sup> = [[ *<sup>V</sup>*¨ ]]<sup>12</sup> = [[ *<sup>V</sup>*¨ ]]<sup>13</sup> <sup>=</sup> 0 and [[ *<sup>V</sup>*¨ ]]<sup>33</sup> <sup>=</sup> <sup>−</sup>[[ *<sup>V</sup>*¨ ]]22, while

with *α*<sup>0</sup> and *β*<sup>0</sup> the values at *τ* = *τ*0. Instead, by analysing Eq. (101)2, we have that **d***sm* = **0**

This kind of micro-wave does not cause any disturbance in the mechanical and thermal fields and the scalar amplitudes *α* and *β* decay to zero as the time interval (*τ* − *τ*0) increases

For the transverse solutions of §7.1, whose amplitude is [[ **V**˙ ]]*tm*, speed *vtm* and jump [[ **u**¨]]*tm* given by equation (82) (while *ρ*˙ and *ϑ*¨ are continuous), the algebraic system of evolution Eqs.

> *dχ<sup>f</sup> dn* **<sup>f</sup>**

*dχ<sup>e</sup>*

− *κ*∗*vtm*

*dn* <sup>+</sup> *σχ<sup>e</sup> vtm* *tm*tr[[ **<sup>V</sup>**¨ ]]. (107)

 **e** +

**d** = −*κ*∗*vsm*

 *dβ dn* <sup>+</sup> *σβ vsm* 

and *β*(*τ*) = *β*<sup>0</sup> exp

*sm* tr[[ **<sup>V</sup>**¨ ]]; (103)

 <sup>−</sup> *<sup>σ</sup> vsm φ* 

**d** = (105)

*dχ<sup>f</sup>*

*λ*5(tr[[ **V**¨ ]])**n** + *λ*6[[ **V**¨ ]]**n**

 **f** 

*dn* <sup>+</sup> *σχ<sup>f</sup> vtm*

*λ*5(tr[[ **V**¨ ]])**n** + *λ*6[[ **V**¨ ]]**n**

(**e** ⊗ **f** + **f** ⊗ **e**)

, (104)

 ,

⊗ **n**, (106)

, (101)

, (102)

wave front at the same time.

*8.1.1. Shear optical case*

 *v*2

[[ *<sup>ρ</sup>*¨]] = *<sup>ρ</sup>*<sup>∗</sup> *dnv*−<sup>1</sup>

*sm* I−C(**n** ⊗ **n**)

*<sup>γ</sup>*<sup>Θ</sup> <sup>=</sup> *<sup>γ</sup>*2*vsmdn* <sup>−</sup> *<sup>κ</sup>*∗*γ*<sup>3</sup> *<sup>v</sup>*<sup>2</sup>

= 2*v*<sup>3</sup> *sm*

[[ *V*¨ ]]<sup>22</sup> and [[ *V*¨ ]]<sup>23</sup> remain undefined,

and thus [[ *ρ*¨]]*sm* = 0 and Θ*sm* = 0.

indefinitely (because *φ* behaves so).

(97)-(100) reduces to the following one:

*tm dn*,

<sup>=</sup> *<sup>κ</sup>*∗*v*<sup>2</sup> *tmλ*<sup>6</sup> *v*2 *tm* + *<sup>v</sup>*<sup>2</sup> *t*

*<sup>γ</sup>*<sup>Θ</sup> <sup>=</sup> *<sup>γ</sup>*2*vtmdn* <sup>−</sup> *<sup>κ</sup>*∗*γ*<sup>3</sup> *<sup>v</sup>*<sup>2</sup>

*tm* I−C(**n** ⊗ **n**)

 *v*2

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

*tm***I** − U(**n** ⊗ **n**)

