3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects

Transcription

1 IOP Conference Series: Materials Science an Engineering 3-D FEM Moeling of fiber/matrix interface eboning in UD composites incluing surface effects To cite this article: A Pupurs an J Varna 2012 IOP Conf. Ser.: Mater. Sci. Eng Relate content - Micromechanics of the fragmentation process in single-fiber composites M Nishikawa, T Okabe, N Takea et al. - Crack opening isplacement etermination in amage cross-ply laminate using electronic speckle pattern interferometry (ESPI) H Zria, M S Loukil, J Varna et al. - Characterization of amage composite laminates by an optical measurement of the isplacement fie M S Loukil, J Varna an Z Ayai View the article online for upates an enhancements. This content was ownloae from IP aress on 27/01/2019 at 04:47

2 3-D FEM Moeling of fiber/matrix interface eboning in UD composites incluing surface effects A Pupurs an J Varna 1 Department of Engineering Sciences an Mathematics, Division of Materials Science, Luleå University of Technology, SE 97187, Luleå, Sween anrejs.pupurs@ltu.se, Janis.Varna@ltu.se Abstract. Fiber/matrix interface ebon growth is one of the main mechanisms of amage evolution in uniirectional (UD) polymer composites. Because for polymer composites the fiber strain to failure is smaller than for the matrix multiple fiber breaks occur at ranom positions when high mechanical stress is applie to the composite. The energy release ue to each fiber break is usually larger than necessary for the creation of a fiber break therefore a partial eboning of fiber/matrix interface is typically observe. Thus the stiffness reuction of UD composite is contribute both from the fiber breaks an from the interface ebons. The aim of this paper is to analyze the ebon growth in carbon fiber/epoxy an glass fiber/epoxy UD composites using fracture mechanics principles by calculation of energy release rate G II. A 3-D FEM moel is evelope for calculation of energy release rate for fiber/matrix interface ebons at ifferent locations in the composite incluing the composite surface region where the stress state iffers from the one in the bulk composite. In the moel iniviual partially ebone fiber is surroune by matrix region an embee in a homogenize composite. 1. Introuction Many engineering structures mae from polymer composites such as win turbine blaes, aircraft wings, etc., are subjecte to thermal an mechanical fatigue loas. Due to overloa or initial manufacturing efects, microamage such as fiber breaks or matrix cracks may occur after certain amount of loa cycles. The presence of microamage not necessarily means that the structure is not suitable for further service. However, ue to further cyclic loaing the propagation of microamage will occur an after a certain number of applie loa cycles the mechanical properties will be significantly egrae leaing to failure of the structure. In polymer composites microamage in form of fiber breaks or matrix cracks is the most common. Uner small-meium stress tension-tension fatigue of a UD polymer composite matrix cracks may occur after some number of loa cycles. In further fatigue loaing the matrix cracks are eflecte an the propagation of amage is in form of growing fiber/matrix interface ebon cracks. In high stress mechanical fatigue loaing, fiber breaks occur in ranom positions after few loaing cycles. Due to energy release uring the breaking of the fibers, partial fiber/matrix interface eboning typically occurs in vicinity of fiber breaks. Also in this case the further propagation of amage will be in form 1 To whom any corresponence shou be aresse. Publishe uner licence by Lt 1

