Homogenization of debonding composites

Transcription

1 Homogenization of debonding composites P. Prochazka, A. Bilek-Larrea Department of Structural Mechanics, Faculty ofmechanical Engineering, Czech Technical University, Thakurova 7, Prague 6, Czech Republic Abstract A local homogenization of microstructures in composites is presented in this paper. Homogenization of composite structures has been solved by many authors, and many methods have been used, particularly when the elastic material behavior of composites was considered. When dealing with linear material properties of such structures, a large background not only in numerical methods but also mathematical theories are successively developed. A little different situation occurs when in some respect nonlinear material is studied. We consider possible debond or change of the fiber - matrix interfacial boundary if the debond appears, and both fiber and matrix to be elastic. The problem is strongly nonlinear. In order to simplify the computation, transformation analysis will be used and influence matrices will be created. Once they are available, Uzawa's algorithm can be applied for solving the debonding process. The idea suggested by Suquet is applied to the unit cell of the pseudoperiodic composite structure, and the system of three types of displacements is used along the outer boundary of the unit cell. These displacements are a direct consequence of unit average (overall) strain tensor. The local stress in the vicinity of the interface will decide whether the debond appears and which part of the interface is debonded. The condition of the debond follows the no tension law. In order to simplify the theory, only this interfacial condition is considered in this paper. More complicated conditions are under study. The Uzawa's algorithms with appropriate accelerators will be applied to the solution of the problem of debonding composite. The theory is applied to 2D problem and the solution by the FEM with linearly distributed displacements is suggested. Simple example shows the ability of the theory suggested.

2 90 Free and Moving Boundary Problems 1 Introduction Unit cell model is suggested in order to study the effect of imperfectly bonded interfaces. For this reason a pseudo-periodical structure of the fibers is assumed. The radial (normal) and the tangential tractions across the interface are continuous. The displacements may be discontinuous along the interface fiber-matrix. The discontinuity is dependant on stress distribution along the interfacial fiber-matrix boundary (or more precisely, the traction being normal to the interface). If the interface shows debonding, the zero traction boundary conditions are invoked along the debonding region. We restrict our considerations to this case of nonlinear mechanical behavior of the composite structure along the interfacial boundary, although the interface conditions may be principally extended, and hence affect the overall material properties of the composite. A possible type of more accurate interface conditions is proposed by Prochazka & Sejnoha^. The composite aggregate exhibits pseudo-periodicity - see Fig. 2 below. The 2D unit cell supplied with proper boundary conditions can be used to represent the composite structure. In the theoretical part we assume 3D composite body, in the applications we only focus on 2D unit cell. The analysis of the unit cell models is performed using the FEM and the Uzawa's algorithm, cf. Prochazka & Sejnoha^. Uzawa's algorithm will also be used in this paper. This is a powerful tool for solving debonding of phases, (a priori localized damage). Multiphase bodies can be solved by procedure suggested by Prochazka & Sejnoha^. The algorithm starts with the idea of transformation field analysis, Dvorak & Prochazka*, to simplify the procedure of computation. The transformation field analysis has originally been suggested for relating overall stress state and eigenparameters. In this paper, it analyzes stress-strain relations along the interfaces (in numerical analysis they call it substructuring). The debonding process is formulated as moving boundary problem, while in Sejnoha, Prochazka & Sejnoha^ the localized damage problem was preferred to the problem considered in this paper. The homogenization technique is concisely described in Suquet^, and it will partly be used for our purpose. One of the most interesting procedures dealing with nonlinear behavior of phases in composite structures is proposed by Ponte Castenada^, where new structures of variational principles are suggested. Our objective is to examine the effects of imperfectly bonded interfaces on local stresses and on the overall response of the composite system. Concerning the procedure used in this paper, we have to distinguish three steps to be solved: - localization problem, - homogenization depending on overall strain state (average strains), - debonding process due to the loading and boundary conditions selected. We will start with basic relations and denotations needed in the following text.

