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Imperfect interlaminar interfaces in laminated composites: interlaminar stresses and strain-energy release rates V.Q. Bui a,*, E. Marechal b, H. Nguyen-Dang a a

Department of Fracture Mechanics, Laboratory of Aeronautic and Spatial Technique (LTAS), University of LieÁge, 21 E. Solvay, B-4000 LieÁge, Belgium b SAMTECH Company, 25 FreÁre-Orban, 4000 LieÁge, Belgium Received 19 April 1999; accepted 30 July 1999

Abstract A numerical procedure with interface ®nite elements has been carried out in order to investigate the in¯uence of imperfect interlaminar interfaces on the local mechanical behaviour of composite laminates. By modelling laminates at a mesoscopic scale, interlaminar interfaces are clearly represented and the eects of interfacial imperfections are thus characterised by displacement discontinuities across each interface. Emphasis is placed on assessing implications of weakened interfaces for the distribution of interlaminar stresses and the change in strain-energy release rates. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: C. Laminates; Imperfect interface; Interface ®nite element

1. Introduction For composites that are laminated through the thickness, the bonding state at interfaces between two adjacent layers clearly plays an important role in determining the mechanical behaviour of composite laminates. In particular, it relates to the likelihood of delamination, which is an interlayer separation damage mode in laminates. From a strictly physical point of view, the existence of a perfect interfacial bond in a real laminated composite seems impossible, since such an idealised interface cannot be delaminated. An appropriate model of composite laminates with imperfect interfaces should therfore be adopted as a more realistic approach to the study of laminates. The signi®cant in¯uence of imperfect laminates on the global mechanical response of laminated structures has recently been recognised and it had also been possible to show the impact of interfacial relaxation on the interlaminar stress distribution [1]. However, most eort has been devoted to the case where the presence of interlaminar stresses is primarily due to warping cross sections in thick laminates, whereas it is well known that interlaminar stresses could also arise from free edges where the existence of geometrical and material discontinuities * Corresponding author.

is found [2]. Moreover, this type of interlaminar stress appears more severe and hence dicult to determine than that referred to earlier, since it might present a singular behaviour. Yet the in¯uence of imperfect interfaces on this particular behaviour has not been investigated. If continuum-mechanics interlaminar stresses are physically responsible for triggering initial failure at an early stage at interfaces, fracture mechanics strainenergy release rate characterising driving forces for crack extension [3] appears to be an important parameter for controlling interlaminar crack growth Ð a common associated mode of interlaminar failure. In addition, these quantities are of practical value in studying the delamination problem, since they possess an energy sound while the near-®eld stresses might have an oscillatory singularity and may not have the usual signi®cance. Therefore, an investigation of strain-energy release rates in the context of imperfect interfaces is likely to be necessary. Toward a treatment of analysing composite laminates in the framework of imperfect interfaces, more complete information on the in¯uence of interfacial imperfections on the local mechanical behaviour of laminates is essential. In particular, a study of the interlaminar stresses and the strain-energy release rates needs to be carried out this; is the principal aim of our present work.

0266-3538/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00101-3

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2. Method of approach Macroscopic models of laminates are generally based on the eective-modulus representation for composite materials and widely used in the analysis of structural composite laminates. Through the well-known homogenisation concept, each ply of the ®bre matrix system is represented as homogeneous anisotropic medium with eective-modulus properties. While this approach allows us to avoid the great diculty in characterising the strong heterogeneity of composite materials, this model inherently introduces an arti®cial material discontinuity between adjacent plies because individual ®bres and matrix material are not well represented. Consequently, it might lead to erroneous conclusions in certain cases, for example point-by-point values of interlaminar stresses near a free edge [4]. To overcome this limitation, it is preferred to model laminates at a ®ner scale so that interfaces will be clearly represented and the arti®cial interlayer discontinuity could consequently be allowed for. In this work, we shall concern ourselves with the so-called meso-scale in which laminates are modelled as a stack of anisotropic homogeneous plies connected by distinct interlaminar adhesive layers (Fig. 1). Such a model, which appears appropriate to represent an actual laminate, has already been employed in the analysis of interlaminar shear stresses [5]. Accordingly, interlaminar shear stresses ts, tt were calculated directly from equilibrium conditions and more importantly, they resulted from sliding displacement discontinuities s , t between adjacent plies. As a limiting case of this model where the adhesive layer thickness tends to zero, we arrive at a mesoscopic model of laminates with zero-thickness interfaces. Of course, we always reserve the displacement-discontinuity property which is an essential point in comparison to the traditional macroscopic model. The latter requires continuity of both stresses and displacements across interfaces. Furthermore, we also postulate the existence of a displacement jump, n , in the normal direction (Fig. 1) and adopt the following linear non-interaction behaviour at interfaces [1]: ti di i

1

where di (i=n, s, t) is the stiness of interfaces.

