On the micro-macro relation for the microdeformation in the homogenization towards micromorphic and micropolar continua

Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

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On the micro-macro relation for the microdeformation in the homogenization towards micromorphic and micropolar continua Geralf Hütter Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Lampadiusstr. 4, Freiberg 09596, Germany

a r t i c l e

i n f o

Article history: Received 2 October 2018 Revised 29 January 2019 Accepted 5 March 2019 Available online 6 March 2019 Keywords: Micromorphic theory Homogenization Cosserat theory Size effects

a b s t r a c t The modeling of size effects requires to employ generalized continuum theories. For instance, the micromorphic theory introduces the microdeformation χ as additional kine i. e. non-classical, matic degree of freedom. Such generalized theories require additional, constitutive relations. The experimental determination of the corresponding non-classical constitutive parameters is cumbersome, even for isotropic and linear-elastic material. Homogenization approaches aim in providing the macroscopic constitutive relations from the behavior of the microscopic constituents of a material. For that purpose, micro-macro relations are required for all kinematic quantities. Though, different micro-macro relations have been used in literature for the microdeformation χ .  The present contribution investigates the effect of the chosen micro-macro relation for χ  on the predicted macroscopic behavior. In particular, a material with pores or inclusions at the microscale is considered for the special case of plane micropolar (Cosserat) elasticity. It is discussed that the coupling modulus should vanish if the material is homogeneous at the micro-scale in order to avoid the artificial prediction of size effects. This goal is reached if the deformation of microscopic heterogeneities is used for the micro-macro relation of the microdeformation. Classical and non-classical micropolar moduli of a material with pores or inclusions are derived in closed form. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Micromorphic continuum theories introduce a tensor of microdeformation χ as additional kinematic degree of freedom  and its gradient as an additional measure of deformation. The latter has the dimension of 1/length. Respective constitutive parameters thus introduce an intrinsic length which is why micromorphic theories can, in contrast to the classical Cauchy– Boltzmann theory, predict size effects. The most prominent representative of this class is the micropolar theory (Cosserat theory) which restricts the microdeformation to a micro-rotation. All micromorphic theories have in common that they require additional constitutive relations and consequently additional constitutive parameters. These additional constitutive parameters can be determined from experiments only indirectly via the size effect (Gauthier and Jahsman, 1975), if possible at all. Homogenization techniques aim in providing these macroscopic constitutive parameters, or even the structure of the macroscopic constitutive equations, from the mechanical behavior at the micro-scale. E-mail address: [email protected] https://doi.org/10.1016/j.jmps.2019.03.005 0022-5096/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Interface between homogeneous and heterogeneous material.

The strain gradient theory can be interpreted as a special case of the micromorphic theory, namely a micromorphic theory where the microdeformation is constrained to the macroscopic deformation gradient. Gologanu et al. (1997) and Kouznetsova et al. (2002) extended Hill’s classical homogenization framework successfully towards the strain gradient theory. Unconstrained micromorphic theories have the difference of microdeformation χ and macroscopic deformation as addi tional strain measure. Corresponding homogenization theories thus require an independent kinematic micro-macro relation for χ . The situation is relatively easy if the material has rotational degrees of freedom at the microscale and shall be ho mogenized towards a micropolar continuum, e.g. if the microstructure consists of beams, shells or particles (Adomeit, 1968; Besdo, 20 09; Chung and Waas, 20 09; Ehlers et al., 20 03; Godio et al., 2017; Hård af Segerstad et al., 2009; Liebenstein et al., 2018; Tauchert et al., 1969; Trovalusci et al., 2017). In this case, the microrotation can be introduced as average over its microscopic counterparts, in analogy to classical theory of homogenization. However, if the material at the micro-scale is a Cauchy continuum, there is no such intuitive relation. Mindlin (1964) introduced χ in his theory to model effects of “micro-structure”. In this sense, Herrmann and Achenbach (1968), ´ Achenbach(1976), Sun et al. (1968), Hlavácek (1975) and Berglund (1982) interpreted the microdeformation as the homogeneous deformation of material heterogeneities. An ad-hoc ansatz was employed for the matrix material, without formulating a boundary-value problem at the microscale, and no gradients of microdeformation were considered. The interpretation of χ via material heterogeneities was adopted recently by Biswas and Poh (2017).  In contrast, Forest and Sab (1998) and Forest and Trinh (2011) proposed to identify χ with the first moment of the  the boundary of a microscopic microscopic displacement field. Since this definition does not involve only displacements at volume element (and neither the definition via material heterogeneities), the microdeformation cannot be prescribed via boundary conditions at the micro-scale. Rather, a polynomial ansatz was employed for the microscopic displacement field by several researchers, e.g. (Addessi et al., 2016; Branke et al., 2009; Chen et al., 2009; Forest and Trinh, 2011; Jänicke et al., 2009). The author prescribed the kinematic micro-macro relation for χ by Forest et al. as global constraint at the micro-scale  ). and established consistent kinetic micro-macro relations (Hütter, 2017a The definition of χ by Forest et al. has the drawback that it requires to have a unique displacement field defined ev erywhere in the microstructure. This is, however, not possible e.g. in pores of foams, though it is known that foams exhibit size effects which can be described heuristically quite-well by micromorphic theories, cf. (Rueger and Lakes, 2016) and references therein. In order to perform a micromorphic homogenization of porous media, the author introduced a weighting function into the kinematic micro-relation of Hütter (2017b, 2019). It was shown that neither the generalized Hill-Mandel condition nor the respective kinetic micro-macro relations impose restriction on the choice of this weighting function. This finding leads to the question how this weighting function shall be chosen and thus how the micro-deformation shall be defined. Weighting functions have been introduced as well recently in a micromorphic homogenisation approach by Rokoš et al. (2019). Closely related is the question how a micromorphic continuum should behave in the limiting case that the material at the micro-scale is homogeneous, i. e., that the material does not have a micro-structure in a strict sense. This question is the topic of an ongoing and controversial debate. Mindlin (1964) proposed to constrain the microdeformation to the macroscopic deformation for “micro-homogeneous material”. By doing so, a strain-gradient continuum is obtained which does exhibit size effects. In contrast, Li (2011) postulated that “strain gradients do not affect constitutive laws if the material is perfectly homogeneous”. For this purpose, Li (2011) performed a “transformation step”, wherein they substracted the strain gradient moduli of homogeneous material from those obtained in the preliminary “homogenization step”. Though, Mühlich et al. (2012) argued that “a Cauchy continuum has to be obtained if [the porosity] f approaches zero everywhere in the body but this cannot be decided by a local measure. That’s why, the strain gradient constants must not vanish only because f goes to zero locally.” Size effects are often related to surface layers or layers at interfaces (Schaefer, 1967) as sketched in Fig. 1. Referring to this sketch, it is expected that near the interface the homogeneous material interacts with the microscopic fields of the heterogeneous material at the top, and that a boundary layer is formed even in the homogeneous material. For modeling this behavior macroscopically, the micro-homogeneous material needs to have micromorphic properties.

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G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

Fig. 2. Volume element for homogenization (Hütter, 2017a).

