JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 4, pp. 789-804, Warsaw 2005 ON HOMOGENIZED MODEL OF PERIODIC STRATIFIED MICROPOLAR ELASTIC COMPOSITES Stanisław J. Matysiak Institute of Hydrogeology and Engineering Geology, Faculty of Geology, University of Warsaw e-mail: s.j. matysiak @uw.edu.pl The paper deals with modelling problems of periodic stratified compo- sites with micropolar elastic components. By using the linear theory of micropolar elasticity and the homogenization method with microlo- cal parameters, a homogenizedmodel accounting certain local effects of stresses and coupled stresses is derived. From the obtained model, sys- tems of equations for the ”first” and the ”second” plane state of strain of the layered composites are presented. Key words: micropolar elasticity, displacement, rotation, periodically layered composite, homogenizedmodel 1. Introduction The theory ofmicropolar elasticity describes elastic bodies as a continuum of orientedparticleswhichmay rotate independently of thedisplacements.The basis of the conceptwas givenbyCosseratE. andF. (1909) and the theorywas developed bymany other authors (see, for instance Dyszlewicz, 2004; Eringen and Suhubi, 1964; Eringen, 1966, 1968; Nowacki, 1974, 1981). The investiga- tions connected with micropolar bodies had principally a theoretical nature, however important experimental results are given by Gauthier and Jahsman (1975), where methods of determinations of micropolar elastic constants are presented. The theory of micropolar elasticity can be applied to modelling of elastic solids with a microstructure, granular media, multimolecular bodies. Recently, the Cosserat theory has been applied to problems of geomechanics (see, for referencesAdhikary andGuo, 2002) and geophysics (Teisseyre, 1995). In the present work, the problem of modelling of periodically layered, mi- cropolar, elastic composites is considered. The basic unit (fundamental layer) 790 S.J.Matysiak is assumed to be composed of (n+1)-different micropolar, elastic, isotropic, homogeneous and centrosymmetric layers. Perfect bonding between the layers is assumed.Theconsiderationsarebasedon the linear theoryofmicropolar ela- sticity (Eringen and Suhubi, 1964; Eringen, 1966, 1968; Nowacki, 1974, 1981) and the homogenization procedure established by Woźniak (1986, 1987a,b), Matysiak and Woźniak (1987). The approach is based on some concepts of nonstandardanalysis combinedwith someapriori postulatedphysical assump- tions.Application of the homogenization procedure leads to equations given in terms of unknownmacrodisplacements, macrorotations as well as some extra unknowns calledmicrolocal parameters. Themacrodisplacements, macrorota- tions describemean values of deformations, and themicrolocal