Micromorphic continuum models provide an approach to describe micro-scale structural and mechanical eects in the continuum description of material behavior [14]. For micromorphic models to be representative, it is important to link the continuum elds with the micro-scale mechanisms. For instance, when we seek descriptions of the collective behavior of a large number of grains, as in the case in which a material's granular microstructural eect need to be modeled, it is necessary to describe the eects of grain and grain-grain interface deformations (termed as micromechanics), which could be highly localized and directional, as in the case of Hertzian contacts [5,6]. These grain deformations could be non-uniform due to the grain-shape and interfacial/surfacial characteristics or due to the grain-neighborhood structure (termed as microstructure). The grains can also experience rotations relative to their neighboring grains further contributing to the overall deformation of the collective system [7,8]. These peculiarities of granular systems renders the modeling of the behavior that emerges at the macro-scale (containing large number of grains) particularly challenging. It is now widely accepted that the classical continuum approach (or the Cauchy format) fails woefully in representing the true nature of the micro-structured material behavior and describe many observed phenomena at the macro-scale, even though it serves well for a number of engineering problems. There is growing realization that to describe the emerging macro-scale behavior of these materials faithfully and with increasing delity, it is necessary to consider continuum models in which the deformations at micro-scale are properly accounted. A key shortcoming of the classical continuum theory is its inability to describe the eects of complex kinematics and distribution of elastic energy in internal deformation modes within the continuum material point. In this regard, it is worthwhile to recall that in the early development of the continuum modeling of deformable media in the works of Navier, Cauchy, Poisson and Piola, the material was viewed as composed of molecules (particles) that attract and repel each other [911]. Many materials, particularly at the scale in which granular microstructures appear, can be treated in a similar sense in which the deformation of an interacting grain-pair can be eectively described in terms of the relative movements of the grain centroids/barycenters regardless of the location of the actual deformation within the grains. Indeed such a treatment can serve as a point of departure for both discrete and continuum models. Examples of discrete models range from atomistic and molecular (see among a very large literature base [12,13]) to more recent large grain models (see among others [14,15]). Continuum models of these materials that proceed from this approach are traced to early part of 20th century (see historical context in the review [16]) to more recent models such as those in [7,17,18] devised for the case of small deformations to account for internal deformation modes.
In discrete models the key kinematic variable is the grain motion, thus by their nature, these models describe the fate of every grain as a result of its interactions with the neighbors and by extension the whole collection. Discrete models, therefore, can result in grain trajectories (that can include grain translations and grain spins) and spatial distributions of deformation energies and their potential decomposition into various deformation modes. The extensive approach is also a principal drawback of discrete model as the detailed data is, in many cases, distinguished by the lack of accurate knowledge of grain locations, shapes and surface type/conditions and every possible grain-pair interaction relationships. Nevertheless, simulations using discrete models [1929] have been suggestive and have led to the recognition of certain micro-scale phenomena, such as localization of energy into small zones of grain/atomic clusters [30,31]or localization bands and vortices [32,33], propensity of micro-rotation [34], identication of oppy modes [35], and so-called 'force-chains' (see for example [14,3638]). Many of these micro-scale phenomena, particularly those related to certain clustering of grain displacements and coherent/incoherent grain rotations, are attested to by experimental measurements of grain motions such as [39,40]. For describing the many relevant phenomena exhibited by grain collections detailed information regarding precise grain trajectory is unnecessary. However, the grain-scale kinematics have profound relevance in the macro-scale description of granular materials representing the emergent collective behavior of large number of grains. Indeed, the practical pathway to the control of macro-scale behavior by accessing the micro-scale lies in their representative linkages based upon predictive theories [41]. The recent realization of metamaterials based upon pantographic motif that link to second gradient continuum description [4250] and chiral granular materials that link to Cosserat continuum are exemplar of such predictive micro-macro theoretical identications [5154]. These works have shown that successful eorts to link micro-to macro-scale lead to generalized continuum theories and these can, therefore provide ecient ways for rational design of (meta)materials (as opposed to trial and error or other ad hoc approaches, see for example the review by [55,56]). Remarkably these micro-macro identications indicate how the stored elastic energy can be distributed within internal deformation modes, including second [5769] and higher [35] gradient modes, grain rotations/spins [7], and nonstandard coupling of shear and rotations [70,71].