Unconstrained micromorphic theories posses the coupling moduli as constitutive parameters, in addition to the gradient parameters. Cowin (1970) pointed out that the (linear-elastic) micropolar theory reduces to classical theory if the coupling number is zero N = 0, i. e., if the coupling modulus vanishes, independent of the gradient parameters. This finding applies to micromorphic theories in general. The reason is that in this case the higher order balance decouples from the classical balances of momentum (Forest and Sievert, 2006). Then, the higher order balance forms a homogeneous PDE. Thus, if all corresponding boundary conditions with respect to microdeformations or double tractions are zero, the microdeformation vanishes everywhere. However, such a decoupled system does still impose resistance to microdeformations at interfaces as sketched in Fig. 1. Following this argumentation, the following requirements for the homogenization towards a micromorphic continuum are formulated: • No size effects are predicted if the material is micro-homogeneous everywhere but the material can resist to microdeformations at its boundary • For a periodic microstructure, the obtained micromorphic continuum description is representative in the sense of Hill (1963), that the macroscopic behavior of a specimen of a specimen of sufficient size under macroscopically uniform loading is correctly predicted. The scope of the present contribution is firstly to investigate the effect of the definition of the micro-deformation at the micro-scale on the resulting homogenized macroscopic constitutive parameters of a micromorphic continuum. In particular, the linear elastic, micro-polar properties of a plane porous medium with idealized microstructure are considered. Secondly, the formulation of periodic boundary conditions within a micromorphic homogenization framework shall be discussed aiming in a representative macroscopic representation in the aforementioned sense. Focusing on these two points, infinitesimal deformations are considered in order to reduce the number of mathematical symbols (an extension to large deformations is straight-forward, cf. (Hütter, 2019)). The present paper is structured as follows: The general theory of homogenization towards a micromorphic continuum from Hütter (2017b, 2019) is briefly recalled in Section 2. Special attention to the adaption of periodic boundary conditions. Section 3 deals with the special case of the micropolar theory (Cosserat theory). This theory is used in Section 4 to compute the micropolar elastic properties of an idealized foam structure in closed form for any choice of the weight functions. Certain possible weight functions are discussed. In Section 5, the obtained micropolar constitutive parameters are employed to evaluate the predicted size effects quantitatively, before the micropolar properties of a material with inclusions are considered in Section 6. Finally, Section 7 closes with a summary and conclusions. The notation of the present contribution follows (Forest and Sab, 1998; Hütter, 2017a). Vectors, dyadics and tensors of third order are denoted by A = Ai bi , B = Bi j bi b j , C = Ci jk bi b j bk , respectively, with bi being the Cartesian base vectors. In particular,   tensor and the permutation tensor, respectively. Multi-fold contractions are I = δi j bi b j and  = i jk bi b j bk are the identity   computed from left to right, e.g. C : B = Ci jk B jk bi . Macroscopic quantities are denoted by capitals or a superimposed bar (◦¯ ),   are denoted by lower-case symbols without bar. For instance, X and x refer to the whereas quantities at the microscale macroscopic and microscopic location vector, respectively, as sketched in Fig. 2. The index of the nabla operator, ∇ X (◦) or ∇ x (◦) indicates whether it is computed with respect to X or x, respectively. 2. Micromorphic theory of homogenization with weight functions 2.1. General theory The balance of linear momentum in the micromorphic theory reads ..

0 = ∇ X ·  + ρ f − ρ U, 

0=

.. ∂ i j +ρfj −ρ Uj ∂ Xi

(1)

G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

65

as in classical theory, and it is amended by a higher-order balance of momentum ..

0 = ∇ X · M +  − σ¯ T + ρ m− χ ·Gρ ,       T

0=

∂ Mi jk .. ρ + k j − σ¯ k j + ρ m jk − χ jm Gmk . ∂ Xi

(2)

Therein, the external stress, the internal stress and the double stress (hyperstress), respectively, are related to the microscopic stress field σ (x ) via the kinetic micro-macro relations  

= 

1

V

ξ  n · σ dS,

σ¯ = σ V ,



∂ V





1 M= V 

ξ n·σ



∂ V

 ξ dS.

(3)

wherein ξ = x − X denotes the location vector relative to the center X = xV of the unit volume cell V(X) as sketched  in Fig. 2. The operator (◦ )V = 1/ V V (◦ ) dV refers to the average over V. Correspondingly, the balance of angular momentum at the microscale implies ¯ T = σ¯ . Furthermore, Eq. (2) involves a  symmetry of the macroscopic intrinsic stress  σ   tensor of micro-inertia Gρ = ρξ ξ , and the moment of volume forces m = ρ f ξ /ρ . The macroscopic density is ρ = ρV . V V   At the kinematic side, the micromorphic continuum has the tensor of microdeformation χ as degree of freedom, in  addition to the displacement U. A generalized Hill-Mandel lemma . .  .   . . . .s . .s σ : ε V =  : ∇ X U + σ¯ T − T : χ + M .. K = σ¯ : E + s : e + M .. K

 







 



 

 

(4)

 

ensures that the macroscopic mechanical power is equal to its microscopic counterpart in average. Due to the symmetry of the double stress M with respect to its first and third index, Eq. (3), only the symmetric part Kisjk = (∂ χi j /∂ Xk + ∂ χk j /∂ Xi )/2 of the gradient ofmicrodeformation enters Eq. (4) as work-conjugate deformation measure to M . Note that the power in Eq. (4) can be expressed equivalently in terms of different measures of stress and strain. An established representation encompasses the relative deformation e = U  ∇ X − χ between microdeformation and macroscopic deformation as work  conjugate kinematic quantity to the difference stress s = T − σ¯ T , in addition to the classical strain tensor E = sym (∇ X U ).     Using the Gauss theorem together with Eq. (3), the micro-macro relation for the difference stress thus reads

s= 

 ∇x · σ



ξ

 V

+

1 V



S||

n · σ   ξ dS. 

(5)

Therein, (◦) denotes a jump at a surface of discontinuity S|| . The kinematic micro-macro relations for the primary degrees of freedom were identified as

  χ = u  ξ M · (Gχ )−1

U = uM ,

wherein the averaging operator

1

(◦ )M = 

  HM ξ V V

(6)





HM

  ξ (◦ ) dV

(7)

V (X )



involves a weighting function HM (ξ ). The center of the latter coincides with the geometric center xM = X, and Gχ = ξ ξ  refers to the weighted second geometric moment. The relation for the macroscopic distorsion

∇ XU =

1 V

 M



n  u dS

(8)

∂ V

remains the same as in classical theory of homogenization. The symmetric part Ks = Ksh + Ksd of the gradient of microde   formation can be split into a “spherical” part Ksh and a “deviatoric” one Ksd :  



Kisjkh =

1 −1 s 1 G K Gmn = n ik m jn 2+n

Kisjkd =

1 4 V



∂ V

1 n V

u j ni ξm dSG−1 + mk



∂ V



u j nm ξm dS − U j G−1 ik

∂ V

u j n k ξm

2 dSG−1 − mi n



(9a)

 u j nm ξm dSG−1 . ik

(9b)

