**9. Conclusions**

22 Will-be-set-by-IN-TECH

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

*pmω*

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

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>

 and ˆ

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.

◦) *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:

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

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,

> *dδ dn* **<sup>n</sup>** <sup>−</sup>

<sup>−</sup>*ω*+*<sup>σ</sup>* <sup>2</sup>*vlm <sup>φ</sup>*

*lm*[[ *<sup>V</sup>*¨ ]]<sup>11</sup> <sup>−</sup> <sup>Υ</sup> <sup>ˆ</sup>

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

*γ*1[[ *ϑ*¨]] = *γ*2*υ*−<sup>1</sup>

using relation (35)4 and by taking its jump.

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

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>*

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

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

Eq. (110)2 must be substituted by

*em***I** − U(**n** ⊗ **n**)

 *vem*  *v*2 *pm*−*v*<sup>2</sup> *l*

*�*, with the scalar amplitudes *�* given

[[ *<sup>V</sup>*¨ ]]22, [[ *<sup>ρ</sup>*¨]]*pm* <sup>=</sup> *<sup>ρ</sup>*∗*v*−<sup>1</sup>

*δ*<sup>0</sup> its value at *τ* = *τ*0; in addition, also

*<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>

*δ*, with Ξ and Υ constants related to constitutive

*<sup>n</sup> dn* <sup>−</sup> *<sup>γ</sup>*2Div [[ **<sup>u</sup>**¨]] <sup>−</sup> *<sup>κ</sup>*∗*γ*3tr[[ **<sup>V</sup>**¨ ]], (113)

**d***em* = (114)

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

*dn* <sup>−</sup> *<sup>κ</sup>*∗*γ*3([[ *<sup>V</sup>*¨ ]]<sup>11</sup> <sup>+</sup> <sup>2</sup>[[ *<sup>V</sup>*¨ ]]22).

 ,

*pmdn* and

*lm dn*

(*�*<sup>0</sup> being the value at *τ* = *τ*0). Thus, it results that

*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

*v*2 *tm*−2*v*<sup>2</sup> *pm*−*λ*<sup>1</sup>

undetermined); moreover, [[ *<sup>V</sup>*¨ ]]<sup>11</sup> <sup>=</sup> <sup>2</sup>*v*<sup>2</sup>

<sup>−</sup>*ω*+*<sup>σ</sup> vpm <sup>φ</sup>*

(*de*)*pm* = (*df*)*pm* <sup>=</sup> 0, while (*dn*)*pm* <sup>=</sup> <sup>Γ</sup>*�* <sup>−</sup> <sup>2</sup>*κ*∗*λ*5*vpm*

discontinuities in the mass and temperature derivatives.

*δ*<sup>0</sup> exp 

*δ*(*τ*) = ˆ

now (*de*)*lm* = (*df*)*lm* = 0, while (*dn*)*lm* = <sup>Ξ</sup> <sup>ˆ</sup>

by *�*(*τ*) = *�*<sup>0</sup> exp

*ω*

constants.

 *v*2

= *κ*∗*vem*

*v*2 *lm*−*v*<sup>2</sup> *sm*+2*λ*<sup>1</sup> ˆ *δ*, with ˆ

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

defined constitutive constants.

and *<sup>γ</sup>*Θ*lm* <sup>=</sup> *<sup>γ</sup>*2*vlmdn* <sup>−</sup> *<sup>κ</sup>*∗*γ*3*v*<sup>2</sup>

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


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 homothermal instance.

At the end, for generalized transverse macro-acceleration waves, only the extensional micro-mode does not occurr, in general.

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