Homogenization methods for interface modeling in damaged masonry

Author manuscript, published in "Advances in Engineering Software 46, 1 (2012) 35-42" DOI : 10.1016/j.advengsoft.2010.09.009 Homogenization methods for interface modeling in damaged masonry A. Rekik1 , F. Lebon2 1 2 Institut PRISME, Polytech’Orléans, 8 rue Léonard de Vinci, 45072 Orléans Cedex 2 LMA CNRS UPR 7051, Université de Provence, 31, Chemin Joseph Aiguier, 13402 Marseille Cedex 20 hal-00652805, version 1 - 16 Dec 2011 Abstract The aim of the present study was to predict damage to masonry by combining structural analysis and homogenization methods. In the case of a masonry structure composed of bricks and mortar, a third material is assumed to exist, which is a mixture of the two materials sandwiched between the other two. This new layer has a small thickness, a low stiffness and a given damage ratio. The mechanical problem set by this masonry, which was initially a 3-D problem, is solved numerically in 2-D terms using finite element methods and modeling the three materials: brick, mortar and the interface material defined above. The properties of the third material are obtained by performing the following three steps: (i) first an exact homogenization of a brick/mortar laminate defining a first homogeneous equivalent medium (HEM-1) is performed. (ii) Secondly, we assume the HEM-1 to be damaged and apply the Kachanov model to assess the global behavior of the damaged HEM-1, thus defining a second equivalent homogeneous medium denoted HEM-2. (iii) Thirdly, asymptotic analysis is performed to model HEM-2 as an interface or a joint. The properties of this joint are deduced from those of the HEM-2 material. This interface is modeled numerically with connector finite elements. This method is applied to two cases: a triplet of full bricks and a triplet of hollow bricks subjected twice to shear loading. The numerical results obtained are compared with experimental data available in the literature. Key words: masonry structure, interface damage, homogenization, asymptotic analysis, finite element method. Preprint submitted to Computers and Structures July 23, 2009 hal-00652805, version 1 - 16 Dec 2011 1. Introduction Traditionally, as in ancient Greece and Egypt, people have always built their homes with natural materials available nearby, such as stones and bricks, in particular. Masonry is therefore one of the oldest construction materials. Masonry is still commonly used nowadays to build houses because of its qualities of strength, solidity, durability and fire resistance, and its elegant appearance, etc. However, masonry, which is not generally thought to be a highly technological material, shows highly complex behaviour, due in particular to the interactions between the components (mortar, bricks, concrete blocks and stones) and the anisotropy induced by the direction of the joints, which are a source of weakness. Masonry structures were classically designed on the basis of empirical rules. Modern virtual methods of design developed only quite recently. In particular, the structures built long ago were extremely stable because they were massive. In modern masonry buildings the walls are very thick for economic reasons and the stability has to be studied from the theoretical point of view, especially when wind or earthquakes are liable to occur. The strength of the masonry then becomes more critical and it is necessary to study the solidity of the structure using fine models and numerical simulations as in the case of concrete and steel structures. Other problems such as cracks are intolerable for the occupants of buildings, and preventing this problem also requires more detailed studies on the design of masonry structures. In this paper, we deal with the resistance of masonry to in-plane shear forces. It is classically held that the seismic vulnerability of masonry buildings depends strongly on their resistance to shear forces. It is therefore of great interest to model and test the shear responses of building components subjected to loading of this kind, especially cyclic loading. These responses have generally been characterized by a peak load, loss of rigidity and energy dissipation. To summarize considerably, two methods of modeling