In the spirit of developing such micro-macro identications in a generalized setting of nite deformations that account for geometric nonlinearity, we focus in this paper upon a micromorphic continuum description of materials in which granular microstructural eects need to be modeled. In these cases, Taylor expansion of only conventional macro-scale kinematic descriptor is not representative and additional kinematic descriptors may be introduced to accurately describe the response as in Cosserat or micromorphic media [17,18,51,7283]. To this end we utilize neighboring grain-pair deformation as the fundamental building block and develop objective kinematic descriptors for relative displacements following Piola's ansatz for micro-macro identication. Micro-scale deformation energy is then introduced in terms of the developed objective relative displacement decomposed into a component along the vector that represents the directors of generic grain-pairs centroids in the system, termed as normal component, and a component in the orthogonal plane, termed as tangential component. For the present work, a quadratic form of the micro-scale deformation energy is utilized to obtain an identication for the case of geometrically non-linear isotropic elasticity. As a result, expressions for elastic constants of a linear novel micromorphic continuum are obtained in terms of the micro-scale parameters. The plan of the paper is the following. Subsequent to this introduction, in Sec. 2, the discrete and continuous models describing a granular system are presented. Particularly, the discrete model, whose kinematics is ultimately specied in terms of a position and a micromorphic deformation gradient for each subsystem a granular aggregate termed here as sub-body is introduced rst. The continuum model is then introduced and discrete-continuum kinematic bridging is performed through the Piola's ansatz. Specialization of the proposed approach to Cosserat and strain-gradient continua through proper restrictions of the micromorphic deformation gradient is subsequently discussed. Section 3 builds on the previous sections, which are exclusively concerned with the kinematics of the studied systems, by introducing the elastic strain energy for the discrete model and the corresponding one for the continuum model as the result of a homogenization procedure based on the Piola's ansatz. More specically, relative deformation measures are introduced followed by the denition of the elastic strain energy function in the nonlinear case. Special emphasis is given to the nonlinear 3D isotropic case for no intergranular micromorphic deformation eects and to the linear 3D isotropic case for general intergranular micromorphic deformations. Finally, conclusions are briey presented. .
2 Discrete and continuous models for granular systems
For a material with granular microstructure, the discrete granular model is illustrated, as a general example, in Fig. 2.1. In the reference conguration we have N sub-bodies. Each sub-body is composed by many grains and is assumed to be a continuum, one point of which is labeled X n ∈ B n with n = 1, . . . N . All other points of the sub-body are labeled X n ∈ B n . When this material undergoes deformation, the point X n is placed, in the present conguration, at x n via the placement function χ n , i.e.,
where t is the time variable. While all the other points X n ∈ B n of the grain n are placed, in the present conguration, at x n via a dierent placement function χ n , i.e.,
in such a way that, evaluating this second placement function at X n = X n , we have
The sub-body B n is assumed to be suciently small that a Taylor's series expansion of χ n centered at X n = X n can be truncated at the rst order, i.e.,
where deformation gradients are dened, and the truncation is equivalent to assuming an ane deformation of each sub-body B n . Thus, the kinematics of the discrete model is completely described, for each sub-body B n , by the following two functions χ n (t) , P n (t) , ∀t ∈ R, ∀n = 1, . . . , N.
The continuum model is described in Fig. 2. In the reference conguration we have a continuum body C * . Each point of this continuum is called X ∈ C * . The same point X is the representative of a micro-structure at a dierent, smaller scale, and may be another continuum. Any point within this micro-structure is called X . The point X is placed, in the present conguration, at x via the placement function χ, i.e.,
Further, the points X of the micro-structure are placed, in the present conguration, at x via a dierent placement function χ', i.e.,
in such a way that, evaluating it at X = X, we have
The micro-structure is assumed to be suciently small that a Taylor's series expansion of the placement function χ (X, X , t) centered at X = X can be truncated at the rst order, i.e.,
where the micromorphic deformation gradient P = P (X, t) of each micro-structure is dened,
and where ( 6) has been considered. The truncation of (7) at rst order is equivalent to assume an ane deformation of each micro-structure. Thus, the kinematics of the continuum model is completely described by the following couple of functions
The deformation gradient, in this case, is dened by two gradients: the gradient ∇χ of the placement function, and the gradient ∇P of the micromorphic deformation gradient
Thus, the Green-Saint-Venant tensor G (or non-linear macro-strain) and the micromorphic Green-Saint-Venant tensor M are dened as in ( 10)
In addition, a relative micro-macro Green-Saint-Venant tensor (or non-linear relative deformation) is dened as
Note that the tensor Υ dened in (11) vanishes when P ≡ F , i.e. when the micromorphic deformation gradient P equals the deformation gradient F . Such a tensor is nothing but a strain measure taking into account the dierential deformations of the continuum element and the microstructure. Further, considering that a polar decomposition holds for both the micromorphic deformation gradient and the macro deformation gradient, it can be concluded that the tensor Υ takes into account also the dierential rotation between the micro-and macro-scale. Remark that, as it will be shown in the sequel, the denition of Υ in ( 11) is only one of the possible non-linear generalizations of the relative deformation γ dened in Mindlin's work [1]. It is also worthwhile to dene the following third order tensors, the so-called rst and second non-linear micro-deformation gradients
where the transpose operator T 13 refers to the rst and to the third indices of the third order tensors Λ and Λ r as better dened in index notation as follows,
Λ r ijh = P T ∇P T13 ijh = P T ∇P hji = P T ha (∇P ) aji = P ah P aj,i .