3 Free and Moving Boundary Problems 91 2 Localization and homogenization 2.1 Linear behavior of the composite Localization and homogenization in general are concisely described in Suquet^. Recall some basic consumptions which we use later in this text. First, we denote quantities used in this text. Two different scales will naturally be introduced. The macroscopic scale, where the homogeneous law (involving the overall material properties) is sought, will be described in coordinate system x = {x\, x^, 3}^, and the microscopic scale - heterogeneous - it is characterized by the system of coordinates y = {y\,3/2,2/3}^- The medium is generally heterogeneous, but locally - in the microscopic scale - is assumed to be pseudo-periodic, so that a representative volume element may be cut out from the structure, and the appropriate boundary conditions can be introduced in this element. In what follows we rather use a unit cell, or even its quarter, see Fig. 2, because of symmetric geometry and either symmetric or antisymmetric loading, see Fig. 3. The unit cell will be described by Q = & U i?, where iw denotes the area of thefiberand Q describes the matrix. The outer boundary of J? is F = Ylk = i ^\ and the interfacial boundary fiber-matrix will be FC, see Fig. 2. The general idea of these two levels is illustrated in Fig. 1 (Suquet^). level Representative volume element Figure 1: Macro- and microstructure of a composite. nun oooooot OOOOOOI oooooot OOOOOOI jtuu Figure 2: Geometry of the model. Figure 3: Fibers structure.

4 92 Free and Moving Boundary Problems Let us distinguish the considered quantities depending on the macroscopic or microscopic scale, in the following manner: displacements in the macroscopic level will be denoted as U = {C/i,(/2,%}^, while the same in the microscopic level as u = {%i,%2, W^. Moreover, in macroscopic level strains denote as E = Eij, i, j, = 1,2,3 and stresses as S = Sij, i, j = 1, 2, 3 In microscopic level stresses denote as <r = cr;y, i,j, = 1,2,3, and strains as e = ij,i,j ~ 1,2,3. Assumingfirstthe linear behavior of the composite aggregate, let us define the corresponding quantities by =< *ij >= meas E,., =<c,, >= - Lpr ;,.,. dl/(y) ==, (1) ' meas $2 Jn where <. > stands for the averages, and meas Q is the volume of the domain Q. The geometry of one quarter of the unit cell is shown in Fig. 2. The fiber has the radius a and each point in Q can be described either by Cartesian coordinates {7/1,2/2}, or polar coordinates {r, 0}. The debond is considered to be between angles 0& and Od- The localization consists of the solution of the system of equilibrium equations on the representative volume element: ^ = 0, < <?,, >=,%. (2) In our case, the boundary conditions will be employed on fg U F^ as follows: u = Ey (3) The boundary conditions on F\ U Ft suffice on the applied loading. If the normal strains are induced by applied boundary displacements (average strains), the symmetry conditions should be applied, i. e. the normal displacements and tangential tractions are equal to zero along F\ UA, whereas if the shear strain is computed from (3), along F\ U A antisymmetric conditions hold: the tangential displacements and normal tractions are equal to zero. Using Green's theorem, the average strain Eij can be expressed as: (4) Since both the matrix and the fibers are assumed to behave physically linearly and are bonded, the localization problem now reads: = 0, #u =< 6^(u) >, (5)

5 Free and Moving Boundary Problems 93 with the boundary conditions (3) and the boundary conditions along TiU/V Eq. (5) is to be taken in the sense of distributions. Stiffness matrix L possesses known symmetries and is defined and given in 2. It is constant in each of the constituents. Let the loading be split into the superposition of influence functions given by unit impulse E^^ = 1 for successive choice of z'o, jo 1,2,3, and the other values of E{j = 0,z"o # i,jo ^ j- The symmetry of E has to be taken into consideration, i.e. if E^j^ 1 for z'o ^ jo, so is Ey^. This process leads to a fourth-order tensor called "concentration factor tensor (or matrix)" Aijki defined as: ij(u(y)) = Aw(y)Eu. (6) The concentration factor matrix is sometimes called tensor of strain localization, since it yields the local strain fc/(u(y)) in terms of the macroscopic strain Eij. Now we can tackle the homogenization, which directly follows from (6): By definition, the homogenized stiffness matrix L* is written as: Comparing (7) and (8), we eventually have: '%' = L*jkiEki. (8) Lijkl < LijaftAapkl > (9) It is worth noting that the homogenized stiffness matrix is symmetric with similar properties as those of the classical stiffness matrices. 2.2 Deboiiding interface Because of the debonding process, the interfacial boundary may split into two parts, and each of these parts may move. Since the stress and strain tensors may not be defined in the interior of the defect, the averaging process (7) which yielded the macroscopic quantities from the microscopic ones must be revised. The defect - void - will be denoted by j?* and its boundaries r' = r;ufl^,seefig. 4. In the case of the stress, the modification is immediate: since the stress tensor (traction vector) vanishes on the boundary of J?*, we can continue the stress tensor by 0 in the interior of the defect. We denote by & this continuation, and we obtain that: meas * Jn /ffijd(2 ' = meas ^r Q J^, / ' <Tij ds2 (10)

