Debonding process in composites using BEM

Boundary Elements XXVII 331 Debonding process in composites using BEM P. Prochazka & M. Valek Czech Technical University, Prague, Czech Republic Abstract The paper deals with the debonding fiber-matrix process in composite materials. A couple of papers has been focused on this problem by the authors of this paper. As usual, the influence of separate pure normal and pure shear energy has been studied for obtaining the overall material properties. Such an approach is a simplification of the problem describing the mechanical behavior of the interface between fibres and the matrix in composite materials, since the standard procedure consisting of the superposition of both normal and shear influences is no longer admissible due to the strongly nonlinear behavior of the process. In this paper a more complex development of the debonding zones is shown, namely, responses of successively applied normal and then shear load and also first shear load and then normal load are observed. The physical behaviour of the interface is non-convex. This assertion immediately follows from the penalty formulation of the problem as published recently by the first author. The penetration of fiber into the matrix is not allowed in every case. In this contribution we start with a definition of the model describing the transfer of elastic stresses from the matrix to the fiber. Then, the contact problem is formulated in a manner leading to a very fast Uzawa s algorithm for its solution. In order to speed up the iterative solution influence matrices are created before the iteration. The approach turns to a similar one known as generalized transformation field analysis. Debonding processes in the interfacial zone are illustrated by examples. Keywords: composites, debonding process, general transformation field method, Uzawa s algorithm. 1 Introduction A 2D unit cell model with diagonal symmetry is considered to study an effect of imperfectly bonded interfaces. This simplification has only formal nature, and is

332 Boundary Elements XXVII introduced for easier calculation of the problem. The generalization is admissible in a very simple way. Also a periodical structure of the fibers is assumed. The radial (normal) and the tangential (shear) tractions across the interface are continuous, but the displacements may suffer from a jump, i.e. discontinuity along the fibermatrix boundary. If the interface exhibits a partial debonding the zero traction boundary conditions are invoked along debonding zone. The composite aggregate exhibits geometrical periodicity, see fig. 1. The 2D unit cell supplied with proper periodical boundary conditions can be used to represent composite structure. The analysis on the unit cell is carried out by BEM and Uzawa s algorithm. The algorithm starts with the idea of Generalized transformation field analysis, [4, 5], to simplify the procedure of computation. Our objective is to examine the effects of imperfectly bonded interfaces on local stresses and on the overall response of the composite system. A possible type of interface conditions is proposed in Achenbach and Zhu [1]. It simulates the interfacial zone with linear behavior and debonding of the fibermatrix system by cracking in the interfacial zone. In Prochazka and ejnoha [8], other contact conditions are assumed. They are more feasible for direct contact between fiber and matrix simulation. In this paper a combination of all these conditions is put forward, but these of [8] are preferred. In the sequel couple of papers being close to the topic presented in this paper are going to be mentioned. Bahey-El-Din et al., [2], modeled damage progression in woven composites for multiscale analysis of structures. The problem is studied on a representative volume element of the woven material, derived from micrographs. The problem is solved by virtue of Transformation field analysis by Dvorak, [4]. In Balasivanandha Prabu et al., [3], a micromechanical interfacial characteristics of metal matrix composites are studied. Effect of volume fraction of SiC and diameter of the fibers on interfacial characteristics of 6061 Al/SiC metal matrix is sought. In the analysis, it was found that the fiber diameter plays an important role in the debonding and the results show that debonding is more pronounced in the interfacial elements near the axis of symmetry. Energetic conditions for interfacial failure in neighborhood of a current crack in brittle matrix, Martin and Leguillon [6], are formulated and used. The problem extends the standard concentration of crack zones only to the matrix-fiber interface. Voronoi cell for multi-scale analysis of composite structures with non-uniform microstructural heterogeneities are solved in Raghavan and Ghosh [9], using coupled modeling. The computational model solves three level hierarchic formulations for obtaining the homogenized solution and comparison with results from optical or scanning electron micrographs. The range of validity of homogenized solution is identified. Effects of fiber debonding and sliding on the fracture behavior of composites is studied on a simplified unidirectional model under plane strain and small-bridging conditions in Zhang et al., [10]. General conception of behavior of composites in described in Prochazka and ejnoha [7].