[[ **<sup>V</sup>**¨ ]] = <sup>−</sup>4*v*<sup>3</sup>

 *dχ<sup>e</sup> dn* **<sup>e</sup>** <sup>+</sup>

*tm*sym

*8.1.2. Transverse micro-wave*

[[ *<sup>ρ</sup>*¨]] = *<sup>ρ</sup>*∗*v*−<sup>1</sup>

 *v*2 *sm* ,

 *dα dn* <sup>+</sup>

*α*(*τ*) = *α*<sup>0</sup> exp

**8.1. Evolution of homothermal waves**

of equations (97)-(100) we get the following consequences:

fields *u*¨, *ρ*˙ and *ϑ*¨ continuous through the wave, we obtain that:

*σα vsm* 

> <sup>−</sup> *<sup>σ</sup> vsm φ*

*sm***I** − U(**n** ⊗ **n**)

 *v*2

 [[ **V**¨ ]] =

In this case the solutions of §7.1 for the amplitude and the speed are [[ **V**˙ ]]*em* and *vem*, respectively, and thus, by applying Eqs (83) and (84), we have:

[[ *<sup>ρ</sup>*¨]] = *<sup>ρ</sup>*∗*dnv*−<sup>1</sup> *em* − *ρ*∗*vem� dδ dn* , *v*2 *em***I** − U(**n** ⊗ **n**) **d** = (110) <sup>=</sup> *<sup>κ</sup>*∗*vem vem* [2*λ*<sup>5</sup> + (*λ*<sup>5</sup> + *λ*6)*ζ*] *v*2 *em* + *v*<sup>2</sup> *l v*2 *em* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *l dδ dn* **<sup>n</sup>** <sup>−</sup> *λ*5(tr[[ **V**¨ ]])**n** + *λ*6[[ **V**¨ ]]**n** , *v*2 *em* **I** − C(**n** ⊗ **n**) [[ **<sup>V</sup>**¨ ]] = <sup>−</sup>*vem* 2 *dδ dn* [*λ*1*<sup>ζ</sup>* <sup>+</sup> *<sup>λ</sup>*2(<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>*)] <sup>+</sup> *vemω*(<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>*)*<sup>δ</sup>* **I** + <sup>+</sup>2*vem <sup>d</sup><sup>δ</sup> dn* (*v*<sup>2</sup> *sm* <sup>−</sup> <sup>2</sup>*v*<sup>2</sup> *tm*)*ζ* − *λ*1(2 + *ζ*) <sup>−</sup> *vemσζδ* (**n** ⊗ **n**) (111) <sup>−</sup>2*vem v*2 *sm dδ dn* <sup>+</sup> *vemσδ* (**e** ⊗ **e** + **f** ⊗ **f**), *<sup>γ</sup>*<sup>Θ</sup> <sup>=</sup> *vem* <sup>2</sup> *<sup>γ</sup> <sup>ϑ</sup>*<sup>ˇ</sup> *<sup>d</sup><sup>δ</sup> dn* <sup>+</sup> *<sup>γ</sup>*2*dn* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *em γ*1*ϑδ*ˇ + *γ*2*vem� dδ dn* <sup>+</sup> *<sup>κ</sup>*∗*γ*<sup>3</sup> tr[[ **<sup>V</sup>**¨ ]] . (112)

The solutions of the system (111) are [[ *<sup>V</sup>*¨ ]]*ij* <sup>=</sup> 0, if *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*, and [[ *<sup>V</sup>*¨ ]]<sup>33</sup> = [[ *<sup>V</sup>*¨ ]]22, while, to determine *δ*, it is necessary to know either of [[ *V*¨ ]]<sup>11</sup> or [[ *V*¨ ]]<sup>22</sup> previously, otherwise it remains undetermined and we cannot say anything about the growth or the decay of this wave; vice versa, if we are able to assign the behaviour of the amplitude *δ*, we can resolve the remaining two jumps. Moreover, from Eqs. (110) and (112), we obtain that (*de*)*em* = (*df*)*em* = 0, while also [[ *ρ*¨]]*em*, (*dn*)*em* and Θ*em* suffer of the same undeterminacies already spoken about. The micro-wave is then accompanied by second and third order discontinuities in macro-mechanical, mass and thermal fields.

