Gianluca Rizzi
email: gianluca.rizzi@insa-lyon.fr

Geralf Hütter

Ionel Hassam Khan

Dumitrel Ghiba

Angela Madeo

Patrizio Neff

Ionel Dumitrel Ghiba

Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

Keywords: generalized continua, torsion, torsional stiffness, characteristic length, size-effect, micromorphic continuum, Cosserat continuum, couple stress model, gradient elasticity, micropolar, relaxed micromorphic model, micro-stretch model, micro-strain model, micro-void model, bounded stiffness

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In this paper we continue our endeavour to find analytical solutions to simple boundary value problems for families of generalized continua [START_REF] Altenbach | Higher Gradient Materials and Related Generalized Continua[END_REF][START_REF] Rizzi | Analytical solutions of the simple shear problem for micromorphic models and other generalized continua[END_REF][START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF]. The focus is on non-homogeneous solutions that on one side activate the additional deformations modes offered by generalized continua (curvature terms) and which may be used, on the other side, in calibrating the additional (many) material parameters. The renewed interest in models of generalized continua comes in part from the fact that for small specimens one may observe size-effects, not accounted for by linear Cauchy elasticity. On the other hand, the description of man-made architecture materials/meta-materials need generalized continua to capture frequency band-gaps in the dynamic range, a prominent example being given by the relaxed micromorphic model [START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2d Cauchy/relaxed-micromorphic interface[END_REF][START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF][START_REF] Madeo | Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design[END_REF][START_REF] Rizzi | Exploring metamaterials' structures through the relaxed micromorphic model: switching an acoustic screen into an acoustic absorber[END_REF][START_REF] Rizzi | Towards the conception of complex engineering meta-structures: relaxedmicromorphic modelling of mechanical diodes[END_REF].

Here, we consider the static St. Venant torsion problem. Since we aim at identifying material parameters, let us first review what can be said for isotropic linear elasticity.

Material parameters in linear elasticity vs. generalized continua

The determination of the two constitutive material parameters in isotropic linear elasticity can be achieved in several different ways. For example the Young's modulus and Poisson's ratio

E macro = 9κ macro µ macro 3κ macro + µ macro , ν macro = 3κ macro -2µ macro 2(3κ macro + µ macro ) , (1)
λ macro = 3κ macro -2µ macro 3 , κ macro = 2µ macro + 3λ macro 3 , (2)
can be uniquely determined from a homogeneous macroscopic tension-compression test. Moreover, the shear modulus µ macro and the Young's modulus E macro can also be identified from the inhomogeneous torsion and bending test, respectively. Indeed, the classical torsional stiffness (per unit length) of a circular rod is given by

T macro = µ macro I p = µ macro πR 4 2 , (3)
and the bending stiffness (per unit length and per unit thickness) in cylindrical bending [START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF] is equivalent to

D macro = h 3 12 E macro (1 -ν 2 macro ) = h 3 12
4µ macro (3κ macro + µ macro ) 3κ macro + 4µ macro . Figure 1: Geometry of the torsion problem: according to the St.Venant principle, we do not consider how the resultant end torque is applied. Furthermore, we assume that each cross-section (orthogonal to x 3 ) rotates as a rigid body with a constant rate of twist ∂Θ ∂x3 = ϑ. Since there is no warping, every cross-section remains in the same plane before and after the deformation. Note that the final solution for linear elasticity satisfies this ansatz only to within first order in the rate of twist ϑ, see Fig. 2.

A third independent identification can be achieved with dynamic measurements, determining the shear wave speed (c s ) and the pressure wave speed (c p )

c s = µ macro ρ , c p = 2µ macro + λ macro ρ . (5)
In reality, all these three methods may lead to slightly different values when used to fit real experiments due to the experimental set up. Nevertheless, they all are useful in complementing the identification procedure. We note that all mentioned tests convey a precise physical meaning to the appearing material parameters and this greatly helps in the mechanical application of linear elasticity to real world structures.

The situation is much more involved when trying to determine material parameters for generalized continua. Even when restricting the attention to linear and isotropic response, the number of additional parameters increase significantly and it is also not clear a priori what the physical meaning of the additional parameters really is. Lakes [START_REF] Lakes | Webpage[END_REF][START_REF] Lakes | Experimental methods for study of Cosserat elastic solids and other generalized elastic continua[END_REF] has prominently investigated the fitting procedure for isotropic Cosserat solids. In the linear isotropic Cosserat model (Section 6) there appear already six independent parameters and a series of experiments with differently sized materials allows to determine the Cosserat constants. A decisive tool for that purpose is the analytical solution for torsion and bending, which is already available in the literature [START_REF] Lakes | Bending of a Cosserat elastic bar of square cross section: Theory and experiment[END_REF][START_REF] Lurie | Bending problems in the theory of elastic materials with voids and surface effects[END_REF][START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF]. The Cosserat model allows to describe size-effects in the sense that more slender specimen have an increased apparent stiffness' in bending and torsion. However, it is observed that the Cosserat model does have an unphysical stiffness singularity in bending [START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF] for a zero slenderness limit, the same appears in general in torsion (Section 6) but can be avoided upon setting to zero some curvature parameters (Sec 6.1). The mentioned stiffness singularity is not only an academic issue, but it concerns the stable identification of the material parameters [START_REF] Neff | Linear Cosserat elasticity, Conformal Curvature and Bounded Stiffness[END_REF]. Yet, in the Cosserat theory, the Young's modulus E macro and the Poisson's ratio ν macro can still be determined in a size-independent manner with a homogeneous tension-compression test. In question are the so-called Cosserat couple modulus µ c ≥ 0 and the three curvature parameters.

A first extension of the Cosserat model is the so-called micro-stretch model, which allows for infinitesimal rotation and volume stretch as independent kinematic fields. For the micro-stretch model we show that the additional kinematic degree of freedom is not activated in the torsion problem.

Another extension of the Cosserat model is given by the recently introduced relaxed micromorphic model [START_REF] Neff | Existence of minimizers for a finite-strain micromorphic elastic solid[END_REF][START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF][START_REF] Sky | A hybrid H 1 ×H(curl) finite element formulation for a relaxed micromorphic continuum model of antiplane shear[END_REF] (Section 4). In its static isotropic version it features only 8 independent material parameters comparing favourably to the large number of constitutive parameters in the classical micromorphic model. While the kinematics of the relaxed micromorphic model coincides with the classical micromorphic model (9 additional degrees of freedom: stretch, shrink, shear, rotations) the curvature term is a direct extension of the Cosserat curvature written in terms of Curl P . An important advantage of the model compared to the Cosserat model is that there is no stiffness singularity in whatsoever situation and four of the eight constants (µ macro , λ macro , µ e , and λ e ) can be determined ab initio from size-independent homogeneous tests [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]. There remains to fit three curvature parameters and the Cosserat couple modulus µ c ≥ 0 (which in some situations may be chosen to be zero since the model remains well-posed) [START_REF] Bauer | Dev-Div-and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions[END_REF][START_REF] Lewintan | Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy[END_REF][START_REF] Lewintan | Nečas-Lions lemma reloaded: An L p -version of the generalized Korn inequality for incompatible tensor fields[END_REF][START_REF] Lewintan | L p -trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions[END_REF][START_REF] Neff | Poincaré meets Korn via Maxwell: extending Korn's first inequality to incompatible tensor fields[END_REF].

Another advantage of the isotropic relaxed micromorphic model is given by the fact that it can replace the isotropic Cosserat model in a straightforward manner without additional costs. Indeed, the Cosserat curvature parameters can be taken as such as well as the Cosserat couple modulus µ c . The only new parameter set is µ micro , λ micro , an estimate of which can be inferred from the small-scale response [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]. Regularity and continuous dependence results for the relaxed micromorphic model have been obtained in [START_REF] Ghiba | Existence results for non-homogeneous boundary conditions in the relaxed micromorphic model[END_REF][START_REF] Ghiba | The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics[END_REF][START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF][START_REF] Owczarek | A note on local higher regularity in the dynamic linear relaxed micromorphic model[END_REF] and first FEM-implementations in H(Curl)-space are presented in [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF].

Next, the micro-strain model [START_REF] Forest | Nonlinear microstrain theories[END_REF] is in a sense complementary to the Cosserat model: it assumes an additional strain like symmetric field S as extra degree of freedom. Here, we recover a simplified microstrain model without mixed terms and a choice for the curvature parameters, see also [START_REF] Shaat | A reduced micromorphic model for multiscale materials and its applications in wave propagation[END_REF] who considers a degenerate micro-strain model in disguise. We recover the analytical solution given by Hütter [START_REF] Hütter | Application of a microstrain continuum to size effects in bending and torsion of foams[END_REF] for the micro-strain torsion problem. It turns out that for bending [START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF], simple shear [START_REF] Rizzi | Analytical solutions of the simple shear problem for micromorphic models and other generalized continua[END_REF] and torsion (Section 10) the micro-strain solution does not show a stiffness singularity either. However, this is not a general feature of the micro-strain model, but only related to the restricted kinematic possibilities: bending and torsion activate prominently rotations, but these are "filtered out" in the micro-strain model. Therefore, bounded stiffness in bending and torsion should come as no surprise. Next, we combine the Cosserat and the micro-strain ansatz in a novel ad-hoc model whose response is nevertheless governed by the Cosserat kinematics.

Lastly, we have the full micromorphic model [START_REF] Eringen | Microcontinuum Field Theories[END_REF][START_REF] Mindlin | Micro-structure in linear elasticity[END_REF]. The kinematics is augmented with a non-symmetric micro-distortion tensor P (as for the relaxed micromorphic model, too) but the curvature energy depends on the full gradient DP of the micro-distortion. For simplicity and for comparison, we consider a subclass without mixed terms and simplified curvature expression. In general, the bending and torsion responses show a stiffness singularity, which can be avoided in torsion by a very special ansatz for the curvature energy. However, nonphysical stiffness singularities cannot, in general, be avoided. Our investigation is complemented by considering the strain-gradient models and its couple-stress subclass. The reason for the singular stiffening behaviour in the other generalized continuum models (except the relaxed micromorphic one) can be connected to their non-redundant formulation of the curvature measure [START_REF] Romano | Micromorphic continua: non-redundant formulations[END_REF].

An alternative method to study the deformation of (finite) elastic cylinders is the semi-inverse method introduced by Ieşan in [START_REF] Ieşan | On Saint-Venant's problem[END_REF][START_REF] Iesan | Saint-Venant's Problem[END_REF], see also [START_REF] Ghiba | Saint-venant's principle[END_REF]. This method was also successfully used to study the deformation of elastic cylinders with microstructures, see [START_REF] Bulgariu | On the thermal stresses in anisotropic porous cylinders[END_REF][START_REF] Ghiba | Semi-inverse solution for Saint-Venant's problem in the theory of porous elastic materials[END_REF][START_REF] Ghiba | On the deformation of transversely isotropic porous elastic circular cylinder[END_REF] and the book [START_REF] Iesan | Classical and Generalized Models of Elastic Rods[END_REF], in which many of Ieşan's results were unified. Regarding the semi-inverse method, all the results obtained in the classical micromorphic theory and all its subclasses (Cosserat, micro-stretch, micro-voids) are obtained by assuming that the internal energy is positive definite in terms of DP . To the contrary, in the framework of the relaxed micromorphic model, the present results are valid also for internal energies which are not positive definite in terms of DP , but rather in terms of Curl P . We recall that an internal energy which is positive definite in terms of Curl P is only semi-positive definite in terms of DP .

The paper is now structured as follows. After fixing our notation in Section 1.2 we shortly dwell on the formulation of the torsion problem in adapted variables, making it clear that we do not revert to express stresses and moments in cylindrical coordinate but we always use a Cartesian expression written in suitable variables. To set the stage we recall the linear isotropic torsion problem, which will then be suitably generalized.

Notation

For vectors a, b ∈ R n , we define the scalar product a, b := n i=1 a i b i ∈ R, the euclidean norm a 2 := a, a and the dyadic product a⊗b := (a i b j ) i,j=1,...,n ∈ R n×n . In the same way, for tensors P , Q ∈ R n×n , we define the scalar product P , Q := n i,j=1 P ij Q ij ∈ R and the Frobenius-norm P 2 := P , P . Moreover,

Sym(3) := {S ∈ R 3×3 | S T = S}.
Using the one-to-one map axl : so(3) → R 3 we have

A b = axl(A) × b ∀ A ∈ so(3) , b ∈ R 3 . ( 6
)
where × denotes the cross product in R 3 . The inverse of axl is denoted by Anti: R 3 → so(3). The Jacobian matrix Du and the curl for a vector field u are defined as

Du =   u 1,1 u 1,2 u 1,3 u 2,1 u 2,2 u 2,3 u 3,1 u 3,2 u 3,3   , curl u = ∇ × u =   u 3,2 -u 2,3 u 1,3 -u 3,1 u 2,1 -u 1,2   . (7)
We also introduce the Curl and the Div operators of the

) T    . (8)
The cross product between a second order tensor and a vector is also needed and is defined row-wise as follow

m × b =   (b × (m 11 , m 12 , m 13 ) T ) T (b × (m 21 , m 22 , m 23 ) T ) T (b × (m 31 , m 32 , m 33 ) T ) T   = m • • b = m ik kjh b h , (9)
where m ∈ R 3×3 , b ∈ R 3 , and is the Levi-Civita tensor. The two indices contraction : is intended as

B : ∇ m = B ip m ijk,p = N jk , B : m = B ij m ijk = b k , (10)
where B and N are second order tensors, m is a third order tensor, and b is a vector.

Cartesian variables expressed through cylindrical variables

To address the torsional problem in its natural environment but with the comfort of the classical Cartesian coordinate system, we introduce the cylindrical set of coordinates which allows us to express the classic Cartesian orthogonal set of coordinates x = {x 1 , x 2 , x 3 } through a more suitable set of variables r = {r, ϕ, z}, without switching completely to a cylindrical coordinate system, i.e., without expressing all the quantities (strains, stresses etc.) in the basis corresponding to the cylindrical coordinates

e r =   cos ϕ sin ϕ 0   , e ϕ =   -sin ϕ cos ϕ 0   , e z =   0 0 1   . (11)
The relations for the coordinates are

x 1 = r cos ϕ , x 2 = r sin ϕ , x 3 = z , (12)
while the relations between the first and the second derivative of a generic vector field f are

∂ f i (r, ϕ, z) ∂ r j = ∂ f i (r, ϕ, z) ∂ x k ∂ x k ∂ r j , ∂ 2 f i (r, ϕ, z) ∂ r j ∂ r k = ∂ 2 f i (r, ϕ, z) ∂ x m ∂ x n ∂ x m ∂ r j ∂ x n ∂ r k + ∂ f i (r, ϕ, z) ∂ x m ∂ 2 x m ∂ r j ∂ r k . (13)
The quantities we want to obtain are ∂ fi(r,ϕ,z) ∂ x k and ∂ 2 fi(r,ϕ,z) ∂ xm∂ xn , which are obtainable thanks to (13) (see Appendix A for full calculations).

It is emphasized again that we will not represent the torsional problem in cylindrical coordinates (namely all the differential operators, the equilibrium equation, and the kinematic fields), but we will use the classical Cartesian coordinates {x 1 , x 2 , x 3 } parameterized in cylindrical variables {r, ϕ, z}.

Structure of the higher-order ansatz

The ansatz for the displacement field for the cylindrical torsion problem, regardless of the treated model, is always given by

u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   . (14)
Here, ϑ is the rate of twist per unit length. It is highlighted that the displacement field has the third component equal to zero since we are studying a cylindrical rod, whose cross-section is not subjected to warping. The most general ansatz for the micro-distortion tensor, which will be used for the full and the relaxed micromorphic model, is

P (r, ϕ, z) = ϑ   0 -x 3 (z) -g 2 (r) x 2 (r, ϕ) x 3 (z) 0 g 2 (r) x 1 (r, ϕ) g 1 (r) x 2 (r, ϕ) -g 1 (r) x 1 (r, ϕ) 0   , (15)
where g 1 , g 2 : [0, ∞) → R. Starting from the form (15) of the ansatz for P , it is possible to obtain the ansatz for the micro-stretch model (A = skew P and ω1 = tr(P )1), the Cosserat model (A = skew P ), the micro-void model (ω1 = tr(P )1), and the micro-strain model (S = sym P ), by taking the skew-symmetric part, the trace of P , or the symmetric part depending on what is needed. Here are reported the symmetric part, the skew-symmetric part and the trace of P S(r, ϕ, z)

= sym P (r, ϕ, z) = ϑ 2   0 0 g m (r) x 2 (r, ϕ) 0 0 -g m (r) x 1 (r, ϕ) g m (r) x 2 (r, ϕ) -g m (r) x 1 (r, ϕ) 0   , (16)
A(r, ϕ, z) = skew P (r, ϕ, z) = ϑ 2   0 -2x 3 (z) -g p (r) x 2 (r, ϕ) 2x 3 (z) 0 g p (r) x 1 (r, ϕ) g p (r) x 2 (r, ϕ) -g p (r) x 1 (r, ϕ) 0   , (17)
were g p (r) = g 1 (r) + g 2 (r), g m (r) = g 1 (r) -g 2 (r), and ω is not reported since the ansatz (15) has a zero trace.

It is highlighted that each section remains "rigid" is not really correct, since the deformation of a cylinder section due to the displacement field [START_REF] Gauthier | A quest for micropolar elastic constants[END_REF] (which is a linear approximation of a rigid rotation) looks like

x + u(x) =   x 1 x 2 0   +   0 -ϑ x 3 0 ϑ x 3 0 0 0 0 0     x 1 x 2 0   =   x 1 -x 2 x 3 ϑ x 2 + x 1 x 3 ϑ 0   . ( 18
) (a) x x + u ( x ) u(x) ϑ x 2 x 1 (b)
= R cos ϑ -R = 1-cos ϑ cos ϑ R = ϑ 2
2 + 5 24 ϑ 4 + h.o.t. R, being of second order in ϑ. Thus, the linear kinematics is correct to within first order in ϑ, as is well-known.

Of course this radial expansion does not contribute energetically under the small displacement hypothesis, and this can be seen from ( 23), in which it is clear that the symmetric strain tensor ε = sym Du does not depend on x 3 ≡ z.

2 Overview of some generalized continuum models and their interconnections

Considering the following notations for the involved quantities:

u : Ω ⊂ R 3 → R 3 , displacement , P : Ω ⊂ R 3 → R 3×3 , micro-distortion , A : Ω ⊂ R 3 → so(3) , micro-rotation , S : Ω ⊂ R 3 → Sym(3) , micro-strain , ω : Ω ⊂ R 3 → R , micro-dilatation
and the orthogonal decomposition

P = dev symP + skew P + 1 3 tr(P ) 1 = dev S + A + ω 1 (19)
we give the following genealogy tree of the generalized continuum models: The strain gradient theory and second gradient theory are equivalent [START_REF] Altenbach | Higher Gradient Materials and Related Generalized Continua[END_REF][START_REF] Mindlin | Micro-structure in linear elasticity[END_REF], and contain additionally the couple stress theory as a special case. Using the Curl as primary differential operator for the curvature terms allows a neat unification of concepts.

Torsional problem for the isotropic Cauchy continuum

In order to set up a comparison with the models we will present in the next sections, we start by presenting the solution of the classical cylindrical torsional problem. The strain energy for a linear elastic isotropic Cauchy continuum is

W (Du) = µ macro sym Du 2 + λ macro 2 tr 2 (Du) , (20)
where λ macro and µ macro are the macroscopic Lamé constants.

In terms of the symmetric Cauchy stress tensor σ = 2 µ macro sym Du + λ macro tr (Du) 1, where ε = sym Du denotes the classical symmetric strain tensor, the equilibrium equation (in the absence of body forces) and the Neumann lateral boundary conditions (at the free surface) are

Div σ = 0 , t(r = R) = σ(r = R) • e r = 0 . ( 21
)
Our aim is to study a state of uniform torsion ϑ for an infinitely extended cylindrical rod. According to the cylindrical reference system shown in Fig. 1, the ansatz for the displacement is

u(x 1 , x 2 , x 3 ) = u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   = ϑ   -z r sin ϕ z r cos ϕ 0   , ( 22
)
where ϑ is the angle of twist per unit length. It is underlined that the third component of the displacement is chosen equal to zero since the cross-section is circular and therefore no warping is expected. The gradient of the displacement and its symmetric part are (the gradient is always taken with respect to the Cartesian coordinate system and then rewritten in the variables {r, ϕ, z})

Du = ϑ   0 -z -r sin ϕ z 0 r cos ϕ 0 0 0   , ε = sym Du = ϑ 2   0 0 -r sin ϕ 0 0 r cos ϕ -r sin ϕ r cos ϕ 0   . ( 23
)
Substituting the ansatz [START_REF] Hassanpour | Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations[END_REF] in the equilibrium equation [START_REF] Hadjesfandiari | Couple stress theory for solids[END_REF], it is easy to verify that they are identically satisfied.

In order to help the geometric interpretation of the torque, see Fig. 3, we present its expression in Cartesian coordinates along with its representation in the cylindrical variables

M c (ϑ) := Γ twisting force per unit area traction σ e 3 ,   -x 2 x 1 0   1 x 2 1 + x 2 2
length of lever arm

x 2 1 + x 2 2 dx 1 dx 2 (24) = Γ x 1 σ 23 -x 2 σ 13 dx 1 dx 2 = 2π 0 R 0 σ e z , e ϕ r r dr dϕ , (25)
where e 3 = e z = (0, 0, 1) is the unit vector aligned with the mid-axis of the cylindrical rod. The torque (or moment of torsion) about the x 3 -axis and energy (per unit length dx 3 ) expressions are

x 1 x 2 0 -x 2 x 1 0 1 x 1 2 + x 2 2 Γ x 2 x 1
M c (ϑ) := 2π 0 R 0 σ e z , e ϕ r r dr dϕ = µ macro πR 4 2 ϑ = µ macro I p ϑ = T macro ϑ , (26)
W tot (ϑ) := 2π 0 R 0 W (Du) r dr dϕ = 1 2 µ macro πR 4 2 ϑ 2 = 1 2 µ macro I p ϑ 2 = 1 2 T macro ϑ 2 ,
where µ macro is the macroscopic shear modulus,

I p = πR 4 2
is the polar moment of inertia, and T macro = µ macro I p is the torsional stiffness. It is also highlighted that

d dϑ W tot (ϑ) = M c (ϑ) = T macro ϑ , d 2 dϑ 2 W tot (ϑ) = T macro . ( 27
)
Here and in the remainder of this work, the elastic coefficients µ i , λ i , κ i are expressed in [MPa], the coefficients a i are dimensionless, the characteristic lengths L c and the radius R in meter [m], the rate of twist ϑ in [1/m].

Torsional problem for the isotropic relaxed micromorphic model

The relaxed micromorphic model, contrary to all the other proposals for generalized continua in the literature, lives on two well-defined and separated scales, each describing linear elastic response: the classical macroscopic response (characteristic length L c → 0, available for experiments with large specimens) is described as usual by

E macro = 9κ macro µ macro 3κ macro + µ macro , ν macro = 3κ macro -2µ macro 2(3κ macro + µ macro ) , (28)
λ macro = 3κ macro -2µ macro 3 , κ macro = 2µ macro + 3λ macro 3 . ( 29
)
The macroscopic parameters can be uniquely determined from a homogeneous macroscopic tension-compression test. However, the shear modulus µ macro and the Young's modulus E macro can also be identified from the inhomogeneous torsion and bending test, respectively. Indeed, the classical torsional stiffness of a circular rod is given by

T macro = µ macro I p = µ macro πR 4 2 . ( 30
)
The microscopic scale (appearing for L c → ∞), representing a surrogate stiffness connected to the smallest meaningful scale of the material is described by the parameters

E micro = 9κ micro µ micro 3κ micro + µ micro , ν micro = 3κ micro -2µ micro 2(3κ micro + µ micro ) , (31)
λ micro = 3κ micro -2µ micro 3 , κ micro = 2µ micro + 3λ micro 3 , (32)
The macroscopic parameters µ macro and λ macro do not directly intervene in the formulation of the relaxed micromorphic model [START_REF] Lakes | Webpage[END_REF], but the connection is necessarily given by the Reuss-like homogenization formula [START_REF] Reuß | Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle[END_REF]

µ macro = µ e µ micro µ e + µ micro ⇐⇒ µ e = µ macro µ micro µ micro -µ macro , (33)
κ macro = κ e κ micro κ e + κ micro ⇐⇒ κ e = κ macro κ micro κ micro -κ macro .

Figure 4: Macro and micro-scale stiffness governed by two springs in series. If µ micro → ∞, this implies that µ macro = µ e . In all suitable cases for our family of considered generalized continua (depending on the kinematics), we use the same/similar lower order energy expression (the energy without curvature).

Note that the Cosserat couple modulus µ c ≥ 0 is not appearing in the homogenization formulas [START_REF] Lakes | Bending of a Cosserat elastic bar of square cross section: Theory and experiment[END_REF]. As a consequence, both parameter sets ( 29)-( 33) can be identified independently of the scale consideration (being particularly careful with the techniques for the micro-parameters identification) and they uniquely determine the meso-scale parameter set µ e , λ e appearing in (33) 2 .

The general expression of the strain energy for the isotropic relaxed micromorphic continuum is where (µ e ,λ e ), (µ micro ,λ micro ), µ c , L c > 0, and (a 1 ,a 2 ,a 3 ) are 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, respectively. This energy expression represents the most general isotropic form possible for the relaxed micromorphic model. It is important to underline that, given the subsequent ansatz [START_REF] Lewintan | L p -trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions[END_REF], it holds that skew Curl P = 0. This reduces immediately the number of curvature parameters appearing in the torsion solution. In the absence of body forces, the equilibrium equations are then Div σ:=

[2µ e sym (Du -P ) + λ e tr (Du -P ) 1 + 2µ c skew (Du -P )] = 0, σ -2µ micro sym P -λ micro tr (P ) 1 -µ L 2 c Curl a 1 dev sym Curl P + The boundary conditions at the lateral free surface are

t(r = R) = σ(r) • e r = 0 R 3 , (traction free) , (36) η
(r = R) = m(r) • • e r = m(r) × e r = 0 R 3×3 , ( moment free)
,

where m = µ L 2 c a 1 dev sym Curl P + a 3 3 tr (Curl P ) 1 (37)
is a generalized non-symmetric second order moment tensor, the (non-symmetric) force-stress tensor σ is given in ( 35), e r is the radial unit vector, and is the Levi-Civita tensor. The vector t(r) ∈ R 3 is the generalised traction and the tensor η(r) ∈ R 3×3 is called the generalized double traction tensor. According to the cylindrical reference system shown in Fig. 1, the ansatz for the displacement and for the microdistortion P is

u(x 1 , x 2 , x 3 ) = u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   , P (x 1 , x 2 , x 3 ) = P (r, ϕ, z) = ϑ   0 -x 3 (z) -g 2 (r) x 2 (r, ϕ) x 3 (z) 0 g 2 (r) x 1 (r, ϕ) g 1 (r) x 2 (r, ϕ) -g 1 (r) x 1 (r, ϕ) 0   , (38)
where x 1 (r, ϕ) = r cos ϕ, x 2 (r, ϕ) = r sin ϕ, and x 3 (z) = z. The Cartesian Du and the Cartesian Curl P expressed in the cylindrical variables (r, ϕ, z) are

Du(r, ϕ, z) = ϑ   0 -z -r sin ϕ z 0 r cos ϕ 0 0 0   , (39)
Curl P (r, ϕ, z) = ϑ   1 -g 2 (r) -r g 2 (r) sin 2 ϕ r g 2 (r) sin ϕ cos ϕ 0 r g 2 (r) sin ϕ cos ϕ 1 -g 2 (r) -r g 2 (r) cos 2 ϕ 0 0 0 -(2 g 1 (r) + r g 1 (r))   .
It can be remarked that Curl P is symmetric.

Inserting the ansatz (38)-( 39) in [START_REF] Lakes | Experimental methods for study of Cosserat elastic solids and other generalized elastic continua[END_REF], the 12 equilibrium equations are reduced to the following 4 ordinary differential equilibrium equations

1 3 ϑ sin ϕ r µ L 2 c ((a 1 -a 3 ) g 1 (r) -(2a 1 + a 3 ) g 2 (r)) + 3µ c (g 1 (r) + g 2 (r) -1) -3 (µ e + µ micro ) (g 1 (r) -g 2 (r)) -3µ e ) + 3µ L 2 c ((a 1 -a 3 ) g 1 (r) -(2a 1 + a 3 ) g 2 (r)) = 0 , 1 3 ϑ cos ϕ r µ L 2 c ((a 3 -a 1 ) g 1 (r) + (2a 1 + a 3 ) g 2 (r)) -3 µ c (g 1 (r) + g 2 (r) -1) +3 (µ e + µ micro ) (g 1 (r) -g 2 (r)) + 3µ e ) + 3 µ L 2 c ((a 3 -a 1 ) g 1 (r) + (2a 1 + a 3 ) g 2 (r)) = 0 , (40) 1 3 ϑ sin ϕ r µ L 2 c ((2a 1 + a 3 ) g 1 (r) + (a 3 -a 1 ) g 2 (r)) -3 (µ c (g 1 (r) + g 2 (r) -1) + (µ e + µ micro ) (g 1 (r) -g 2 (r))) -3µ e ) + 3 µ L 2 c ((2a 1 + a 3 ) g 1 (r) + (a 3 -a 1 ) g 2 (r)) = 0 , 1 3 ϑ cos ϕ r µ L 2 c ((a 1 -a 3 ) g 2 (r) -(2a 1 + a 3 ) g 1 (r)) + 3 (µ c (g 1 (r) + g 2 (r) -1) + (µ e + µ micro ) (g 1 (r) -g 2 (r))) + 3µ e ) + 3 µ L 2 c ((a 1 -a 3 ) g 2 (r) -(2a 1 + a 3 ) g 1 (r)) = 0 .
It is important to underline that (35) 1 is identically satisfied, and that from the entire set of four equilibrium equations [START_REF] Madeo | Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design[END_REF] It is also pointed out that the two remaining linearly independent equations (40) 1,3 can be uncoupled and are of the Bessel ODE type (see Appendix B). Indeed, if we take their sum and difference, while being careful of substituting g p (r) = g 1 (r) + g 2 (r) and g m (r) = g 1 (r) -g 2 (r) along with their derivatives, we deduce

ϑ sin ϕ a 1 µ L 2 c (3 g m (r) + r g m (r)) -2 r µ e (g m (r) + 1) -2 r g m (r) µ micro = 0 , (41) 1 3
ϑ sin ϕ 6 r µ c (g p (r) -1) -µ L 2 c (a 1 + 2a 3 ) 3 g p (r) + r g p (r) = 0 .