∂ V

Therein, n is the  dimension of space (n = 2 or n = 3). The terms “spherical” and “deviatoric” refer to the second geometric moment G = ξ ξ which is a spherical tensor for simple geometric shapes of the unit cell like rectangle or cube. In this V  d G = 0. Note that the kinematic macro-micro relation for the deviatoric part sense, the deviatoric part has the property Kisjk ik in Eq. (9b) does contain only surface integrals, but not the macroscopic displacement U. The primary kinematic quantities U and χ are defined via weighted volume averages, Eq. (6), which is why they cannot  microscale. Rather, these volume averages have to be enforced by Lagrange be prescribed via boundary conditions at the

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ξ2

n−

n+ ξ1

Fig. 3. Microscopic volume element with periodic boundary conditions.

multipliers. The Hill-Mandel condition (4) shows that these Lagrange multipliers can be identified with the respective workconjugate stress measures. Consequently, the strong form at the microscale reads:

 

ξ   s · (Gχ )−1 · ξ − λU .   HM ξ V

∇x · σ =  

HM

(10)

2.2. Boundary conditions Corresponding static boundary conditions are obtained by enforcing the remaining kinematic micro-macro relations (8) and (9) as:

n i σi j = n i i j +

1 1 ni Mi jp G−1 ni ξi Mn jp G−1 pm ξm − np 2 2 (2 + n )

on

∂ V (X )

(11)

Global equilibrium of the cell V determines the Lagrange multiplier λUj = − (2+n )1H  Mi jk G−1 to point into the direction ik M V of the spherical part of the double stress. This fact is directly related to the appearance of volume averages in the workconjugate kinematic micro-macro relation (9a). Alternatively, kinematic boundary conditions

u = U + ξ · ∇ X U + ξ · Ks · ξ 

on

∂ V (X )

(12) (1997).1

can be presribed with a quadratic term as in the strain-gradient theory of Gologanu et al. For extending the concept of periodic boundary conditions to the micromorphic theory, a fluctuation field u(ξ ) is amended to the kinematic boundary condition (12)

u = U + ξ · ∇ X U + ξ · K s · ξ + u 

  ξ

on

∂ V (X )

(13)

The fluctuations are assumed to be periodic

    u ξ + = u ξ − +

(14)

−

with ξ ∈ ∂ Ω + and ξ ∈ ∂ Ω − being the locations (relative to the center X) of “homologous” points at the boundary ∂ V(X), i. e., points with opposite normal directions n+ = −n− , Fig. 3. These conditions ensure that the classical micromacro relation (8) is satisfied. However, this periodicity does not satisfy per se relation (9) for the gradient of microdeformation, but additional requirements are necessary. Lacking an explicit micro-macro relation for the second gradient, Kouznetsova et al. (2002) imposed an integral constraint to the fluctuation field. This integral constraint forms an implicit kinematic micro-macro relation, cf. (Hütter, 2019). In contrast, explicit micro-macro relations (6) and (9) are available for displacements, microdeformations and their gradients in the present micromorphic theory. For formulating a boundary-value problem with respect to the local displacements, the fluctuation field u(ξ ) is favorably eliminated from the periodicity condition (14) by Eq. (13), yielding



u

     ξ + − u ξ − = ξ + − ξ − · ∇ X U + ξ + · Ks · ξ + − ξ − · Ks · ξ − 



on

∂ V (X ).

(15)

The remaining micro-macro relations (6) and (9) are imposed as constraints in integral form. For a hyperelastic material (σ = ∂ W (ε )/∂ ε ), the corresponding Lagrangian thus reads   

1 V

L =W V −

 + λi jk

     

λi (ξ + ) ui ξ + − ui ξ − − Ui, j ξ j+− ξ j− + K sjik (ξ j+ ξk+ − ξ j− ξk− ) dS

∂ V +

Kisjk−

1

4 V



∂ V

ξ

ni u j p dS G−1 + pk



∂ V

ξ

−1 p u j nk dS Gip

2 − 2+n



 ξ

u j n p p dS G−1 ik

(16)

∂ V

1 Eq. (12) reduces to the boundary condition from Gologanu et al. (1997) if the kinematic constraint χ = 1/2U  ∇ X is imposed. Furthermore, Gologanu et al. (1997) did not impose a kinematic micro-macro relation on the displacement, Eq. (6), which  is why their double stress tensor is deviatoric, cf. (Hütter, 2019).

G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

+

67



     1   −1 χ −1  u j Gik + λUj U j − u j − si j χi j − ui ξk M G jk . M M 2+n +

It has to be minimized with respect to the microscopic displacement field u(ξ ) and to the Lagrange multipliers λi (ξ ), λi jk ,

λUj , and sij . Thereby, the assignment ξ − = f (ξ + ) of homologous points at the boundary has to be given. The stationarity condition to this variational problem are the enforced kinematic constraints (6) and (9), as well as the local equilibrium condition (10) in the domain and

ni σi j = ±λ j

  1 ξ + ni λ i jp G−1 pm ξm − 2

1 ni ξi λ n jp G−1 np 2 (2 + n )

(17)

at the boundary. Thereby, the plus sign in the first term refers to ξ ∈ ∂ Ω + and the minus to ξ ∈ ∂ Ω − . Thus, the tractions resulting from λ j (ξ ) are anti-periodic. However, Eq. (17) comprises additional contributions of double stress-type which are linear both in normal n and in relative location ξ . From these double stress loadings, there are both periodic and antiperiodic contributions. This finding complies with the requirement of Forest and Trinh (2011) that the “anti-periodicity condition must be abandoned in the presence of overall stress and strain gradients”. The work associated with Eqs. (10) and (17) is

⎡ ⎤ ⎡ ⎤  .  .  . .s 1 1 + − + + − − ξ λξ − ξ λξ dS⎦.. K σ :ε V = ⎣ ξ − ξ  λ dS ⎦ : ∇ X U + ⎣ λ + V V     ∂ V + ∂ V +  . 

  −1 T − u  ξ · ( Gχ ) :  − σ¯ T M .

.s







(18)

The co-factors of ∇ X U and K coincide with the external stress  and double stress M computed from the respective  kinetic micro-macro relations(3). Together with the kinematic micro-macro relation (6)2for the last term in Eq. (18), the Hill-Mandel condition (4) is thus satisfied. For irreversible material behavior, the stationarity conditions are used without existence of a Lagrangian L (principle of virtual power). − + For a centro-symmetric problem ξ = −ξ , the double stress M coincides with the Lagrange multiplier λ and  Eq. (17) can be interpreted as superposition between classical, anti-periodic tractions and a linear term with doublestress M as in static boundary conditions (11).  3. Micropolar theory of homogenization (Cosserat theory) The continuum theory of the Cosserat brothers extends the Cauchy–Boltzmann theory towards moment-type stresses (double stresses) together with independent rotational degrees of freedom. In the context of micromorphic theories, theories of this type are denoted as micropolar theories and are obtained by assuming the microdeformation χ to be skew symmetric. Thus, χ can be written in terms of an axial vector of microrotation r as 

χ = −r ·  . 

(19)



Consequently, only the projection of the higher-order balance (2) on the permutation tensor  is required in the micropolar  theory. In view of the symmetry of the internal stress, Eq. (3)2 , the balance of angular momentum thus reads r

.. r

0 = ∇ X · Mr +  :  + f −  · Gr .    

(20)

Thereby, the polar double stress is related to the microscopic stress state via

1 Mr = V 



ξ  ξ × (n · σ ) dS = −M :  . 