For a ®nite-element analysis, the interlaminar interface is conveniently represented by continuous interface elements. These speci®c ®nite elements are formed by degenerating continuously to zero the thickness of a thin layer ®nite element. By this way, layer strains tend to constant values across the thickness, rotate themselves with the direction of discontinuity and become proportional to relative displacements [6]. Therefore, the desired mechanical behaviour (1) for interlaminar interfaces can be achieved exactly. Note that in the mesoscopic model, the plies can be modelled by using traditional multilayered elements, which are available for example in the SAMCEF code [7]. From its speci®c formulation, the utilisation of interface ®nite elements for the determination of the interlaminar stresses, ti, is straightforward with some prominent advantages. Firstly, the interface element can estimate values of interlaminar stresses right at interfaces thanks to its zero thickness. Consequently, no further eort is required, for example, an extrapolation or averaging procedure of stress values calculated at interior numerical integration points to interfaces as in a pure displacement ®nite-element formulation. Secondly, the continuity condition of interlaminar stresses across interfaces is always guaranteed. This interlayer stress continuity is found to be very important in obtaining an accurate estimation of interlaminar stresses. Take note also that interlaminar stresses given by interface elements are recovered from equilibrium equations, which are more accurate than from constitution relations. Thirdly, the free transverse stress condition on boundaries can also be somewhat satis®ed. Since interlaminar tractions of interface elements are in direct relation to displacement jumps and thus nodal displacements, an appropriate constraint to the corresponding nodal displacements ensures traction-free conditions. Finally, interface elements allow the calculation of interlaminar stresses with dierent degrees of interlaminar imperfect bonding. A suciently precise estimation of interlaminar stresses by interface elements oers an attractive possibility to estimate directly the Irwin's virtual crack closure integral relating to strain energy release rates Gi: Gi lim

a !0

1 2a

a 0

ti r; 0i a ÿ r; dr

2

where ti(r; =0) is the stress ahead of crack tips and i (a ÿ r; ) relative displacements behind crack tips corresponding to an in®nitesimally virtual crack extension a. Once the strain energy release rate components GI, GII and GIII corresponding to opening and sliding modes are determined, the total strain energy release rate is deduced: Fig. 1. Modelling of composite laminates with distinct interfaces.

G GI GII GIII

3

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133

Fig. 2. Two run stage procedure for calculation of strain energy release rates.

Physically, the Irwin's integral indicates the work necessary to close the crack back to its original length. The following procedure of two run stages is, therefore, appropriate to calculate numerically this integral (Fig. 2). For the sake of simpli®cation, we will illustrate this procedure through a ®nite element analysis with elements of 1st order. The application for the case of higher order elements is quite similar. Furthermore, we suppose that a Lobatto schema will be employed in all numerical integration relating to interface ®nite elements. This kind of integration scheme is found to be appropriate, particularly when a rather high value of interfacial stiness is used, while the employment of the traditional Gauss scheme could lead to erroneous oscillations of stresses. Hence, output values of interlaminar stresses will be given at nodal points of interface elements. In the ®rst run, the crack is closed by holding the nodes e, f, and i, j together. The interlaminar stresses ti at these nodes are evaluated. In the second run, the structure is loaded in the same manner but the nodes e, f and i, j are released by simply setting the stiness of the interface element right at the crack tip to zero. The relative displacements i at these nodes are recorded. With a numerical integration scheme with p sample points whose the positions are exactly at the element nodes, the Irwin's integration can be approximately estimated by the following manner: Gi