Since g 1 (r) := gp(r)+gm(r) 2 and g 2 (r) := gp(r)-gm(r)

2

, the solution in terms of g 1 (r) and g 2 (r) of ( 41) is

g 1 (r) = 1 2   1 - iA 1 I 1 r f2 Lc -A 2 Y 1 -i r f2 Lc + iB 1 I 1 r f1 Lc -B 2 Y 1 -i r f1 Lc r - µ e µ e + µ micro   , g 2 (r) = 1 2   1 + iA 1 I 1 r f2 Lc -A 2 Y 1 -i r f2 Lc -iB 1 I 1 r f1 Lc + B 2 Y 1 -i r f1 Lc r + µ e µ e + µ micro   , ( 42
)
f 1 := 6µ c (a 1 + 2a 3 ) µ , f 2 := 2(µ e + µ micro ) a 1 µ ,
where I n (•) is the modified Bessel function of the first kind, Y n (•) is the Bessel function of the second kind (see appendix B for the formal definitions), and

A 1 , B 1 , A 2 , B 2 are integration constants.
The values of A 1 , B 1 are determined from the boundary conditions [START_REF] Lewintan | Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy[END_REF], while, due to the divergent behaviour of the Bessel function of the second kind at r = 0, we have to set A 2 = 0 and B 2 = 0 in order to have a continuous solution. The fulfilment of the boundary conditions [START_REF] Lewintan | Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy[END_REF] allows us to find the expressions of the integration constants

A 1 = i L c 3f 1 R z 1 I 0 R f1 Lc -2L c I 1 R f1 Lc f 2 L c I 0 R f2 Lc I 1 R f1 Lc + f 1 z 1 I 0 R f1 Lc L c I 1 R f2 Lc -2f 2 R I 0 R f2 Lc µ micro µ e + µ micro , (43)
B 1 = i L c f 2 R I 0 R f2 Lc -2L c I 1 R f2 Lc f 2 L c I 0 R f2 Lc I 1 R f1 Lc + f 1 z 1 I 0 R f1 Lc L c I 1 R f2 Lc -2f 2 R I 0 R f2 Lc µ micro µ e + µ micro , z 1 := a 1 + 2a 3 3a 1 .
The classical torque, the higher-order torque, and energy (per unit length dz) expressions are

M c (ϑ) := 2π 0 R 0 σ e z , e ϕ r r dr dϕ =     4µ c f 2 I 2 R f1 Lc I 2 R f2 Lc Lc R 2 µ e f 1 f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc Lc R -f 2 I 0 R f2 Lc I 1 R f1 Lc Lc R + f 1 z 1 I 0 R f1 Lc 24I 1 R f2 Lc Lc R 3 -12f 2 I 0 R f2 Lc Lc R 2 -f 2 2 I 1 R f2 Lc Lc R + 2f 3 2 I 0 R f2 Lc f 2 2 f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc R Lc -f 2 I 0 R f2 Lc I 1 R f1 Lc R Lc - I 1 R f1 Lc 16I 1 R f2 Lc Lc R 4 -8f 2 I 0 R f2 Lc Lc R 3 + f 3 2 I 0 R f2 Lc Lc R f 2 2 f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc R Lc -f 2 I 0 R f2 Lc I 1 R f1 Lc R Lc   µ e µ micro µ e + µ micro   I p ϑ = T c ϑ , M m (ϑ) :=

R f2

Lc

q 1 I 0 R f1 Lc Lc R 2 -q 2 I 1 R f1 Lc Lc R 3 3 f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc Lc R -f 2 I 0 R f2 Lc I 1 R f1 Lc Lc R 4µ µ micro 3 (µ e + µ micro )   I p ϑ = T m ϑ , W tot (ϑ) := 2π 0 R 0 W (Du, P , CurlP ) r dr dϕ = 1 2     4µ c f 2 I 2 R f1 Lc I 2 R f2 Lc Lc R 2 µ e f 1 f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc Lc R -f 2 I 0 R f2 Lc I 1 R f1 Lc Lc R + f 1 z 1 I 0 R f1 Lc 24I 1 R f2 Lc Lc R 3 -12f 2 I 0 R f2 Lc Lc R 2 -f 2 2 I 1 R f2 Lc Lc R + 2f 3 2 I 0 R f2 Lc f 2 2 f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc R Lc -f 2 I 0 R f2 Lc I 1 R f1 Lc R Lc - I 1 R f1 Lc 16I 1 R f2 Lc Lc R 4 -8f 2 I 0 R f2 Lc Lc R 3 + f 3 2 I 0 R f2 Lc Lc R f 2 2 f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc R Lc -f 2 I 0 R f2 Lc I 1 R f1 Lc R Lc + 4µ I 2 R f2 Lc q 1 I 0 R f1 Lc Lc R 2 -q 2 I 1 R f1 Lc Lc R 3 3 3µ e f 1 z 1 I 0 R f1 Lc 2f 2 I 0 R f2 Lc -I 1 R f2 Lc Lc R -f 2 I 0 R f2 Lc I 1 R f1 Lc Lc R   µ e µ micro µ e + µ micro   I p ϑ 2 = 1 2 T w ϑ 2 , q 1 := 3a 1 f 1 f 2 z 1 , q 2 := 2f 2 (a 1 -a 3 ) . Again it holds d dϑ W tot (ϑ) = M c (ϑ) + M m (ϑ) , d 2 dϑ 2 W tot (ϑ) = T c + T m = T w . (45)
Both quantities M c and W tot are immediately accessible in any higher order generalized continuum model. However, the precise form of M m is difficult to guess. The latter identity can, therefore, also be seen as an implicit definition of the higher order moment M m . In the Appendix we will provide an independent way of obtaining the notation for M m starting form considerations done on the Cosserat model (see Appendix C). We provide again the homogenization relations between the macro-parameters, the meso-(with index (•) e ), and the micro-parameters [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF][START_REF] Neff | A unifying perspective: the relaxed linear micromorphic continuum[END_REF] µ macro = µe µmicro µe+µmicro , κ macro = κe κmicro κe+κmicro , with

κ i = 2µi+3λi 3 , i = {e, micro, macro} , (46)
which can be used to define the following torsional stiffnesses

T macro = µ macro I p = µ micro µ e µ micro + µ e I p , T micro = µ micro I p , T e = µ e I p . (47)
The plots of the torsional stiffness for the classical torque (light blue), the higher-order torque (red), and the torque energy (green) for µ c = {0, 1/2, ∞} while varying L c is shown in Fig. 5.

µ micro = 1/4, a 1 = 1/5, a 2 = 1/6, a 3 = 1/7, R = 1.
It is here highlighted that the torsional stiffness obtainable from the energy T w is the only stiffness available experimentally.

Limits

The relaxed micromorphic model with symmetric force stresses (µ

c → 0) W tot (ϑ) := 2π 0 R 0 W (Du, P , CurlP ) r dr dϕ = 1 2   f 2 I 0 Rf2 Lc v 2 µ micro L 2 c R 2 + µ e (µ e + µ micro ) f 2 I 0 Rf2 Lc -2v 1 I 1 Rf2 Lc Lc R (48)
- Note that the torsional stiffness at the micro-scale T micro is here independent of the Cosserat couple modulus µ c , see [START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF]. In the particular case for which the parameter a 3 is equal to zero the elastic energy turns into

2I 1 Rf2 Lc v 2 µ micro L 2 c R 2 + v 1 µ e (µ e + µ micro ) Lc R f 2 I 0 Rf2 Lc -2v 1 I 1 Rf2 Lc Lc R   µ micro (µ e + µ micro ) 2 I p ϑ 2 , v 1 := a 1 + 2a 3 a 1 + 8a 3 , v 2 := 24a 1 a 3 µ a 1 + 8a 3 . 0 2.5 5
T macro T micro R/L c T w 0 ∞ μ c
W (Du, P , Curl P ) = µ e sym (Du -P ) 2 + λ e 2 tr 2 (Du -P ) + µ c skew (Du -P ) 2 (49)
+

µ micro sym P 2 + λ micro 2 tr 2 (P ) + µ L 2 c 2 a 1 dev sym Curl P 2 .
In this case, the torsional stiffness at the micro scale, namely for

L c → ∞ (R → 0) 1 , depends also on µ c T := lim Lc→∞ T w = µ micro (9µ c + µ e ) (9µ c + µ e ) + µ micro I p . (50)
For µ c → 0 we obtain a linear elastic model with stiffness T macro , for µ c → ∞ it is recovered a model that has T micro at the micro-scale, while for intermediate values of 0 < µ c < ∞ a torsional stiffness between T macro and T micro appears.

µ micro = 1/4, a 1 = 2, R = 1.
In this case, varying µ c influences the torsional stiffness also for small specimen size.

We may consider a further limit in [START_REF] Neff | A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy[END_REF]. It holds

T := lim µmicro→∞ T = (9µ c + µ e ) I p = (9µ c + µ macro ) I p , (51)
where the last relation for which we have µ e = µ macro is obtained from (33) 2 taking µ micro → ∞.

The Cosserat model as a limit of the relaxed micromorphic model (µ micro → ∞)

The Cosserat model can be obtained from the relaxed micromorphic model by formally letting µ micro → ∞ and κ micro → ∞. 2 From the homogenization formula [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF] it is possible to see that for µ micro → µ macro we have µ e → ∞, while µ macro = µ e for µ micro → ∞, which is the stiffness at the macro-scale for the Cosserat model. Sensitivity study for the relaxed micromorphic model while varying a 1 and a 3 independently. The parametric study represented in Fig. 9 has not been carried out for the limit a 1 → 0 and a 3 → 0 since we would have had an indeterminate form for L c → ∞, and that is why we used the symbol ∼ 0. The solution of the problem while having a 3 = 0 a priori is analyzed carefully in Section 4.1.2, and the solution of the problem while having a 1 = 0 a priori make the relaxed model collapse into a classical linear elastic model with torsional stiffness T macro .

0 2.5 5
T macro R/L c T w ∞ μ micro C o ss er a t m o d el (a) 0 2.5 5
T macro T=T  R/L c T w T w relax T w Coss (b)
0 2.5 5
T macro T micro R/L c T w a 1 ∼0 ∞ (a) 0 2.5 5
T macro T micro R/L c T w a 3 ∼0 ∞ (b)

Finite element simulations

Using finite element analysis as a tool of comparison, in this section we will i) test the validity of the solution in terms of the hypothesis of small deformations (i.e., small twist rate); ii) discuss the validity of the St.Venant principle for the relaxed micromorphic model.

In this analysis we take a finite-size cylindrical rod and we apply opposite and equal finite-rotation at both of its ends (z = ±L/2). Accordingly, the boundary conditions are

u(z = ±L/2) =   cos ±Θ sin ±Θ 0 -sin ±Θ cos ±Θ 0 0 0 1     x 1 x 2 L   -   x 1 x 2 L   , (52)
P (z = ±L/2) × e 1 =   -P 12 P 11 0 -P 22 P 21 0 -P 32 P 31 0   =   sin ±Θ cos ±Θ -1 0 1 -cos ±Θ sin ±Θ 0 0 0 0   = Du(z = ±L/2) × e 1 ,
where P (z) × e 1 = Du(z) × e 1 are the consistent boundary conditions on the tangential part for the micro-distortion tensor P . In Fig. 11 it is possible to see how the non identically zero components of the micro-distortion P vary across the diameter aligned to the x 1 -axis (ϕ = π/2) of the cross-section placed in the middle of the cylindrical rod (z = 0). We chose the middle section in order to study the solution far away enough from the disturbance region on which the boundary conditions have been applied. The values of the components of P of Fig. 11 (twist rate ϑ = π/50) are perfectly in agreement with the analytical solution, confirming the validity of the small-deformation solution obtained in Section 4. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

xx xx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
µ c = 1, µ e = 1, µ micro = 1, a 1 = 1, a 3 = 1, R = 1, ϑ = π/50, L = 10.
Furthermore, in Fig. 12 we show how the solution obtained while applying consistent boundary conditions converges to the one obtained analytically in a distance from the boundary which is more or less between one radius and one diameter. This is the pinnacle expression of the Saint Venant principle: we have applied not only a finite-rotation instead of a linearized one, but we have also used consistent boundary conditions for P which we know are different from the correct values that the tangential part of P should have, and we obtained nevertheless the analytical linearized solution after a rather small boundary layer. We describe in particular the component P 31 (other than the component P 13 ) since, due to the consistent boundary conditions, it is forced to start from zero at the lateral boundary. In Fig. 12 we plot this component which are evaluated for the length of the rod on the external surface (ϕ = π/2 and r = R).

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

µ micro = 1, a 1 = 1, a 3 = 1, R = 1, ϕ = π/2.
As it can be seen, the solution does not converge stably and not perfectly symmetrically (the mesh is not symmetric) to the analytical one, but nevertheless it converges rapidly.

In Fig. 13 is reported how the component P 13 vary on the cross section centered in the origin of the reference system (z = 0).

µ micro = 1, a 1 = 1, a 3 = 1, R = 1, ϕ = π/2, L c = 1.
The implications of this results are of great value in the context of the identification of the elastic material parameters: it is clear how to apply consistent boundary conditions on a real sample in a laboratory (Dirichlet hard), and thanks to this results, we now know that our analytical solution is taking place far away enough from the boundary layer.

5 Torsional problem for the isotropic micro-stretch model in dislocation format

In the micro-stretch model in dislocation format [START_REF] Cicco | Torsion and flexure of microstretch elastic circular cylinders[END_REF][START_REF] Kirchner | Mechanics of extended continua: modeling and simulation of elastic microstretch materials[END_REF][START_REF] Neff | A unifying perspective: the relaxed linear micromorphic continuum[END_REF][START_REF] Neff | Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity[END_REF][START_REF] Scalia | Extension, bending and torsion of anisotropic microstretch elastic cylinders[END_REF], contrary to the relaxed micromorphic model, the micro-distortion tensor is devoid from the deviatoric component dev sym P = 0 ⇔ P = A + ω1, A ∈ so(3), ω ∈ R. The expression of the strain energy for this model in dislocation format can be written as [START_REF] Neff | A unifying perspective: the relaxed linear micromorphic continuum[END_REF]:

W (Du, A, ω, Curl (A -ω1)) = µ macro dev sym Du 2 + κ e 2 tr 2 (Du -ω1) + µ c skew (Du -A) 2 + 9 2 κ micro ω 2 (53)
+ µ L 2 c 2 a 1 dev sym Curl A 2 + a 2 skew Curl (A + ω1) 2 + a 3 3 tr 2 (Curl A) ,
since Curl (ω1) ∈ so(3). The equilibrium equations, in the absence of body forces, are then Div

σ:= [2µ macro dev sym Du + κ e tr (Du -ω1) 1 + 2µ c skew (Du -A)] = 0 , 2µ c skew (Du -A) -µ L 2 c skew Curl a 1 dev sym Curl A + a 2 skew Curl (A + ω1) + a 3 3 tr (Curl A) 1 = 0 , (54)
tr 2µ macro dev sym Du

+κ e tr (Du -ω1) 1 -κ micro tr (ω1) 1 -µ L 2 c a 2 Curl skew Curl (ω1 + A) = 0 .
The boundary conditions at the free surface are

t(r = R) = σ(r) • e r = 0 R 3 , η(r = R) = skew (m(r) • • e r ) = skew (m(r) × e r ) = 0 R 3×3 , (55)
γ(r = R) = 1 3 tr (m(r) • • e r ) = 1 3 tr (m(r) × e r ) = 0 .
According with the reference system shown in Fig. 1, the ansatz for the displacement and micro-distortion fields is

u(x 1 , x 2 , x 3 ) = u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   , ω = 0 , A(x 1 , x 2 , x 3 ) = A(r, ϕ, z) = ϑ 2   0 -2x 3 (z) -g p (r) x 2 (r, ϕ) 2x 3 (z) 0 g p (r) x 1 (r, ϕ) g p (r) x 2 (r, ϕ) -g p (r) x 1 (r, ϕ) 0   . (56)
Since the ansatz requires ω = 0, the micro-stretch model coincides with the Cosserat model which will be presented in the next section.

