# Rayleigh waves in isotropic elastic materials with micro-voids

Emilian Bulgariu and Ionel-Dumitrel Ghiba and Hassam Khan
and Patrizio Neff  Emilian Bulgariu, Department of Exact Sciences, Faculty of
Horticulture, Iaşi University of Life Sciences (IULS), Aleea Mihail Sadoveanu
no.3, 700490 Iaşi, Romania, email<EMAIL_ADDRESS>Ionel-Dumitrel Ghiba,
Department of Mathematics, Alexandru Ioan Cuza University of Iaşi, Blvd. Carol
I, no. 11, 700506 Iaşi, Romania; and Octav Mayer Institute of Mathematics of
the Romanian Academy, Iaşi Branch, 700505 Iaşi, email:
<EMAIL_ADDRESS>Khan, Department of Mathematics and Statistics,
Institute of Business and Management Sciences (IOBM), Karachi. Pakistan,
email<EMAIL_ADDRESS>Neff, Head of Lehrstuhl für Nichtlineare
Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-
Essen, Thea-Leymann Str. 9, 45127 Essen, Germany, email: patrizio.neff@uni-
due.de

(Dedicated to Yibin Fu on the occasion of his 60th birthday)

###### Abstract

In this paper, we show that a general method introduced by Fu and Mielke
allows to give a complete answer on the existence and uniqueness of a subsonic
solution describing the propagation of surface waves in an isotropic half
space modelled with the linear theory of isotropic elastic materials with
micro-voids. Our result is valid for the entire class of materials admitting
real wave propagation which include auxetic materials (negative Poisson’s
ration) and composite materials with negative-stiffness inclusions (negative
Young’s modulus). Moreover, the used method allows to formulate a simple and
complete numerical strategy for the computation of the solution.

Keywords: relaxed micromorphic model, materials with micro-voids, Riccati
equation, Rayleigh waves, secular equation, existence and uniqueness, micro-
voids model, Fu and Mielke’s method.

AMS 2020 MSC: 74J15, 74M25, 74H05, 74J05

###### Contents

  1. 1 Introduction
  2. 2 The micro-voids model – a particular case of the relaxed micromorphic model
    1. 2.1 The relaxed micromorphic model
    2. 2.2 The micro-voids model
  3. 3 Real plane waves in the model of isotropic materials with voids
  4. 4 Rayleigh waves in isotropic elastic materials with micro-voids
  5. 5 The algebraic analysis of a Riccati-type equation
  6. 6 The ansatz for the solution and the limiting speed
    1. 6.1 The ansatz
    2. 6.2 The existence of the limiting speed in the model
  7. 7 Existence and uniqueness of Rayleigh waves
  8. 8 Numerical implementation

##  1 Introduction

Starting with [60] and [51] we have re-investigated the general micromorphic
model [21, 56] and we have introduced the relaxed micromorphic model which is
general enough to incorporate all known subclasses of micromorphic models
(Cosserat, microstretch, micro-voids, etc) but in the same time most simple to
be considered in practice and for fitting the constitutive parameters. The
relaxed micromorphic model can be used for engineering metamaterials design
and metamaterial shields for inner protection and outer tuning [17, 72, 74,
73, 70, 71]. In the micromorphic model
$P=(P_{ij}):\Omega\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$ describes the
substructure of the material which can rotate, stretch, shear and shrink,
while $u=(u_{i}):\Omega\times[0,T]\rightarrow\mathbb{R}^{3}$ is the
displacement of the macroscopic material points. Due to the coercive
inequalities proved by Neff, Pauly and Witsch [63, 61, 62] and by Bauer, Neff,
Pauly and Starke [3, 2] (see also [45]) we have shown [29, 59] that the
relaxed linear micromorphic model is well-posed for non-symmetric symmetric
Cauchy force stresses (non-vanishing Cosserat couple modulus) as well as for
the important case of symmetric Cauchy force stresses (zero Cosserat couple
modulus), even if the internal energy density is not positive definite in
terms of the strain tensors previously used in the literature. The new
inequalities given by Lewintan, Müller and Neff [47, 48] and by Gmeineder,
Lewintan, and Neff [31, 30] lead to existence results even in more degenerated
cases (less partial derivatives involved in the model).

When the micromorphic theory is linked to the theory of dislocations, there
are many reasons to assume that the Cosserat couple modulus vanishes and the
Cauchy force stresses are symmetric, see the paper by Kröner and [60]. The
particular relaxed micromophic model constructed for
$P=\zeta\cdot{{\mathbbm{1}}}$ will be called micro-voids model in the rest of
the paper. We show that the obtained initial-boundary value problem is
identical to the micro-voids theory given by Goodman and Cowin [32] and by
Nunziato and Cowin [66, 15, 37, 38]. Therefore, from two different
perspectives and approaches we are arriving at the same total energy of the
model (see [38, 28]) given by111For $a,b\in\mathbb{R}^{3\times 3}$ we let
$\langle{a,b}\rangle_{\mathbb{R}^{3}}$ denote the scalar product on
$\mathbb{R}^{3}$ with associated vector norm $\|a\|^{2}=\langle{a,a}\rangle$.
We denote by $\mathbb{R}^{3\times 3}$ the set of real $3\times 3$ second order
tensors, written with capital letters. Matrices will be denoted by bold
symbols, e.g. ${X}\in\mathbb{R}^{3\times 3}$, while $X_{ij}$ will denote its
component. The standard Euclidean product on $\mathbb{R}^{3\times 3}$ is given
by $\langle{{X},{Y}}\rangle_{\mathbb{R}^{3\times 3}}={\rm tr}({X}\,{Y}^{T})$,
and thus, the Frobenious tensor norm is
$\|{X}\|^{2}=\langle{{X},{X}}\rangle_{\mathbb{R}^{3\times 3}}$. In the
following we omit the index $\mathbb{R}^{3},\mathbb{R}^{3\times 3}$. The
identity tensor on $\mathbb{R}^{3\times 3}$ will be denoted by
${{\mathbbm{1}}}$, so that ${\rm
tr}({X})=\langle{{X},{{\mathbbm{1}}}}\rangle$. We let ${\rm Sym}$ denote the
set of symmetric tensors. We adopt the usual abbreviations of Lie-algebra
theory, i.e., $\mathfrak{so}(3):=\\{{A}\,\in\mathbb{R}^{3\times
3}|{A}^{T}=-{A}\\}$ is the Lie-algebra of skew-symmetric tensors and
$\mathfrak{sl}(3):=\\{{X}\,\in\mathbb{R}^{3\times 3}|{\rm tr}({X})=0\\}$ is
the Lie-algebra of traceless tensors. For all ${X}\in\mathbb{R}^{3\times 3}$
we set ${\rm sym}\,{X}=\frac{1}{2}({X}^{T}+{X})\in{\rm Sym},\,{\rm
skew}{X}=\frac{1}{2}({X}-{X}^{T})\in\mathfrak{so}(3)$ and the deviatoric
(trace-free) part ${\rm dev}\,{X}={X}-\frac{1}{3}{\rm
tr}({X})\in\,\mathfrak{sl}(3)$ and we have the orthogonal Cartan-decomposition
of the Lie-algebra $\mathfrak{gl}(3)=\\{\mathfrak{sl}(3)\cap{\rm
Sym}(3)\\}\oplus\mathfrak{so}(3)\oplus\mathbb{R}\cdot{{\mathbbm{1}}},\
{X}={\rm dev}\,{\rm sym}\,{X}+{\rm skew}\,{X}+\frac{1}{3}{\rm
tr}({X})\,{{\mathbbm{1}}}.\ $

$\displaystyle\mathcal{L}=\int_{\Omega}\bigg{(}$
$\displaystyle\frac{1}{2}\varrho_{0}\,\|u_{,t}\|^{2}+\frac{1}{2}\varrho_{0}\,\varkappa\,\|\zeta_{,t}\|^{2}+\mu_{\rm
e}\,\|{\rm sym}\,{\rm D}u\|^{2}+\frac{\lambda_{\rm e}}{2}\,[{\rm tr}({\rm
D}u)]^{2}+\frac{\xi}{2}\,\zeta^{2}+\frac{\alpha}{2}\,\|{\rm
D}\zeta\|^{2}+\beta\,{\rm tr}({\rm D}u)\,\zeta\bigg{)}\,dv,$ (1.1)

with $\mu_{\rm e},\lambda_{\rm e},\xi,\alpha$, $\beta$ the constant
constitutive coefficients and $\varkappa$ a micro-inertia coefficient.
However, from our modelling process, we are able to observe that the
coefficients $\xi$ and $\beta$ from above are defined by the elastic bulk
modulus $\kappa_{\rm e}=\frac{2\mu_{\rm e}+3\lambda_{\rm e}}{3}$ and the
microscopic bulk modulus $\kappa_{\rm micro}$ through $\beta=-3\kappa_{\rm
e},$ $\xi=9(\kappa_{\rm e}+\kappa_{\rm micro})$. When the microscopic bulk
modulus $\kappa_{\rm micro}\to\infty$, see Section 2), $\xi\to\infty$, while
$\mu_{\rm e}$ and $\lambda_{\rm e}$ tend to the macroscopic classical Lamé
coefficients and we recover classical linear elasticity. Moreover, in this
limit case, from the boundedness of the internal energy density it follows
that $\zeta\to 0$ and the total energy is that from classical linear
elasticity.

When it comes to understanding how waves travel on surfaces, a significant
focus is placed on studying seismic waves. Such waves, crucial for modelling
the waves generated by earthquakes on Earth’s surface, are known as Rayleigh
waves [69], characterized by a mixture of horizontal (longitudinal) and
vertical (transverse) movements. The horizontal displacement components align
with the direction of wave propagation, while the vertical components extend
into the surrounding half-space. The investigation of Rayleigh waves has
garnered considerable interest among scientists due to their practical
applications in fields such as seismology and near-surface geophysical
exploration.

There are several methods available for developing solutions that describe the
propagation of Rayleigh waves in classical and generalized models of elastic
materials. For example, Hayes (referenced in [35, 4]) introduced the
”directional-ellipse method” to systematically study all inhomogeneous plane
waves that can propagate in classical linear elasticity. In classical
elasticity of isotropic materials, the solution for the Rayleigh surface wave
speed, also known as the ”secular equation,” has been explored numerically (as
seen in references like [69, 36]). Hayes and Rivlin also delved into the
secular equation [34]. For a more comprehensive understanding of this problem,
additional details can be found in the book [1] and other works like [64, 65,
68, 53, 49, 19, 81, 82, 83, 84, 85]. Another method, known as the ”Stroh
formalism,” was introduced in [76]. Stroh, in [77], simplified the complex
secular equation originally derived by Synge [78], providing a real expression
for it. Currie’s work [16] also addresses this. In classical elasticity, the
Stroh formalism has successfully solved the problem of finding explicit
secular equations for certain types of material, such as anisotropic and
inhomogeneous materials. Relevant references for this include [79, 57, 18, 19,
20, 80, 46]. However, in the context of generalized models, the Stroh
formalism does not yield an explicit analytical form of the secular equation.

It is important to note that in the process of constructing the solution,
including the displacement and other unknown variables in the generalized
model, all methods impose certain restrictions, which form an admissible set
for the Rayleigh wave solution. These restrictions must be satisfied by the
Rayleigh surface wave speed, which is the solution of the secular equation.
For instance, in the case of classical isotropic materials, the Rayleigh
surface wave speed ($v_{s}$) must adhere to the inequality $0\leq
v<v_{s}=\sqrt{\frac{\lambda_{\rm e}+2\mu_{\rm e}}{\varrho_{0}}}$.

In some generalized continuum models, the secular equation, which provides the
surface wave speed, is still not explicitly derived. Even when the secular
equation is available in explicit analytical form, authors often do not
demonstrate the existence and uniqueness of a solution within the admissible
set. In most cases, at best, this critical aspect is only speculated upon or
numerically explored for a few specific materials [8, 10, 11, 5, 27].

Regarding the model discussed in this paper, the existence of an admissible
solution has been established in [8] under certain hypotheses, i.e.,

${v}_{t}^{2}\leq{v}_{s}^{2}-\displaystyle\frac{{\beta}^{2}}{{\varrho_{0}}\,\xi}\
\ \text{ and }\ \
{v}_{t}^{2}\leq\frac{\alpha}{{\varrho_{0}}\,\varkappa},\quad\text{where}\quad{v}_{t}=\sqrt{\frac{\mu_{\rm
e}}{\varrho_{0}}}.$ (1.2)

However, the uniqueness of such an admissible solution has not been addressed.

It is important to note that the conditions presented in (1.2) have their
limitations. On one hand, when $\alpha$ approaches zero, the constraints in
(1.2) indicate that the limit of the group/phase velocity of the acoustic
branch of the shear-rotational wave as $\omega$ approaches zero (or as $k$
approaches zero), which is denoted as ${v}_{t}$, tends to zero. This
contradicts the principles of classical elasticity. On the other hand, when
$\xi$ tends to infinity (corresponding to the model converging to classical
linear elasticity), the restriction in (1.2)1 implies that the result
established in [8] is valid only when ${c}_{t}^{2}\leq{c}_{s}^{2}$ (assuming
$\mu_{\rm e}>0$ and $\lambda_{\rm e}+\mu_{\rm e}\geq 0$).

To address the shortcomings mentioned earlier, our current study explores an
alternative approach for examining Rayleigh waves in elastic materials
containing voids. This method was initially introduced by Fu and Mielke [25,
54] and was inspired by the prior work of Mielke and Sprenger [55], which has
stronger ties to control theory [41] than surface wave propagation. Notably,
this approach stands out conceptually from other methods and offers a
straightforward numerical algorithm for computing Rayleigh wave solutions.
What distinguishes the Fu and Mielke’s method further is that it provides an
elegant proof of existence and uniqueness while offering practical
computational advantages. It allows us to derive the explicit form of the
secular equation without the need for a priori knowledge of analytical
expressions (as functions of wave speed) for certain eigenvalues and their
associated eigenvectors, which most other methods require. Indeed, calculating
these symbolic expressions for eigenvalues and eigenvectors as functions of
wave speed poses a primary challenge in many generalized models, with the
exception of a few models where this task is relatively straightforward, such
as classical isotropic linear elasticity [1, 36], albeit under certain
restrictive conditions imposed on the constitutive coefficients.

In a related development, it has been observed in [40] that the method
initially introduced in [54, 25] within the context of classical elasticity is
applicable to generalized elastic materials as well. By leveraging properties
of an ”algebraic Riccati equation” [54, 25], we deduce the form of the secular
equation for isotropic elastic materials with micro-voids. Additionally, we
establish that this secular equation possesses a solution within the
admissible set and demonstrate its uniqueness.

Our result is valid for materials characterized by constitutive parameters
satisfying222In terms of the Young’s modulus and the Poisson’s ratio these
restrictions imply $(E<0$ and $\nu<-1$ or $(E>0$ and $(-1<\nu<\frac{1}{2}$ or
$\nu>1)),$ while in terms of the constitutive parameters of the relaxed
micromorphic model the inequality (1.3)4 reads $(\lambda_{\rm e}+2\mu_{\rm
e})(\kappa_{\rm e}+\kappa_{\rm micro})>\kappa_{\rm e}^{2}.$

$\displaystyle\mu_{\rm e}>0,\qquad\lambda_{\rm e}+2\mu_{\rm
e}>0,\qquad\alpha>0,\qquad(\lambda_{\rm e}+2\mu_{\rm e})\,\xi>\beta^{2}.$
(1.3)

Hence, the approach given in this paper

  * •

is the most comprehensive, similar to the findings in [8], and results in a
straightforward numerical algorithm,

  * •

is applicable to the entire spectrum of materials that allow real wave
propagation, encompassing.

    * –

auxetic materials (negative Poisson’s ratio) [43, 42, 7, 75, 75, 23, 39, 6,
33, 86, 24],

    * –

composite materials with negative-stiffness inclusions (negative Young’s
modulus) [44].

##  2 The micro-voids model – a particular case of the relaxed micromorphic
model

In this subsection we provide the initial-boundary value problem of a
particular relaxed micromorphic isotropic model [60, 51, 50, 52, 59] when the
micro-distortion tensor $P$ is assumed to a priori have the special structure
${P}=\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}}\in\mathbb{R}^{3\times 3}$. Such a
micro-distortion tensor is not able to describes the general micro-rotation,
micro-stretching or micro-shearing, but uniform stretching and shrinking of
the microstructure (micro-void). The body occupying the domain
$\overline{\Omega}\subset\mathbb{R}^{3}$ is referred to a fixed system of
rectangular Cartesian axes $Ox_{i}$ $(i=1,2,3)$, $\\{e_{1},e_{2},e_{3}\\}$
being the unit vectors of these axes. We denote by $n$ the outward unit normal
on $\partial\Omega$.