parameters are connectedwith some local values of deformation gradients, stresses and couple stresses in every component of composites. Thehomogenization procedurewas applied tomodelling of periodically layered fluid-saturated porous solids (Ka- czyński andMatysiak, 1988;Matysiak, 1992) anddiffusionprocesses in layered composites (Matysiak and Mieszkowski, 2001). The approach is summarized inMatysiak (2001). Starting fromequations ofmicropolar elasticity, a homogenizedmodelwith microlocal parameters for a three dimensional case is derived. The model is described by linear partial differential equations with constant coefficients for macrodisplacements andmacrorotations as well as by a system of linear alge- braic equations for microlocal parameters. Equations for plane problems of periodic two-layered composites are de- rived from three-dimensional models. In the considered case, microlocal pa- rameters are eliminated, and the plane problems are expressed in terms of macrodisplacements and macrorotaions. Finally, derivation of equations of homogeneous micropolar bodies, periodically layered elastic composites and homogeneous elastic solids is presented. 2. Preliminaries Consider anonhomogeneousmicrolocal elastic bodywhichoccupies a regu- lar region B in theEuclidean3-space referred to thefixedCartesian coordinate system x=(x1,x2,x3). The body in a natural (undeformed) configuration is composed of periodically repeated (n+1)-different homogeneous, isotropic, centrosymmetrical layers, see Fig.1. Let h1, . . . ,hn+1 be the thickness of each basic unit of the body, and h=h1+ . . .+hn+1. The axis x1 is assumed to be normal to the layering. Let α(r), β(r), λ(r), µ(r), γ(r), ε(r), r=1, . . . ,n+1 be On homogenized model of periodic... 791 thematerial constants of the subsequent layers. By ρ(r), J(r), r=1, . . . ,n+1, we denote themass densities and the densities of rotational inertia of the lay- ers. Let σ (r) ij ,µ (r) ij , i,j =1,2,3 be the stress tensors and coupled stress tensors in the layer of the rth kind. Let t denote time, u(x, t) = (u1,u2,u3)(x, t), andϕ(x, t)= (ϕ1,ϕ2,ϕ3)(x, t), denote the displacement and rotation vectors, respectively. Let γij, χij, i,j =1,2,3 be the components of the strain tensor and the curvature-twist tensor. Perfect bonding between the layers being the components of the composite is assumed. This assumption implies continuity of the displacement and rotation vectors, stress vector and the coupled stress vector on the interfaces (planes between the subsequent layers). Fig. 1. A scheme of the fundamental layer (basic unit) The system of equations of motion for the micropolar, isotropic, centro- symmetric, elastic layer of the rth kind takes the following form1 (Dyszlewicz, 2004; Nowacki, 1974, 1981) σ (r) ji,j+ρ (r)Xi−ρ (r)üi =0 i,j,k=1,2,3 εijkσ (r) jk +µ (r) ji,j +ρ (r)Yi−J (r)ϕ̈i =0 r=1, . . . ,n+1 (2.1) where εijk denotes the permutation symbol (Ricci’s alternator). The constitutive relations in the considered case of homogeneous, isotropic, centrosymmetric elastic layers being the composite components can bewritten as follows (Dyszlewicz, 2004; Nowacki, 1974, 1981) σ (r) ji =(µ (r)+α(r))γji+(µ (r)−α(r))γij +λ (r)γkkδij (2.2) µ (r) ji =(γ (r)+ε(r))χji+(γ (r)−ε(r))χij +β (r)χkkδij 1Summation convention holds with respect to the repeated indices, and ϕ,i ≡ ∂ϕ/∂xi, φ̇≡ ∂φ/∂t. 792 S.J.Matysiak where γji =ui,j −εkjiϕk χji =ϕi,j (2.3) and δij denotes the Kronecker delta. By using equations (2.1)-(2.3), the equations of motion can be expressed in the following integral (weak) form n+1∑ r=1 ∫ Br { [(µ(r)+α(r))(ui,j −εkjiϕk)+(µ (r)−α(r))(uj,i−εkijϕk)+ +λ(r)uk,kδij]νi,j −ρ (r)Xiνi+ρ (r)üiνi } dB =0 (2.4) n+1∑ r=1 ∫ Br { εijk[(µ (r)+α(r))(uk,j −εmjkϕm)+(µ (r)−α(r))(uj,k−εmkjϕm)+ +λ(r)um,mδkj]νi+ρ (r)Yiνi−J (r)üiνi+ − [(γ(r)+ε(r))ϕi,j +(γ (r)−ε(r))ϕj,i+β (r)ϕm,mδij]νi,j } dB=0 for all test functions νi(·) such that νi(·)|∂B =0, andwhereBr, r=1, . . . ,n+1 denotes the part of the region B occupied by the rth material. Since the body is assumed to be periodic, the material coefficients are h-periodic functions taking constant values in the subsequent layers of the body. 3. Homogenized models of periodic micropolar elastic composites To obtain a homogenized model of periodic stratified micropolar elastic composites described in Section 2, the homogenization procedure with be ap- plied.This approach, presented inpapersbyWoźniak (1986, 1987a,b) for ther- moelastic composites, is based on some concepts of the nonstandard analysis and some a priori postulated physical assumptions. In this paper, we shall derive equations of homogenized models omitting the presentation of mathematical assumptions and detailed calculations. Si- milarly to papers by Matysiak (1992), Woźniak (1987a), the components of the displacement vector ui(·) and rotation vector ϕi(·) are assumed in the form On homogenized model of periodic... 793 ui(x, t)=Ui(x, t)+fa(x1)qai(x, t) (3.1) ϕi(x, t)=Φi(x, t)+fa(x1)Qai(x, t) i=1,2,3 a=1, . . . ,n where fa(·) : R→R are know a priori h-periodic functions, called the shape functions, given inMatysiak andWoźniak (1987) fa(x1)=    x1− 1 2 δa for 0¬x1 ¬ δa δa(x1−h) δa−h − 1 2 δa for δa ¬x1 ¬h (3.2) fa(x1+h)= fa(x1) x1 ∈R and δa =h1+ . . .+ha a=1, . . . ,n (3.3) h=h1+ . . .