Torsional problem for the isotropic Cosserat continuum

The strain energy for the isotropic Cosserat continuum in dislocation tensor format (curvature expressed in term of CurlA) can be written as [START_REF] Cosserat | Théorie des corps déformables[END_REF][START_REF] Izadi | Torsional characteristics of carbon nanotubes: Micropolar elasticity models and molecular dynamics simulation[END_REF][START_REF] Jeong | A numerical study for linear isotropic Cosserat elasticity with conformally invariant curvature[END_REF][START_REF] Neff | The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric[END_REF][START_REF] Neff | A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy[END_REF][START_REF] Rizzi | Analytical solutions of the simple shear problem for micromorphic models and other generalized continua[END_REF][START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF][START_REF] Rueger | Strong Cosserat elasticity in a transversely isotropic polymer lattice[END_REF]]

W (Du, A, Curl A) = µ macro sym Du 2 + λ macro 2 tr 2 (Du) + µ c skew (Du -A) 2 (57)
+ µ L 2 c 2 a 1 dev sym Curl A 2 + a 2 skew Curl A 2 + a 3 3 tr 2 (Curl A) ,
where A ∈ so(3). It is underlined that for the ansatz (61), which will be presented later in this section, it holds that skew (Curl A) = 0 (see calculation [START_REF] Lurie | Bending problems in the theory of elastic materials with voids and surface effects[END_REF] 2 ). The equilibrium equations, in the absence of body forces, are therefore the following Div

σ:= [2µ macro sym Du + λ macro tr (Du) 1 + 2µ c skew (Du -A)] = 0 , 2µ c skew (Du -A) -µ L 2 c skew Curl a 1 dev sym Curl A + a 3 3 tr (Curl A) 1 = 0 . (58)
The boundary conditions at the free surface are

t(r = R) = σ(r) • e r = 0 R 3 , (59) η
(r = R) = skew (m(r) • • e r ) = skew (m(r) × e r ) = 0 R 3×3 ,
where the second order moment stress tensor is now given by

m = µ L 2 c a 1 dev sym Curl A + a 3 3 tr (Curl A) 1 , (60)
the expression of σ is in ( 58), e r is the radial unit vector, and is the Levi-Civita tensor.

According to the reference system shown in Fig. 1, the ansatz for the displacement field and the microrotation is

u(x 1 , x 2 , x 3 ) = u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   , A(x 1 , x 2 , x 3 ) = A(r, ϕ, z) = ϑ 2   0 -2x 3 (z) -g p (r) x 2 (r, ϕ) 2x 3 (z) 0 g p (r) x 1 (r, ϕ) g p (r) x 2 (r, ϕ) -g p (r) x 1 (r, ϕ) 0   , (61)
where, in relation to the ansatz (38), we define g p (r) := g 1 (r) + g 2 (r), so that there is only one unknown function to be determined. Substituting the ansatz ( 61) in ( 58) the 6 equilibrium equations are equivalent to 1 6 ϑ sin ϕ 6r µ c (g p (r) -1) -µ L 2 c (a 1 + 2a 3 ) 3g p (r) + r g p (r) = 0 ,

ϑ cos ϕ 6r µ c (g p (r) -1) -µ L 2 c (a 1 + 2a 3 ) 3g p (r) + r g p (r) = 0 .

It is important to underline that (58) 1 is identically satisfied, and that between the two equilibrium equations ( 62) there is only one independent equation since (62) 1 = tan ϕ (62) 2 . The solution of (62) is

g p (r) = 1 - i A 1 I 1 r f1 Lc r + A 2 Y 1 -i r f1 Lc r , f 1 := 6µ c (a 1 + 2a 3 ) µ , (63)
where I n (•) is the modified Bessel function of the first kind, Y n (•) is the Bessel function of the second kind (see appendix B for the formal definitions), and A 1 , A 2 are integration constants. The value of A 1 is determined from to the boundary conditions [START_REF] Rizzi | Exploring metamaterials' structures through the relaxed micromorphic model: switching an acoustic screen into an acoustic absorber[END_REF], where, due to the divergent behaviour of the Bessel function of the second kind at r = 0, we have to set A 2 = 0 in order to have a continuous solution. The fulfilment of the boundary conditions (59) allows us to find the expressions of the integration constants

A 1 = - i R L c f 1 R z 1 I 0 R f1 Lc + I 2 R f1 Lc + z 2 L c I 1 R f1 Lc , z 1 := a 1 + 2a 3 3a 1 , z 2 := 4a 3 -a 1 3a 1 . (64)
The classical torque, the higher-order torque, and energy (per unit length dz) expressions are

M c (ϑ) := 2π 0 R 0 σ e z , e ϕ r r dr dϕ =   µ macro + 4µ c I 2 R f1 Lc L 2 c R 2 f 1 2 f 1 z 1 I 0 R f1 Lc + (z 2 -2z 1 ) I 1 R f1 Lc Lc R   I p ϑ = T c ϑ , M m (ϑ) := 2π 0 R 0 skew(m × e z )e ϕ , e r -skew(m × e z )e r , e ϕ r dr dϕ =   2µ 3a 1 f 1 z 1 I 0 R f1 Lc L 2 c R 2 -2(a 1 -a 3 ) I 1 R f1 Lc L 3 c R 3 6f 1 z 1 I 0 R f1 Lc -3I 1 R f1 Lc Lc R   I p ϑ = T m ϑ , (65)
W tot (ϑ) :

= 2π 0 R 0 W (Du, A, CurlA) r dr dϕ = 1 2   µ macro + 4µ c I 2 R f1 Lc L 2 c R 2 f 1 2 f 1 z 1 I 0 R f1 Lc + (z 2 -2z 1 ) I 1 R f1 Lc Lc R + 2µ 3a 1 f 1 z 1 I 0 R f1 Lc L 2 c R 2 -2(a 1 -a 3 ) I 1 R f1 Lc L 3 c R 3 6f 1 z 1 I 0 R f1 Lc -3I 1 R f1 Lc Lc R   I p ϑ 2 = 1 2 T w ϑ 2 .
The validity of ( 65) 2 for M m will be shown in the Appendix C. The plot of the torsional stiffness for the classical torque (light blue), the higher-order torque (red), and the torque energy (green) while varying L c is shown in Fig. 14.

Cosserat conformal curvature case -bounded stiffness in torsion

In the particular case for which the parameter a 3 is equal to zero the elastic energy turns into

W (Du, A, Curl A) = µ macro sym Du 2 + λ macro 2 tr 2 (Du) + µ c skew (Du -A) 2 (66)
+ µ L 2 c 2 a 1 dev sym Curl A 2 .
In terms of φ = axl(A), the curvature energy can be written as

µ L 2 c
2 a 1 dev sym D axl (A) 2 which is the conformal curvature case [START_REF] Neff | Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature[END_REF]. In this special case, the torsional stiffness remains bounded as L c → ∞ (R → 0), namely T := (9µ c + µ macro ) I p , which is consistent with the results in (51). Here, the Cosserat couple modulus µ c is clearly related to the value of the stiffness for small specimen size.

6.2 Cosserat limit case µ c → ∞ (indeterminate couple stress model)

lim µc→∞ M c (ϑ) = µ macro + a 1 µ L 2 c R 2 I p ϑ = T c ϑ , lim µc→∞ M m (ϑ) = 2a 1 µ L 2 c R 2 I p ϑ = T m ϑ , ( 67
) lim µc→∞ W tot (ϑ) = 1 2 µ macro + 3a 1 µ L 2 c R 2 I p ϑ 2 = 1 2 T w ϑ 2 .
It is highlighted that there is not a one to one correspondence between the torque obtained as a limit from the Cosserat model ( 67) and the one obtained using the indeterminate couple stress model from the beginning (78), but of course the energy (or the sum of the two torques) coincides and thus the total torque stiffness T w coincides as well. There is not a one to one correspondence between the respective torque but the energy coincides.

0 2.5 5
T macro 0 R/L c T c , T m ,T w T c T m T w (a) 0 2.5 5
T macro 0 R/L c T c , T m ,T w T c T m T w (b)

Cosserat limit case µ

c → 0. lim µc→0 M c (ϑ) = µ macro I p ϑ = T c ϑ , lim µc→0 M m (ϑ) = 24µ a 1 a 3 a 1 + 8a 3 L 2 c R 2 I p ϑ = T m ϑ , (68)
lim µc→0 W tot (ϑ) = 1 2 µ macro + 24µ a 1 a 3 a 1 + 8a 3 L 2 c R 2 I p ϑ 2 = 1 2 T w ϑ 2 .
It is highlighted that the Cosserat model does not collapse into a classical linear elastic model for µ c → 0, but it remains proportional to (L c /R) 2 eq.( 68). In this case, the Cosserat model behaves similarly to the indeterminate couple stress model eq.( 67) or eq.( 78), and it collapses into this model (both the energy and the torques) by formally letting a 3 → ∞ as it can be seen from equations ( 67) and (68). for µ c → ∞ we recover the indeterminate couple stress model, while for µ c → 0 we still have a non linear relation between T w and R/L c since a classical linear elastic model is not attained (see eq.( 68)). The values of the material parameter used are µ = 1, µ e = 1/10, a 1 = 12, a 3 = 1/20, R = 1.

0 2.5 5
T macro 0 R/L c T c , T m ,T w T c T m T w 0 2.5 5
T macro R/L c
6.4 Sensitivity of the Cosserat model with respect to the curvature parameters a 1 and a 3 .

Here, we study the sensitivity for the Cosserat model while varying a 1 and a 3 independently. The parametric study represented in Fig. 18 has not been carried out for the limit a 1 → 0 and a 3 → 0 since we would have had an indeterminate form for L c → ∞, and that is why we used the symbol ∼ 0. The solution of the problem while having a 3 = 0 a priori is analyzed carefully in Section 6.1, and the solution of the problem while having a 1 = 0 a priori makes the relaxed micromorphic model collapse into a classical linear elastic model with torsional stiffness T macro . It is also highlighted that the Cosserat model collapses into the indeterminate couple stress model for a 3 → ∞ also in this more general case for which µ c is arbitrary.

0 2.5 5
T macro R/L c T w ∼0 a 1 ∞ (a) 0 2.5 5
T macro R/L c T w ∼0 a 3 ∞ (b)

Torsional problem for the isotropic micro-void model in dislocation tensor format

The strain energy for the isotropic micro-void continuum in dislocation tensor format can be written as [START_REF] Cowin | Linear elastic materials with voids[END_REF][START_REF] Rizzi | Analytical solutions of the simple shear problem for micromorphic models and other generalized continua[END_REF] W (Du, ω, Curl (ω1)) = µ macro dev sym Du

2 + κ e 2 tr 2 (Du -ω1) + κ micro 2 tr 2 (ω1) (69) + µ L 2 c 2 a 2 Curl (ω1) 2 .
Here, ω : R 3 → R is the additional scalar micro-void degree of freedom [START_REF] Cowin | Linear elastic materials with voids[END_REF]. The equilibrium equations, in the absence of body forces, are 3 Div σ:=

[2µ macro dev sym Du + κ e tr (Du -ω1) 1] = 0,

(70) 1 3 tr σ -κ micro tr (ω1) 1 -µ L 2 c a 2 Curl Curl (ω1) = 0.
The boundary conditions at the free surface are

t(r = R) = σ(r) • e r = 0 R 3 , (71)
η(r = R) = 1 3 tr (m(r) • • e r ) = 1 3 tr (m(r) × e r ) = 0 .
According with the reference system shown in Fig. 1, the ansatz for the displacement field and the function ω have to be

u(x 1 , x 2 ) =   -x 2 x 3 x 1 x 3 0   , ω (x 2 ) 1 =   0 0 0 0 0 0 0 0 0   . ( 72
)
which clearly reduce the model to a classical linear elastic one. No further calculation will be carried on and the reader is referred to Section 3.

Torsional problem for the isotropic couple stress continuum

The indeterminate couple stress model [START_REF] Ghiba | 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[END_REF][START_REF] Hadjesfandiari | Comparison of theoretical elastic couple stress predictions with physical experiments for pure torsion[END_REF][START_REF] Koiter | Couple stresses in the theory of elasticity: I and II[END_REF][START_REF] Neff | Correct traction boundary conditions in the indeterminate couple stress model[END_REF][START_REF] Neff | On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of AR Hadjesfandiari and GF Dargush[END_REF][START_REF] Tong | Size effects of hair-sized structures-torsion[END_REF][START_REF] Tsiatas | A new microstructure-dependent Saint-Venant torsion model based on a modified couple stress theory[END_REF] appears by letting formally the Cosserat couple modulus µ c → ∞. This implies the constraint A = skew Du ∈ so(3). It is highlighted that for the torsional problem, we do not have any unknown fields in this model since the displacement u is known a priori. 4 Since tr(Curl skew Du) = skew Curl skew Du 2 = 0 in terms of the ansatz (77), the indeterminate couple stress elastic energy for the torsion can be written as

W (Du, Curl skew Du) = µ macro sym Du 2 + λ macro 2 tr 2 (Du) + µ L 2 c 2 a 1 sym Curl skew Du 2 . ( 73
)
The equilibrium equations, in the absence of body forces, are 5 Div 2µ macro sym Du + λ macro tr (Du)

1 + µ L 2 c skew Curl (a 1 dev sym Curl skew Du ) = 0 , (74)
while the (highly non-trivial) boundary traction conditions on the free surface are (for more details see [START_REF] Hadjesfandiari | Comparison of theoretical elastic couple stress predictions with physical experiments for pure torsion[END_REF][START_REF] Neff | Correct traction boundary conditions in the indeterminate couple stress model[END_REF])

t(r = R) = ± σ - 1 2 Anti (Div m) • e r - 1 2 e r × D [ e r , sym m • e r ] - 1 2 D [Anti ((1 -e r ⊗ e r ) • m • e r ) • (1 -e r ⊗ e r )] : (1 -e r ⊗ e r ) = 0 , ( 75
) (1 -e r ⊗ e r ) • η(r = R) = ± (1 -e r ⊗ e r ) • Anti [(1 -e r ⊗ e r ) • m • e r ] • e r = 0 , π(r = R) = ± Anti [(1 -e r ⊗ e r ) • m • e r ] + -Anti [(1 -e r ⊗ e r ) • m • e r ]
-• e ϕ = 0 , 3 Where κe = 2µe +3λe 3 and κmicro = 2µ micro +3λ micro 3

are the meso-and the micro-scale 3D bulk modulus. 4 Since we can show that the classical torsion displacement solution satisfies the external balance equation (74) as well as the higher order traction boundary conditions (75). 5 Using Nye's formula [START_REF] Ghiba | 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[END_REF] According to the reference system shown in Fig. 1, the ansatz for the displacement field and consequently the skew-symmetric part of the gradient of the displacement are

u(x 1 , x 2 ) = ϑ   -x 2 x 3 x 1 x 3 0   ⇒ skew Du = ϑ 2   0 -2x 3 -x 2 2x 3 0 x 1 x 2 -x 1 0   . ( 77
)
Since the ansatz is completely known, it is possible to check that both the equilibrium equations ( 74) and the boundary conditions ( 75) are identically satisfied 6 , and it is possible then to evaluate directly the classical torque, the higher-order torque, and the energy. The classical torque, the higher-order torque, and energy (per unit length dz) expressions are

= 3a 1 µ L 2 c R 2 I p ϑ = T m ϑ , (78)
W tot (ϑ) := 2π 0 R 0 W (Du, Curl skew Du) r dr dϕ = 1 2 µ macro + 3a 1 µ L 2 c R 2 I p ϑ 2 = 1 2 T w ϑ 2 .
The plot of the torsional stiffness for the classical torque, the higher-order torque, and the torque energy while varying L c is shown in Fig. 19.