1 a lim ti i p wp 2a a !0 p p 2

constraint to stien the structure. The second one refers to a rapid variation of interlaminar stresses near a freeedge. The later is also called the free-edge eect and more important than the former. 3.1. Interlaminar stresses due to warping The classical laminated plate theory is based on several assumptions, and the most important of them is to neglect transverse shear and transverse normal strain eects in the thickness direction. Consequently, the error of such a theory increases with the augmentation of the plate thickness. As will be shown in the following problem of laminates in cylindrical bending, the existence of interlaminar stresses due to transverse eorts by warping the cross-section is quite evident. Consider a two layer unidirectional composite beam of length a, which is simply supported at both ends (Fig. 3). Each layer of the composite laminate has identical thickness h/2 and its elastic constants are given in Table 1. The structure is subjected to a sinusoidal tension q=q0sin(mx=a) at the top surface. In our analysis, the constants q0 and m are ®xed to unit values. Also, a span ratio a/h=20 will be taken for numerical calculation. The laminated beam is in a cylindrical bending state and 2D modelling can be involved to carry out the analysis. For ®nite element computation, all the elementary plies are discretised by pure displacement quadrangular elements of degree 2 with plane strain

4

where wp is the weighted factor at point p. 3. In¯uence of imperfect interface on interlaminar stresses Problems relating to interlaminar stresses could be in general distinguished into two types. The ®rst one concerns to transverse shear deformations that introduce a

Fig. 3. Laminated plate subjected to a transversal sinusoid load.

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assumption (Fig. 4). Between adjacent plies, line interface elements are inserted to represent the unidirectional interlaminar interface. The whole analysis is thus performed with the help of the SAMCEF code. The in¯uence of imperfect interfaces on the static bending response is clearly shown elsewhere and it is not our main interest in this investigation. However, it is also brie¯y represented here, for the sake of evaluating the validity of our modelling. As shown in Table 2, if the transverse de¯ection w(a/2,0), which is normalised by the following relation: w

100E2 h3 w a=2; 0 q0 a4

5

is equal to 0.551 in a perfect bonding condition, this de¯ection will be increased by 3.656 times in the case of a pure shear slip and multiplied by 7.312 times when the interface is completely debonded. A very close agreement to the results issued by Liu et al. [1] is observed. Our primary interest is now to investigate the importance of an imperfect interface on the interlaminar normal stress tn in the middle of the span (x=a/2 and z=0) and the interlaminar shear stress ts at the end (x=0, z=0) of the laminate. The normalised values as de®ned below will be employed in the resulting representation following:

Table 1 Mechanical properties of ply Young's modulus (106 psi) Shear modulus ( 106 psi) Poisson's ratio

E1=25 G12=0.5 v12=0.25

E2=1 G23=0.2 v23=0.25

Fig. 4. Discretisation of a half section of a composite plate.

Table 2 Comparison of transverse de¯ection with dierent imperfect states at interface

w-

Liu et al. Present

Perfect interface

Imperfect interface

dn!1 and ds!1

dn!1 and ds!0

dn!0 and ds!0

0.552 0.551

3.684 times 3.656 times

7.360 times 7.312 times

tn

tn a=2; 0 q0

ts

ts 0; 0 q0

6

In the case that the normal bonding rests always perfect (dn!1), the interlaminar shear stress ts at the end beam reduced quickly from its maximum to a zero value as the shear bonding condition ds changed from perfect to a complete slip (Fig. 5). The same trend is also observed for the interlaminar normal stress tn in the case of a mixed mode of imperfect interface. In this later case, the interfacial shear stiness is ®xed to a value of ds=5103 psi/in while the normal bonding dn varies from a perfect state to totally debonding one (Fig. 6). Again, we obtain a good accord between the present results and those given by Liu et al [1]. 3.2. Interlaminar stresses due to free-edge Because of material and geometrical discontinuity at free edges, certain interlaminaire stresses will rise in magnitude near the free edge region of a laminated structure as will be seen clearly in the following problem of laminates under uniform axial extension. Examine a symmetric cross-ply laminate of four layers subjected to a prescribed uniform in-plane normal strain e0 in the axial x-direction (Fig. 7). Thickness of each ply is supposed identical and will be denoted by h and the total thickness of a laminate is hence 4h. In the present analysis, a width of laminates 2b=16h is taken and consequently the ratio of the width to thickness of the 4-ply laminate will be 2b/4h=4. Two con®gurations of laminates stacked in [0/90]s and [90/0]s sequence are examined. The ply elastic constants of unidirectional composite are given in Table 3. The analysis is performed in the y±z plane and owing to the symmetry about the z-axis, only half of the crosssection needs to be involved in a ®nite element analysis (Fig. 8). Symmetry about the axis z=0 could be used to reduce the analysis to a quarter of the cross-section. However, this is not done here, because interface elements need to be introduced at interface z=0 in order to give the value of interlaminar stresses at this interface. Following a ®nite element procedure and since it is expected that stresses vary rapidly near the boundaries, rather ®ne elements are required near the free-edge to approximate the exact solution. Away from the ends where the loads are applied, the displacement components in x, y and z direction at any plane of x=constant can be generally expressed as [2]: 8 < u x; y; z e0 U x; y v x; y; z V y; z 7 : w x; y; z W y; z where U, V and W are functions of y and z only. The laminate can therefore, be, considered in a general plane