###  2.1 The relaxed micromorphic model

In the relaxed micromorphic model, in which a positive Cosserat modulus
$\mu_{\rm c}>0$ is related to the isotropic Eringen-Claus model for
dislocation dynamics [13, 22, 14], the free energy is given by333Let us recall
some other useful notations for the present work. For a regular enough
function $f(t,x_{1},x_{2},x_{3})$, $f_{,t}$ denotes the derivative with
respect to the time $t$, while $\frac{\partial\,f}{\partial\,x_{i}}$ denotes
the $i$-component of the gradient ${\rm D}f$. For vector fields
$u=\left(u_{1},u_{2},u_{3}\right)^{T}$ with $u_{i}\in{\rm
H}^{1}(\Omega)\,=\,\\{u_{i}\in{\rm L}^{2}(\Omega)\,|\,{\rm D}\,u_{i}\in{\rm
L}^{2}(\Omega)\\},$ $i=1,2,3$, we define $\mathrm{D}u:=\left({\rm
D}\,u_{1}\,|\,{\rm D}\,u_{2}\,|\,{\rm D}\,u_{3}\right)^{T}.$ The corresponding
Sobolev-space will be also denoted by ${\rm H}^{1}(\Omega)$. In addition, for
a tensor field ${P}$ with rows in ${\rm H}({\rm div}\,;\Omega)$, i.e.,
${P}=\footnotesize\footnotesize\begin{pmatrix}{P}^{T}.e_{1}\,|\,{P}^{T}.e_{2}\,|\,{P}^{T}\,e_{3}\end{pmatrix}^{T}$
with $({P}^{T}.e_{i})^{T}\in{\rm H}({\rm div}\,;\Omega):=\,\\{v\in{\rm
L}^{2}(\Omega)\,|\,{\rm div}\,v\in{\rm L}^{2}(\Omega)\\}$, $i=1,2,3$, we
define ${\rm Div}\,{P}:=\footnotesize\footnotesize\begin{pmatrix}{\rm
div}\,({P}^{T}.e_{1})^{T}\,|\,{\rm div}\,({P}^{T}.e_{2})^{T}\,|\,{\rm
div}\,({P}^{T}\,e_{3})^{T}\end{pmatrix}^{T}$ while for tensor fields ${P}$
with rows in ${\rm H}({\rm curl}\,;\Omega)$, i.e.,
${P}=\footnotesize\footnotesize\begin{pmatrix}{P}^{T}.e_{1}\,|\,{P}^{T}.e_{2}\,|\,{P}^{T}\,e_{3}\end{pmatrix}^{T}$
with $({P}^{T}.e_{i})^{T}\in{\rm H}({\rm curl}\,;\Omega):=\,\\{v\in{\rm
L}^{2}(\Omega)\,|\,{\rm curl}\,v\in{\rm L}^{2}(\Omega)\\}$, $i=1,2,3$, we
define ${\rm Curl}\,{P}:=\footnotesize\footnotesize\begin{pmatrix}{\rm
curl}\,({P}^{T}.e_{1})^{T}\,|\,{\rm curl}\,({P}^{T}.e_{2})^{T}\,|\,{\rm
curl}\,({P}^{T}\,e_{3})^{T}\end{pmatrix}^{T}.$

$\displaystyle W_{\rm relax}$ $\displaystyle=\mu_{\rm e}\|{\rm
sym}(\mathrm{D}u-{P})\|^{2}+\mu_{\rm c}\|{\rm
skew}(\mathrm{D}u-{P})\|^{2}+\frac{\lambda_{\rm e}}{2}\,[{\rm
tr}(\mathrm{D}u-{P})]^{2}+\mu_{\rm micro}\|{\rm
sym}\,{P}\|^{2}+\frac{\lambda_{\rm micro}}{2}[{\rm tr}({P})]^{2}$
$\displaystyle\quad\quad+\frac{\mu_{\rm e}L_{\rm c}^{2}}{2}\left[{a_{1}}\|{\rm
dev}\,{\rm sym}\,{\rm Curl}\,{P}\|^{2}+{a_{2}}\|{\rm skew}\,{\rm
Curl}\,{P}\|+\frac{a_{3}}{3}\,{\rm tr}({\rm Curl}\,{P})^{2}\right],$ (2.1)

where $(\mu_{\rm e},\lambda_{\rm e})$, $(\mu_{\rm micro},\lambda_{\rm
micro})$, $\mu_{\rm c},L_{\rm c}$ and $(a_{1},a_{2},a_{3})$ are the elastic
moduli representing the parameters related to the meso-scale, the parameters
related to the micro-scale, the Cosserat couple modulus, the characteristic
length, and the three general isotropic curvature parameters (weights),
respectively. Formally, letting $L_{\rm c}\to\infty$ means a “zoom” into the
micro-structure while $L_{\rm c}\to 0$ means considering arbitrary large
bodies while retaining the size of the unit-cell or keeping the dimensions of
the body fixed while reducing the dimensions of the unit cell to zero, or, in
other words, “no special effects of the microstructure taking into account”
(classical elasticity).

The complete system of linear partial differential equations in terms of the
kinematical unknowns $u$ and $P$ is given by

$\displaystyle\varrho_{0}\,u_{,tt}$ $\displaystyle={\rm
Div}[\underbrace{2\,\mu_{\rm e}\,{\rm sym}(\mathrm{D}u-{P})+2\,\mu_{\rm
c}\,{\rm skew}(\mathrm{D}u-{P})+\lambda_{\rm e}\,{\rm
tr}(\mathrm{D}u-{P}){\cdot}{{\mathbbm{1}}}}_{\textrm{$=:\\!\sigma$ \ the non-
symmetric force-stress tensor}}]+f\,,$ (2.2)
$\displaystyle\varrho_{0}\,\eta\,\tau_{\rm c}^{2}\,\,P_{,tt}$
$\displaystyle=-{\mu_{\rm e}L_{\rm c}^{2}}\,{\rm Curl}[\underbrace{a_{1}\,{\rm
dev}\,{\rm sym}\,{\rm Curl}\,{P}+a_{2}\,{\rm skew}\,{\rm
Curl}\,{P}+\frac{a_{3}}{3}\,{\rm tr}({\rm
Curl}\,{P}){\cdot}{{\mathbbm{1}}}}_{=:\,m\,\textrm{ the second-order moment
stress tensor }}]$ $\displaystyle\quad\ +2\,\mu_{\rm e}\,{\rm
sym}(\mathrm{D}u-{P})+\lambda_{\rm e}{\rm
tr}(\mathrm{D}u-{P}){\cdot}{{\mathbbm{1}}}-\underbrace{(2\,\mu_{\rm
micro}\,{\rm sym}\,{P}+\lambda_{\rm micro}{\rm
tr}({P}){\cdot}{{\mathbbm{1}}})}_{=:\,\sigma_{\rm micro}\ \text{the symmetric
microstress tensor}}+{M}\,\ \ \ \text{in}\ \ \ \Omega\times[0,T],$

where $f:\Omega\times[0,T]\rightarrow\mathbb{R}^{3}$ describes the external
body force, ${M}:\Omega\times[0,T]\rightarrow\mathbb{R}^{3\times 3}$ describes
the external body moment, $\varrho_{0}$ is the mass density and
$\eta\,\tau_{\rm c}^{2}$ is the inertia coefficient, with $\eta>0$ a weight
parameter and $\tau_{\rm c}$ the internal characteristic time [21, page 163].

To the system of partial differential equations of this model we adjoin the
natural boundary conditions

$\displaystyle{u}({x},t)=\widetilde{u}({x},t),\ \ \
\quad\quad{P}_{i}({x},t)\times n(x)=\widetilde{p}({x},t),\ \ \ i=1,2,3,\ \ \ \
({x},t)\in\partial\Omega\times[0,T],$ (2.3)

and the nonzero initial conditions

$\displaystyle{u}({x},0)={u}_{0}(x),\quad\quad\quad\dot{u}({x},0)=\dot{u}_{0}(x),\quad\quad\quad{P}({x},0)={P}_{0}(x),\quad\quad\quad\dot{P}({x},0)=\dot{P}_{0}(x),\
\ \text{\ \ }{x}\in\bar{\Omega}\,,$ (2.4)

where $\widetilde{u}$, $\widetilde{p}$, ${u}_{0},\dot{u}_{0},{P}_{0}$ and
$\dot{P}_{0}$ are prescribed functions, satisfying $u_{0}(x)=\widetilde{u}$
and $P_{0i}(x)\times n(x)=\widetilde{p}$ on $\partial\Omega$.

The internal energy is positive definite in terms of the independent
constitutive variables $\mathrm{D}u-{P}$, ${\rm sym}\,{P}$, ${\rm Curl}\,{P}$
if and only if

$\displaystyle\mu_{\rm e}$ $\displaystyle>0,\qquad\qquad\quad\ \,\kappa_{\rm
e}:=\frac{2\,\mu_{\rm e}+3\,\lambda_{\rm e}}{3}>0,\qquad\qquad\qquad\ \
\mu_{\rm c}>0,$ (2.5) $\displaystyle\mu_{\rm micro}$
$\displaystyle>0,\qquad\qquad\kappa_{\rm micro}:=\frac{2\,\mu_{\rm
micro}+3\,\lambda_{\rm micro}}{3}>0,\qquad\qquad a_{1}>0,\qquad\qquad
a_{2}>0,\qquad\qquad a_{3}>0.$

However, existence and uniqueness of the solution is guaranteed for the weaker
conditions

$\displaystyle\mu_{\rm e}$ $\displaystyle>0,\qquad\qquad\quad\ \,\kappa_{\rm
e}:=\frac{2\,\mu_{\rm e}+3\,\lambda_{\rm e}}{3}>0,\qquad\qquad\qquad\ \
\mu_{\rm c}\boldsymbol{\geq}0,$ (2.6) $\displaystyle\mu_{\rm micro}$
$\displaystyle>0,\qquad\qquad\kappa_{\rm micro}:=\frac{2\,\mu_{\rm
micro}+3\,\lambda_{\rm micro}}{3}>0,\qquad\qquad a_{1}>0,\qquad\qquad
a_{2}\boldsymbol{\geq}0,\qquad\qquad a_{3}\boldsymbol{\geq}0,$

due to the new inequalities given by Lewintan, Müller and Neff [47, 48] and by
Gmeineder, Lewintan, and Neff [31, 30].

We say that there exists real plane waves444The plane wave is called “real”
since it is defined by real values of $\omega$. in a given direction
$\xi=(\xi_{1},\xi_{2},\xi_{3})$, $\lVert{\xi}\rVert^{2}=1$, if for every wave
number $k$ the system of partial differential equations (2.2) admits a plane
wave ansatz solution only for real frequencies $\omega\in\mathbb{R}$. It is
proven in [58] that the dynamic relaxed micromorphic model (2.2) admits real
planar waves if and only if

$\displaystyle\mu_{\rm e}>0,\qquad\mu_{\rm micro}>0,\qquad\mu_{\rm c}\geq
0,\qquad\kappa_{\rm e}+\kappa_{\rm micro}>0,\qquad 2\,\mu_{\rm
macro}+\lambda_{\rm macro}>0,$ (2.7)

where

$\displaystyle\kappa_{\rm macro}:=\frac{\kappa_{\rm e}\,\kappa_{\rm
micro}}{\mu_{\rm e}+\kappa_{\rm micro}},\qquad\mu_{\rm macro}:=\frac{\mu_{\rm
e}\,\mu_{\rm micro}}{\mu_{\rm e}+\mu_{\rm micro}},\qquad\lambda_{\rm
macro}:=\frac{3\,\kappa_{\rm macro}-2\,\mu_{\rm macro}}{3}.$ (2.8)

As remarked in [58] if $\mu_{\rm e}+\mu_{\rm micro}>0$ and $\kappa_{\rm
e}+\kappa_{\rm micro}>0$, the macroscopic parameters are less or equal than
respective microscopic parameters, namely

$\displaystyle\kappa_{\rm micro}$ $\displaystyle\geq\kappa_{\rm
macro},\qquad\qquad\mu_{\rm micro}\geq\mu_{\rm macro},$
$\displaystyle\kappa_{\rm e}$ $\displaystyle\geq\kappa_{\rm
macro},\qquad\qquad\quad\ \ \mu_{\rm e}\geq\mu_{\rm macro},$ (2.9)

while letting $\mu_{\rm micro}\rightarrow+\infty$ and $\kappa_{\rm
micro}\rightarrow+\infty$ (or $\mu_{\rm micro}\rightarrow+\infty$ and
$\lambda_{\rm micro}>\mathrm{const}$) generates the limit condition for real
wave velocities ($\mu_{\rm e}\rightarrow\mu_{\rm macro}$)

$\displaystyle\mu_{\rm macro}>0,\qquad\mu_{\rm c}\geq 0,\qquad 2\,\mu_{\rm
macro}+\lambda_{\rm macro}>0$ (2.10)

which coincides, up to the Cosserat couple modulus $\mu_{\rm c}$, with the
strong ellipticity condition in isotropic linear elasticity, and it coincides
fully with the condition for real wave velocities in Cosserat elasticity [40].

###  2.2 The micro-voids model

In this subsection we consider a particular relaxed micromorphic model and we
show that the obtained system of partial differential equation coincide (after
an identification of parameters) with that from the micro-void model.

To this aim, let us consider the particular case of (2.2) in which we assume a
priori the kinematical constraint $P=\zeta{\cdot}{{\mathbbm{1}}}$. The
boundary condition which follows from the tangential boundary condition (2.3)
is then the strong anchoring condition

$\displaystyle\zeta(x,t)=\widetilde{\zeta}({x},t)\quad\quad\text{on}\quad\partial\Omega\times[0,T].$
(2.11)

Note that for all differentiable functions
$\zeta:\mathbb{R}\rightarrow\mathbb{R}$ on $\Omega$ we have555We use the
canonical identification of $\mathbb{R}^{3}$ with $\mathfrak{so}(3)$, and, for
${A}=\footnotesize\begin{pmatrix}0&-a_{3}&a_{2}\\\ a_{3}&0&-a_{1}\\\
-a_{2}&a_{1}&0\end{pmatrix}\in\mathfrak{so}(3)$ we consider the operators
${\textrm{Anti}}:\mathbb{R}^{3}\rightarrow\mathfrak{so}(3)$ through
$({\textrm{Anti}}(v))_{ij}=-\epsilon_{ijk}\,v_{k},\
\forall\,v\in\mathbb{R}^{3}$, where $\epsilon_{ijk}$ is the totally
antisymmetric third order permutation tensor. ${\rm
Curl}(\zeta{\cdot}{{\mathbbm{1}}})=-{\textrm{Anti}}\,({\rm
D}\zeta)\in\mathfrak{so}(3)$ and we obtain that the total energy can be
written as

$\displaystyle\mathcal{L}(u_{,t},\zeta_{,t},\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}},\mathrm{D}\,\zeta)=\int_{\Omega}\bigg{(}$
$\displaystyle\frac{1}{2}\,\varrho_{0}\,\|u_{,t}\|^{2}+\,\frac{1}{2}\,\varrho_{0}\,\eta\,\tau_{\rm
c}^{2}\,\,\,\|(\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})_{,t}\|^{2}$ (2.12)
$\displaystyle+\mu_{\rm e}\|{\rm
sym}(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})\|^{2}+\mu_{\rm
c}\|{\rm skew}\,\mathrm{D}u\|^{2}+\frac{\lambda_{\rm e}}{2}\,[{\rm
tr}(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})]^{2}$
$\displaystyle+\mu_{\rm micro}\|{\rm
sym}(\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})\|^{2}+\frac{\lambda_{\rm
micro}}{2}[{\rm tr}(\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})]^{2}+\frac{\mu_{\rm
e}L_{\rm c}^{2}}{2}{a_{2}}\|{\rm skew}\,{\rm
Curl}(\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})\|^{2}\bigg{)}\,dv.$

Since $\|{\rm skew}\,\mathrm{D}u\|^{2}$ alone is not infinitesimally frame-
invariant, $\mu_{\rm c}$ has to vanish, and the total energy is given by

$\displaystyle\mathcal{L}(u_{,t},\zeta_{,t},$
$\displaystyle\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}},\mathrm{D}\,\zeta)=\int_{\Omega}\bigg{(}\frac{1}{2}\,\varrho_{0}\,\|u_{,t}\|^{2}+\,\frac{3}{2}\,\varrho_{0}\,\eta\,\tau_{\rm
c}^{2}\,\zeta_{,t}^{2}+\mu_{\rm e}\|{\rm
sym}(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})\|^{2}+\frac{\lambda_{\rm
e}}{2}\,[{\rm tr}(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})]^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{3}{2}(2\,\mu_{\rm
micro}+3\,\lambda_{\rm micro})\,\zeta^{2}+\mu_{\rm e}L_{\rm
c}^{2}{a_{2}}\|\mathrm{D}\,\zeta\|^{2}\bigg{)}\,dv\,$
$\displaystyle=\int_{\Omega}\bigg{(}$
$\displaystyle\frac{1}{2}\,\varrho_{0}\,\|u_{,t}\|^{2}+\,\frac{3}{2}\,\varrho_{0}\,\eta\,\tau_{\rm
c}^{2}\,\,\,\zeta_{,t}^{2}+\mu_{\rm e}\|{\rm
sym}\,\mathrm{D}u\|^{2}-2\,\mu_{\rm e}{\rm tr}(\mathrm{D}u)\,\zeta+3\,\mu_{\rm
e}\zeta^{2}$ $\displaystyle+\frac{\lambda_{\rm e}}{2}\,[{\rm
tr}(\mathrm{D}u)]^{2}-2\,\frac{\lambda_{\rm e}}{2}\,{\rm
tr}(\mathrm{D}u)\,3\,\zeta+\frac{\lambda_{\rm
e}}{2}\,9\,\zeta^{2}+\frac{3}{2}(2\,\mu_{\rm micro}+3\,\lambda_{\rm
micro})\zeta^{2}+\mu_{\rm e}L_{\rm
c}^{2}{a_{2}}\|\mathrm{D}\,\zeta\|^{2}\bigg{)}\,dv\,$ (2.13)
$\displaystyle=\int_{\Omega}\bigg{(}$
$\displaystyle\frac{1}{2}\,\varrho_{0}\,\|u_{,t}\|^{2}+\,\frac{3}{2}\,\varrho_{0}\,\eta\,\tau_{\rm
c}^{2}\,\,\,\zeta_{,t}^{2}+\mu_{\rm e}\|{\rm dev}\,{\rm
sym}\,\mathrm{D}u\|^{2}+\frac{3\lambda_{\rm e}+2\mu_{\rm e}}{6}\,[{\rm
tr}(\mathrm{D}u)]^{2}$ $\displaystyle-(2\,\mu_{\rm e}+3\,\lambda_{\rm
e})\,{\rm tr}(\mathrm{D}u)\,\zeta+\frac{3}{2}(2\,\mu_{\rm e}+3\,\lambda_{\rm
e}+2\,\mu_{\rm micro}+3\,\lambda_{\rm micro})\zeta^{2}+\mu_{\rm e}L_{\rm
c}^{2}{a_{2}}\|\mathrm{D}\,\zeta\|^{2}\bigg{)}\,dv\,$

and the power functional is $\Pi(t)=\int_{\Omega}(\displaystyle\langle
f,{u}_{,t}\rangle+\langle
M,\zeta_{,t}\cdot{{\mathbbm{1}}}\rangle)\,dv=\int_{\Omega}(\displaystyle\langle
f,{u}_{,t}\rangle+\underbrace{{\rm
tr}(M)}_{:=\ell}\,\zeta_{,t}\rangle)\,dv\,.$

A direct identification of the coefficients, by comparing the energies in the
dislocation form (written with ${\rm Curl}$) with the specific internal energy
and the kinetic energy in the Cowin-Nunziato form for the micro-voids model
(1.1), shows that the coefficient of the micro-voids theory can be expressed
in terms of our constitutive coefficients as

$\displaystyle\varkappa$
$\displaystyle=3\,\eta\,\tau_{c}^{2},\quad\quad\quad\quad\alpha=2\,\mu_{\rm
e}L_{c}^{2}a_{2},\quad\quad\quad\quad\beta=-(2\,\mu_{\rm e}+3\lambda_{\rm
e})=-3\kappa_{\rm e},$ (2.14) $\displaystyle\xi$ $\displaystyle=3(2\mu_{\rm
e}+3\lambda_{\rm e}+2\mu_{\rm micro}+3\lambda_{\rm micro})=9(\kappa_{\rm
e}+\kappa_{\rm micro}).$

Let us notice some differences between the micro-voids model and the micro-
voids theory proposed by Cowin and Nunziato [15]:

  * •

in our version of the micro-voids model the constitutive parameter $\beta$ is
not an independent coefficient, as it is in the micro-voids model. It depends
on the elastic Lamé coefficients $\lambda_{\rm e}$ and $\mu_{\rm e}$.

