Multiscale asymptotic homogenization analysis of thermo-diffusive composite materials
aa r X i v : . [ m a t h - ph ] D ec Multiscale asymptotic homogenization analysisof thermo-diffusive composite materials
A. Bacigalupo ∗ , L. Morini , and A. Piccolroaz IMT Institute for Advanced Studies, Lucca, Italy Department of Civil, Environmental and Mechanical Engineering, University of Trento, Italy
October 20, 2017
Abstract
In this paper an asymptotic homogenization method for the analysis of composite ma-terials with periodic microstructure in presence of thermodiffusion is described. Appropriatedown-scaling relations correlating the microscopic fields to the macroscopic displacements, tem-perature and chemical potential are introduced. The effects of the material inhomogeneitiesare described by perturbation functions derived from the solution of recursive cell problems.Exact expressions for the overall elastic and thermodiffusive constants of the equivalent firstorder thermodiffusive continuum are derived. The proposed approach is applied to the case ofa two-dimensional bi-phase orthotropic layered material, where the effective elastic and ther-modiffusive properties can be determined analytically. Considering this illustrative exampleand assuming periodic body forces, heat and mass sources acting on the medium, the solutionperformed by the first order homogenization approach is compared with the numerical resultsobtained by the heterogeneous model.
Keywords:
Periodic microstructure, Asymptotic homogenization, Thermodiffusion, Overallmaterial properties.
Composite materials are extensively used in industrial practice. Indeed, many advanced engineer-ing applications, such as aerospace, aircraft, green building, biomedical, energetics and electronicsrequire the design and the use of heterogeneous multiphase materials. Due to the microstructuraleffects as well as the interaction between their constituents, these materials may present severalfavorable physical properties, as for example high stiffness, improved strength and toughness, en-hanced thermal conductivity, mass diffusivity or electrical permittivity.Recently, multiphase composite materials have been largely used in the design and fabricationof battery devices, in particular of lithium-ion batteries and solid oxide fuel cells (Nakajo et al.,2012; Dev et al., 2014; Ellis et al., 2012). Since high operational temperatures can be reached and ∗ Corresponding author. Tel.: +39 0583 4326613, email address: [email protected] ε of the microstructure. This expansion dependsboth on the macroscopic strains, temperature and chemical potential gradients and on unknownperturbation functions accounting for the effects of the heterogeneities. Perturbation functionsrepresenting the effects of the material microstructures on the displacement, temperature, chemicalpotential and on the coupling effects between these fields are introduced. These perturbationfunctions, depending only on the properties of the microstructure, are obtained through the solutionof non-homogeneous problems on the cell with periodic boundary conditions.Similarly to the procedure proposed in Smyshlyaev and Cherednichenko (2000) and Bacigalupo(2014), averaged field equations of infinite order are obtained, and their formal solution is performedby representing the macroscopic displacements, temperature and chemical potential in terms ofpower series. Field equation for the homogenized first order thermodiffusive continuum are derived,and exact expressions for the overall elastic and thermodiffusive constants of this equivalent mediumare obtained. The proposed formulation is applied to the case of a two-dimensional bi-phase or-thotropic layered material. The effective elastic and thermodiffusive constants corresponding tothis example are determined analytically using the general expressions derived by the homogeniza-tion procedure. The solution performed by the proposed approach is compared with the numericalresults obtained by the heterogeneous model assuming periodic body forces, heat and mass sourcesacting on the considered bi-phase layered composite.The article is organized as follows: in Section 2 the geometry of the considered thermodiffusivecomposite material with periodic microstructure is illustrated, and the corresponding constitutiverelations and balance equations are introduced. The developed multiscale asymptotic homogeniza-tion technique is described in Section 3, based on down-scaling relations correlating the microscopicfields to the macroscopic displacements, temperature and chemical potential. The unknown per-turbation functions describing the effects of the material heterogeneities are defined as solutionsof the corresponding non-homogeneous cell problems. In the same Section, averaged field equa-tions of infinite order are obtained, and a solution scheme based on asymptotic expansion of themacroscopic displacements, temperature and chemical potential field is reported. Field equationsand explicit expressions for the overall elastic and thermodiffusive constants of the equivalent firstorder homogeneous continuum are derived in Section 4. As just anticipated, the proposed approachis applied for studying overall properties of two-dimensional bi-phase orthotropic layered materialsin Section 5. Finally, a critical discussion about the obtained results is reported together withconclusions and future perspectives in Section 6. Let us consider an heterogeneous composite material having periodic micro-structure and subjectto stresses induced by temperature changes, mass diffusion and body forces. The two-dimensionalgeometry shown in Fig. 1 is assumed for the system. Considering small strains approximation,the constituent elements of the medium are modelled as a linear thermodiffusive elastic Cauchycontinua. The material point is identified by position vector x = x e + x e referred to a system of3igure 1: (a) Heterogeneous material – Periodic domain L ; (b) Periodic cell A and periodicityvectors.coordinates with origin at point O and orthogonal base { e , e } . The periodic cell A = [0 , ε ] × [0 , δε ] with characteristic size ε is illustrated in Fig. 1b. The entire periodic medium can be obtainedspanning the cell A by the two orthogonal vectors v = d e = ε e , v = d e = δε e .According to the periodicity of the material, A is the elementary cell period of the elasticitytensor C ( m,ε ) ( x ) : C ( m,ε ) ( x + v i ) = C ( m,ε ) ( x ) , i = 1 , , ∀ x ∈ A , (1)where the superscript m stands for microscopic field. Similarly, the heat conduction tensor K ( m,ε ) ( x ) and the thermal dilatation tensor α ( m,ε ) ( x ) are defined as follows K ( m,ε ) ( x + v i ) = K ( m,ε ) ( x ) , α ( m,ε ) ( x + v i ) = α ( m,ε ) ( x ) , i = 1 , , ∀ x ∈ A , (2)and then the mass diffusion tensor D ( m,ε ) ( x ) and diffusive expansion tensor β ( m,ε ) ( x ) become D ( m,ε ) ( x + v i ) = D ( m,ε ) ( x ) , β ( m,ε ) ( x + v i ) = β ( m,ε ) ( x ) , i = 1 , , ∀ x ∈ A . (3)The tensors (1), (2) and (3) are commonly referred to as A− periodic functions.The system is subject to body forces b ( x ) , heat source r ( x ) and mass source s ( x ) which areassumed to be L− periodic with period L = [0 , L ] × [0 , δL ] and to have vanishing mean values on L . Since L is a large multiple of ε , then L can be assumed to be a representative portion of theoverall body. This means that the body forces, heat sources and mass sources are characterized bya period much greater than the microstructural size ε .Following the procedure reported in Bacigalupo (2014), a non-dimensional unit cell Q = [0 , × [0 , δ ] that reproduces the periodic microstructure by rescaling with the small parameter ε is in-troduced. Two distinct scales are represented by the macroscopic (slow) variables x ∈ A andthe microscopic (fast) variable ξ = x /ε ∈ Q (see for example Bakhvalov and Panasenko (1984);Smyshlyaev and Cherednichenko (2000) and Bacigalupo (2014)). The constitutive tensors (1),(2) and (3) are functions of the microscopic variable, whereas the body forces, heat sources andmass sources depend by the slow macroscopic variable. Consequently, the mapping of both theelasticity and thermodiffusive tensors may be defined on Q as follows: C ( m,ε ) ( x ) = C m ( ξ = /ε ) , K ( m,ε ) ( x ) = K m ( ξ = x /ε ) , α ( m,ε ) ( x ) = α m ( ξ = x /ε ) , D ( m,ε ) ( x ) = D m ( ξ = x /ε ) , β ( m,ε ) ( x ) = β m ( ξ = x /ε ) , respectively.The relevant micro-fields are the micro-displacement u ( x ) , the microscopic temperature θ ( x ) = T ( x ) − T ( T stands for the temperature of the natural state) and the microscopic chemical potential η ( x ) . The micro-stress σ ( x ) , the microscopic heat and mass fluxes q ( x ) and j ( x ) are defined bythe following constitutive relations: σ ( x ) = C m (cid:16) x ε (cid:17) ε ( x ) − α m (cid:16) x ε (cid:17) θ ( x ) − β m (cid:16) x ε (cid:17) η ( x ) , (4) q ( x ) = − K m (cid:16) x ε (cid:17) ∇ θ ( x ) , j ( x ) = − D m (cid:16) x ε (cid:17) ∇ η ( x ) , (5)where ε ( x ) = sym ∇ u ( x ) is the micro-strain tensor which is assumed to be zero at the fundamentalstate of the system.Note that, in eqs. (5) describing the heat and mass fluxes, we confine ourselves to the essentialeffects and neglect coupling terms, which is an assumption generally accepted in the quasi-statictheory of thermodiffusion, see for instance Nowacki (1974).The micro-stresses (4) and the microscopic fluxes (5) satisfy the local balance equations on thedomain A ∇ · σ ( x ) + b ( x ) = , ∇ · q ( x ) − r ( x ) = 0 , ∇ · j ( x ) − s ( x ) = 0 . (6)Substituting expressions (4)-(5) in equations (6) and remembering the symmetry of the elasticitytensor, the resulting set of partial differential equations is written in the form ∇ · (cid:16) C m (cid:16) x ε (cid:17) ∇ u ( x ) (cid:17) − ∇ · (cid:16) α m (cid:16) x ε (cid:17) θ ( x ) (cid:17) − ∇ · (cid:16) β m (cid:16) x ε (cid:17) η ( x ) (cid:17) + b ( x ) = (7) ∇ · (cid:16) K m (cid:16) x ε (cid:17) ∇ θ ( x ) (cid:17) + r ( x ) = 0 , ∇ · (cid:16) D m (cid:16) x ε (cid:17) ∇ η ( x ) (cid:17) + s ( x ) = 0 . (8)Moreover, at the interface Σ between two different phase of the material, the microscopic fieldssatisfy the following interface conditions: [[ u ( x )]] | x ∈ Σ = 0 , hh(cid:16) C m (cid:16) x ε (cid:17) ∇ u ( x ) − α m (cid:16) x ε (cid:17) θ ( x ) − β m (cid:16) x ε (cid:17) η ( x ) (cid:17) n ii(cid:12)(cid:12)(cid:12) x ∈ Σ = 0 , (9) [[ θ ( x )]] | x ∈ Σ = 0 , hh K m (cid:16) x ε (cid:17) ∇ θ ( x ) · n ii(cid:12)(cid:12)(cid:12) x ∈ Σ = 0 , (10) [[ η ( x )]] | x ∈ Σ = 0 , hh D m (cid:16) x ε (cid:17) ∇ η ( x ) · n ii(cid:12)(cid:12)(cid:12) x ∈ Σ = 0 , (11)where the notation [[ f ]] = f i (Σ) − f j (Σ) denotes the difference between the values of a function f at the interface Σ separating the phase i from the phase j .The micro-displacement, microscopic temperature and chemical potential may be seen in theform u ( x , ξ = x /ε ) , θ ( x , ξ = x /ε ) , η ( x , ξ = x /ε ) as functions of both the slow and the fast variable.It is important to note that since u ( x , ξ ) , θ ( x , ξ ) and θ ( x , ξ ) are assumed to be Q− periodicsmoothing functions with respect to the variable x , the interface conditions (9)-(11) can be expresseddirectly in function of the fast variable ξ (Bakhvalov and Panasenko, 1984).The solution of microscopic field equations (7), (8) is computationally very expensive and pro-vides too detailed results to be of practical use, so that it is convenient to replace the heterogeneous5odel with an equivalent homogeneous one to obtain equations whose coefficients are not rapidlyoscillating while their solutions are close to those of the original equations.Further in the paper, assuming that the size of the microstructure ε is sufficiently small withrespect to the structural size L , an equivalent classical first order thermodiffusive continuum isconsidered. The overall elastic moduli, thermal and diffusion expansion tensors, thermal and diffu-sive conduction tensors of a homogeneous continuum equivalent to periodic heterogeneous materialreported in Fig. 1 are derived by means of asymptotic homogenization techniques based on thegeneralization of down-scaling relations. The overall elastic and thermodiffusive properties of thehomogeneous continuum are expressed in terms of geometrical, mechanical, thermal and