PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of procaThe Royal Society PublishingProceedings AAboutBrowse by SubjectAlertsFree Trial
 
Proc Math Phys Eng Sci. 2017 January; 473(2197): 20160790.
PMCID: PMC5312136

Real wave propagation in the isotropic-relaxed micromorphic model

Abstract

For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.

Keywords: ellipticity, positive definiteness, real wave velocity, rank-one convexity, acoustic tensor, relaxed micromorphic model

1. Introduction

Investigations of real wave propagation and ellipticity, in principle, are not new. Indeed, it is textbook knowledge for linear elasticity that positive definiteness of the elastic energy implies real wave velocities (phase velocities) v=ω/k, where ω[rads1] is the angular frequency and k[radm1]R is the wavenumber of planar propagating waves. In classical elasticity, having real wave velocities is equivalent to rank-one convexity (strong ellipticity or Legendre–Hadamard ellipticity). Moreover, ellipticity is equivalent to the positive definiteness of the acoustic tensor. For anisotropic linear elasticity, see [1], whereas for anisotropic nonlinear elasticity we refer the reader to [25].

The same question of ellipticity and real wave velocities in generalized continuum mechanics has been discussed for micropolar models, e.g. in [6] and for elastic materials with voids in [7]. For the isotropic micromorphic model, results can be found with respect to positive-definite energy and/or real wave velocity in Nowacki [8], Smith [9], Mindlin [10, 11] and Eringen [12, pp. 277–280]. These latter results present conditions which are neither easily verifiable nor are truly transparent. This is due to the very high number of material coefficients of the Eringen–Mindlin theory that are strongly reduced in the relaxed micromorphic model [13]. Indeed, the implication that positive definiteness of the energy always implies real wave velocities is not directly established and demonstrated. In this paper, we investigate the relaxed micromorphic model in terms of conditions for real wave velocities for planar waves and establish a necessary and sufficient conditions for this to happen.

This paper is organized as follows. We shortly recall the basics of the relaxed micromorphic model and discuss the wave propagation problem for propagating planar waves. Because we deal with an isotropic model, we can, without loss of generality, assume wave propagation in one specific direction only. The dispersion relations are then obtained, and real wave velocities, under the assumption of uniform positiveness of the elastic energy, are established.

We next present a set of necessary and sufficient conditions for real wave velocities in the relaxed micromorphic model which is weaker than the positivity of the energy, as is the strong ellipticity condition with respect to positive definiteness of the energy in the case of linear elasticity. Then, for didactic purposes, we repeat the analysis for isotropic linear elasticity in order to see relations of our necessary and sufficient condition to the strong ellipticity condition in linear elasticity. Similarly, we discuss micropolar elasticity and establish the necessary and sufficient conditions for real wave propagation. We finally show that strong ellipticity in micropolar and micromorphic models is not sufficient for having real wave velocities, when dealing with plane waves.

2. The relaxed micromorphic model

The relaxed micromorphic model has been recently introduced into continuum mechanics in [14]. In subsequent works [1518], the model has shown its wider applicability compared with the classical Mindlin–Eringen micromorphic model in diverse areas [1012, 19].

The dynamic relaxed micromorphic model counts only eight constitutive parameters in the (simplified) isotropic case, namely five elastic moduli μe, λe, μmicro, λmicro, μc [Pa], one characteristic length Lc [m], the average macroscopic inertia ρ [kg] and the microinertia η [kg m−1]. The simplification consists of assuming one scalar microinertia parameter η and a uniconstant curvature expression. The characteristic length, Lc, is intrinsically related to non-local effects due to the fact that it weights a suitable combination of first-order space derivatives of the microdistortion tensor in the strain energy density (2.1). For a general presentation of the features of the relaxed micromorphic model in the anisotropic setting, we refer the reader to [20].

(a) Elastic energy density

The relaxed micromorphic model couples the macroscopic displacement uR3, and an affine substructure deformation attached at each macroscopic point is encoded by the microdistortion field PR3×3. Our novel relaxed micromorphic model endows the Mindlin–Eringen representation of linear micromorphic models with the second-order dislocation density tensor α=CurlP instead of the full gradient P.1 In the isotropic hyperelastic case, the elastic energy density reads

W=μesym(uP)2+λe2(tr(uP))2+μcskew(uP)2+μmicrosymP2+λmicro2(trP)2+μeLc22CurlP2=μedevsym(uP)2+2μe+3λe3(tr(uP))2isotropicelasticenergy+μcskew(uP)2rotationalelasticcoupling+μmicrodevsymP2+2μmicro+3λmicro3(trP)2microselfenergy+μeLc22CurlP2simplifiedisotropiccurvature,
2.1

where the parameters and the elastic stress are analogous to the standard Mindlin–Eringen micromorphic model. The model is well posed in the statical and dynamic case even for zero Cosserat couple modulus μc=0; see [21, 22]. In that case, it is non-redundant in the sense of [23]. Well-posedness results for the statical and dynamic cases have been provided in [14], making decisive use of recently established new coercive inequalities, generalizing Korn's inequality to incompatible tensor fields [2428].

Decisive for the relaxed micromorphic formulation is the definition of the elastic energy in terms of suitable strain tensors. Because u is the macroscopic displacement gradient and P is the microdistortion, it appears possible to use the non-symmetric relative (elastic) strain tensor uP as the basic building block in the energy. Using the Cartan–Lie orthogonal decomposition, we may introduce

μedevsym(uP)2+2μe+3λe3(tr(uP))2+μcskew(uP)2.
2.2

The microstructure contribution based on P alone is restricted, by infinitesimal frame indifference, to

μmicrodevsymP2+2μmicro+3λmicro3(trP)2+μeLc22CurlP2.
2.3

Strict positive definiteness of the potential energy is equivalent to the following simple relations for the introduced parameters [14]:

μe>0,μc>0,2μe+3λe>0,μmicro>0,2μmicro+3λmicro>0andLc>0.
2.4

As for the kinetic energy density, we consider that it takes the following (simplified) form:

J=ρ2u,t2+η2P,t2,simplifiedmicroinertia
2.5

where ρ>0 is the value of the averaged macroscopic mass density of the considered material, whereas η>0 is its microinertia density.

For very large sample sizes, a scaling argument shows easily that the relative characteristic length scale Lc of the micromorphic model must vanish. Therefore, we have a way of comparing a classical first-gradient formulation with the relaxed micromorphic model and to offer an a priori relation between the microscopic parameters λe,λmicro,μe,μmicro, on the one side, and the resulting macroscopic parameters λmacro,μmacro, on the other side [20, 29, 30]. We have

(2μmacro+3λmacro)=(2μe+3λe)(2μmicro+3λmicro)(2μe+3λe)+(2μmicro+3λmicro)andμmacro=μeμmicroμe+μmicro,
2.6

where μmacro,λmacro are the moduli obtained for Lc0.