  * •

for elastic materials having a positive definite internal energy, the
constitutive parameter $\beta$ is always negative. In some numerical
simulation from the literature $\beta$ is taken positive in the context of the
micro-voids model [67] since the physical relevance of this parameter was not
properly understood.

  * •

in our version of the micro-voids model we show how the constitutive parameter
$\xi$ is depending on the micro-, and macro-constitutive parameters. This
gives them a physical interpretation.

  * •

in our version of the micro-voids model there are no ad hoc defined
quantities, e.g., $h$ and $g$ in the micro-voids model. All the quantities are
defined by the force stress tensor $\sigma$, the moment stress tensor $m$ and
by the microstress tensor $\sigma_{\rm micro}$.

######  Remark 2.1.

For the micro-voids model to be operative, we need $a_{2}>0$, i.e., the
presence of $\|{\rm skew}\,{\rm Curl}P\|^{2}$ in the expression of the
internal energy density.

We introduce the action functional of the considered system by

$\displaystyle\mathcal{A}=\int_{0}^{T}\int_{\Omega}\bigg{(}$
$\displaystyle\frac{1}{2}\,\varrho_{0}\,\|u_{,t}\|^{2}+\,\frac{3}{2}\,\varrho_{0}\,\eta\,\tau_{\rm
c}^{2}\,\zeta_{,t}^{2}-\mu_{\rm e}\|{\rm
sym}(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})\|^{2}-\frac{\lambda_{\rm
e}}{2}\,[{\rm tr}(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})]^{2}$
$\displaystyle-\frac{3}{2}(2\,\mu_{\rm micro}+3\,\lambda_{\rm
micro})\,\zeta^{2}-\mu_{\rm e}L_{\rm
c}^{2}{a_{2}}\|\mathrm{D}\zeta\|^{2}\bigg{)}\,dv+\int_{0}^{T}\int_{\Omega}(\displaystyle\langle
f,{u}_{,t}\rangle+{\rm tr}(M)\,\zeta_{,t}\rangle)\,dv\,dt.$ (2.15)

The corresponding Euler-Lagrange equation for $u$ and $\zeta$ are

$\displaystyle\varrho_{0}\,u_{,tt}$ $\displaystyle={\rm
Div}[\underbrace{2\,\mu_{\rm e}\,{\rm
sym}\,(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}})+\lambda_{\rm
e}\,{\rm
tr}(\mathrm{D}u-\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}}){\cdot}{{\mathbbm{1}}}}_{=:\sigma\
\text{the symmetric force-stress tensor}}+f\,,$ (2.16)
$\displaystyle\varrho_{0}\,3\,\eta\,\tau_{\rm c}^{2}\,\zeta_{,tt}$
$\displaystyle={\rm
div}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{[2\,\mu_{\rm
e}L_{\rm c}^{2}\,{a_{2}}\,{\rm D}\zeta]}_{=:h\ \text{the equilibrated micro-
stress
vector}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{-3(2\mu_{\rm
e}+3\lambda_{\rm e}+2\mu_{\rm micro}+3\lambda_{\rm micro})\zeta-(2\,\mu_{\rm
e}+3\lambda_{\rm e}){\rm div}\,u}_{=:g\ \text{the equilibrated micro-
force}}+{\rm tr}\,{M}\,,$

in $\Omega\times[0,T]$, which, in view of our parameter identification (2.14)
and setting $\ell:={\rm tr}(M)$, is in complete agreement with the equations
proposed in the Cowin-Nunziato theory [66], i.e.,

$\displaystyle\mu_{\rm e}\sum_{l=1}^{3}\frac{\partial^{2}u_{r}}{\partial
x_{l}^{2}}+(\lambda_{\rm e}+\mu_{\rm
e})\displaystyle\sum_{l=1}^{3}\frac{\partial^{2}u_{l}}{\partial x_{l}\partial
x_{r}}+\beta\,\frac{\partial\zeta}{\partial x_{r}}+f$
$\displaystyle=\varrho_{0}\,\frac{\partial^{2}\,u_{r}}{\partial\,t^{2}},\qquad
r=1,2,3,\vspace{1.2mm}$
$\displaystyle\alpha\sum_{l=1}^{3}\frac{\partial^{2}\zeta}{\partial
x_{l}^{2}}-\xi\zeta-\,\beta\,\sum_{l=1}^{3}\frac{\partial u_{l}}{\partial
x_{l}}+\ell$
$\displaystyle=\varrho_{0}\,\varkappa\,\frac{\partial^{2}\,\zeta}{\partial\,t^{2}}.$
(2.17)

Hence, in the case of isotropic materials we have

$\displaystyle\sigma$ $\displaystyle=2\,\mu_{\rm
e}\,\varepsilon+\beta\,\zeta\,{{\mathbbm{1}}}+\lambda_{\rm e}\,{\rm
tr}(\varepsilon)\,{{\mathbbm{1}}}=2\,\mu_{\rm e}\,{\rm sym}\,{\rm
D}u+\lambda_{\rm e}\,({\rm
div}\,u)\,{{\mathbbm{1}}}+\beta\,\zeta\,{{\mathbbm{1}}},$ (2.18)

while the natural Neumann-boundary conditions are

$\displaystyle\sigma.n=\widetilde{\sigma}\qquad\text{and}\qquad
h.n=\widetilde{h},$ (2.19)

where $h=\alpha\,{\rm D}\zeta$ is the only reminiscent part of the moment
stress tensor, describing the internal forces inside the microstructure (the
equilibrated stress vector from the Cowin-Nunziato theory), and
$\widetilde{\sigma}$ and $\widetilde{h}$ are given functions.

Note that the positivity conditions for the Cowin-Nunziato theory with micro-
voids are

$\displaystyle 3\lambda_{\rm e}+2\mu_{\rm e}>0,\qquad\mu_{\rm
e}>0,\qquad\xi>0,\qquad\alpha>0,\qquad({3\lambda_{\rm e}+2\mu_{\rm
e}})\,\xi>3\,\beta^{2}.$ (2.20)

while in our version of the micro-voids model in dislocation format the
positivity conditions are obvious

$\displaystyle\mu_{\rm e}>0,\quad\quad\quad 2\mu_{\rm e}+3\lambda_{\rm
e}>0,\quad\quad\quad 2\mu_{\rm micro}+3\lambda_{\rm micro}>0,\quad\quad\quad
a_{2}>0\,.$ (2.21)

In any case, in view of the identifications the systems (2.14), (2.20) and
(2.21) are equivalent. Let us remark that

$\displaystyle 3\lambda_{\rm e}+2\mu_{\rm e}>0,\qquad\mu_{\rm e}>0,$ (2.22)

are equivalent to

$\displaystyle E>0,\qquad-1<\nu<\frac{1}{2},$ (2.23)

where $E=\frac{\mu_{\rm e}\,(3\lambda_{\rm e}+2\mu_{\rm e})}{\lambda_{\rm
e}+\mu_{\rm e}}$ and $\nu=\frac{\lambda_{\rm e}}{2(\lambda_{\rm e}+\mu_{\rm
e})}$, represent the Young’s modulus and the Poisson’s ratio, respectively.

In the following we assume $\varrho_{0}>0$ and $\varkappa>0$ without
mentioning these conditions in the hypothesis of our results.

Note that in the relaxed micromophic model for
$P=\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}}$ we have only four parameters,
because $2\mu_{\rm micro}+3\lambda_{\rm micro}$ can be regarded as a single
parameter, instead of five considered by Cowin and Nunziato [15]. Letting
$\mu_{\rm micro}\rightarrow+\infty$ and $\kappa_{\rm micro}\rightarrow+\infty$
(or $\mu_{\rm micro}\rightarrow+\infty$ and $\lambda_{\rm
micro}>\mathrm{const}$) we have that $\mu_{\rm e}\rightarrow\mu_{\rm macro}$,
$\kappa_{\rm e}\rightarrow\kappa_{\rm macro}$, $\lambda_{\rm
e}\rightarrow\lambda_{\rm macro}$ and that the internal energy density remains
bounded only for vanishing $\zeta$. In this limit case the total energy is

$\displaystyle\mathcal{L}_{\rm macro}=\int_{\Omega}\bigg{(}$
$\displaystyle\frac{1}{2}\varrho_{0}\|u_{,t}\|^{2}+\mu_{\rm macro}\,\|{\rm
sym}\,{\rm D}u\|^{2}+\frac{\lambda_{\rm macro}}{2}\,{\rm tr}({\rm
D}u)^{2}\bigg{)}\,dv,$ (2.24)

i.e., we arrive in a transparent way from the relaxed micromorphic model to
the framework of classical linear elasticity, via the model of micro-voids.

Regarding the existence of real plane waves, we will not use yet the general
conditions for real planar waves (if and only if in (2.7)), since due to the
simplification of the system of partial differential equations some other
cases may occur in the micro-voids model.

##  3 Real plane waves in the model of isotropic materials with voids

We say that there exists real plane waves in the direction
$m=(m_{1},m_{2},m_{3})$, $\lVert m\rVert^{2}=1$, if for every wave number
$k>0$ the system of partial differential equations (2.17) admits a solution in
the form

$\displaystyle u(x_{1},x_{2},x_{3},t)$
$\displaystyle=\underbrace{\footnotesize\begin{pmatrix}\widehat{u}_{1}\\\
\widehat{u}_{2}\\\ \widehat{u}_{3}\end{pmatrix}}_{=:\,\widehat{u}}\,e^{{\rm
i}\,\left(k\langle m,\,x\rangle_{\mathbb{R}^{3}}-\,\omega\,t\right)}\,,$ (3.1)
$\displaystyle\zeta(x_{1},x_{2},x_{3},t)$ $\displaystyle={\rm
i}\,\widehat{\zeta}e^{{\rm i}\,\left(k\langle
m,\,x\rangle_{\mathbb{R}^{3}}-\,\omega\,t\right)},\quad\widehat{u}\in\mathbb{C}^{3},\quad\widehat{\zeta}\in\mathbb{C},\quad(\widehat{u},\widehat{\zeta})^{T}\neq
0\,,$

only for real frequencies $\omega\in\mathbb{R}$, where ${\rm i}\,=\sqrt{-1}$
is the complex unit. The plane wave is called “real” since it is defined by
real values of $\omega$. The speed of propagation is given by
$v=\frac{\omega}{k}$. For the definition of $\zeta$ we take ${\rm
i}\,\widehat{\zeta}$ since this choice will lead us in the end to deal only
with real valued matrices.

From (2.17) and (3.1) it follows that
$\widehat{u}_{1},\widehat{u}_{2},\widehat{u}_{3}$ and $\widehat{\zeta}$ have
to be the solutions of the following linear algebraic system

$\displaystyle-\omega^{2}\varrho_{0}\,\widehat{u}_{r}$
$\displaystyle=-k^{2}\mu_{\rm
e}\,\displaystyle{\widehat{u}_{r}}-k^{2}\,(\mu_{\rm e}+\lambda_{\rm
e})\displaystyle\sum_{l=1}^{3}\widehat{u}_{l}\,m_{l}\,m_{r}-\beta\,k\,\widehat{\zeta}\,m_{r},\vspace{1.2mm}\qquad
r=1,2,3,$ (3.2) $\displaystyle-{\rm
i}\,\omega^{2}\varrho_{0}\,\varkappa\,\widehat{\zeta}$ $\displaystyle=-{\rm
i}\,\alpha\,k^{2}\,\widehat{\zeta}-{\rm i}\,\xi\,\widehat{\zeta}-{\rm
i}\,\beta\,k\,\sum_{l=1}^{3}\widehat{u}_{l}\,m_{l}.$

Since our formulation is for isotropic materials, by demanding real plane
waves in any direction $m=(m_{1},m_{2},m_{3})$, $\lVert m\rVert=1$, it is
equivalent to demand real plane waves with $m_{1}=1$ and $m_{2}=m_{3}=0$. In
this setting the system (3.2) reads

$\displaystyle-\omega^{2}\varrho_{0}\,\widehat{u}_{1}$
$\displaystyle=-k^{2}\mu_{\rm
e}\,\displaystyle{\widehat{u}_{1}}-k^{2}\,(\mu_{\rm e}+\lambda_{\rm
e})\displaystyle\widehat{u}_{1}-\beta\,k\,\widehat{\zeta},\vspace{1.2mm}$
$\displaystyle-\omega^{2}\varrho_{0}\,\widehat{u}_{2}$
$\displaystyle=-k^{2}\mu_{\rm
e}\,\displaystyle{\widehat{u}_{2}},\vspace{1.2mm}$ (3.3)
$\displaystyle-\omega^{2}\varrho_{0}\,\widehat{u}_{3}$
$\displaystyle=-k^{2}\mu_{\rm
e}\,\displaystyle{\widehat{u}_{3}},\vspace{1.2mm}$
$\displaystyle-\omega^{2}\varrho_{0}\,\varkappa\,\widehat{\zeta}$
$\displaystyle=-\alpha\,k^{2}\,\widehat{\zeta}-\xi\,\widehat{\zeta}-\beta\,k\,\widehat{u}_{1}.$

We split the above system in the following two systems

$\displaystyle\left[{Q}_{1}(k)-\omega^{2}\widehat{{{\mathbbm{1}}}}_{2}\,\right]\footnotesize\begin{pmatrix}\widehat{u}_{1}\\\
\widehat{\zeta}\end{pmatrix}=0,$ (3.4)

and

$\displaystyle\left(\frac{\mu_{\rm
e}\,k^{2}}{\varrho_{0}}-\omega^{2}\,\right)\widehat{u}_{2}=0,\qquad\left(\frac{\mu_{\rm
e}\,k^{2}}{\varrho_{0}}-\omega^{2}\,\right)\widehat{u}_{3}=0,\vspace{1.2mm}$
(3.5)

where

$\displaystyle{{Q}_{1}}(k)$
$\displaystyle=\footnotesize\begin{pmatrix}(\lambda_{\rm e}+2\,\mu_{\rm
e})k^{2}&\beta\,k\vspace{2mm}\\\
\beta\,k&\alpha\,k^{2}+\xi\end{pmatrix},\qquad\widehat{{{\mathbbm{1}}}}_{2}=\footnotesize\begin{pmatrix}\varrho_{0}&0\\\
0&\varrho_{0}\,\varkappa\,\end{pmatrix}.$ (3.6)

In order to have an eigenvalue problem, we multiply (3.4) from the left with
$\widehat{{{\mathbbm{1}}}}_{2}^{-1/2}$ and make a substitution to obtain

$\displaystyle\left[\widehat{{{\mathbbm{1}}}}_{2}^{-1/2}{{Q}_{1}}(k)\widehat{{{\mathbbm{1}}}}_{2}^{-1/2}-\omega^{2}{{\mathbbm{1}}}\right]\,{d}=0,\qquad\qquad{d}=\widehat{{{\mathbbm{1}}}}_{2}^{1/2}\footnotesize\begin{pmatrix}\widehat{u}_{1}\\\
\widehat{\zeta}\end{pmatrix}.$ (3.7)

Hence, the system (3.4) is equivalent to the eigenvalue problem

$\displaystyle\left[\widetilde{{Q}}(k)-\omega^{2}\,{{\mathbbm{1}}}\,\right]{d}=0,\qquad\qquad\text{where}\qquad\qquad\widetilde{{Q}}(k)=\widehat{{{\mathbbm{1}}}}_{2}^{-1/2}{{Q}_{1}}(k)\widehat{{{\mathbbm{1}}}}_{2}^{-1/2}\in{\rm
Sym}(2).$ (3.8)