We remark that the tensors dened in (10), (11) and ( 12) are objective as shown in the following. Let Q be a general orthogonal matrix giving a change of the frame of reference, let [F ] be the matrix representation of the deformation gradient F , [P] be that of the micromorphic deformation gradient P and [∇P ] be that of its gradient. This leads to
where F , P and ∇P are the matrix representations of the same tensors F , P and ∇P , respectively, in the rotated, via Q, frame of reference. It is straightforward to show that the matrix representation G of the Green-Saint-Venant tensor in this rotated frame of reference is the same as that in the initial frame of reference, denoted by [G],
where [I] is the identity matrix. The same frame indierence can be demonstrated for the micromorphic Green-Saint-Venant tensor M , using the matrix representations M and [M ] as
for the relative micro-macro Green-Saint-Venant tensor Υ, using the matrix representations Υ and
Figure 2: Continuum model. The reference conguration of the continuum body is C * . Each point of it is called X ∈ C * and its placement is χ (X, t). Such a point X is the representative of a micro-structure, e.g. the cube in the gure. Within the micro-structure (thus, within the cube in the gure) two placements are dened. The rst, according to (4), is the same placement χ (X, t) of the point X. The second, according to (5), placement χ (X, X , t) dene that of any other points X of the micro-structure.
for the rst relative micro-deformation gradient using with the matrix representations Λ and [Λ]
(or with that of its transpose 1 -3 counterparts Λ T13 and Λ T13 ) as
and, for the second relative micro-deformation gradient, using the matrix representations Λ r and
and Λ T13 r ) as,
.
In the continuum-discrete models identication, we follow Piola's ansatz, such that
The (15) implies that the placements χ i (t) and the micro-deformation P i (t), with i = 1, ..., N , of the N sub-bodies B n in the discrete model illustrated in Fig. 2.1 correspond to the placement χ (X, t) and the micro-deformation P (X, t), evaluated respectively at the points X i with i = 1, ..., N , of the body C * in the continuous model given in Fig. 2. With this in mind, we will utilize the discrete model only as a guiding justication for the constitutive assumptions postulated in the following Section 3. Needless to say, the content of this paper refers to the continuous model of Fig. 2.
The connection with the discrete model is only suggestive of possible micro-scale mechanism that could be revealed through the Piola's ansatz (15) and is useful for the introduction of the indicated constitutive assumptions. We note that no attempt is made here to give an evolution equation of each grain as one would for a completely discrete description.
