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Bending of debonding fiber reinforced composite beams

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  • Save International Journal of Composite Materials 2014, 4(2): 130-134 DOI: 10.5923/j.cmaterials.20140402.12 Bending of a Fiber-Reinforced Composite Beam with Debonded Fibers Yahya Berrehili Equipe de Modélisation et Simulation Numérique, Université Mohamed 1er, Ecole Nationale des Sciences Appliquées, Oujda, 60000, Maroc Abstract The paper is devoted to the illustration of effective behavior on a classical bending problem of a fibred-reinforced composite beam with debonded fibers (but in contact with the matrix). We show that when the period of the microstructure is small enough, the displacement field solution is approximated by an explicit displacement field which depends not only of loading, volume fraction of fibers in the matrix and the homogenized stiffness tensor but also new homogenized coefficients ignored in the literature, characterizing the composite structure with debonded fibers (see [Berrehili, Y. and J.-J. Marigo, 2013, The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers, Mathematics and Mechanics of Complex Systems, sous presse.]) Keywords Homogenization, Composite beam, Debonded fibers, Modeling, Behavior, Bending problem 1. Introduction The microstructure of the composite materials in the aeronautical sector is so small that it is very difficult or even impossible to study directly, by finite element, this type of materials: since it requires discretizing fine enough its various components, mainly in the contact area (fibers-matrix) for sensing the great stress gradients present in this area. This may cause then a exceeding of the capacity of a computer (memory and time required for the calculation). To overcome these difficulties, multi-scale methods (including homogenization method in particular) were introduced to approach the real problem by a limit problem, in tending to zero the parameter characterizing the fineness of the microstructure [1]. The results are well established on the effective behavior of composite materials with perfectly bonded fibers (see [2] [3] [4] [5] [6] [7] and [8] [9] [10] [11] [12]), but not many studies have been made in the case where the constituents are debonded (see [13] [14] [15] [16] and [17] [18] [19]). The objective of this work is to illustrate the effective behavior on a classical elastostatic problem of bending of a fiber-reinforced composite beam with debonded fibers but in contact with the matrix. Although simple, the bending problem, one of the three famous classical problems (traction/compression, bending and torsion), is of paramount importance for practical applications. For solving this probllem, we will try, with * Corresponding author: (Yahya Berrehili) Published online at Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved intuitive remarks, to guess the analytical form of a part of the solution which will serve as a starting assumption. We then calculate the full solution using the equilibrium equations and the boundary conditions, i.e. to determine all the constants in a unique way. The uniqueness of these constants ensures, thanks to the Lax-Milgram theorem, the uniqueness of the sought solution of the bending problem considered. Specifically, the paper is organized as follows. The next section is devoted to the setting of the problem which consists to study the bending problem of a fiber-reinforced composite beam with debonded fibers but in contact with the matrix. The third section is devoted to a brief general recall of homogenization results obtained in [20] and adaptation of these results to our bending problem of a “debonded” composite beam by formulating the homogenized problem associated with our bending problem which consists to solve an equilibrium equations system (coupled not classic). We solve therefore this system in the fourth section. And we end with a conclusion in the fifth section. 