The system (3.5) admits a non-zero solution if and only if
$\omega^{2}=k^{2}\,\frac{\mu_{\rm e}}{\varrho_{0}}$, while the system (3.4)
(i.e., (3.8)) has a non-zero solution if and only if
$\det{\left[\widetilde{{Q}}(k)-\omega^{2}\,{{\mathbbm{1}}}\,\right]}=0$.
Having real waves implies that

$\displaystyle\mu_{\rm e}>0$ (3.9)

and that $\omega^{2}$ are positive real eigenvalues of $\widetilde{Q}$, i.e.,

$\displaystyle\widetilde{{{Q}}}(k)$
$\displaystyle=\footnotesize\begin{pmatrix}\frac{(\lambda_{\rm e}+2\,\mu_{\rm
e})k^{2}}{\varrho_{0}}&\frac{\beta\,k}{\varrho_{0}\,\sqrt{\varkappa}}\vspace{2mm}\\\
\frac{\beta\,k}{\varrho_{0}\,\sqrt{\varkappa}}&\frac{\alpha\,k^{2}+\xi}{\varrho_{0}\,\varkappa}\end{pmatrix}$
(3.10)

must be positive definite. But $\widetilde{{Q}}(k)$ is positive definite if
and only if

$\displaystyle\lambda_{\rm e}+2\mu_{\rm
e}>0,\qquad\alpha>0,\qquad(\lambda_{\rm e}+2\mu_{\rm e})\,\xi>\beta^{2}.$
(3.11)

In conclusion, we have obtained:

######  Proposition 3.1.

There exists real plane waves in the linear model of isotropic elastic
materials with micro-voids if and only if

$\displaystyle\mu_{\rm e}>0,\qquad\lambda_{\rm e}+2\mu_{\rm
e}>0,\qquad\alpha>0,\qquad(\lambda_{\rm e}+2\mu_{\rm e})\xi>\beta^{2}.$ (3.12)

In terms of the constitutive parameters of the relaxed micromorphic model the
conditions (3.12) read

$\displaystyle\mu_{\rm e}>0,\qquad\lambda_{\rm e}+2\mu_{\rm e}>0,\qquad
a_{2}>0,\qquad(\lambda_{\rm e}+2\mu_{\rm e})(\kappa_{\rm e}+\kappa_{\rm
micro})>\kappa_{\rm e}^{2}.$ (3.13)

As we observe, comparing to the analogue conditions for the parental relaxed
micromorphic model, it follows that

$\displaystyle\kappa_{\rm e}+\kappa_{\rm micro}>0.$ (3.14)

In the micro-voids framework the real plane waves existence do not imply that
$\mu_{\rm micro}>$ or $\mu_{\rm e}+\mu_{\rm micro}>0$.

The conditions (3.12) are equivalent with the _strong ellipticity_ [9] of the
elastic material with micro-voids, i.e., with the condition

${W}\left(m\otimes\eta,\theta,{\vartheta}\right)>0,\text{ \ \ for all }\ \ \
m,\eta,\vartheta\in\mathbb{R}^{3}\setminus\\{{0}\\},\theta\in\mathbb{R},$
(3.15)

where

$\displaystyle{W}\left(X,c,b\right)=\mu_{\rm e}\,\|{\rm
sym}X\|^{2}+\frac{\lambda_{\rm e}}{2}\,{\rm
tr}(X)^{2}+\frac{\xi}{2}\,c^{2}+\frac{\alpha}{2}\,\|b\|^{2}+b\,{\rm
tr}(X)\,c.$ (3.16)

Moreover, for an isotropic elastic material with micro-voids, the progressive
wave (3.1) can propagate in the direction $m=(m_{1},m_{2},m_{3})\neq 0$ with
the speeds $v_{1}=v_{2}$, $v_{3}$ and $v_{4}$, see [9], where

$\displaystyle v_{1}^{2}$ $\displaystyle=v_{2}^{2}=v_{t}^{2}:=\frac{\mu_{\rm
e}}{{\varrho_{0}}},$ $\displaystyle\displaystyle v_{3}^{2}$
$\displaystyle=\frac{1}{2{\varrho_{0}}\varkappa}\left\\{\xi+\alpha+\varkappa\left(\lambda_{\rm
e}+2\mu_{\rm e}\right)+\sqrt{\left[\xi+\alpha-\varkappa\left(\lambda_{\rm
e}+2\mu_{\rm
e}\right)\right]^{2}+4\,\varkappa{\beta}^{2}}\right\\},\vspace{1mm}$ (3.17)
$\displaystyle\displaystyle v_{4}^{2}$
$\displaystyle=\frac{1}{2{\varrho_{0}}\varkappa}\left\\{\xi+\alpha+\varkappa\left(\lambda_{\rm
e}+2\mu_{\rm e}\right)-\sqrt{\left[\xi+\alpha-\varkappa\left(\lambda_{\rm
e}+2\mu_{\rm e}\right)\right]^{2}+4\,\varkappa{\beta}\,^{2}}\right\\}.$

The progressive waves propagating with speeds
$v=v_{1}=v_{2}=\sqrt{\displaystyle\frac{\mu_{\rm e}}{{\varrho_{0}}}}$ are
transverse waves and the characteristic space (the solutions of (3.2)) is
generated by the linear independent vectors
$\widehat{u}^{\left(1\right)}=\left(-m_{3},0,m_{1}\right)$ and
$\qquad\widehat{u}^{\left(2\right)}=\left(-m_{2},m_{1},0\right).$ For
$v=v_{3}$ and $v=v_{4}$ the progressive waves are longitudinal waves with the
corresponding solution of the algebraic system (3.2) given by
$\widehat{u}^{(3)}=\left(m_{1}\mathfrak{c},m_{2}\mathfrak{c},m_{3}\mathfrak{c},{\beta}\right),$
and
$\widehat{u}^{(4)}=\left(m_{1}\mathfrak{a},m_{2}\mathfrak{a},m_{3}\mathfrak{a},{\beta}\right),$
respectively, where $\mathfrak{c}={\varrho_{0}}\varkappa v_{3}^{2}-\xi-\alpha$
and $\mathfrak{a}={\varrho_{0}}\varkappa v_{4}^{2}-\xi-\alpha.$

In the next sections we show that there exists a unique Rayleigh wave solution
for all isotropic elastic materials with micro-voids satisfying (3.12), i.e.,
in the same range of values for the constitutive coefficients for which real
progressive plane waves exist.

##  4 Rayleigh waves in isotropic elastic materials with micro-voids

In this section we consider that the material with micro-voids occupying the
half-space

$\Sigma:=\\{(x_{1},x_{2},x_{3})\,|\,x_{1},x_{3}\in\mathbb{R},\,\,x_{2}\geq
0\\}$

is homogeneous and isotropic and its boundary is free of surface traction,
i.e.,

$\displaystyle{\sigma}.\,n=\sigma_{i3}=0,\qquad\qquad{h}.\,n=h_{2}=0\qquad\qquad\textrm{for}\quad
x_{2}=0.$ (4.1)

Without loss of generality we will study the waves propagating along the
$x_{1}-$axis. In the context of Rayleigh wave propagation, the surface
particles move in the planes normal to the surface $x_{2}=0$ and parallel to
the direction of propagation. In consequence, we consider the following plane
strain ansatz as a first step in our process of construction of the solution

$\displaystyle u=u(x_{1},x_{2},t)$
$\displaystyle=\footnotesize\begin{pmatrix}u_{1}(x_{1},x_{2},t)\\\
u_{2}(x_{1},x_{2},t)\\\
0\end{pmatrix},\qquad\qquad\zeta=\zeta(x_{1},x_{2},t).$ (4.2)

Beside the boundary condition (4.1) we require that the solutions be
attenuated in the direction $x_{2}$, so that they are decaying with distance
from the plane surface $x_{2}=0$, that is we require that

$\displaystyle\lim_{x_{2}\rightarrow\infty}\\{u_{1},u_{2},{\zeta},\sigma_{12},\sigma_{21},\sigma_{22},h_{2}\\}(x_{1},x_{2},t)=0\qquad\quad\forall\,x_{1}\in\mathbb{R},\quad\forall\,t\in[0,\infty).$
(4.3)

The further aim of this paper is to give an explicit solution
$(u_{1},u_{2},\zeta)$ of the system of partial differential equations which
results from (2.17) by considering the plane strain ansatz (4.2), i.e., a
solution of

$\displaystyle\varrho_{0}\,\frac{\partial^{2}{u}_{1}}{\partial\,t^{2}}$
$\displaystyle=(\lambda_{\rm e}+2\,\mu_{\rm
e})\,\frac{\partial^{2}u_{1}}{\partial\,x_{1}^{2}}+(\lambda_{\rm e}+\mu_{\rm
e})\,\frac{\partial^{2}u_{2}}{\partial\,x_{2}\partial\,x_{1}}+\mu_{\rm
e}\,\frac{\partial^{2}u_{1}}{\partial\,x_{2}^{2}}+\beta\,\frac{\partial\,\zeta}{\partial\,x_{1}},$
$\displaystyle\varrho_{0}\,\frac{\partial^{2}{u}_{2}}{\partial\,t^{2}}$
$\displaystyle=\mu_{\rm
e}\,\frac{\partial^{2}u_{2}}{\partial\,x_{1}^{2}}+(\lambda_{\rm e}+\mu_{\rm
e})\,\frac{\partial^{2}u_{1}}{\partial\,x_{2}\,\partial\,x_{1}}+(\lambda_{\rm
e}+2\mu_{\rm
e})\,\frac{\partial^{2}u_{2}}{\partial\,x_{2}^{2}}+\beta\,\frac{\partial\,\zeta}{\partial\,x_{2}},$
(4.4)
$\displaystyle\varrho_{0}\,\varkappa\,\frac{\partial^{2}{{\zeta}}}{\partial\,t^{2}}$
$\displaystyle=\alpha\,\frac{\partial^{2}\zeta}{\partial\,x_{1}^{2}}+\alpha\,\frac{\partial^{2}\zeta}{\partial\,x_{2}^{2}}-\xi\zeta-\beta\,\frac{\partial\,u_{1}}{\partial\,x_{1}}-\beta\,\frac{\partial\,u_{1}}{\partial\,x_{2}},$

which satisfies the boundary conditions at $x_{2}=0$

$\displaystyle\mu_{\rm e}\,\frac{\partial\,u_{1}}{\partial\,x_{2}}+\mu_{\rm
e}\,\frac{\partial\,u_{2}}{\partial\,x_{1}}=\,0,\qquad(\lambda_{\rm
e}+2\mu_{\rm e})\,\frac{\partial\,u_{2}}{\partial\,x_{2}}+\lambda_{\rm
e}\,\frac{\partial\,u_{1}}{\partial\,x_{1}}+\beta\zeta=\,0,\qquad\alpha\,\frac{\partial\,\zeta}{\partial\,x_{2}}=\,0,$
(4.5)

and has the asymptotic behaviour (4.3).

##  5 The algebraic analysis of a Riccati-type equation

Let ${\mathcal{X}},{\mathcal{Y}}\in\mathbb{R}^{3\times 3}$ be positive
definite matrices and ${\mathcal{Z}}\in\mathbb{R}^{3\times 3}$.

######  Definition 5.1.

By the limiting speed associated to
${\mathcal{X}},{\mathcal{Y}}\in\mathbb{R}^{3\times 3}$ and
${\mathcal{Z}}\in\mathbb{R}^{3\times 3}$ we understand a speed
$\widehat{v}>0$, such that for all wave speeds satisfying $0\leq
v<\widehat{v}$ (subsonic speeds) the solutions $r$ of

$\displaystyle\det\,[r^{2}{\mathcal{X}}+r({\mathcal{Z}}+{\mathcal{Z}}^{T})+{\mathcal{Y}}-k^{2}v^{2}{{{\mathbbm{1}}}}]=0,$
(5.1)

are not real.

In other words, if the roots of the characteristic equation (5.1) are not real
then they correspond to wave speeds $v$ satisfying $0\leq v<\widehat{v}$.

In this paper, we will use a matrix algebraic analysis of a Riccati-type
equation, see (5.3), presented by Fu and Mielke in [26, 25] (see also the work
by Mielke and Sprenger [55]), i.e.,

######  Theorem 5.2.

(Fu and Mielke in [26, 25]) Let
${\mathcal{X}},{\mathcal{Y}}\in\mathbb{R}^{3\times 3}$ be positive definite
matrices and ${\mathcal{Z}}\in\mathbb{R}^{3\times 3}$.

  1. 1.

If $0\leq v<\widehat{v}$, then the matrix problem

$\displaystyle{\mathcal{X}}{\mathcal{E}}^{2}-{\rm
i}\,({\mathcal{Z}}+{\mathcal{Z}}^{T})\mathcal{E}-{\mathcal{Y}}+k^{2}v^{2}{{{\mathbbm{1}}}}=0,\qquad\textrm{\rm
Re\,spec\,}{\mathcal{E}}>0,$ (5.2)

where $\textrm{\rm Re\,spec\,}{\mathcal{E}}$ means the real part of spectra of
${\mathcal{E}}$ has a unique solution for ${\mathcal{E}}$ and
${{\mathcal{M}}}:=-(-{\mathcal{X}}{\mathcal{E}}+{\rm i}\,{\mathcal{Z}}^{T})$
is Hermitian.

  2. 2.

If $0\leq v<\widehat{v}$, then the unique solution $\mathcal{M}$ of the
algebraic Riccati equation

$\displaystyle({\mathcal{M}}-{\rm
i}\,{\mathcal{Z}}){\mathcal{X}}^{-1}({\mathcal{M}}+{\rm
i}\,\mathcal{R^{T}})-{\mathcal{Y}}+k\,v^{2}\,{{{\mathbbm{1}}}}=0,\qquad{{\mathcal{M}}}\,y(0)=0.$
(5.3)

that satisfies $\text{\rm Re\,spec}\,({\mathcal{X}}^{-1}({\mathcal{M}}+{\rm
i}\,{\mathcal{Z}}^{T}))>0$ is given explicitly by

$\displaystyle{\mathcal{M}}=\Big{(}\int_{0}^{\pi}{\mathcal{X}}_{\theta}^{-1}\,d\theta\Big{)}^{-1}\Big{(}\pi{{\mathbbm{1}}}-{\rm
i}\,\int_{0}^{\pi}{\mathcal{X}}_{\theta}^{-1}{{\mathcal{Z}}}_{\theta}^{T}\,d\theta\Big{)},$
(5.4)

where we write
$\widetilde{{\mathcal{Y}}}={\mathcal{Y}}-k\,v^{2}\,{{{\mathbbm{1}}}}$,
$\theta$ is an arbitrary angle, while the matrices ${\mathcal{X}}_{\theta}$,
${\mathcal{Y}}_{\theta}$ and ${\mathcal{Z}}_{\theta}$ are obtained by rotation
of the old coordinate system666We remark that ${\mathcal{X}}$ and
${\mathcal{Y}}$ remain symmetric and that
$\widetilde{{\mathcal{Y}}}_{\theta}$, ${\mathcal{X}}_{\theta}$ and
${\mathcal{Z}}_{\theta}$ are periodic in $\theta$ with periodicity $\pi$ and
$\displaystyle\widetilde{{\mathcal{Y}}}_{\theta}(\theta+\frac{\pi}{2})={\mathcal{X}}_{\theta},\qquad\quad{\mathcal{Z}}_{\theta}(\theta+\frac{\pi}{2})=-{\mathcal{Z}}_{\theta}^{T},\qquad\quad{\mathcal{X}}_{\theta}(\theta+\frac{\pi}{2})=\widetilde{{\mathcal{Y}}}_{\theta}.$
(5.5) In addition, according to the definition of the limiting speed,
regarding Proposition 6.2 and Proposition 6.4, the limiting velocity
$\widehat{v}$ is in fact the lowest velocity for which the matrices
$\widetilde{{\mathcal{Y}}}_{\theta}$ and $T(\theta)$ become singular for some
angle $\theta$ and $\widetilde{{\mathcal{Y}}}_{\theta}$ is positive definite
for $0\leq v<\widehat{v}$, see the proof of Proposition 6.2 for more details.
Hence, from (5.5) we have that ${\mathcal{X}}_{\theta}$ is positive definite,
too. Thus by the definition of the limiting speed $\widehat{v}$ both
$\widetilde{{\mathcal{Y}}}_{\theta}$ and ${\mathcal{X}}_{\theta}$ are positive
definite or positive semi-definite depending on $\theta$ (for $v=\widehat{v}$
there is at least one $\theta$ at which ${\mathcal{X}}(\theta)$ has an
eigenvalue 0, and likewise $\widetilde{{\mathcal{Y}}}_{\theta}$). about
$e_{3}$ by an angle $\theta$

$\displaystyle{\mathcal{X}}_{\theta}=\cos^{2}\theta\,{\mathcal{X}}-\sin\theta\cos\theta\,({\mathcal{Z}}+{\mathcal{Z}}^{T})+\sin^{2}\theta\,\widetilde{{\mathcal{Y}}},\vspace{2mm}$
$\displaystyle{\mathcal{Z}}_{\theta}=\cos^{2}\theta\,{\mathcal{Z}}+\sin\theta\cos\theta\,({\mathcal{X}}-\widetilde{{\mathcal{Y}}})-\sin^{2}\theta\,{\mathcal{Z}}^{T},\vspace{2mm}$
(5.6)
$\displaystyle\widetilde{{\mathcal{Y}}}_{\theta}=\cos^{2}\theta\,\widetilde{{\mathcal{Y}}}+\sin\theta\cos\theta\,({\mathcal{Z}}+{\mathcal{Z}}^{T})+\sin^{2}\theta\,{\mathcal{X}}.$

  3. 3.