∂ V

(21)

 

Eq. (21) coincides with the kinetic micro-macro relation in (Ehlers and Bidier, 2018). It shows directly, that the sub-symmetry of the double stress due to definition (3)3 results in a vanishing trace .

Mr : I = −M .. = 0   

(22) Mr .

Krd

of the polar double stress Consequently, only the deviatoric part of the gradient of microrotation (curvature) Kr =    r ∇ X  contributes to the power density. The corresponding Hill-Mandel relation thus reads

 . . r . rd  . σ : ε V =  : ∇ X U −  ·  + Mr : K  









(23)

and allows to identify the micropolar strain

Er = ∇ X U −  ·   

r

(24)

as work-conjugate quantity to the macroscopic stress . Regarding the transition from the general micromorphic theory  to the special case of a micropolar theory, the question arises, how constraint (19) on the microdeformation is transferred

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G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

to the microscale. The question is in particular how to deal with those quantities which do not appear anymore in the micropolar theory, i. e., with the symmetric part of the microdeformation and its gradient. In order to satisfy the Hill-Mandel lemma (23), there are two opportunities: either the respective kinematic quantities are set to zero or their work-conjugate stresses are relaxed. Regarding the micro-deformation, only the skew-symmetric part of Eq. (19) is retained, yielding the kinematic micro-macro relation

   1    1 r = −  : u  ξ M · (Gχ )−1 = ξ · (Gχ )−1 × u M 

2

(25)



2

for the micro-rotation. The symmetric part of the gradient of microdeformation is computed for the micropolar theory (21) as

Kisjk =

 1 r  K r +  jmi Kmk 2 jmk mi

(26)

Inserting this expression to Eq. (12) yields kinematic boundary conditions

u = U + ξ · ∇ XU +



 ξ · Kr × ξ 

on

∂ V

(27)

for the homogenization towards a micropolar continuum. Note, that the spherical part of Kr does not contribute to (27), in  consistency with Eq. (22). The quadratic boundary conditions (27) are identical to those specified by Forest (1998) (for a constrained micropolar theory). Different coefficients of the quadratic terms were employed in (Bouyge et al., 2001; Branke et al., 2009; Chen et al., 2009). For Ks at the left-hand side of Eq. (26), the kinematic micro-macro relation (9) is available for the full micromorphic theory.  For the right-hand side, an isotropic mapping between Ks and Kr can favorably be employed. Unfortunately, such a   mapping is not unique, even under consideration of the symmetries of the respective tensors. A mapping

Kirdj =



2 sd 2 sd K + G−1 G pq K pqk 3 imk n + 2 im

 mk j

(28)

shall be used here, which involves only the devatoric part Ksd of the gradient of microdeformation. Inserting the micro macro relation (9b) for Ksd to the right-hand side of Eq. (28) finally yields 



Kirdj

1

=−

6 V



∂ V

2 ni um p G−1 +nk um p G−1 − G−1 pi pk n + 2 im

ξ

ξ



ξ

ξ

ξ

n p uq G pq G−1 mr r +nm p u p +n p p um





dS

mk j .

(29)

The pragmatic choice, Eq. (28), has the advantage that resulting the micro-macro relation (29) for the gradient of microrotation can be expressed solely by surface integrals over the volume element, without macroscopic displacement U. The boundary-value problem at the micro-scale is obtained again via concept of minimal loading conditions, i. e., by enforcing the micropolar kinematic micro-macro relations (25) and (29), in addition to the classical ones, Eqs. (8) and (6)1 . This leads to an equilibrium condition

   ξ 1   k j −  jk Gχkm−1 ξm , 2 HM ξ V

σi j,i = 

HM

(30)

together with static boundary conditions on ∂ V(X)

ni σi j =ni i j +

1 r M 6 im



ni G−1 ξ +nk G−1 ξ − ip p kp p

2 n p ξ p G−1 ik 2+n

    2 mk j + G−1 nq G jq G−1 np ξ p + ξ j nn nkm . ik 2+n

In Eq. (31), it was already taken into account that those terms, which are associated with Mr , are self-equilibrating so  U that the Lagrange multiplier for the macroscopic displacement vanishes λ = 0. This is a consequence of Eq. (29), which does not involve volume integrals. The non-classical deformation modes of the micropolar theory are sketched in Fig. 4 for both types of boundary conditions. Fig. 4a and b show that the off-diagonal components of Kr and Mr induce bending-type deformation modes of the   volume element. Fig. 4c illustrates the effect of a diagonal component of Kr which twists the opposite surfaces of the vol ume element against each other. The skew-symmetric part of  drives an internal twisting of the volume element as can  be seen in Fig. 4d. Periodic boundary conditions can be implemented as discussed in Section 2.2 for the general micromorphic theory by amending a periodic fluctuation field u(ξ ) to the right-hand side of Eq. (27) under global enforcement of the kinematic micro-macro relations (6)1 , (25) and (29), respectively. Under the conditions discussed above, this corresponds to a superposition of classical periodic boundary conditions and the double stress contribution to the static boundary conditions (31).

G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

69

Fig. 4. Non-classical micropolar deformation modes of the volume element V: (a) kinematic boundary condition (27) with Kr = b1 b3 , (b) static bound ary condition (31) with Mr = b1 b3 , (c) kinematic boundary condition (27) with Kr = b1 b1 , (d) static boundary condition (31)  = b1 b2 − b2 b1 (for     HM ξ =const.).

Fig. 5. Unit cell of a plane porous material.

4. Micropolar elastic properties of porous media As a simplest case, a periodic arrangement of circular pores shall be investigated. The corresponding volume element

V can be approximated by a circle as sketched in Fig. 5. Such a circular volume element has the advantage that firstly its geometry is isotropic. If the constitutive behavior of the matrix is isotropic as well, then the macroscopic constitutive

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G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

Fig. 6. Load cases for the micropolar homogenization: (a) dilatation, (b) pure shear, (c) bending, (d) internal twist (skew-symmetric loading).

relations will consequently exhibit isotropy. In particular, linear-elastic behavior σ = λIε : I + 2με shall be considered. Sec    ondly, certain analytical solutions can be found for this geometry. The constitutive equations of a linear-elastic, isotropic micropolar continuum

 = λ(eff) Er : II + (μr + κ )Er + μr (Er )T 

Mr = 

 



(31)



α Kr : II + β (Kr )T + γ Kr 





(32)



are specified typically (e.g. Eringen, 1966; Gauthier and Jahsman, 1975) in terms of the six Lamé-type constants μr , κ , α , β , γ . Therein, the Lamé parameters μr and λ are known from the classical theory of elasticity. However, note that μr does not correspond to the shear modulus. Rather, the effective modulus as ratio between shear stress and shearing amounts to μ(eff) = μr + κ /2 and involves additionally the so-called coupling modulus κ associated with skew-symmetric parts. Alternatively to the Lamé-type constants, a linear-elastic isotropic micropolar medium can be characterized by engineering constants (Gauthier and Jahsman, 1975; Lakes, 1983). These engineering constants appear in solutions for simple loading cases and can thus be related to respective experiments. In addition to Young’s modulus and Poisson ratio, these are the coupling number N, polar ratio  and the characteristic lengths lb and lt under bending and torsion, respectively:

β +γ = , α+β +γ



N=

λ(eff) ,

κ

2μ(eff) + κ



,

lt =

β +γ , 2μ(eff)



lb =

γ

4μ(eff)

.