2. Position of the Real Problem We consider a cylindrical fibrous composite beam with circular cross section which occupies, in its natural reference configuration, the solid cylinder Ω=Dx]0, L [ of IR3 where D is the disk with center O and radius R. We denote by (e1,e2,e3) the canonical basis of IR3 and (x1,x2,x3) the coordinates of a point x of Ω. The beam is assumed of axis Ox3 and with two circular straight sections Σ0 and ΣL located respectively in the planes x3=0 and x3=L. The two sections Σ0 and ΣL are submitted respectively to a density of surface forces International Journal of Composite Materials 2014, 4(2): 130-134 131 F=−βx1e3 and −F=βx1e3 (see Figure 1.), β being a given arbitrary real constant. With this type of loading, one says that the cylinder considered is subjected to simple bending with axis Ox3. The lateral surface force applied on ΣLAT is supposed null and the density of the volume forces applied on Ω is assumed negligible. where we conventionally denote by δij the Kronecker symbol equal to 1 if i=j and 0 if i≠j. λ and µ are the Lamé coefficients, defined in the periodic cell V=Vf∪Vm (reunion of the fiber part Vf and the matrix part Vm) by (λ(y), µ(y))=(λf,µf) if y∈ Vf and (λ(y), µ(y))= (λm,µm) if y∈Vm (see Figure 2). Figure 1. The cylindrical composite beam The fibers of the composite beam are supposed all debonded from the matrix. We assume further that during the deformation, the fibers remain in contact with the matrix and can sliding without friction. This express that the normal displacement field is continuous and the shear is null on the interfaces fibers-matrix Iε(ε denotes the period of the microstructure destined to tending to zero[21][22]). The problem consists in seeking for the displacement and the associated stress fields, denoted respectively by uε and σε so that the dependence in ε be explicit, solutions to the following bending problem: Div σε = 0 in Ω\Iε, (1) σε = Αεε(uε) in Ω\Iε, (2) ε(uε) = 1 (▼uε + ▼Tuε) in Ω\Iε, (3) 2 σεn=−βx1e3 on Σ0, (4) σεn=βx1e3 on ΣL, (5) σεn=0 on ΣLat, (6) [uε]•n=0, [σε]n = 0, σεn∧n = 0 sur Iε, (7) where Aε denote the elasticity tensor of the composite beam, ε(uε) the strain field associated with the displacement field uε, n the outer normal of Σ0(n=−e3), ΣL(n=e3) or ΣLat(n=n1e1+ n2e2 with n12+n22=1) and [uε] the jump of the displacement field uε across the interface Iε. The three relations of the last line reflect the continuity of normal displacement field, the continuity of the stress vector and the nullity of the shear on the debonded interfaces Iε. It is assumed that the two materials, fibers and matrix, constituting the composite structure, are elastic homogeneous and isotropic, of Lamé coefficients respective (λf,µf) and (λm,µm). The coefficients of the elasticity tensor of the composite structure, Aεijkl, 1≤i,j,k,l≤3, are given therefore by: Αε(x)ijkl = λ( x ε )δijδkl+µ( x ε )(δikδjl+δilδjk), (8) Figure 2. The unit periodic cell V=Vf∪Vm 3. Homogenized Macroscopic Problem Following the classical two-scale procedure in homogenization theory of periodic media [1][23][20], we assume that uε can be expanded as follows: uε(x)=u0(x,y1,y2)+εu1(x,y1,y2)+ ε2u2(x,y1,y2)+... (9) where x is the macroscopic variable(x∈ Ω) and, y1=x1/ε and y2=x2/ε are the coordinates of the microscopic variable y describing the unit cell V\Γ, with V=]−1/2,+1/2[x]−1/2,+1/2 [and Γ={(y1,y2∈V / y12+y22=α2} where 0<α<1/2 (Vf=πα2 denotes the volume fraction of fibers in the matrix). And the ui=(ui1,ui2,ui3), i ≥ 0, are the V\Γ-periodic fields with respect to the variable microscopic y. We have shown in [20] that the result obtained differs in general from the usual property of the homogenization theory. Indeed, because of debonding of the fibers from the matrix, the