If $0\leq v<\widehat{v}$, then the solution ${\mathcal{M}}_{v}$ obtained from
(5.3) has the following properties

    1. (a)

${\mathcal{M}}_{v}$ is Hermitian,

    2. (b)

$\frac{d{{\mathcal{M}}}_{v}}{dv}$ is negative definite,

    3. (c)

${\rm tr}({{\mathcal{M}}}_{v})\geq 0$, and $\langle
w,{{\mathcal{M}}}_{v}\,w\rangle\geq 0$ for all real vectors $w$ for all $0\leq
v\leq\widehat{v}$,

    4. (d)

${\mathcal{M}}_{v}$ is and positive definite for all $0\leq v<\widehat{v}$.

  4. 4.

The secular equation

$\displaystyle\det{{\mathcal{M}}}_{v}=0,$ (5.7)

where ${\mathcal{M}}_{v}$ is obtained from $\eqref{120}$, has a unique
admissible solution $0\leq v<\widehat{v}$.

##  6 The ansatz for the solution and the limiting speed

###  6.1 The ansatz

We look for a solution of (4) and (4.5) having the form777We take ${\rm
i}\,z_{3}$ since this choice leads us, in the end, only to deal with real
matrices.

$\displaystyle\mathcal{U}(x_{1},x_{2},t)=\footnotesize\begin{pmatrix}u_{1}(x_{1},x_{2},t)\\\
u_{2}(x_{1},x_{2},t)\\\ \zeta(x_{1},x_{2},t)\end{pmatrix}={\rm
Re}\left[\footnotesize\begin{pmatrix}z_{1}(x_{2})\\\ z_{2}(x_{2})\\\ {\rm
i}\,z_{3}(x_{2})\end{pmatrix}e^{{\rm i}\,k\,(x_{1}-vt)}\right],$ (6.1)

where $v$ is the propagation speed (the phase velocity). If the Cauchy problem
given by the following system

$\displaystyle\footnotesize\begin{pmatrix}\mu_{\rm e}&0&0\\\ 0&\lambda_{\rm
e}+2\,\mu_{\rm e}&0\\\
0&0&\alpha\end{pmatrix}\footnotesize\begin{pmatrix}z_{1}^{\prime\prime}(x_{2})\\\
z_{2}^{\prime\prime}(x_{2})\\\ \ z_{3}^{\prime\prime}(x_{2})\end{pmatrix}+{\rm
i}\,\footnotesize\begin{pmatrix}0&k\,(\lambda_{\rm e}+\mu_{\rm e})&0\\\
k\,(\mu_{\rm e}+\lambda_{\rm e})&0&\beta\\\
0&\beta&0\end{pmatrix}\footnotesize\begin{pmatrix}z_{1}^{\prime}(x_{2})\\\
z_{2}^{\prime}(x_{2})\\\ \ z_{3}^{\prime}(x_{2})\end{pmatrix}\vspace{2mm}$
$\displaystyle\qquad\qquad\ \ -\footnotesize\begin{pmatrix}k^{2}\,(2\,\mu_{\rm
e}+\lambda_{\rm e})-\varrho_{0}\,k^{2}v^{2}&0&\beta\,k\\\ 0&k^{2}\,\mu_{\rm
e}-\varrho_{0}\,k^{2}v^{2}&0\\\
\beta\,k&0&k^{2}\,\alpha+\xi-\varrho_{0}\,\varkappa\,k^{2}v^{2}\end{pmatrix}\footnotesize\begin{pmatrix}z_{1}(x_{2})\\\
z_{2}(x_{2})\\\ z_{3}(x_{2})\end{pmatrix}=0,$ (6.2)

and the boundary conditions

$\displaystyle\footnotesize\begin{pmatrix}\mu_{\rm e}&0&0\\\ 0&2\,\mu_{\rm
e}+\lambda_{\rm e}&0\\\
0&0&\alpha\end{pmatrix}\footnotesize\begin{pmatrix}z_{1}^{\prime}(0)\\\
z_{2}^{\prime}(0)\\\ z_{3}^{\prime}(0)\end{pmatrix}+{\rm
i}\,\footnotesize\begin{pmatrix}0&k\,\mu_{\rm e}&0\\\ k\,\lambda_{\rm
e}&0&\beta\\\ 0&0&0\end{pmatrix}\footnotesize\begin{pmatrix}z_{1}(0)\\\
z_{2}(0)\\\ z_{3}(0)\end{pmatrix}=0,$ (6.3)

has the solution $z_{i}$, $i=1,2,3$, where $\cdot^{\prime}$ denotes the
derivative with respect to $x_{2}$, then $\mathcal{U}$ given by the ansatz
(6.1) satisfies (4) and (4.5). In a more compact notation, the above equations
reads

$\displaystyle\frac{1}{k^{2}}\,{{T}}\,z^{\prime\prime}(x_{2})+{\rm
i}\,\frac{1}{k}\,({{R}}+{{R}}^{T})\,z^{\prime}(x_{2})-{{Q}}\,z(x_{2})+k^{2}\,v^{2}\,\hat{{\mathbbm{1}}}\,z(x_{2})=\,0,\qquad\frac{1}{k^{2}}\,{{T}}\,z^{\prime}(0)+{\rm
i}\,\frac{1}{k}\,{{R}}^{T}\,z(0)=$ $\displaystyle\,0,$ (6.4)

where the matrices ${{T}}\,\,,{{R}},\,{Q}$ and
$\widehat{{{\mathbbm{1}}}}^{-\frac{1}{2}}$ are defined by

$\displaystyle{{T}}=k^{2}\footnotesize\begin{pmatrix}\mu_{\rm e}&0&0\\\
0&2\,\mu_{\rm e}+\lambda_{\rm e}&0\\\
0&0&\alpha\end{pmatrix},\qquad{{R}}=k\footnotesize\begin{pmatrix}0&k\lambda_{\rm
e}&0\\\ k\,\mu_{\rm e}&0&0\\\ 0&\beta&0\end{pmatrix},\hskip 42.67912pt$ (6.5)
$\displaystyle{{Q}}=\footnotesize\begin{pmatrix}k^{2}\,(2\,\mu_{\rm
e}+\lambda_{\rm e})&0&\beta\,k\\\ 0&k^{2}\,\mu_{\rm e}&0\\\
\beta\,k&0&k^{2}\,\alpha+\xi\end{pmatrix},\qquad\widehat{{{\mathbbm{1}}}}^{-1/2}=\footnotesize\begin{pmatrix}\frac{1}{\sqrt{\varrho_{0}}}&0&0\\\
0&\frac{1}{\sqrt{\varrho_{0}}}&0\\\
0&0&\frac{1}{\sqrt{\varrho_{0}\,\varkappa}}\end{pmatrix}.$

To find the solution $z(x_{2})$ of (6.4) is equivalent to finding a solution
$y(x_{2}):=\widehat{{{\mathbbm{1}}}}^{1/2}\,z(x_{2})$ of

$\displaystyle\frac{1}{k^{2}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,{{T}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,y^{\prime\prime}(x_{2})+{\rm
i}\,\frac{1}{k}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,({{R}}+{{R}}^{T})\widehat{{{\mathbbm{1}}}}^{-1/2}\,y^{\prime}(x_{2})-\widehat{{{\mathbbm{1}}}}^{-1/2}\,{{Q}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,y(x_{2})+k^{2}\,v^{2}\,{{\mathbbm{1}}}\,y(x_{2})$
$\displaystyle=\,0,$ (6.6)
$\displaystyle\frac{1}{k^{2}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,{{T}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,y^{\prime}(0)+{\rm
i}\,\frac{1}{k}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,{{R}}^{T}\,\widehat{{{\mathbbm{1}}}}^{-1/2}\,y(0)$
$\displaystyle=\,0.$

With the help of the modified matrices

$\displaystyle{\mathcal{T}}:=\widehat{{{\mathbbm{1}}}}^{-1/2}\,{{T}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}:=\,k^{2}\footnotesize\begin{pmatrix}\frac{\mu_{\rm
e}}{\varrho_{0}}&0&0\\\ 0&\frac{2\,\mu_{\rm e}+\lambda_{\rm
e}}{\varrho_{0}}&0\\\
0&0&\frac{\alpha}{\varrho_{0}\,\varkappa}\end{pmatrix},\qquad{\mathcal{R}}:=\widehat{{{\mathbbm{1}}}}^{-1/2}\,{{R}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}:=k\,\footnotesize\begin{pmatrix}0&k\,\frac{\lambda_{\rm
e}}{\varrho_{0}}&0\\\ k\,\frac{\mu_{\rm e}}{\varrho_{0}}&0&0\\\
0&\frac{\beta}{\varrho_{0}\sqrt{\varkappa}}&0\end{pmatrix},$ (6.7)
$\displaystyle{\mathcal{Q}}:=\widehat{{{\mathbbm{1}}}}^{-1/2}\,{{Q}}\,\widehat{{{\mathbbm{1}}}}^{-1/2}:=\footnotesize\begin{pmatrix}\frac{2\,\mu_{\rm
e}+\lambda_{\rm
e}}{\varrho_{0}}k^{2}&0&\frac{\beta\,k}{\varrho_{0}\sqrt{\varkappa}}\\\
0&\frac{\mu_{\rm e}}{\varrho_{0}}k^{2}&0\\\
\frac{\beta\,k}{\varrho_{0}\sqrt{\varkappa}}&0&\,\frac{\alpha\,k^{2}+\xi}{\varrho_{0}\,\varkappa}\end{pmatrix}\hskip
99.58464pt$

the system (6.4) turns into

$\displaystyle\frac{1}{k^{2}}{\mathcal{T}}\,y^{\prime\prime}(x_{2})+{\rm
i}\,\frac{1}{k}({\mathcal{R}}+{\mathcal{R}}^{T})\,y^{\prime}(x_{2})-{\mathcal{Q}}\,y(x_{2})+k^{2}\,v^{2}\,{{\mathbbm{1}}}\,y(x_{2})=\,0,\qquad\frac{1}{k^{2}}{\mathcal{T}}\,y^{\prime}(0)+{\rm
i}\,\frac{1}{k}\,{\mathcal{R}}^{T}\,y(0)=$ $\displaystyle\,0.$ (6.8)

######  Lemma 6.1.

If the constitutive coefficients satisfy the conditions $\mu_{\rm e}>0,\ \
\lambda_{\rm e}+2\mu_{\rm e}>0,\ \ \alpha>0$ and $(\lambda_{\rm e}+2\mu_{\rm
e})\,\xi>\beta^{2}$ then the matrices ${\mathcal{Q}}$ and ${\mathcal{T}}$ are
symmetric and positive definite.

###### Proof.

The symmetry is clear. The matrix ${T}$ is positive definite if and only if
$\mu_{\rm e}>0,\,\lambda_{\rm e}+2\mu_{\rm e}>0,\,\alpha>0$ and so the matrix
${\mathcal{T}}$ is positive definite, being a product of positive definite
matrices.

In addition, ${{Q}}$ is positive-definite if and only if the principal minors
are positive, namely

$\displaystyle{{Q}}_{11}$ $\displaystyle=k^{2}(\lambda_{\rm e}+2\,\mu_{\rm
e}),\qquad{{Q}}_{11}{{Q}}_{22}-{{Q}}_{12}{{Q}}_{21}=k^{4}\,\mu_{\rm
e}(\lambda_{\rm e}+2\,\mu_{\rm e})\,,$ (6.9) $\displaystyle\det({{Q}})$
$\displaystyle=k^{4}\,\mu_{\rm e}\left[(\lambda_{\rm e}+2\,\mu_{\rm
e})\alpha\,k^{2}+(\lambda_{\rm e}+2\,\mu_{\rm e})\xi-\beta^{2}\right]\,,$

i.e., under the hypothesis of the lemma. Since ${Q}$ is positive definite, so
it is ${\mathcal{Q}}$ defined by (6.7), and the proof is complete. ∎

###  6.2 The existence of the limiting speed in the model

We now seek a solution $y$ of the differential system (6.8) in the form

$\displaystyle y(x_{2})=\footnotesize\begin{pmatrix}d_{1}\\\ d_{2}\\\
d_{3}\end{pmatrix}\,e^{{\rm i}\,r\,k\,x_{2}},\qquad\text{Im}\,r>0,$ (6.10)

where $r\,\in\mathbb{C}$ is a complex parameter,
$d=\footnotesize\begin{pmatrix}d_{1},&d_{2},&d_{3}\end{pmatrix}^{T}\in\mathbb{C}^{3}$,
$d\neq 0$ is the amplitude and $\text{Im}\,r$ is the coefficient of the
imaginary part of $r$.

From (6.10) and $\eqref{n11}$ we obtain the system

$\displaystyle[r^{2}{\mathcal{T}}+r\,({\mathcal{R}}+{\mathcal{R}}^{T})+{\mathcal{Q}}-\
k^{2}v^{2}{{{\mathbbm{1}}}}]\,d=0,\qquad\qquad[r\,{\mathcal{T}}+{\mathcal{R}}^{T}]\,d=0.$
(6.11)

Because (6.11)1 is a characteristic equation corresponding to an eigenvalue
problem, the eigenvectors
$d=\footnotesize\begin{pmatrix}d_{1},&d_{2},&d_{3}\end{pmatrix}^{T}\neq 0$ if

$\displaystyle\det\,[r^{2}{\mathcal{T}}+r({\mathcal{R}}+{\mathcal{R}}^{T})+{\mathcal{Q}}-k^{2}v^{2}{{{\mathbbm{1}}}}]=0,$
(6.12)

which is an equation of order 6 with the unknown $r$. We will search the
conditions for having solutions for (6.11) with $\text{Im}\,r>0$, which
ensures the asymptotic decay condition (4.3).

The limiting speed in the micro-voids model is the limiting speed associated
to ${\mathcal{T}},{\mathcal{Q}}\in\mathbb{R}^{3\times 3}$ and
${\mathcal{R}}\in\mathbb{R}^{3\times 3}$, see Definition 5.1 and Theorem 5.2.

######  Proposition 6.2.

If the constitutive coefficients satisfy the conditions $\mu_{\rm
e}>0,\,\lambda_{\rm e}+2\mu_{\rm e}>0,\,\alpha>0$ and $(\lambda_{\rm
e}+2\mu_{\rm e})\,\xi>\beta^{2}$, then there exists a limiting speed
$\widehat{v}>0$ and moreover if one root $r_{v}$ of the characteristic
equation (6.12) is real then it corresponds to a speed $v\geq\widehat{v}$
(non-admissible).

###### Proof.

Assume that there exists a real $r_{v}$ as solution of the characteristic
equation (6.12), then $\exists\,\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$ such
that $r_{v}=\tan\theta$. Since

$\displaystyle e^{ik\,(x_{1}+rx_{2}-vt)}=e^{ik\,(x_{1}+\tan\theta
x_{2}-vt)}=e^{\frac{ik}{\cos\theta}(\cos\theta x_{1}+\sin\theta
x_{2}-\cos\theta vt)},$ (6.13)

and since the solution $\mathcal{U}$ given by (6.1) can be written using
(6.10) in the form

$\displaystyle\mathcal{U}(x_{1},x_{2},t)$
$\displaystyle=\footnotesize\begin{pmatrix}u_{1}(x_{1},x_{2},t)\\\
u_{2}(x_{1},x_{2},t)\\\ \zeta(x_{1},x_{2},t)\end{pmatrix}={\rm
Re}\left[\footnotesize\begin{pmatrix}a_{1}\\\ a_{2}\\\ {\rm
i}\,\,a_{3}\end{pmatrix}e^{\frac{ik}{\cos\theta}(\cos\theta x_{1}+\sin\theta
x_{2}-\cos\theta vt)}\right],$ (6.14)

where
$(a_{1},a_{2},a_{3})^{T}=\widehat{{{\mathbbm{1}}}}^{-1/2}(d_{1},d_{2},d_{3})^{T}$,
we are looking to $\mathcal{U}(x_{1},x_{2},t)$ as to a non-trivial plane body
wave solution with wave number $k_{\theta}=\frac{k}{\cos\theta}$, the speed
$v_{\theta}=v\cos\theta$, the same frequencies
$\omega_{\theta}=k_{\theta}v_{\theta}=kv=\omega$ and propagation in the
direction ${n}_{\theta}$ where ${n}_{\theta}=(\cos\theta,\sin\theta,0)$.