+hn+1 The functions Ui,Φi are unknown functions interpreted as the components of macrodispacement,macrorotation.Theunknown functions qai(·),Qai(·) stand for the kinematical and rotational microlocal parameters, and they are related with the periodic structure of the body. Since |fa(x1)| < h for every x1 ∈ R, then for small h the underlined terms in equations (3.1) are small andwill be neglected (for exact explanation in terms of the nonstandard analysis see papers by Woźniak, 1986, 1987a,b). It is emphasized that f ′a(·) are not small and the terms involving f ′ a(·) cannot be neglected. So, we have ui,1 ≈Ui,1+f ′ a(x1)qai ui,β ≈Ui,β ϕi,1 ≈Φi,1+f ′ a(x1)Qai ϕi,β ≈Φi,β β=2,3 (3.4) Taking into account the tested functions in the form νi(x, t)=Vi(x, t)+fb(x1)Zbi(x, t) i=1,2,3 b=1, . . . ,n (3.5) and substituting equations (3.5) into (2.4) after some calculations similar to those given in Matysiak and Woźniak (1987), Matysiak ((1992) and Woźniak 794 S.J.Matysiak (1987a), the equations of the homogenized model with microlocal parameters are obtained in the form (i,j,k,m=1,2,3, a=1, . . . ,n) 〈µ+α〉Ui,jj + 〈λ+µ−α〉Uj,ij + 〈(µ+α)f ′ a(x1)〉qai,1+ + 〈(µ−α)f ′a(x1)〉qaj,jδ1i+ 〈λf ′ a(x1)〉qa1,i−〈µ+α〉εkjiΦk,j+ −〈µ−α〉εkijΦk,j + 〈ρ〉Xi−〈ρ〉Üi =0 (3.6) εijk[〈µ+α〉Uk,j + 〈µ−α〉Uj,k+ 〈(µ+α)f ′ a(x1)〉qakδ1j + + 〈(µ−α)f ′a(x1)〉qajδ1k −〈µ+α〉εmjkΦm−〈µ−α〉εmkjΦm]+ + 〈γ+ε〉Φi,jj + 〈γ−ε+β〉Φj,ji+ 〈(γ+ε)f ′ a(x1)〉Qai,1+ + 〈(γ−ε)f ′a(x1)〉Qaj,jδi1+ 〈βf ′ a(x1)〉Qa1,i+ 〈ρ〉Yi−〈J〉Φ̈i =0 and 〈(µ+α)f ′b(x1)〉Ui,1+ 〈(µ−α)f ′ b(x1)〉U1,i+ 〈λf ′ b(x1)〉Uk,kδi1+ + 〈(µ+α)f ′a(x1)f ′ b(x1)〉qai+ 〈(µ−α+λ)f ′ a(x1)f ′ b(x1)〉qa1δ1i+ −〈(µ+α)f ′b(x1)〉εk1iΦk−〈(µ−α)f ′ b(x1)〉εki1Φk =0 (3.7) 〈(γ+ε)f ′b(x1)〉Φi,1+ 〈(γ−ε)f ′ b(x1)〉Φ1,i+ 〈βf ′ b(x1)〉Φm,mδi1+ + 〈(γ+ε)f ′a(x1)f ′ b(x1)〉Qai+ 〈(γ−ε+β)f ′ a(x1)f ′ b(x1)〉Qa1δi1 =0 where the symbol 〈g〉 denotes 〈g〉 ≡ 1 h h∫ 0 g(x1) dx1 (3.8) for any h-periodic integrable function g(·). Equations (3.6) and (3.7) constitute a system of linear algebraic and par- tial differential equations for 6(n+1) unknowns Ui, Φi, qai, Qai, i = 1,2,3, a=1, . . . ,n. Equations (3.7) stand for a system of linear algebraic equations for microlocal parameters qai, Qai. By using equations (3.7), the microlocal parameters can be eliminated from equations (3.6), which leads to 6 linear partial differential equations with constant coefficients for the unknown ma- crodisplacements Ui andmacrorotations Φi. Using formulae (3.2), (3.3) and (3.8) for an arbitrary h-periodic function g(·) taking a constant value gr in the layer of the rth kind, r=1, . . . ,n+1, we have On homogenized model of periodic... 795 〈g〉= n+1∑ r=1 ηrgr 〈gf ′ a(x1)〉= a∑ r=1 ηrgr −ωa n+1∑ r=a+1 ηrgr (3.9) 〈gf ′a(x1)f ′ b(x1)〉= b∑ r=1 ηrgr−ωb a∑ r=b+1 ηrgr−ωaωb n+1∑ r=a+1 ηrgr b¬ a where ηr ≡ δr h ωa ≡ η1+ . . .+ηa 1− (η1+ . . .