diffusiveproperties of the microstructure by means of an asymptotic expansion for the microscopic fields.The asymptotic expansion is performed in terms of the parameter ε that keeps the dependence onthe slow variable x separate from the fast one ξ = x /ε such that two distinct scales are represented.In the equivalent homogenized continuum, the macro-displacement U ( x ) of component U i , themacroscopic temperature Θ ( x ) and chemical potential Υ ( x ) are defined at a point x in the reference ( e i , i = 1 , . The displacement gradient is given by ∇ U ( x ) = ∂U i ∂x j e i ⊗ e j = H ij e i ⊗ e j = H ( x ) ,andthen the macroscopic strain is E ( x ) = sym ∇ U ( x ) . The macro-stress Σ ( x ) associate to E ( x ) aredefined as Σ ( x ) = Σ ij e i ⊗ e j with Σ ij = Σ ji , and the macroscopic heat and mass fluxes arerespectively: Q ( x ) = Q i e i and J ( x ) = J i e i . Following the approaches developed in Bakhvalov and Panasenko (1984); Smyshlyaev and Cherednichenko(2000); Bacigalupo and Gambarotta (2014) and Bacigalupo (2014) for purely elastic problems inperiodic heterogeneous media, the microscopic displacement, temperature and chemical potentialfields are represented through an asymptotic expansion with respect to the parameter ε , whoseterms depend on macroscopic fields and perturbation functions: u k (cid:16) x , ξ = x ε (cid:17) = U k ( x ) + + ∞ X l =1 ε l X | q | = l N ( l ) kpq ( ξ ) ∂ l U p ( x ) ∂x q ++ + ∞ X l =1 ε l X | q | = l − (cid:18) ˜ N ( l ) kq − ( ξ ) ∂ l − Θ( x ) ∂x q + ˆ N ( l ) kq − ( ξ ) ∂ l − Υ( x ) ∂x q (cid:19) ξ = x /ε = U k ( x ) + ε (cid:18) N (1) kpq ( ξ ) ∂U p ( x ) ∂x q + ˜ N (1) k ( ξ )Θ( x ) + ˆ N (1) k ( ξ )Υ( x ) (cid:19) ξ = x /ε ++ ε (cid:18) N (2) kpq q ( ξ ) ∂ U p ( x ) ∂x q ∂x q + ˜ N (2) kq ( ξ ) ∂ Θ( x ) ∂x q + ˆ N (2) kq ( ξ ) ∂ Υ( x ) ∂x q (cid:19) ξ = x /ε + · · · , (12)6 (cid:16) x , ξ = x ε (cid:17) = Θ( x ) + + ∞ X l =1 ε l X | q | = l (cid:18) M ( l ) q ( ξ ) ∂ l Θ( x ) ∂x q (cid:19) ξ = x /ε = Θ( x ) + ε (cid:18) M (1) q ( ξ ) ∂ Θ( x ) ∂x q (cid:19) ξ = x /ε + ε (cid:18) M (2) q q ( ξ ) ∂ Θ( x ) ∂x q ∂x q (cid:19) ξ = x /ε + · · · , (13) η (cid:16) x , ξ = x ε (cid:17) = Υ( x ) + + ∞ X l =1 ε l X | q | = l (cid:18) W ( l ) q ( ξ ) ∂ l Υ( x ) ∂x q (cid:19) ξ = x /ε = Υ( x ) + ε (cid:18) W (1) q ( ξ ) ∂ Υ( x ) ∂x q (cid:19) ξ = x /ε + ε (cid:18) W (2) q q ( ξ ) ∂ Υ( x ) ∂x q ∂x q (cid:19) ξ = x /ε + · · · . (14)In equations (12), (13) and (14) (commonly known as down-scaling relations), q = q , · · · , q l is a multi-index and ∂ l ( · ) /∂x q = ∂ l ( · ) /∂x q · · · ∂x q l . Due to their dependence on the slow spacevariable x , the macroscopic fields U k , Θ and Υ are L− periodic functions. N ( l ) kpq , M ( l ) q and W ( l ) q are the mechanical, thermal and diffusive fluctuation functions, respectively, whereas ˜ N ( l ) kq and ˆ N ( l ) kq denote the additional fluctuation functions corresponding to the contribution of the thermodiffusionto local displacement. All these perturbation functions depend on the fast space variable ξ = x /ε , and moreover, as it will be shown in Section 3.2, they are Q− periodic. Similarly to theprocedure reported in Smyshlyaev and Cherednichenko (2000) and Bacigalupo (2014), the meanvalue of the fluctuation functions is assumed to vanish on the unit cell Q , this means that thefollowing normalization conditions are satisfied: D N ( l ) kpq E = 1 δ ˆ Q N ( l ) kpq ( ξ ) d ξ = 0 , D ˜ N ( l ) kq E = 1 δ ˆ Q ˜ N ( l ) kq ( ξ ) d ξ = 0 , D ˆ N ( l ) kq E = 1 δ ˆ Q ˆ N ( l ) kq ( ξ ) d ξ = 0 , D M ( l ) q E = 1 δ ˆ Q M ( l ) q ( ξ ) d ξ = 0 , D W ( l ) q E = 1 δ ˆ Q W ( l ) q ( ξ ) d ξ = 0 . (15)Introducing a new variable ζ ∈ Q and a vector ε ζ ∈ A , which represents the translations ofthe medium with respect to the L− periodic body forces b ( x ) , heat sources r ( x ) and mass sources s ( x ) (Bacigalupo and Gambarotta, 2014), it can be shown that any Q− periodic function g ( ξ + ζ ) satisfies the following invariance property: h g ( ξ + ζ ) i = 1 δ ˆ Q g ( ξ + ζ ) d ζ = 1 δ ˆ Q g ( ξ + ζ ) d ξ . (16)According to the invariance property (16) and to the normalization conditions (15), the macroscopicfields can be defined as the mean values of the microscopic quantities (12), (13) and (14) evaluatedon the unit cell Q : U k ( x ) . = D u k (cid:16) x , x ε + ζ (cid:17)E , Θ( x ) . = D θ (cid:16) x , x ε + ζ (cid:17)E , Υ( x ) . = D η (cid:16) x , x ε + ζ (cid:17)E , (17)Expressions (17) are commonly known as up-scaling relations. More details regarding the structureof the down-scaling relations (12), (13) and (14) are provided in Appendix D.7 .2 First-order asymptotic analysis and derivation of the corresponding first-order cell problems In order to derive exact expressions for the fluctuation functions affecting the behavior of the micro-scopic fields u k , θ, η , the down-scaling relations (12), (13) and (14) are substituted into the micro-scopic field equations (7), (8). Remembering the property ∂∂x j f ( x , ξ = x ε ) = (cid:16) ∂f∂x j + ε ∂f∂ξ j (cid:17) ξ = x /ε = (cid:16) ∂f∂x j + f ,j ε (cid:17) ξ = x /ε , equation (7) become to the first order approximation ε − (cid:26)(cid:20)(cid:16) C εijkl N (1) kpq ,l (cid:17) ,j + C εijpq ,j (cid:21) H pq ( x ) + (cid:20)(cid:16) C εijkl ˜ N (1) k,l (cid:17) ,j − α εij,j (cid:21) Θ( x )+ (cid:20)(cid:16) C εijkl ˆ N (1) k,l (cid:17) ,j − β εij,j (cid:21) Υ( x ) (cid:27) + · · · · · · + b i ( x ) = 0 , i = 1 , , (18)where H pq = ∂U p /∂x q are the components of the macroscopic displacement gradient tensor pre-viously defined. Equations (8) assume the following form ε − (cid:20)(cid:16) K εij M (1) q ,j (cid:17) ,i + K εiq ,i (cid:21) ∂ Θ ∂x q + · · · · · · + r ( x ) = 0 , (19) ε − (cid:20)(cid:16) D εik W (1) q ,j (cid:17) ,i + D εiq ,i (cid:21) ∂ Υ ∂x q + · · · · · · + s ( x ) = 0 . (20)In order to transform the field equation (18), (19) and (20) in a PDEs system with constantcoefficients, in which the unknowns are the macroscopic quantities U k ( x ) , Θ( x ) and Υ ( x ) , thefluctuation functions have to satisfy non-homogeneous equations ( first-order cell problems ) reportedbelow.At the order ε − from the equation (18) we derive: (cid:16) C εijkl N (1) kpq ,l (cid:17) ,j + C εijpq ,j = n (1) ipq , (cid:16) C εijkl ˜ N (1) k,l (cid:17) ,j − α εij,j = ˜ n (1) i , (cid:16) C εijkl ˆ N (1) k,l (cid:17) ,j − β εij,j = ˆ n (1) i , (21)whereas from thermodiffusion equations (19) and (20) we obtain: (cid:16) K εij M (1) q ,j (cid:17) ,i + K εiq ,i = m (1) q , (cid:16) D εij W (1) q ,j (cid:17) ,i + D εiq ,i = w (1) q , (22)where: n (1) ipq = h C εijpq ,j i = 0 , ˜ n (1) i = −h α εij,j i = 0 , ˆ n (1) i = −h β εij,j i = 0 ,m (1) q = h K εiq ,i i = 0 , w (1) q = h D εiq ,i i = 0 . (23)The properties (23) are consequence of the Q− periodicity of the components C εijpq , α εij , β εij , K εiq and D εiq . Note that in equations (18)–(23) the derivatives should be understood in the generalizedsense.The perturbation functions characterizing the down-scaling relations (12), (13), and (14) areobtained by the solution of the previously defined cells problems, derived by imposing the normal-ization conditions (15). 8 Homogenized thermodiffusive Cauchy continuum: field equa-tions and overall properties
The field equations of the first order homogeneous continuum can be obtained by the zero orderterms (equations (63) and (64)) of the sequence of PDEs derived applying the asymptotic analysisto the averaged field equation, see Appendix A. This implies that the macroscopic displacement,temperature and chemical potential are approximated as follows: U p ( x ) ≈ U (0) p ( x ) , Θ( x ) ≈ Θ (0) ( x ) , Υ( x ) ≈ Υ (0) ( x ) . (24)Alternatively, the field equations of the equivalent Cauchy continuum can be derived consideringonly the terms of order ε in the equations (59), (60) and (61).The field equations of an homogeneous first order continuum in presence of thermodiffusion aregiven by C iq pq ∂ U p ∂x q ∂x q − α iq ∂ Θ ∂x q − β iq ∂ Υ ∂x q + b i = 0 , (25) K q q ∂ Θ ∂x q ∂x q + r = 0 , D q q ∂ Υ ∂x q ∂x q + s = 0 , (26)where C iq pq are the components of the overall elastic tensor, α iq and β iq are respectively thecomponents of the overall thermal dilatation and diffusive expansion tensors, K q q denotes thecomponents of the overall heat conduction tensor and D q q represents the components of the overallmass diffusion tensor. Remembering the approximation (24), the macroscopic field equations (25)–(26) can be compared to the zero order terms of the averaged field equation (63) and (64) fordetermining the overall properties of the thermodiffusive Cauchy continuum. In order to relate thecoefficients n (2) ipq q , ˜ n (2) iq , ˆ n (2) iq , m (2) q q , w (2) q q contained in the equations (63) and (64) to the overallelastic and thermodiffusive constants of the media C iq pq , α iq , β iq , K q q , D q q , the symmetriesof the tensors of components n (2) ipq q , ˜ n (2) iq , ˆ n (2) iq , m (2) q q , w (2) q q , and the ellipticity of the field equations(63) and (64) are required. A demonstration of these properties is reported in Appendix C. As aconsequence of these properties, it can be observed that: n (2) ipq q = ( C iq pq + C iq pq ) , ˜ n (2) iq = α iq , ˆ n (2) iq = β iq , m (2) q q = K q q and w (2) q q = D q q . In particular, comparing the field equation (25) to(63), and remembering the relationship between n (2) ipq q and C iq pq , it is easy to note that due to therepetition of the indexes q and q : C iq pq ∂ U p ∂x q ∂x q = n (2) ipq q ∂ U p ∂x q ∂x q = ( C iq pq + C iq pq ) ∂ U p ∂x q ∂x q .The overall elastic and thermodiffusive tensors, obtained in terms of fluctuation functions, andthe components of microscopic elastic and thermodiffusive tensors, take the form (see Appendix Afor details): C iq pq = 14 D C εrjkl (cid:16) N (1) riq ,j + δ ri δ jq + N (1) rq i,j + δ rq δ ij (cid:17) (cid:16) N (1) kpq ,l + δ kp δ q ,l + N (1) kq p,l + δ kq δ lp (cid:17)E α iq = D C εiq kl ˜ N (1) k,l − α εiq E , β iq = D C εiq kl ˆ N (1) k,l − β εiq E ,K q q = D K εij ( M (1) q ,j + δ jq )( M (1) q ,i + δ iq ) E , D q q = D D εij ( W (1) q ,j + δ jq )( W (1) q ,i + δ iq ) E . (27)The components C iq pq , K q q and D q q of the overall constitutive tensors of the material co-incide with those derived by asymptotic homogenization techniques applied to uncoupled static9lastic (Bakhvalov and Panasenko, 1984; Smyshlyaev and Cherednichenko, 2000; Bacigalupo, 2014)and heat conduction problems (Zhang et al., 2007) in media with periodic microstructures. Thecomponents α iq and β iq of the coupling thermodiffusive tensors have been obtained by meansof a consistent generalization of the down-scaling relations (12) (13) and (14). These expressionsrelate the microscopic displacement field to the macroscopic displacements, temperature, chemicalpotential and their higher order gradients. The general results obtained are now applied to the case of a bi-phase layered material in presence ofthermodiffusion. Exact analytical expressions for the overall elastic and thermodiffusive constantsare derived. Considering a two-dimensional infinite thermodiffusive medium subject to periodicbody forces, heat and/or mass sources, the solution obtained applying the proposed homogenizedmodel is compared with the results provided by the analysis of the corresponding heterogeneousproblem.Figure 2: (a) Heterogeneous and homogenized models with L -periodic body force b j , heat sources r ( x j ) and mass sources s ( x j ) ; (b) Periodic cell and constituents: bi-phase layered material. Let us consider a layered body obtained as an unbounded d − periodic arrangement of two differentlayers having thickness a and b , where d = ε = a + b and ζ = a/b are defined. The phasesare assumed homogeneous and orthotropic, with an orthotropic axis coincident with the layeringdirection e , the geometry of the system is shown in Fig. 2. The orthotropic symmetry is assumedfor both the elastic and thermodiffusive tensors. The micro-fluctuation functions N (1) riq , ˜ N (1) k , ˆ N (1) k , M (1) q and W (1) q are analytically obtained through the solution of the cell problems formulated inSection 3.2 (see equations (21) and (22) and conditions (23)). Due to the particular properties ofsymmetry of the microstructure, these functions depend only on the fast variable ξ . This variable10s perpendicular to the