For future use, we define the elastic bulk modulus κe, the microscopic bulk modulus κmicro and the macroscopic bulk modulus κmacro, respectively,

κe=2μe+3λe3,κmicro=2μmicro+3λmicro3andκmacro=2μmacro+3λmacro3.
2.7

In terms of these moduli, strict positive definiteness of the energy is equivalent to

μe>0,μc>0,κe>0,μmicro>0,κmicro>0,Lc>0.
2.8

If strict positive definiteness (2.8) holds, we can write the macroscopic consistency conditions as

κmacro=κeκmicroκe+κmicroandμmacro=μeμmicroμe+μmicro,
2.9

and, again under condition (2.8),

κe=κmicroκmacroκmicroκmacro,κmicro=κeκmacroκeκmacro,μe=μmicroμmacroμmicroμmacroandμmicro=μeμmacroμeμmacro.}
2.10

Here, strict positivity (2.8) implies that

κe+κmicro>0,μe+μmicro>0,κe>κmacro,κmicro>κmacroandμe>μmacro,μmicro>μmacro.}
2.11

Because it is useful in what follows, we explicitly remark that

2μe+λe=43μe+2μe+3λe3=43μe+κe=4μe+3κe3and2μmicro+λmicro=4μmicro+3κmicro3.}
2.12

With these relations, it is easy to show how μe>0 and κe>0 imply 2μe+λe>0. Moreover, as shown in appendix A (equations (A 2) and (A 3)), we note here that if only μe+μmicro>0 and κe+κmicro>0, then the macroscopic parameters are less than or equal to respective microscopic parameters, namely

κeκmacro,κmicroκmacro,μeμmacroandμmicroμmacro,
2.13

and, moreover, the following inequalities are satisfied:

2μe+λe2μmacro+λmacro,2μmicro+λmicro2μmacro+λmacroand4μmacro+3κe32μmacro+λmacro.}
2.14

Note that the Cosserat couple modulus μc [31] does not appear in the introduced scale between micro and macro.

(b) Dynamic formulation

The dynamic formulation is obtained by defining a joint Hamiltonian and assuming stationary action. The dynamic equilibrium equations are

ρu,tt=Div[2μesym(uP)+2μcskew(uP)+λetr(uP)𝟙]andηP,tt=μeLc2CurlCurlP+2μesym(uP)+2μcskew(uP)+λetr(uP)𝟙[2μmicrosymP+λmicrotr(P)𝟙].}
2.15

We note here that the presence of the Curl P in the energy generates a non-local term Curl Curl P in the equation of motion, whereas the possibility of band gaps is still present; see [15]. The presence of the Curl P term is essential to simultaneously allow us to describe the non-localities and band gap in an enriched continuum mechanics framework.

Sufficiently far from a source, dynamic wave solutions may be treated as planar waves. Therefore, we now want to study harmonic solutions travelling in an infinite domain for the differential system (2.15). To do so, we define

PS:=13tr(P),P[ij]:=(skewP)ij=12(PijPji),PD:=P11PS,P(ij):=(symP)ij=12(Pij+Pji)andPV:=P22P33}
2.16

and we introduce the unknown vectors

v1=(u1,PD,PS)longitudinal,vτ=(uτ,P(1τ),P[1τ]),transversalτ=2,3andv4=(P(23),P[23],PV).uncoupled
2.17

The definition of the unknown vectors was made considering the coupling of the variables in the equations of motion; see [1518, 3236]. More particularly, it has been shown in these previous works that three sets of equations can be isolated: one involving only longitudinal quantities, one involving only transverse quantities and one of three completely uncoupled equations. We suppose that the space dependences of all introduced kinematic fields are limited to a direction defined by a unit vector ξ~R3, which is the direction of propagation of the wave and which is assumed given. Hence, we look for solutions of (2.15) in the form

v1=β1ei(kξ~,xR3ωt),vτ=βτei(kξ~,xR3ωt),τ=2,3andv4=β4ei(kξ~,xR3ωt),
2.18

where β1=(β11,β21,β31)TC3, βτ=(β1τ,β2τ,β3τ)TC3 and β4=(β14,β24,β34)TC3 are the unknown amplitudes of the considered waves, C3 is the space of complex constant three-dimensional vectors,2 k is the wavenumber and ω is the wave frequency. Because our formulation is isotropic, we can, without loss of generality, specify the propagation direction ξ~=e1. Then, X=e1,xR3=x1, and we obtain that the space dependences of all introduced kinematic fields are limited to the component X, which is now the direction of propagation of the wave.3 This means that we look for solutions in the form

v1=β1ei(kXωt),vτ=βτei(kXωt),τ=2,3andv4=β4ei(kXωt).
2.19

Replacing these expressions in equations (2.15), it is possible to express the system (see [15, 16]) as

A1β1=0,Aτβτ=0,τ=2,3andA4β4=0,
2.20

with

A1(ω,k)=(ω2+cp2k2ik2μeρik(2μe+3λe)ρik43μeηω2+13k2cm2+ωs223k2cm213ik(2μe+3λe)η13k2cm2ω2+23k2cm2+ωp2),
2.21

A2(ω,k)=A3(ω,k)=(ω2+k2cs2ik2μeρikηρωr2ikμeηω2+cm22k2+ωs2cm22k2i2ωr2kcm22k2ω2+cm22k2+ωr2)
2.22

and

A4(ω,k)=(ω2+cm2k2+ωs2000ω2+cm2k2+ωr2000ω2+cm2k2+ωs2).
2.23

Here, we have defined

cm=μeLc2η,cs=μe+μcρ,cp=2μe+λeρ,ωs=2(μe+μmicro)η,ωp=(2μe+3λe)+(2μmicro+3λmicro)η,ωr=2μcη,ωl=2μmicro+λmicroη,ωt=μmicroη.

Let us next define the diagonal matrix

diag1=(ρ000i6η2000i3η).
2.24

Considering γ=diag1β and the matrix A¯1(ω,k)=diag1A1(ω,k)diag11, it is possible to formulate the problem (2.20) equivalently as4

A¯1γ=(ω2+cp2k2263kμeρη33k(2μe+3λe)ρη263kμeρηω2+13k2cm2+ωs223k2cm233k(2μe+3λe)ρη23k2cm2ω2+23k2cm2+ωp2)(γ1γ2γ3)=0.
2.25

Analogously considering

diag2=(ρ00i2η00i2η),
2.26

it is possible to obtain A¯2(ω,k)=A¯3(ω,k)=diag2A2(ω,k)diag21

A¯2(ω,k)=A¯3(ω,k)=(ω2+k2cs2k2μeρηk2μcρηk2μeρηω2+cm22k2+ωs2cm22k2k2μcρηcm22k2ω2+cm22k2+ωr2).
2.27

In order to have non-trivial solutions of the algebraic systems (2.20), one must impose that

detA¯1(ω,k)=0,detA¯2(ω,k)=detA¯3(ω,k)=0anddetA4(ω,k)=0,
2.28

the solution of which allows us to determine the so-called dispersion relations ω=ω(k) for the longitudinal and transverse waves in the relaxed micromorphic continuum (figure 1).5 The solutions of the eigenvalue problem obtained via the proposed decomposition are the same as the ones obtained via the standard formulation shown in appendix Aa with the full 12×12 matrix; for more details, see [34]. For estimates on the isotropic moduli, we refer the reader to [17, 33] and, for a comparison with other micromorphic models, to [32, 36]. For solutions ω=ω(k) of (2.28), we define

phase velocity:v=ωkandgroup velocity: dω(k)dk.
2.29

Real wavenumbers kR correspond to propagating waves, whereas complex values of k are associated with waves whose amplitude either grows or decays along the coordinate X. In linear elasticity, phase velocity and group velocity coincide, because there is no dispersion, and both are real; see section 3.

Because, in this paper, we are interested only in real k (outside the band-gap region), the wave velocity (phase velocity) is real, if and only if ω is real.