6 94 Free and Moving Boundary Problems 4 Figure 4: Geometry of the debond. Figure 5: ID schema a e relation. This is a typical definition of average stress when dealing with damage problems. In our case we rather use the following definition: meas More attention should be paid to the treatment of strain, since the microscopic displacement does not vanish on the boundary of fi*. We will admit that it can be continuated in a regular manner in the inside of i?*. We denote by u this continuation, and we obtain: meas meas (11) In order to fulfill the condition of successive application of components of the average strain E, in the sense of (11) instead of (4) we write: The last integral may be split as: f o2meas / Jp-r*-r* 1 2 meas meas 4- df- - ^ ). (12a) * The last expression deserves more accurate explanation. Our aim in the localization and homogenization is to apply such a loading (in our case the loading is given by the appropriate displacement vector along the boundary TS U A), which induces average strains (and arbitrary stresses) in the composite aggregate. In the case of elastic bond, the term e* disappears, and there is a unique choice of boundary displacements to induce only one component of the average strain. For example, after positioning the origin

7 Free and Moving Boundary Problems 95 of the coordinate system 0%/i2/2 in 2D problem, depicted in Fig. 2, the displacements Uyi = 2 meas Qy\ and Uy^ = 0, being prescribed along PS U 7~4, induce E^y, 1 (we considered b\ b^ ~ 1, see Fig. 2). From (4) we immediately obtain Ey^y^ Ey^y^ = 0. Note that meas Q 1. This is no longer valid for debonding composite aggregate. The disturbing term is 6*, and we have to study it very carefully. Having on mind the definition of this term, and applying say Uy^ y\ along the outer boundary, we get Furthermore it holds: 1 n I (%m, + Ujrii) dr*. 2 meas f? meas J/V#*+#* From the symmetry, it immediately follows that: / 1/2^2 dr* = / (r/iw2-i-1/2^1) dr* = o, Jn*rn+n* Jntn+n/ where C\ is a constant. When applying Uy^ along the outer boundary, the situation is quite the same, as this case is the latter after rotation by 90. Now we only have to discuss the effect of average shear strain. We can proceed in the same manner as in the case of normal direction quantities, and get the result that only 6^ = C*2, where 62 is a constant, and the other average strains are equal to zero (antisymmetry has been used). Lene^ even proved that in the case of damage, the constants C\ and Ci also disappear. Generally, the procedure for the localization and homogenization may be stated as follows: Let us apply successively the outer boundary displacements a.) %i =2/i,%y2 = 0 b.) %%i = 0,%, = 1/2 c.) %%/i = 2/2, %2 = 2/1 From a.) we get Ey^y^ = \ + C\, from b.), similarly, Ey^^ \ -f C\, and from c.) Ey^y^ -f C%. In each case the average strains, which have not been introduced, are zeros. This result is very important and enables us to homogenize heterogeneous materials also in the case when localized debonding appears. Recall that the averaged stresses from a.), b.), and c.) create columns in the overall (effective) stiffness matrix L*, reduced by a certain constant, see (8). This standard procedure was described in Section 2.1, where elastic problem of the localization and homogenization was solved. In the case of

8 96 Free and Moving Boundary Problems the debonding composite the overall nonlinear material behavior has to be taken into account, see Fig. 5, although the fiber and the matrix separately behave linearly. The values of boundary displacements have to change, and depending on their values the stresses also change, so do their averages. L* change with the change of E. 3 Formulation of the debonding problem Recall that the displacements are described by the vector function u = {1*1,^2} = {un,u>t} of the variable y = (1/1,2/2) E f?, the domain i? & U Q. The index n means projection of a vector to the outward unit normal with respect to fij and the index t denotes the projection to the tangential direction to the appropriate boundary. The normal and the tangential directions create a positive counterclockwise coordinate system. The restriction of any function to & or Q is denoted, respectively, by index / or m, e. g. u/1?/ = u/ and u/j?^ = u. The tractionfieldon F is denoted as p = {pi,p2}, or after projection to the normal and tangential directions, respectively, as p = {pn,pt}- Let us denote the set of admissible displacements on J?, satisfying the essential boundary conditions and being equal to u/ on & and u on Q, ast/. Assuming the "small deformation" theory, it may be satisfactory to formulate the essential contact conditions on the interface as (Signorini's conditions): [u]n = w;f -?C < 0 a.e. on fc. (13) Let us denote K = {ug V] [u]n < 0 a.e. on FC}. 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 Q$ and Q, retaining the stress and strain states in these subdomains. Then the vector of contact tractions p = (pn>pt) must be introduced. The total energy J of both subdomains being considered reads separately as: J(u,p) = n(u)-/(p,u), (14) where D(u) = ia(u,u), (14o) /(P,u)= Pn(u]+drc, (146) JFc o(u,u)= / (e^)^l^c^d^+ / (e^l^e^dfi, Jni Jf> (14c)