+ηa) r=1, . . . ,n+1 a=1, . . . ,n (3.10) Employing equations (3.9), all material constants in equations (3.6), (3.7) can be calculated by substituting the h-periodic functions α, β, λ, µ, γ, ε, ρ, J for function g(·). The components of the stress tensor σ (r) ji and the couple stress tensor µ (r) ji in the layers of the rth kind can be determinedby using equations (2.2), (2.3), (3.3) and (3.4). Thus, we have σ (r) ji =(µ (r)+α(r))[Ui,j +f ′ a(x1)qaiδ1j −εmjiΦm]+ + (µ(r)−α(r))[Uj,i+f ′ a(x1)qajδi1−εmijΦm]+λ (r)[Um,m+f ′ a(x1)qa1]δij (3.11) µ (r) ji =(γ (r)+ε(r))[Φi,j +f ′ a(x1)Qaiδ1j]+(γ (r)−ε(r))[Φj,i+f ′ a(x1)Qajδ1i]+ +β(r)[Φk,k+f ′ a(x1)Qai]δij a=1, . . . ,n; r=1, . . . ,n+1; i,j,k=1,2,3. 4. Plane problems of two-layered periodic micropolar composites 4.1. The ”first” plane state of strain Consider now amicropolar elastic stratified composite composed of perio- dically repeated two different layers. Moreover, we confine our attention on plane problems described by displacement and rotation vectors in the form u(x1,x2, t)= (u1(x1,x2, t),u2(1,x2, t),0) (4.1) ϕ(x1,x2, t)= (0,0,ϕ3(x1,x2, t)) 796 S.J.Matysiak In the considered case n= 1, so the set of shape functions is reduced to the following function f1(x1)=    x1− 1 2 h1 for 0¬x1 ¬h1 − ηx1 1−η + h1 1−η − 1 2 h1 for h1 ¬x1 ¬h (4.2) where η= h1 h (4.3) Using equations (3.8), (3.10), (3.11) and (4.3), the coefficients in equations (3.6) and (3.7) can be written as follows (α̃, β̃, λ̃, µ̃, γ̃, ε̃, ρ̃, J̃)≡ (〈α〉,〈β〉,〈λ〉,〈µ〉,〈γ〉,〈ε〉,〈ρ〉,〈J〉) = = η(α1,β1,λ1,µ1,γ1,ε1,ρ1,J1)+(1−η)(α2,β2,λ2,µ2,γ2,ε2,ρ2,J2) ([α], [β], [λ], [µ], [γ], [ε]) ≡ ≡ (〈αf ′1(x1)〉,〈βf ′ 1(x1)〉,〈λf ′ 1(x1)〉,〈µf ′ 1(x1)〉,〈γf ′ 1(x1)〉,〈εf ′ 1(x1)〉)= = η(α1−α2,β1−β2,λ1−λ2,µ1−µ2,γ1−γ2,ε1−ε2) (4.4) (α̂, β̂, λ̂, µ̂, γ̂, ε̂)≡ (〈α(f ′1(x1)) 2〉,〈β(f ′1(x1)) 2〉,〈λ(f ′1(x1)) 2〉, 〈µ(f ′1(x1)) 2〉,〈γ(f ′1(x1)) 2〉,〈ε(f ′1(x1)) 2〉)= = η(α1,β1,λ1,µ1,γ1,ε1)+ η2 1−η (α2,β2,λ2,µ2,γ2,ε2) Equations (3.6), (3.7) for the plane problems of the periodically two-layered micropolar composite (see, equations (4.1), (3.1), (3.4)) take the following form (δ=1,2) (µ̃+ α̃)U1,δδ +(λ̃+ µ̃− α̃)Uδ,δ1+([λ]+ [µ]− [α])q11,1+([µ]− [α])q1δ,δ + +2α̃Φ3,2+ ρ̃X1− ρ̃Ü1 =0 (µ̃+ α̃)U2,δδ +(λ̃+ µ̃− α̃)Uδ,δ2+([µ]+ [α])q12,1+[λ]q11,2−2α̃Φ3,1+ + ρ̃X2− ρ̃Ü2 =0 (4.5) (γ̃+ ε̃)Φ3,δδ +2α̃(U2,1−U1,2)−4α̃Φ3+2[α]q12+([γ]+ [ε])Q13,1+ + ρ̃Y3− J̃Φ̈3 =0 and (λ̂+2µ̂)q11 =−2[µ]U1,1− [λ]Uδ,δ (µ̂+ α̂)q12 =−([µ]+ [α])U2,1− ([µ]− [α])U1,2+2[α]Φ3 (4.6) (γ̂+ ε̂)Q13 =−([γ]+ [ε])Φ3,1 On homogenized model of periodic... 797 Byusing equations (4.6), themicrolocal parameters q11, q12,Q13 can be elimi- nated fromequations (4.5). It leads to the following equations for theunknown macrodisplacements U1, U2 andmacrotation Φ3 A1U1,11+A2U1,22+A3U2,21+A4Φ3,2+ ρ̃X1− ρ̃Ü1 =0 B1U2,11+B2U2,22+B3U1,12+B4Φ3,1+ ρ̃X2− ρ̃Ü2 =0 (4.7) C1Φ3,11+C2Φ3,22+C3Φ3+C4U2,1+C5U1,2+ ρ̃Y3− J̃Φ̈3 =0 where A1 = λ̃+2µ̃− ([λ]+2[µ])([λ]+2[µ]−2[α]) λ̂+2µ̂ A2 = µ̃+ α̃− ([µ]− [α])2 µ̂+ α̂ A4 =2 ( α̃− [α]([µ]− [α]) µ̂+ α̂ ) A3 = λ̃+ µ̃− α̃− [λ]([λ]+ [µ]− [α]) λ̂+2µ̂ − [µ]2− [α]2 