#### *8.1.4. λ*<sup>1</sup> + *λ*<sup>2</sup> = 0 *case.*

In this peculiar subcase we observed in §7.1:

#### 22 Will-be-set-by-IN-TECH 82 Wave Processes in Classical and New Solids Linear Wave Motions in Continua with Nano-Pores <sup>23</sup>

i) *A purely transverse micro-wave* of amplitude [[ **V**˙ ]]*pm* and speed *vpm*, the other jumps being given by Eqs. (85) and (86). By performing same developments of previous solutions, we obtain that [[ *<sup>V</sup>*¨ ]]*ij* <sup>=</sup> 0, if *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*, and [[ *<sup>V</sup>*¨ ]]<sup>33</sup> = [[ *<sup>V</sup>*¨ ]]<sup>22</sup> (which remain undetermined); moreover, [[ *<sup>V</sup>*¨ ]]<sup>11</sup> <sup>=</sup> <sup>2</sup>*v*<sup>2</sup> *pmω v*2 *tm*−2*v*<sup>2</sup> *pm*−*λ*<sup>1</sup> *�*, with the scalar amplitudes *�* given by *�*(*τ*) = *�*<sup>0</sup> exp <sup>−</sup>*ω*+*<sup>σ</sup> vpm <sup>φ</sup>* (*�*<sup>0</sup> being the value at *τ* = *τ*0). Thus, it results that (*de*)*pm* = (*df*)*pm* <sup>=</sup> 0, while (*dn*)*pm* <sup>=</sup> <sup>Γ</sup>*�* <sup>−</sup> <sup>2</sup>*κ*∗*λ*5*vpm v*2 *pm*−*v*<sup>2</sup> *l* [[ *<sup>V</sup>*¨ ]]22, [[ *<sup>ρ</sup>*¨]]*pm* <sup>=</sup> *<sup>ρ</sup>*∗*v*−<sup>1</sup> *pmdn* and *<sup>γ</sup>*Θ*pm* <sup>=</sup> *<sup>γ</sup>*2*vpmdn* <sup>−</sup>2*κ*∗*γ*3*v*<sup>2</sup> *pm*[[ *<sup>V</sup>*¨ ]]<sup>22</sup> <sup>−</sup> <sup>Π</sup> *�*, where <sup>Γ</sup> and <sup>Π</sup> are constants related to previous defined constitutive constants.

Hence, with this simple change of constants, the conclusions about the evolution of amplitudes of the macro-acceleration waves remain the same as in the corresponding

◦) *Generalized Transverse Case:* This last peculiar wave, both homothermal and homentropic, has the same shear optical and transverse solutions as the homothermal case (see §7.2) and

The extensional micro-wave, and so comments about its amplitude evolution, occurrs only

a) *Linear thermo-dynamic theory.* We derived the linear theory of a thermoelastic solid with nano-pores which includes inelastic surface effects associated with changes in the deformation of the holes in the vicinity of void boundaries and which generalizes classical voids theories. In order to get the fields equations we used the principles of objectivity and

b) *Micro-vibrations.* The first application to micro-vibrations in absence of dissipation gives origin to three admissible results: a dilatational micro-thermal oscillation and two solutions, both with no thermal vibrations, with the same frequency and with null trace: a

c) *Plane waves.* Here we presented the solutions of secular equations governing the propagation of harmonic plane waves in the porous thermoelastic medium: there can exist two shear optical micro-elastic waves, two coupled transverse elastic waves and four coupled longitudinal thermo-elastic waves. The exact or approximate values of the phase speeds, specific losses, attenuation factors and amplitude ratios are discussed for large and