9 Free and Moving Boundary Problems 97 and L is the matrix of elastic constants. In (14b) we could subtract a subset of PC between the angles 0& and 0<j, where zero tractions are reached (the flaw). Let us introduce the space A of admissible tractions on PC for the Problem: A 5 {pis a quadratically integrable vector-function on PC, if pn > cr+ then pn = Pt = 0}, where a+ is the tension strength given for the interface fiber-matrix. Now we can formulate the problem in terms of Lagrange's multipliers p: find a pair {u, p}, such that u E K and p E A and and J(u, p) < J(w, p), for all w G K, (16) /(*!, u) < /(p, u), for all rj G A, (17) This formulation leads to a very fast Uzawa's algorithm, see e. g. Prochazka & Sejnoha^, which we also use here in an improved version, following the idea of the transformationfieldanalysis, Dvorak & Prochazka*. 4 Example Several examples were tested by the FEM. In all examples thefiberpossessed the following material properties E* =210 GPa, v* In the case of matrix E - 17 GPa, i/ = 0.3. The fiber volume ration was 0.6, i. e. the radius a = The tension strength <r+ = 3GPa. When no debond appeared along the interfacial boundary, the homogenized stiffness matrix L* had the following components (measured in GPa): L* = From the above matrix one can conclude that the responses on normal unit strains are computed with high accuracy while the results from shearing strains are less accurate. In the case of the largest debond the components of homogenized stiffness matrix were very close to the stifness matrix of material coefficients of the matrix, i. e. the fiber had very low influence on the overall stiffness: L* =

10 98 Free and Moving Boundary Problems In the case of normal strains, there were nearly no changes in values of the components of L* starting from the approximate value of boundary displacement Uy^ = mm (from the symmetry also at Uy^ = mm). In the case of shear strain at Uy^ Uy^ mm approximately constant values of the stiffness commenced. The change of the material overall properties is schematically shown in Fig. 5 above for ID problem. 5 Conclusions In this paper a homogenization procedure of debonding composite structure was studied. In the homogenization it was shown that the additional displacement field is in effect only in the direction of applied displacement along the outer boundary. The other components disappear. Consequently, the appropriate overall strains may be split into exactly one nonzero component and the other vanish. This is very importante when applying the standard microstructure homogenization. Starting with "Minmax" formulation of the debonding problem, the procedure for solving this problem followed Uzawa's algoritmus, which provides very fast, precise and reliable iterative schema. References 1. Dvorak, G. J. & Prochazka, P.: Thick-walled composite cylinders with optimal fiber prestress. Composites, Part B, 27B, pp , Lene, F.: Comportement macroscopique de materiaux elastiques comportant des inclusions. C. R. Acad. Sc. Paris 286, A, pp , Ponte Castenada, P.: The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids, 39, j_, pp , Prochazka, P. & Sejnoha, J.: Behavior of composites on bounded domain. BE Communications, 7, 1_, pp. 6-8, Prochazka, P. & Sejnoha, M.: Development of debond region of lag model. Computers & Structures, 55, 2, pp , Suquet, P. M.: Homogenization Techniques for Composite Media. Lecture Notes in Physics 272, (eds. E. Sanchez-Palencia and A. Zaoui), Springer-Verlag Berlin, pp , Sejnoha, J., Prochazka, P. & Sejnoha, M.: Evaluation of localfieldsin damaged composites. Localized damage'96, Fukuoka, CMP, ACKNOWLEDGEMENT: This paper was supported by GACR, grant project # 103/99/0756.