This is the reason why, in the following, we examine the conditions under
which we observe ”real” bulk waves only for positive real values of
$\omega^{2}$ and the presence of a non-trivial solution of (2.17) having the
form

$\displaystyle u(x_{1},x_{2},t)$
$\displaystyle=\footnotesize\begin{pmatrix}u_{1}(x_{1},x_{2},t)\\\
u_{2}(x_{1},x_{2},t)\\\
0\end{pmatrix}=\footnotesize\begin{pmatrix}\widehat{u}_{1}\\\
\widehat{u}_{2}\\\ 0\end{pmatrix}\,e^{{\rm i}\,\left(k\langle
m,\,x\rangle_{\mathbb{R}^{3}}-\,\omega\,t\right)},\qquad\zeta(x_{1},x_{2},t)=i\,\widehat{\zeta}\,e^{{\rm
i}\,\left(k\langle m,\,x\rangle_{\mathbb{R}^{3}}-\,\omega\,t\right)},$ (6.15)
$\displaystyle\qquad\qquad\qquad\qquad(\widehat{u}_{1},\widehat{u}_{2},\widehat{\zeta})^{T}\in\mathbb{C}^{3},\quad(\widehat{u}_{1},\widehat{u}_{2},\widehat{\zeta})^{T}\neq
0\,$

for a wave propagation direction $m=(m_{1},m_{2},0)^{T}$ with
$\lVert{m}\rVert^{2}=1$ and for every wave number $k>0$. With this conditions,
system (2.17) becomes

$\displaystyle\left[{Q}_{2}(k)-\omega^{2}\widehat{{{\mathbbm{1}}}}\,\right]\footnotesize\begin{pmatrix}\widehat{u}_{1}\\\
\widehat{u}_{2}\\\
\widehat{\zeta}\end{pmatrix}=\footnotesize\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix},$ (6.16)

where

$\displaystyle{{Q}_{2}}(k)$
$\displaystyle=\footnotesize\begin{pmatrix}(\lambda_{\rm e}+2\mu_{\rm
e})k^{2}m_{1}^{2}+\mu_{\rm e}\,k^{2}m_{2}^{2}&(\lambda_{\rm e}+\mu_{\rm
e})k^{2}m_{1}m_{2}&\beta\,k\,ma_{1}\vspace{2mm}\\\ (\lambda_{\rm e}+\mu_{\rm
e})k^{2}m_{1}m_{2}&(\lambda_{\rm e}+2\mu_{\rm e})k^{2}m_{2}^{2}+\mu_{\rm
e}\,k^{2}m_{1}^{2}&\beta\,k\,m_{2}\vspace{2mm}\\\
\beta\,k\,m_{1}&\beta\,k\,m_{2}&\alpha\,k^{2}+\xi\end{pmatrix},\qquad\widehat{{{\mathbbm{1}}}}=\footnotesize\begin{pmatrix}\varrho_{0}&0&0\\\
0&\varrho_{0}&0\\\ 0&0&\varrho_{0}\,\varkappa\,\end{pmatrix}.$ (6.17)

An equivalent form of (6.17) is

$\displaystyle\left[{\widetilde{Q}_{2}}(k)-\omega^{2}\,{{\mathbbm{1}}}\,\right]\footnotesize\begin{pmatrix}\widetilde{u}_{1}\\\
\widetilde{u}_{2}\\\
\widetilde{\zeta}\end{pmatrix}=\footnotesize\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix},\qquad(\widetilde{u}_{1},\widetilde{u}_{2},\widetilde{\zeta})^{T}=\widehat{{{\mathbbm{1}}}}^{1/2}(\widehat{u}_{1},\widehat{u}_{2},\widehat{\zeta})^{T}.$
(6.18)

The matrix ${{Q}_{2}}(k)$ is positive definite for all $k>0$ if and only if

$\displaystyle m_{1}^{2}(\lambda_{\rm e}+2\mu_{\rm e})+m_{2}^{2}\mu_{\rm
e}>0,\quad\mu_{\rm e}(\lambda_{\rm e}+2\mu_{\rm e})>0,\qquad
k^{2}\left[\alpha\mu_{\rm e}(\lambda_{\rm e}+2\mu_{\rm
e})\right]+\left[\mu_{\rm e}\left(\xi(\lambda_{\rm e}+2\mu_{\rm
e})-\beta^{2}\right)\right]>0,$ (6.19)

for all $k>0$. It is easy to remark that the above inequalities are satisfied
if and only if $\mu_{\rm e}>0,\,\lambda_{\rm e}+2\mu_{\rm e}>0,\,\alpha>0$ and
$(\lambda_{\rm e}+2\mu_{\rm e})\,\xi>\beta^{2}$. But ${{Q}_{2}}(k)$ is
positive definite if and only if
${\widetilde{{Q}}_{2}}(k)=\widehat{{{\mathbbm{1}}}}^{-1/2}{{Q}_{2}}(k)\,\widehat{{{\mathbbm{1}}}}^{-1/2}$
is positive definite, which further is equivalent to the fact that the
equation
$\det\left[{\widetilde{Q}_{2}}(k)-\omega^{2}\,{{\mathbbm{1}}}\,\right]=0$ has
only real positive solutions for $\omega^{2}$. Hence, we have proven that for
every direction of propagation of the form $m=(m_{1},m_{2},0)^{T}$, $\lVert
m\rVert^{2}=1$, the necessary and sufficient conditions for existence of a non
trivial solution of the form (6.15) for the system of partial differential
equations (2.17) are $\mu_{\rm e}>0,\,\lambda_{\rm e}+2\mu_{\rm
e}>0,\,\alpha>0$ and $(\lambda_{\rm e}+2\mu_{\rm e})\,\xi>\beta^{2}$

By knowing this, we may proceed further to our analysis. A direct substitution
of (6.14) into (3.2) and using the same idea as in the case of obtaining the
formulas (6.6)-(6.8) we have that there exists a non-trivial solution
$\footnotesize\begin{pmatrix}d_{1},&d_{2},&d_{3}\end{pmatrix}\neq 0$ of the
algebraic system written in matrix form

$\displaystyle\left[\sin^{2}\theta\,{\mathcal{T}}+\sin\theta\cos\theta({\mathcal{R}}+{\mathcal{R}}^{T})+\cos^{2}\theta\,{\mathcal{Q}}-k^{2}v^{2}\cos^{2}\theta\,{{{\mathbbm{1}}}}\right]\footnotesize\begin{pmatrix}d_{1}\\\
d_{2}\\\ d_{3}\end{pmatrix}=0,$ (6.20)

with notations from (6.7). Let us remark that the equation (6.20) is the
propagation condition for plane waves in isotropic materials with micro-voids,
in the direction ${n}_{\theta}=(\cos\theta,\sin\theta,0)$. According to the
information of the previous paragraph, for the direction
${n}_{\theta}=(\cos\theta,\sin\theta,0)$ in particular, the system of partial
differential equations (2.17) admits a non trivial solution in the form given
by (6.15) only for real positive values $\omega^{2}$. Therefore, to each
$\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$ we can associate the real
frequencies $\omega_{\theta}$ satisfying

$\displaystyle\det\,\\{\sin^{2}\theta{\mathcal{T}}+\sin\theta\cos\theta({\mathcal{R}}+{\mathcal{R}}^{T})+\cos^{2}\theta{\mathcal{Q}}-\omega^{2}_{\theta}\,{{{\mathbbm{1}}}}\\}=0,$
(6.21)

and further a propagation speed $v_{\theta}=v\cos\theta$ such that
$\omega_{\theta}=k\,v_{\theta}$ that verify the equation

$\displaystyle\det\,\\{\sin^{2}\theta{\mathcal{T}}+\sin\theta\cos\theta({\mathcal{R}}+{\mathcal{R}}^{T})+\cos^{2}\theta{\mathcal{Q}}-k^{2}v^{2}_{\theta}\,{{{\mathbbm{1}}}}\\}=0.$
(6.22)

We define the limiting speed $\widehat{v}$ as the minimum of the values of
$v_{\theta}=v\,\cos\theta$ for all $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$

$\displaystyle\widehat{v}=\inf_{\theta\in(-\frac{\pi}{2},\frac{\pi}{2})}v_{\theta}.$
(6.23)

In conclusion, for a real $r_{v}$ there exists a
$\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that $r_{v}=\tan\theta$. For
this $\theta$ we have the real frequencies $\omega_{\theta}$ satisfying (6.21)
and the propagation speed $v_{\theta}$ corresponding to $r_{v}$ is satisfying
(6.22) with $v_{\theta}\geq\widehat{v}$. Thus, if $v$ is such that $0\leq
v<\widehat{v},$ then $r_{v}$ can not be real and if $r_{v}$ is real, then
$v\geq\widehat{v}$. ∎

######  Proposition 6.3.

If the constitutive coefficients satisfy the conditions $\mu_{\rm
e}>0,\,\lambda_{\rm e}+2\mu_{\rm e}>0,\,\alpha>0$ and $(\lambda_{\rm
e}+2\mu_{\rm e})\,\xi>\beta^{2}$, then for all
$\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $k>0$, the tensor
${\mathcal{Q}}_{\theta}:=\sin^{2}\theta{\mathcal{T}}+\sin\theta\cos\theta({\mathcal{R}}+{\mathcal{R}}^{T})+\cos^{2}\theta{\mathcal{Q}}$
is positive definite.

###### Proof.

The proof is straightforward since from the equation (6.20) we have that the
tensor ${\mathcal{Q}}_{\theta}$ satisfies

$\displaystyle\left[{\mathcal{Q}}_{\theta}-k^{2}v^{2}\cos^{2}\theta\,{{{\mathbbm{1}}}}\right]\footnotesize\begin{pmatrix}d_{1}\\\
d_{2}\\\ d_{3}\end{pmatrix}=0,$ (6.24)

which means that if we take the particular propagation direction
$m=(\cos\theta,\sin\theta,0)$ we obtain that
${\widetilde{Q}_{2}}(k)={\mathcal{Q}}_{\theta}$ is positive definite. ∎

######  Proposition 6.4.

If the constitutive coefficients satisfy the conditions $\mu_{\rm
e}>0,\,\lambda_{\rm e}+2\mu_{\rm e}>0,\,\alpha>0$ and $(\lambda_{\rm
e}+2\mu_{\rm e})\,\xi>\beta^{2}$, then for all
$\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, $k>0$ and $0\leq
v<\widehat{v}$, the tensor
$\widetilde{{Q}}_{\theta}:={\mathcal{Q}}_{\theta}-k^{2}v^{2}\cos^{2}\theta\,{{{\mathbbm{1}}}}$
is positive definite.

###### Proof.

We have to prove that all eigenvalues ${\mathcal{Q}}_{\theta}$ are larger than
those of the matrix $k^{2}v^{2}\cos^{2}\theta\,{{{\mathbbm{1}}}}$, i.e., than
$k^{2}v^{2}\cos^{2}\theta$. Note that since
${\mathcal{Q}}_{\theta}:=\sin^{2}\theta{\mathcal{T}}+\sin\theta\cos\theta({\mathcal{R}}+{\mathcal{R}}^{T})+\cos^{2}\theta{\mathcal{Q}}$
is positive definite, it admits only positive eigenvalues. Assuming that there
exist $\theta_{0}\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and
$v_{0}\in[0,\widehat{v})$ for which there is an eigenvalue
$\Lambda_{\theta_{0}}$ of ${\mathcal{Q}}_{\theta}$ such that
$\Lambda_{\theta_{0}}<k^{2}v^{2}_{0}\cos^{2}\theta_{0}$, then

$\displaystyle
v_{\theta_{0}}:=\sqrt{\frac{\Lambda_{\theta_{0}}}{k^{2}\,\cos^{2}\theta_{0}}}<v_{0}<\widehat{v}$
(6.25)

is solution of (6.22), i.e., for fixed $\theta_{0}$ we have that
$v_{\theta_{0}}<\widehat{v}$ verifies

$\displaystyle\det\,\\{\sin^{2}{\theta_{0}}\,{\mathcal{T}}+\sin{\theta_{0}}\,\cos{\theta_{0}}\,({\mathcal{R}}+{\mathcal{R}}^{T})+\cos^{2}{\theta_{0}}\,{\mathcal{Q}}-k^{2}v^{2}_{{\theta_{0}}}\,\cos^{2}{\theta_{0}}\,{{{\mathbbm{1}}}}\\}=0.$
(6.26)

This contradicts the definition of the limiting speed and Proposition 6.23,
since $\widehat{v}$ is the smallest speed having this property. Therefore, it
remains that for all $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$,
$k>0$ and for all $0\leq v<\widehat{v}$, all the eigenvalues of
${\mathcal{Q}}_{\theta}$ are larger than $k^{2}v^{2}\cos^{2}\theta$ and the
proof is complete. ∎

Gathering together the calculations done for the construction of the Rayleigh
wave solution given by Chirită and Ghiba [8], we have the explicit form of the
solutions of (6.12) given as

$\displaystyle
r_{1}^{2}=k^{2}\left(1-\frac{c^{2}}{v_{t}^{2}}\right),\quad\qquad
r_{2,3}^{2}=k^{2}\left(1-\displaystyle\frac{T_{\mp}}{k^{2}}\right)\qquad\text{where}\quad
v_{t}=\sqrt{\frac{\mu_{\rm e}}{{\varrho_{0}}}}$ (6.27)

and

$\displaystyle T_{\mp}(\omega^{2})$
$\displaystyle=\displaystyle\frac{1}{2\alpha(\lambda_{\rm e}+2\mu_{\rm
e})}\Bigg{\\{}{\beta}^{2}-(\xi-{\varrho_{0}}\varkappa\omega^{2})(\lambda_{\rm
e}+2\mu_{\rm e})+\alpha\omega^{2}{\varrho_{0}}$
$\displaystyle\qquad\qquad\qquad\quad\mp\sqrt{4\,\alpha\,\omega^{2}{\beta}^{2}{\varrho_{0}}+[-{\beta}^{2}+\alpha\omega^{2}{\varrho_{0}}+(\xi-{\varrho_{0}}\varkappa\omega^{2})(\lambda_{\rm
e}+2\mu_{\rm e})]^{2}}\Bigg{\\}}.$ (6.28)

In view of the definition of the limiting speed, from Theorem 5.2 we conclude:

######  Proposition 6.5.

If the constitutive coefficients satisfy the conditions $\mu_{\rm
e}>0,\,\lambda_{\rm e}+2\mu_{\rm e}>0,\,\alpha>0$ and $(\lambda_{\rm
e}+2\mu_{\rm e})\,\xi>\beta^{2}$ then the roots $r_{v}$ of the characteristic
equation (6.12) are not real if and only if

$v^{2}<\min\\{v_{s}^{2},{v}_{m}^{2}\\}.$ (6.29)

where

$\displaystyle{v}_{s}^{2}$ $\displaystyle:={\displaystyle\frac{2\,\mu_{\rm
e}+\lambda_{\rm e}}{\varrho_{0}}},$ $\displaystyle v_{m}^{2}$
$\displaystyle:=\displaystyle\frac{1}{2k^{2}{\varrho_{0}}\varkappa}\Big{\\{}\xi+k^{2}(\lambda_{\rm
e}+2\mu_{\rm e})\varkappa+k^{2}\alpha\vspace{1mm}$ (6.30)
$\displaystyle\qquad\qquad\ \ -\sqrt{(\xi+k^{2}(\lambda_{\rm e}+2\mu_{\rm
e})\varkappa+k^{2}\alpha)^{2}-4k^{2}\varkappa[\xi(\lambda_{\rm e}+2\mu_{\rm
e})-{\beta}^{2}+k^{2}\alpha(\lambda_{\rm e}+2\mu_{\rm e})]}\Big{\\}}.$

This defines the limiting speed in isotropic elastic materials with micro-
voids to be

$\displaystyle\widehat{v}:=\inf_{\theta\in(-\frac{\pi}{2},\frac{\pi}{2})}v_{\theta}\equiv\min\\{v_{s}^{2},{v}_{m}^{2}\\}.$
(6.31)

##  7 Existence and uniqueness of Rayleigh waves

Following the ideea by Fu and Mielke [25] we are now looking at (6.8) as to an
initial value problem and we search the solution in the form

$\displaystyle y(x_{2})=e^{-k\,x_{2}\,{\mathcal{E}}}y(0),$ (7.1)

where ${\mathcal{E}}\in\mathbb{C}^{3\times 3}$ is to be determined as solution
(see (7.1) and (6.11)) of

$\displaystyle[{\mathcal{T}}{\mathcal{E}}^{2}-{\rm
i}\,({\mathcal{R}}+{\mathcal{R}}^{T}){\mathcal{E}}-{\mathcal{Q}}+\
k^{2}v^{2}{{{\mathbbm{1}}}}]\,y(x_{2})=0,\qquad\qquad\qquad(-{\mathcal{T}}{\mathcal{E}}+{\rm
i}\,\,{\mathcal{R}}^{T})\,y(0)=0.$ (7.2)

In order to have a proper decay, the eigenvalues of ${\mathcal{E}}$ have to be
such that their real part is positive. The unknowns in (7.2) are
${\mathcal{E}}\in\mathbb{C}^{3\times 3}$ and $v$.

We are doing a change of variable by introducing the so called surface
impedance matrix

$\displaystyle{{\mathcal{M}}}=-(-{\mathcal{T}}{\mathcal{E}}+{\rm
i}\,{\mathcal{R}}^{T})\qquad\iff\qquad{\mathcal{E}}={\mathcal{T}}^{-1}({\mathcal{M}}+{\rm
i}\,\,{\mathcal{R}}^{T}),$ (7.3)

which, by substituting $\eqref{10}_{2}$ into $\eqref{09}_{1}$, must be a
solution of the algebraic Riccati equation

$\displaystyle({\mathcal{M}}-{\rm
i}\,{\mathcal{R}}){\mathcal{T}}^{-1}({\mathcal{M}}+{\rm
i}\,\mathcal{R^{T}})-{\mathcal{Q}}+k\,v^{2}\,{{{\mathbbm{1}}}}=0,\qquad\qquad{{\mathcal{M}}}\,y(0)=0.$
(7.4)

Since we are interested in a nontrivial solution $y$, we impose $y(0)\neq 0$,
so that the matrix ${\mathcal{M}}$ has to satisfy

$\displaystyle{\rm det}\,{\mathcal{M}}=0.$ (7.5)

The equation (7.5) is called secular equation for the micro-voids model in
terms of the impedance matrix ${\mathcal{M}}$.