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135

Fig. 5. Variation of interlaminar shear stress ts (dn=1010 psi/in).

Fig. 6. Variation of interlaminar normal stress tn (ds=5103 psi/in).

strain state. For the cross-ply laminates in question, the elasticity axes of orthotropic layers are along the y and z directions, the warping function U(y,z) should be zero. Consequently, the corresponding displacement ®eld can be exactly presented by using the axisymmetric multilayered elements that are available in the SAMCEF code. In the case of perfect interfaces, Figs. 9±11 show the interlaminar stresses distribution along z=0 at the midplane interface and z=h at the (0/90) interface obtained by the present interface ®nite element and by conventional displacement-type ®nite elements of Wang et al. [2]. In general, the results are close to each other. A fairly good accord in the distribution of tn along z=0. Especially, the rising trend toward the free edge is well-detected (Fig. 9). Also, the results on the tn distribution along the interface z=h agree very well

Fig. 7. Laminate subjected to uniform in-plane strain e0.

throughout the whole region y/b, except that our solution gives a slightly higher value at the free edge. Wang et al. [2] found a value of 0.43 at the co-ordinate y/ b=0.9985 in the case of [0/90]s laminate, while the interface element give a value of 0.48 right at the edge

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Table 3 Mechanical properties of ply Young's modulus (106 psi) Shear modulus (106 psi) Poisson's ratio

E1=20.0 E2=E3=2.1 G12=G23=G13=0.85 V12=23=13=0.21

Fig. 8. Half section modelling of a plate.

y/b=1 (Fig. 10). The marked dierence is, therefore, quite reasonable. Fig. 11 shows the distribution of tt along z=h. Again, it is observed that good agreement is for most of the region y/b. Note that, in the case of constant-strain triangular elements which predict a constant stress ®eld within each element, the traction-free conditions at free edges can be approximately satis®ed in an average manner. However, such requirement can be exactly satis®ed in the interface element approach. The high stress gradient of tn at the interface z=h in the free-edge region y/b=1 in the case of the [0/90]s laminate suggests the possibility of a singularity of the normal stress tn . Indeed, with a much ®ner mesh, we will obtain a greater value of 1.365 that is not so far from the value of 1.24 issued from a series solution of Wang

Fig. 9. tn-Distribution along laminate centre line z=0.

Fig. 10. tn-Distribution along (90/0) interface, z=h.

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Fig. 11. tt-Distribution along (90/0) interface, z=h.

Fig. 12. Shift of stress solution from non-convergence to convergence.

et al. (Fig. 12) [8]. Therefore a convergence of stresses seems impossible to achieve because of the singularity. Examine now this singularity behaviour in the context of imperfect interfaces. To achieve it, the interfacial normal stiness dn needs to be varied in order to represent dierent bonding conditions at interfaces. When it is done, we observe, as it could be guessed, a rapid reduction of the stress peak for both types of meshes (Fig. 12) according to the relaxation at interfaces. More importantly, the non-convergence of stress solution shifts to a stable convergence when the interfacial normal stiness dn ranges from a rather high value to a more moderate one, for example dn<106 psi/in in this case.

Fig. 13. Con®guration of DCB test.