Figure 1.
Dispersion relations ω=ω(k) [17] for the relaxed micromorphic model with non-vanishing Cosserat couple modulus μc>0. Uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRO, transverse rotational optic; ...

Because ω2 appears on the diagonal only, the problem (2.28) can be analogously expressed as an eigenvalue problem

det(B1(k)ω2𝟙)=0,det(B2(k)ω2𝟙)=0anddet(B3(k)ω2𝟙)=0,det(B4(k)ω2𝟙)=0,}
2.30

where

B1(k)=(cp2k2263kμeρη33k(2μe+3λe)ρη263kμeρη13k2cm2+ωs223k2cm233k(2μe+3λe)ρη23k2cm2+23k2cm2+ωp2),
2.31

B2(k)=B3(k)=(k2cs2k2μeρηk2μcρηk2μeρηcm22k2+ωs2cm22k2k2μcρηcm22k2cm22k2+ωr2)
2.32

andB4(k)=(cm2k2+ωs2000cm2k2+ωr2000cm2k2+ωs2).
2.33

Note that B1(k), B2(k), B3(k) and B4(k) are real symmetric matrices and, therefore, the resulting eigenvalues ω2 are real. Obtaining real wave velocities is tantamount to having ω20 for all solutions of (2.30).

(c) Necessary and sufficient conditions for real wave propagation

We show next that all the eigenvalues ω2 of B1(k), B2(k) and B3(k) are real and positive for every k0 and non-negative for k=0, provided certain conditions on the material coefficients are satisfied. Sylvester's criterion states that a Hermitian matrix M is positive definite if and only if the leading principal minors are positive [38]. For the matrix B1, the three principal minors are

(B1)11=2μe+λeρ,
2.34

(Cof(B1))33=k23ηρ[6(2μe+λe)μmicro+6μeκe+(2μe+λe)μeLc2k2]=k23ηρ[2(4μmacro+3κe)(μe+μmicro)+(2μe+λe)μeLc2k2]
2.35

and

det(B1)=k2η2ρ[6κeκmicro(μe+μmicro)+8μeμmicro(κe+κmicro)+(2μe+λe)(2μmicro+λmicro)μeLc2k2]=k2η2ρ[6(κe+κmicro)(μe+μmicro)(2μmacro+λmacro)+(2μe+λe)(2μmicro+λmicro)μeLc2k2].
2.36

The three principal minors of B1 are clearly positive for k0 if6

μe>0,μmicro>0,κe+κmicro>0,2μmacro+λmacro>0and4μmacro+3κe>0,2μe+λe>0,2μmicro+λmicro>0.}
2.37

Similarly, for the matrix B2, the three principal minors are

(B2)11=μe+μcρ,
2.38

(Cof(B2))33=k22ηρ[4(μeμc+μmicro(μe+μc))+(μe+μc)μeLc2k2]
2.39

anddet(B2)=k2η2ρ[4μmicroμcμe+(μe+μc)μmicroμeLc2k2].
2.40

For the matrix B2(k)=B3(k), considering positive η,ρ and separating terms in the brackets by looking at large and small values of k, we can state the necessary and sufficient conditions for strict positive definiteness of B2(k) at arbitrary k0,

μe>0,μmicro>0andμc0.
2.41

Because B4(k) is diagonal, it is easy to show that positive definiteness is tantamount to the set of necessary and sufficient conditions for k0

μe>0,μe+μmicro>0,μc0.
2.42

On the other hand, considering the case k=0, we obtain that the matrices reduce to

B1(0)=(0000ωs2000ωp2),B2(0)=B3(0)=(0000ωs2000ωr2)andB4(0)=(ωs2000ωr2000ωs2).
2.43

Because the matrices are diagonal for k=0, it is easy to show that positive semi-definiteness is tantamount to the set of necessary and sufficient conditions

μe0,μe+μmicro0,μc0andκe+κmicro0.
2.44

Hence, we can state a simple sufficient condition for real wave velocities for all real k

μe>0,μmicro>0,κe+κmicro>0,2μmacro+λmacro>0and4μmacro+3κe>0,2μe+λe>0,2μmicro+λmicro>0.}
2.45

In order to see a set of global necessary conditions for positivity at arbitrary k0, we consider first large and small values of k0 separately. For k+, we must have

2μe+λe>0,(2μe+λe)μeLc2>0,(2μe+λe)(2μmicro+λmicro)μeLc2>0,
2.46

or analogously

2μe+λe>0,μeLc2>0and2μmicro+λmicro>0,
2.47

while, for k0, we must have

2μe+λe>0,(4μmacro+3κe)(μe+μmicro)>0and(κe+κmicro)(μe+μmicro)(2μmacro+λmacro)>0.}
2.48

Because from (2.41) we have necessarily μe>0, μmicro>0, and from (2.44) we get κe+κmicro0, and considering together the two limits for k, we obtain the necessary condition

2μe+λe>0,2μmicro+λmicro>0,4μmacro+3κe>0,κe+κmicro>0andμe>0,μmicro>0,μc0,2μmacro+λmacro>0.}
2.49

Inspection shows that (2.49) is our proposed sufficient condition (2.37). From μe>0 and μmicro>0, it follows that μmacro>0. Therefore, condition (2.49) is necessary and sufficient. We have shown our main proposition as follows.

Proposition (real wave velocities). —

The dynamic relaxed micromorphic model (equation (2.15)) admits real planar waves if and only if

μc0,μe>0,2μe+λe>0,μmicro>0,2μmicro+λmicro>0,(μmacro>0),2μmacro+λmacro>0andκe+κmicro>0,4μmacro+3κe>0.}
2.50

In (2.50), the requirement μmacro>0 is redundant, because it is already assumed that μe,μmicro>0. It is clear that positive definiteness of the elastic energy (2.4) implies (2.50). We remark that, as shown in appendix Aa, the set of inequalities (2.50) is already implied by

μe>0, μmicro>0,μc0,κe+κmicro>0 and 2μmacro+λmacro>0.
2.51

Finally letting μmicro+ and κmicro+ (or μmicro+ and λmicro>const.) generates the limit condition for real wave velocities (μeμmacro)

μmacro>0,μc0and2μmacro+λmacro>0,
2.52

which coincides, up to μc, with the strong ellipticity condition in isotropic linear elasticity (see §3) and it coincides fully with the condition for real wave velocities in micropolar elasticity; see §4. A condition similar to (2.52) can be found in [10, equation 8.14 p. 26] where Mindlin requires that μmacro>0,2μmacro+λmacro>0 (in our notation),7 which are obtained from the requirement of positive group velocity at k=0

dωacoustic,long(0)dk>0anddωacoustic,trans(0)dk>0.
2.53

Let us emphasize that our method is not easily generalized to two immediate extensions. First, one could be interested in the isotropic-relaxed micromorphic model with weighted inertia contributions and weighted curvatures [34]. Second, one could be interested in the anisotropic setting [20]. In the second case, the block structure of the problem will be lost, and one has to deal with the full 12×12 case; see equation (A 23) in appendix A. Nonetheless, we expect positive definiteness to always imply real wave propagation.

In [34], we show that the tangents of the acoustic branches in k=0 in the dispersion curves are

cl=dωacoustic,long(0)dk=2μmacro+λmacroρandct=dωacoustic,trans(0)dk=μmacroρ.
2.54

The tangents coincide with the classical linear elastic response if the latter has Lamé constants μmacro and λmacro, as shown in figure 2.