leading term u0 of the asymptotic displacement field expansion depends in general on the microscopic coordinates y. Moreover a new macroscopic field enters in the effective kinematics of the composite structure. Specifically, we obtain a classical vector field representing the macroscopic displacement of the matrix u and additional macroscopic scalar fields δ and ω interpreted respectively as the sliding and the rotation of the fibers with respect to the matrix. Consequently we obtain a generalized homogenized problem (not classic). It contains in additional of the homogenized stiffness tensor (of the debonded composite structure) Ahom, new homogenized tensors K, T and Σ. K and T are interpreted as the effective rigidity tensors respectively to the extension and to the torsion (of the debonded fibers), and Σ as the effective stress tensor. These three tensors are obtained by solving new cell problems [20] ignored in the existent literature. It appears in the equilibrium equations of the effective problem as coupling coefficients of these three macroscopic fields, u, δ and ω. 132 Yahya Berrehili: Bending of a Fiber-Reinforced Composite Beam with Debonded Fibers The leading term u0(x,y) of the asymptotic displacement field expansion uε(x) can be written as in [20]: u0(x,y) = u(x) + χf(y)(δ(x)e3 + ω(x)e3^y) (10) where χf(y), with y∈ V, is the characteristic function of Vf (equal to 1 on Vf and 0 on Vm). But the forces applied are not working in rotation (since it is a problem of simple bending), we will obtain therefore simply ω = 0 and T = 0. u0 is written then: u0(x,y) = u(x) + χf(y)δ(x)e3, (11) where the couple of macroscopic fields (u,δ) is a solution of the following homogenized problem(posed on the same domain Ω, but which one has replaced the composite structure by an equivalent homogeneous medium(see Figure hom 3.), characterized by the tensors A , K and Σ(see [20]): ∂δ Div (σ + ∂x 3 Σ ) = 0 in Ω (12) ∂ ∂δ ∂x 3 ( Σ.ε (u) + K ∂x 3 ) = 0 in Ω (13) hom σ(x) = Α ε(u)(x) in Ω (14) ε(u)(x)= 1 (▼u + ▼Tu)(x) in Ω (15) 2 ∂δ ( σ + ∂x 3 Σ )n(x)=βx1e3 on Σ0 (16) ∂δ ( σ + ∂x 3 Σ )n(x)= βx1e3 on ΣL (17) ∂δ ( σ + ∂x 3 Σ )n(x)=0 on ΣLat (18) ∂δ ( Σ.ε (u) + K ∂x 3 )(x)n3(x)=Vfβx1 on Σ0 (19) ∂δ ( Σ.ε (u) + K ∂x 3 )(x)n3(x)=Vfβx1 on ΣL (20) ∂δ ( Σ.ε (u) + K ∂x 3 )(x)n3(x)=0 on ΣLat (21) The first equation is a three-dimensional equilibrium. The second one is a family of scalar equations, identified by the indices (x1,x2). It is the equations of beam-type (K ∂ δ ∂x 3 represents the normal effort). We see that, in the first ∂δ equation, the term ∂x 3 Σ plays the role of a pre-stressed field whereas in the second one, the term Σε(u) plays the role of a pre-tensioning field of the beam composite. This system of equations is completed by boundary conditions (16)-(21). Figure 3. The homogenized composite beam 4. Resolution It should be noted that the problem is symmetrical: we have a geometrical and loading symmetries with respect to the plans x2=0 and x3=L/2. The couple solution (u,δ), with u=(u1,u2,u3), can be then searched under the following form: u(x)=(a1(x12−x22)+a2x32)e1 + 2a1x1x2e2−2a2x1x3e3 (22) and δ(x) = a3x1. (23) i.e. u1(x)=a1(x12 − x22)+a2x32 (24) u2(x)=2a1x1x2 (25) u3(x)=−2a2x1x3 (26) δ(x) = a3x1 (27) where a1, a2 and a3 are real numbers to seeking for. The strain field associated with the displacement field u is given by,  2a1x1 0 0  ε(u)(x)=  0 2a1x1 0  . (28)  0 0 − 2a 2x1  The form of the new stress tensor Σ, which we have obtained in [20], is given as follows: Σ =  Σ11 0  0 Σ11 0 0   . (29) 0 0 Σ33  In the case of a disposition of fibers at the vertices of a square lattice(as in our case), the homogenized stiffness tensor Ahom is given, in the axis system (O;e1,e2,e3), by the following matrix[8]: A1h1o1m1 A1h1o2m2  hom A=   A1h1o1m1     − SYM −   A1h1o3m3 0 0 0 0  A1h1o3m3 0 0 0 0  A3h3o3m3 0 0 2A3h1o3m1 0 0  0 2A3h1o3m1  0 