masonry structures have been used so far. The first method involves macroscopic models, in which the wall is assumed to be a single structural element characterized by a non linear response when it is exposed to shear forces [1, 2, 3, 4, 5, 6, 7, 37]. These models, which are generally based on the use of homogenization techniques, usually take the properties of the materials and the bonding between the various components into account [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. On the other hand, some models have been developed for predicting the 2 hal-00652805, version 1 - 16 Dec 2011 evolution of damage to the interface between two initially bonded deformable bodies [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 36, 38]. These models are based mostly on phenomenological approaches. In this paper, an original multi-level model for interfaces based on homogenization techniques is presented, which takes the characteristics of the masonry into account. The first part of this paper deals with the mechanical modeling approach used, and in particular with the model developed. The multi-level approach used is described. This approach takes the mechanical characteristics of the mortar and bricks, the presence of micro-cracks and the thickness of the interface into account. In the second part, the numerical procedure used and implemented using a finite element software program is presented and some numerical examples are given and compared with experimental data [11, 6]. Details of the computations are given in appendices A and B. 2. The method of analysis proposed In order to model interface damage to masonry structures, the present method based on homogenization theories, asymptotic techniques and finite element methods was developed. The main steps involved in this method will be described below. Most studies on masonry structures have dealt with only two materials: brick and mortar. In the present work, we assume the existence of a third material: a mixture of brick and mortar with a crack density ρ. To obtain the effective properties of the damaged intermediate material, three steps are performed. First we calculate the exact effective properties of the material devoid of cracks using homogenization techniques for laminate composites, and thus define a first homogeneous equivalent medium, which will be referred below as HEM-1. In the second step, we attribute the crack density ρ to the material HEM-1. To model the macroscopic behavior of the cracked material HEM-1, we use the Kachanov model and then define a new homogeneous equivalent medium HEM-2. This material sandwiched between the brick and mortar is taken to have a small thickness, and its mechanical behavior can therefore be obtained using asymptotic techniques to shift from the micro to macro level. With this interface law, the masonry structure problem can be solved using finite element methods. The following overall scheme (Fig.1) describes the principles underlying the model proposed: 3 hal-00652805, version 1 - 16 Dec 2011 Figure 1: Principle of the proposed model 2.1. First step: undamaged stratified composite homogenization The behavior of brick and mortar is assumed to be isotropic and linear, as described by the local law: 4 hal-00652805, version 1 - 16 Dec 2011 i i εijk = Sjkln σln , for j, k, l, n = 1, 2, 3 (1) Where S i is the compliance tensor of the phase i = b, m. The indexes b and m denote the brick and mortar phases respectively. Each fourth-order local 1 1 homogeneous phase compliance tensor S i reads: S i = i J + i K, where 3k 2µ Ei i the compressibility and the shear coefficients are given by: k = 3(1 − 2ν i ) i E and µi = , respectively. E i and ν i are the Young’s modulus and the 2(1 + ν i ) Poisson ratio, respectively, of each phase i . The fourth-order tensors J and K are given by the relations: J = 31 i ⊗ i and K = I − J, respectively, where i is the second-order identity tensor and I is the fourth-order one. The macroscopic behavior law of the laminate brick/mortar reads: X X ε̄ = S̃ 0 : σ̄ where ε̄ = f i εi = f iS i : σi (2) i=b, m i=b, m where f i (i = b, m) denotes the volume fractions of the phases, S̃ 0 is the effective fourth-order compliance tensor of the transversely isotropic crackfree laminate. According