(33)

Note that the vanishing trace of the double stress, Eq. (22), determines the parameter α and the polar ratio evaluates to  = 3/2, independent of the particular values of β and γ . This value of  leads to a bounded stiffness of torsion specimens of arbitrarily small size as discussed by Neff (2006). For the plane case under consideration, Fig. 5, the vector of microrotation r = r (X1 , X2 )b3 is normal to the plane (i. e., in direction of the base vector b3 ) and only the parameter γ is relevant for the double-stress contributions.2 The four remaining parameters λ(eff) , μr , κ and γ can be determined by homogenization via the boundary-value problem at the microscale from Section 3. Periodic boundary conditions are employed in order to obtain a representative macroscopic theory. Favorably, four decoupled load cases are considered as sketched in Fig. 6. The classical cases of dilatation and shear can be treated as usual. The solution is provided in Appendix A. The elastic problem for obtaining the double stress modulus, Fig. 6, can be solved by elementary methods as well, see Appendix B, resulting in

γ = 2 μR 2

1 − c2 3 − 4ν + c 2

(34)

for the plain srain case. Therein, c = R2void /R2 refers to the porosity of the material. Fig. 7a shows that γ has a finite value at c = 0 and decreases with increasing c and vanishes for c → 1. However, the intrinsic length lb remains finite since the effective shear modulus μ(eff) tends to zero as well, see Fig. 7b. The Poisson ratio ν of the matrix material has a notable effect on γ and thus on lb . The focus shall be on the role of the weight function HM (ξ ) which appears in the micro-macro relation (25) for the microrotation and consequently in the microscopic equilibrium condition (30) in conjunction with the skew-symmetric part  skw = (12 − 21 )/2 of the macroscopic stress. For objectivity reasons, the weight function needs to be isotropic, i. e., it can be only a function HM (r) of the distance r = |ξ | from the center. Consequently, the microscopic boundary-value problem for loading by  skw , Fig. 6d, is axi-symmetric3 and can be solved in closed form. The respective Lamé equation (see e.g. 2 The parameter β determines the out-of-plane double stresses M3r i . However, these components do not depend on X3 and thus do not contribute to the balance of angular momentum (20). 3 The geometry of the microscopic volume element is isotropic and the loading  skw is of axial nature. Consequently, the resulting boundary-value problem at the microscale needs to be axi-symmetric, and skew-symmetric with respect to an inversion in order to be objective. In this sense, the approach of Chen et al. (2009) with a boundary condition in form of a cubic polynomial is not objective.

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Fig. 7. Effect of porosity on (a) double stress modulus and (b) characteristic length scale.

Fig. 8. Effect of skew-symmetric stress for definition of micro-rotation over heterogeneity (HM (r ) = δ(r − Rvoid )).

Kachanov et al., 2003) to Eq. (30) for the circumferencial displacement uϕ (r) reads

   

1 3 uϕ

r = r r3 R

Rvoid

R2 r 3 HM (r )dr

 skw H (r ). μ M

(35)

Therein, the prime () refers to the derivative with respect to r. Eq. (35) can be integrated twice yielding two constants of integration. One of them represents an irrelevant rigid rotation, the other one is determined by the condition of a tractionfree pore surface at r = Rvoid . Since the problem is axi-symmetric, it makes no difference for the resulting coupling modulus, whether kinematic boundary conditions (27), or static ones (31) are applied for the skew-symmetric parts of ∇ X U and ,  respectively. The latter are satisfied identically by the solution of Eq. (35) due to global equilibrium of the unit cell. Subsequently, the kinematic micro-macro relations (8) and (25) are used to compute the skew-symmetric part of the micropolar strain Er , Eq. (24). A comparison of the resulting relation with the general elastic law (31) allows to identify the coupling  modulus κ for any choice of the weight function HM (r). Regarding the choice of HM (r), the first requirement is that HM (r ) = 0 for r < Rvoid . Otherwise, the kinematic micro-macro relation (25) would require to define a displacement field artificially within the pore, compare (Hütter, 2017b). Secondly, it seems reasonable to require that HM (ξ ) is a local quantity in the sense that it has the same value whenever the material at the micro-scale is the same. Thus, the author sees two potential choices: either a homogeneous weighting of the matrix HM (r ) = 1 for Rvoid ≤ r ≤ R or a focused weighting HM (r ) = δ(r − Rvoid ) by means of the Dirac distribution at the pore surface. The latter is the only heterogeneity at the micro-scale of the problem under consideration. In the former case, the skew-symmetric stress  skw acts like a circumferencial volume force whose magnitude scales linearly with r as shown schematically in Fig. 6d. The second case HM = δ(r − Rvoid ) corresponds to circumferencial tractions at the pore surface as sketched in Fig. 8. The resulting coupling moduli for both cases are

κ = 6μ

(1 + c )2 (1 − c )(1 + 3c )

for HM = 1.

(36a)

κ = 2μ

c 1−c

for HM = δ(r − Rvoid ).

(36b)

The plot in Fig. 9a shows that κ tends to infinity with high porosities c → 1 for both cases. However, for homogeneous material c = 0, the coupling modulus vanishes for HM = δ(r − Rvoid ) whereas it has a finite value for HM = 1. Correspondingly, the coupling number N in Fig. 9b tends to one for high porosities c → 1 for both cases, i. e., towards the limiting

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G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

Fig. 9. Effect of choice of weighting function HM (r) on (a) coupling modulus and (b) coupling number (ν = 0.3).

ˆs U

X2

h

Fig. 10. Simple shearing of an infinitely wide layer.

case of couple-stress theory. However, for low to medium values of porosity, the predicted √ coupling number differs tremendously between the two cases. For a uniform weighting HM = 1, N remains larger than 3/2 ≈ 0.87 even for homogeneous material c = 0. In contrast, N = 0 is reached for c = 0 if the microrotation is defined via the rotation of the pore surface HM = δ(r − Rvoid ). This means that the PDEs (1) and (20) for the macroscopic fields of displacements U(X) and microrotation r (X) decouple. Consequently, the solution for the displacements is identical to the classical Cauchy solution but the medium can still resist to micro-rotations or double tractions which are applied at its boundary. Apparently, the definition of the micro-rotation

r =

1 |S|| |



S||

 1 ξ  ξ dS. ξ · (Gχ )−1 × u dS with Gχ = || |S |  



(37)