layering direction e (see Fig. 2). The non-vanishing micro-fluctuationfunctions N (1) riq , ˜ N (1) k and ˆ N (1) k , obtained by solving the cell problem of order ε − (21) are: N (1) _ a = C b − C a C a − ζC b ξ a ; N (1) _ b = ζ C a − C b C a − ζC b ξ b ; N (1) _ a = C b − C a C a − ζC b ξ a ; N (1) _ b = ζ C a − C b C a − ζC b ξ b ; N (1) _ a = N _ a = C b − C a C a − ζC b ξ a ; N (1) _ b = N _ b = ζ C a − C b C a − ζC b ξ b ; (28) ˜ N (1) _ a = − α b − α a C a + ζC b ξ a ; ˜ N (1) _ b = ζ α b − α a C a + ζC b ξ b ; (29) ˆ N (1) _ a = − β b − β a C a + ζC b ξ a ; ˆ N (1) _ b = ζ β b − β a C a + ζC b ξ b ; (30)where ξ a ∈ h − ζ ζ +1) , ζ ζ +1) i and ξ b ∈ h − ζ +1) , ζ +1) i are non-dimensional vertical coordinatescentered in each layer. The non-vanishing fluctuation functions associate with the thermodiffusionequations, derived by the solution of the cell problems of order ε − (22) are: M (1) _ a = − K a − K b K a − ζK b ξ a ; M (1) _ b = ζ K a − K b K a − ζK b ξ b ; (31) W (1) _ a = − D a − D b D a − ζD b ξ a ; W (1) _ b = ζ D a − D b D a − ζD b ξ b . (32)Note that the superscripts a,b denote that the elastic and thermodiffusive constants are referredrespectively to the phases a and b .In order to derive the overall elastic and thermodiffusive constants corresponding to a firstorder equivalent continuum, the fluctuation functions (28), (29), (30), (31) and (32) are used intoexpressions (27). The components of the overall elastic tensor C iq pq take the form: C = ζ C a C b + ζ ( C b C b − ( C a ) + 2 C a C b − ( C b ) + C a C a ) + C b C a ( ζ + 1)( C a + ζC b ) ; C = ( ζ + 1) C a C b C a + ζC b , C = ( ζ + 1) C a C b C a + ζC b , C = C b C a + ζC a C b C a + ζC b . (33)The non-vanishing components of the thermal dilatation tensor α iq and diffusive expansion tensor β iq are respectively given by α = − ζ ( C b α b − C b α a − C b α b − C a α b + C a α a − C a α a ) − ζ C b α a − C a α b ( ζ + 1)( C a + ζC b ) ; α = ζC b α a + α b C a C a + ζC b ; (34)11 = − ζ ( C b β b − C b β a − C b β b − C a β b + C a β a − C a β a ) − ζ C b β a − C a β b ( ζ + 1)( C a + ζC b ) ; β = ζC b β a + β b C a C a + ζC b . (35)The non-vanishing components of the heat conduction tensor K iq and mass diffusion tensor D iq take the form K = K b + ζK a ζ + 1 , K = ( ζ + 1) K a K b K a + ζK b ; (36) D = D b + ζD a ζ + 1 , D = ( ζ + 1) D a D b D a + ζD b . (37)Considering the case of isotropic phases, the components of the elasticity tensor become C ς = C ς = ˜ E ς − ˜ ν ς , C ς = ˜ ν ς ˜ E ς − ˜ ν ς , C ς = ˜ E ς ν ς ) , (with ς = a, b ), where for plane-strain: ˜ E ς = E ς − ν ς , ˜ ν ς = ν ς − ν ς , whereas for plane-stress: ˜ E ς = E ς , ˜ ν ς = ν ς , being E ς the Young’s modulus and ν ς the Poisson’s ratio, respectively. The components of the thermal dilatation and diffusive expansiontensors take respectively the forms: α ς = α ς = α ς , β ς = β ς = β ς (note that the coefficients α ς and β ς can be expressed in terms of the linear isotropic thermal and diffusive expansion coefficientsand the elastic moduli (Nowacki, 1974, 1986). The components of the heat conduction and massdiffusion tensors finally become K ς = K ς = K ς and D ς = D ς = D ς . The overall elastic andthermodiffusive constants for the case of isotropic phases are reported in Appendix C.By an asymptotic expansion of the constants (33), (34), (35), (36) and (37) in terms of theconcentration of the two constituents phases (not reported here for conciseness), it can be easilyshown that, if the concentration of the phase a vanishes, the overall elastic and thermodiffusiveconstants of the bi-phase layered material tend to the values corresponding to phase b . Conversely,if the concentration of the phase a tends to one, the same expressions tend to the elastic andthermodiffusive constants of the phase a .In order to simplify the required computations, for the illustrative examples both the phasesare assumed to be isotropic, and then the overall elastic and thermodiffusive constants reported inAppendix C are used. These constants can be represented in the non-dimensional form: ˜ C iq pq ( ρ C , ζ, ˜ ν a , ˜ ν b ) = C iq pq ˆ C iq pq , ˜ α iq ( ρ C , ρ α , ζ, ˜ ν a , ˜ ν b ) = α iq ˆ α iq , ˜ β iq ( ρ C , ρ β , ζ, ˜ ν a , ˜ ν b ) = β iq ˆ β iq , ˜ K iq ( ρ K , ζ ) = K iq ˆ K iq , ˜ D iq ( ρ D , ζ ) = D iq ˆ D iq , (38)where ρ C = ˜ E a / ˜ E b , ρ α = ˜ α a / ˜ α b , ρ β = ˜ β a / ˜ β b , ρ K = ˜ K a / ˜ K b , ρ D = ˜ D a / ˜ D b , and ˆ C iq pq = ( C aiq pq + C biq pq ) / , ˆ α iq = ( α aiq + α biq ) / , ˆ β iq = ( β aiq + β biq ) / , ˆ K iq = ( K aiq + K biq ) / , ˆ D iq = ( D aiq + D biq ) / . It is important to note that if the Poisson’s coefficients of the two phases are identical(i.e. ν a = ν b ), the non-dimensional overall elastic and thermodiffusive constants (38) possess thefollowing property: ˜ C iq pq ( ρ C , ζ, ˜ ν ) = ˜ C iq pq ( ρ − C , ζ − , ˜ ν ) , ˜ α iq ( ρ C , ρ α , ζ, ˜ ν ) = ˜ α iq ( ρ − C , ρ − α , ζ − , ˜ ν ) , ˜ β iq ( ρ C , ρ α , ζ, ˜ ν ) = ˜ β iq ( ρ − C , ρ − α , ζ − , ˜ ν ) , ˜ K iq ( ρ K , ζ ) = ˜ K iq ( ρ − K , ζ − ) , D iq ( ρ K , ζ ) = ˜ D iq ( ρ − K , ζ − ) . (39)The variation of the normalized components of the overall elasticity tensor ˜ C and ˜ C withthe ratio ρ C is reported in Figs. 3 / ( a ) and 4 / ( a ) , respectively. The same value of the Poisson’scoefficient ˜ ν = 0 . has been assumed for both the phases, and several values of the non-dimensionalgeometrical parameter ζ has been considered for the computations. It can be observed that for ρ C = 1 , corresponding to the case of two isotropic phases having identical elastic properties, the non-dimensional components of the overall elastic tensor assume the value ˜ C iq pq = 1 (i. e. C iq pq = C aiq pq = C biq pq ). In Figs. 3 / ( b ) and 4 / ( b ) the components ˜ C and ˜ C are plotted asfunctions of ζ for different values of ρ C considering the fixed Poisson’s coefficient ˜ ν = 0 . identicalfor both the phases. For ζ → , the thickness of the phase a vanishes. Consequently, the valuesof the overall elastic constants tends to those of the phase b : C iq pq = C biq pq , and the limitvalues assumed by the normalized components of the elastic tensor reported in the figures are ˜ C iq pq = 2 / (1 + ρ C ) . Conversely, for ζ → + ∞ the thickness of the phase b tends to zero, and then C iq pq = C aiq pq and the non-dimensional constants plotted in Figs. 3 / ( b ) -4 / ( b ) assume the limitvalues ˜ C iq pq = 2 ρ C / (1 + ρ C ) . Results (not reported here) show that the normalized components ˜ C and ˜ C have a behaviour very similar to that of the component ˜ C .Figure 3: (a) Dimensionless constant ˜ C vs. the ratio ρ C for ˜ ν a = ˜ ν b = 0 . and for differentvalues of the geometric ratio ζ : ζ = 1 / green line, ζ = 1 / blue line, ζ = 1 / red line, ζ = 1 / black line, ζ = 2 red points, ζ = 5 blue points, ζ = 10 green points. (b) Dimensionless constant ˜ C vs. the geometric ratio ζ for ˜ ν a = ˜ ν b = 0 . and for different values of the ratio ρ C : ρ C = 2 red line, ρ C = 5 blue line, ρ C = 10 green line, ρ C = 30 black line, ρ C = 50 violet line.13igure 4: (a) Dimensionless constant ˜ C vs. the ratio ρ C for ˜ ν a = ˜ ν b = 0 . and for differentvalues of the geometric ratio ζ : ζ = 1 / green line, ζ = 1 / blue line, ζ = 1 / red line, ζ = 1 / black line, ζ = 2 red points, ζ = 5 blue points, ζ = 10 green points. (b) Dimensionless constant ˜ C vs. the geometric ratio ζ for ˜ ν a = ˜ ν b = 0 . and for different values of the ratio ρ C : ρ C = 2 red line, ρ C = 5 blue line, ρ C = 10 green line, ρ C = 30 black line, ρ C = 50 violet line.The three-dimensional plots reported in Fig. 5 show the variation of the normalized componentsof the overall thermal dilatation tensor ˜ α and ˜ α as functions of ρ c and ρ α , assuming ˜ ν = 0 . for both the phases and ζ = 1 . In Figs. 6 / ( a ) and 7 / ( a ) the variation of ˜ α and ˜ α with thenon-dimensional ratio ρ C is reported for several values of ζ assuming ˜ ν a = ˜ ν b = ˜ ν = 0 . and ρ α = 2 .For ρ C = 1 , corresponding to the case of two isotropic phases with identical elastic constants butdifferent thermal dilatation properties, the normalized components of the overall thermal dilatationtensor tend to the values ˜ α iq = 2( ζρ α + 1) / [( ρ α + 1)( ζ + 1)] (i.e. α iq = ( α aiq ζ + α biq ) / ( ζ + 1) ).In the case where ρ C = 1 and also ρ α = 1 , both the elastic and thermal dilatation tensors of thetwo phases are identical, and then ˜ α iq = 1 . In Figs. 6 / ( b ) and 7 / ( b ) the same constants ˆ α and ˆ α are plotted as functions of ζ for ˜ ν = 0 . , ρ α = 2 and several different values of ρ C . For ζ → , the thickness for the phase a vanishes, and the elements of the overall thermal dilatationtensor tends to those of the phase b (i.e. α iq = α biq ). As it can be observed in the figures, inthis case the normalized constants tend to a limit value which is the same for any value of ρ C (i.e. ˜ α iq = 2 / (1 + ρ α ) ). This value can be easily derived by using expressions for α and α reportedin Appendix C. Conversely, for ζ → + ∞ , the thickness of the layer b tends to zero, the effectivethermal dilatation constants tend to those of the phase a (i.e. α iq = α aiq ) and the normalizedcomponents ˜ α and ˜ α reported in Figs. 6 / ( b ) and 7 / ( b ) assume the values ˜ α iq = 2 ρ α / (1 + ρ α ) .The properties of the normalized elements of the overall diffusive expansion tensor ˜ β and ˜ β aresimilar to those of ˜ α and ˜ α , and can be easily studied substituting the non-dimensional ratio ρ α with ρ β .The variation of the normalized components of the overall heat conduction tensor ˜ K and ˜ K with the non-dimensional ratio ρ K are shown in Figs. 8 / ( a ) and 9 / ( a ) . Several values of ζ have been assumed for the computations. It can be observed that for ρ K = 1 , we have ˜ K iq = 1 .This is due to the fact that the value ρ K = 1 corresponds to the case where the heat conductionof the two phases are identical, and then K aiq = K biq . In Figs. 8 / ( a ) and 9 / ( b ) the same non-dimensional components ˜ K and ˜ K are reported as functions of ζ for different values of ρ K . Asit is shown by these figures, for ζ → , ˜ K and ˜ K tends to a finite value depending on ρ K . Thislimit correspond to the case of vanishing thickness of the layer a , where K iq = K biq , and then14igure 5: Dimensionless component ˜ α (a), ˜ α (b) vs. the ratios ρ C and ρ α for ˜ ν a = ˜ ν b = 0 . and ζ = 1 / .Figure 6: Dimensionless component ˜ α vs. the ratio ρ C for ˜ ν a = ˜ ν b = 0 . , ρ α = 2 and for differentvalues of the geometric ratio ζ : ζ = 1 / green line, ζ = 1 / blue line, ζ = 1 / red line, ζ = 1 / black line, ζ = 2 red points, ζ = 5 blue points, ζ = 10 green points. (b) Dimensionless component α vs. the ratio ρ C for ˜ ν a = ˜ ν b = 0 . , ρ α = 2 and for different values of the geometric ratio ρ C : ρ C = 2 red line, ρ C = 5 blue line, ρ C = 10 green line, ρ C = 30 black line, ρ C = 50 violet line. ˜ K iq = 2 / (1 + ρ K ) . In the limit ζ → + ∞ , for which the thickness of the layer b vanishes and K iq = K aiq , the normalized components of the overall heat conductivity tensor tend to a finitevalue given by ˜ K iq = 2 ρ K / (1 + ρ K ) . The non-dimensional components of the overall mass diffusiontensor ˜ D and ˜ D are characterized by properties similar to those of ˜ K and ˜ K , and can bestudied substituting the non-dimensional ratio ρ K with ρ D .15igure 7: Dimensionless component ˜ α vs. the ratio ρ C for ˜ ν a = ˜ ν b = 0 . , ρ α = 2 and for differentvalues of the geometric ratio ζ : ζ = 1 / green line, ζ = 1 / blue line, ζ = 1 / red line, ζ = 1 / black line, ζ = 2 red points, ζ = 5 blue points, ζ = 10 green points. (b) Dimensionless component α vs. the ratio ρ C for ˜ ν a = ˜ ν b = 0 . , ρ α = 2 and for different values of the geometric ratio ρ C : ρ C = 2 red line, ρ C = 5 blue line, ρ C = 10 green line, ρ C = 30 black line, ρ C = 50 violet line.Figure 8: Dimensionless component ˜ K vs. the ratio ρ K for different values of the geometric ratio ζ : ζ = 1 / green line, ζ = 1 / blue line, ζ = 1 / red line, ζ = 1 / black line, ζ = 2 red points, ζ = 5 blue points, ζ = 10 green points. (b) Dimensionless component ˜ K vs. the ratio ρ K fordifferent values of the ratio ρ K : ρ K = 2 red line, ρ K = 5 blue line, ρ K = 10 green line, ρ K = 30 black line, ρ K = 50 violet line. 16igure 9: Dimensionless component ˜ K vs. the ratio ρ K for different values of the geometric ratio ζ : ζ = 1 / green line, ζ = 1 / blue line, ζ = 1 / red line, ζ = 1 / black line, ζ = 2 red points, ζ = 5 blue points, ζ = 10 green points. (b) Dimensionless component ˜ K vs. the ratio ρ K fordifferent values of the ratio ρ K : ρ K = 2 red line, ρ K = 5 blue line, ρ K = 10 green line, ρ K = 30 black line, ρ K = 50 violet line. 17 .2 Comparative analysis: homogenized model vs heterogeneous mate-rial In order to study the capabilities of the proposed homogenization procedure, the two-dimensionalbi-phase orthotropic layered material shown in Fig. 2 is assumed to be subjected to L -periodicharmonic body forces b i , directed along the orthotropy direction x j (see Fig. 2) and L -periodic heatand mass sources r ( x j ) and s ( x j ) : b j ( x j ) = B j e i πmx j L j , r ( x j ) = Re i πnx j L j , s ( x j ) = Se i πpx j L j , (40)where: j = 1 , ; L = L = L ; B i , R, S ∈ C ; m, n, p ∈ Z ; and i = − .This problem is analyzed by applying the homogenized first-order model with overall elastic andthermodiffusive constants derived from the homogenization of the periodic cell through the approachdeveloped in previous Sections. The obtained results are then compared with those derived by meansof a fully heterogeneous modelling procedure. Due to the periodicity of the heterogeneous material,body forces, heat and mass sources considered, only an horizontal (or vertical) characteristic portionof length L of the heterogeneous model is analyzed (Fig. 2b). In order to assess the reliability ofthe homogenized model, the macroscopic displacement, temperature and chemical potential fieldsare compared to the corresponding fields in the heterogeneous model by means of the up-scalingrelations (17).The overall elastic and thermodiffusive constants involving the fluctuation functions are obtainedin exact analytical forms via expressions (33), (34), (35), (37), and (36). Conversely, the solutionof the heterogeneous problem with L− periodic harmonic body forces is computed via FE analysiswith periodic boundary conditions on the displacement temperature and chemical potential fields.For the considered two-dimensional body subject to body forces along the orthotropy axes, heatand mass sources, the homogenized field equations (25)–(26) take the form: C jjjj ∂ U j ∂x j − α jj ∂ Θ ∂x j − β jj ∂ Υ ∂x j + b j = 0 , K jj ∂ Θ ∂x j + r = 0 , D jj ∂ Υ ∂x j + s = 0 , (41)where j = 1 , are not summed indexes. Equations (41) describe an extensional problem in pres-ence of thermodiffusion. Considering body forces, heat and mass sources of the form (40), themacroscopic displacements, temperature and chemical potential fields are given by U j ( x j ) = B j L C jjjj (2 πm ) e i πmxjLj − i (cid:20) Rα jj L C jjjj K jj (2 πn ) e i πnxjLj + Sβ jj L C jjjj D jj (2 πp ) e i πpxjLj (cid:21) , (42) Θ( x j ) = RL K jj (2 πn ) e i πnxjLj , Υ( x j ) = SL D jj (2 πp ) e i πpxjLj , (43)where j = 1 , are still not summed indexes. In order to compare the behavior of the derivedanalytical solution with the numerical results provided by the heterogeneous model, only the realpart of macroscopic fields (42), (43) is accounted. Moreover, the imaginary part of the amplitudes B j , R and S is assumed to be zero. The real part of expressions (42), (43) can be written in thenon-dimensional form: U ∗ j ( x j ) = cos (cid:18) πmx j L j (cid:19) + Ξ αjj m πn sin (cid:18) πnx j L j (cid:19) + Ξ βjj m πp sin (cid:18) πpx j L j (cid:19) , (44)18 ∗ ( x j ) = cos (cid:18) πnx j L j (cid:19) , Υ ∗ ( x j ) = cos (cid:18) πpx j L j (cid:19) , (45)where U ∗ j = U j C jjjj (2 πm ) B j L j , Θ ∗ = Θ K jj (2 πn ) RL j , Υ ∗ = Υ D jj (2 πp ) SL j ; and Ξ αjj = α jj RL j K jj B j and Ξ βjj = β jj SL j D jj B j , j = 1 , are still not summed indexes.The amplitude functions Ξ αjj and Ξ βjj are associated respectively with the thermal expansionand mass diffusion contribution to the macroscopic displacement (44) along the direction e j . Inorder to study the influence of the geometrical, elastic and thermodiffusive properties of the phaseson Ξ αjj and Ξ βjj , the following non-dimensional form for these functions is introduced: ˜ Ξ αjj ( ρ C , ρ α , ρ K , ζ, ˜ ν a , ˜ ν b ) = Ξ αjj ˆ Ξ αjj , ˜ Ξ βjj ( ρ C , ρ β , ρ K , ζ, ˜ ν a , ˜ ν b ) = Ξ βjj ˆ Ξ βjj , with j = 1 , , (46)where ˆ Ξ αjj = RL j ( α a + α b ) /B j ( K a + K b ) and ˆ Ξ βjj = RL j ( β a + β b ) /B j ( D a + D b ) . In the case wherethe two phases possess the same value of the Poisson’s coefficient ( ˜ ν a = ˜ ν b ), the following propertyis verified for the normalized amplitude functions: ˜ Ξ αjj ( ρ C , ρ α , ρ K , ζ, ˜ ν ) = ˜ Ξ αjj ( ρ − C , ρ − α , ρ − K , ˜ ν ) , ˜ Ξ αjj ( ρ C , ρ β , ρ K , ζ, ˜ ν ) = ˜ Ξ βjj ( ρ − C , ρ − α , ρ − K , ˜ ν ) . (47)The three-dimensional plot reported in Fig. 10 / ( a ) shows the variation of the normalized ampli-tude component ˜ Ξ α with ρ α and ρ K for ˜ ν a = ˜ ν b = 0 . , ρ C = 10 and ζ = 1 . In Fig. 10 / ( b ) thesame component ˜ Ξ α , which represents the contribution of the thermal expansion to the macro-scopic displacement along the direction e , is plotted as a function of ρ C assuming ˜ ν a = ˜ ν b = 0 . , ρ α = ρ K = 10 and several values of the dimensionless ratio ζ . The variation of the normalizedcomponent ˜ Ξ α , corresponding to the contribution of the thermal expansion to the macroscopicdisplacement along e (see Fig. 2), is reported in Fig. 11 / ( a ) as a function of ρ α and ρ K for ˜ ν a = ˜ ν b = 0 . , ρ C = 10 and ζ = 1 , and in Fig. 11 / ( b ) as a function of ρ C assuming ˜ ν a = ˜ ν b = 0 . , ρ α = ρ K = 10 . Observing the curves reported in the figures, it can be noted that the normalizedamplitude ˜ Ξ α , associate to the component of the macroscopic displacement U ∗ ( x ) parallel tothe stratification direction, is greater than the amplitude ˜ Ξ α which correspond to the component U ∗ ( x ) parallel to the stratification direction. The dimensionless amplitude ˜ Ξ β and ˜ Ξ β , associ-ated to the contributions of the mass diffusion respectively to U ∗ ( x ) and U ∗ ( x ) , are characterizedby the same properties of ˜ Ξ α and ˜ Ξ α , and their behavior can be easily studied substituting thenon-dimensional ratios ρ α and ρ K with ρ β and ρ D .19igure 10: (a) Dimensionless amplitude ˜Ξ α vs. the ratios ρ α and ρ K for ˜ ν a = ˜ ν b = 0 . , ρ C =10and ζ = 1 . (b) Dimensionless amplitude ˜Ξ αjj vs. the ratios ρ C for ˜ ν a = ˜ ν b = 0 . , ρ α = ρ K = 10 and ζ = 1 for different values of the geometric ratio ζ : ζ = 1 / red line, ζ = 1 / blue line, ζ = 1 blackline, ζ = 5 blue line points, ζ = 10 red line points.Figure 11: (a) Dimensionless amplitude ˜Ξ α vs. the ratios ρ α and ρ K for ˜ ν a = ˜ ν b = 0 . , ρ C =10and ζ = 1 . (b) Dimensionless amplitude ˜Ξ α vs. the ratios ρ C for ˜ ν a = ˜ ν b = 0 . , ρ α = ρ K = 10 and ζ = 1 for different values of the geometric ratio ζ : ζ = 1 / red line, ζ = 1 / blue line, ζ = 1 blackline, ζ = 5 blue line points, ζ = 10 red line points.20he analytical solution (44) and (45), derived by the solution of the homogenized field equations(41) is now compared with the results obtained by the finite element analysis of the heterogeneousproblem corresponding to the bi-phase layered material reported in Fig. 2 subject to harmonic bodyforces, heat and mass sources. More precisely, finite element analysis of the heterogeneous problem,performed by means of the program COMSOL Multiphysics, provides the local fields u j , θ , η whichare used together with the up-scaling relations (17) for obtaining the macro-scopic fields U j , Θ and Υ . These macro-scopic quantities are compared with the analytical expressions (44) and (45).Plane stress condition has been assumed for both the solution of the homogenized equations andthe heterogeneous problem, and two isotropic phases with the same value of the Poisson’s coefficient ν a = ν b = 0 . have been considered.In Fig. 12, the macroscopic displacement component U ∗ and temperature Θ ∗ evaluated usinganalytical expressions (44) and (45) are reported as functions of the normalized spatial coordinate x /L (continuous lines in the figure) and compared with the numerical results obtained by theheterogeneous model assuming periodic body forces b ( x ) and heat sources r ( x ) and consideringthe value of the amplitude ˜ Ξ α = 1 (diamonds in the figure). The following values for the geometricalparameters, the ratios between the elastic and of thermodiffusive constants have been assumed: L/ε = 10 , ρ C = 10 , ρ α = 10 , ρ β = 0 ρ K = 10 , ρ D = 0 , the effects of the mass diffusion have beenneglected in this example. The macroscopic displacement and temperature fields are plotted for thecharacteristic portion of length L = L , corresponding to x /L = 1 (i. e. for ≤ x /L ≤ ), andseveral values for the wave numbers m, n ∈ Z have been considered. Observing the curves, for boththe quantities U ∗ ( x /L ) and Θ ∗ ( x /L ) a good agreement is detected between the results derived bymeans of the first order homogenization approach and those obtained by the heterogeneous model.Results for the macroscopic displacement component U ∗ and temperature in Θ ∗ along the char-acteristic portion of length L = L in direction parallel to e (not reported here for conciseness)show a good agreement between the solution obtained by means of the first-order asymptotic homog-enization method and the values obtained by means of finite element analysis of the heterogeneousproblem.Figure 12: Macroscopic displacement component U ∗ and temperature fields Θ ∗ due to harmonicbody force b and temperature source r . The heterogeneous model (diamonds) is compared withthe homogenized first order model. (a) Dimensionless macro displacement U ∗ vs. the ratio x /L for Ξ α = 1 for different values of wave number n , m ( n = 1 , m = 1 red line; n = 2 , m = 1 blueline; n = 1 , m = 2 green line). (b) Macro temperature Θ ∗ vs. the ratio x /L for different values ofwave number n ( n = 1 red line; n = 2 , blue line).21igure 13: Macroscopic displacement component U ∗ , temperature Θ ∗ , and chemical potential Υ ∗ due to harmonic body force b and temperature and mass sources r , s respectively. The heteroge-neous model (diamonds) is compared with the homogenized first order model. Dimensionless macrodisplacement U ∗ vs. the ratio x /L for Ξ α = 1 , Ξ β = 1 and wave number m = 1 (a), m = 2 (b)for different values of wave number n , p ( n = 1 , p = 1 red line; n = 2 , p = 1 blue line; n = 2 , p = 2 green line). (c) Macro temperature Θ ∗ vs. the ratio x /L for different values of wave number n ( n = 1 red line; n = 2 , blue line). (d) Macro concentration Υ ∗ vs. the ratio x /L for differentvalues of wave number p ( p = 1 red line; p = 2 , blue line).In Fig. 13, the variation of the normalized component of the macroscopic displacement U ∗ ( x /L ) ,temperature Θ ∗ ( x /L ) and chemical potential Υ ∗ ( x /L ) along the characteristic portion of length L = L is plotted as a function of x /L . Two isotropic phases having the same Poisson’s coefficient ν = 0 . have been assumed, and the same values of the previous example have been assigned to thegeometrical, elastic and thermal parameters. The amplitude of the mass diffusion contribution tothe displacement is assumed to be ˜ Ξ β = 1 , and ρ K = 10 . Similarly to the previous case, for thefinite element analysis of the heterogeneous elastic thermodiffusive problem harmonic body forces b ( x ) , heat and mass sources r ( x ) and s ( x ) have been introduced. The reported curves show agood agreement between the results obtained by the asymptotic homogenization (continuous linesthe figure) and those provided by the heterogeneous elastic thermodiffusive model (diamonds inthe figure). The good agreement between the results coming from the two different approachescan be observed for U ∗ ( x /L ) in Fig. 13 / ( a ) , ( b ) , for Θ ∗ ( x /L ) in Fig. 13 / ( b ) and for Υ ∗ ( x /L ) inFig. 13 / ( d ) . Similar results are obtained for the macroscopic displacement U ∗ , temperature Θ ∗ andchemical potential Υ ∗ along the characteristic portion of length L = L in direction parallel to e Conclusions
A general asymptotic homogenization approach for describing the static elastic, thermal and diffu-sive properties of periodic composite materials in presence of thermodiffusion is proposed.