µ̂+ α̂ B1 = µ̃+ α̃− ([µ]+ [α])2 µ̂+ α̂ B2 = λ̂+2µ̃− [λ]2 λ̂+2µ̂ B3 = λ̃+ µ̃− α̃− [µ]2− [α]2 µ̂+ α̂ − [λ]([λ]+2[µ]) λ̂+2µ̂ (4.8) B4 =2 ([µ]([µ]+ [α]) µ̂+ α̂ − α̃ ) C1 = γ̃+ ε̃− ([γ]+ [ε])2 γ̂+ ε̂ C2 = γ̃+ ε̃ C3 =−4 ( α̃− [α]2 µ̂+ α̂ ) C4 =2 ( α̃− [α]([µ]+ [α]) µ̂+ α̂ ) C5 =−2 ( α̃+ [α]([µ]− [α]) µ̂+ α̂ ) The components of stress and couple stress tensors in the layers of the rth kind, r = 1,2, can be obtained by using equations (3.11), (4.1), (4.2) and (4.6). Thus, we have σ (r) 11 =D1U1,1+D2U2,2 σ (r) 22 = b (r) 1 U1,1+b (r) 2 U2,2 σ (r) 12 =E1U1,2+E2U2,1+E3Φ3 σ (r) 33 = c (r) 1 U1,1+ c (r) 2 U2,2 σ (r) 21 = a (r) 1 U1,2+a (r) 2 U2,1+a (r) 3 Φ3 µ (r) 13 =F1Φ3,1 µ (r) 31 = d (r) 1 Φ3,1 µ (r) 23 = d (r) 2 Φ3,2 µ (r) 32 = d (r) 3 Φ3,2 (4.9) 798 S.J.Matysiak where (r=1,2) D1 =(λ (1)+2µ(1)) ( 1− [λ]+2[µ] λ̂+2µ̂ ) D2 =λ (1)− [λ](λ(1)+2µ(1)) λ̂+2µ̂ E1 =− µ(1)+α(1) µ̂+ α̂ ([µ]− [α])+µ(1)+α(1) E2 =(µ (1)+α(1)) ( 1− [µ]+ [α] µ̂+ α̂ ) E3 =(µ (1)+α(1)) ( 2[α] µ̂+ α̂ −1 ) +µ(1)−α(1) F1 =(γ (1)+ε(1)) ( 1− [γ]+ [ε] γ̂+ ε̂ ) (4.10) a (r) 1 =µ (r)+α(r)− µ(r)−α(r) µ̂+ α̂ ([µ]− [α])f(r) a (r) 2 =(µ (r)−α(r)) ( 1− [µ]+ [α] µ̂+ α̂ ) f(r) a (r) 3 =2 ( α(r)+ [α] µ̂+ α̂ (µ(r)−α(r))f(r) ) c (r) 1 = b (r) 1 b (r) 1 =λ (r) ( 1− [λ]+2[µ] λ̂+2µ̂ f(r) ) c (r) 2 =λ (r) ( 1− [λ] λ̂+2µ̂ f(r) ) b (r) 2 =λ (r)+2µ(r)− [λ]λ(r) λ̂+2µ̂ f(r) d (r) 2 = γ (r)+ε(r) d (r) 1 =(γ (r)−ε(r)) ( 1− [γ]+ [ε] γ̂+ ε̂ f(r) ) d (r) 3 = γ (r)−ε(r) and f(r) =    1 for r=1 − η 1−η for r=2 (4.11) Equations (4.7) and (4.9) with the constant coefficients described by (4.8), (4.10) constitute the governing systemof equations for thehomogenizedmodel withmicrolocal parameters ofmicropolar layered composites in theplane state of strain. On homogenized model of periodic... 799 Remark. It shouldbe emphasized that the continuity conditions on interfaces of the stress vector (σ (r) 11 ,σ (r) 12 ,0) and the couple stress vector (0,0,µ (r) 13 ), r=1,2, are satisfied (see, equations (4.9)). 4.2. The ”second” state of strain Consider now the ”second” state of strain described by the displacement and rotation vectors in the form u(x, t)= (0,0,u3(x1,x2, t)) (4.12) ϕ(x, t)= (ϕ1(x1,x2, t),ϕ2(x1,x2, t),0) Using equations (3.6), (3.7), (4.3), (4.4), (4.12) and (3.1), we obtain the follo- wing equations of motion for the considered case of the strain state (µ̃+ α̃)(U3,11+U3,22)+([µ]+ [α])q13,1+2α̃(Φ2,1−Φ1,2)+ ρ̃X3− ρ̃Ü3 =0 (2γ̃+ β̃)Φ1,11+(γ̃+ ε̃)Φ1,22+(γ̃− ε̃+ β̃)Φ2,12+2α̃U3,2−4α̃Φ1+ +(2[γ]+ [β])Q11,1+([γ]− [ε])Q12,2+ ρ̃Y1− J̃Φ̈1 =0 (4.13) (γ̃+ ε̃)Φ2,11+(2γ̃+ β̃)Φ2,22+(γ̃− ε̃+ β̃)Φ1,12−2α̃U3,1−4α̃Φ2+ +([γ]+ [ε])Q12,1+[β]Q11,2+ ρ̃Y2− J̃Φ̈2 =0 and q13 = 1 µ̂+ α̂ { −([µ]+ [α])U3,1−2[α]Φ2 } Q11 = 1 2γ̂+ β̂ { −(2[γ]+ [β])Φ1,1− [β]Φ2,2 } (4.14) Q12 = 1 γ̂+ β̂ { −([γ]+ [ε])Φ2,1− ([γ]− [ε])Φ1,2 } Eliminating