Torsional problem for the modified and the "pseudo"-consistent isotropic couple stress model

The modified couple stress model [START_REF] Ghiba | 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[END_REF][START_REF] Madeo | A new view on boundary conditions in the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model[END_REF][START_REF] Münch | 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[END_REF][START_REF] Neff | Correct traction boundary conditions in the indeterminate couple stress model[END_REF][START_REF] Neff | Subgrid interaction and micro-randomness -Novel invariance requirements in infinitesimal gradient elasticity[END_REF] consists in choosing a 1 > 0, a 2 = 0 and leads to a symmetric couple stress tensor while the ("pseudo")-consistent couple stress model [START_REF] Hadjesfandiari | Couple stress theory for solids[END_REF] appears for a 1 = 0, a 2 > 0 and leads to a skew symmetric stress tensor m.

Since tr(Curl skew Du) = skew Curl skew Du 2 = 0, the term dev sym Curl skew Du 2 is the only non zero component in the curvature energy, the form of the energy remains the same. This implies that, for the torsion problem, the modified couple stress model coincides with the indeterminate couple stress model, and that the ("pseudo")-consistent couple stress model reduces to a classical linear elastic model without size-effects. According to the notation [START_REF] Hadjesfandiari | Couple stress theory for solids[END_REF], the constitutive law can be written as

7 σ = 2µ macro sym Du + λ macro tr (Du) 1 , m = η (D curl u) T + η D curl u . ( 79
)
where according to the classical Cosserat notation (see Appendix C)

η = β = µ macro L 2 c 2 a 1 -a 2 2 , η = γ = µ macro L 2 c 2 a 1 + a 2 2 . ( 80
)
In this notation, the modified couple stress model appears for η = η and the "pseudo"-consistent couple stress model appears for η = -η .

Torsional problem for the classical isotropic micromorphic continuum without mixed terms

The strain energy for the isotropic micromorphic continuum without mixed terms ( sym P , sym (Du -P ) , etc.) and simplified isotropic curvature can be written as

W (
+ µ L 2 c 2 a 1 D (dev sym P ) 2 + a 2 D (skew P ) 2 + 2 9
a 3 D (tr (P ) 1) 2 .

The equilibrium equations, in the absence of body forces, are the following Div σ:=

[2µ e dev sym (Du -P ) + κ e tr (Du -P ) 1 + 2µ c skew (Du -P )] = 0 , σ -2µ micro dev sym P -κ micro tr (P ) 1

(82) +µL 2 c a 1 dev sym ∆P + a 2 skew ∆P + 2 9 a 3 tr (∆P ) 1 = 0 ,
where ∆P ∈ R 3×3 is taken component-wise. The boundary conditions at the external surfaces are

t(r = R) = σ(r) • e r = 0 R 3 , η(r = R) = m(r) • e r = 0 R 3×3 , (83)
where

m = µ L 2 c a 1 D (dev sym P ) + a 2 D (skew P ) + 2 9
a 3 D (tr (S) 1)

(84)
is the third order moment stress tensor, the expression of σ is given in (82), and e r is the radial unit vector. According with the reference system shown in Fig. 1, the ansatz for the displacement field and the micro-distortion is

u(x 1 , x 2 , x 3 ) = u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   , P (x 1 , x 2 , x 3 ) = P (r, ϕ, z) = ϑ   0 -x 3 (z) -g 2 (r) x 2 (r, ϕ) x 3 (z) 0 g 2 (r) x 1 (r, ϕ) g 1 (r) x 2 (r, ϕ) -g 1 (r) x 1 (r, ϕ) 0   . ( 85
)
7 Setting a1 µmacro L 2 c = 8η we obtain the rigidity as

Tw := d 2 dϑ 2 W tot (ϑ) = µmacro 1 + 24 η µmacro 1 R 2
Ip. In [START_REF] Neff | Existence of minimizers for a finite-strain micromorphic elastic solid[END_REF] of Hadjesfandiari and Dargush [START_REF] Hadjesfandiari | Comparison of theoretical elastic couple stress predictions with physical experiments for pure torsion[END_REF] we have the relation 2 = η µmacro , while in [START_REF] Neff | On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of AR Hadjesfandiari and GF Dargush[END_REF] we have the formula Tw = µmacro 1 + 24 R 2 Ip = µmacro 1 + 24

η µmacro 1 R 2 Ip.
Substituting the ansatz (85) in (82) the 12 equilibrium equations are equivalent to

1 2 ϑ sin ϕ ρ µL 2 c ((a 1 -a 2 )g 1 (ρ) -(a 1 + a 2 )g 2 (ρ)) + 2µ c (g 1 (ρ) + g 2 (ρ) -1) -2 (µ e + µ micro ) (g 1 (ρ) -g 2 (ρ)) -2µ e ) + 3µL 2 c ((a 1 -a 2 )g 1 (ρ) -(a 1 + a 2 )g 2 (ρ)) = 0 , 1 2 ϑ cos ϕ ρ µL 2 c ((a 2 -a 1 )g 1 (ρ) + (a 1 + a 2 )g 2 (ρ)) -2µ c (g 1 (ρ) + g 2 (ρ) -1) +2 (µ e + µ micro ) (g 1 (ρ) -g 2 (ρ)) + 2µ e ) + 3µL 2 c ((a 2 -a 1 )g 1 (ρ) + (a 1 + a 2 )g 2 (ρ)) = 0 , (86) 1 2 ϑ sin ϕ ρ µL 2 c ((a 1 + a 2 )g 1 (ρ) + (a 2 -a 1 )g 2 (ρ)) -2 (µ c (g 1 (ρ) + g 2 (ρ) -1) + (µ e + µ micro ) (g 1 (ρ) -g 2 (ρ))) -2µ e ) + 3µL 2 c ((a 1 + a 2 )g 1 (ρ) + (a 2 -a 1 )g 2 (ρ)) = 0 , 1 2 ϑ cos ϕ ρ µL 2 c ((a 1 -a 2 )g 2 (ρ) -(a 1 + a 2 )g 1 (ρ)) + 2 (µ c (g 1 (ρ) + g 2 (ρ) -1) + (µ e + µ micro ) (g 1 (ρ) -g 2 (ρ))) + 2µ e ) + 3µL 2 c ((a 1 -a 2 )g 2 (ρ) -(a 1 + a 2 )g 1 (ρ)) = 0 .
It is important to underline that (82) 1 is identically satisfied, and that between the four equilibrium equations (86) there are only two that are linearly independent since: (86) 1 = tan ϕ (86) 2 and (86

) 3 = tan ϕ (86) 4 .
It is also pointed out that the two remaining linearly independent equations (86) 1,3 can be uncoupled 8 and have the form of the Bessel ODE if we take their sum and difference, while being careful of substituting g p (r) := g 1 (r) + g 2 (r) and g m (r) := g 1 (r) -g 2 (r) along with their derivatives:

ϑ sin ϕ a 1 µL 2 c (3g m (ρ) + ρ g m (ρ)) -2ρµ e (g m (ρ) + 1) -2ρ g m (ρ)µ micro = 0 , (87)
ϑ sin ϕ 2ρ µ c (g p (ρ) -1) -a 2 µL 2 c 3g p (ρ) + ρ g p (ρ) = 0 .
Since g 1 (r) := gp(r)+gm(r) 2 and g 2 (r) := gp(r)-gm(r)

2

, the solution in terms of g 1 (r) and g 2 (r) of ( 87) is

g 1 (r) = 1 2   1 - iA 1 I 1 r f2 Lc -A 2 Y 1 -i r f2 Lc + iB 1 I 1 r f1 Lc -B 2 Y 1 -i r f1 Lc r - µ e µ e + µ micro   , g 2 (r) = 1 2   1 + iA 1 I 1 r f2 Lc -A 2 Y 1 -i r f2 Lc -iB 1 I 1 r f1 Lc + B 2 Y 1 -i r f1 Lc r + µ e µ e + µ micro   , (88) f 1 := 2µ c a 2 µ , f 2 := 2(µ e + µ micro ) a 1 µ ,
where I n (•) is the modified Bessel function of the first kind, Y n (•) is the Bessel function of the second kind (see appendix B for the formal definitions), and A 1 , B 1 , A 2 , B 2 are integration constants. The values of A 1 , B 1 are determined thanks to the boundary conditions (83), while, due to the divergent behaviour of the Bessel function of the second kind at r = 0, we have to set A 2 = 0 and B 2 = 0 in order to have a continuous solution. The fulfilment of the boundary conditions (83) allows us to find the expressions of the integration constants

A 1 = 2iL c µ e f 2 (µ e + µ micro ) I 0 Rf2 Lc + I 2 Rf2 Lc , B 1 = - 2iL c f 1 I 0 Rf1 Lc + I 2 Rf1 Lc . ( 89
)
The classical torque, the higher-order torque, and energy (per unit length dz) expressions are

M c (ϑ) := 2π 0 R 0 σ e z , e ϕ r r dr dϕ =     8µ c I 2 Rf1 Lc f 2 1 I 0 Rf1 Lc + f 2 1 I 2 Rf1 Lc + 8µ 2 e I 2 Rf2 Lc (µ e + µ micro ) f 2 2 I 0 Rf2 Lc + f 2 2 I 2 Rf2 Lc   L 2 c R 2 + µ e µ micro µ e + µ micro   I p ϑ = T c ϑ , M m (ϑ) := 2π 0 R 0 (m e z ) e ϕ , e r -(m e z ) e r , e ϕ r dr dϕ = 4a 2 µ L 2 c R 2 I p ϑ = T m ϑ ,
8 That this uncoupling takes place at all seems to be connected to the chosen form of the curvature energy. It remains unclear at present whether this feature holds for the most general isotropic curvature expression as well.

W tot (ϑ) :

= 2π 0 R 0 W (Du, P , DP ) r dr dϕ = 1 2 8µ c I 2 Rf1 Lc f 2 1 I 0 Rf1 Lc + f 2 1 I 2 Rf1 Lc + 8µ 2 e I 2 Rf2 Lc (µ e + µ micro ) f 2 2 I 0 Rf2 Lc + f 2 2 I 2 Rf2 Lc L 2 c R 2 (90) + µ e µ micro µ e + µ micro µmacro +4a 2 µ L 2 c R 2 I p ϑ 2 = 1 2 T w ϑ 2 ,
and again it holds,

d dϑ W tot (ϑ) = M c (ϑ) + M m (ϑ) , d 2 dϑ 2 W tot (ϑ) = T c + T m = T w . (91)
It is underlined that the boundary conditions for the micromorphic model are consistent with the relaxed micromorphic model's one, being careful of changing m e z with m×e z . The plot of the torsional stiffness for the classical torque (light blue), the higher-order torque (red), and the torque energy (green) while varying L c is shown in Fig. 20. The classical micromorphic model with reduced curvature energy (a 2 = 0) collapses into the micro-strain model (Section 10 with symP ) thus becoming independent with respect to the Cosserat couple modulus µ c (see [START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF] for the different stiffnesses expressions).

µ e = 1/3, µ micro = 1/4, a 1 = 2, a 3 = 1/20, R = 1.
In this case, the stiffness for arbitrary small sample size is governed by T e and not T micro . The reason for this is explained in Appendix D.

10 Torsional problem for the micro-strain model without mixed terms

The micro-strain model [START_REF] Forest | Nonlinear microstrain theories[END_REF][START_REF] Hütter | Micromorphic homogenization of a porous medium: elastic behavior and quasi-brittle damage[END_REF] is a particular case of the classical Mindlin-Eringen model, in which it is assumed a priori that the micro-distortion remains symmetric, P = S ∈ Sym(3). 9A torsion solution for a more general case with mixed terms has been derived in [START_REF] Hütter | Application of a microstrain continuum to size effects in bending and torsion of foams[END_REF], but here we employ a reduced isotropic curvature expression to make the calculations more manageable.

It is underlined that the micro-strain model cannot be obtained as a limit case of the relaxed micromorphic model and vice versa, although there are some similarities. The strain energy which we consider is

W (Du, S, DS) = µ e dev (sym Du -S) 2 + κ e 2 tr 2 (Du -S) + µ micro dev S 2 + κ micro 2 tr 2 (S) (92) + µ L 2 c 2 a 1 D (dev S) 2 + 2 9
a 3 D (tr (S) 1) 2 .

The chosen 2-parameter curvature expression represents a simplified isotropic curvature (the full isotropic curvature for the micro-strain model would still counts 8 parameters [START_REF] Barbagallo | Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics[END_REF]). In this form, the micro-strain model can be obtained from the classical micromorphic model (Section 9), in general, by setting µ c = 0 and a 2 = 0. For the torsion problem, the condition a 2 = 0 alone is sufficient.

It is underlined that for the ansatz (96), which will be presented later in this section, it holds tr (S) = 0. The equilibrium equations, in the absence of body forces, are therefore the following Div σ:=

[2µ e dev (sym Du -S) + κ e tr (Du -S) 1] = 0, 2µ e dev (sym Du -S) + κ e tr (Du -S) 1 -2µ micro dev S -κ micro tr (S) 1

(93) + µ L 2 c sym Div a 1 D (dev S) + 2 9
a 3 D (tr (S) 1) = 0 .