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4. In¯uence of imperfect interface on strain energy release rates Problems concerning the interlaminar interfaces of laminated composites is generally complex in nature and dicult to solve, since it involves not only geometric and material discontinuities, but also the inherently coupled modes I, II and III of fracture in an anisotropic layered material system. Interlaminar fracture is usually associated with crack growth and hence its control and prediction are accomplished through strain energy release rates. 4.1. Pure modes of fracture Consider a double cantilever beam (DCB) specimen of thickness 2H (Fig. 13). The beam of total length l contains an initial crack of length a between its arms. In most applications, the choice (l-a)>>h is taken and hence the in¯uence of the crack on the free-end of the specimen will be negligible. The specimen is limited in our analysis to unidirectional laminates and its engineering constants are given in Table 4. In such a con®guration of test and under the action of the applied force P, there exists only one normal component tn of interlaminar stresses ahead of the crack. This normal stress has a tendency of opening the crack tips and hence the interlaminar crack growths in mode I of fracture. Our interest is to calculate the available energy for this crack extension. Table 4 Mechanical properties of ply Young's modulus (GPa) Shear modulus (GPa) Poisson's ratio

E1=3.0 E2=E3=1.0 G12=G12=G12=0.533 v12=v13=0.3 v23=0.55

For ®nite element computation, the specimen is assumed in a plane strain state and again it can be modelled by pure displacement quadrangular elements of degree 1. Line interface elements are inserted between two arms to represent the unidirectional interlaminar interface. By taking dn=ds=107 N/mm3, a perfect interface is supposed to exist in the specimen. In this case, our modelling with interface elements appears quite adequate, since a variation of the de¯ection w at the point of load application to the ratio a/h is found in good agreement (Fig. 14) to the closed form solution of Penado [9]. Also, a result of strain energy release rate by using a virtual crack extension a=0.05 mm seems very satisfying (Fig. 15). It is important to note that Penado's solution is employed here in a plane strain version and with a coecient relating to foundation modulus f=2.7 in stead of f=4 as proposed by Penado. Note that if the DCB specimen is treated simply as a built-in-cantilever beam, i.e. elastic foundation eect at crack tips is neglected, one would have: G0I

P2 12 2 a E 1 b2 h3

8

With some relaxation in the interfacial normal stiness dn, an imperfect interface can now be examined. To the authors' knowledge, a result of strain energy release rate in the presence of imperfect interfaces is not available in the literature. Hence, we will try to exploit here the Penado's solution on the energy release rate of the DCB specimen with an adhesive layer with some appropriate interpretation (see Appendix) relating the adhesive thickness to the interfacial normal stiness. When it is done, we obtain a ``closed form solution'' which allows the estimation of the validation of our numerical results.

Fig. 14. De¯ection at the point of load application.

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Consider for example the case a/h =5. A good trend in agreement between the results is found (Fig. 16) for a large range of dn variation. Clearly, an increasing relaxation at interfaces leads to an enlarging deformation at the interlaminar crack tip and hence this helps to explain the augmentation of the energy available for crack extension. In both cases of perfect and imperfect interfaces, the strain energy release rate is recovered by direct computation of interlaminar stresses in stead of nodal forces. The obtained results show the validation of the approach calculating directly the Irwin's integral by using interface elements.

139

4.2. Mixed modes of fracture Examine now a simply supported composite beam (Fig. 17) where the laminate has a cross-ply con®guration (905/05/905). The orthotropic material properties of each ply are given in Table 5. The thickness of the beam Table 5 Mechanical properties of ply Young's modulus (106 psi) Shear modulus (106 psi) Poisson's ratio

Fig. 15. Strain energy release rate at the crack tip.

Fig. 16. Strain energy release in the case of an imperfect interface (a/h=5).

E1=17.4 G12=0.76 v12=0.3

E2=1.43

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is h=0.0861 in and the length of the beam between two supports is 2l=2 in. An initial transverse edge crack located at the centre of the bottom ply and perpendicular to 0 ®bres is created in order to induce an interlaminar crack of length 2a. In such a condition, there exists both normal and tangential interlaminar stresses at the crack tips and this interlaminar crack undergoes essentially a combination of opening and sliding modes of fracture, i.e. a mixed mode I/II. One wishes now to evaluate the strain energy release rate that plays the role of driving forces for the crack extension. Owing to the symmetry of the structure, only half of the cross-section needs to be involved in a ®nite element analysis. With a virtual crack extension a=0.005 in, we ®nd a fairy good agreement of total strain energy release rate G=GI+GII to the analytical result of Sun et al. [3] in the case of a perfect interface (ds=dn=1012 psi/ in). The fundamental mode contribution GI, GII is also favourably compared with the ®nite element result of Liu et al. [10] (Fig. 18). From this the result on mode separation, we observe clearly the domination of the opening mode I to the sliding mode II during the crack growth.