d) *Macro-acceleration waves.* Last investigation regarded the propagation conditions and the growth equations which govern the motion of particular weak singularities, called macro-acceleration waves, for which only jumps of the derivatives of the macro- and micro-displacement of order 2 and of the temperature of order 1 are of interest in the theory. We observed that, for a linear conducting homogeneous centrosymmetric isotropic material with nano-pores, every macro-acceleration wave is homothermal and only three speeds of propagation are possible: i) one related to two shear-optical micro-modes completely decoupled from the mass and thermoelastic macro-properties of the matrix material and which decay to zero when the time interval increases; ii) the second velocity associated to two transverse micro-modes coupled with a transverse macro-acceleration wave, spreading without perturbing mass and thermal fields: the micro-modes decay still to zero, while the associated macro-ones have a constant part and an added contribution that decays still to zero; iii) the third one linked to one extensional micro-wave coupled with a longitudinal macro-wave and with discontinuities in the second and third order of

Instead in the non-conducting case every wave is homentropic, but we obtained the same number of propagation velocities and of macro-acceleration waves as in the previous

equipresence, besides the compatibility with the Clausius-Duhem inequality.

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

*em*) = *γ*2[2*λ*<sup>5</sup> + (*λ*<sup>5</sup> + *λ*6)*ζ*].

Linear Wave Motions in Continua with Nano-Pores 83

so the solutions of the evolution equations have the same behaviour.

if the condition (95) is satisfied, *i.e.*, *γ*3(2 + *ζ*)(*v*<sup>2</sup>

The results showed in this chapter can be outlined as it follows:

shear mode and an extensional mode with constant volume.

homothermal case.

**9. Conclusions**

small frequencies.

derivatives of mass and thermal fields.

homothermal instance.

Hence the third order discontinuity of the longitudinal macro-wave, induced by the purely transverse micro-wave, has a first part that decays to zero as the time interval (*τ* − *τ*0) go to infinity and a second one related to the elastic properties of nano-pores, as well as discontinuities in the mass and temperature derivatives.

ii) *A purely longitudinal micro-wave* of amplitude [[ **V**˙ ]]*lm* and speed of propagation *vlm* for which we have that all [[ *<sup>V</sup>*¨ ]]*ij* <sup>=</sup> 0, if *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*, [[ *<sup>V</sup>*¨ ]]<sup>11</sup> remains undefined and [[ *<sup>V</sup>*¨ ]]<sup>33</sup> = [[ *<sup>V</sup>*¨ ]]<sup>22</sup> <sup>=</sup> *ω v*2 *lm*−*v*<sup>2</sup> *sm*+2*λ*<sup>1</sup> ˆ *δ*, with ˆ *δ*(*τ*) = ˆ *<sup>δ</sup>*<sup>0</sup> exp <sup>−</sup>*ω*+*<sup>σ</sup>* <sup>2</sup>*vlm <sup>φ</sup>* and ˆ *δ*<sup>0</sup> its value at *τ* = *τ*0; in addition, also now (*de*)*lm* = (*df*)*lm* = 0, while (*dn*)*lm* = <sup>Ξ</sup> <sup>ˆ</sup> *<sup>δ</sup>* <sup>−</sup> *<sup>κ</sup>*∗*vlm*(*λ*<sup>5</sup> <sup>+</sup> *<sup>λ</sup>*6)[[ *<sup>V</sup>*¨ ]]11, [[ *<sup>ρ</sup>*¨]]*lm* <sup>=</sup> *<sup>ρ</sup>*∗*v*−<sup>1</sup> *lm dn* and *<sup>γ</sup>*Θ*lm* <sup>=</sup> *<sup>γ</sup>*2*vlmdn* <sup>−</sup> *<sup>κ</sup>*∗*γ*3*v*<sup>2</sup> *lm*[[ *<sup>V</sup>*¨ ]]<sup>11</sup> <sup>−</sup> <sup>Υ</sup> <sup>ˆ</sup> *δ*, with Ξ and Υ constants related to constitutive constants.