S||

meets the requirements which were formulated in Section 1 and will thus be used in the following. A kinematic micro-macro relation as integral over an inclusion was employed recently by Biswas and Poh (2017). However, these author did not incorporate the gradients of the skew-symmetric part of microdeformation, which is why their approach cannot account for micropolar effects. Lakes and Drugan (2015), Rueger and Lakes (2016), Ha et al. (2016) and Rueger et al. (2017) identified the Cosserat parameters of different foam-like materials by regression of the measured size effects. They found indeed that a polar ratio  = 3/2 matches quite well. For commercial foams, they obtained a coupling number N well below unity (Lakes and Drugan, 2015) but also a value N = 0.99 (Rueger and Lakes, 2016). Recently, they employed artificially printed foams for which they obtained again values of N close to unity and concluded that lower values were attributed to surface layers which were damaged during the manufacturing of the specimens. Thus, the predictions of the present theory comply with these experimental findings. 5. Size effect in simple shear In order to investigate the predictions of the presented model, the simple shearing of an infinitely wide layer of height h of the homogenized micromorphic material is considered as sketched in Fig. 10. This one-dimensional problem can be solved analytically which is why it is often used as a benchmark to study size effects predicted by generalized continuum theories, e.g. in (Aifantis, 1987; Aifantis and Willis, 2005; Diebels and Steeb, 2003; Forest, 2013; Iltchev et al., 2015; Kruch and Forest, 1998; Liebenstein et al., 2018; Mazière and Forest, 2015; Tekog˘ lu and Onck, 2008). Also for the micro-polar theory, an analytical solution can be found as outlined in Appendix C. Finally, the resulting stress

21 =

Uˆs h

μ(eff)

  √ 2 2lb h 1−N tanh N √ h 2 2l b

(38)

G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

73

Fig. 11. Size effect in simple shearing: comparison with direct numerical simulations.

Fig. 12. Direct numerical simulation of simple shearing of an infinitely wide layer (c = 0.8, ν = 0.23).

is obtained which corresponds the resistance to the displacement Uˆs . Obviously, the tanh () term reflects the size effect with lb as characteristic intrinsic length. Eq. (38) shows again that the size effect vanishes if the coupling number N is zero, even for arbitrary values of lb . ∞ = μ(eff)U ˆs /h Fig. 11 visualizes the size effect. For this purpose, the reaction shear stress  21 is related to its limit 21 for thick layers h/lb → ∞ and plotted versus height h normalized by the intrinsic length R for ν = 0. In this representation, ∞ =  . In contrast, the micropolar theory predicts a size classical Cauchy–Boltzmann theory predicts a horizontal line 21 21 effect as deviation from this horizontal line for N > 0, i. e., for c > 0. The figure shows that the size effect increases considerably with increasing porosity c and saturates for c ࣡ 0.8. In addition, Fig. 11 compares the predictions of the present micropolar approach to respective FEM simulations of foam-like materials with discretely resolved microstructure (“direct numerical simulation”, DNS). In particular, Jänicke (2010) investigated honeycomb microstructures, cmp. Fig. 5, whereas Tekog˘ lu et al. (Tekog˘ lu et al., 2011; Tekog˘ lu and Onck, 2008) simulated plane random Voronoi microstructures of beams, both for porosities c ࣡ 0.9. For the direct numerical simulations, an equivalent intrinsic length R was defined as radius of a circle of equal area of the (average of the) discrete cells. The simulations of Jänicke (2010) cover only a regime h ࣡ 7R, where the deviation from the classical theory amounts to less than 30%. That is why, own DNS of a honeycomb structure with circular pores have been performed as depicted in Fig. 12. Fig. 11 shows that the predictions of the present micropolar model comply quite well with the direct numerical simulations from (Tekog˘ lu et al., 2011; Tekog˘ lu and Onck, 2008). However, the present DNS with circular pores (and thus with struts with changing cross section) exhibits a weaker size effect than those with prismatic struts from the aforementioned references. The mode of deformation in Fig. 12 gives rise to the suspicion that the compliance is mainly governed by the narrowest part of the strut. In this sense, the present circular volume element with circular pore, Fig. 6, has a constant strut width as it is the case with the beam models in (Tekog˘ lu et al., 2011; Tekog˘ lu and Onck, 2008). A systematic study of the effect of the foam topology on the size effect seems to be an important future task. 6. Material with inclusion As a second example, a material is considered which contains an inclusion (shear modulus μi and Poisson ratio ν i ), instead of the pore. The micro-rotation is defined via the heterogeneity, Eq. (37), i. e., via the interface between inclusion and

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G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

Fig. 13. Inclusion at micro-scale: (a) micro-scopic problem and (b) resulting coupling number (ν = νi = 0.3).

Fig. 14. Effect of the phase contrast on (a) the double stress modulus and (b) the resulting characteristic length scale of bending (ν = νi = 0.3).

matrix. The problem to be solved at the microscale is sketched in Fig. 13a. This axisymmetric problem yields the differential Eq. (35) with vanishing right-hand side, both for the matrix and for the inclusion. The solution for the later is a pure rigid rotation. Thus, the inclusion remains stress-free and the solution for the matrix is the same as in the case with the pore. Consequently, the resulting coupling modulus κ is the same, Eq. (36b). ´ Remarkably, it was assumed ad hoc by Hlavácek (1975) and Berglund (1982) that the inclusion undergoes a pure rotation. Though, both authors assumed ad hoc also a linear circumferencial displacement field for the matrix which does not satisfy the differential Eq. (35) for the matrix. Similarly, Forest (1998) prescribed a “relative rotation of the microstructure with respect to the material lines” to determine the “resistance to inner rotation”, though in a direct numerical simulation of the microstructure. Although the coupling modulus κ is independent of the modulus μi of the inclusion, the effective shear modulus μ(eff) increases with increasing μi (see Eq. (50) in Appendix). Consequently, lower values of the coupling number N are obtained compared to a material with pores, as plotted in Fig. 13b for different values of the phase contrast μi /μ. Fig. 14a shows the values of the double stress modulus γ for different values of the phase contrast μi /μ. As expected, γ increases with increasing μi /μ. However, so does the effective shear modulus as well. The characteristic length lb quantifies the ratio of both quantities, Eq. (33). Fig. 14b shows, that a value lb ≈ 0.5R is obtained with weak influences of c and μi /μ. Only for very soft inclusion or pores μi /μ → 0, about twice this value can be reached for c → 1. In this context it is recalled that the predicted size effect at the macroscale depends strongly on N and lb , and that lower values of both quantities lead to a less distinct size effect, compare e.g. Eq. (38). Thus, the predictions of the present theory comply with the experimental results of Gauthier and Jahsman (1975) and Lakes (1986), who found only a weak size effect for hard particles in a soft matrix, in contrast to the strong size effect observed by Lakes et al. for foams (Ha et al., 2016; Lakes and Drugan, 2015; Lakes, 1986; Rueger and Lakes, 2016; Rueger et al., 2017). Bigoni and Drugan (2006) demonstrated that material with hard inclusions can even exhibit softening size effects under bending-type loading. Softening size effects have been reported also for foam-like materials with incomplete layers of cells at free surfaces (Tekog˘ lu et al., 2011; Tekog˘ lu and Onck, 2008; Wheel et al., 2015). Such softening size effects cannot be described by the micro-polar theory at all. An important question is the prediction of a homogenization theory for homogeneous material at the micro-scale. For the volume element with inclusion, micro-homogeneous material can be obtained by two ways. Either, by vanishing volume fraction c = 0, or by a vanishing phase contrast (μi = μ and νi = ν ). For the classical homogenization, the result is the same,

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namely that the macroscopic properties correspond to their microscopic counterparts, compare e.g. Eq. (50). Regarding the coupling modulus, the proposed homogenization procedure predicts κ = 0 for c = 0 but not for a vanishing phase contrast, although c becomes meaningless in this case (which is why this case is not plotted in Fig. 13b). It has to be concluded that the proposed homogenization procedure can be applied only to materials with sufficient phase contrast. This problem is common to all homogenization procedure which are based on the deformation of an inclusion, e.g. (Berglund, 1982; Biswas ´ and Poh, 2017; Forest, 1998; Hlavácek, 1975).