Fig. 17. Three-point bending beam with an edge crack.

However, the situation becomes more complex with the presence of a certain imperfection at the interface. Indeed, suppose that the normal bonding rests always perfect (dn=1012 psi/in) and the shear bonding is weakened for some reason. We observe a strong variation of mode contribution depending on the value of the interfacial shear stiness ds. More precisely, there exists an interval ds(105±107) psi/in where the energy devoting to the sliding action is really major (Fig. 19), whereas beyond of this limit, this energy turns out to be minor. If the shear bonding is now maintained to be perfect (dn=1012 psi/in) and the normal bonding is relaxed, we obtain a quite dierent image of mode contribution. For a rather large range of the interfacial normal stiness (dn<106 psi/in), the GII rests nearly constant while the GI growths increasingly (Fig. 20). Beyond of this limit, the GII increases slightly and GI reduces weakly before reaching ®nally to a steady plateau at a rather great value of dn. 5. Discussions In¯uence of imperfect interfaces on interlaminar stresses is signi®cant. The stress singularity in the case of perfect interfaces shifts to a stress concentration once interfaces are imperfect. As a result, the peaky interlaminar free-edge stresses are evidently reduced and the existence of a ®nite value of interlaminar stresses is hence expected rather than a theoretically in®nite value. Elsewhere, it is reported that the calculated normal stress tn at the free edge with the assumption of a perfect interface can be considerably higher than the experimental value [11]. The presence of imperfect interfaces appears to give, therefore, a more realistic result. Furthermore, although the existence of an interlaminar perfect interface

Fig. 18. Variation of strain energy release rates to dierent crack lengths.

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141

Fig. 19. In¯uence of normal bonding to strain energy releases.

Fig. 20. In¯uence of shear bonding to strain energy release.

is somewhat idealised, in reality these ``interlaminar perfect bonds'', even if they were able to exist in a laminate, would probably relax themselves immediately under the eect of stress singularities resulting in a weakened bonding. The net eect is that stress redistribution occurs and instead of the stress singularity in a very narrow region, a stress concentration will be present in the larger zone at real interfaces. It is argued somewhere that interlaminar stresses at free edges should be ®nite rather than singular. This has been explained by the use of eective modulus from the homogenised assumption for composite material in the stress analysis of structural composite laminates [4]. However, the present analysis for the ®rst time suggests

the important role of interfacial bonding in the vanishing of the stress singularity. As a ®rst application of this consequence, if a point stress criterion is meaningless due to the stress singularity with the traditional assumption on a perfect interface, it appears quite acceptable in the framework of imperfect interfaces. It suggests, therefore, in stead of an average value of stresses over a characteristic length, point values can be directly used in a stress-based criterion devoted to delamination onset prediction. Secondly, with the presence of oscillatory singularities in the case of a perfect interface assumption, it is clear that the mode separation Gi of strain energy release rates does not converge to a de®nite value as a!0

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fracture mechanics. Therefore, also in the framework of imperfect interfaces further work is currently performed with the damage mechanics in order to examine damage evolution during a debonding process. Acknowledgements Fig. 21. DCB specimen with an adhesive layer.