to the modified Voigt notation [23], the macroscopic law reads:     0  0 0 S̃1111 S̃1122 S̃1133 0 0 0 ε̄11 σ̄11  S̃ 0 0 0  ε̄22  σ̄22  0 0 0   1122 S̃2222 S̃1133        0 0 0     ε̄ σ̄ S̃ S̃ S̃ 0 0 0 33 33    √  =  1133 1133 3333   √  0  2ε̄23    2σ̄23   0 0 0 0 0 2 S̃ 1313  √   √  0  2ε̄13    2σ̄13    0 0 0 0 2 S̃ 0 1313 √ √ 0 2ε̄12 2σ̄12 0 0 0 0 0 2S̃1212 (3) 0 The unknown components of the compliance tensor S̃ are determined in Appendix A by applying four independent loads: axial tension, axisymmetric loading, plane traction combined with a plane shear load and a longitudinal shear load. With each loading condition, it is necessary to fulfill two conditions at the brick/mortar interface as regards: (i) the continuity of the normal stress vector σ.e3 , i.e.: b m σj3 = σj3 for every index j = 1, 2, 3 5 (4) and (ii) the continuity of the in-plane deformation components εbkj = εm kj for the couplets (k, j) = (1, 1), (2, 2) and (1, 2) (5) hal-00652805, version 1 - 16 Dec 2011 In [39], the macroscopic description of a periodic stratified composite is obtained differently, using the asymptotic expansion technique [40] up to third ³ ´−1 order. The expression for the macroscopic elastic tensor C̃ 0 = S̃ 0 is written in terms of the elastic properties of the layers and its volume fractions in the periodic cell. 2.2. Second step: homogenization of the damaged composite In this section, the material HEM-1 is assumed to contain an arbitrary crack orientation distribution. Kachanov and Mauge [15] provided an accurate approximation of the effective behaviour of the cracked material HEM-1. Assuming that cracks are rectilinear and located in the plane (e1 , e3 ), the initial 3−D problem can be reduced to a 2−D problem. In this case, it has been established [15] that the overall strain ε̄ in a solid with N cracks can be written in the form N 1X ε̄ = S̃ : σ̄ = S̃ : σ̄ + (n⊗ < b > + < b > ⊗n)(k) l(k) = (S̃ 0 + ∆S̃) : σ̄ A k=1 (6) 0 where S̃ is the effective compliance of the cracked material, S̃ is the compliance of the crack-free matrix, n(k) , 2l(k) and < b >(k) are the normal unit vector, the length of the kth crack and the average displacement discontinuity (COD) vector at the kth crack; and A is the representative averaging area. The potential of the cracked material is therefore the sum of two terms: 0 N 1 1 1X f (σ̄) = ε̄ : S̃ : ε̄ = σ̄ : S̃ 0 : σ̄ + (n.σ̄. < b >)(k) l(k) = f 0 (σ̄) + ∆f 2 2 A k=1 f 0 is the potential of the crack-free material and ∆f is the change of f due to the presence of cracks. The problem of finding the additional compliance ∆S̃ (relation (6)) due to cracks is reduced to finding the COD tensor B in terms of the average stress σ̄. The latter is defined by the relation [16, 17]: < b >= n.σ̄.B 6 (7) The components of B depend on the compliance matrix S̃ 0 and the orientation of the crack with respect to the matrix anisotropy axes. In the crack coordinate system (t, n), where t is a unit vector which is tangential to the crack so that: t = cosφ e1 + sinφ e3 n = −sinφ e1 + cosφ e3 hal-00652805, version 1 - 16 Dec 2011 B = Bnn n ⊗ n + Btt t ⊗ t + Bnt n ⊗ t + Btn t ⊗ n (8) (e1 , e3 ) are the principal axes of the matrix orthotropy and φ is the angle between the crack line and the e1 -principal axis. The expressions for the COD tensor (7) and (8) are used to obtain the change in f due to the presence of a family of non-interacting cracks for which t and n are the same: ∆f = ρσ̄ : n. {Bnn n ⊗ n + Btt t ⊗ t + Bnt (n ⊗ t + t ⊗ n)} .n : σ̄ (9) P where ρ = A1 k (l(k) )2 is the conventional scalar crack density parameter. In the considered case of an orthotropic matrix with one arbitrarily oriented family of parallel cracks (− π2 ≤ φ ≤ π2 ), B can be expressed in the following form, using the solution given by Lekhnitsky [19] and Savin [20].   Btt = C(1 − Dcos2φ)l Bnn = C(1 + Dcos2φ)l (10)  Btn = CD(sin2φ)l where  ! 