Figure 2.
Dispersion relations ω=ω(k) for the longitudinal acoustic wave LA, and the transverse acoustic TA in the relaxed micromorphic model (a) and in a classical Cauchy medium (b).

3. A comparison: classical isotropic linear elasticity

For classical linear elasticity with isotropic energy density and kinetic energy density

W(u)=μmacrosymu2+λmacro2(tr(u))2,J=ρ2u,t2.
3.1

The positive definiteness of the energy is equivalent to

μmacro>0,2μmacro+3λmacro>0.
3.2

It is easy to see that our homogenization formula (2.6) implies (3.2) under the condition of positive definiteness of the relaxed micromorphic model.

The dynamic formulation is obtained by defining a joint Hamiltonian and assuming stationary action. The dynamic equilibrium equations are

ρu,tt=Div[2μmacrosym(u)+λmacrotr(u)𝟙].
3.3

As before, in our study of wave propagation in micromorphic media, we limit ourselves to the case of plane waves travelling in an infinite domain. We suppose that the space dependence of all introduced kinematic fields is limited to a direction defined by a unit vector ξ~R3, which is the direction of propagation of the wave. Therefore, we look for solutions of (3.3) in the form

u(x,t)=u^ei(kξ~,xR3ωt),u^C3,ξ~2=1.
3.4

Because our formulation is isotropic, we can, without loss of generality, specify the direction ξ~=e1. Then, X=e1,xR3=x1, and we obtain

u(x,t)=u^ei(kXωt)andu^C3.
3.5

With this ansatz, it is possible to write (3.3) as

A5(e1,ω,k)u^=0(B(e1,k)ω2𝟙)u^=0,
3.6

where

A5(e1,ω,k)=(2μmacro+λmacroρk2ω2000μmacroρk2ω2000μmacroρk2ω2)
3.7

and

B(e1,k)=k2ρ(2μmacro+λmacro000μmacro000μmacro).
3.8

Here, we observe that A5(e1,ω,k) is already diagonal and real. Requesting real wave velocities means ω20. For k0, this leads to the classical so-called strong ellipticity condition

μmacro>0,2μmacro+λmacro>0,
3.9

which is implied by positive definiteness of the energy (3.2).

In classical (linear or nonlinear) elasticity, the condition of real wave propagation (3.9) is equivalent to strong ellipticity and rank-one convexity. Indeed, rank-one convexity amounts to set (ξ=kξ~ with ξ2=1)

d2dt2t=0W(u+tu^ξ)0C(u^ξ),u^ξR3×30,
3.10

where C is the fourth-order elasticity tensor. Condition (3.10) then reads

02μmacrosym(u^ξ)2+λmacro(tr(u^ξ))2=μmacrou^2ξ2+(μmacro+λmacro)u^,ξR32.
3.11

We may express (3.11) given ξR3 as a quadratic form in u^R3, which results in

μmacrou^2ξ2+(μmacro+λmacro)u^,ξR32=D(ξ)u^,u^R3,
3.12

where the components of the symmetric and real 3×3 matrix D(ξ) read

D(ξ)=((2μmacro+λmacro)ξ12+μmacro(ξ22+ξ32)(λmacro+μmacro)ξ1ξ2(λmacro+μmacro)ξ1ξ2(2μmacro+λmacro)ξ22+μmacro(ξ12+ξ32)(λmacro+μmacro)ξ1ξ3(λmacro+μmacro)ξ1ξ2(λmacro+μmacro)ξ1ξ3(λmacro+μmacro)ξ2ξ3(2μmacro+λmacro)ξ32+μmacro(ξ12+ξ22)).
3.13

The three principal invariants are independent of the direction ξ owing to isotropy and are given by

tr(D(ξ))=ξ2(4μmacro+λmacro)=k2(4μmacro+λmacro),tr(CofD(ξ))=ξ4μmacro(5μmacro+2λmacro)=k4μmacro(5μmacro+2λmacro)anddet(D(ξ))=ξ6μmacro2(2,μmacro+λmacro)=k6μmacro2(2μmacro+λmacro).}
3.14

Because D(ξ) is real and symmetric, its eigenvalues are real. The eigenvalues of the matrix D(ξ) are k2(2μmacro+λmacro) and k2μmacro (of multiplicity 2) such that positivity at k0 is satisfied, if and only if8

μmacro>0,2μmacro+λmacro>0,
3.15

which are the usual strong ellipticity conditions. We note here that the latter calculations also show that B(e1)=(1/ρ)k2D(e1). Alternatively, one may directly form the so-called acoustic tensor B(ξ)R3×3 by

B(ξ)u^:=[C(u^ξ)]ξ,u^R3;
3.16

in indices we have (B(ξ))ij=Cikjlu^ku^lC(ξξ). With (3.16), we obtain9

u^,B(ξ)u^3=[(u^ξ)]=:B^3×3ξ,u^3=B^ξ,u^=B^(ξu^),𝟙3×3=B^,(ξu^)T3×3=B^,u^ξ3×3=(u^ξ),u^ξ3×3,
3.17

and we see that strong ellipticity C(u^ξ),u^ξR3×3>0 is equivalent to the positive definiteness of the acoustic tensor B(ξ).

4. A further comparison: the linear Cosserat model

In the isotropic hyperelastic case, the elastic energy density and the kinetic energy density of the Cosserat model read

W=μmacrosymu2+μcskew(uA)2+λmacro2(tr(u))2+μmacroLc22CurlA2andJ=ρ2u,t2+η2A,t2.}
4.1

Introducing the canonical identification of R3 with so(3), A can be expressed as a function of aR3 as

A=anti(a)=(0a3a2a30a1a2a10).
4.2

Here, we assume for clarity a uniconstant curvature expression in terms of only CurlA2. Strict positive definiteness of the potential energy is equivalent to the following simple relations for the introduced parameters:

2μmacro+3λmacro>0,μmacro>0,μc>0andLc>0.
4.3

The dynamic formulation is obtained by defining a joint Hamiltonian and assuming stationary action. The dynamic equilibrium equations are

ρu,tt=Div[2μmacrosym(uA)+2μcskew(uA)+λmacrotr(uA)𝟙]andηA,tt=μmacroLc2skew(CurlCurlA)+2μcskew(uA);}
4.4

see also [4043] for formulations in terms of axial vectors. Note that, for zero Cosserat couple modulus μc=0, the coupling of the two fields (u,A) is absent, in opposition to the relaxed micromorphic model (equation (2.15)). Considering plane and stationary waves of amplitudes u^ and a^, it is possible to express this system as

A6(ω,k)(u^1a^1)T=0,A7(ω,k)(u^2a^3)T=0,A7(ω,k)(u^3a^2)T=0,
4.5

where

A6(ω,k)=(k2(2μmacro+λmacro)ρω200(2μmacroLc2k2+2μc)ηω2)
4.6

and

A7(ω,k)=(k2(μmacro+μc)ρω22ikμcρikμcη(k2μmacroLc2+4μc)(2η)ω2).
4.7

As done in the case of the relaxed micromorphic model, it is possible to express equivalently the problem with A6(ω,k) and the following symmetric matrix:

A¯7(k)=diag7A7(ω,k)diag71=(k2(μmacro+μc)ρω22kμcρη2kμcρη(k2μmacroLc2+4μc)(2η)ω2),
4.8

where

diag7=(ρ00i2η).
4.9

Because ω2 appears only on the diagonal, the problem can be analogously expressed as the following eigenvalue problems:

det(B6(k)ω2𝟙)=0anddet(B7(k)ω2𝟙)=0,
4.10

where

B6(k)=(k2(2μmacro+λmacro)ρ00(2μmacroLc2k2+2μc)η2)
4.11

and

B7(k)=(k2(μmacro+μc)ρ2kμcρη2kμcρη(k2μmacroLc2+4μc)(2η))
4.12

are the blocks of the acoustic tensor B

B(k)=(B6000B7000B7).
4.13

The eigenvalues of the matrix B6(k) are simply the elements of the diagonal; therefore, we have

ωacoustic,long(k)=k2μmacro+λmacroρandωoptic,long(k)=2μmacroLc2k2+2μcη,
4.14

whereas for B7(k) it is possible to find

ωacoustic,trans(k)=a(k)a(k)2b2k2andωoptic,trans(k)=a(k)+a(k)2b2k2,
4.15

where we have set

a(k)=4μc+μmacroLc2k2η+2μmacro+μcρk2andb2=8μmacro(4μc+k2Lc2(μmacro+μc))ρη.
4.16

The acoustic branches are those curves ω=ω(k) as solutions of (4.9) that satisfy ω(0)=0. We note here that the acoustic branches of the longitudinal and transverse dispersion curves have, as tangent in k=0,10

cl=dωacoustic,long(0)dk=2μmacro+λmacroρandct=dωacoustic,trans(0)dk=μmacroρ,
4.17

respectively. Moreover, the longitudinal acoustic branch is non-dispersive, i.e. a straight line with slope (4.17)1. The matrix B6(k) is positive definite for arbitrary k0 if

2μmacro+λmacro>0,μmacro>0,μc0.
4.18

Using the Sylvester criterion, B7(k) is positive definite, if and only if the principal minors are positive, namely

(B7)11=k2(μmacro+μc)ρ>0anddet(B7)=k22ηρ(4μmacroμc+k2μmacroLc2(μmacro+μc))>0,}
4.19

from which we obtain the condition

μmacro+μc>0,μmacro>0andμc0.
4.20

Considering these two sets of conditions, it is possible to state a necessary and sufficient condition for the positive definiteness of B6(k) and B7(k) and therefore of the acoustic tensor B(k)

2μmacro+λmacro>0,μmacro>0andμc0,
4.21

which are implied by the positive definiteness of the energy (4.3). Eringen [12, p. 150] also obtains correctly (4.18) and (4.20) (in his notation μc=κ/2,μmacro=μEringen+κ/2).

In [44, 45], strong ellipticity for the Cosserat micropolar model is defined and investigated. In this respect, we note that ellipticity is connected to acceleration waves, whereas our investigation concerns real wave velocities for planar waves. Similarly to [46], it is established in [44, 45] that strong ellipticity for the micropolar model holds if and only if (the uniconstant curvature case in our notation)

2μmacro+λmacro>0andμmacro+μc>0.
4.22

We conclude that, for micropolar material models (and therefore also for micromorphic materials), strong ellipticity (4.22) is too weak to ensure real planar waves because it is implied by, but does not imply, (4.21). This fact seems to have been appreciated also in the study of the Cosserat model [4751].

5. Conclusion

In this paper, we derive the set of necessary and sufficient conditions that have to be imposed on the constitutive parameters of the relaxed micromorphic model in order to guarantee

  •  positive definiteness;
  •  real wave velocity; and
  •  Legendre–Hadamard strong ellipticity condition.

We show that if, on the one hand, definite positiveness implies real wave propagation, on the other hand, real wave propagation is not guaranteed by the strong ellipticity condition.

We conclude that in strong contrast to the case of classical isotropic linear elasticity, where the three concepts are known to be equivalent, in the case of the relaxed micromorphic continua only definite positiveness of the strain energy density can be considered to be a good criterion to guarantee real wave speeds in the considered media. The proposed considerations can be extended to all generalized continua where the equivalence between the three notions is far from being straightforward.

Acknowledgments

We thank Victor A. Eremeyev for helpful clarification.

Appendix A

(a) Inequality relations between material parameters

The formulae in §2a are based on the harmonic mean of two numbers κe and κmicro (or μe and μmicro). If the two numbers are positive, it is easy to see that

κmacromin(κe,κmicro).
A 1

Here, we show that the same conclusion still holds if we merely assume that κe+κmicro>0. This allows for either κe<0 or κmicro<0. Therefore, considering that κe+κmicro>0, even if the energy is not strictly positive, it is possible to derive that

κmacro=κmicroκeκe+κmicro=κmicroκe+κe2κe2κe+κmicro=κeκmicro+κeκe+κmicroκe2κe+κmicro=κeκe2κe+κmicro0κeandκmacro=κmicroκeκe+κmicro=κmicroκe+κmicro2κmicro2κe+κmicro=κmicroκmicro+κeκe+κmicroκmicro2κe+κmicro=κmicroκmicro2κe+κmicro0κmicro.}
A 2

Considering similarly μe+μmicro>0, it is possible to obtain

μmacro=μmicroμeμe+μmicro=μmicroμe+μe2μe2μe+μmicro=μeμmicro+μeμe+μmicroμe2μe+μmicro=μeμe2μe+μmicro0μeandμmacro=μmicroμeμe+μmicro=μmicroμe+μmicro2μmicro2μe+μmicro=μmicroμmicro+μeμe+μmicroμmicro2μe+μmicro=μmicroμmicro2μe+μmicro0μmicro.}
A 3

Therefore, if μe+μmicro>0 and κe+κmicro>0, the macroscopic parameters are less than or equal to the respective microscopic parameters, namely

κeκmacro,κmicroκmacro,μeμmacro,μmicroμmacro,
A 4

and it is possible to show that

2μe+λe=13(4μe+3κe)13(4μmacro+3κmacro)=2μmacro+λmacro>0,2μmicro+λmicro=13(4μmicro+3κmicro)13(4μmacro+3κmacro)=2μmacro+λmacro>0,(2μe+λe)+(2μmicro+λmicro)2(2μmacro+λmacro)>0and4μmacro+3κe4μmacro+3κmacro=3(2μmacro+λmacro)>0.}
A 5

Therefore, the set of inequalities (2.50) is implied from the smaller set

μe>0,μmicro>0,μc0,κe+κmicro>0and2μmacro+λmacro>0.
A 6

We note here that 3(2μe+λe)4μmacro+3κe3(2μmacro+λmacro), because

3(2μe+λe)=4μe+3κe4μmacro+3κe4μmacro+3κmacro=3(2μmacro+λmacro).
A 7

(b) The 12×12 acoustic tensor for arbitrary direction

We suppose that the space dependence of all introduced kinematic fields is limited to a direction defined by a unit vector ξ, which is the direction of propagation of the wave. Therefore, we look for solutions of

ρu,tt=Div[2μesym(uP)+2μcskew(uP)+λetr(uP)𝟙]andηP,tt=μeLc2CurlCurlP+2μesym(uP)+2μcskew(uP)+λetr(uP)𝟙[2μmicrosymP+λmicrotr(P)𝟙],}
A 8

in the form

u(x,t)=u^ei(kξ,x3ωt)s(x,t)/ scalar,u^3,ξ2=1andP(x,t)=P^ei(kξ,x3ωt)s(x,t)/ scalar,P^3×3,}
A 9

where u^ is the polarization vector and P^ is the polarization matrix. We start by remarking that, considering A,BR3×3, we have