The strategy to solve the system of nonlinear equations (7.4)1 and (7.5) is
the following. We find an admissible domain for $v$ such that, for fixed $v$
in this domain we may define the mapping $v\mapsto{{\mathcal{M}}}_{v}$, where
${\mathcal{M}}_{v}$ is the solution of (7.4)1. Note that not any
${\mathcal{M}}_{v}$ is admissible because ${\mathcal{M}}_{v}$ has to be such
that $\text{Re(spec}\,{\mathcal{E}}{\rm)}$ is positive, where “$\text{\rm
Re\,(spec\,}{\mathcal{E}}{\rm)}$” means the “real part of spectra of
${\mathcal{E}}$”. In the end, having ${\mathcal{M}}_{v}$ we find $v$ as
solution of (7.5). An important aspect is that we have to be sure that the
solution $v$ belongs to the admissible domain considered for having a suitable
${\mathcal{M}}_{v}$ and that the solution is unique, since otherwise the
uniqueness of the matrix ${\mathcal{M}}_{v}$ is violated.

Since ${\mathcal{T}}$ and ${\mathcal{Q}}$ are symmetric and positive definite
matrices, many aspects from the above discussions are purely mathematical
questions, and they are not specific to the elastic materials with micro-
voids. Indeed, since ${\mathcal{T}}$, ${\mathcal{Q}}$ and
$\widehat{{{\mathbbm{1}}}}$ are symmetric real matrices, the results
established in [54, 25] remains valid in the framework of the elastic
materials with micro-voids.

First, in view of Lemma (6.1) and the item i) of Theorem 5.2, if the
constitutive coefficients satisfy the conditions $\mu_{\rm e}>0,\,\lambda_{\rm
e}+2\mu_{\rm e}>0,\,\alpha>0$ and $(\lambda_{\rm e}+2\mu_{\rm
e})\,\xi>\beta^{2}$ and $0\leq v<\widehat{v}$, the matrix problem

$\displaystyle{\mathcal{T}}{\mathcal{E}}^{2}-{\rm
i}\,\,({\mathcal{R}}+{\mathcal{R}}^{T}){\mathcal{E}}-{\mathcal{Q}}+k^{2}v^{2}{{{\mathbbm{1}}}}=0,\qquad\text{\rm
Re\,(spec\,}{\mathcal{E}}{\rm)}>0,$ (7.6)

has a unique solution for ${\mathcal{E}}_{v}$ and the corresponding matrix
${\mathcal{M}}_{v}$ obtained from $\eqref{10}_{2}$ is Hermitian. Hence, we
know that for all $0\leq v<\widehat{v}$ there exists a unique solution
${\mathcal{M}}_{v}$ of the Riccati equation (7.4), defined by the unique
matrix ${\mathcal{E}}_{v}$ indicated in the item i) of Theorem 5.2, i.e., we
can consider the mapping which associates to each $0\leq v<\widehat{v}$ the
Hermitian matrix ${\mathcal{M}}_{v}$ satisfying the equation (7.4). In
addition, from item ii) of Theorem 5.2 we know the full representation of the
unique admissible solution ${\mathcal{M}}_{v}$ of the algebraic Riccati
equation (7.4) to be given by

$\displaystyle{\mathcal{M}}_{v}={H}_{v}^{-1}+{\rm
i}\,\,{H}_{v}^{-1}\,{S}_{v},\qquad\text{with}\qquad{H}_{v}=\frac{1}{\pi}\displaystyle\int_{0}^{\pi}{\mathcal{T}}_{\theta}^{-1}\,d\theta,\qquad{S}_{v}=-\frac{1}{\pi}\int_{0}^{\pi}{\mathcal{T}}_{\theta}^{-1}{\mathcal{R}}_{\theta}^{T}\,d\theta,$
(7.7)

where

$\displaystyle{\mathcal{T}}_{\theta}=\cos^{2}\theta\,{\mathcal{T}}-\sin\theta\cos\theta\,({\mathcal{R}}+{\mathcal{R}}^{T})+\sin^{2}\theta\,\widetilde{{\mathcal{Q}}},\vspace{2mm}$
$\displaystyle{\mathcal{R}}_{\theta}=\cos^{2}\theta\,{\mathcal{R}}+\sin\theta\cos\theta\,({\mathcal{T}}-\widetilde{{\mathcal{Q}}})-\sin^{2}\theta\,{\mathcal{R}}^{T},\vspace{2mm}$
(7.8)
$\displaystyle\widetilde{{\mathcal{Q}}}_{\theta}=\cos^{2}\theta\,\widetilde{{\mathcal{Q}}}+\sin\theta\cos\theta\,({\mathcal{R}}+{\mathcal{R}}^{T})+\sin^{2}\theta\,{\mathcal{T}}$

and $\widetilde{{\mathcal{Q}}}={\mathcal{Q}}-k\,v^{2}\,{{{\mathbbm{1}}}}$,
while $\theta$ is an arbitrary angle. Let us point out that (7.7) allows us to
obtain the explicit form of the secular equation $\det{\mathcal{M}}_{v}=0$
without a priori knowing the analytical expressions (as function of the wave
speed) of the eigenvalues that satisfy (6.12) and the associated eigenvector,
which is the major difficulty in almost all the generalised models when other
methods are used. After having the admissible solution of the secular
equation, we will go back to the task of finding the eigenvalues that satisfy
(6.12) and the associated eigenvector. But, by doing so this task becomes a
purely numerical task, avoiding symbolic (analytical) computations, since all
the involved quantities will be known, as numbers.

Then, after replacing the solution ${\mathcal{M}}_{v}$ of (7.4) into (7.9),
the pair $(v,{{\mathcal{M}}}_{v})$ must also be a solution of the secular
equation (7.5), i.e.,

$\displaystyle{\rm det}\,{\mathcal{M}}_{v}=0.$ (7.9)

The unique unknown is now $v$ and we have to check if the resulting equation
will lead to a unique wave speed belonging to the interval $[0,\widehat{v})$.
This is necessary because under this assumption we have constructed the
mapping $v\mapsto{{\mathcal{M}}}_{v}$ with ${{\mathcal{M}}}_{v}$ admissible.
But, according to item iii) of Theorem 5.2, the matrix ${\mathcal{M}}_{v}$
determined by (7.7) satisfies the conditions

  1. 1.

${\mathcal{M}}_{v}$ is Hermitian,

  2. 2.

$\frac{d{{\mathcal{M}}}_{v}}{dv}$ is negative definite,

  3. 3.

${\rm tr}({{\mathcal{M}}}_{v})\geq 0$, and $\langle
w,{{\mathcal{M}}}_{v}\,w\rangle\geq 0$ for all real vectors $w$ for all $0\leq
v\leq\widehat{v}$,

  4. 4.

${\mathcal{M}}_{v}$ is and positive definite for all $0\leq v<\widehat{v}$.

and all these imply the existence of a unique subsonic solution of the secular
equation. Summarizing, using Theorem 5.2, we have

######  Theorem 7.1.

Assume the constitutive coefficients satisfy the conditions $\mu_{\rm
e}>0,\,\lambda_{\rm e}+2\mu_{\rm e}>0,\,\alpha>0$ and $(\lambda_{\rm
e}+2\mu_{\rm e})\,\xi>\beta^{2}$ and ${\mathcal{M}}_{v}$ is given by
$\eqref{explMielke}$, then the secular equation $\det{{\mathcal{M}}}_{v}=0$
has a unique admissible solution $0\leq v<\widehat{v}$.

The Theorem 7.1 serves as the final checkpoint in our algorithm, essentially
validating the feasibility of our strategy for solving the nonlinear equations
(7.4)1 and (7.5). It conclusively demonstrates that there is indeed an unique
Rayleigh wave propagating within isotropic elastic materials containing micro-
voids, and that this wave is unique.

##  8 Numerical implementation

In this section we follow the theoretical solution strategy given in the
previous sections to give the numerical solution for the material design
properties for structural steel S235, for the following values of the Poisson
ratio and the Young modulus for the macroscopic structure

$\displaystyle\nu_{\rm macro}=0.3,\qquad E_{\rm
macro}=210\,\textrm{(GPa)},\qquad\varrho_{0}=7850\,\textrm{($kg/m^{3}$)},$
(8.1)

and we take the wave number $k=1\,\textrm{($m^{-1}$)}.$ Accordingly, the
considered Lamé coefficient of the macroscopic structure are given by

$\displaystyle\lambda_{\text{macro}}$ $\displaystyle=\frac{E_{\rm
macro}}{2(\nu_{\rm
macro}+1)}=121.154\,\textrm{(GPa)},\qquad\qquad\mu_{\text{macro}}=\frac{\nu_{\rm
macro}E_{\rm macro}}{(\nu_{\rm macro}+1)(1-2\,\nu_{\rm
macro})}=80.7692\,\textrm{(GPa)},$ (8.2) $\displaystyle\kappa_{\text{macro}}$
$\displaystyle=\frac{1}{3}\left(3\,\lambda_{\text{macro}}+2\,\mu_{\text{macro}}\right)=161.538\,\textrm{(GPa)}.$

We assume

$\displaystyle\lambda_{\text{micro}}=:2\,\lambda_{\text{macro}}=242.308\,\textrm{(GPa)},\qquad\mu_{\text{micro}}=:2\,\mu_{\text{macro}}=161.538\,\textrm{(GPa)},$
(8.3)

and we find the following values of the elastic constitutive coefficients

$\displaystyle\mu_{e}$
$\displaystyle=-\frac{\mu_{\text{macro}}\,\mu_{\text{micro}}}{\mu_{\text{macro}}-\mu_{\text{micro}}}=242.308\,\textrm{(GPa)},\qquad\kappa_{e}=-\frac{\kappa_{\text{macro}}\,\kappa_{\text{micro}}}{\kappa\kappa_{\text{macro}}-\kappa_{\text{micro}}}=323.077\,\textrm{(GPa)},$
(8.4) $\displaystyle\lambda_{e}$
$\displaystyle=\frac{1}{3}\left(3\,\kappa_{e}-2\,\mu_{e}\right)=161.538\,\textrm{(GPa)},$

and

$\displaystyle\beta=-3\,\kappa_{e}=-969.231\,\textrm{(GPa)},\qquad\xi=9\left(\kappa_{e}+\kappa_{\text{micro}}\right)=5815.38\,\textrm{(GPa)}.$
(8.5)

We choose the values of $\eta$, $L_{\rm c}$, $a_{2}$ and $\tau_{\rm c}$ such
that

$\displaystyle\varkappa$
$\displaystyle=3\,\eta\,\tau_{c}^{2}=1.5\,\textrm{($m^{2}$)},\quad\quad\quad\quad\alpha=2\,\mu_{\rm
e}L_{c}^{2}a_{2}=10\,\textrm{(GPa $m^{2}$)}.$ (8.6)

For the chosen material, using (6.31) the value of the limiting speed is found
to be $\widehat{v}=0.242876\,(m/s)$. An admissible solution of the secular
equation is a positive wave speed less then $\widehat{v}$. As we have
analytically proven such a solution exists and it is unique. We find the
unique solution of the secular equation using the interpolating process on a
set of 100 values in the interval $[0,\widehat{v})$. We compute numerically
the integrals in (7.7) since we do not have the symbolic values of them. Using
interpolation we find a function $v\mapsto f(v)$ that approximates the
function $v\mapsto\det\mathcal{M}_{v}$, on $[0,\widehat{v})$, see Figure 1.
The root of the approximation function $f$ on $[0,\widehat{v})$ is
$v_{0}=0.153005\,(m/s)$. Our numerical interpolation algorithm of the function
$v\mapsto\det{{\mathcal{M}}}_{v}$ is made on 100 points just to exemplify the
method. One can increase the number of points and use more precision options
in order to increase the precision of the results.

Figure 1: A plot of $\det\mathcal{M}_{v}$ with respect to the surface waves
speed $v$ for a set of 100 equidistant values in the interval
$[0,\widehat{v})$. It is illustrated that $\det\mathcal{M}_{v}$ is a
decreasing function of the wave speed $v$.

Having the wave speed $v_{0}$, the approximation of the solution of the
secular equation we find $y(0)$ as solution of the algebraic Riccati equation
(7.4) in the form

$\displaystyle y(0)=\left(\begin{array}[]{c}y_{1}\\\ -1.263037839859903454{\rm
i}\,y_{1}\\\ 0.20925703711388041\,y_{1}\\\ \end{array}\right),\qquad
y_{1}\in\mathbb{C}.$ (8.10)

We also have computed $\mathcal{M}_{v_{0}}$ to be

$\displaystyle\mathcal{M}_{v_{0}}=\left(\begin{array}[]{ccc}0.030152425181168009480&-0.023503969977247586063{\rm
i}&-0.0022270300650370918782\\\ 0.023503969977247580765\,{\rm
i}&0.018790427909898263031&0.0010945940135284871257\,{\rm i}\\\
-0.0022270300650370918782&-0.0010945940135284891285\,{\rm
i}&0.017249300922842580276\end{array}\right)$ (8.14)

(its determinant $\det\mathcal{M}_{v_{0}}=-4.135543508749075155*10^{-13}$ is
approximately zero as we expected from Theorem 7.1) and the matrix
${\mathcal{E}_{v_{0}}}={\mathcal{T}}^{-1}({\mathcal{M}_{v_{0}}}+{\rm
i}\,\,{\mathcal{R}}^{T})$ is numerically approximated by

$\displaystyle{\mathcal{E}_{v_{0}}}=\left(\begin{array}[]{ccc}0.976843&0.238546\,{\rm
i}&-0.0721487\\\ 0.535545\,{\rm i}&0.228281&-1.21145\,{\rm i}\\\
-2.62233&-1.28888\,{\rm i}&20.3111\\\ \end{array}\right).$ (8.18)

Having the matrix ${\mathcal{E}_{v_{0}}}$, using (7.1) we are able to find the
function $y(x_{2})$. In this way, going back further with the changes of
variables $y(x_{2}):=\widehat{{{\mathbbm{1}}}}^{1/2}\,z(x_{2})$ we find
$z(x_{2})$ and one step back, using (6.1), we find the solution
$\mathcal{U}(x_{1},x_{2},t)$, see Figure 2.

(a) Plot of the $u_{1}$-component of the displacement.

(b) Plot of the $u_{2}$-component of the displacement.

(c) Plot of the volume fraction field $\zeta$ from
$P=\zeta\\!\\!\cdot\\!\\!{{\mathbbm{1}}}$.

Figure 2: The plot of the solution at time $t=1$ and for the choice
$y_{1}=-{\rm i}\,$.

$R(c)$function from [8]our approach

Figure 3: The plots of the approximation of $v\mapsto\det{{\mathcal{M}}}_{v}$
(blue curve) and of $v\mapsto R(v)10^{-8}$ (red curve), where $R(v)$ from
(8.19) defines the secular equations from [8], with respect to the surface
waves speed $v$ for the material we considered. These two different functions
have the same (unique) root in the interval $[0,\widehat{v})$. We have
multiplied $R(v)$ with $10^{-8}$ to have the same range plot interval of the
values for both functions.

In [8] the secular equation has the form $R(c)=0$, where $c$ is the wave speed
and

$\displaystyle R(v)\equiv$
$\displaystyle\displaystyle\left[\left(2-\frac{v^{2}}{v_{t}^{2}}\right)^{2}-4\sqrt{\left(1-\frac{v^{2}}{v_{t}^{2}}\right)\left(1-\frac{X_{1}^{2}}{k^{2}}\right)}\
\right]\sqrt{1-\frac{X_{2}^{2}}{k^{2}}}\left(-\frac{v^{2}}{v_{s}^{2}}+\frac{X_{2}^{2}}{k^{2}}\right)\vspace{1mm}$
$\displaystyle-\displaystyle\left[\left(2-\frac{v^{2}}{v_{t}^{2}}\right)^{2}-4\sqrt{\left(1-\frac{v^{2}}{v_{t}^{2}}\right)\left(1-\frac{X_{2}^{2}}{\kappa^{2}}\right)}\
\right]\sqrt{1-\frac{X_{1}^{2}}{k^{2}}}\left(-\frac{v^{2}}{v_{s}^{2}}+\frac{X_{1}^{2}}{k^{2}}\right),$
(8.19)

where

$\displaystyle X_{1,2}^{2}=X_{1,2}^{2}(\omega^{2})=$
$\displaystyle\displaystyle\frac{1}{2\alpha(\lambda_{\rm e}+2\mu_{\rm
e})}\Bigg{\\{}{\beta}^{2}-(\xi-{\varrho_{0}}\varkappa\omega^{2})(\lambda_{\rm
e}+2\mu_{\rm e})+\alpha\omega^{2}{\varrho_{0}}$
$\displaystyle\mp\sqrt{4\alpha\omega^{2}{\beta}^{2}{\varrho_{0}}+[-{\beta}^{2}+\alpha\omega^{2}{\varrho_{0}}+(\xi-{\varrho_{0}}\varkappa\omega^{2})(\lambda_{\rm
e}+2\mu_{\rm e})]^{2}}\Bigg{\\}}.$ (8.20)

In [8] the uniqueness of the solution was not demonstrated and the existence
of the solution of the secular equation was proven assuming the conditions
(1.2). These conditions are more restrictive in comparison with the
assumptions (6.29). Our analysis is valid for all materials admitting planar
real wave propagation and do not enter into conflict with the necessary
assumptions for classical linear elasticity. With less restrictive conditions
on the constitutive parameters (connected with conditions usually imposed by
the engineering community) we have proven that the solution exists and it is
unique. More than that the algorithm for computing the Rayleigh wave solution
presented in this paper is clear and simple without knowing in advance the
analytical expressions of some eigenvalues and their associated eigenvectors
(the main difficulty to obtain a complete algorithm and the proof of existence
and uniqueness in the other methods).