21 p p à  0 0 0  E + E ν 2 π 1  3  p1 − 2 130 + p 0 0  C= 0 0 0 4 G E1 Ep E1 E3 13 1 E3 p 0 0   E1 − E3    D=p 0 p 0 E1 + E3 (11) 0 E10 , E30 , ν13 and G013 are the elastic engineering constants of the crack-free material HEM-1. These constants are deduced from the effective elastic 7 hal-00652805, version 1 - 16 Dec 2011 compliances S̃ 0 [19] as follows:  1 0   S̃1111 = 0   E1    1  0   S̃3333 = 0 E3 0 ν13  0  S̃1133 = −   E10    1  0   S̃1313 = 0 G13 (12) We note here the dependance of the sign of D on the anisotropy ratio E10 /E30 . Substituting B from (10) into (9), we obtain the elastic potential f , and therefore, all the moduli of the material HEM-2 (i.e. HEM-1 with cracks). On the principal axes e1 , e3 , the effective properties of HEM-2 are therefore given by:  E1 1   0 =  2 2  E1 1 + 2ρsin φ(Btt cos φ + Bnn sin2 φ − Bnt sin2φ)E10    1 E 3    0 = E3 1 + 2ρcos2 φ(Btt sin2 φ + Bnn cos2 φ + Bnt sin2φ)E30 (13) G 1 13   =   G013 1 + ρ(Bnn sin2 2φ + Btt cos2 2φ − Bnt sin4φ)G013     ν0 ν   13 = 130 E1 E1 The stiffness tensor C̃ of the material HEM−2 is obtained by performing inversion on the compliance tensor S̃ deduced from the elastic engineering constants E1 , E3 , ν13 and G13 (giving a relation similar to (12)). For the following applications, we assume the cracks to be parallel to the e1 -principal axis, i.e., φ = 0. In this case, the engineering constants defining the behavior of HEM-2 read: 0 E1 = E10 , E3 = E30 /(1 + 2ρBnn E30 ), G13 = G013 /(1 + ρBtt G013 ) and ν13 = ν13 where, in this case, Bnn = C(1 + D)l and Btt = C(1 − D)l. 2.3. Third step: asymptotic analysis and interface law In this section, we consider a thin joint composed of the material defined above, which is sandwiched between brick and mortar. Since the joint is 8 hal-00652805, version 1 - 16 Dec 2011 thin and soft, it is natural to use asymptotic techniques, to study the limit problem (when the thickness is equal to zero) and to replace the joint by an interface law defined along the limit surface. We take ε to denote the thickness of the joint, which is assumed to be constant and S to denote the limit surface of the joint (a line in 2D), corresponding to a thickness equal to zero. In what follows, we work in 2-dimensional terms (direction 1 and 3) in order to simplify the computations (see fig. 2). The details of the computations are given in appendix B. We take a to denote the elasticity tensor of the joint and lim aijkl /ε = āijkl . The limits are assumed to exist. Figure 2: the principle of the asymptotic techniques in a 2-D case We take [ ] to denote the jump along S. We obtain £ ¤ 0 σi3 = āi3i3 u0i We find an interface law which links the stress vector to the jump in the displacement via a diagonal matrix. In our case, the terms CN and CT 9 hal-00652805, version 1 - 16 Dec 2011 in this matrix (corresponding to the normal and tangential jumps in the displacement) are given by    CN = C̄3333 (ε → 0) where C̄3333 = C̃3333 ε (14)  C̃ 1313  C = C̄ (ε → 0) where C̄ T 1313 1313 = ε Using expressions ((13), (12)) and since the crack density scalar can be expressed in the form: ρ = l2 /ε L0 , where L0 denotes the joint length, it can be established that the normal and tangential joint stiffnesses read:  L0   CN = 2C(1 + Dcos2φ) l3 (15) L0   CT = 4C(1 − Dcos2φ) l3 The present model takes the evolution of the micro-crack into account by taking a variable crack half length l depending on the load. For the following applications and for the sake of simplicity, we first assume that the half length l depends only on the predominant tangential stress τ and we neglect its dependance on the normal stress. For its evolution, we assume that l remains constant (l = ξ1 L0 , where ξ1 is a scalar such as 0 < ξ1 < 1) until a certain value τ0 of the shear stress has been reached. From this value, the crack half length l evolves linearly with respect to the shear stress τ up to a second value of the crack length ξ2 L0 (ξ1 < ξ2 < 1) reached at the maximum shear stress value τmax . The following figure (Fig. 3) describes the evolution of the half crack length with respect to the shear stress τ : The first step (where l is constant) corresponds to a stable state of the interface material in which crack propagation occurs. The second step (where l evolves linearly as a function of the shear stress τ ) includes the crack propagation, which leads to the failure of the interface. The values of the shear stresses τ0 and τmax are fixed in advance, based on experimental ’stress-displacement’ diagrams obtained on various masonries subjected to shear conditions. The values of the crack evolution law parameters (ξ1 and ξ2 ) result from the minimization of the difference between the numerical and experimental ’stress-displacement’ diagrams. 