Boundary Elements XXVII 333 Figure 1: Structure of the fibers. Figure 2: Geometry of the model. 2 Formulation of the problem Consider a periodical composite structure in coordinate system 0y 1 y 2 under loading of normal overall stresses σ 01,orσ 02 in, respectively, y 1 and y 2 directions, or shear stresses τ 0 12, see fig. 1 where one quarter of the unit cell is considered. Under this assumptions, a problem of two elastic bodies (fiber and matrix) in the unit cell, the geometry of the first quarter of which is shown in fig. 2, may be formulated. In fig. 2, α istheangleidentifyingthepointsontheinterfacialboundary, α c is a fixed angle connected with the point, the debond in which ends, the zone on Γ C between α c and α p describes a slipping zone, and the rest of Γ C belongs to the bonded part. The body of the particular unit cell considered in this paper is in undeformed state described by the domain Ω Ω f Ω m. The first body (fiber) in undeformed state occupies a circular domain Ω f and the second body (matrix) occupies domain Ω m. The common part of the boundaries of both domains is denoted by Γ C.The boundary conditions are defined later on, according to the load imposed, considered periodicity, and particular shape and position of the body. Let n be the unit outward normal to Ω C and t be the tangent to Γ C with respect to Ω C, see fig 2. Quantities u i r and u i t,i= m, f, defined on the contact Γ C are, respectively, the magnitudes of projections of the displacement vectors to n and to t. Superscript m denotes the quantities connected with the matrix and superscript f denotes the quantities defined in the fiber. Similarly, p n and p t are, respectively, the magnitudes of projections of tractions p to n and to t; it holds p = p = p m, p =(p n,p t ). The transfer of elastic stresses from the matrix to the fiber has been assumed by the following cases of debonding rules in some previous papers, e.g. Prochazka et al., [8]: if p t p n F + κ( p n )τ b then there exists positive real k t such that p t = k t [u] t, if p n >σ + or p t > p n F + κ( p n )τ b then p n = p t =0, where σ + is given strength in tension, F is the tangent of internal friction, τ b is the cohesion, κ is the Heaviside function (being introduced because of physical

334 Boundary Elements XXVII meaning of the item 2, i.e. Mohr-Coulomb hypotheses), [.] i,i = n, t, is the jump in the argument on interfacial boundary. 3 Formulation of the contact problem The restrictions of any function to Ω f or Ω m are denoted, respectively, by superscripts f or m, e.g., u/ω f = u f and u/ω m = u m. The boundary Γ of both fiber and matrix is decomposed into disjoint parts Γ = Γ C 4 i=1 Γ i, see fig. 2. The boundary tractions are prescribed according to the way of loading, see fig. 3 and fig. 4. Figure 3: Unit y 1 -load. Figure 4: Unit shear load. In the case of normal overall loading, being derived from overall unit stress, the loading and boundary conditions are seen from fig. 3. In the case of normal stress, along the boundaries Γ 1 and Γ 2, no displacement in the normal direction are permitted because of symmetry. Along both Γ 1 and Γ 2 the tangential traction p t is equal to zero. On the boundaries Γ 3 and Γ 4 p t =0;onΓ 3 u 1 = C 1, C 1 is a constant computed from the condition Γ 3 σ 01 dγ = 1 2, σ 01 is a given external normal load (average overall stress) in the y 1 -direction. Along Γ 4 u 2 is uniform (boundary condition) calculated from the solution of the problem. In the case of applied shear stress the displacements and tractions are obvious from fig. 4. The axis of symmetry, where symmetry boundary conditions are applied, normal tractions are zero while shear tractions p t have to obey the condition ( Γ 3 + Γ 4 )σ 012 dγ = 1 2. The distribution of normal displacements along the entire external boundary of the unit cell have linear distribution. Axes of symmetry are diagonals of the unit cell. Denote the set of admissible displacements on Ω satisfying the essential boundary conditions and being equal to u f on Ω f and u m on Ω m by V. Assuming the small deformation theory, it may be satisfactory to formulate the essential contact conditions on the interface as follows (Signorini s conditions): [u] r u f r um r 0 a.e. on Γ C. (1)