Down-scaling relations associating the displacements, temperature and chemical potential at the micro-scale to the corresponding fields at the macro-scale are introduced. Perturbation functions aredefined for representing the effects of the microstructures on the microscopic displacement, tem-perature, chemical potential and on the coupling effects between these fields. These perturbationfunctions are obtained through the solution of non-homogeneous problems on the cell definingperiodic boundary conditions and normalization conditions ( up-scaling relations).Averaged field equations of infinite order are derived for the considered class of periodic thermod-iffusive materials, and an original formal solution is performed by means of power series expansionof the macroscopic displacements, temperature and chemical potential fields. Field equation for thehomogenized Cauchy thermodiffusive continuum are derived, and exact expressions for the overallelastic and thermodiffusive constants of this equivalent first order medium are obtained.An example of application of the developed general method to the illustrative case of a two-dimensional bi-phase orthotropic layered material is provided. The effective elastic and thermodif-fusive constants of this particular composite material are determined using the general expressionsderived by the asymptotic homogenization procedure. Analytical expressions for the macroscopicfields derived by the solution of the homogenized equations corresponding to the first order equiv-alent continuum. Finite element analysis of the corresponding heterogeneous model is performedassuming periodic body forces, heat and mass sources acting on the considered bi-phase layeredcomposite. In order to compare the analytical solution of the homogenized equations with thenumerical results obtained by the heterogeneous model, the microscopic fields computed by finiteelements techniques are used to estimate the macroscopic displacements, temperature and chemicalpotential fields by means of the up-scaling relations defined in the paper. The good agreementdetected between the solution derived by the homogenized first order equations and the numer-ical results obtained by the heterogeneous model through the up-scaling relations represents animportant validation of the accuracy of the proposed asymptotic homogenization approach.Thanks to the great versatility of the asymptotic homogenization techniques and to the pro-posed general rigorous formulation, the method developed in the paper can be adopted for studyingeffective elastic and thermodiffusive properties of many composite materials, without any other as-sumption regarding the geometry of the microstructures in addition to the periodicity. In particu-lar, the proposed asymptotic homogenization procedure can have relevant applications in modellingmechanical and thermodiffusive properties of energy devices with layered configurations, such aslithium ions batteries and solid oxide fuel cells. In standard operative situations, the componentsof these devices are commonly subject to severe thermomechanical and diffusive stress which cancause damages and crack formation compromising their performances. Consequently, evaluating theoverall elastic and thermodiffusive properties of these battery devices through the asymptotic ho-mogenization approach illustrated in the paper can represent an important issue in order to predictdamaging phenomena and to improve the efficient design and manufacturing of these systems.Multi-scale homogenization techniques such as that proposed in the paper provide an accu-rate description of the macroscopical mechanical and thermodiffusive properties of heterogeneousmaterials through the derivation of effective constants of the first order equivalent continuum. Nev-ertheless, first order homogenization procedures are not enough accurate to model size-effects andnon-local phenomena connected to the microstructural scale length. As a consequence, the devel-23ped first order homogenization approach does not provide a precise description of the behavior ofthermodiffusive composite materials in presence of high gradients of stresses, deformations, temper-ature, chemical potential, heat and mass fluxes, as well as of non-local phenomena such as wavesdispersion. In order to overcome these limits in the accuracy, non-local higher order homogenizationtechniques can be used. These methods provide constitutive relations of equivalent higher ordercontinuum media including characteristic scale-lengths associate to the microstructural effects. Us-ing the rigorous and general approach illustrated in the paper, a better approximation of the elasticand thermodiffusive behavior of composite materials in presence of strong gradients can be obtainedthrough the solution of higher order cells problems involving the coefficients of the averaged fieldequations of infinite order reported in Appendix A. As it is shown in the same Appendix, theseequations can be formally solved by means of a double asymptotic expansion performed in termsof the microstructural size.
Acknowledgments
AB and LM gratefully acknowledge financial support from the Italian Ministry of Education, Uni-versity and Research in the framework of the FIRB project 2010 "Structural mechanics modelsfor renewable energy applications". AP would like to acknowledge financial support from theEuropean Union’s Seventh Framework Programme FP7/2007-2013/ under REA Grant agreementnumber PCIG13-GA-2013-618375-MeMic.
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A Higher-order analysis and averaged equations of infiniteorder
In this Appendix, explicit expressions for the higher order cells problems associated to the down-scaling relations (12), (13) and (14) are reported. Moreover, the averaged field equations of infiniteorder are derived, and a formal solution is obtained by means of an asymptotic expansion of themacroscopic fields in terms of the microstructural size.
A.1 Higher-order asymptotic analysis and derivation of the correspond-ing cell problems
In order to derive exact expressions for the fluctuation functions affecting the behavior of the micro-scopic fields u k , θ, η , the down-scaling relations (12), (13) and (14) are substituted into the micro-scopic field equations (7), (8). Remembering the property ∂∂x j f ( x , ξ = x ε ) = (cid:16) ∂f∂x j + ε ∂f∂ξ j (cid:17) ξ = x /ε = (cid:16) ∂f∂x j + f ,j ε (cid:17) ξ = x /ε , equation (7) become ε − (cid:26)(cid:20)(cid:16) C εijkl N (1) kpq ,l (cid:17) ,j + C εijpq ,j (cid:21) H pq ( x ) + (cid:20)(cid:16) C εijkl ˜ N (1) k,l (cid:17) ,j − α εij,j (cid:21) Θ( x )+ (cid:20)(cid:16) C εijkl ˆ N (1) k,l (cid:17) ,j − β εij,j (cid:21) Υ( x ) (cid:27) + ε (cid:26)(cid:16) C εijkl N (2) kpq q ,l (cid:17) ,j + 12 (cid:18)(cid:16) C εijkq N (1) kpq (cid:17) ,j + C εiq pq + C εiq kl N (1) kpq ,l + (cid:16) C εijkq N (1) kpq (cid:17) ,j + C εiq pq + C εiq kl N (1) kpq ,l (cid:19) κ pq q ( x )+ (cid:20)(cid:16) C εijkl ˜ N (2) kq ,l (cid:17) ,j + (cid:16) C εijkq ˜ N (1) k (cid:17) ,j + C εiq kl ˜ N (1) k,l − α εiq − (cid:16) α εij M (1) q (cid:17) ,j (cid:21) ∂ Θ ∂x q + (cid:20)(cid:16) C εijkl ˆ N (2) kq ,l (cid:17) ,j + (cid:16) C εijkq ˆ N (1) k (cid:17) ,j + C εiq kl ˆ N (1) k,l − β εiq − (cid:16) β εij W (1) q (cid:17) ,j (cid:21) ∂ Υ ∂x q (cid:27) + · · · · · · + b i ( x ) = 0 , i = 1 , , (48)27here H pq = ∂U p /∂x q are the components of the macroscopic displacement gradient tensor pre-viously defined, and κ pq q = ∂ U p /∂x q ∂x q are the elements of the macroscopic second gradienttensor. Equations (8) assume the following form ε − (cid:20)(cid:16) K εij M (1) q ,j (cid:17) ,i + K εiq ,i (cid:21) ∂ Θ ∂x q + ε (cid:20)(cid:16) K εij M (2) q q ,j (cid:17) ,i + 12 (cid:18)(cid:16) K εiq M (1) q (cid:17) ,i + K εq q + K εq j M (1) q ,j + (cid:16) K εiq M (1) q (cid:17) ,i + K εq q + K εq j M (1) q ,j (cid:19)(cid:21) ∂ Θ ∂x q ∂x q + · · · · · · + r ( x ) = 0 , (49) ε − (cid:20)(cid:16) D εik W (1) q ,j (cid:17) ,i + D εiq ,i (cid:21) ∂ Υ ∂x q + ε (cid:20)(cid:16) D εij W (2) q q ,j (cid:17) ,i + 12 (cid:18)(cid:16) D εiq W (1) q (cid:17) ,i + D εq q + D εq j W (1) q ,j + (cid:16) D εiq W (1) q (cid:17) ,i + D εq q + D εq j W (1) q ,j (cid:19)(cid:21) ∂ Υ ∂x q ∂x q + · · · · · · + s ( x ) = 0 . (50)In order to transform the field equations (48), (49) and (50) in a PDEs system with constantcoefficients, in which the unknowns are the macroscopic quantities U k ( x ) , Θ( x ) and Υ ( x ) , thefluctuation functions have to satisfy non-homogeneous equations ( cell problems ) reported below.At the order ε − from the equations (48), (49), (50) we derive the first-order cell problems reported in Sec. 3.2 in the text of the paper, equations (21) and (22).At the order ε , equation (48) yields the following second-order cell problems (cid:16) C εijkl N (2) kpq q ,l (cid:17) ,j + 12 (cid:20)(cid:16) C εijkq N (1) kpq (cid:17) ,j + C εiq pq + C εiq kl N (1) kpq ,l + (cid:16) C εijkq N (1) kpq (cid:17) ,j + C εiq pq + C εiq kl N (1) kpq ,l (cid:21) = n (2) ipq q , (cid:16) C εijkl ˜ N (2) kq ,l (cid:17) ,j + (cid:16) C εijkq ˜ N (1) k (cid:17) ,j + C εiq kl ˜ N (1) k,l − α εiq − (cid:16) α εij M (1) q (cid:17) ,j = ˜ n (2) iq , (cid:16) C εijkl ˆ N (2) kq ,l (cid:17) ,j + (cid:16) C εijkq ˆ N (1) k (cid:17) ,j + C εiq kl ˆ N (1) k,l − β εiq − (cid:16) β εij W (1) q (cid:17) ,j = ˆ n (2) iq (51)At the same order, from (49), (50) we derive the second-order thermodiffusive cell problems : (cid:16) K εij M (2) q q ,j (cid:17) ,i + 12 (cid:20)(cid:16) K εiq M (1) q (cid:17) ,i + K εq q + K εq j M (1) q ,j + (cid:16) K εiq M (1) q (cid:17) ,i + K εq