10 hal-00652805, version 1 - 16 Dec 2011 Figure 3: Function describing the evolution of the crack half length with respect to the applied shear stress 3. Results and discussion The numerical evaluation of the model described above is restricted to the case of 2D plane stress in masonry structures subjected to shear loading. The obtained numerical results are then compared with experimental data available in the literature [36]. In this paper, two cases are considered: a triplet of full bricks and a triplet of hollow bricks, both subjected twice to the same nominal loading (with no lateral confinement) as shown in figure (4). Table (1) lists the material properties of the mortar and full bricks used to simulate a triplet of full bricks 11 Young’s moduli Poisson ratio of Young’s moduli Poisson ratio of Young’s moduli Poisson ratio of (MPa) of full brick full brick (MPa) of hollow brick hollow brick (MPa) of mortar mortar 9438 0.13 6059 0.13 4000 0.3 hal-00652805, version 1 - 16 Dec 2011 Table 1: Mechanical properties of the three-fold masonry constituents [36] under the shear loading conditions described (Fig. (4)). The properties of the full bricks given in this table were based on experiments by Fouchal et al [36]. Those of the hollow bricks were deduced using the self-consistent estimate [21, 22], based on the fact that the porosity ratio of the bricks equals 30%. These homogenized properties are in good agreement with the results of experiments performed in [36]. Case of the triplet of full bricks. The properties of the undamaged material are those of the equivalent behavior of the mortar and full brick sandwich (see table 1), assuming the existence of an equal volume fraction between these two materials (fm = fb = 0.5). In the case of this sandwich configuration, the constants αi (23, 25), β i (32, 33), γ i (37) are equal to: αb = −0.229, αm = 0.229, βb = −0.884, βm = −0.115, γb = 1.461, γm = 0.538 The compliance tensor S̃ 0 therefore reads:  1.478 −0.271 −0.348 0 0 0  −0.271 1.478 −0.348 0 0 0   −0.348 −0.348 1.639 0 0 0 10−4   0 0 0 4.444 0 0   0 0 0 0 4.444 0 0 0 0 0 0 3.499         (16) The engineering constants defining this undamaged homogeneous material (HEM-1) are therefore: 0 Ẽ10 = Ẽ20 = 6762.57 MPa, G̃012 = 5714.57 MPa, and ν̃12 = 0.183 0 0 Ẽ30 = 6099.53 MPa, G̃013 = G̃023 = 4497.12 MPa, and ν̃13 = ν̃23 = 0.236 12 To evaluate the engineering constants defining the homogeneous damaged material (HEM-2), we need to determine the coefficients C and D (11) which are independent of the crack length parameter 2 l: C = 4.223 10−4 and D = 0.0258 Lastly, the normal and tangential stiffness of the interface are given by the following expressions, respectively: hal-00652805, version 1 - 16 Dec 2011 CN = 242365/l3 (N/mm2 ) and CT = 255199/2 l3 (N/mm2 ) The experimental responses (blue dashed and pink dashed-dotted curves) are used to identify the parameters ξ1 and ξ2 as follows: ½ for the blue dashed curve: ξ1 L0 = 1.47 µm, ξ2 L0 = 1.484 µm for the pink dashed-dotted curve: ξ1 L0 = 1.397 µm, ξ2 L0 = 1.47 µm (17) The shear stresses values taken from the experimental ’stress-displacement’ curves and serving to construct a simplified ’two-slope’ crack evolution law, as depicted in Figure (3), are: ½ for the blue dashed curve: τ0 = 0.7 M P a, τmax = 0.83 M P a for the pink dashed-dotted curve: τ0 = 1.26 M P a, τmax = 1.6 M P a (18) Case of the triplet of hollow bricks. With this triplet of hollow bricks, the coefficients αi , β i and γ i , i = b, m, are equal to: αb = −0.168, αm = 0.168, βb = −0.717, βm = −0.282, γb = 1.270, γm = 0.729 The compliance tensor S̃ 0 of this homogeneous undamaged material HEM-1 is written as follows:   1.973 −0.396 −0.456 0 0 0  −0.396 1.973 −0.456 0 0 0     1.985 0 0 0  −4  −0.456 −0.456  10  (19)  0 0 0 5.115 0 0    0 0 0 0 5.115 0  0 0 0 0 0 4.740 13 The engineering constants are therefore: 0 Ẽ10 = Ẽ20 = 5066.57 MPa, G̃012 = 4219.43 MPa, and ν̃12 = 0.2 0 Ẽ30 = 5037.2 MPa, G̃013 = 3910.1 MPa, and ν̃13 = 0.23 The constants C and D (11) used to evaluate the components of the COD tensor B are: C = 5.232 10−4 and D = 14.53 10−4 The normal and tangential stiffness of the interface sandwiched between mortar and hollow brick is given by the following expressions: hal-00652805, version 1 - 16 Dec 2011 CN = 200396/l3 (N/mm2 ) and CT = 200980/2 l3 (N/mm2 ) The blue dashed experimental curve, which shows that failure occurs at the interface and not across the thickness of the mortar or across the hollow brick, makes it possible to identify the following two parameters ξ1 and ξ2 : ξ1 L0 = 1.60 µm and ξ2 L0 = 1.61 µm, obtained for the shear stresses τ0 = 0.63 M P a and τmax = 0.67 M P a, respectively. Fig.