The boundary conditions at the external free surfaces are

t(r = R) = σ(r) • e r = 0 R 3 , η(r = R) = sym (m(r) • e r ) = 0 R 3×3 , (94)
where

m = µ L 2 c a 1 D (dev S) + 2 9 a 3 D (tr (S) 1) (95)
is the third order moment stress tensor, the expression of σ is in ( 93), e r is the radial unit vector. According with the reference system shown in Fig. 1, the ansatz for the displacement field and the micro-distortion is

u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   , (96)
S(r, ϕ, z) = ϑ 2   0 0 g m (r) x 2 (r, ϕ) 0 0 -g m (r) x 1 (r, ϕ) g m (r) x 2 (r, ϕ) -g m (r) x 1 (r, ϕ) 0   ,
where, in relation to the ansatz [START_REF] Lewintan | L p -trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions[END_REF], g m (r) := g 1 (r) -g 2 (r). Substituting the ansatz (96) in (93) the 9 equilibrium equation are equivalent to

1 2 ϑ sin ϕ a 1 µ L 2 c (3g m (r) + r g m (r)) -2r µ e (g m (r) + 1) -2r g m (r) µ micro = 0 , (97)
- 1 2 ϑ cos ϕ a 1 µ L 2 c (3g m (r) + r g m (r)) -2r µ e (g m (r) + 1) -2r g m (r) µ micro = 0 .
Between the two equilibrium equations (97) there is only one independent equation since (97) 1 = -tan ϕ (97) 2 . The solution of (97) is

g m (r) = A 2 Y 1 -irf1 Lc -iA 1 I 1 rf1 Lc r - µ e µ e + µ micro , f 1 := 2(µ e + µ micro ) a 1 µ , (98)
where I n (•) is the modified Bessel function of the first kind, Y n (•) is the Bessel function of the second kind (see appendix B for the formal definitions), and A 1 , A 2 are integration constants. The value of A 1 is determined thanks to the boundary conditions (94), while, due to the divergent behaviour of the Bessel function of the second kind at r = 0, we have to set A 2 = 0 in order to have a continuous solution. The fulfilment of the boundary conditions (94) allows us to find the expressions of the integration constants

A 1 = 2i L c I 0 Rf1 Lc + I 2 Rf1 Lc µ e f 1 (µ e + µ micro ) . (99)
The classical torque, the higher-order torque, and the energy (per unit length dz) expressions are

M c (ϑ) := 2π 0 R 0 σ e z , e ϕ r r dr dϕ =   µ e µ micro µ e + µ micro + µ 2 e µ a 1 (µ e + µ micro ) 2 4 I 2 R f1 Lc I 0 R f1 Lc + I 2 R f1 Lc L 2 c R 2   I p ϑ = T c ϑ , M m (ϑ) := 2π 0 R 0 sym(me z )e ϕ , e r -sym(m e z )e r , e ϕ r dr dϕ = 0 , (100)
W tot (ϑ) := 2π 0 R 0 W (Du, S, DS) r dr dϕ = 1 2     µ e µ micro µ e + µ micro µmacro + µ 2 e µ a 1 (µ e + µ micro ) 2 4 I 2 R f1 Lc I 0 R f1 Lc + I 2 R f1 Lc L 2 c R 2     I p ϑ 2 = 1 2 T w ϑ 2 .
The plot of the torsional stiffness for the classical torque, the higher-order torque, and the torque energy while varying L c is shown in Fig. 14. Since the higher-order torque is zero and the following relation holds

d dϑ W tot (ϑ) = M c (ϑ) + M m (ϑ) = M c (ϑ) , d 2 dϑ 2 W tot (ϑ) = T c + T m = T c = T w , (101)
only the plot of the energy (per unit length dz) while changing L c is shown in Fig. The energy of the model remains bounded, as for the shear and bending problem [START_REF] Rizzi | Analytical solutions of the simple shear problem for micromorphic models and other generalized continua[END_REF][START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF], since for both problems the higher-order moments are zero, and this does not create a conflict with the boundary condition as L c → ∞ (see [START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF] for the different stiffnesses expressions). Note that there is no simple way to a-priori guess that the small size torsional response is given by T e since S ∈ Sym(3) is not easily seen to be zero. In Appendix D we show that the variational limit for L c → ∞ is indeed realized by S(x) = S = 0 and this shows that the limit stiffness is given by T e .

Torsional problem for the second gradient continuum

The expression of the most general isotropic strain energy for the second gradient continuum is [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF][START_REF] Neff | Subgrid interaction and micro-randomness -Novel invariance requirements in infinitesimal gradient elasticity[END_REF] W Du, D where, since the boundary surface is smooth, one set of boundary condition is identically satisfied (see [START_REF] Madeo | A new view on boundary conditions in the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model[END_REF][START_REF] Mindlin | Micro-structure in linear elasticity[END_REF] for all the details). According to the reference system shown in Fig. 1, the ansatz for the displacement field and consequently the gradient of the displacement are

u(x 1 , x 2 ) = ϑ   -x 2 x 3 x 1 x 3 0   ⇒ Du = ϑ 2   0 -2x 3 -2x 2 2x 3 0 2x 1 0 0 0   . (106)
Since the ansatz is completely known, it is possible to check that the equilibrium equation ( 104) and the boundary conditions (106) are identically satisfied and it is possible to evaluate directly the classical torque, the higher-order torque, and the energy. The classical torque, the higher-order torque, and energy (per unit length dz) expressions are

= 2µ(a 1 + 3a 2 ) L 2 c R 2 I p ϑ = T m ϑ , W tot (ϑ) := 2π 0 R 0 W Du, D 2 u r dr dϕ = 1 2 µ macro + 2µ(a 1 + 3a 2 ) L 2 c R 2 I p ϑ 2 = 1 2 T w ϑ 2 . ( 107
)
The plot of the torsional stiffness for the classical torque (light blue), the higher-order torque (red), and the torque energy (green) while varying L c is shown in Fig. 24.

where ∆ (Du) ∈ R 3×3 is taken component-wise. 12 Ad-hoc model containing Cosserat and micro-strain effects

Given S ∈ Sym(3) and A ∈ so(3), the strain energy which we consider now is

t(r = R) = σ(r) • e r = 0 R 3 , (112) η b (r = R) = skew (m(r) • • e r ) = skew (m(r) × e r ) = 0 R 3×3 , η a (r = R) = sym (m(r) • e r ) = 0 R 3×3 , where m = µ L 2 c a 1 dev sym Curl A + a 3 3 tr (Curl A) 1 , (113)
m = µ L 2 c a 4 D (dev S)
, is the second and third order moment stress tensor respectively, the expression of σ is in (111), e r is the radial unit vector. According with the reference system shown in Fig. 1, the ansatz for the displacement field and the micro-distortion is

u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   , A(r, ϕ, z) = ϑ 2   0 -2x 3 (z) -g p (r) x 2 (r, ϕ) 2x 3 (z) 0 g p (r) x 1 (r, ϕ) g p (r) x 2 (r, ϕ) -g p (r) x 1 (r, ϕ) 0   , (114)
S(r, ϕ, z) = ϑ 2   0 0 g m (r) x 2 (r, ϕ) 0 0 -g m (r) x 1 (r, ϕ) g m (r) x 2 (r, ϕ) -g m (r) x 1 (r, ϕ) 0   ,
where, in relation to the ansatz [START_REF] Lewintan | L p -trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions[END_REF], g m (r) := g 1 (r) -g 2 (r) and g p (r) := g 1 (r) + g 2 (r). Substituting the ansatz (114) in (111) the 15 equilibrium equation are equivalent to

1 6 ϑ sin ϕ 6r µ c (g p (r) -1) -µ L 2 c (a 1 + 2a 3 ) 3g p (r) + r g p (r) = 0 , 1 6 ϑ cos ϕ 6r µ c (g p (r) -1) -µ L 2 c (a 1 + 2a 3 ) 3g p (r) + r g p (r) = 0 , (115)
g p (r) = 1 - i A 1 I 1 r f1 Lc r + A 2 Y 1 -i r f1 Lc r , f 1 := 6µ c (a 1 + 2a 3 ) µ , (116)
g m (r) = A 2 Y 1 -irf2 Lc -iA 1 I 1 rf2 Lc r - µ e µ e + µ micro , f 2 := 2(µ e + µ micro ) a 4 µ , (117)
where I n (•) is the modified Bessel function of the first kind, Y n (•) is the Bessel function of the second kind (see appendix B for the formal definitions), and A 1 , A 2 , A 3 , A 4 are integration constants. The values of A 1 and A 2 are determined thanks to the boundary conditions (112), while, due to the divergent behaviour of the Bessel function of the second kind at r = 0, we have to set A 2 = A 4 = 0 in order to have a continuous solution. The fulfilment of the boundary conditions (112) allows us to find the expressions of the integration constants

A 1 = - i R L c f 1 R z 1 I 0 R f1 Lc + I 2 R f1 Lc + z 2 L c I 1 R f1 Lc , z 1 := a 1 + 2a 3 3a 1 , A 3 = 2i L c I 0 Rf2 Lc + I 2 Rf2 Lc µ e f 2 (µ e + µ micro ) , z 2 := 4a 3 -a 1 3a 1 . ( 118
)
The classical torque, the higher-order torque, and the energy (per unit length dz) expressions are

M c (ϑ) := 2π 0 R 0 σ e z , e ϕ r r dr dϕ =   4µ c I 2 R f1 Lc L 2 c R 2 f 1 2 f 1 z 1 I 0 R f1 Lc + (z 2 -2z 1 ) I 1 R f1 Lc Lc R + µ e µ micro µ e + µ micro + µ 2 e µ a 4 (µ e + µ micro ) 2 4 I 2 R f2 Lc I 0 R f2 Lc + I 2 R f2 Lc L 2 c R 2   I p ϑ = T c ϑ , M m (ϑ) :=
  2µ 3a 1 f 1 z 1 I 0 R f1 Lc L 2 c R 2 -2(a 1 -a 3 ) I 1 R f1 Lc L 3 c R 3 6f 1 z 1 I 0 R f1 Lc -3I 1 R f1 Lc Lc R   = T m ϑ , (119)
W tot (ϑ) := 2π 0 R 0 W (Du, A, S, Curl A, DS) r dr dϕ = 1 2   4µ c I 2 R f1 Lc L 2 c R 2 f 1 2 f 1 z 1 I 0 R f1 Lc + (z 2 -2z 1 ) I 1 R f1 Lc Lc R + µ e µ micro µ e + µ micro + µ 2 e µ a 4 (µ e + µ micro ) 2 4 I 2 R f2 Lc I 0 R f2 Lc + I 2 R f2 Lc L 2 c R 2 + 2µ 3a 1 f 1 z 1 I 0 R f1 Lc L 2 c R 2 -2(a 1 -a 3 ) I 1 R f1 Lc L 3 c R 3 6f 1 z 1 I 0 R f1 Lc -3I 1 R f1 Lc Lc R   I p ϑ 2 = 1 2 T w ϑ 2 .
It is highlighted that, like for the micro-strain model (Section 10), the higher order torque contribution (m e z ) e ϕ , e ϕ is equal to zero. The plot of the torsional stiffness for the classical torque, the higher-order torque, and the torque energy while varying L c is shown in Fig. 26. Again, it holds

d dϑ W tot (ϑ) = M c (ϑ) + M m (ϑ) , d 2 dϑ 2 W tot (ϑ) = T c + T m = T w . (120)

Summary and conclusions

We have derived the analytical expressions of the torsional rigidity for a family of generalized continua capable of modelling size-dependence in the sense that more slender specimens are comparatively stiffer. We only consider (simplified) isotropic expressions so as to better compare the different models with each other. For example, a strain gradient continuum, by construction, does not have mixed energy terms. Therefore, we omitted these terms in all models. Excluding the mixed terms like sym Du, sym Du -P also simplifies considerably the investigation of positive definiteness. Indeed, all presented models are positive definite if the usual relations

µ macro > 0 , κ macro = 2µ macro + 3λ macro 3 > 0 , µ micro > 0 , κ micro = 2µ micro + 3λ micro 3 > 0 , (121)
µ micro > µ macro =⇒ µ e > 0 , κ e = 2µ e + 3λ e 3 > 0 .

are satisfied together with individual positivity of all curvature parameters. In all cases, the displacement follows the classical pure torsion solution. Despite the conceptual simplicity of the models, we observe already a delicate interplay between the used kinematics and the assumed curvature energy expression. For example, let us compare the relaxed micromorphic model with the micro-strain model (Section 10). Both models have a similar looking lower order energy term (if the Cosserat couple modulus µ c ≡ 0), but different kinematics and different curvature energy. For arbitrary slender specimens, the torsional stiffness of the micro-strain model is governed by µ e , whereas the torsional stiffness of the relaxed micromorphic model is determined by µ micro . Thus, the physical interpretation of the material parameters in both models is completely different. This is surprising at first glance but the reason for this response is finally explained in Appendix D.

In the end, the more restricted the used kinematics, the less viable a model may become. In this respect, only the full micromorphic kinematics degree of freedom (12 DOFS) can be advised. In addition, the curvature energy should not intervene too strongly. For example, penalizing a full gradient DP in the curvature energy of the classical micromorphic model leads to a stiffness singularity for arbitrary slender specimens, while penalizing only Curl P in the relaxed micromorphic model does not show the same singular response. Moreover, in the relaxed micromorphic model the interpretation of the lower order material parameters (µ e , µ micro , µ micro , etc.) does not in principle change when different curvature energies are considered. In the end, it is therefore the relaxed micromorphic model that produces sensible and consistent response in all considered cases. It remains to be investigated if, together with the previously developed solution for bending and shear [START_REF] Rizzi | Analytical solutions of the simple shear problem for micromorphic models and other generalized continua[END_REF][START_REF] Rizzi | Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua[END_REF], the present analytical solution allows to identify the complete set of micromorphic parameters of a material from bending, shear and torsion experiments at specimens with different diameters. ∇xu(x) = (∇r u(r)) Q (1) , ∇xP (x) = ∇r P (r) Q (1) , (122)

u i,j = u i,α Q (1) αj , P ij,k = P ij,α Q (1)
αk , (123) where

Q (1) = (∇rx(r)) -1 =   cos ϕ sin ϕ 0 -sin ϕ r cos ϕ r 0 0 0 1   , (124)
and

∇ 2 x u(x) = ∇ 2 r u(r) : Q (2) + (∇r u(r)) Q (3) , ∇xP (x) = ∇ 2 r P (r) : Q (2) + ∇r P (r) Q (3) . ( 125
) u i,jk = u i,αβ Q αβjk + u i,α Q αjk , P ij,kl = P ij,αβ Q αβkl + P ij,α Q αkl . (126)
Q (2) =                   cos 2 ϕ sin ϕ cos ϕ 0 sin ϕ cos ϕ sin 2 ϕ 0 0 0 0      -sin ϕ cos ϕ r cos(2ϕ) 2r 0 cos(2ϕ) 2r sin ϕ cos ϕ r 0 0 0 0      0 0 cos ϕ 2 0 0 sin ϕ 2 cos ϕ 2 sin ϕ 2 0      -sin ϕ cos ϕ r cos(2ϕ) 2r 0 cos(2ϕ) 2r sin ϕ cos ϕ r 0 0 0 0       sin 2 ϕ r 2 -sin ϕ cos ϕ r 2 0 -sin ϕ cos ϕ r 2 cos 2 ϕ r 2 0 0 0 0      0 0 -sin ϕ 2r 0 0 cos ϕ 2r -sin ϕ 2r cos ϕ 2r 0     0 0 cos ϕ 2 0 0 sin ϕ 2 cos ϕ 2 sin ϕ 2 0     0 0 -sin ϕ 2r 0 0 cos ϕ 2r -sin ϕ 2r cos ϕ 2r 0     0 0 0 0 0 0 0 0 1 sin 2 ϕ r -sin ϕ cos ϕ r 0       -sin ϕ cos ϕ r cos 2 ϕ r 0      0 0 0      sin(2ϕ) r 2 - cos(2ϕ) r 2 0       - cos(2ϕ) r 2 - sin(2ϕ) r 2 0      0 0 0     0 0 0     0 0 0     0 0 0                  . (127)

B Bessel functions

The Bessel functions are the solutions y(x) of the Bessel differential equation [START_REF] Watson | A Treatise on the Theory of Bessel Functions[END_REF] x 2 d 2 y dx 2 + x dy dx + (x 2 -α 2 )y = 0 .