3 of growing interface ebon cracks. In this paper only the latter scenario of amage initiation an propagation is stuie. Characterizing the growth of amage propagation is important to estimate the lifetime an serviceability of the structure. Analytical [1] an numerical [2] moels have been earlier evelope to characterize the growth of ebon crack when the broken an partially ebone fiber is in the bulk of the composite. The main objective of the present paper is to analyze the fiber/matrix ebon growth as epenent on the fiber position with respect to the specimen surface. 2. Mechanics of ebon propagation Analyzing cyclic loaing Paris law, well known for single crack growth in metals, may be suitable for characterization of the ebon growth along fiber/matrix interface as a function of applie number of loa cycles N. Paris law expression escribes the increase of the ebon surface area A as: A N = B ( G II ) m (1) where: B an m are unknown parameters, which have to be etermine experimentally; G II is the energy release rate ifference between values corresponing to ε max an ε min in a cycle. The cyclic loaing type consiere in this stuy is mechanical tension-tension fatigue (ε max > 0 an ε min > 0). Notation G II with inex II is use since only Moe II ebon crack propagation is relevant - for the stuie polymer composites the matrix Poisson s ratio is higher than for the fibers an therefore in purely mechanical tension loaing the raial stresses on fiber surface will always be compressive, the interface crack is close an pure Moe II crack propagation will occur. Debon length increment for circular fiber is relate to ebon surface increment as ( ) 1. l = A 2π r f Interface ebon growth relate energy release rate G II can be calculate by various analytical an numerical methos as escribe in Section 3 of this paper. The values of two parameters in Paris law ( B an m in (1)), however, can only be etermine experimentally. Having a reliable moel for energy release rate G II calculation in han, an performing experimental measurements of ebon length as a function of applie loa cycles N wou allow fining the values of Paris law parameters B an m from the best fit. However, performing experimental measurements of fiber/matrix interface ebon crack growth in the bulk of UD composite is a challenging task. Still it cou be one using, for example, microtomography techniques. Since Paris law parameters for interface ebon growth can be consiere as fiber/matrix interface property, a more convenient metho wou be measuring the ebon crack growth in a single fiber composite. Observing single fiber embee in a transparent block of resin is convenient an can be performe in an optical microscope. In [3,4], for example, ebon crack growth in single fiber composites was measure uring quasi-static loaing. Other option wou be to measure ebon growth on a composite specimen surface, as, for example, in [5], which can be one in an optical microscope as well. However, the ebon growth in vicinity of UD composite surface is ifferent than in the bulk of composite ue to ege effects. The high scatter in the experimental measurements in [5] cou be an inication that the interface ebon growth is very sensitive to the actual istance from the observe fiber to the surface of composite. Therefore, one of the aims of this paper is to calculate the epenency of energy release rate G II on the istance from fiber to composite surface. The change in energy release rate G II is expecte ue to change in the stress state aroun the partially ebone fiber when it is in vicinity of composite surface. It is expecte that the stress state aroun ebone fiber will have higher eviation from axisymmetric stress state, which we have when the fiber is in the bulk of composite, when it is closer to the surface 2

4 of composite. In such case 3-D numerical moels have to be use for calculation of energy release rate G II. However, at a certain istance from the composite surface the eviation from axisymmetric stress state will become negligible an more convenient axisymmetric moels evelope in [1] an [2] can be use for G II calculation. 3. Strain energy release rate for long an short ebons The ebon crack propagation can be consiere as self-similar when the tip of the fiber/matrix ebon crack is far away from the fiber break where it was initiate an also far from another ebon which may be approaching from the other en of the fiber. If the fiber is also sufficiently far from the surface of composite, the stress state aroun the fiber is axisymmetric an analytical solution for energy release rate G II can be foun. In [1] expressions of analytical moel for calculation of strain energy are presente. In case of short ebons, the ebon crack tip is close to the fiber break where it initiate from an the stress perturbation regions relate to fiber break an ebon crack tip interact. Due to interaction, G II is magnifie. The shorter is the ebon the higher will be the interaction an thus G II will be higher. Since the stress states interact, the self-similar conitions are no longer in power an analytical close form solution is not possible. Therefore finite element metho FEM calculations are useful for etermination of G II for short ebon lengths. As an output from FEM calculations stress an isplacement istributions are available an G II can then be easily calculate, for example, using the virtual crack closure technique [6] as escribe further. Virtual crack closure technique was use, for example, in [2] to calculate magnification of G II at short ebon lengths. 4. Material properties In this paper two ifferent UD composites were stuie carbon fiber/epoxy resin composite (CF/EP) an glass fiber/epoxy resin composite (GF/EP). The properties of composite constituents are given in Table 1. The accuracy of the values liste in Table 1 are important for correct calculations of the elastic properties of the composite. Importance of this accuracy is iscusse in [1]. Table 1. Elastic properties of constituents. Material E L E T G LT ν 12 ν 23 [GPa] [GPa] [GPa] [-] [-] CF GF EP Properties of CF/EP an GF/EP uniirectional composites were calculate using the Concentric Cyliner Assembly (CCA) moel introuce by Hashin [7,8] for the bone case. The transverse shear moulus was obtaine using the Christensen s self consistent moel [9]. Only fiber volume fraction equal to V f = 0.5 was stuie in this paper. Fiber raius for all cases was equal to r f = 4 µm. 5. FEM moel 5.1. Geometry of the moel For analysis of ebon growth in the bulk of composite, axisymmetric FEM moels have been evelope previously [2]. However, for the case when the fiber is in vicinity of composite surface, axisymmetric moel may not be applicable ue to ege (surface) effects. In this stuy a 3-D FEM moel consisting of partially ebone fiber surroune by a matrix cyliner an embee in a large block of effective composite was generate using finite element coe ANSYS [10]. The geometry of the 3-D FEM moel is shown in figure 1. As it is shown in figure 1 3