2A1h2o1m2   (30) International Journal of Composite Materials 2014, 4(2): 130-134 133 So, by calculating the stress tensor σ, given by the constitutive law (14), we obtain: σ(x)=   σ11 0 0 σ22 0 0   . (31)  0 0 σ33  with σ11 = σ22 = 2a1x1( A1h1o1m1 − A1h1o2m2 )2a2x1 A1h1o3m3 (32) and σ33 = 2a1x1 A1h1o3m3 2a2x1 A hom 3333 . (33) ● We verify without difficulty, taking account of (28) and (29), that the equilibrium equations (13) and the condition imposed on ΣLat (21) are satisfied. ● The equilibrium equation (12) gives by cons a relation linking the three constants, a1, a2 and a3, to determine: 2a1( A1h1o1m1 + A1h1o2m2 ) 2a2 A1h1o3m3 = a3Σ11 (34) ● In expliciting now the boundary conditions (16), (17) and (18) defined respectively on Σ0, ΣL et ΣLat, we obtain: 4a1 A1h1o3m3 2a2 A3h3o3m3 = −βa3Σ33 (35) ● Similarly as above, for the boundary conditions (19) and (20) defined respectively on the Σ0 et ΣL, we deduce the following condition: 4a1Σ11+a2 Σ33= βVf+Κa3 (36) In summary, one has then to solve the following system of three equations with three unknowns, giving the constants a1, a2 and a3: − 4a1Σ11 + a 2Σ33 − Ka 2 = βVf 4a1A1h1o3m3 − 2a 2 A hom 3333 + a 3Σ33 = −β (37)  2a1 (A1h1o1m1 + A1h1o2m2 ) − 2a 2 A1h1o3m3 + a 33Σ11 = 0 which admits a unique solution: a1 = 1 4CΣ11 [(−βCVf + GΣ33) − AB − CD AE − CF (EΣ33 + KC)] (38) a2 = G C − E(AB − CD) C(AE − CF) , (39) a3 = AB − CD , AE − CF (40) where we have posed: A=8( A1h1o3m3 )2 4A hom 3333 (A1h1o1m1 + A1h1o2m2 ) (41) B=4C(Vf A1h1o3m3 Σ11) (42) C=4(Σ33 A1h1o3m3 2Σ11 A hom 3333 ) (43) D= (A1h1o1m1 + A1h1o2m2 ) (44) E=4(K A1h1o3m3 + Σ11Σ33) (45) F=2Σ33 (A1h1o1m1 + A1h1o2m2 ) 2Σ11 A1h1o3m3 (46) G=4C(Vf A1h1o3m3 Σ11) (47) The displacement field, defined explicitly by, u0(x,y) = u(x) + χf(y)δ(x)e3= (a1(x12−x22) +a2x32)e1+2a1x1x2e2+(χf(y)a3x1−2a2x1x3)e3 (48) (where a1, a2 and a3 are given by (38)-(40)), is then an approached solution, in first order, of the bending problem (1)-(7). Any other solution is obtained by adding to the expression (48) a rigid displacement, without consequences for the mechanical problem considered. 5. Conclusions We have shown in this paper, by using the results obtained in [20] and homogenization theory of periodic media that the bending problem of a fiber-reinforced composite beam with debonded fibers (but in contact with the matrix) admits a unique solution approached, in first order, by an explicit displacement field, i.e. leading term u0 of the asymptotic expansion of uε given by the expression (48). This is because the problem considered is symmetrical. This field depends on two displacement fields, the vector field u representing the macroscopic displacement of the matrix and the scalar field δ representing the longitudinal sliding of the fibers with respect to the matrix. These two fields, given explicitly by the expressions (22) and (23), are solutions of the coupled homogenized problem (12)-(21). Moreover, unlike that found in the literature, this field depends on the fineness of the microstructure, i.e. on the small parameter ε related to the size of the microstructure (presence of microscopic variable y, with y1=x1/ε and y2=x2/ε, in its expression (48)). It also depends on the density of surface forces F, the volume fraction of fibers Vf, the classical homogenized tensor Ahom and the new homogenized tensors K and Σ(calculated once and for all in [20]) characterizing the ”debonded” composite. REFERENCES [1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis of Periodic Structures. (1978), North Holland. [2] M.Z. Asik, S. Tezacan, A mathematical model for the behavior of laminated glass beams. Comp. Struct., 83 (2005) 1742-1753. [3] A. Brillard, M. El Jarroudi, Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibres. IMA J. Appl. Math. 66 (2001) 567-590. [4] G. Chatzigeorgiou, N. Charalambakis, F. 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