Therefore, also the pure micro-wave of compaction or distention is accompanied by a third order discontinuities in the mechanical field with a first part that decays to zero with the increasing of the time interval (*τ* − *τ*0) and a second one related to nano-pores properties.

#### **8.2. Homentropic and generalized transverse evolution instances**

◦) *Homentropic macro-acceleration waves:* When the solid with nano-pores does not conduct heat (see §7.2), we have to substitute evolution Eq. (100) with the following one:

$$
\gamma\_1 \|\left[\ddot{\theta}\right] = \gamma\_2 \upsilon\_n^{-1} d\_n - \gamma\_2 \text{Div} \left[\left[\ddot{\mathbf{u}}\right] - \kappa\_\* \gamma\_3 \text{tr} \left[\ddot{\mathbf{V}}\right] \right] \tag{113}
$$

which is obtained by deriving with respect to the time *<sup>τ</sup>* the energy balance *<sup>η</sup>*˙ <sup>=</sup> *<sup>θ</sup>*−<sup>1</sup> <sup>∗</sup> *<sup>λ</sup>*, by using relation (35)4 and by taking its jump.

As we observed in §7.2, discussions about this subcase follow closely those carried out for the homothermal one with respect to shear optical and transverse macro-acceleration waves; instead, for the extensional, purely transverse and purely longitudinal ones the only change consists in the choose of constants *<sup>v</sup>*˜*<sup>l</sup>* and *<sup>λ</sup>*˜ <sup>5</sup> in place of *vl* and *<sup>λ</sup>*5: for example, Eq. (110)2 must be substituted by

$$\begin{cases} \boldsymbol{v}\_{em}^{2}\mathbf{I} - \mathcal{U}(\mathbf{n}\otimes\mathbf{n})\Big{[}\mathbf{d}\_{em} =\\ \boldsymbol{v} = \kappa\_{\ast}\boldsymbol{v}\_{em}\left\{\boldsymbol{v}\_{em}\left[2\tilde{\lambda}\_{5} + (\tilde{\lambda}\_{5} + \lambda\_{6})\tilde{\zeta}\right]\frac{\boldsymbol{v}\_{em}^{2} + \boldsymbol{v}\_{I}^{2}}{\boldsymbol{v}\_{em}^{2} - \tilde{\boldsymbol{v}}\_{I}^{2}}\frac{d\delta}{d\boldsymbol{n}}\mathbf{n} - \left[\lambda\_{5}(\text{tr}\left[\dot{\mathbf{V}}\right]\_{em})\mathbf{n} + \lambda\_{6}\llbracket\dot{\mathbf{V}}\right]\_{em}\mathbf{n}\right\}\end{cases},\tag{114}$$

while Eq. (113) gives *<sup>γ</sup>*1[[ *<sup>ϑ</sup>*¨]]*em* <sup>=</sup> *<sup>γ</sup>*2*v*−<sup>1</sup> *em* (*dn*)*em* <sup>−</sup> *<sup>γ</sup>*2*�*˜ *vem <sup>d</sup><sup>δ</sup> dn* <sup>−</sup> *<sup>κ</sup>*∗*γ*3([[ *<sup>V</sup>*¨ ]]<sup>11</sup> <sup>+</sup> <sup>2</sup>[[ *<sup>V</sup>*¨ ]]22). Hence, with this simple change of constants, the conclusions about the evolution of amplitudes of the macro-acceleration waves remain the same as in the corresponding homothermal case.

◦) *Generalized Transverse Case:* This last peculiar wave, both homothermal and homentropic, has the same shear optical and transverse solutions as the homothermal case (see §7.2) and so the solutions of the evolution equations have the same behaviour.

The extensional micro-wave, and so comments about its amplitude evolution, occurrs only if the condition (95) is satisfied, *i.e.*, *γ*3(2 + *ζ*)(*v*<sup>2</sup> *<sup>l</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *em*) = *γ*2[2*λ*<sup>5</sup> + (*λ*<sup>5</sup> + *λ*6)*ζ*].