We note that restrictions on the micro-deformation P = P (X, t) dene dierent type of microstructural continua. First of all, no restriction on P denes a micromorphic continua, but (i) an orthogonal micro-deformation P dene from (10) Cosserat continua,
and (ii) second gradient continua are obtained when the micro-deformation P is identied with the deformation gradient F , P = F = (∇χ) , that implies (iia) zero non-linear relative deformation Υ from ( 11), (iib) identication of the Green-Saint-Venant tensor G and of the micromorphic Green-Saint-Venant tensor M from (10),
(iic) identication of the 13-transpose second relative micro-deformation gradient Λ r and non-linear macro-strain-gradient tensor ∇G,
and (iid) the following relation between the rst relative micro-deformation gradient Λ and the non-linear macro-strain-gradient ∇G,
where the left Cauchy-Green deformation tensor C is dened,
3 Elastic strain energy
Let us now assume that two sub-bodies, n and p, respectively placed in the reference conguration at X n and X p , are neighboring ones, that their distance is L in the reference conguration and that the unit vector ĉ is dened as follows,
In the reference conguration, therefore, the vector attached to the position X n and pointing to the position X p is ĉL and given in (16). Further, let us restrict the present model to the case in which the sub-bodies, n and p, place and deform similarly in the present conguration, and therefore the following Taylor's series expansions are possible and yield
We can now dene the following 3 objective tensors that may be utilized to represent the material deformation that are traceable to the micro-scale grain-pair relative displacements
where the superscripts n and p refers to the microstructures placed at X n and at X p . We call the tensor g np in (20) the macro deformation, the tensor m np in (21) the micro deformation and the tensor γ np in ( 22) the micro-macro deformation. The proof of their objectivity is analogous with that derived at the end of subsection 2.2.
By insertion of the Taylor's series expansions (18)(19) into the 3 denitions (20-21-22), yield, respectively,
The use of objective Green-Saint-Venant tensors in ( 10), ( 11) and ( 12) into ( 23), ( 24) and ( 25), yield,
The last two equations are derived easily in index notation as follows
, where the denitions ( 13) and ( 14) have been considered.
Thus, we dene the objective relative displacement, i.e. the macro-relative displacement, with (26),
the micro-macro-relative displacement, with (28),
and the micro-relative displacement, with (27),
that, in index notation, are
The half projection of the objective relative displacements on the unit vector ĉ, dened in ( 16), is the so called normal displacements u η . In the same way d η is dened as the normal micro-macrorelative displacement and r η is dened as the normal micro-relative displacement,
Their squares are
and therefore
-L 3 M ij Λ abc ĉi ĉj ĉh ĉa ĉb ĉc -L 4 Λ r ijh Λ abc ĉi ĉj ĉh ĉa ĉb ĉc .
The tangent displacement u τ is dened
as well as its square
The tangent micro-macro-relative displacement d τ and the tangent micro-relative displacement r τ are dened
Thus, their squares are calculated as follows
+4L 3 M ij Λ r abc [ĉ j ĉa ĉc δ ib -ĉi ĉj ĉa ĉb ĉc ] + L 4 Λ r ijh Λ r abc [ĉ i δ jb ĉh ĉa ĉc -ĉi ĉj ĉh ĉa ĉb ĉc ] .
The elastic energy function for a given couple of sub-bodies, say the couple n -p considered in Section 3.1, is assumed to be a quadratic form of normal and tangent components of the macrorelative displacement (29), of the micro-macro-relative displacement (30) and of the micro-relative displacement (31),
where k η , k τ , k dη , k dτ , k rη , k rτ , k ud , k ur and k rd are 9 elastic constitutive coecients of the present formulation. In principle, in the anisotropic case they all are a function of the unit vector ĉ, i.e. they are 9 orientation distribution function of the stiness of the continuum. In particular, k η and k τ are the normal and tangent stiness dened and used in [84]. Here the kinematic characterization of the material is more complicated and we have also the normal k dη and tangent k dτ micro-macrorelative stiness and the normal k rη and tangent k rτ micro-relative stiness. Besides, the presence of three scalar invariants u η , d η and r η makes possible three kinds of elastic interactions, i.e. the displacement-micro-macro-relative interaction with the homonymous stiness k ud , the displacementmicro-relative interaction with the homonymous stiness k ur and the micro-macro-micro-relative interaction with the homonymous stiness k rd . We also note that the quadratic assumption in ( 46) is a rst step. Other potential functions can be introduced that can lead to material nonliearity, and for the case of asymmetric tension-compression response evolving anisotropy (see for example [85]) and chirality [84] can emerge at the macro-scale when subjected to loading. Insertion of (34-41-35-44-36-45-37-38-39) into (46) and integrating over all the orientations of the unit circle S 1 in the 2D case or over the unit sphere S 2 in the 3D case, yields
or, in a compact form we have
where the elastic stiness C, B, A, D, D r , D rr , F, F r , G, C r , A r and B r are identied in (47) as follows, with the symmetrization induced by the symmetry of the strain tensors G and M
k τ (δ ik ĉj ĉl + δ il ĉj ĉk + δ jk ĉi ĉl + δ jl ĉi ĉk ) ,
k ur ĉi ĉj ĉk ĉl ĉm ,
k rτ (δ ik ĉj ĉl + δ il ĉj ĉk + δ jk ĉi ĉl + δ jl ĉi ĉk ) ,
(k rη -k rτ -2k rd ) ĉi ĉj ĉh ĉa ĉb ĉc + k rτ ĉi δ jb ĉh ĉa ĉc , (57)
k rd ĉi ĉj ĉk ĉl .