q + K εq j M (1) q ,j (cid:21) = m (2) q q , (52)28 D εij W (2) q q ,j (cid:17) ,i + 12 (cid:20)(cid:16) D εiq W (1) q (cid:17) ,i + D εq q + D εq j W (1) q ,j + (cid:16) D εiq W (1) q (cid:17) ,i + D εq q + D εq j W (1) q ,j (cid:21) = w (2) q q , (53)where: n (2) ipq q = 12 (cid:28) C εiq pq + C εiq kl N (1) kpq ,l + C εiq pq + C εiq kl N (1) kpq ,l (cid:29) , ˜ n (2) iq = (cid:28) C εiq kl ˜ N (1) k,l − α εiq (cid:29) , ˆ n (2) iq = (cid:28) C εiq kl ˆ N (1) k,l − β εiq (cid:29) ,m (2) q q = 12 (cid:28) K εq q + K εq j M (1) q ,j + K εq q + K εq j M (1) q ,j (cid:29) ,w (2) q q = 12 (cid:28) D εq q + D εq j W (1) q ,j + D εq q + D εq j W (1) q ,j (cid:29) . (54)In general, for the m = 1 , , · · · , the m − order cell problems associate to equation (48) assume theform: (cid:16) C εijkl N ( m ) kpq ··· q m ,l (cid:17) ,j + 1 m ! X P ( q ) (cid:20)(cid:16) C εijkq m N ( m − kpq ··· q m − (cid:17) ,j + C εiq m kq m − N ( m − kpq ··· q m − + C εiq m kl N ( m − kpq ··· q m − ,l (cid:21) = n ( m ) ipq ··· q m , (cid:16) C εijkl ˜ N ( m ) kq ··· q m − ,l (cid:17) ,j + 1 m ! X P ( q ) (cid:20)(cid:16) C εijkq m − ˜ N ( m − kq ··· q m − (cid:17) ,j + C εiq m − kq m − ˜ N ( m − kq ··· q m − + C εiq m − kl ˜ N ( m − q ··· q m − ,l − α εiq m − M ( m − q ··· q m − − (cid:16) α εij M ( m − q ··· q m − (cid:17) ,j (cid:21) = ˜ n ( m ) iq ··· q m − , (cid:16) C εijkl ˆ N ( m ) kq ··· q m − ,l (cid:17) ,j + 1 m ! X P ( q ) (cid:20)(cid:16) C εijkq m − ˆ N ( m − kq ··· q m − (cid:17) ,j + C εiq m − kq m − ˆ N ( m − kq ··· q m − + C εiq m − kl ˆ N ( m − q ··· q m − ,l − β εiq m − W ( m − q ··· q m − − (cid:16) β εij W ( m − q ··· q m − (cid:17) ,j (cid:21) = ˆ n ( m ) iq ··· q m − , (55)whereas the m − order thermodiffusive cell problems corresponding to equations (49) and (50) are: (cid:16) K εij M ( m ) q ··· q m ,j (cid:17) ,i + 1 m ! X P ( q ) (cid:20)(cid:16) K εiq m M ( m − q ··· q m − (cid:17) ,i + K εq m q M ( m − q ··· q m − + K εq m j M ( m − q ··· q m − ,j (cid:21) = m ( m ) q ··· q m , (56) (cid:16) D εij W ( m ) q ··· q m ,j (cid:17) ,i + 1 m ! X P ( q ) (cid:20)(cid:16) D εiq m W ( m − q ··· q m − (cid:17) ,i + D εq m q W ( m − q ··· q m − + D εq m j W ( m − q ··· q m − ,j (cid:21) = w ( m ) q ··· q m , (57)29here the symbol P ( q ) denotes all possible permutations of the multi-index q , and the constants n ( m ) ipq ··· q m , ˜ n ( m ) iq ··· q m − , ˆ n ( m ) iq ··· q m − , m ( m ) q ··· q m , w ( m ) q ··· q m are defined as follows: n ( m ) ipq ··· q m = 1 m ! X P ( q ) (cid:28) C εiq m kq m − N ( m − kpq ··· q m − + C εiq m kl N ( m − kpq ··· q m − ,l (cid:29) , ˜ n ( m ) iq ··· q m − = 1 m ! X P ( q ) (cid:28) C εiq m − kq m − ˜ N ( m − kq ··· q m − + C εiq m − kl ˜ N ( m − q ··· q m − ,l − α εiq m − M ( m − q ··· q m − (cid:29) , ˆ n ( m ) iq ··· q m − = 1 m ! X P ( q ) (cid:28) C εiq m − kq m − ˆ N ( m − kq ··· q m − + C εiq m − kl ˆ N ( m − q ··· q m − ,l − β εiq m − W ( m − q ··· q m − (cid:29) ,m ( m ) q ··· q m = 1 m ! X P ( q ) (cid:28) K εq m q M ( m − q ··· q m − + K εq m j M ( m − q ··· q m − ,j (cid:29) ,w ( m ) q ··· q m = 1 m ! X P ( q ) (cid:28) D εq m q W ( m − q ··· q m − + D εq m j W ( m − q ··· q m − ,j (cid:29) . (58)The perturbation functions characterizing the down-scaling relations (12), (13), and (14) areobtained by the solution of the previously defined cells problems, derived by imposing the normaliza-tion conditions (15). According to Bakhvalov and Panasenko (1984) and Smyshlyaev and Cherednichenko(2000), the constants (54) and (58) are determined by imposing that the non-homogeneous terms inequations (55), (51), (52), (53), (56) and (57) (associated to the auxiliary body forces (Bacigalupo,2014), heat and mass sources) possess vanishing mean values over the unit cell Q . This impliesthe Q− periodicity of the perturbations functions N ( m ) kpq , ˜ N ( m ) kq , ˆ N ( m ) kq , M ( m ) q , W ( m ) q , and then thecontinuity and regularity of the microscopic fields (micro-displacements, micro-temperature andmicro-concentration) at the interface between adjacent cells are guaranteed. A.2 Averaged field equation of infinite order and its formal solution
Using the cell problems (21), (22), (55), (51), (52), (53), (56) and (57) together with the constantsdefinitions (23), (54) and (58) into the microscopic field equations (18), (19) and (20), the averagedequations of infinite order are derived: n (2) ipq q ∂ U p ∂x q ∂x q + ˜ n (2) iq ∂ Θ ∂x q + ˆ n (2) iq ∂ Υ ∂x q + + ∞ X n =0 ε n +1 X | q | = n +3 n ( n +3) ipq ∂ n +3 U p ∂x q + + ∞ X n =0 ε n +1 X | q | = n +2 ˜ n ( n +2) iq ∂ n +2 Θ ∂x q + + ∞ X n =0 ε n +1 X | q | = n +2 ˆ n ( n +2) iq ∂ n +2 Υ ∂x q + b i = 0 (59) m (2) q q ∂ Θ ∂x q ∂x q + + ∞ X n =0 ε n +1 X | q | = n +3 m ( n +3) q ∂ n +3 Θ ∂x q + r = 0 , (60) w (2) q q ∂ Υ ∂x q ∂x q + + ∞ X n =0 ε n +1 X | q | = n +3 w ( n +3) q ∂ n +3 Υ ∂x q + s = 0 , (61)30here q is a multi-index, ∂ n + j ( · ) /∂x q = ∂ n + j ( · ) /∂x q · · · ∂x q n + j with j ∈ N , n ( n +3) ipq = n ipq ··· q n +3 , ˜ n ( n +2) iq = ˜ n iq ··· q n +2 , ˆ n ( n +2) iq = ˆ n iq ··· q n +2 , m ( n +3) q = m ( n +3) q ··· q n +3 and w ( n +3) q = w ( n +3) q ··· q n +3 .A formal solution of the averaged field equations of infinite order (59), (60) and (61) is ob-tained by means of an asymptotic expansion of the macroscopic fields U i , Θ and Υ in terms of themicrostructural size ε , i.e. U i ( x ) = + ∞ X j =0 ε m U ( m ) i ( x ) , Θ( x ) = + ∞ X j =0 ε m Θ ( m ) ( x ) , Υ( x ) = + ∞ X j =0 ε m Υ ( m ) ( x ) . (62)By substituting the series (62) into (59), (60) and (61), a sequence of equations for determiningthe terms of the asymptotic expansion U ( m ) i , Θ ( m ) and Υ ( m ) is obtained. At the order ε , from theequation (59) we derive: n (2) ipq q ∂ U (0) p ∂x q ∂x q + ˜ n (2) iq ∂ Θ (0) ∂x q + ˆ n (2) iq ∂ Υ (0) ∂x q + b i = 0 . (63)whereas thermodiffusion equations (60) and (61) yield respectively m (2) q q ∂ Θ (0) ∂x q ∂x q + r = 0 , w (2) q q ∂ Υ (0) ∂x q ∂x q + s = 0 . (64)At the generic order m from (59) we obtain n (2) ipq q ∂ (2) U ( m ) p ∂x q ∂x q + ˜ n (2) iq ∂ Θ ( m ) ∂x q + ˆ n (2) iq ∂ Υ ( m ) ∂x q + m +2 X r =3 X | q | = r n ( r ) ipq ∂ r U ( m +2 − r ) p ∂x q + m +2 X r =3 X | q | = r − ˜ n ( r ) iq ∂ r − Θ ( m +2 − r ) ∂x q + m +2 X r =3 X | q | = r − ˆ n ( r ) iq ∂ r − Υ ( m +2 − r ) ∂x q = 0 , (65)and (60) and (61) are given by m (2) q q ∂ Θ ( m ) ∂x q x q + m +2 X p =3 X | h | = p m ( p ) h ∂ p Θ ( m +2 − p ) ∂x h = 0 , (66) w (2) q q ∂ Υ ( m ) ∂x q x q + m +2 X p =3 X | h | = p w ( p ) h ∂ p Υ ( m +2 − p ) ∂x h = 0 , (67)where h and q are multi-indexes. The solution of equations (63)-(67) requires that the followingnormalization conditions are satisfied: δL ˆ L U ( m ) p ( x ) d x = 0 , δL ˆ L Θ ( m ) ( x ) d x = 0 , δL ˆ L Υ ( m ) ( x ) d x = 0 , (68)where the L− periodic domain is the same defined in previous Section as L = [0 , L ] × [0 , δL ] .The averaged field equation (59), (60) and (61) (or alternatively the sequence of PDEs (63)-(67)), obtained by means of the proposed rigorous asymptotic procedure, are used in Sec. 4 of the31ext of the paper for deriving the field equation of the first order (Cauchy) homogeneous continuumequivalent to the considered periodic thermodiffusive material.The approximation of the average field equations (59)-(61) yielded by solution of homogenizeddifferential problems of generic order m (65) is more accurate with respect to that obtained bythe assumption (24). This implies also a more precise approximation of the solution of the micro-scopic field equation (7)-(8) by means of the down-scaling relation (18), (19) and (20) involvingthe macroscopic field (62). As it is explained for periodic elastic composites in Peerlings and Fleck(2004) and Bacigalupo and Gambarotta (2012), the truncation of the average equations of infiniteorder (59)-(61) at a generic order m with the aim to derive higher order field equations for gener-alized thermodiffusive continua may lead to problems in which the symmetries of the higher orderelastic and thermodiffusive constants is not guaranteed. Moreover a loss of ellipticity of the govern-ing equations can be observed. Asymptotic-variational homogenization techniques similar to thoseillustrated in Smyshlyaev and Cherednichenko (2000) and Bacigalupo and Gambarotta (2012) rep-resent an appropriate and powerful tool in order to avoid these problems. The generalization ofthese methods to the case of elastic materials in presence of thermodiffusion is still missing inliterature. B Symmetry and positive definiteness of elastic and thermod-iffusive tensors