7. Summary and conclusions The general theory of homogenization towards a micromorphic theory, which was developed by the author in previous papers, contains a weight function HM (ξ ) in the kinematic micro-macro relation for the microdeformations χ . If this  kinematic micro-macro relation is imposed as global constraint at the micro-scale, the generalized Hill-Mandel condition is satisfied for any choice of this weight function. The present study addressed the question how this weight function should be chosen. For this purpose, certain criteria were discussed. The choice of this weight function should be objective and secondly the function value HM (ξ ) should be related to a local quantity of the microstructure. Furthermore, it was discussed that size-effects should diminish at the macro-scale for micro-homogeneous material. This requires that the macroscopic coupling moduli vanish, but not necessarily the double-stress moduli. Two potential choices of the weight function were considered. For a uniform weight function, the microdeformation corresponds to the first moment of the complete microscopic displacement field. Alternatively, the microdeformation can be defined via the deformations of a material heterogeneity. The similarities of both definitions with approaches, which have been adopted in the literature, were pointed out. As an example, the homogenization of a linear-elastic material with a circular pore or inclusion towards a plane micropolar continuum was considered. The resulting boundary problem at the micro-scale was solved analytically for periodic boundary conditions. Periodic boundary conditions ensure that the resulting macroscopic theory is representative in the sense that the behavior of a sufficiently large sample under macroscopically uniform loading is represented exactly. The axi-symmetric problem to determine the coupling modulus of a plane structure with inclusion or pore turned out to be a suitable benchmark problem for micromorphic homogenization procedures, due to its simplicity. It was found that the definition of the micro-rotation via a material heterogeneity yields a vanishing coupling modulus for micro-homogeneous material. Consequently, size effects are predicted to vanish if the material is micro-homogeneous everywhere. This definition complies with Mindlin’s intention that the microdeformation χ is associated with the deformations of the “micro-structure”  Drugan, 2006; Li, 2011), the present approach is not limited (Mindlin, 1964). In contrast to previous approaches (Bigoni and to linear-elastic material, but non-linear and irreversible behavior of the microscopic constituents can be incorporated. The predicted non-classical parameters, i. e., coupling modulus and double stress modulus, comply with the experimental findings from literature that materials with pores, like foams, exhibit much more pronounced size effects than materials with inclusions. It remains open apply the homogenization procedure to more realistic 3D structures and to non-linear problems, and to compare it with corresponding experiments quantitatively.

Appendix A. Classical effective elastic moduli Bulk modulus The kinematic and kinetic micro-macro relations for the axisymmetric problem of in-plane dilatation, Fig. 6a, read

2 E v := E : I = u(R ),   R

1 2 

 h :=  : I = σrr (r = R )

(39)

Due to the axisymmetry, it makes no difference for the homogenization whether (39)1 or (39)2 are applied as kinematic or static boundary condition, respectively. The resulting microscopic fields of radial stresses and displacements, respectively, for the Lamé problem are well known and read

σrr

h



R2 = 1−c 2 1−c r



,

u (r ) =

h 1−c



r c R2 + 2 (μ + λ ) 2μ r



.

(40)

Consequently, a relation

 h = (μ + λ )

1−c +λ 1 + c μμ

Ev

(41)

between the macroscopic values is obtained. A comparison with the macroscopic constitutive law (31) shows that the cofactor of Ev on the right-hand side of Eq. (41) can be identified as effective in-plane bulk modulus λ(eff) + μ(eff) .

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G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

Shear modulus For obtaining the exact solution for the effective shear modulus under kinematic or periodic boundary conditions (Fig. 6b), an ansatz

F = 2μg(r )Ed : ξ ξ . 

(42)

for the Airy function F is adopted. The pre-factor 2μ is introduced for convenience. In order to apply the respective terms of the kinematic boundary conditions (12), it is favorable to switch to a coordinate system b1 –b2 which is aligned with the principal axes of the deviatoric strain Ed , i. e. Ed = E d (b1 b1 − b2 b2 ). Thus, in polar coordinates the scalar product in   Eq. (42) becomes Ed : ξ ξ = E d r 2 cos(2ϕ ). The radial function g(r) 

g(r ) =C1 r 2 +

C2 C3 + 4 + C4 . r2 r

(43)

involves respective terms of the Mitchell series. The traction-free pore surface requires

g (r = Rvoid ) = 0.

g(r = Rvoid ) = 0,

(44)

The displacement field which belongs to the Mitchell series can be found in (Barber, 2010). For the particular ansatz (42) with Eq. (43) it reads





C2 2C3 + 3 − 2C4 r cos(2ϕ ) r r   C2 2C3 d 3 uϕ = E 2(3 − 2ν )C1 r − 2(1 − 2ν ) + 3 + 2C4 r sin(2ϕ ). r r ur = E d −4νC1 r 3 + 4(1 − ν )

(45)

In polar coordinates, the kinematic boundary conditions (12) for loading by Ed read 

ur ( r = R ) =

E d R cos(2ϕ ).

uϕ (r = R ) = −E d R sin(2ϕ ).

(46)

Equating Eqs. (45) with (46) yields two equations which, together with Eq. (44), allow to determine the coefficients C1 to C4 . Finally, the kinetic micro-macro relation (3) is evaluated, with stress field σ = F I − ∇ ∇ F from Eq. (42), as  

  1  = 2μ −2g(R ) − g (R ) Ed . 

(47)



2

Obviously, the prefactor of Ed on the right-hand side corresponds to the macroscopic effective shear modulus   

4 − 4ν − ( 1 − c )3 ( 1 − c )

μ(eff) = μ

(3 − 4ν )(1 + (3 − 4ν )c + c4 ) + c(4c2 − 6c + 3 )

.

(48)

In polar coordinates, periodic boundary conditions (15) read for the present loading case

ur (R, ϕ ) + ur (R, ϕ + π ) =

2E d R cos(2ϕ ),

uϕ (R, ϕ ) + uϕ (R, ϕ + π ) = −2E d R sin(2ϕ ).