(58)
Let us assume that the micro deformation m np do not have a role in contributing to the elastic deformation energy (47). In this case, we can see from ( 31) that the micro-relative displacement r np does not contribute to the elastic deformation energy (47). The consequence is that micro-relative displacement r η and r τ play no role in the elastic energy expression (47), as seen from ( 33) and (43). Therefore, we assume that the corresponding stiness constants with subscript r, are null, that is
Thus from ( 52), ( 53), (54) 2 , (55) 2 , ( 56), ( 57) and ( 58) we have that the corresponding stiness tensors with superscripts r,
are null and therefore the elastic energy ( 47) is reduced to be in a form that is the analogous of that in eq. ( 5.1) Mindlin [1],
We will prove in Subsection 3.4 that ( 59) is nothing else than a possible non-linear geometric generalization of eq. (5.1) in Mindlin [1]. Besides, in the isotropic case Mindlin [1] in eq. ( 5.4) has given, among the isotropic identication D = 0 and F = 0 (at the end of page 15 in Mindlin [1]), the following representations
+a 4 δ jk δ il δ mn + a 5 δ jk δ im δ nl + a 6 δ jk δ in δ lm +a 7 δ ki δ jl δ mn + a 8 δ ki δ jm δ nl + a 9 δ ki δ jn δ lm +a 10 δ il δ jm δ kn + a 11 δ jl δ km δ in + a 12 δ kl δ im δ jn +a 13 δ il δ jn δ km + a 14 δ jl δ kn δ im + a 15 δ kl δ in δ jm , with those conditions that are made explicit at the end of page 16 in Mindlin [1], i.e.,
Insertion of ( 64) into ( 60), ( 61), ( 62) and ( 63) into the compact form of the strain energy (59) we have
or, expanding the Kronecker symbols, it yields a geometrical non-linear generalization of eq. (5.5) in Mindlin [1],
The aim of this Subsection is to identify the corresponding 18 isotropic micromorphic constitutive coecients, i.e., λ, µ, b 1 , b 2 , b 3 , g 1 , g 2 , a 1 , a 2 , a 3 , a 4 , a 5 , a 8 , a 10 , a 11 , a 13 , a 14 and a 15 . To do this, we impose the isotropic condition by assuming no dependence of the 5 elastic stiness k η , k τ , k dη , k dτ and k ud with respect to the orientation ĉ (or, in the present 3D case, to the co-latitude θ and to the longitude ϕ), i.e.,
where kη , kτ , kdη , kdτ and kud are the averaged stiness over the unit sphere S 2 , that are dened in the general anisotropic case as follows,
Insertion of (66-67) into ( 48), ( 61), ( 62) and ( 63) yield the following and desired identication:
From the denition of the deformation gradient F = ∇χ in (9) 1 and of the micromorphic deformation gradient P in (8), we dene the displacement gradient H and the transpose micromorphic displacement gradient Ψ,
Thus, the rst non-linear micro-deformation gradient, for small displacement approximations, is simplied from (13),
and the second non-linear micro-deformation gradient, for small displacement approximations, is simplied from ( 14),
Λ r ijh = P ah P aj,i = (δ ah + Ψ ha ) P aj,i ∼ = δ ah P aj,i = P hj,i = Ψ jh,i = κ ijh ,
that means that, in the linear approximation, the rst Λ and the second Λ r non-linear microdeformation gradients are the same third-order tensor κ, that is called the micro-deformation gradient. Besides, the Green-Saint-Venant tensor G (or non-linear macro-strain), the micromorphic Green-Saint-Venant tensor M and the micro-macro Green-Saint-Venant tensors Υ (or non-linear relative deformation), for small displacement approximations, are simplied from ( 10) and ( 11)
that means from (80) that, in the linear approximation, the micromorphic Green-Saint-Venant tensor M depends upon the Green-Saint-Venant tensor G and upon the micro-macro Green-Saint-Venant tensor Υ and it is not anymore an independent strain measure. Besides, the non-linear macro-strain G is simplied from (78) in the macro strain and the non-linear relative deformation Υ is simplied from (79) in the relative deformation γ. Thus, the three strain measure from (78), ( 79) and ( 76) are the same dened in Mindlin [1] respectively in eqns. (1.10), (1.11) and (1.12), viz.,
Insertion of the linear approximations (76-80) into the general form of the elastic energy (47), yields
where new constitutive tensors (with the super-script n) are dened in terms of that dened in (48)(49)(50)(51)(52)(53)(54)(55)(56)(57)(58),
where the symmetrization and skew-symmetrization rules
have been used in (84)(85)(86)(87)(88)(89). Insertion of (48-58) into (84-89) yields the explicit identication of the new constitutive tensors,
(k η -4k τ + 4k rη -4k rτ + 2k ur ) ĉi ĉj ĉk ĉl (90)