For the particular case in which α = n is an integer, the solution of (128) can be expressed as a linear combination of the Bessel function of the first Jn(x) and second Yn(x) kind y(x) = A 1 Jn(x) + A 2 Yn(x) , (129) whose definitions are

Jn(x) = π 0 cos(n τ -x sin(τ )) dτ , Yn(x) = Jn(x) cos(nπ) -J -n (x) sin(nπ) . (130)
Moreover, the modified Bessel functions of the first kind is defined as In(x) = i -n Jn(ix) .

C Classical Cosserat formulation in micro-rotation vector format

An overview of the different classical notations for the Cosserat model has been given in [START_REF] Hassanpour | Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations[END_REF]. In [START_REF] Neff | A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy[END_REF] we have presented the correspondence between the Cosserat model expressed in dislocation format (Curl of the skew symmetric micro-distortion tensor) and in its classical formulation (gradient of the micro-rotation vector φ). The relation between the coefficients in the two notations is

α = µ L 2 c 2 1 3 (4a 3 -a 1 ) , β = µ L 2 c 2 a 1 -a 2 2 , γ = µ L 2 c 2 a 1 + a 2 2 , (131)
a 1 = γ + β µ L 2 c , a 2 = γ -β µ L 2 c , a 3 = 3α + β + γ 4µ L 2 c .
Setting φ := axl(A) and taking into account (131), the expression of the strain energy for the isotropic Cosserat continuum can be equivalently expressed as

W (Du, A, Curl A) = µmacro sym Du 2 + λmacro 2 tr 2 (Du) + µc skew Du -A 2 + µ L 2 c 2 a 1 dev sym Curl A 2 + a 2 skew Curl A 2 + a 3 3 tr 2 (Curl A) dislocation tensor format (132) = W (Du, φ, Dφ) = µmacro sym Du 2 + λmacro 2 tr 2 (Du) + µc 2 curlu -2φ 2 1 2 α tr 2 (Dφ) + β Dφ T , Dφ + γ Dφ 2 classical micro-rotation vector format , since skew Du -A 2 = 2 axl(skew Du -Anti(φ)) 2 = 2 1 2 curlu -φ 2 = 1 2 curlu -2φ 2 . ( 133
)
The equilibrium equations, in the absence of body forces, in the classical notation are

Div [2µmacro sym Du + λmacrotr (Du) 1] -µc curl [curl u -2φ] = 0 , (134)
Div α tr (Dφ) 1 + β (Dφ) T + γ Dφ + 2µc (curl u -2φ) = 0 .

The boundary conditions at the free surface are

t(r = R) = σ(r) • er = 0 R 3 , η(r = R) = m(r) • er = 0 R 3×3 , (135)
where

σ = 2µmacro sym Du + λmacrotr (Du) 1 + 2µc (skew Du -Anti(φ)) , (136)
er is the radial unit vector, and the second-order moment stress tensor

m = α tr (Dφ) 1 + β (Dφ) T + γ Dφ . (137)
According to the reference system shown in Fig. 1, the ansatz for the displacement field and the micro-rotation vector turns into The last relation is a pure algebraic relation valid for all m, m related by (138) and er, eϕ, ez are given in [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]. Thus, we have shown that skew (m × ez) eϕ, er -skew (m × ez) er, eϕ = mzz .

u(x 1 , x 2 , x 3 ) = u(r, ϕ, z) = ϑ   -x 2 (r, ϕ) x 3 (z) x 1 (r, ϕ) x 3 (z) 0   , φ(x 1 , x 2 , x 3 ) = φ(r, ϕ, z) = ϑ 2   -gp(r) x 1 (r, ϕ) -gp(r) x 2 (r, ϕ) 2x 3 (z)   , (138)
(150)

This solution is valid for a generic second order tensor and for a generic vector triplet. Using (149), we finally see that The ratio Ω between the Cosserat torsional stiffness and the classical value that can be found e.g. in [START_REF] Anderson | Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam[END_REF][START_REF] Gauthier | A quest for micropolar elastic constants[END_REF][START_REF] Lakes | Experimental methods for study of Cosserat elastic solids and other generalized elastic continua[END_REF] is In Fig. 28 we report how the torsional stiffness divided by the radius of the cylindrical rod squared ( Tw/R 2 ) vary with respect to the radius squared R 2 for the Cosserat model and the relaxed micromorphic model where

Ω = 1 + 6 t R 2 1 -4/3Ψ χ 1 -Ψ χ , 2 t = β + γ 2µmacro , Ψ = β + γ α + β + γ , (152)
a = µ L 2 c 12π a 1 a 3 a 1 + 8a 3 = π(β + γ)(3α + β + γ) 2α + β + γ , b = µ L 2 c 3 2 π a 1 = 3 2 π(β + γ) . (154)
It is highlighted that the Cosserat model do not tent to a classical linear elastic model for µc → 0 as it can bee seen from eq.(153) or eq.( 68). It is underlined that for the relaxed micromorphic model the stiffness is bounded by the one obtained for Lc → 0 (macro) and Lc → ∞ (micro): the macro-stiffness (Lc → 0) is the limit to which all curves with finite Lc tend asymptotically to for R 2 → ∞ (this limit has been cut in order make possible to distinguish all the curves), while the micro-stiffness (Lc → ∞) is the limit to which all the curves tend asymptotically to for R 2 → 0.

From (165) it is possible to see that the symmetric part of the integral of Du on the circular cross-section is zero, while the skew-symmetric part is zero only if the domain is symmetric with respect z.

For the Cosserat model, letting Lc → ∞ still implies that A(x) = A = const. must be constant. The same calculations as before yield

A = skew 1 |Ω| Ω Du dV = 1 |Ω| Ω Du dV , since Du ∈ so(3) . (166)
For Ω = [0, L] × Γ we have

We remark that the same limit Lc → ∞ in the relaxed micromorphic model yields a linear elastic response with stiffness Cmicro since CurlP = 0 does not imply that P = const. but P = ∇ζ for some ζ : Ω ∈ R 3 → R 3 , see [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF].

Figure 2 :

 2 Figure 2: In the linear approximation, sections of the cylindrical rod are not only rotated, but also expanded radially for non zero rate of twist ϑ. With (b) we see that the change of radius δR = R cos ϑ -R = 1-cos ϑ cos ϑ R =

WWW

  (Du, P , DP ) micro-strain (P = S) (Du, P , CurlP ) micro-stretch (P = A + ω1) min u,A,ω W (Du, A, ω1, Curl (A + ω1)) Cosserat (P = A) min u,A W (Du, A, Curl A) couple stress (A = skew Du) min u W (Du, Curl (skew Du)) skew symmetric couple stress min u W (Du, skew Curl (skew Du)) modified couple stress min u W (Du, sym Curl (skew Du)) micro-void (P = ω1) min u,ω W (Du, ω, Curl (ω1)) ad-hoc model min u,A,S W (Du, A, S, Curl A, DS) second gradient (P = Du) min u Du, D 2 u

Figure 3 :

 3 Figure 3: Calculation of the classical torque.

  P ) 1 = 0.

  only two are not redundant since (40) 1 = tan ϕ (40) 2 and (40) 3 = tan ϕ (40) 4 .

  e z ) e ϕ , e r -(m × e z ) e r , e ϕ r dr dϕ (

Figure 5 :

 5 Figure 5: (Relaxed micromorphic model) Torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c for (a) µ c → 0, (b) µ c = 1/2, and (c) µ c → ∞. The torsional stiffness remains bounded as L c → ∞ (R → 0). The values of the parameters used are: µ = 1, µ e = 1/10, µ micro = 1/4, a 1 = 1/5, a 2 = 1/6, a 3 = 1/7, R = 1.

Figure 6 :

 6 Figure 6: (Relaxed micromorphic model) Torsional stiffness for the torque energy while varying L c , for different values of µ c = {0, 1/30, 1/10, 1/5, 1, ∞}. The torsional stiffness remains bounded as L c → ∞ (R → 0) and the model does not collapse into a linear elastic one. The values of the other parameters used are: µ = 1, µ e = 1/3, µ micro = 1/4, a 1 = 10, a 3 = 1/50, R = 1. Here, varying µ c does not intervene with T macro and T micro .

4. 1 . 2

 12 The relaxed micromorphic model with conformal curvature energy (a 3 = 0) while varying the Cosserat couple modulus µ c

Figure 7 :

 7 Figure 7: (Relaxed micromorphic model with conformal curvature energy) Torsional stiffness for the torque energy while varying L c , for different values of µ c = {0, 1/30, 1/10, 1/5, 1, ∞}. The torsional stiffness remains bounded as L c → ∞ (R → 0) and the model does not collapse in a linear elastic one beside the case µ c = 0. The values of the other parameters used are: µ = 1, µ e = 1/3, µ micro = 1/4, a 1 = 2, R = 1.In this case, varying µ c influences the torsional stiffness also for small specimen size.

Figure 8 :

 8 Figure 8: (a) (Relaxed micromorphic model with full curvature) Torsional stiffness for the torque energy while varying L c , for different values of µ micro = {0, 1/20, 1/7, 1/4, 1/2, ∞}. The torsional stiffness becomes unbounded as L c → ∞ (R → 0) when µ micro → ∞. The values of the other parameters used are: µ = 1, µ macro = 1/10, µ c = 1/2, a 1 = 1/5, a 3 = 1/7, R = 1. The Cosserat solution appears for µ micro → ∞. (b) (Cosserat model and relaxed micromorphic model with conformal curvature). Torsional stiffness for the torque energy while varying L c . The torsional stiffness is bounded as L c → ∞ (R → 0). For the Cosserat model we chose µ c = 1/9 while for the relaxed micromorphic model µ c = 1/2 and µ micro = 3 in order to have the same upper bound T = T . The values of the other parameters used are: µ = 1, µ macro = 1, a 1 = 5, R = 1.

Figure 9 :

 9 Figure 9: (Relaxed micromorphic model) Response of the relaxed micromorphic model while varying (a) the curvature parameter a 1 having a 3 = 20 and (b) the curvature parameter a 3 having a 1 = 20. The values of the other parameters are µ = 1, µ c = 1/5, µ e = 1, µ micro = 1/9, R = 1.

Figure 10 :

 10 Figure 10: (a) boundary conditions scheme for a cylindrical rod of length L = 10 and radius R = 1; (b) deformed rod from the finite-element simulation on which it is mapped how the component of the gradient of the displacement u 1,3 changes.

-

Figure 11 :

 11 Figure 11: Plots of the components (a) P 12 , (b) P 13 , (c) P 21 , and (d) P 31 of the micro-distortion tensor P at the cross-section z = 0. The purple line corresponds to L c = 0, the gray one to L c = 1, and the green one to L c = ∞. The values of the other parameters used are µ = 1, µ c = 1, µ e = 1, µ micro = 1, a 1 = 1, a 3 = 1, R = 1, ϑ = π/50, L = 10.

Figure 12 :

 12 Figure 12: (a) Plot of how the component P 13 ; (b) and P 31 vary along a line on the external boundary (ϕ = π/2 and r = R): the solid lines are the analytical solution while the marker are the numerical values obtained thanks to a finite-element analysis. The purple line has been obtained for L c = 0, the gray one for L c = 1, and the green one for L c = ∞. The values of the other parameters used are µ = 1, µ c = 1, µ e = 1,µ micro = 1, a 1 = 1, a 3 = 1, R = 1, ϕ = π/2.As it can be seen, the solution does not converge stably and not perfectly symmetrically (the mesh is not symmetric) to the analytical one, but nevertheless it converges rapidly.

Figure 13 :

 13 Figure 13: Plots of the component P 13 across the section placed at z = 0 obtained (a) analytically and (b) with the finite-element analysis. The two results are in perfect agreement. The values of the other parameters used are µ = 1, µ c = 1, µ e = 1, µ micro = 1, a 1 = 1, a 3 = 1, R = 1, ϕ = π/2, L c = 1.

Figure 14 ::= L relax c √ 2 .

 142 Figure 14: (a)(Cosserat model with full curvature) Torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c . The torsional stiffness is unbounded as L c → ∞ (R → 0). (b)(Cosserat model with full curvature vs relaxed micromorphic model) Torsional stiffness for the torque energy (T w ) while varying L c . Observe that the torsional stiffness remains bounded for the relaxed micromorphic model while it blows up for the Cosserat model as L c → ∞ (R → 0). For best comparison, the characteristic length scale of the Cosserat model has been chosen L Coss c

Figure 15 :

 15 Figure 15: (Cosserat model with conformal curvature) Torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c . The torsional stiffness is bounded as L c → ∞ (R → 0). The values of the parameters used are: µ = 1, µ c = 1/2, µ macro = 1/2, a 1 = 5, R = 1.Here, the Cosserat couple modulus µ c is clearly related to the value of the stiffness for small specimen size.

Figure 16 :

 16 Figure 16: (Cosserat model vs indeterminate couple stress model) Comparison of the torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c for (a) the Cosserat model with µ c → ∞ and for (b) the indeterminate couple stress model. There is not a one to one correspondence between the respective torque but the energy coincides.

Figure 17 :

 17 Figure 17: (Cosserat model) (a) Torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c for the limit µ c → 0. The model does not collapse into a classical linear elastic one. The values of the material parameter used are µ = 1, µ e = 1/10, a 1 = 1/5, a 3 = 1/7, R = 1. (b) Sensitivity study on how the Cosserat model behaves while varying µ c = {0, 1/3, 1, ∞}:for µ c → ∞ we recover the indeterminate couple stress model, while for µ c → 0 we still have a non linear relation between T w and R/L c since a classical linear elastic model is not attained (see eq.(68)). The values of the material parameter used are µ = 1, µ e = 1/10, a 1 = 12, a 3 = 1/20, R = 1.

Figure 18 :

 18 Figure 18: (Cosserat model) Response of the Cosserat model while varying (a) the curvature parameter a 1 having a 3 = 20 and (b) the curvature parameter a 3 having a 1 = 20. The values of the other parameters are µ = 1, µ c = 1/5, µ macro = 1/10, R = 1.

1 2

 1 Curl A = tr (D axlA) T -(D axlA) T for A ∈ so(3) we can rewrite Curl skew Du = -(D axl (skew Du)) T = D curl u, since tr (Curl skew Du) = 0. where σ = 2 µ e sym Du + λ e tr (Du) 1 is the symmetric force stress tensor, e r is the radial unit vector, and the non-symmetric second order moment stress is m = µ L 2 c (a 1 dev sym Curl skew Du + a 2 skew Curl skew Du) . (76) The term (Anti [(1 -e r ⊗ e r ) • m • e r ] + -Anti [(1 -e r ⊗ e r ) • m • e r ] -) is the measure of the discontinuity of Anti [(1 -e r ⊗ e r ) • m • e r ] across the boundary.

  z , e ϕ r r dr dϕ = µ e I p ϑ = T c ϑ , M m (ϑ) := 2π 0 R 0 (m × e z ) e ϕ , e r -(m × e z ) e r , e ϕ + (m × e r ) e ϕ , e z -(m × e r ) e z , e ϕ r dr dϕ

Figure 19 :

 19 Figure 19: (Indeterminate couple stress model) Torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c . The torsional stiffness is unbounded as L c → ∞ (R → 0). The values of the parameters used are: µ = 1, µ e = 1/3, a 1 = 1/5, R = 1.