Boundary Elements XXVII 335 Denote K {u V ;[u] r 0 a.e. on Γ C }. The set K is a cone of admissible displacements satisfying the essential boundary and contact conditions. Let us split the unit cell under consideration into Ω f and Ω m, retaining the stress and strain states in these subdomains. Then the vector of contact tractions p =(p n,p t ) must be introduced. The total energy J of both subdomains being considered separately reads: J(u, p) =Π(u) I(p, u), (2) where Π(u) = 1 a(u, u), 2 (2a) I(p, u) = (p n [u] + r + p t[u] t ) dγ, [u] t = u f t u m t, (2b) Γ C Γ θc a(u, u) = (ɛ f ) T L f ɛ f dω + (ɛ m ) T L m ɛ m dω, (2c) Ω f Ω { m ur ɛ = ; u t ; 1 ( ur + u )} T t, (3) x r x t 2 x t x r and L is the matrix of elastic constants and Γ θc is a subset of Γ C where zero tractions are reached. Let us introduce the space Λ of admissible tractions on Γ C for the Problem: Λ {p is a quadratically integrable vector-function on Γ C }, if u m r = u f r then p n < 0; p t p n F + κ( p n )τ b if p t p n F + κ( p n )τ b then there is positive λ<0; p t = λ[u] t, if p n >σ + then p n = p t =0, (4) Now we can formulate the problem in terms of Lagrange s multipliers p: finda pair {u, p}, such that u K and p Λ and and J(u, p) J(w, p), for all w K, (5) I(η, u) I(p, u), for all η Λ, (6) This formulation leads to a very fast Uzawa s algorithm, see e.g. Dvorak and Prochazka [5], which we use in an improved version also here, following the idea of the Transformation field analysis, Bahei-El-Din et al., [2]. The solvability and uniqueness of solution of the Problem remain open. Some basic lemmas were proved, but the general uniqueness and solvability theorem has still not been proved. On the other hand, the practical experiments give reasonable results.

336 Boundary Elements XXVII 4 Boundary element solution From the above paragraph we can conclude that the problem is linear in each domain Ω f and Ω m, respectively. The linearity fails along the contact surface Γ C, where the condition of continuity of displacements is no more valid necessarily on entire Γ C, only the balance condition p f = p m holds. The solution of this problem after discretization of the boundaries of both domains into the element-wise linear distribution of both displacements and tractions leads to the solution of two linear algebraic systems given by square matrices K f and K m : K f u f = λ f = g f + f f, K m u m = λ m = g m + f m (7) or (from the linearity of the problem on each domain separately) K f u f g = g f, K f u f f = f f, K m u m g = g m, K m u m f = f m (8) The vectors g and f are the loading vectors responding to the influences of interface tractions (till now not known) and external loading, respectively. It is necessary to note that the discretization of boundaries of both domains Ω f and Ω m must be constrained by the condition of conforming nodes along the common contact boundary Γ C, i.e. it has to exist a univalent mapping between the nodes from the discretization of Γ C with respect to Ω f and Ω m. After introducing geometrical boundary conditions the stiffness matrices become regular, so that there exists unique solution to each of problems eqs (8), for arbitrary r.h.s. Note that the components of the vectors g f and g m are equal to zero except of the components connected with the nodal points of the contact Γ C. The vector f f = 0, as there is no external load applied to fiber. Instead of f m, there is the displacement fields along the boundaries Γ 3 and Γ 4. They produce the vector u m g while u f g = 0. From each vector of eqs (8), the components related to the direction of coordinate axes on the boundary Γ C can be extracted. Let us renumerate the numbers of nodal points in that way that the first numbers in the order will be connected with the nodes belonging to Γ C. For example, g and g r and g t are related in the following sense: gi r = g 2i 1 and gi t = g 2i for all admissible i. The components of the vector g r are the nodal tractions in the normal direction at the nodal points of the contact Γ C, while the components of the vector g t are nodal tractions in the tangential direction at the nodal points of Γ C. As the solution u f f and um f may be computed in advance, also [u f ] r and [u f ] t is known in each iteration steps. In the first iteration step we take g r = g t =[u g ] r =[u g ] t = 0. Let us choose some appropriate positive number ρ fixed, and put, to get the current contact tractions on the contact Γ C : g r := P Λ [g r + ρ([u f ] r +[u g ] r )] g t := P Λ [g t + ρ([u f ] t +[u g ] t )] (9)