In the isotropic case, among ( 66) and ( 67), we assume also the independence of the remaining stiness with respect to the unit vector ĉ, i.e.,
In this case the Lame's constant in (68) These expressions for the stiness parameters in 90 to 95 provide an essential seed for an initial estimation of all the elastic parameters that characterize a micromorphic continuum. It is remarkable that these rst estimates indicate that such materials are described by several characteristic lengths, which can be multiples of relevant grain-size, and represent the inuence of grain-scale micro-mechanisms on the emergent behavior at the macro-scale. These micromechanisms may include those that resemble the oppy behavior of pantograph, best described by second-gradient macro-scale continua analyzed in [35,86], or other mechanisms that require additional kinematical descriptors to capture the deformation energy of grain-pair [7,17,51,71]. It is also noteworthy that it is possible to estimate the elastic parameters a micromorphic continuum in terms of a few parameters that link to the micro-mechanisms without recourse to ad hoc prescriptions or a priori (over) simplications. For certain, relatively simple micro-mechanisms and structures, such linkages can indeed be identied, synthesized and experimentally characterized as discussed in [51,52]and [4250,8790].
For accurate and tractable description of the mechanical behavior of a large class of materials which at some spatial scale possess granular microstructure, rened models, such as the micromorphic models, are needed. Such models are particularly signicant for bridging, in a heuristic way, across spatial scales ranging from that at grain interactions to collective behavior of large numbers of grains. At meso-scales of few grains to tens of thousands of grains, discrete simulations can be conceived that provide the trajectories and distribution of grain-scale deformation energies. It is worthwhile to note here that although discrete models have proliferated over the past several decades, their systematic validation through experimentally measured particle trajectories and grain-scale energy distributions have been characteristically sparse (absent to the knowledge of these authors). At the macro-scales consisting of large number of grains of various sizes, interfaces/surfaces, composition and arrangement (collectively micro-mechano-morphology), discrete models could be intractable. In these cases, micromorphic continuum models can serve as eective reduced-order models that can capture many essential aspects of the grain-scale mechanisms. This paper describes an approach to construct such micromorphic models in the framework of nite (geometrically nonlinear) deformations using the concepts of granular micromechanics. The key aspect of the described approach is the identication of the appropriate kinematic measures that describe the macro-deformation and link it to the micro-deformation, formulation of the deformation energies in terms of these measures and the application of energy methods to identify the constitutive relations. Such an approach permits potential identication of inner deformation modes that store elastic energy contributing to the emergent behavior at the macro-scale, and indicates the pathway to access these modes with the view of rational design of (meta) materials.
Furthermore, we would like to identify a number of potential outlook of the presented approach. First, the isotropic identication we have shown can be extended to an anisotropic one by the use of proper non constant orientation distribution function instead of (66-67-96). Second, the truncation of the Taylor's series expansions (18)(19) up to the rst order in terms of the kinematic descriptors results in a rst-grade continuum theory. Such a limitation can be removed to obtain higher order gradient continuum theories without unduly augmenting the number of the constitutive coecients that need to be experimentally identied. Third, the quadratic assumption (46) can be generalized, such as with Leonard-Jones type potential to take into account elastic-hardening eects and tensioncompression asymmetry that can lead to emergence of anisotropy. Fourth, dissipative phenomena such as damage [84] and plasticity [91] can be included by using for example an hemivariational approach or by assuming dissipation energy in terms of additional entropic irreversible kinematical descriptors. It is further remarkable that plastic deformation in the present micromorphic form can give rise to inelastic microstructural rotation.