Figure 20 :9. 1 Limits 9 . 1 . 1 Figure 21 :

 20191121 Figure 20: (Micromorphic model, classical case) Torsional stiffness for the classical torque T c , the higherorder torque T m , and the torque energy T w while varying L c . The torsional stiffness is unbounded as L c → ∞ (R → 0). The values of the parameters used are: µ = 1, µ e = 1/3, µ micro = 1/4, µ c = 1/5, a 1 = 1/5, a 2 = 1/6, R = 1.

Figure 22 :

 22 Figure 22: (Micromorphic model) Torsional stiffness for the torque energy while varying L c , for different values of µ c = {0, 1/30, 1/10, 1/5, 1, ∞}. The torsional stiffness remains bounded as L c → ∞ (R → 0) and the model does not collapse in a linear elastic one. The values of the other parameters used are: µ = 1,µ e = 1/3, µ micro = 1/4, a 1 = 2, a 3 = 1/20, R = 1.In this case, the stiffness for arbitrary small sample size is governed by T e and not T micro . The reason for this is explained in Appendix D.

23

 23

Figure 23 :

 23 Figure23:(Micro-strain model) Torque energy T w while varying L c . Observe that the torsional stiffness remains bounded as L c → ∞ (R → 0). The values of the parameters used are: µ e = 1/3, µ micro = 1/4, µ = 1, a 1 = 1/5. In this case, the stiffness for arbitrary small sample size is governed by T e and not T micro .

2 + 2 9 a 3 2 . 1 ( 104 ) -µL 2 c a 1 9 a 3

 2293211042193 D tr (Du) 1 The equilibrium equation, in the absence of body forces, is Div 2µ macro sym Du + λ macro tr (Du) dev sym ∆ (Du) + a 2 skew ∆ (Du) + 2 tr (∆ (Du)) 1 = 0 , where ∆ (Du) ∈ R 3×3 is taken component-wise. The non-trivial boundary conditions at the free surface are t(r = R) = σ e r + [(e r ⊗ e r ) : ∇m] e r -2 [(1 -e r ⊗ e r ) : ∇m] e r (105) + [(1 -e r ⊗ e r ) : ∇e r ] (e r ⊗ e r ) -(1 -e r ⊗ e r ) (∇e r ) T : m = 0 R 3 , η(r = R) = (e r ⊗ e r ) : m = 0 R 3 , 10

0 R 0 (

 00 z , e ϕ r r dr dϕ = µ macro I p ϑ = T c ϑ , M m (ϑ) := 2π m e z ) e r , e ϕ -(m e z ) e ϕ , e r + (m e r ) e z , e ϕ -(m e ϕ ) e z , e r r dr dϕ

Figure 24 : 2 + a 3 2 . 2 c a 1

 2423221 Figure 24: (Second gradient model) Torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c . The torsional stiffness is unbounded as L c → ∞ (R → 0). The values of the parameters used are: µ = 1, µ macro = 1/4, a 1 = 1/5, a 3 = 1/6, R = 1.

2 0Figure 25 :

 225 Figure25: The purplish curves show how the micro-strain model particularises to the strain gradient model for µ e → ∞ (the following set has been used µ e = {1/3, 1/2, 2/3, ∞}). The greenish curves show how the choice of a 2 = 0 guarantees the formal equivalence (which is however always substantially true) between the second gradient model and the strain gradient model[START_REF] Mindlin | Micro-structure in linear elasticity[END_REF] (the following set has been used a 2 = {1/4, 1/10, 0}). The values of the other parameters used are: µ = 1, µ macro = 1/7, a 1 = 1/5, R = 1.

2

  = D (tr (S) 1) 2 = 0 in terms of the ansatz (114). The equilibrium equations, in the absence of body forces, are the following Div σ:= [2µ e (sym Du -S) + λ e tr (Du -S) 1 + 2µ c (skew Du -A)] = 0, 2µ c skew (Du -A) -µ L 2 c skew Curl a 1 dev sym Curl A + a 3 3 tr (Curl A) 1 = 0 (111) 2µ e (sym Du -S) + λ e tr (Du -S) 1 -2µ micro S -λ micro tr (S) 1 + µ L 2 c a 4 sym ∆ (dev S) = 0 . The boundary conditions at the external free surfaces are

1 2 ϑ 2 ϑ

 122 sin ϕ a 4 µ L 2 c (3g m (r) + r g m (r)) -2r µ e (g m (r) + 1) -2r g m (r) µ micro = 0 , -1 cos ϕ a 4 µ L 2 c (3g m (r) + r g m (r)) -2r µ e (g m (r) + 1) -2r g m (r) µ micro = 0 . Between the two equilibrium equations (115) there are only two independent equation since (115) 1 = -tan ϕ (115) 2 and (115) 3 = tan ϕ (115) 4 . The solution of (115) is

  e z )e ϕ , e r -sym(m e z )e r , e ϕ + skew(m × e z )e ϕ , e r -skew(m × e z )e r , e ϕ Cosserat component r dr dϕ =

Figure 26 :

 26 Figure 26: (Ad-hoc model) Torsional stiffness for the classical torque T c , the higher-order torque T m , and the torque energy T w while varying L c . The torsional stiffness is unbounded as L c → ∞ (R → 0) due to the Cosserat effects. The values of the parameters used are: µ = 1, µ c = 1/2, µ e = 1/3, µ micro = 1/4, a 1 = 1/5, a 3 = 1/6, a 4 = 1/7, R = 1.

2 ϑ 2 ϑ 3 tr(m) 2 1 , ( 1 -Since 1 , 1 = 3 , 1 , 3 tr(m) 2 1 , ( 1 - 3 tr(m) 2 [ 3

 223211131321323 Substituting the ansatz (138) in (134) the equilibrium equations are equivalent to -1 cos ϕ 4ρ µc (g(ρ) -1)-(α + β + γ) 3g (ρ) + ρg (ρ) = 0 ,(139)-1 sin ϕ 4ρ µc (g(ρ) -1) -(α + β + γ) 3g (ρ) + ρg (ρ) = 0 , which are completely equivalent to (62) in Section 6 once used the relations (131). Since also the boundary conditions (135) are equivalent to the boundary condition (59) in Section 6, further calculations are avoided. Here, we recall the relations between the two moment stress tensors expressed in the classical format (m) and in the dislocation format (m) dev sym m = -dev sym m , skew m = skew m , m = skew m = 0 for the torsional problem, and m = µ L 2 c a 1 dev sym Curl A + a 3 3 tr (Curl A) 1 , m = α tr (D(axl(A))) 1 + β (D(axl(A))) T + γ D(axl(A)) , (141) It is also interesting to show the relation between the two higher-order torques expressed in terms of m and m, respectively. First, we observe skew (m × ez) eϕ, er -skew(m × ez)er, eϕ = (m × ez) eϕ, er -(m × ez) er, eϕ ,(142)which does not holds component-wise.11 Using that the cross product between two unit vectors gives the third one, and(m × v) w = m (v × w) ∀v, w ∈ R 3 and ∀m ∈ R 3×3 ,(143)it is possible to write (m × ez) eϕ, er -(m × ez) er, eϕ = m (eϕ × ez) , erm (er × ez) , eϕ = -[ m er, er + m eϕ, eϕ ] .(144)Since (er ⊗ er + eϕ ⊗ eϕ + ez ⊗ ez) = 1 we may convert the double dot-product into a dyadic product as follows-[ m er, er + m eϕ, eϕ ] = -m, (er ⊗ er + eϕ ⊗ eϕ) = -m, (1 -ez ⊗ ez) . (145)Substituting the relation (140) between m and m we havem, (1 -ez ⊗ ez) = --dev m + skew m + 1 ez ⊗ ez) . (ez ⊗ ez) = 1,and m is decomposed into its three orthogonal components (except for multiplying factors) we can write --dev m + skew m + 1 ez ⊗ ez) m, (1 -ez ⊗ ez) -1 -1] = -dev m, ez ⊗ ez -1 3 tr(m) = (148) -dev m, ez ⊗ ez -1 3 tr(m) 1, ez ⊗ ez = -dev m + 1 3 tr(m)1, ez ⊗ ez = m, ez ⊗ ez = -m ez, ez = -mzz . (149)11 The values of the four terms for a generic second order tensor m are: skew(m×ez)eϕ, er = 1 2 (m11 + m22) , skew(m×ez)er, eϕ = -1 2 (m11 + m22) , (m × ez) eϕ, er = m22 sin 2 ϕ + m11 cos 2 ϕ + (m12 + m21) sin ϕ cos ϕ , (m × ez) er, eϕ = -m22 cos 2 ϕ -m11 sin 2 ϕ + (m12 + m21) sin ϕ cos ϕ. Note that skew (m × ez) eϕ, er = (m × ez) eϕ, er and skew (m × ez) er, eϕ = (m × ez) er, eϕ .

Γ

  mzz r dr dϕ = Γ -skew (m × ez) eϕ, er -skew (m × ez) er, eϕ r dr dϕ .(151)

  .where t is the characteristic length for torsion, Ψ is the polar ratio, N is the Cosserat coupling number, α, β, and γ are the curvature coefficients in the classical Cosserat formulation, µmacro is the classical Cauchy shear modulus, µc is the Cosserat couple modulus, and In is the modified Bessel function of the first kind of order n.

Figure 27 : 3 Lc → 6 Lc → 1 2 LFigure 28 :

 2736228 Figure 27: For comparison, the figure has been taken from Lakes [35]. Here, G = µ macro denotes the classical shear modulus and N → 1 corresponds to µ c → ∞. To go from (65) to (152) we have to use the relations (131), while remembering to incorporate the term µ L 2 c (the terms not reported do not change between the two notations) Ω = 1 + 6 t R

Figure 29 :

 29 Figure 29: (a) Cosserat model; (b) relaxed micromorphic model. The values of the coefficient used are: µ = 1, µ macro = 1/14, µ micro = 1/4, a 1 = 2, a 3 = 1/50, L c = 3. In the Cosserat model, the solution for µ c → ∞ (the indeterminate couple stress model) shows a jump.

D+ κe 2 tr 2 2

 22 Ad-hoc minimization for L c → ∞ in the full micromorphic model and in the micro-strain model Looking at the curvature energy of the full micromorphic model (or the micro-strain model) it is clear that for Lc → ∞ the micro-distortion tensor field P must be constant P = P , provided all curvature coefficients are strictly positive. We calculate this constant in the following. Thus we consider min u,P Ω µe dev sym Du -P 2 Du -P + µc skew Du -P dV . The weak form is given by Ω 2µe dev sym Du -P , -δP + κe tr Du -P 1, -δP + 2µc skew P , -δP (156) +2µmicro dev sym P , δP + κmicro tr P 1, δP dV = 0 ∀ δP . Ω 2µe dev sym Du -P + κe tr Du -P 1 + 2µc skew Du -P (157) -2µmicro dev sym P -κmicro tr P 1, δP dV = 0 ∀ δP . For constant δP this can be rewritten as Ω 2µe dev sym Du -P + κe tr Du -P 1 + 2µc skew Du -P (158) -2µmicro dev sym P -κmicro tr P 1 dV, δP = 0 ∀ δP . Since δP is arbitrary, this implies that Ω 2µe dev sym Du -P + κe tr Du -P 1 + 2µc skew Du -P -2µmicro dev sym P -κmicro tr P 1 dV = 0 , (159) or Ω 2µe dev sym Du + κe tr (Du) 1 + 2µc skew Du dV = (160) = Ω 2µe dev sym P + κe tr P 1 + 2µc skew P + 2µmicro dev sym P + κmicro tr P 1 dV . Using the orthogonality of dev sym•, skew• and tr(•) 1 we obtain Ω 2µe dev sym Du dV = Ω 2µe dev sym P + 2µmicro dev sym P dV , (161) Ω κe tr (Du) dV = Ω κe tr P + κmicro tr P dV , Ω 2µc skew Du dV = Ω 2µc skew P dV , and since P is constant we can write dev sym Since dev sym, skew, and tr are linear operators, we obtain equivalently dev sym P = µe µeSubstituting the ansatz (85) into (163) we obtain P = 0. Analogous calculations can be carried out for the micro-strain model for which skew P = 0 and µc = the ansatz (96) into (164) we obtain S = 0.The integral on the circular cross-section Γ of the gradient of the displacement isΓ Du(x)dV =

  2 u = µ macro sym Du χ iik χ kjj + a 2 χ ijj χ ikk + a 3 χ iik χ jjk + a 4 χ ijk χ ijk + a 5 χ ijk χ kji , where χ = D 2 u (χ ijk = ∂ 2 u k ∂xi ∂xj). The expression we are going to use in the following is a simplified isotropic strain energy with three curvature parameters W Du, D 2 u = µ macro sym Du

	+	2 + λ macro 2 a 1 D dev sym Du λ macro 2 tr 2 (Du) tr 2 (Du) + a 1 2 + µ L 2 2 c 2 + a 2 D skew Du	(102) (103)

  W (Du, A, S, Curl A, DS) = µ e sym Du -S 2 + λ e 2 tr 2 (Du -S) + µ c skew (Du -A)

								2
		+ µ micro dev S	2 +	κ micro 2	tr 2 (S)	(110)
		+	µ L 2 c 2	a 1 dev sym Curl A	2 +	a 3 3	tr 2 (Curl A) + a 4 D (dev S)	2 ,
	since skew Curl A	2 = D (skew S)

Looking at the analytical solution obtained in[START_REF] Madeo | Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design[END_REF] we see that the expression R Lc solely determines the response. Therefore, we can either fix R > 0 and send Lc → ∞, or fix Lc and send R → 0, having the same effect.

For the torsion problem, κmicro does not intervene.

In Hadjesfandiari and Dargush[START_REF] Hadjesfandiari | Comparison of theoretical elastic couple stress predictions with physical experiments for pure torsion[END_REF] the discussion of higher traction boundary conditions seems to be missing some terms in (75), letting the authors erroneously conclude that the classical displacement pure torsion solution does not satisfy the higher order boundary conditions.

Shaat [66] uses the micro-strain model with mixed terms and a degenerate curvature expression in DS, omitting S11,1, S22,2, and S33,3.

In index notation (1 -er ⊗ er) : ∇m = (δip -ninp)m ijk,p .

Acknowledgements. Angela Madeo acknowledges support from the European Commission through the funding of the ERC Consolidator Grant META-LEGO, N°101001759. Angela Madeo and Gianluca Rizzi acknowledge funding from the French Research Agency ANR, "METASMART" (ANR-17CE08-0006). Angela Madeo and Gianluca Rizzi acknowledge support from IDEXLYON in the framework of the "Programme Investissement d'Avenir" ANR-16-IDEX-0005. Hassam Khan acknowledges the support of the German Academic Exchange Service (DAAD) and the Higher Education Commission of Pakistan (HEC). Ionel Dumitrel Ghiba acknowledges support from a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P1-1.1-TE-2019-0348, Contract No. TE 8/2020, within PNCDI III. Patrizio Neff acknowledges support in the framework of the DFG-Priority Programme 2256 "Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials", Neff 902/10-1, Project-No. 440935806.

A Cylindrical coordinates

The relation between the derivatives for a vector field u(x), u(r) and a second order tensor P (x), P (r) with respect an orthogonal set of coordinates x = {x 1 , x 2 , x 3 } and a cylindrical set of coordinates r = {r, ϕ, z} are the following