Curl(AB)=LB(A)+ACurl(B),
A 10

where LB:R27R3×3 is a linear operator with constant coefficients defined by the appropriate product rule of differentiation. Therefore, we obtain

Curl(P^s(x,t))=Curl(P^𝟙s(x,t))=P^Curl(𝟙s(x,t)),
A 11

where

Curl(𝟙s(x,t))=(03s(x,t)2s(x,t)3s(x,t)01s(x,t)2s(x,t)1s(x,t)0)so(3).
A 12

The derivatives of s(x,t) can be evaluated by considering

xs(x,t)=(1s(x,t)2s(x,t)3s(x,t))=ei(kξ,xR3ωt)(ikξ1ikξ2ikξ3)=ei(kξ,xR3ωt)ikξ=ikξs(x,t).
A 13

It can be noted that

Curl(s(x,t)𝟙)=anti(s(x,t))=ei(kξ,xR3ωt)ikanti(ξ)=s(x,t)ikanti(ξ).
A 14

Therefore, it is possible to evaluate the CurlCurlP as

CurlCurl(P^s(x,t))=Curl(P^anti(ξ)so(3)iks(x,t))=ikCurl([P^anti(ξ)]𝟙s(x,t))=ikP^anti(ξ)Curl(𝟙s(x,t))=ikikP^anti(ξ)anti(ξ)s(x,t)=k2P^anti(ξ)anti(ξ)ei(kξ,xR3ωt).
A 15

On the other hand, the second derivative of P with respect to time is

P,tt=t2(P^ei(kξ,xR3ωt))=ω2P^ei(kξ,xR3ωt))=ω2P^s(x,t).
A 16

Analogously for u, it is possible to evaluate the gradient and the derivatives with respect to time as

xu=iks(x,t)u^ξ,u,tt=ω2u^s(x,t).
A 17

The sym, skew and tr of uP can then be expressed as

sym(uP)=sym(iku^ξP^)s(x,t)=(iksym(u^ξ)symP^)s(x,t),skew(uP)=skew(iku^ξP^)s(x,t)=(ikskew(u^ξ)skewP^)s(x,t)andtr(uP)=tr(iku^ξP^)s(x,t)=(iku^,ξR3trP^)s(x,t).}
A 18

Therefore, we have

Divsym(uP)=Div[(iksym(u^ξ)symP^)s(x,t)]=(iksym(u^ξ)symP^)xs(x,t)=(iksym(u^ξ)symP^)(ikξs(x,t))=(k2sym(u^ξ)ξ+iksymP^ξ)s(x,t),Divskew(uP)=Div[(ikskew(u^ξ)skewP^)s(x,t)]=(ikskew(u^ξ)skewP^)xs(x,t)=(ikskew(u^ξ)skewP^)(ikξs(x,t))=(k2skew(u^ξ)ξ+ikskewP^ξ)s(x,t)andDiv(tr(uP)𝟙)=Div[((iku^,ξR3trP^)𝟙)s(x,t)]=(iku^,ξR3trP^)𝟙xs(x,t)=(iku^,ξR3trP^)𝟙(ikξs(x,t))=(k2u^,ξR3+iktrP^)ξs(x,t).}
A 19

Here, we have considered that, given a generic BR3×3 and a scalar s(x,t), we have

Div[Bs(x,t)]=Div[B]=0s(x,t)+Bxs(x,t).
A 20

With all the formulae obtained, it is possible to write (A 8) simplifying s(x,t) everywhere as

ρω2u^=[2μe(k2sym(u^ξ)ξ+iksymP^ξ))+2μc(k2skew(u^ξ)ξ+ikskewP^ξ)+λe(k2u^,ξR3+iktrP^)ξ]andηω2P^=μeLc2k2P^anti(ξ)anti(ξ)+2μe(iksym(u^ξ)symP^)+2μc(ikskew(u^ξ)skewP^)+λe(iku^,ξR3trP^)𝟙[2μmicrosymP^+λmicrotr(P^)𝟙],}
A 21

or analogously

ρω2u^+k2(2μesym(u^ξ)ξ+2μcskew(u^ξ)ξ+λeu^,ξR3ξ)+ik(2μesymP^ξ+2μcskewP^ξ+λetrP^ξ)=0andηω2P^μeLc2k2P^anti(ξ)anti(ξ)+2(μe+μmicro)symP^+2μcskewP^+(λe+λmicro)tr(P^)𝟙2μeiksym(u^ξ)2μcikskew(u^ξ)λeiku^,ξR3𝟙=0.}
A 22

At given ξR3, this is a linear system in (u^,P^)C12 which can be written in 12×12 matrix format as

(A~(ξ,ω,k))(u^1u^2u^3P^11P^12P^13P^21P^22P^23P^31P^32P^33)=0,(B~(ξ,k)ω2𝟙)(u^1u^2u^3P^11P^12P^13P^21P^22P^23P^31P^32P^33)=(000000000000).
A 23

Here, B~(ξ,k) is the 12×12 acoustic tensor. The columns of A~ are

A~i1=(ρω2k2(λe+2μe)ξ12k2(μc+μe)(ξ22+ξ32)k2(λeμc+μe)ξ1ξ2k2(λeμc+μe)ξ1ξ3ik(λe+2μe)ξ1ik(μc+μe)ξ2ik(μc+μe)ξ3ik(μcμe)ξ2ikλeξ10ik(μcμe)ξ30ikλeξ1),A~i2=(k2(λeμc+μe)ξ1ξ2ρω2k2(λe+2μe)ξ22k2(μc+μe)(ξ12+ξ32)k2(λeμc+μe)ξ2ξ3ikλeξ2ik(μcμe)ξ10ik(μc+μe)ξ1ik(λe+2μe)ξ2ik(μc+μe)ξ30ik(μcμe)ξ3ikλeξ2),

A~i3=(k2(λeμc+μe)ξ1ξ3k2(λeμc+μe)ξ2ξ3ρω2k2(λe+2μe)ξ32k2(μc+μe)(ξ12+ξ22)ikλeξ30ik(μcμe)ξ10ikλeξ3ik(μcμe)ξ2ik(μc+μe)ξ1ik(μc+μe)ξ2ik(λe+2μe)ξ3),A~i4=(ik(λe+2μe)ξ1ikλeξ2ikλeξ3ηω2(2(μe+μmicro)+λe+λmicro)k2μeLc2(ξ22+ξ33)k2μeLc2ξ1ξ2k2μeLc2ξ1ξ30(λe+λmicro)000(λe+λmicro)),

A~i5=(ik(μc+μe)ξ2ik(μcμe)ξ10k2μeLc2ξ1ξ2ηω2(μc+μe+μmicro)k2μeLc2(ξ12+ξ32)k2μeLc2ξ1ξ200000),A~i6=(ik(μc+μe)ξ30ik(μcμe)ξ1k2μeLc2ξ1ξ3k2μeLc2ξ2ξ3ηω2(μc+μe+μmicro)k2μeLc2(ξ12+ξ22)000μcμeμmicro00),

A~i7=(ik(μcμe)ξ2ik(μc+μe)ξ100μcμeμmicro0ηω2(μc+μe+μmicro)k2μeLc2(ξ22+ξ32)k2μeLc2ξ1ξ2k2μeLc2ξ1ξ3000),A~i8=(ikλeξ1ik(2μe+λe)ξ2ikλeξ3λeλmicro00k2μeLc2ξ1ξ2ηω2(μc+μe+μmicro)k2μeLc2(ξ12+ξ32)k2μeLc2ξ2ξ300λeλmicro),