Boundary Elements XXVII 337 The components of the vector g r are the nodal tractions in the normal direction at the nodal points of the contact Γ C, while the components of the vector g t are the nodal tractions in the tangential direction at the nodal points of Γ C. At each nodal point of Γ C, both g r and g t must belong to Λ, i.e. the physical law must be satisfied (this is the impact of linear distribution of discretized tractions λ). This is why the operator P Λ is introduced in eq (9). It projects the current values of g to Λ. Note that we do not need to test the geometrical condition, eq (1), it follows from the algorithm. There are several ways of such projection. The most simple one appears to be as follows: at each point i M, M being the set of admissible indices of nodes on the boundary Γ C, test the following steps, Scheme: 1. If g r i 0 then gr := g r + ρ([u f ] r +[u g ] r ) g t := g t + ρ([u f ] t +[u g ] t ) else g r = k r [u] r. 2. If g r i σ+ then g r i = gt i =0. 3. If g t i F gr i + τ b κ(g r i ) thengt i := (F gr i + τ b κ(g r i )) sign gt i. 4. For the next nodal point i M go to 1. As the equilibrium on the contact holds, put g f = g m = g. Next, solve the equations (8) for u f g and u m g while both u f f and um f remain unchanged. Repeat the above procedure till the change of both vectors g r and g t is negligible. This procedure can be made principally faster using the idea of the Transformation field analysis, [2]. We can prepareapriorimatrices with eliminated influences of the boundary out of the contact and using unit force impulses in both normal and tangential directions at the point M we obtain inverse matrices to that of eq (7), i.e. matrices K. The relation between the old conception and the new one is clear from the following consideration. Denote u on g and gon the displacements and tractions, respectively, at the nodal points on the contact Γ,and u out g the displacements at the nodal points out of the contact Γ. Then we can write [ ]( ) ( ) K f 11 K f 12 (u on g ) f (g on ) f K f 21 K f = (10) 22 (u out g )f 0 [ ]( ) ( ) K m 11 K m 12 (u on g ) m (g on ) m = (11) K m 21 K m 22 (u out g )m 0 After eliminating u out g where from the above equations we get (u on g )f = P f (g on ) f P f =(K f 11 Kf 12 (Kf 22 ) 1 K f 21 ) 1 (u on g )m = P m (g on ) m (12) P m =(K m 11 K m 12(K m 22) 1 K m 21) 1 The relation (12) enables one to avoid the solution of the large system of equations (8) but according to eq (12) only multiply the influence matrices P f and P m by the appropriate tractions on Γ C.