(4, (a)) shows the initial geometrical configuration of the three-fold masonry structure. The deformation and the ’stress-displacement’ relation of the triplet of full bricks under shear loading conditions are shown in Fig.((4, (b)) and Fig.(5), respectively. The numerical results obtained with the present model match the experimental data (curves denoted by 1-f and 2-f) and are in good agreement particularly with the test data obtained in 1 (curve 1-f). Fig.(6) shows the numerical results and the experimental data obtained on a triplet of hollow bricks under shear loading conditions. In the case of the experimental data denoted by 1-h, 2-h and 3-h, the failure occurred only across the mortar phase (figure (7), (b)) or the brick and mortar phases (figure (7), (a)). These experimental tests (1-h, 2-h and 3-h) do not show the failure occurring across the brick/mortar interface and were therefore not used to obtain our numerical results. The experimental data denoted 4-h show the ability of the present model to predict the failure process across the brick/mortar interface. The numerical results provided by the present model accurately describe the ’stress-displacement’ relationship in a masonry structure where the failure occurs at the brick-mortar interface. 14 hal-00652805, version 1 - 16 Dec 2011 Figure 4: Initial geometrical configuration and loading conditions imposed on the triplet of full bricks (a), deformation of the triplet of full bricks in a shear test (b) . 4. Conclusion The multi-scale model presented here was successfully used to simulate the experimental tests in which failure occurred at the brick/mortar interface presented in [36], which provided the coefficients required to model the interface, namely the stiffness parameters and the length of the micro-cracks. The model is sensitive to these characteristics but the results obtained are in line with the experimental data. The present method was also tested on a more complex structure presented in the literature (the case of a full wall subjected to diagonal compression loading [7]). The results obtained show that the model presented here can be used to describe the behavior of structures of this kind. From the practical point of view, an optimization routine is needed to sys15 hal-00652805, version 1 - 16 Dec 2011 Figure 5: Stress-displacement diagrams of the triplet of full bricks under shear, numerical and experimental results. tematically determine the values of the parameters describing the evolution of the crack length (ξ1 and ξ2 ). This idea will be applied to more complex masonry structures in a future study. It is proposed in the future to establish an optimization routine for systematically determining the values of the parameters describing the evolution of the crack length (ξ1 and ξ2 ), which have depended up to now only on the shear stress. In a future study, it is plained to also examine the dependence on the normal stress. More complex structures such as modern masonry buildings will also be 16 hal-00652805, version 1 - 16 Dec 2011 Figure 6: Stress-displacement diagrams of the triplet of hollow bricks under shear loading conditions: numerical and experimental results. studied to improve the model presented here, by accounting more efficiently for the evolution of the micro-cracks more efficiently into account, in particular. References [1] Orduna A., Lourenço P.B., Three-dimensional limit analysis of rigid blocks assemblages. Part II: Load-path following solution procedure and validation, Int. J. of Sol. and Struct., 42, 5161-5180 (2005). 17 (a) (b) hal-00652805, version 1 - 16 Dec 2011 Figure 7: Experimental data on a masonry structures composed of hollow bricks under shear stress loading conditions. 