A~i9=(0ik(μc+μe)ξ3ik(μc+μe)ξ2000k2μeLc2ξ1ξ3k2μeLc2ξ2ξ3ηω2(μc+μe+μmicro)k2μeLc2(ξ12+ξ22)0μcμeμmicro0),A~i10=(ik(μcμe)ξ30ik(μc+μe)ξ100μcμeμmicro000ηω2(μc+μe+μmicro)k2μeLc2(ξ22+ξ32)k2μeLc2ξ1ξ2k2μeLc2ξ1ξ3),

A~i11=(0ik(μcμe)ξ3ik(μc+μe)ξ200000μcμeμmicrok2μeLc2ξ1ξ2ηω2(μc+μe+μmicro)k2μeLc2(ξ12+ξ32)k2μeLc2ξ2ξ3),A~i12=(ikλeξ1ikλeξ2ik(λe+2μe)ξ3λeλmicro000λeλmicro0k2μeLc2ξ1ξ3k2μeLc2ξ2ξ3ηω2(μc+μe+μmicro)k2μeLc2(ξ12+ξ22)).

It is clear that, even with the aid of up-to-date computer algebra systems, it is practically impossible to determine the positive definiteness of the 12×12 acoustic tensor B~ as dependent on the given material parameters. In the main body of our paper, we succeed by choosing immediately the propagation direction ξ=e1 and by considering a set of new variables (2.16). This allows us to obtain a certain pre-factorization of B~(e1,k) in 3×3 blocks. Because the formulation is isotropic, choosing ξ=e1 is no restriction, as argued before.

Footnotes

1The dislocation tensor is defined as αij=(CurlP)ij=Pih,kϵjkh, where ϵ is the Levi–Civita tensor.

2Here, we understand that, having found the (in general, complex) solutions of (2.19), only the real or imaginary parts separately constitute actual wave solutions which can be observed in reality.

3In an isotropic model, it is clear that there is no direction dependence. More specifically, let us consider an arbitrary direction ξ~R3. Now we consider an orthogonal spatial coordinate change Qe1=ξ~ with QSO(3). In the rotated variables, the ensuing system of PDEs (2.15) is form invariant; see [37].

4It is possible to face the problem in two more equivalent ways. The first one is to consider from the start that the amplitudes of the microdistortion field are multiplied by the imaginary unit i, i.e. β=(β1,iβ2,iβ3)TC3, as done in [10, p. 24, equation 8.6]. Doing so, we obtain a real matrix that can be symmetrized with diag1=(ρ0006η20003η). On the other hand, it is also possible to consider from the beginning β=(ρβ1,i(6η/2)β2,i3ηβ3)TC3, obtaining directly a real symmetric matrix.

5The formal limit η+ shows no dispersion at all giving two pseudo-acoustic linear curves, longitudinal and transverse with slopes cp=(2μe+λe)/ρ and cs=(μe+μc)/ρ, respectively.

6We note here that 4μmacro+3κe>02μe+λe>43(μeμmacro)2μmacro+λmacro>κmacroκe. Furthermore, if μe+μmicro>0 and κe+κmicro>0, we have 3(2μe+λe)4μmacro+3κe3(2μmacro+λmacro); see appendix A.

7Mindlin explains that such parameters ‘are less than those that would be calculated from the strain-stiffnesses [of the unit cell]. This phenomenon is due to the compliance of the unit cell and has been found in a theory of crystal lattices by Gazis & Wallis [39]’.

8The eigenvalues of D(ξ) are independent of the propagation direction ξR3, which makes sense for the isotropic formulation at hand.

9The term [C(u^ξ)](u^ξ) that in index notation reads Cijklu^kξlu^jξm is different from C[(u^ξ)(u^ξ)], i.e. Cijklu^kξmu^mξl.

10To obtain the slopes in 0, it is possible to search for a solution of the type ω=ak and then evaluate the limit for a0; see [34] for a thorough explanation in the relaxed micromorphic case.

Author's contributions

All the authors contributed equally to this work.

Competing interests

The authors have no conflict of interests to declare.

Funding

The work of I.-D.G. was supported by a grant from the Romanian National Authority for Scientific Research and Innovation, CNCSUEFISCDI, project no. PN-II-RU-TE-2014-4-1109. A.M. thanks INSA-Lyon for the funding of the BQR 2016 ‘Caractérisation mécanique inverse des métamatériaux: modélisation, identification expérimentale des paramètres et évolutions possibles’, as well as the PEPS CNRS-INSIS.