In this Appendix, the symmetry properties of the tensors of components n (2) ipq q , m (2) q q , w (2) q q , andthe ellipticity of the field equations (63) and (64) are demonstrated. B.1 Symmetry and positive definiteness of tensor of components n (2) ipq q (vs. C iq pq ) Let us consider the cell problem (21) , remembering that n (1) ipq = 0 , it becomes (cid:16) C mijkl N (1) kpq ,l (cid:17) ,j + C mijpq ,j = 0 , (69)where C mijkl are Q− periodic functions. The weak form of equation (69), using N (1) riq as Q− periodictest function, is given by (cid:28)(cid:16) C mijkl N (1) kpq ,l + C mijpq (cid:17) ,j N (1) riq (cid:29) = 0 , (70)applying the divergence theorem to (70), and remembering that for the Q− periodicity of C mijkl and N (1) riq the path integrals evaluated on the boundary of the unit cell Q vanish, we obtain: D(cid:16) C mijkl N (1) kpq ,l + C mijpq (cid:17) N (1) riq ,j E = 0 . (71)32sing the result (71), expression (54) can be written in the equivalent form: n (2) ipq q = 12 (cid:28) (cid:16) C εiq pq + C εiq kl N (1) kpq ,l (cid:17) + (cid:16) C εiq pq + C εiq kl N (1) kpq ,l (cid:17) (cid:29) = 12 D C εiq pq + C εiq kl N (1) kpq ,l + (cid:16) C mijkl N (1) kpq ,l + C mijpq (cid:17) N (1) riq ,j + C εiq pq + C εiq kl N (1) kpq ,l + (cid:16) C mrjkl N (1) kpq ,l + C mrjpq (cid:17) N (1) riq ,j E = 12 (cid:20) D C mrjkl (cid:16) N (1) riq ,j + δ ri δ jq + N (1) rq i,j + δ rq δ ij (cid:17) · (cid:16) N (1) kpq ,l + δ kp δ lq + N (1) kq p,l + δ kq δ lp (cid:17)E +14 D C mrjkl (cid:16) N (1) riq ,j + δ ri δ jq + N (1) rq i,j + δ rq δ ij (cid:17) · (cid:16) N (1) kpq ,l + δ kp δ q l + N (1) kq p,l + δ kq δ lp (cid:17)Ei , (72)as a consequence, we can observe that: n (2) ipq q = 12 ( C iq pq + C iq pq ) , (73)where the components C iq pq of the overall elastic tensor take the form: C iq pq = 14 D C mrjkl (cid:16) N (1) riq ,j + δ ri δ jq + N (1) rq i,j + δ rq δ ij (cid:17) · (cid:16) N (1) kpq ,l + δ kp δ lq + N (1) kq p,l + δ kq δ lp (cid:17)E , (74)Observing expression (74), it is easy to deduce that the tensor of components C iq pq is symmetricand positive definite. B.2 Symmetry and positive definiteness of tensors of components m (2) q q and w (2) q q (vs. K q q and D q q ) Remembering that m (1) q = 0 , the cell problems (22) , possesses the form (cid:16) K mij M (1) q ,j (cid:17) ,i + K miq ,i = 0 (75)where K mij are Q− periodic functions. The weak form of equation (75), using M (1) q as Q− periodictest function, is given by (cid:28)(cid:16) K mij M (1) q ,j + K miq (cid:17) ,i M (1) q (cid:29) = 0 , (76)applying the divergence theorem to (76), and remembering that for the Q− periodicity of K mij and M (1) q the path integrals evaluated on the boundary of the unit cell Q vanish, we obtain: D(cid:16) K mij M (1) q ,j + K miq (cid:17) M (1) q ,i E = 0 . (77)33sing the result (77), expression (54) (4) can be written in the equivalent form: m (2) q q = 12 (cid:28) ( K mq q + K mq j M (1) q ,j ) + ( K mq q + K mq j M (1) q ,j ) (cid:29) = 12 D K mq q + K mq j M (1) q ,j + (cid:16) K mij M (1) q ,j + K miq (cid:17) M (1) q ,i + K mq q + K mq j M (1) q ,j + (cid:16) K mij M (1) q ,j + K miq (cid:17) M (1) q ,i E = 12 hD K mij ( M (1) q ,i + δ iq )( M (1) q ,j + δ q j ) E + D K mji ( M (1) q ,i + δ iq )( M (1) q ,j + δ q j ) Ei = D K mij ( M (1) q ,i + δ iq )( M (1) q ,j + δ q j ) E (78)as a consequence, we can observe that m (2) q q = K q q , i.e. K q q = D K mij ( M (1) q ,i + δ iq )( M (1) q ,j + δ q j ) E . (79)Observing expression (79), it is easy to deduce that the tensor of components K q q is symmetricand positive definite. Since the the equations of heat and mass diffusion possess an identical form,the components of the tensors K q q and D q q have the same properties, and then the resultsobtained for the components of the overall heat conduction tensor can be extended to the case ofthe overall mass diffusion tensor of components D q q . These components are given by the followingexpression: D q q = D D mij ( W (1) q ,i + δ iq )( W (1) q ,j + δ q j ) E . (80) C Overall elastic and thermodiffusive constants for bi-phaseisotropic layered materials
In this Appendix the explicit expressions for the overall elastic and thermodiffusive constant of abi-phase layered material with isotropic phases are reported. The components of the overall elastictensor take the form: C = − ζ ˜ E a ˜ E b + ζ [( ˜ E a ˜ ν b ) − E a ˜ ν a ˜ E b ˜ ν b + ( ˜ E b ˜ ν a ) − ( ˜ E a ) − ( ˜ E b ) ] − ˜ E a ˜ E b ( ζ + 1)[ ζ ( ˜ E b (˜ ν a ) − ˜ E b ) + ˜ E a (˜ ν b ) − ˜ E a ] ; C = − ( ζ + 1) ˜ E a ˜ E b ζ ( ˜ E b (˜ ν a ) − ˜ E b ) + ˜ E a (˜ ν b ) − ˜ E a ; C = ( ζ + 1) ˜ E a ˜ E b
2[ ˜ E a + ˜ E a ˜ ν b + ζ ( ˜ E b ˜ ν a + ˜ E b )] ; C = − ˜ E a ˜ E b (˜ ν b + ζ ˜ ν a ) ζ ( ˜ E b (˜ ν a ) − ˜ E b ) + ˜ E a (˜ ν b ) − ˜ E a . (81)34he components of the overall thermal dilatation tensor and diffusive expansion tensor are respec-tively given by α = A α a − B α a ∆ ; α = ζ ( ˜ E b (˜ ν a ) − ˜ E b ) α a + ˜ E a ((˜ ν b ) − α b ζ ˜ E b ((˜ ν a ) −
1) + ˜ E a ((˜ ν b ) −
1) ; (82) β = A β a − B β a ∆ ; β = ζ ( ˜ E b (˜ ν a ) − ˜ E b ) β a + ˜ E a ((˜ ν b ) − β b ζ ˜ E b ((˜ ν a ) −
1) + ˜ E a ((˜ ν b ) −
1) ; (83)where: A = ζ [ ˜ E b (˜ ν a ) − ˜ E b ] + ζ [ ˜ E a (˜ ν b ) − ˜ E b ˜ ν b + ˜ E b ˜ ν b (˜ ν a ) + ˜ E a ˜ ν a − ˜ E a ˜ ν a (˜ ν b ) − ˜ E a ]; B = ζ [ ˜ E a ˜ ν a (˜ ν b ) − ˜ E b + ˜ E b (˜ ν a ) + ˜ E b ˜ ν b − ˜ E b ˜ ν b (˜ ν a ) − ˜ E a ˜ ν a ] + ˜ E a (˜ ν b ) − ˜ E a ;∆ = ( ζ + 1)[ ζ ˜ E b ((˜ ν a ) −
1) + ˜ E a ((˜ ν b ) − . (84)Finally, the components of the overall heat conduction and mass diffusion tensors become K = K b − ζK a ζ + 1 , K = ( ζ + 1) K a K b K a + ζK b ; (85) D = D b − ζD a ζ + 1 , D = ( ζ + 1) D a D b D a + ζD b . (86) D Down-scaling relations vs cells problems
In this Appendix, we provide more details regarding the structure of the down-scaling relations(12), (13) and (14) and of the related cells problems. Following the approaches proposed byBensoussan et al. (1978); Bakhvalov and Panasenko (1984); Allaire (1992); Boutin and Auriault(1993); Meguid and Kalamkarov (1994) and Boutin (1996), the microscopic fields can be repre-sented through an asymptotic expansion in the general form: u h (cid:16) x , x ε (cid:17) = + ∞ X l =1 ε l u ( l ) h (cid:16) x , x ε (cid:17) = u (0) h (cid:16) x , x ε (cid:17) + εu (1) h (cid:16) x , x ε (cid:17) + ε u (2) h (cid:16) x , x ε (cid:17) + · · · , (87) θ (cid:16) x , x ε (cid:17) = + ∞ X l =1 ε l θ ( l ) (cid:16) x , x ε (cid:17) = θ (0) (cid:16) x , x ε (cid:17) + εθ (1) (cid:16) x , x ε (cid:17) + ε θ (2) (cid:16) x , x ε (cid:17) + · · · , (88) η (cid:16) x , x ε (cid:17) = + ∞ X l =1 ε l η ( l ) (cid:16) x , x ε (cid:17) = η (0) (cid:16) x , x ε (cid:17) + εη (1) (cid:16) x , x ε (cid:17) + ε η (2) (cid:16) x , x ε (cid:17) + · · · . (89)35ubstituting expressions (87), (88) and (89) into the microscopic field equations (7), (8), and re-membering the property ∂∂x j f ( x , ξ = x ε ) = (cid:16) ∂f∂x j + ε ∂f∂ξ j (cid:17) ξ = x /ε = (cid:16) ∂f∂x j + f ,j ε (cid:17) ξ = x /ε , we obtain ε − (cid:16) C εijhl u (0) h,l (cid:17) ,j + ε − (" C εijhl ∂u (0) h ∂x l + u (1) h,l ! ,j + (cid:16) C εijhl u (0) h,l (cid:17) ,j − (cid:16) α εij θ (0) (cid:17) ,j − (cid:16) β εij η (0) (cid:17) ,j ) + ε (" C εijhl ∂u (1) h ∂x l + u (2) h,l ! ,j + ∂∂x j " C εijhl ∂u (0) h ∂x l + u (1) h,l ! − (cid:16) α εij θ (1) (cid:17) ,j − ∂∂x j (cid:16) α εij θ (0) (cid:17) − (cid:16) β εij η (1) (cid:17) ,j − ∂∂x j (cid:16) β εij η (0) (cid:17) ) + ε (" C εijhl ∂u (2) h ∂x l + u (3) h,l ! ,j + ∂∂x j " C εijhl ∂u (1) h ∂x l + u (2) h,l ! − (cid:16) α εij θ (2) (cid:17) ,j − ∂∂x j (cid:16) α εij θ (1) (cid:17) − (cid:16) β εij η (2) (cid:17) ,j − ∂∂x j (cid:16) β εij η (1) (cid:17) ) + · · · · · · + b i = 0 , i = 1 , , (90) ε − (cid:16) K εij θ (0) ,j (cid:17) ,i + ε − (cid:26)(cid:20) K εij (cid:18) ∂θ (0) ∂x j + θ (1) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:16) K εij θ (0) ,j (cid:17)(cid:27) + ε (cid:26)(cid:20) K εij (cid:18) ∂θ (1) ∂x j + θ (2) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:20) K εij (cid:18) ∂θ (0) ∂x j + θ (1) ,j (cid:19)(cid:21)(cid:27) + ε (cid:26)(cid:20) K εij (cid:18) ∂θ (2) ∂x j + θ (3) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:20) K εij (cid:18) ∂θ (1) ∂x j + θ (2) ,j (cid:19)(cid:21)(cid:27) + · · · · · · + r = 0 , (91) ε − (cid:16) D εij η (0) ,j (cid:17) ,i + ε − (cid:26)(cid:20) D εij (cid:18) ∂η (0) ∂x j + η (1) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:16) D εij η (0) ,j (cid:17)(cid:27) + ε (cid:26)(cid:20) D εij (cid:18) ∂η (1) ∂x j + η (2) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:20) D εij (cid:18) ∂η (0) ∂x j + η (1) ,j (cid:19)(cid:21)(cid:27) + ε (cid:26)(cid:20) D εij (cid:18) ∂η (2) ∂x j + η (3) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:20) D εij (cid:18) ∂η (1) ∂x j + η (2) ,j (cid:19)(cid:21)(cid:27) + · · · · · · + s = 0 . (92)At the order ε − from equation (90) we derive: (cid:16) C εijhl u (0) h,l (cid:17) ,j = f (0) i ( x ) , (93)whereas from heat conduction and mass diffusion equations (91) and (92) we get respectively: (cid:16) K εij θ (0) ,j (cid:17) ,i = g (0) ( x ) , (cid:16) D εij η (0) ,j (cid:17) ,i = h (0) ( x ) . (94)The interface conditions (9)-(11), expressed with respect to ξ , become: [[ u (0) h ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh C εijhl u (0) h,l n j ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (95)36 [ θ (0) ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh K εij θ (0) ,j n i ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (96) [[ η (0) ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh D εij η (0) ,j n i ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (97)where Σ is the representation of the interface Σ bewteen two different phases of the material inthe non-dimensional space of the variable ξ .At the order ε − equation (90) yields " C εijhl ∂u (0) h ∂x l + u (1) h,l ! ,j + ∂∂x j (cid:16) C εijhl u (0) h,l (cid:17) ,j − (cid:16) α εij θ (0) (cid:17) ,j − (cid:16) β εij η (0) (cid:17) ,j = f (1) i ( x ) , (98)at the same order, from equations (91) and (92) we obtain: (cid:20) K εij (cid:18) ∂θ (0) ∂x j + θ (1) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:16) K εij θ (0) ,j (cid:17) = g (1) ( x ) , (99) (cid:20) D εij (cid:18) ∂η (0) ∂x j + η (1) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:16) D εij η (0) ,j (cid:17) = h (1) ( x ) , (100)and the interface conditions are given by [[ u (1) h ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0; "" C εijhl ∂u (0) h ∂x l + u (1) h,l ! − α εij θ (0) − β εij η (0) ! n j ξ ∈ Σ = 0 , (101) [[ θ (1) ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0; (cid:20)(cid:20) K εij (cid:18) θ (1) ,j + ∂θ (0) ∂x j (cid:19) n i (cid:21)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (102) [[ η (1) ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0; (cid:20)(cid:20) D εij (cid:18) η (1) ,j + ∂η (0) ∂x j (cid:19) n i (cid:21)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 . (103)At the order ε , the cells problems associate to equation (90) assume the form: " C εijhl ∂u (1) h ∂x l + u (2) h,l ! ,j + ∂∂x j " C εijhl ∂u (0) h ∂x l + u (1) h,l ! − (cid:16) α εij θ (1) (cid:17) ,j − ∂∂x j (cid:16) α εij θ (0) (cid:17) − (cid:16) β εij η (1) (cid:17) ,j − ∂∂x j (cid:16) β εij η (0) (cid:17) = f (2) i ( x ) , (104)whereas the cells problems correspoding to equations (91) and (92) are: (cid:20) K εij (cid:18) ∂θ (1) ∂x j + θ (2) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:20) K εij (cid:18) ∂θ (0) ∂x j + θ (1) ,j (cid:19)(cid:21) = g (2) ( x ) , (105) (cid:20) D εij (cid:18) ∂η (1) ∂x j + η (2) ,j (cid:19)(cid:21) ,i + ∂∂x i (cid:20) D εij (cid:18) ∂η (0) ∂x j + η (1) ,j (cid:19)(cid:21) = h (2) ( x ) , (106)37nd the interface conditions assume the form: [[ u (2) h ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , "" C εijhl ∂u (1) h ∂x l + u (2) h,l ! − α εij θ (1) − β εij η (1) ! n j ξ ∈ Σ = 0 , (107) [[ θ (2) ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (cid:20)(cid:20) K εij (cid:18) θ (2) ,j + ∂θ (1) ∂x j (cid:19) n i (cid:21)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (108) [[ η (2) ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (cid:20)(cid:20) D εij (cid:18) η (2) ,j + ∂η (1) ∂x j (cid:19) n i (cid:21)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 . (109)At the order ε − , the solvibility conditions in the class of the functions Q− periodic with respectto the fast variable ξ implies that f (0) i ( x ) = g (0) ( x ) = h (0) ( x ) = 0 , then the cell problems (93)-(94)become: (cid:16) C εijhl u (0) h,l (cid:17) ,j = 0 , (cid:16) K εij θ (0) ,j (cid:17) ,i = 0 , (cid:16) D εij η (0) ,j (cid:17) ,i = 0 , (110)as a consequence, the solution of problems (110) does not depend by the fast variable ξ and then u (0) h ( x , ξ ) = U h ( x ) , θ (0) ( x , ξ ) = Θ( x ) and η (0) ( x , ξ ) = Υ( x ) .At the order ε − , the solvability conditions in the class of the functions Q− periodic with respectto the fast variable ξ together with the interface conditions (101)-(103) yield to h C εijhl,j i ∂U h ∂x l − h α εij,j i Θ (1) − h β εij,j i Υ = f (1) i ( x ) , (111) h K εij,i i ∂ Θ ∂x j = g (1) ( x ) , h D εij,i i ∂ Υ ∂x j = h (1) ( x ) . (112)For the Q− periodicity of the functions C εijhl , α εij , β εij , K εij and D εij , we have f (1) i ( x ) = g (1) ( x ) = h (1) ( x ) = 0 , and then at this order the solution of the fields equations assumes the form: u (1) h ( x , ξ ) = N (1) hpq ( ξ ) ∂U p ∂x q + ˜ N (1) h ( ξ )Θ( x ) + ˆ N (1) h ( ξ )Υ( x ) , (113) θ (1) ( x , ξ ) = M (1) q ( ξ ) ∂ Θ ∂x q , η (1) ( x , ξ ) = W (1) q ( ξ ) ∂ Υ ∂x q , (114)where N (1) hpq , ˜ N (1) h , ˆ N (1) h , M (1) q and W (1) q are the same fluctuations functions introduced in Section 3.Substituting expressions (113)-(114) into the cell problems (111)-(112) and considering the interfaceconditions (101)-(103), we derive: (cid:16) C εijhl N (1) hpq ,l (cid:17) ,j + C εijpq ,j = 0 , (cid:16) C εijhl ˜ N (1) h,l (cid:17) ,j − α εij,j = 0 , (cid:16) C εijhl ˆ N (1) h,l (cid:17) ,j − β εij,j = 0 , (115) (cid:16) K εij M (1) q ,j (cid:17) ,i + K εiq ,i = 0 , (cid:16) D εij W (1) q ,j (cid:17) ,i + W εiq ,i = 0 , (116)and then the interface conditions (101)-(103) become; [[ N (1) hpq ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh C εijhl (cid:16) N (1) hpq ,l + δ hp δ lq (cid:17) n j ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (117)38 [ ˜ N (1) h ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh(cid:16) C εijhl ˜ N (1) h,l − α εij (cid:17) n j ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (118) [[ ˆ N (1) h ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh(cid:16) C εijhl ˆ N (1) h,l − β εij (cid:17) n j ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (119) [[ M (1) q ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh K εij (cid:16) M (1) q ,j + δ q j (cid:17) n i ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (120) [[ W (1) q ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh D εij (cid:16) W (1) q ,j + δ q j (cid:17) n i ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 . (121)The solution of the cell problems (115)-(116) taking into account the interface conditions (117)-(121)provides the Q− periodic perturbation functions N (1) hpq , ˜ N (1) h , ˆ N (1) h , M (1) q and W (1) q .Taking into account the solvability conditions in the class of the functions Q− periodic withrespect to the fast variable ξ and the interface conditions (107)-(109), the cell problems (104)-(106)associate to the ε become: h C εiq pl + C εilhj N (1) hpq ,j i ∂ U p ∂x q ∂x l − h C εilhj ˜ N (1) h,j − α εil i ∂ Θ ∂x l − h C εilhj ˆ N (1) h,j − β εil i ∂ Υ ∂x l = f (2) i ( x ) , (122) D K εij (cid:16) M (1) q ,j + δ jq (cid:17)E ∂ Θ ∂x i ∂x q = g (2) ( x ) , D D εij (cid:16) W (1) q ,j + δ jq (cid:17)E ∂ Υ ∂x i ∂x q = h (2) ( x ) . (123)these cell problems possess a solution satisfying the conditions (107)-(109) in the form: u (2) h ( x , ξ ) = N (2) hpq q ∂ U p ∂x q ∂x q + ˜ N (2) hq ∂ Θ ∂x q + ˆ N (2) hq ∂ Υ ∂x q , (124) θ (2) ( x , ξ ) = M (2) q q ∂ Θ ∂x q ∂x q , η (2) ( x , ξ ) = W (2) q q ∂ Υ ∂x q ∂x q , (125)where N (2) hpq q , ˜ N (2) hq ˆ N (2) hq , M (2) q q and W (2) q q are second order fluctuations functions just introducedin Section 3. As a consequence, from the cell problems (122)-(123) we derive: (cid:16) C εijhl N (2) hpq q ,l (cid:17) ,j + (cid:16) C εijhq N (1) hpq (cid:17) ,j + C εiq pq + C εiq hj N (1) hpq ,j = h C εiq pq + C εiq hj N (1) hpq ,j i , (126) (cid:16) C εijhl ˜ N (2) hq ,l (cid:17) ,j + (cid:16) C εijhq ˜ N (1) h (cid:17) ,j + C εiq kj ˜ N (1) h,j − (cid:16) α εij M (1) q (cid:17) ,j − α εiq = h C εiq hj ˜ N (1) h,j − α εiq i , (127) (cid:16) C εijhl ˆ N (2) hq ,l (cid:17) ,j + (cid:16) C εijhq ˆ N (1) h (cid:17) ,j + C εiq kj ˆ N (1) h,j − (cid:16) β εij W (1) q (cid:17) ,j − β εiq = D C εiq hj ˆ N (1) h,j − β εiq E , (128) (cid:16) K εij M (2) q q ,j (cid:17) ,i + (cid:16) K εiq M (1) q (cid:17) ,i + K εiq (cid:16) M (1) q ,j + δ jq (cid:17) = D K εq j (cid:16) M (1) q ,j + δ jq (cid:17)E , (129) (cid:16) D εij W (2) q q ,j (cid:17) ,i + (cid:16) D εiq W (1) q (cid:17) ,i + D εiq (cid:16) W (1) q ,j + δ jq (cid:17) = D D εq j (cid:16) W (1) q ,j + δ jq (cid:17)E , (130)and then the interface conditions (107)-(109) become; [[ N (2) hpq q ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh(cid:16) C εijhl N (2) hpq q ,l + C εijhq N (1) hpq (cid:17) n j ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (131)39 [ ˜ N (2) hq ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh(cid:16) C εijhl ˜ N (2) hq ,l + C εijhq ˜ N (1) h − α εij M (1) q (cid:17) n j ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (132) [[ ˆ N (2) hq ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh(cid:16) C εijhl ˆ N (2) hq ,l + C εijhq ˆ N (1) h − β εij W (1) q (cid:17) n j ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (133) [[ M (2) q q ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh K εij (cid:16) M (2) q q ,j + M (1) q δ jq (cid:17) n i ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , (134) [[ W (2) q q ]] (cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 , hh S εij (cid:16) W (2) q q ,j + W (1) q δ jq (cid:17) n i ii(cid:12)(cid:12)(cid:12) ξ ∈ Σ = 0 . (135)The solution of the cell problems (126)-(130) taking into account the interface conditions (131)-(135)provides the Q− periodic perturbation functions N (2) hpq q , ˜ N (2) hq , ˆ N (2) hq , M (2) q q and W (2) q q .The general procedure here reported can be applied to higher order cell problems for derivingthe averaged field equations of infinite order, which assume the form: h C εiq pl + C εilhj N (1) hpq ,j i ∂ U p ∂x q ∂x l − h C εilhj ˜ N (1) h,j − α εil i ∂ Θ ∂x l − h C εilhj ˆ N (1) h,j − β εil i ∂ Υ ∂x l + h C εiq hq N (1) hpq + C εiq hl N (2) hpq q ,l i ∂ U p ∂x q ∂x q ∂x q + h C εiq hq ˜ N (1) h + C εiq hl ˜ N (2) hq ,l − α εiq M (1) q i ∂ Θ ∂x q ∂x q + h C εiq hq ˆ N (1) h + C εiq hl ˆ N (2) hq ,l − β εiq W (1) q i ∂ Υ ∂x q ∂x q + · · · · · · + b i ( x ) = 0 , (136) D K εq j (cid:16) M (1) q ,j + δ jq (cid:17)E ∂ Θ ∂x q ∂x q + D K εq q M (1) q + K εq j M (2) q q ,j E ∂ Θ ∂x q ∂x q ∂x q + · · · · · · + r ( x ) = 0 , (137) D D εq j (cid:16) W (1) q ,j + δ jq (cid:17)E ∂ Υ ∂x q ∂x q + D D εq q W (1) q + D εq j W (2) q q ,j E ∂ Υ ∂x q ∂x q ∂x q + · · · · · · + s ( x ) = 0 . (138)Note that applying the permutation of the saturated indexes, (136), (137) and (138) become iden-tical to the averaged field equations derived in Appendix A. The structure of the down-scalingrelations is defined by the solutions of the various cells problems associate to the different orders ofthe asymptotic expansion. These down-scaling relations assume the form: u h (cid:16) x , x ε (cid:17) = (cid:20) U h ( x ) + ε (cid:18) N (1) hpq ( ξ ) ∂U p ( x ) ∂x q + ˜ N (1) h ( ξ )Θ( x ) + ˆ N (1) h ( ξ )Υ( x ) (cid:19) ++ ε (cid:18) N (2) hpq q ( ξ ) ∂ U p ( x ) ∂x q ∂x q + ˜ N (2) hq ( ξ ) ∂ Θ( x ) ∂x q + ˆ N (2) hq ( ξ ) ∂ Υ( x ) ∂x q (cid:19) + · · · (cid:21) ξ = x /ε , (139) θ (cid:16) x , x ε (cid:17) = (cid:20) Θ( x ) + εM (1) q ( ξ ) ∂ Θ( x ) ∂x q + ε M (2) q q ( ξ ) ∂ Θ( x ) ∂x q ∂x q + · · · (cid:21) ξ = x /ε (140) η (cid:16) x , x ε (cid:17) = (cid:20) Υ( x ) + εW (1) q ( ξ ) ∂ Υ( x ) ∂x q + ε W (2) q q ( ξ ) ∂ Υ( x ) ∂x q ∂x q + · · · (cid:21) ξ = x /ε ..