5 only one half of the moel was generate in FEM coe ANSYS, taking avantage of the symmetry conitions. Figure 1. Geometry fragment of the 3-D FEM moel. In figure 1 z, x are the axes in Cartesian system an ϕ is polar angle, l is the variable interface ebon crack length, L s is the total length of moel (½ of fiber fragment), l c is the variable coorinate of epth of the fiber (istance from fiber center axis to composite surface), c is the size of the effective composite surrouning the fiber an matrix in the y-irection. The values of geometrical parameters L s an c were taken from convergence analysis performe in [2] where it was foun that moel length L s = 90 r f (where r f is the fiber raius) an the effective composite size c = 5 r f are efficient for moeling accuracy to represent a long partially ebone fiber in an infinite composite. Contact elements were use on the surfaces of fiber an matrix in the ebone region. The effect of friction was neglecte. The moel was meshe with mesh refinements near the tip of the ebon crack in orer to obtain accurate solution of stress an isplacement istributions which are neee for G II calculations. Figure 2. Detail of the FEM moel use for G II calculations. F fiber, M matrix, C effective composite. 4

6 Since the virtual crack closure technique was use for G II calculations, the finite element mesh refinement was optimize for the chosen integration length. Figure 2 shows the etail of the FEM moel, incluing the length of integration l. Following the convergence analysis performe in [2] the length of integration in all calculations was set equal to l =1 r f Virtual crack closure technique The crack closure technique states that the energy release ue to ebon crack growth by A is equal to the work which is require to close the newly create surface from size A + A back to size A. Figure 2 can be use for geometric representation of the further statements. Closing the ebon crack by length l (from l + l to l ) by applying tangential tractions, points z l ; l + l which have relative tangential isplacement at the ebone surface in the region [ ] u l +l (z) = u l +l l fz (z) u +l mz (z) (2) are move back to coinciing positions. Here an in following the upper inex for stress an isplacement shows the length of the ebon use in calculations. l At the en of this proceure the shear stress in point z is equal to σ xz (z), which is the shear stress in front of the crack with size l. For the case when fiber is in the bulk of composite an the stress state aroun it is axisymmetric, the work require to close the crack by l can be expresse as: 1 π z (3) + + W ( ) = 2 rf u ( z) σ xz ( z) 2 In (3) σ xz is the shear stress component relevant to Moe II crack propagation. For a general 3-D stress state, which may not be axisymmetric the expression (3) can be rewritten as: 2π l W ( ) = u ( z) σ ( ) ϕ z xz z rf z (4) 2 0 Within the virtual crack closure technique it is assume that ue to small value of l the relative sliing isplacement at the tip of the crack with size l + l is the same as at the tip of the ebon crack with size l : u + z) = u ( z l ) (5) ( The usefulness of this assumption is that only one stress state calculation for a given ebon length is require. The energy release rate is efine as: W G II = (6) A 5

7 where A = 2πr l. Thus G f II equations (4), (5) an (6): can be easily calculate from a single stress state by combining G 2π l II = lim u ( z l, ϕ) σ (, ϕ) ϕ 0 z xz z rf z 4πr l (7) 0 l f As mentione, in all calculations performe in this stuy the length of integration was equal to l =1 r f, which as foun in [2] can be assume as sufficiently small to satisfy conitions of (7) (where l 0) an at the same time sufficiently large to obtain accurate stress an isplacement istributions. Distributions of isplacement u z an shear stress σ xz were obtaine by performing post-processing (path operations) in ANSYS. Since non-axisymmetric stress state was stuie, pre-efine paths were generate along the whole circumference of the fiber/matrix surface. To facilitate the path operations, the finite element mesh was uniform along the circumference. Coe comprising (7) in incremental form was generate for convenience of calculations as an input. Although the stress state aroun partially ebone fiber which is close to composite surface may not be axisymmetric, the crack growth was consiere as uniform the shape of the interface ebon crack front was assume to be circular with same z coorinates for all points of the tip of the cylinrical crack. 6. Results an iscussion 6.1. G II calculations Figures 3 an 4 show results of interface ebon growth relate energy release rate G II calculations for CF/EP an GF/EP composites. In all cases uniform axial isplacement u z = 0.01 L s was applie at the en surface of the FEM moel (see figure 2). Results in figures 3 an 4 are presente as functions of normalize ebon length l n = l r f 1. The calculations were performe for ifferent cases by varying the fiber epth parameter l c (istance from composite surface). For CF/EP calculations were one for three cases: l c = 10, 15 an 20 µm as shown in figure 3. For GF/EP calculations were one for two cases: l c = 10 an 15 µm as shown in figure 4. First of all it can be note from both cases (CF/EP in figure 3 an GF/EP in figure 4) that the values of energy release rate are higher for shorter ebon lengths. It is expecte that for shorter ebon lengths ue to interaction of stress states the energy release rate will be magnifie. However, when the ebon length is sufficiently long, the growth of the ebon crack becomes self-similar as it was shown in [2]. In the present paper a range of ebon lengths from l n =1.5 r f 4 r f has been stuie. As it can be seen from the trens in figure 3 an figure 4, this range of ebon lengths correspons to interaction region an a further increase of ebon length will result in smaller values of G II approaching the asymptotic self-similar value. Concerning the epenency of G II on fiber istance from surface (parameter l c ) it can be note for both CF/EP in figure 3 an for GF/EP in figure 4 that G II is higher when the fiber is closer to the surface of composite. Since the energy release rate is higher, it follows that the growth of the interface ebon crack will be faster for the fibers which are closer to the composite surface compare to the fibers in the bulk of composite. From calculation results for CF/EP in figure 3 it can be note that there is no significant ifference between results for cases when fiber istance from surface l c = 15 6

8 µm or l c = 20 µm. This istance l c = 15 µm, at which energy release rate G II becomes inepenent of istance from composite surface inicates the bounary between the close-to-surface region an bulk of composite region. Figure 3. Energy release rate G II as a function of normalize ebon length l n for CF/EP. l c values are in µm. Figure 4. Energy release rate G II as a function of normalize ebon length l n for GF/EP. l c values are in µm. To analyze the stress state aroun the broken an partially ebone fiber an how it epens on the istance from composite surface ( l c ) a comparison between the shear stress σ xz an axial isplacement u istributions was performe. As it can be seen from results shown in figures 5 an 6 (for CF/EP an GF/EP respectively), where stress istributions are shown for the case when angular coorinate ϕ = 0, there is a minimal epenence on the istance l c within the observe range from l c = 10 to 20 µm. In fact, the ifferences in calculate energy release rate G II values in this range were also relatively small (less than 4%), therefore small ifferences in stress istributions were expecte. 7

9 Regaring the axisymmetry of the stress istribution, epenency of shear stress σ xz istribution on angular coorinate ϕ was investigate. In figures 7 an 8 stress istributions at 3 ifferent angular irections ϕ = 0, 45 an 90 are presente for CF/EP an GF/EP respectively. The results in both figures 7 an 8 inicate that there is no significant epenence of shear stress σ xz istribution on the angular coorinate ϕ, which means that an axisymmetric approximation is applicable. The same can be conclue from analysis of isplacement istributions as shown in figure 9. Figure 5. Shear stress σ xz istribution in front of the tip of the ebon crack for CF/EP. z = 0 is location of the ebon crack tip. Angular coorinate in all cases ϕ = 0, l c values are in µm. Figure 6. Shear stress σ xz istribution in front of the tip of the ebon crack for GF/EP. z = 0 is location of the ebon crack tip. Angular coorinate in all cases ϕ = 0, l c values are in µm. 8

10 Figure 7. Shear stress σ xz istribution in front of the tip of the ebon crack for CF/EP. z = 0 is location of the ebon crack tip. Distance from surface in all cases l c =10 µm. Figure 8. Shear stress σ xz istribution in front of the tip of the ebon crack for GF/EP. z = 0 is location of the ebon crack tip. Distance from surface in all cases l c =10 µm. 9

11 Figure 9. Axial isplacement u z istribution behin the tip of the ebon crack for CF/EP. z = 0 is location of the ebon crack tip, l c values are in µm. The results show that within the observe region of istances from composite surface, the changes in energy release rate G II values are approximately 1.5-2%. This means that in the selecte range of istances from composite surface l c, the stress state is very close to the stress state in the bulk of composite. Using the present FEM moel geometry (figure 1) the minimal istance from composite surface l c is limite to approximately 2 r f an new geometry moel with explicit heterogeneous microstructure of the surrouning composite has to be generate to analyze cases when the istance l c is smaller. 7. Conclusions The calculation results prove epenency of energy release rate G II on istance from composite surface for both CF/EP an GF/EP composites. The energy release rate is larger for ebons closer to the specimen surface. However, within the observe range ( l c = 10 to 20 µm) the changes in absolute values of G II are small. It was prove that ifferences ue to fiber location in shear stress an axial isplacement istributions in front an behin the crack tip were even smaller. Furthermore, epenency of stress istributions on angular coorinate ϕ was also small, therefore, it can be conclue that axisymmetric approximation can be use in the stuie region. 8. References [1] Pupurs A an Varna J 2011 Mech. Av. Mater. Struct. Accepte [2] Pupurs A, Krasnikovs A an Varna J 2011 Mech. Av. Mater. Struct. Accepte [3] Kim B W an Nairn J A 2002 J. Compos. Mater [4] Kim B W an Nairn J A 2002 J. Mater. Sci [5] Gamstet E K 2000 J. Appl. Polymer Sci [6] Irwin G R 1958 Fracture, Hanbuch er Physik (Berlin: Springer) [7] Hashin Z an Rosen B W 1964 J. Appl. Mech [8] Hashin Z 1983 J. Appl. Mech [9] Christensen R M an Lo K H 1979 J. Mech. Phys. Soli [10] ANSYS Release