For the coefficients considered in this paper, the conditions (1.2) from [8]
are not verified
(${c}_{t}^{2}\leq{c}_{s}^{2}-\displaystyle\frac{{\beta}^{2}}{{\varrho_{0}}\xi}$
is true, but ${c}_{t}^{2}\leq\frac{\alpha}{{\varrho_{0}}\varkappa}$ is not
valid). However, as one can see in Figure 3 (the red curve) the secular
equation from [8] is well defined and it still has a unique solution in the
interval $[0,\widehat{v})$. However, for the values of the constitutive
coefficients considered, the existence and uniqueness cannot be guaranteed by
the results established in [8]. From Figure 3 we can see that the same value
of the propagation speed is obtained, from both forms of the secular equation
(our vs. [8]). This represents another check that the algorithm proposed by us
is viable and the computations are correct.

The same value $v_{0}=0.113175\,(m/s)$ is obtained as limit case in our
analysis of the micro-voids model for structural steel S235 and large values
of the micro-constitutive parameter $\mu_{\rm micro}$ (i.e., a large $\xi$),
e.g.,
$\mu_{\text{micro}}=10^{7}\,\mu_{\text{macro}}=10^{7}*161.538\,\textrm{(GPa)},$
which indicates the convergence of the Rayleigh wave solution from the micro-
void model to the Rayleigh wave solution from classical elasticity when
$\mu_{\rm micro}\to\infty$.

## References

  * [1] J.D. Achenbach.  Wave Propagation in Elastic Solids.  North-Holland Publishing Company, Amsterdam, 1973.
  * [2] S. Bauer, P. Neff, D. Pauly, and G. Starke.  New Poincaré-type inequalities.  Compte Rendus Acad. Sci. Paris, Ser. Math., 352(2):163–166, 2014\.
  * [3] S. Bauer, P. Neff, D. Pauly, and G. Starke.  Dev-Div and DevSym-DevCurl inequalities for incompatible square tensor fields with mixed boundary conditions.  ESAIM: Control, Optimisation and Calculus of Variations, 22(1):112–133, 2016.
  * [4] Ph. Boulanger and M. Hayes.  Bivectors and Waves in Mechanics and Optics, volume 4.  CRC Press, 1993.
  * [5] A. Bucur.  Rayleigh surface waves problem in linear thermoviscoelasticity with voids.  Acta Mechanica 227.4: 1199-121, 2016.
  * [6] B. Brandel and R.S. Lakes.  Negative Poisson’s ratio polyethylene foams.  Journal of Materials Science, 36:5885–5893, 2001.
  * [7] S. Burns.  Negative Poisson’s ratio materials.  Science, 238(4826):551–551, 1987.
  * [8] S. Chiriţă and I.D. Ghiba.  Inhomogeneous plane waves in elastic materials with voids.  Wave Motion, 47:333–342, 2010.
  * [9] S. Chiriţă and I.D. Ghiba.  Strong ellipticity and progressive waves in elastic materials with voids.  Proceedings of the Royal Society A: Mathematical, Physical, 466:439–458, 2010.
  * [10] S. Chiriţă.  Thermoelastic surface waves on an exponentially graded half-space.  Mechanics Research Communications 49: 27-35, 2013.
  * [11] S. Chiriţă and A. Danescu.  Surface waves problem in a thermoviscoelastic porous half-space.  Wave Motion, 54: 100-114, 2015.
  * [12] S. Chiriţă and A. Arusoaie.  Thermoelastic waves in double porosity materials.  European Journal of Mechanics-A/Solids, 86: 104177, 2021.
  * [13] W.D. Claus and A.C. Eringen.  Three dislocation concepts and micromorphic mechanics.  In Developments in Mechanics, Proceedings of the 12th Midwestern Mechanics Conference, volume 6, pages 349–358. Midwestern, 1969\.
  * [14] W.D. Claus and A.C. Eringen.  Dislocation dispersion of elastic waves.  International Journal of Engineering Science, 9:605–610, 1971.
  * [15] S.C. Cowin and J.W. Nunziato.  Linear elastic materials with voids.  Journal of Elasticity, 13:125–147, 1983.
  * [16] P.K. Currie.  Rayleigh waves on elastic crystals.  The Quarterly Journal of Mechanics and Applied Mathematics, 27(4):489–496, 1974.
  * [17] F. Demore, G. Rizzi, M. Collet, P. Neff, and A. Madeo.  Unfolding engineering metamaterials design: Relaxed micromorphic modeling of large-scale acoustic meta-structures.  Journal of the Mechanics and Physics of Solids, 168:104995, 2022\.
  * [18] M. Destrade.  The explicit secular equation for surface acoustic waves in monoclinic elastic crystals.  The Journal of the Acoustical Society of America, 109(4):1398–1402, 2001.
  * [19] M. Destrade.  Seismic Rayleigh waves on an exponentially graded, orthotropic half-space.  Proceedings of the Royal Society A: Mathematical, Physical, 463(2078):495–502, 2007.
  * [20] M. Destrade, P. Martin, and T.C.T. Ting.  The incompressible limit in linear anisotropic elasticity, with applications to surface waves and elastostatics.  Journal of the Mechanics and Physics of Solids, 50(7):1453–1468, 2002.
  * [21] A.C. Eringen.  Microcontinuum Field Theories. Springer, Heidelberg, 1999.
  * [22] A.C. Eringen and W.D. Claus.  A micromorphic approach to dislocation theory and its relation to several existing theories.  In J.A. Simmons, R. de Wit, and R. Bullough, editors, Fundamental Aspects of Dislocation Theory., volume 1 of Nat. Bur. Stand. (U.S.), Spec. Publ., pages 1023–1040. Spec. Publ., 1970.
  * [23] K.E. Evans and B.D. Caddock.  Microporous materials with negative Poisson’s ratios. II. Mechanisms and interpretation.  Journal of Physics D: Applied Physics, 22(12):1883, 1989.
  * [24] E.A. Friis, R.S. Lakes, and J.B. Park.  Negative Poisson’s ratio polymeric and metallic foams.  Journal of Materials Science, 23:4406–4414, 1988.
  * [25] Y.B. Fu and A. Mielke.  A new identity for the surface–impedance matrix and its application to the determination of surface-wave speeds.  Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 458 (2026):2523–2543, 2002.
  * [26] Y.B. Fu and R.W. Ogden.  Nonlinear Elasticity: Theory and Applications, volume 281\.  Cambridge University Press, 2001.
  * [27] C. Galeş and S. Chiriţă.  Wave propagation in materials with double porosity.  Mechanics of Materials, 149: 103558, 2020.
  * [28] I. D. Ghiba.  On the temporal behaviour in the bending theory of porous thermoelastic plates.  Zeitschrift für Angewandte Mathematik und Mechanik, 93:284–296, 2013.
  * [29] I.D. Ghiba, P. Neff, A. Madeo, L. Placidi, and G. Rosi.  The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics.  Mathematics and Mechanics of Solids, 20:1171–1197, 2015.
  * [30] F. Gmeineder, P. Lewintan, and P. Neff.  Korn-Maxwell-Sobolev inequalities for general incompatibilities.  arXiv preprint arXiv:2212.13227, 2022.
  * [31] F. Gmeineder, P. Lewintan, and P. Neff.  Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions.  Calculus of Variations and Partial Differential Equations, 62(6):1–33, 2023.
  * [32] M.A. Goodman and S.C. Cowin.  A continuum theory for granular materials.  Archive for Rational Mechanics and Analysis, 44(4):249–266, 1972\.
  * [33] J.N. Grima, R. Gatt, N. Ravirala, A. Alderson, and K.E. Evans.  Negative Poisson’s ratios in cellular foam materials.  Materials Science and Engineering: A, 423(1-2):214–218, 2006.
  * [34] H. Hayes and R. Rivlin.  A note on the secular equation for Rayleigh waves.  Zeitschrift für angewandte Mathematik und Physik, 13:80–83, 1962.
  * [35] M. Hayes.  Inhomogeneous plane waves.  In The Breadth and Depth of Continuum Mechanics, pages 247–285. Springer, 1986.
  * [36] M. Hayes and R.S. Rivlin.  A note on the secular equation for Rayleigh waves.  Zeitschrift für angewandte Mathematik und Physik, 13(1):80–83, 1962.
  * [37] D. Ieşan.  A theory of thermoelastic materials with voids.  Acta Mechanica, 60:67–89, 1986.
  * [38] D. Ieşan and M. Ciarletta.  Non-Classical Elastic Solids.  Longman Scientific and Technical, Harlow, Essex, UK, Inc., New York, 1993.
  * [39] J.W. Jiang and H.S. Park.  Negative Poisson’ ratio in single-layer black phosphorus.  Nature Communications, 5(1):4727, 2014.
  * [40] H. Khan, I.D. Ghiba, A. Madeo, and P. Neff.  Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects.  Wave Motion, 110:102898, 2022.
  * [41] H.W. Knobloch, A. Isidori, and D. Flockerzi.  Topics in Control Theory, volume 22.  Birkhäuser, 2012.
  * [42] R.S. Lakes.  Foam structures with a negative Poisson’s ratio.  Science, 235(4792):1038–1040, 1987.
  * [43] R.S. Lakes.  Advances in negative Poisson’s ratio materials.  Advanced Materials, 5(4):293–296, 1993.
  * [44] R.S. Lakes, T. Lee, A. Bersie, and Y.C. Wang.  Extreme damping in composite materials with negative-stiffness inclusions.  Nature, 410(6828):565–567, 2001.
  * [45] J. Lankeit, P. Neff, and D. Pauly.  $\text{Uniqueness of integrable solutions to }{\bigtriangledown\zeta}={G}{\,\zeta,\ \zeta|_{\Gamma}=0}\text{ for integrable tensor coefficients }{G}$ and applications to elasticity.  Zeitschrift für Angewandte Mathematik und Physik, 64:1679–1688, 2013.
  * [46] M. Lazar and H. Kirchner.  Cosserat (micropolar) elasticity in Stroh form.  International Journal of Solids and Structures, 42(20):5377–5398, 2005.
  * [47] P. Lewintan, S. Müller, and P. Neff.  Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy.  Calculus of Variations and Partial Differential Equations, 60:1–46, 2021.
  * [48] P. Lewintan and P. Neff.  Nečas–Lions lemma revisited: An $L^{p}$-version of the generalized Korn inequality for incompatible tensor fields.  Mathematical Methods in the Applied Sciences, 44(14):11392–11403, 2021.
  * [49] X.-F. Li.  On approximate analytic expressions for the velocity of Rayleigh waves.  Wave Motion, 44:120–127, 2006.
  * [50] A. Madeo, P. Neff, I. D. Ghiba, L. Placidi, and G. Rosi.  Band gaps in the relaxed linear micromorphic continuum.  Zeitschrift für Angewandte Mathematik und Mechanik, 95(9):880–887, 2015.
  * [51] A. Madeo, P. Neff, I. D. Ghiba, L. Placidi, and G. Rosi.  Wave propagation in relaxed linear micromorphic continua: modelling metamaterials with frequency band-gaps.  Continuum Mechanics and Thermodynamics, 27:551–570, 2015.
  * [52] A. Madeo, P. Neff, I.D. Ghiba, and G. Rosi.  Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model.  Journal of the Mechanics and Physics of Solids, 95:441–479, 2016\.
  * [53] P.G. Malischewsky.  Comment to “A new formula for velocity of Rayleigh waves” by D. Nkemzi [Wave Motion 26 (1997) 199-205].  Wave Motion, 31:93–96, 2000.
  * [54] A. Mielke and Y.B. Fu.  Uniqueness of the surface-wave speed: a proof that is independent of the Stroh formalism.  Mathematics and Mechanics of Solids, 9(1):5–15, 2004.
  * [55] A. Mielke and P. Sprenger.  Quasiconvexity at the boundary and a simple variational formulation of Agmon’s condition.  Journal of Elasticity, 51(1):23–41, 1998.
  * [56] R.D. Mindlin.  Micro-structure in linear elasticity.  Archive for Rational Mechanics and Analysis, 16:51–77, 1964.
  * [57] V.G. Mozhaev.  Some new ideas in the theory of surface acoustic waves in anisotropic media.  In IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, pages 455–462, 1995.
  * [58] P. Neff, A. Madeo, G. Barbagallo, M.V. d’Agostino, R. Abreu, and I.D. Ghiba.  Real wave propagation in the isotropic-relaxed micromorphic model.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2197):20160790, 2017.
  * [59] P. Neff, I. D. Ghiba, M. Lazar, and A. Madeo.  The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations.  The Quarterly Journal of Mechanics and Applied Mathematics, 68:53–84, 2015.
  * [60] P. Neff, I. D. Ghiba, A. Madeo, L. Placidi, and G. Rosi.  A unifying perspective: the relaxed linear micromorphic continuum.  Continuum Mechanics and Thermodynamics, 26:639–681, 2014.
  * [61] P. Neff, D. Pauly, and K.J. Witsch.  A canonical extension of Korn’s first inequality to ${\rm H(Curl)}$ motivated by gradient plasticity with plastic spin.  Compte Rendus Acad. Sci. Paris, Ser. Math., 349:1251–1254, 2011.
  * [62] P. Neff, D. Pauly, and K.J. Witsch.  Maxwell meets Korn: a new coercive inequality for tensor fields in $\mathbb{R}^{N\times N}$ with square-integrable exterior derivative.  Mathematical Methods in the Applied Sciences, 35:65–71, 2012.
  * [63] P. Neff, D. Pauly, and K.J. Witsch.  Poincaré meets Korn via Maxwell: Extending Korn’s first inequality to incompatible tensor fields.  Journal of Differential Equations, 258:1267–1302, 2015.
  * [64] D. Nkemzi.  A new formula for the velocity of Rayleigh waves.  Wave Motion, 26:199–205, 1997.
  * [65] D. Nkemzi.  A simple and explicit algebraic expression for the Rayleigh wave velocity.  Mechanics Research Communications, 35:201–205, 2008.
  * [66] J.W. Nunziato and S.C. Cowin.  A nonlinear theory of elastic materials with voids.  Archive for Rational Mechanics and Analysis, 72:175–201, 1979.
  * [67] P. Puri and S.C. Cowin.  Plane waves in linear elastic materials with voids.  Journal of Elasticity, 15(2):167–183, 1985.
  * [68] M. Rahman and J.R. Barber.  Exact expression for the roots of the secular equation for Rayleigh waves.  ASME Journal of Applied Mechanics, 62:250–252, 1995.
  * [69] Lord Rayleigh.  On waves propagated along the plane surface of an elastic solid.  Proceedings of the London Mathematical Society, 17:4–11, 1885.
  * [70] G. Rizzi, M.V. d’Agostino, P. Neff, and A. Madeo.  Boundary and interface conditions in the relaxed micromorphic model: Exploring finite-size metastructures for elastic wave control.  Mathematics and Mechanics of Solids, 27(6):1053–1068, 2022.
  * [71] G. Rizzi, P. Neff, and A. Madeo.  Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach.  Philosophical Transactions of the Royal Society A, 380(2231):20210400, 2022.
  * [72] G. Rizzi, D. Tallarico, P. Neff, and A. Madeo.  Towards the conception of complex engineering meta-structures: Relaxed-micromorphic modelling of low-frequency mechanical diodes/high-frequency screens.  Wave Motion, 113:102920, 2022.
  * [73] G. Rizzi, J. Voss, P. Neff, and A. Madeo.  Modeling a labyrinthine acoustic metamaterial through an inertia-augmented relaxed micromorphic approach.  Mathematics and Mechanics of Solids, 28 (10):2177-2201, 2023.
  * [74] G. Rizzi, L.A. Perez Ramirez and A. Madeo.  Multi-element metamaterials design through the relaxed micromorphic model.  In Sixty Shades of Generalized Continua: Dedicated to the 60th Birthday of Prof. Victor A. Eremeyev, pages 579–600. Springer, 2023.
  * [75] L. Rothenburg, A.I. Berlin, and R.J. Bathurst.  Microstructure of isotropic materials with negative Poisson’s ratio.  Nature, 354(6353):470–472, 1991.
  * [76] A.N. Stroh.  Dislocations and cracks in anisotropic elasticity.  Philosophical Magazine, 3(30):625–646, 1958.
  * [77] A.N. Stroh.  Steady state problems in anisotropic elasticity.  Journal of Mathematical Physics, 41(1-4):77–103, 1962.
  * [78] J.L. Synge.  Elastic waves in anisotropic media.  Journal of Mathematical Physics, 35(1-4):323–334, 1956.
  * [79] R.M. Taziev.  Dispersion relation for acoustic waves in an anisotropic elastic half-space.  Akustičeskij Žurnal, 35(5):922–928, 1989.
  * [80] T.C.T. Ting.  An explicit secular equation for surface waves in an elastic material of general anisotropy.  The Quarterly Journal of Mechanics and Applied Mathematics, 55(2):297–311, 2002.
  * [81] T.C.T. Ting.  Secular equations for Rayleigh and Stoneley waves in exponentially graded elastic materials of general anisotropy under the influence of gravity.  Journal of Elasticity, 105:331–347, 2011.
  * [82] T.C.T. Ting.  Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity.  Wave Motion, 48:335–344, 2011.
  * [83] P.C. Vinh and P.G. Malischewsky.  An approach for obtaining approximate formulas for the Rayleigh wave velocity.  Wave Motion, 44:549–562, 2007.
  * [84] P.C. Vinh and P.G. Malischewsky.  Improved approximations of the Rayleigh wave velocity.  Journal of Thermoplastic Composite Materials, 21:337–352, 2008.
  * [85] P.C. Vinh and R.W. Ogden.  On formulas for the Rayleigh wave speed.  Wave Motion, 39:191–197, 2004.
  * [86] Y. Yang, D. Cormier, H. West, O. Harrysson, and K. Knowlson.  Non-stochastic ti–6al–4v foam structures with negative Poisson’s ratio.  Materials Science and Engineering: A, 558:579–585, 2012.