when the Irwin's integral is performed, despite the fact that the total energy release rate G is always well de®ned [3]. Indeed, the strain energy release rate G represents a global concept, since the global energy release rate of the whole structure is always considered, even if crack onset is investigated at a local point. However, it is not the case for mode separations Gi that depends closely to the local cinematic movement of crack tips and, therefore, representing a local concept. In the absence of such singularities at weakened interfaces, the non-convergence of fundamental rupture mode Gi is expected to shift to a convergence value. Hence, the obtained value Gi could be used with a high con®dence in a fracture energy-based criterion devoted to delamination growth prediction. It is worth noting that a direct estimation of the Irwin's integral requires a precise knowledge of stress distribution, which is dicult to achieve near the crack tip with the presence of singularities. Therefore, an indirect estimation of this integral is preferably performed with the so-called virtual crack close technique through the use of nodal forces in stead of stresses [3]. From the disappearance of singularities, this diculty is no longer existent. This suggests, therefore, a direct estimation of mode separation Gi as already done in our work. 6. Conclusions Imperfection at interlaminar interfaces of laminated composites is numerically found to have a signi®cant in¯uence on the local interlaminar stress distribution and mode separation of strain energy release rates. Especially, the well-known stress singularity existing with the presence of a perfect bonding could be shifted to a stress concentration once the interlaminar bonding is weakened. The consideration of imperfect interfaces appears to give a more realistic result to the real world of composite laminates. Furthermore, it oers an evident advantage in the delamination analysis with the ®nite element methods, since we can avoid the serious problem of mesh dependence in studying local mechanical behaviour at interfaces by the continuum mechanics and the

This work is supported by Walloon Regional Government through ``Mobilisation-MultiMaterials'' and ``European COST-512'' projects. The technical collaboration of Samtech S.A. is also appreciated. Appendix Consider at ®rst an adhesively bonded DCB specimen where 2e is the total adhesive thickness (Fig. 21). According to Penado's solution, the strain energy release rate at the crack tip is given by the following relation: GI

P2 12 P2 3 la 12 2 2 3 E1 b l h G13 b2 2h

with l4

3k E1 bh3

where k is a foundation modulus. Once the stines of the adherend arm kaherend and of the adhesive layer kadhesive are known: kadherend f

E3 b h

kadhesive

Ea b e 1 ÿ v2a

the foundation modulus k can be estimated by assuming that the adherend kaherend and the adhesive kadhesive act as springs in series: 1 1 1 k kadherend kadhesive Now, in laminates modelling with distinguished interfaces, we can assume the existence of an additional layer of equivalent isotropic resin Ea, a between adjacent plies. It is in the limit of degenerating to zero the thickness 2e of a moderate thickness ®nite element that the interface ®nite element of zero-thickness can be obtained [6]. Morever, the stiness di of this interface element can be calculated by the following relations (a =0):

V.Q. Bui et al. / Composites Science and Technology 60 (2000) 131±143

dn

Ea 2e

ds

Ga 2e

It is clear now that a relation between kadhesive and dn, ds can be etablished through the mechanical properties Ea and a . For example, in the case where the DCB specimen is in a plane strain state (b=1), we achieve the following very simple relations: dn

kadhesive 2

ds

dn 2

If these relations are respected, the result from the modeling laminates with interface elements is quite comparable with the result issued by the close solution of Penado. References [1] Liu D, Xu L, Lu X. Stress analysis of imperfect composite laminates with an interlaminar bonding theory. Int J Numer Meth Eng 1994;37:2819±39.

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[2] Wang ASD, Crossman FW. Some new results on edge eect in symmetric composite laminates. J Comp Mater 1977;11:92±106. [3] Sun CT, Manoharan MG. Growth of delamination cracks due to bending in a [905/05/905] laminate. Comp Sci Technol 1989;34:365±77. [4] Lin Ye. Some characteristics of distributions of free-edge interlaminar stresses in composite laminates. Int J Solids Structures 1990;26:331±51. [5] Puppo AH, Evensen HA. Interlaminar shear in laminated composites under generalized plane stress. J Comp Mater 1970;4:204± 20. [6] Hohberg J-M, Schweiger HF. On the penalty behaviour of thinlayer elements. In: Pande GN, Pietruszczak S, editors. Numerical models in geomechanics. Rotterdam: Balkema, 1992. [7] SAMCEF, User Manuals, release 8.0, 1999. [8] Wang JTS, Dickson JN. Interlaminar stresses in symmetric composite laminates. J Comp Mater 1978;12:390±402. [9] Penado FE. A closed form solution for the energy release rate of the double cantilever beam specimen with an adhesive layer. J Comp Mater 1993;27:383±407. [10] Liu S, Kutlu Z, Chang FK. Matrix cracking and delamination in laminated composite beam subjected to a transverse concentrated load. J Comp Mater 1993;27:436±70. [11] Kim RY, Sony SR. Experimental and analytical studies on the onset of delamination in laminated composites. J Comp Mater 1984;18:70±80.

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