themicrolocal parameters q13,Q11,Q12 from equations (4.13) by using (4.14), we obtain a system of equations for the macrodisplacement U3 andmacrorotations Φ1, Φ2 in the form A∗1U3,11+A ∗ 2U3,22+A ∗ 3Φ1,2+A ∗ 4Φ2,1+ ρ̃X3−ρÜ3 =0 B∗1Φ1,11+B ∗ 2Φ2,12+B ∗ 3Φ1,22+B ∗ 4U3,2+B ∗ 5Φ1+ ρ̃Y1− J̃Φ̈1 =0 (4.15) C∗1Φ2,11+C ∗ 2Φ1,12+C ∗ 3Φ2,22+C ∗ 4U3,1+C ∗ 5Φ2+ ρ̃Y2− J̃Φ̈2 =0 800 S.J.Matysiak where A∗1 = µ̃+ α̃− ([µ]+ [α])2 µ̂+ α̂ A∗2 = µ̃+ α̃ A∗4 =−2 (([µ]+ [α])[α] µ̂+ α̂ − α̃ ) A∗3 =−2α̃ B∗1 =2γ̃+ β̃− (2[γ]+ [β])2 2γ̂+ β̂ B∗4 =2α̃ B∗2 = γ̃− ε̃+ β̃− (2[γ]+ [β])[β] 2γ̂+ β̂ − [γ]2− [ε]2 γ̂+ ε̂ B∗5 =−4α̃ B∗3 = γ̃+ ε̃− ([γ]− [ε])2 γ̂+ ε̂ C∗2 =B ∗ 2 C∗1 = γ̃+ ε̃− ([γ]+ [ε])2 γ̂+ ε̂ C∗4 =−2α̃ C∗3 =2γ̃+ β̃− [β]2 2γ̂+ β̂ C∗5 =−4α̃ (4.16) The components of stress and couple stress tensors in the layers of the rth kind, r = 1,2, for the ”second” plane state of strains, can be obtained by using equations (3.11), (4.3), (4.4), (4.11). Thus we have σ (r) 13 =D ∗ 1U3,1+D ∗ 2Φ2 σ (r) 31 = a ∗(r) 1 U3,1+a ∗(r) 2 Φ2 σ (r) 23 = b ∗(r) 1 U3,2+ b ∗(r) 2 Φ1 σ (r) 32 = c ∗(r) 1 U3,2+ c ∗(r) 2 Φ1 µ (r) 11 = d ∗ 1Φ1,1+d ∗ 2Φ2,2 µ (r) 12 = e ∗ 1Φ1,2+e ∗ 2Φ2,1 µ (r) 21 = f ∗(r) 1 Φ1,2+f ∗(r) 2 Φ2,1 µ (r) 22 = g ∗(r) 1 Φ1,1+g ∗(r) 2 Φ2,2 (4.17) where D∗1 ≡ (µ (1)+α(1)) ( 1− [µ]+ [α] µ̂+ α̂ ) b ∗(r) 1 =µ (r)+α(r) D∗2 ≡ 2α (1) − 2[α](µ(1) +α(1)) µ̂+ α̂ b ∗(r) 2 =−2α (r) a ∗(r) 1 =(µ (r)−α(r)) ( 1−f(r) [µ]+ [α] µ̂+ α̂ ) c ∗(r) 1 =µ (r)−α(r) a ∗(r) 2 =−2 ( µ(r)+ (µ(r)−α(r))[α] µ̂+ α̂ ) c ∗(r) 2 =2α (r) On homogenized model of periodic... 801 d∗1 =(2γ (1) +β(1)) ( 1− 2[γ]+ [β] 2γ̂+ β̂ ) d∗2 =β (1)− (2γ(1)+β(1))[β] 2γ̂+ β̂ (4.18) e∗1 ≡ γ (1)−ε(1)− [γ]− [ε] γ̂+ ε̂ (γ(1)+ε(1)) e∗2 ≡ (γ (1)+ε(1)) ( 1− [γ]+ [ε] γ̂+ ε̂ ) f ∗(r) 1 = γ (r)+ε(r)− (γ(r)−ε(r)) [γ]− [ε] γ̂+ ε̂ f(r) f ∗(r) 2 =(γ (r)−ε(r)) ( 1− [γ]+ [ε] γ̂+ ε̂ f(r) ) g ∗(r) 2 =2γ (r)+β(r)− (2[γ]+ [β])β(r) 2γ̂+ β̂ f(r) g ∗(r) 1 =β (r) ( 1− 2[γ]+ [β] 2γ̂+ β̂ f(r) ) r=1,2, and f(r) is in (4.11). Remark. It shouldbe emphasized that the continuity conditions on interfaces of the stress vector (0,0,σ (r) 13 ), and the couple stress vector (µ (r) 11 ,µ (r) 12 ,0), r=1,2, are satisfied (see, equations (4.16) and (4.17)). 5. Final remarks and conclusions Wehave investigated theproblemofmodelling of periodically layered com- posites composed of different, homogeneous, isotropic, centrosymmetrical lay- ers.Theobtainedhomogenizedmodel is given in terms ofmacrodisplacements, macrorotations as well as kinematical and rotational microlocal parameters. The microlocal parameters are determined by a system of linear algebraic equations (3.7), and they can be expressed by the macrodisplacements and macrorotations (for the ”first” plane problem we obtained equations (4.6), and equations (4.14) for the ”second” plane problem). Thus, the boundary value problems for the considered composites can be determined in terms of themacrodisplacements andmacrorotations described by a system of 6 linear partial differential equations with constant coefficients. 