References

1. Chiriţă S, Danescu A, Ciarletta M 2007. On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87, 1–27. (doi:10.1007/s10659-006-9096-7)
2. Balzani D, Neff P, Schröder J, Holzapfel GA 2006. A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43, 6052–6070. (doi:10.1016/j.ijsolstr.2005.07.048)
3. Merodio J, Neff P 2006. A note on tensile instabilities and loss of ellipticity for a fiber-reinforced nonlinearly elastic solid. Arch. Mech. 58, 293–303.
4. Schröder J, Neff P 2002. Application of polyconvex anisotropic free energies to soft tissues. In Proc. 5th World Congress on Computational Mechanics, Vienna, Austria, 7–12 July 2002. Barcelona, Spain: IACM.
5. Schröder J, Neff P, Ebbing V 2008. Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J. Mech. Phys. Solids 56, 3486–3506. (doi:10.1016/j.jmps.2008.08.008)
6. Smith AC. 1967. Waves in micropolar elastic solids. Int. J. Eng. Sci. 5, 741–746. (doi:10.1016/0020-7225(67)90019-5)
7. Chiriţă S, Ghiba I-D 2009. Strong ellipticity and progressive waves in elastic materials with voids. Proc. R. Soc. A 466, 439–458. (doi:10.1098/rspa.2009.0360)
8. Nowacki W. 1985. Theory of asymmetric elasticity. Oxford, UK: Pergamon Press.
9. Smith AC. 1968. Inequalities between the constants of a linear micro-elastic solid. Int. J. Eng. Sci. 6, 65–74. (doi:10.1016/0020-7225(68)90020-7)
10. Mindlin RD. 1963. Microstructure in linear elasticity. Technical report no. 50. Office of Naval Research, Arlington, VA, USA.
11. Mindlin RD. 1964. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78. (doi:10.1007/BF00248490)
12. Eringen AC. 1999. Microcontinuum field theories. New York, NY: Springer.
13. Maugin GA. 2016. Non-classical continuum mechanics, vol. 51 Berlin, Germany: Springer.
14. Neff P, Ghiba I-D, Madeo A, Placidi L, Rosi G 2014. A unifying perspective: the relaxed linear micromorphic continuum. Contin. Mech. Thermodyn. 26, 639–681. (doi:10.1007/s00161-013-0322-9)
15. Madeo A, Neff P, Ghiba I-D, Placidi L, Rosi G 2014. Band gaps in the relaxed linear micromorphic continuum. Z. Angew. Math. Mech. 95, 880–887. (doi:10.1002/zamm.201400036)
16. Madeo A, Neff P, Ghiba I-D, Placidi L, Rosi G 2015. Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Contin. Mech. Thermodyn. 27, 551–570. (doi:10.1007/s00161-013-0329-2)
17. Madeo A, Barbagallo G, d'Agostino MV, Placidi L, Neff P 2016. First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model. Proc. R. Soc. A 472, 20160169 (doi:10.1098/rspa.2016.0169) [PMC free article] [PubMed]
18. Madeo A, Neff P, Ghiba I-D, Rosi G 2016. Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model. J. Mech. Phys. Solids 95, 441–479. (doi:10.1016/j.jmps.2016.05.003)
19. Greene S, Gonella S, Liu WK 2012. Microelastic wave field signatures and their implications for microstructure identification. Int. J. Solids Struct. 49, 3148–3157. (doi:10.1016/j.ijsolstr.2012.06.011)
20. Barbagallo G, , d'Agostino MV, , Abreu R, , Ghiba I-D, , Madeo A, , Neff P. 2016 Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics. ( http://arxiv.org/abs/1601.03667)
21. Neff P, Ghiba I-D, Lazar M, Madeo A 2015. The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations. Q. J. Mech. Appl. Math. 68, 53–84. (doi:10.1093/qjmam/hbu027)
22. Ghiba I-D, Neff P, Madeo A, Placidi L, Rosi G 2014. The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. Math. Mech. Solids 20, 1171–1197. (doi:10.1177/1081286513516972)
23. Romano G, Barretta R, Diaco M 2016. Micromorphic continua: non-redundant formulations. Contin. Mech. Thermodyn. 28, 1659–1670. (doi:10.1007/s00161-016-0502-5)
24. Neff P, Pauly D, Witsch K-J 2015. Poincaré meets Korn via Maxwell: extending Korn's first inequality to incompatible tensor fields. J. Differ. Equ. 258, 1267–1302. (doi:10.1016/j.jde.2014.10.019)
25. Neff P, Pauly D, Witsch K-J 2012. Maxwell meets Korn: a new coercive inequality for tensor fields in Rn×n with square-integrable exterior derivative. Math. Methods Appl. Sci. 35, 65–71. (doi:10.1002/mma.1534)
26. Neff P, Pauly D, Witsch K-J 2011. A canonical extension of Korn's first inequality to H(Curl) motivated by gradient plasticity with plastic spin. C. R. Math. 349, 1251–1254. (doi:10.1016/j.crma.2011.10.003)
27. Bauer S, Neff P, Pauly D, Starke G 2014. New Poincaré-type inequalities. C. R. Math. 352, 163–166. (doi:10.1016/j.crma.2013.11.017)
28. Bauer S, Neff P, Pauly D, Starke G 2016. Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions. ESAIM 22, 112–133. (doi:10.1051/cocv/2014068)
29. Neff P. 2004. On material constants for micromorphic continua. In Trends in Applications of Mathematics to Mechanics (eds Y Wang, K Hutter), pp. 337–348. Aachen, Germany: Shaker.
30. Neff P, Forest S 2007. A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 87, 239–276. (doi:10.1007/s10659-007-9106-4)
31. Neff P. 2006. The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. Z. Angew. Math. Mech. 86, 892–912. (doi:10.1002/zamm.200510281)
32. Madeo A, Neff P, d'Agostino MV, Barbagallo G 2016. Complete band gaps including non-local effects occur only in the relaxed micromorphic model. C. R. Mech. 344, 784–796. (doi:10.1016/j.crme.2016.07.002)
33. Madeo A, , Collet M, , Miniaci M, , Billon K, , Ouisse M, , Neff P. 2016 Modeling real phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia. ( http://arxiv.org/abs/1610.03878)
34. d'Agostino MV, , Barbagallo G, , Ghiba I-D, , Madeo A, , Neff P. 2016 A panorama of dispersion curves for the weighted isotropic relaxed micromorphic model. ( http://arxiv.org/abs/ 1610.03296)
35. Madeo A, , Neff P, , Aifantis EC, , Barbagallo G, , d'Agostino MV. 2016 On the role of micro-inertia in enriched continuum mechanics. ( http://arxiv.org/abs/1607.07385)
36. Madeo A, , Neff P, , Barbagallo G, , d'Agostino MV, , Ghiba I-D. 2016 A review on wave propagation modeling in band-gap metamaterials via enriched continuum models. ( http://arxiv.org/abs/1609.01073)
37. Münch I, , Neff P. 2016 Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy. ( http://arxiv.org/abs/1603.06153)
38. Gilbert GT. 1991. Positive definite matrices and Sylvester's criterion. Am. Math. Month. 98, 44–46. (doi:10.2307/2324036)
39. Gazis DC, Wallis RF 1962. Extensional waves in cubic crystals plates. In Proc. 4th U.S. National Congress of Applied Mechanics, Berkeley, CA, 18–21 June, 1962, pp. 161–168. New York, NY: American Society of Mechanical Engineers.
40. Jeong J, Neff P 2010. Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids 15, 78–95. (doi:10.1177/1081286508093581)
41. Jeong J, Ramézani H, Münch I, Neff P 2009. A numerical study for linear isotropic Cosserat elasticity with conformally invariant curvature. Z. Angew. Math. Mech. 89, 552–569. (doi:10.1002/zamm.200800218)
42. Neff P, Jeong J 2009. A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. Z. Angew. Math. Mech. 89, 107–122. (doi:10.1002/zamm.200800156)
43. Neff P, Jeong J, Fischle A 2010. Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature. Acta Mech. 211, 237–249. (doi:10.1007/s00707-009-0230-z)
44. Altenbach H, Eremeyev VA, Lebedev LP, Rendón LA 2010. Acceleration waves and ellipticity in thermoelastic micropolar media. Arch. Appl. Mech. 80, 217–227. (doi:10.1007/s00419-009-0314-1)
45. Eremeyev VA. 2005. Acceleration waves in micropolar elastic media. Doklady Phys. 50, 204–206. (doi:10.1134/1.1922562)
46. Neff P. 2008 Relations of constants for isotropic linear Cosserat elasticity. Technical report, Fachbereich Mathematik, Technische Universituat Darmstadt, Darmstadt, Germany. See http://www.unidue.de/%7ehm0014/Cosserat_files/web_coss_relations.pdf (note: typographical errors in equations 2.8, 2.9, 2.10).
47. Bigoni D, Gourgiotis PA 2016. Folding and faulting of an elastic continuum. Proc. R. Soc. A 472, 20160018 (doi:10.1098/rspa.2016.0018) [PMC free article] [PubMed]
48. Gourgiotis PA, Bigoni D 2016. Stress channelling in extreme couple-stress materials. Part I: strong ellipticity, wave propagation, ellipticity, and discontinuity relations. J. Mech. Phys. Solids 88, 150–168. (doi:10.1016/j.jmps.2015.09.006)
49. Ghiba I-D, , Neff P, , Madeo A, , Münch I. 2016 A variant of the linear isotropic indeterminate couple-stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and orthogonal boundary conditions. Math. Mech. Solids. ()
50. Münch I, , Neff P, , Madeo A, , Ghiba I-D. 2015 The modified indeterminate couple stress model: why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. ( http://arxiv.org/ abs/1512.02053)
51. Madeo A, Ghiba I-D, Neff P, Münch I 2016. A new view on boundary conditions in the Grioli–Koiter–Mindlin–Toupin indeterminate couple stress model. Eur. J. Mech. A, Solids 59, 294–322. (doi:10.1016/j.euromechsol.2016.02.009)

Articles from Proceedings. Mathematical, Physical, and Engineering Sciences are provided here courtesy of The Royal Society