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[37] Sacco E., A nonlinear homogenization procedure for periodic masonry, European Journal of Mechanics A/Solids, Vol. 28 (2), pp. 609 -222 (2009) [38] Sejnoha J., Sejnoha M., Zeman J., Sykora J., Vorel J., Mesoscopic study on historic masonry, Structural Engineering and Mechanics, 612 . 30, Issue 1, pp. 99-117, 2008 [39] Boutin C., Microstructural effects in elastic composites, International Journal of Solids and Structures, Vol. 33, pp. 1023-1051, 614 6 [40] Sanchez P., In: Homogenization Techniques for composites media, Lecture Note in Physics, 272, pp. 122-189, Springer, Berlin, 616 5 21 Appendix A: Effective properties of the undamaged stratified brick/mortar: HEM-1 Axial tension. The brick/mortar laminate is subjected to an axial stress: σ̄ = σ̄33 e3 ⊗ e3 (20) In this case, relation (3) involves the equalities: 0 σ̄33 ε̄11 = ε̄22 = S̃1133 0 ε̄33 = S̃3333 σ̄33 ε̄23 = ε̄13 = ε̄12 = 0 (21) hal-00652805, version 1 - 16 Dec 2011 In each phase, we assume the local stress induced by the macroscopic loading to be transversely isotropic with the form: σ i = σ̄33 (e3 ⊗ e3 + αi (e1 ⊗ e1 + e2 ⊗ e2 )), (22) where αi (i = b, m) are real constants determined so thatP(4) and (5) are P fulfilled. The average stress σ̄ = i=b,m f i σ i means that i=b,m f i αi = 0. The real constant αm can then be expressed as a function of the primary constant αb . It reads: fb αm = − m αb (23) f Under the present loading conditions (20), the local deformation reads i i i i i εikj = Skjln σln = (Skj33 + αi (Skj11 + Skj22 ))σ̄33 , i = b, m (24) Applying relations (5) and (24) to the couplet (1, 1) gives the constant αb , which reads: m b fm (S1133 − S1133 ) b α = (25) m m b b fb (S1111 + S1122 ) + fm (S1111 + S1122 ) 0 To identify the effective compliance S̃1133 of the material HEM-1, we then use the relations: ½ 0 ε̄11 = S̃ (a) P P1133 σ̄33 i i i i i + S1122 ))σ̄33 668ox(b) + αi (S1111 ε̄11 = i=b,m f ε11 = i=b,m f i (S1133 (26) 0 and S̃1133 therefore reads: X £ ¤ 0 i i (27) S̃1133 = f i S1133 + αi (S1111 + S1122 ) i=b,m 22 We proceed in a similar way with the macroscopic strain component ε̄33 in 0 order to determine the compliance component S̃3333 as follows: X £ ¤ 0 i i i S̃3333 = f i S3333 + αi (S3311 + S3322 ) (28) hal-00652805, version 1 - 16 Dec 2011 i Axisymmetric loading. It is now proposed to subject the laminate to the following axisymmetric loading 1 (29) σ̄ = σ̄33 (e3 ⊗ e3 − (e1 ⊗ e1 + e2 ⊗ e2 )) 2 Based on the overall constitutive relation (3), the macroscopic strain response components read:  0 0 0 + S̃1122 ) + S̃1133 )σ̄33 (a)  ε̄11 = ε̄22 = (− 21 (S̃1111 0 0 (30) ε̄33 = (−S̃1133 + S̃3333 )σ̄33 (b)  ε̄23 = ε̄13 = ε̄12 = 0 The local stress induced by the macroscopic stress (29) is assumed to take the form: σ i = σ̄33 (e3 ⊗ e3 + β i (e1 ⊗ e1 + e2 ⊗ e2 ))) (31) where β i (i = b, m) are real constants determined so that the conditions specified above in (4) and (5) are met. X 1 The average stress over the laminate gives the expression: f iβ i = − . 2 i=b,m The constant β m can then be expressed as a function of β b as follows: βm = −1/2 − f b β b fm (32) It is now only necessary to express the constant β b . The first continuity condition (4) is ensured by the assumed form of the local stress (31), whatever are the values of the constants β i . The continuity condition (5) is used to express β b : βb = b m m m ) − 12 (S1111 − S1133 fm (S1133 ) + S1122 m m b b f1 (S1111 + S1122 ) + f2 (S1111 + S1122 ) (33) The average strain component ε̄11 given by relations (30, (a)) and (26, (b)) 0 0 gives a first algebraic equation for the compliance components S̃1111 and S̃1122 : X £ ¤ 0 0 0 i i i (34) S̃1111 + S̃1122 = 2S̃1133 −2 fi S1133 + β i (S1111 + S1122 ) i=b,m 23 Mixed load: plane traction and plane shear. In this paragraph, the macroscopic stress applied to the stratified crack-free composite reads: 1 σ̄ = σ̄12 (e1 ⊗ e2 + e2 ⊗ e1 ) + (σ11 − σ22 )(e1 ⊗ e1 − e2 ⊗ e2 ) 2 (35) The assumed local stress reads: hal-00652805, version 1 - 16 Dec 2011 σ i = γ i σ̄, i = b, m (36) where γ i (i = b, m) has to be determined in order to meet the continuity conditions (4) The average value of the local stress solution (36) P and (5). i i means that i=b,m f γ = 1. The local strain continuity condition (5) can be used to determine the real primary constant γ b , which reads: γb = m m m S1212 − S1122 S1111 = m b m m b b fb S1212 + fm S1212 fb (S1111 − S1122 ) + fm (S1111 − S1122 ) (37) Based on Hooke’s law, the local strain reads: εi = S i σ i = γ i S i σ̄. With this kind of macroscopic load (35), the couplet (1, 2) in relation (3) can be used 0 to identify the effective compliance component S̃1212 as follows: X i 0 f i γ i S1212 S̃1212 = i=b,m The average local strain components ε11 and ε22 in the stratified composite give the following algebraic relations: P i i 0 0 ) (a) − S1122 = i=b,m fi γ i (S1111 − S̃1122 S̃1111 P (38) i i i 0 0 S̃1122 − S̃2222 = i=b,m fi γ (S2211 − S2222 ) (b) Expressions (34) and (38, (a)) are used to determine the HEM-1 compliance 0 0 components S̃1111 and S̃1122 . Relation (38, (b)) is then used to obtain the 0 compliance component S̃2222 . Longitudinal shear loading. In this case, the macroscopic stress applied is: σ̄ = σ̄13 (e1 ⊗ e3 + e3 ⊗ e1 ) + σ̄23 (e2 ⊗ e3 + e3 ⊗ e2 ) The local stress solution reads σ i = δ i σ̄, where δ i (i = b, m) are such that the local stress meets the continuity conditions ((4), (5)). The average stress 24 P i i gives the following relation between the constants δ i : i f δ = 1. The continuity of the force vector at the brick/mortar interface (condition (4)) means that: δ b = δ m = 1. With the couplet (2, 3), relations (3) and (1) are 0 , which reads: used to obtain the effective compliance component S̃2323 X 0 i S̃2323 = f i S2323 i=b,m hal-00652805, version 1 - 16 Dec 2011 Appendix B: asymptotic analysis, elements of proof The idea underlying matched asymptotic expansions is to find two expansions of the displacement uε and the stress σ ε in powers of ε, i.e., an external expansion in the case of the bodies and an internal one in that of the joint, and to combine these two expansions in order to obtain the same limit. In what follows, we work in 2-dimensional terms in order to simplify the computations. The relations obtained in the internal expansions will be expressed using values occurring in the external expansions. External asymptotic expansions. The external expansion is a classical expansion in powers of ε uε (x1 , x3 ) = u0 (x1 , x3 ) + εu1 (x1 , x3 ) + ..., eij (uε )(x1 , x3 ) = e0ij + εe1ij + ..., ∂ul ∂ul elij = 21 ( ∂xji + ∂xji ), σijε (x1 , x2 ) = σij0 (x1 , x3 ) + εσij1 (x1 , x3 ) + ... (39) Internal asymptotic expansions. In the internal expansions, we perform a rescaling of the second variable. Let y3 = xε3 . The internal expansion gives uε (x1 , x3 ) = v 0 (x1 , y3 ) + εv 1 (x1 , y3 ) + ..., 0 1 eij (uε )(x1 , y3 ) = ε−1 e−1 ij + eij + εeij + ..., l ∂v el11 = ∂x11 , el33 = el13 = σijε (x1 , y3 ) = ε σij,j = ∂v2l+1 , ∂y3 l ∂v1l+1 ∂v 1 2 ( + ), 2 ∂x1 ∂y3 −1 −1 ε τij (x1 , y3 ) + τij0 (x1 , y3 ) ∞ P ∂τ l+1 ∂τ l εl ( ∂xi11 + ∂yi33 ). l=−2 25 (40) + ετij1 (x1 , y3 ) + ..., Conventionally v l = 0, l < 0, τ l = 0, l < −1. (41) Continuity conditions. The third step in this method consists in combining the two expansions. We adopted some intermediate lines defined by x3 = ±θεt , 0 < t < 1, θ ∈]0, +∞[. When ε tends to zero, x3 tends to 0± and y3 = x3 /ε tends to ±∞. The principle of the method consists in assuming that the two expansions both give the same asymptotic limits, that is hal-00652805, version 1 - 16 Dec 2011 (i) v 0 (x1 , ±∞) = u0 (x1 , 0± ), (ii) τ −1 (x1 , ±∞) = 0, (iii) τ 0 (x1 , ±∞) = σ 0 (x1 , 0± ). (42) At order zero, these conditions give the continuity of the displacements (u0 and v 0 ) and stresses (σ 0 and τ 0 ) along the surface S. Elasticity of the thin layer. We take a to denote the elasticity tensor of the joint and lim aijkl /ε = āijkl . This limit is assumed to exist. τij−1 = 0 τij0 = āijkl e−1 kl (43) in the thin layer. Equilibrium. The equilibrium at −2 gives ∂τi3−1 =0 ∂y3 The equilibrium at −1 gives ∂τi20 =0 ∂y3 The stress vector does not depend on the second variable in the thin layer. We add the connectivity conditions vi0 = u0i (x1 , 0) 0 τi30 = σi3 (x1 , 0) 26 Using the constitutive equations, we have τi30 = āi2i2 ∂vi0 ∂y3 for i = 1, 3. By integration (the stress is constant) into [0, 1]. £ ¤ τi30 = āi3i3 vi0 Using the connection method, we obtain hal-00652805, version 1 - 16 Dec 2011 £ ¤ 0 σi3 = āi3i3 u0i 27