802 S.J.Matysiak From the obtained homogenized model of micropolar composites, we can pass to the following cases of elastic bodies: Case 1. Homogeneous micropolar bodies Assuming that the considered body is homogeneous, so α(r) =α β(r) =β γ(r) = γ λ(r) =λ µ(r) =µ ε(r) = ε ρ(r) = ρ J(r) = J (5.1) r=1, . . . ,n+1 and substituting (5.1) for the functions g(·) in equations (3.9) and (3.10), we obtain 〈α〉=α 〈β〉=β 〈γ〉= γ 〈λ〉=λ 〈µ〉=µ 〈ε〉= ε 〈ρ〉= ρ 〈J〉= J (5.2) and (a=1, . . . ,n) 〈αf ′a(x1)〉=0 〈βf ′ a(x1)〉=0 〈γf ′ a(x1)〉=0 〈µf ′a(x1)〉=0 〈λf ′ a(x1)〉=0 〈εf ′ a(x1)〉=0 (5.3) Thus, from equations (5.3), (3.7), it follows that qai =0 Qai =0 a=1, . . . ,n i=1,2,3 (5.4) and fromequations (5.4), (5.2), (3.6) and (3.11) we obtain equations ofmotion and constitutive relations for homogeneous, micropolar, isotropic and centro- symmetric bodies (Dyszlewicz, 2004; Nowacki, 1974, 1981). Case 2. Periodically layered elastic bodies In the case when α(r) =0 β(r) =0 γ(r) =0 ε(r) =0 J(r) =0 r=1, . . . ,n+1 (5.5) fromthe obtained results given in (3.6), (3.7), (3.11), we obtain a homogenized model of periodically layered, elastic composites composed of (n+1) different isotropic, homogeneous layers (see the result ofMatysiak andWoźniak, 1987). Also, from equations (4.7), (4.8) and (5.5) we pass to a system of equations for the plane state of strain for periodically two-layered composites (Kaczyński andMatysiak, 1987, 1988). On homogenized model of periodic... 803 Case 3. Homogeneous elastic bodies If we assume that λ(r) =λ µ(r) =µ ρ(r) = ρ (5.6) we obtain from (5.5), (5.6) and (3.6), (3.7), (3.11), equations of the classical theory of elasticity. The derived homogenized model with microlocal parameters for periodi- cally layered, micropolar composites creates a basis for considerations of bo- undary value problems for nonhomogeneous bodies. 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Sci., 35, 133-143 O homogenizowanym modelu periodycznych warstwowych sprężystych kompozytów mikropolarnych Streszczenie Praca dotyczy zagadnieńmodelowania periodycznych warstwowych kompozytów o składnikachmikropolarnych.Wykorzystując liniową teorię mikopolarnej sprężysto- ści i metodę homogenizacji z parametramimikrolokalnymi wyprowadzonomodel ho- mogenizowany uwzględniający pewne efekty lokalne w naprężeniach i naprężeniach momentowych. Z otrzymanegomodelu otrzymano układy równań dla ”pierwszego” i ”drugiego”płaskiego stanu odkształcenia dla periodyczniewarstwowychkompozytów mikropolarnych. Manuscript received March 14, 2005; accepted for print May 30, 2005