(49)

For the particular displacement field (45), the periodic boundary conditions (49) coincide with kinematic boundary conditions (46). Thus, for the circular volume element under consideration, periodic and kinematic boundary conditions yield the same effective shear modulus, Eq. (48). If the pore is replaced by an inclusion with shear modulus μi and Poisson ratio ν i , then the ansatz (42) can be adapted therein r < Rvoid as well (corresponding coefficients are marked by an additional subscript ()i ). From the radial function g(r), coefficients C2i and C3i have to vanish within the inclusion for the fields to remain regular. The remaining six coefficients (four for the matrix and two for the inclusion), are determined by two boundary conditions (46) together with four continuity conditions at r = Rvoid (two for the tractions and two for the displacements). Evaluation of the kinetic micro-macro relation (47) yields the effective shear modulus

μ

(eff)

=

μ

 2 a1 + a2 μμi + a3 μμi  2 1 + a4 μμi + a5 μμi

a1 = ( 1 − c ) a2 = 2

with

4 − 4ν − ( 1 − c )3 (3 − 4ν )(1 + (3 − 4ν )c + c4 ) + c(4c2 − 6c + 3 )

(3 − 2(ν + νi ))c4 + 2c(1 − 2νi )(3c − 2ν (1 + c2 )) + (3 − 4ν )(5 − 6(ν + νi ) + 8νi ν )   (3 − 4νi ) (3 − 4ν )(1 + (3 − 4ν )c + c4 ) + c(4c2 − 6c + 3 )

a3 = ( 1 − c )

(3 − 4ν )(3 − 4ν + c3 ) + (15 − 28ν + 16ν 2 )c(1 + c ) − 6c   (3 − 4νi ) (3 − 4ν )(1 + (3 − 4ν )c + c4 ) + c(4c2 − 6c + 3 )

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77

a4 = 2 ( 1 − c )

(3 − 4ν )((3 − 2(ν + νi ))(3 + c3 ) − 4 + 8νi ν ) + (3 − 14ν + 8ν 2 )c(1 + c ) + 6c(c + νi (1 − c ))   (3 − 4νi ) (3 − 4ν )(1 + (3 − 4ν )c + c4 ) + c(4c2 − 6c + 3 )

a5 = ( 1 − c )2

( 3 − 4ν )2 ( 1 + c + c 2 ) − 3c  (3 − 4νi ) (3 − 4ν )(1 + (3 − 4ν )c + c4 ) + c(4c2 − 6c + 3 )



(50)

For μi = 0, the result μ(eff) = a1 μ from Eq. (48) is recovered. For a very stiff inclusion, the effective shear modulus amounts to

lim

μi /μ→∞

μ(eff) = μ

a3 μ (3 − 4ν )(3 − 4ν + c3 ) + (15 − 28ν + 16ν 2 )c(1 + c ) − 6c = . a5 1−c ( 3 − 4ν )2 ( 1 + c + c 2 ) − 3c

(51)

Appendix B. Effective double stress modulus The boundary-value problem at the micro-scale to obtain the effective double stress modulus γ (Fig. 6c), shall also be solved via an Airy stress function F(ξ ). For symmetry reasons, the Airy function has to have the structure

F = g(r )ξ ·  · Mr . 

(52)

Therein, Mr = Mr · b3 and  =  · b3 refer to the projections of the double stress and permutation tensor (both being axial   in nature) to the out-of-plane  direction b3 . The corresponding radial function from the Mitchell series are

g(r ) =C1 r 2 +

C2 + C3 ln(r ) + C4 . r2

(53)

The coefficient C4 does not contribute to the stresses. Furthermore, it is required that C3 vanishes since the term C3 ln (r) is related to a displacement field which is not periodic in tangential direction, compare §31 in (Timoshenko and Goodier, 1951). The resulting stress field is

σ = F I − ∇ ∇ F = 2 





3C1 +



4C2 C2 I− ξξ r4  r6

  

C ξ ·  · Mr − 2 C1 − 42  · Mr  ξ + ξ   · Mr 

r





(54)

The static boundary condition (31) and a traction free pore surface require

g (R ) =

1 , R

g (Rvoid ) = 0,

(55)

respectively, and allow to compute the remaining coefficients C1 and C2 . The resulting macroscopic constitutive equation can then obtained via the kinematic micro-macro relation (29) which requires to compute the displacement field belonging to Eq. (54). Instead of this lengthy procedure, E shall be computed  here via Castigliano’s method as Kr = ∂ W /∂ Mr . For this purpose, the macroscopic complementary strain energy needs to be computed which is equal the strain energy W = 1/2σ : ε V for the linear-elastic material ε = (σ − νσ : II )/(2μ ) under       consideration (plain strains). In particular for the stress field (54) with coefficients determined as described above, a strain energy

W =

1 1 3 − 4ν + c 2 r M · Mr 2 2 μR 2 1 − c2



(56)



=1/γ

is obtained. The double stress modulus γ can be extracted from the co-factor in Eq. 56. For a material with an elastic inclusion, the ansatz (52) and (53) is adopted for the inclusion r ≤ Rvoid , too. Respective terms are denoted by a subscript ()i in the following. Regarding Eq. (53), only the first coefficient C1i can be different from zero for the solution to remain regular. Boundary condition (55)1 remains valid but (55)2 has to be replaced by a continuity condition g (Rvoid ) = g i (Rvoid ). Furthermore, continuity of the displacements at the interface, taking into account a potential relative translation between the centers of matrix and inclusion, requires [(3 − 4ν )R2voidC1 + C2 /R2void ]/μ = (3 − 4νi )R2voidC1i /μi . These three equations allow to determine C1 , C2 and C1i . Subsequently, the double stress modulus can be computed by Castiagliano’s method as

 μ ( 1 − c 2 ) ( 3 − 4 νi ) + 1 + 3 c 2 − 4 ν c 2 i μ γ =2μR2   μ 3 + c2 − 4ν (3 − 4νi ) + (1 − c2 )(3 − 4ν ) i μ

(57)

For a very stiff inclusion, a limit value

lim

μi /μ→∞

is obtained.

γ = 2 μR 2

1 + 3c 2 − 4ν c 2 (1 − c2 )(3 − 4ν )

(58)

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G. Hütter / Journal of the Mechanics and Physics of Solids 127 (2019) 62–79

Appendix C. Micropolar solution of the simple shear problem For an analytical solution to the simple shear problem, the ansatz

r = r (X2 )b3

U = U (X2 )b1 ,

(59)

is adopted that the horizontal displacement as well as the rotation depend solely on the vertical coordinate X2 . Therein, b1 , b2 and b3 denote the unit base vectors. Based on these field, the deformation are obtained as





Er = U + r b2 b1 − r b1 b2 , 



Kr = (r ) b2 b3 

(60)

whereby the prime () refers to the derivative with respect to X2 . Firstly, these deformations are inserted into the constitutive relations (31) and (32) to get the stress fields  and Mr  



 = (μr + κ )U + κ r b2 b1 + μrU − κ r b1 b2 ,



Mr = (r ) (β b3 b2 + γ b2 b3 ) 

respectively. Subsequently, the equilibrium conditions (1) and (20) yield two second order ODEs for the two functions U(X2 ) and r (X2 ), respectively. These ODEs can be solved e.g. by elimination of U(X2 ). In addition to the classical displacement boundary conditions U (0 ) = 0 and U (h ) = Uˆs , boundary conditions for the microrotations r have to be specified. Here, so-called micro-clamped boundary conditions r (0 ) = r (h ) = 0 are prescribed in order to address size effects. Finally, the stress  21 can be computed, see Eq. (38). References Achenbach, J., 1976. Generalized continuum theories for directionally reinforced solids. Arch. Mech. 28 (3), 257–278. Addessi, D., Bellis, M.L., Sacco, E., 2016. A micromechanical approach for the Cosserat modeling of composites. Meccanica. 51 (3), 569–592. doi:10.1007/ s11012-015- 0224- y. Adomeit, G., 1968. Determination of elastic constants of a structured material. In: Kröner, E. (Ed.), Mechanics of Generalized Continua. 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