338 Boundary Elements XXVII Summary of the numerical procedure: 1. Compute the influence function matrices P f and P m by direct calculation from the BEM. It means that we do not use the procedure described in eqs (10) and (11), but apply the unit forces at i M respectively in normal and tangential direction to Γ C. The responses on displacements at points j M creates the matrices P f P f ij and Pm Pij m. 2. Set [u g ] r =[u g ] t = g r = g t = 0. 3. For given f f and f m solve eq (8) to get u f f and um f. 4. From u f f and um f select the components connected with the contact Γ C; compute [u f ] r and [u f ] t. Note that these vectors remain constant during the iteration. 5. Compute g r and g t from eq (9). 6. Test the admissibility of the tractions and following the Schema improvethe values of the tractions on Γ C. 7. Create the vectors u f g and um g from eq (12) using the influence function matrices P f and P m. Form the vectors [u g ] r and [u g ] t. 8. Test the convergence (for example test a change of [u g ] r and [u g ] t in current and previous iteration steps, using standard norm). If the convergence is satisfactory, stop, else repeat the iteration from step 5. The procedure is generally very simple, but it is strictly dependant on the choice of the relaxation parameter ρ in eq (9). In particular problems, the convergence can be dramatically influenced by the choice of ρ. If the parameter is too small, the convergence is ensured but it is very slow (the system is undermodulated). If the parameter is too large, the system is overmodulated, the convergence oscillates and can even diverge. From the theoretical point of view, this algorithm can be regarded as a method of finding a fixed point on the set Λ. 5 Example Several examples were tested by the BEM. In all examples the fiber possessed the following material properties E f =772 GPa, ν f = 0.25 while the epoxy matrix was considered, for which E m = 96.5 GPa, ν m = 0.3. The interface had the coefficient of Coulomb friction F = 0.22 and the shear bond strength τ b = 8.5 MPa. The unit cell was formed as a square 1mm 1mm, volume fraction of the fibers was 0.4. The unit cell is loaded by normal and shear tractions applied on the boundaries Γ 3 and Γ 4 under the condition that their normal displacements are either uniform, or linear. The results are shown in fig. 5 for normal unit overall stress, in fig. 6 for unit overall shear stress, and in fig. 7 the combination of both latter load are assumed. The last case is described only in the first quarter, although the problems is symmetric with respect to the origin of the unit cell. The denotation follows that of fig. 2. Γ denotes the debonding matrix.

Boundary Elements XXVII 339 Figure 5: Structure of the fibers. Figure 6: Geometry of the model. Figure 7: Structure of the fibers. 6 Conclusions In this paper development of debonding and sliding zones due to normal, shear and combined unit load is described. Influence matrices are created from unit interfacial tractions in order to speed up the iterative solution. Uzawa s algorithm is used, which follows from variational formulation of the problem. The higher the tensile strength is, the shorter zone of debonding appears and the sliding zones extend. Generally, the problem can be solved on the entire cell with standard periodic

340 Boundary Elements XXVII boundary conditions, but in our case symmetric cell and fiber are studied. This enables us to simplify the geometry in particular cases of loading. Acknowledgment This paper has been supported by GAR, grant project # 103/03/1178. References [1] Achenbach, J.D. & Zhu, H. Effect of interfacial zone on mechanical behavior and failure of fibers-reinforced composites. JMPS 17(3), 381-393, 1989. [2] Bahei-El-Din, Y.A., Rajendran, A.M. & Zikry, M.A. A micromechanical model for damage progression in woven composite systems. Int. J. Solids & Structures 41 2307-2330, 2004. [3] Balasivanandha Prabu, S., Karunamoorthy, L. & Kandasami, G.S. A finite element analysis study of micromechanical interfacial characteristics of metal matrix composites. Journal of Materials Processing Technology. To appear in 2004. [4] Dvorak, G.J. Transformation field analysis of inelastic composite material. Proc. Roy. Soc. London A437, 311 327, 1992. [5] Dvorak, G.J. & Prochazka, P. Thin-walled composite cylinders with optimal prestress. Composites Part B 27B, 643 649, 1996. [6] Martin, E. & Leguillon, D. Energetic conditions for interfacial failure in the vicinity of a matrix crack in brittle composites. Int. J. Solids & Structures 41, 6937-6948, 2004. [7] Prochazka, P. & ejnoha, J. Behavior of composites on bounded domain. BE Communications 7(1), 6 8, 1996. [8] Prochazka, P. & ejnoha, M. Development of debond region of lag model. Computers & Structures 55(2), 249-260, 1995. [9] Raghavan, P., Li, S. & Ghosh, S. Two scale response and damage modeling of composite materials. Finite Elements in Analysis and Design 40(12), 1619 1640, 2004. [10] Zhang, X., Liu, H-Y & Mai, Y-W. Effects of fibre debonding and sliding on the fracture behaviour of fibre-reinforced composites. Composites Part A: Applied Science and Manufacturing 13(11), 1313 1323, 2004.