Modelling of damage in composite materials using interface elements

Transcription

1 5 th European LS-DYNA Users Conference Coposites Modelling of daage in coposite aterials using interface eleents Authors: W.G. Jiang, Departent of Aerospace Engineering, University of Bristol S.R. Hallett, Departent of Aerospace Engineering, University of Bristol M.R. Wisno, Departent of Aerospace Engineering, University of Bristol Correspondence: W.G. Jiang, Departent of Aerospace Engineering, University of Bristol, Queens Building, University Walk, Bristol, BS8 1TR Tel: Fax: Keywords: delaination, progressive failure analysis, interface eleents

2 Coposites 5 th European LS-DYNA Users Conference ABSTRACT A siple and robust ipleentation of an interface eleent has been addressed in this paper. A new state variable is introduced to trace the extent of daage accuulated at the interface. The eleent not only siulate ixed ode delaination propagation in coposite aterials but also satisfactorily deals with ode ratio change during the debonding process. The interface odel is ipleented in LS-DYNA code. The odel has been applied to scaled open hole tension tests. Coparison between nuerical results and experients shows good correlation. NTRODUCTON The effect decreasing strength with increasing specien size in coposite aterials is a well docuented phenoenon [1]. With the increasing use of large coposite coponents in aerospace applications coes the need for reliable design ethods to be able to extrapolate data fro sall scale laboratory tests to full size coponents, thus reducing the need for full-scale testing. Holes are required in aircraft coponents for joints, access, or to allow the passage of wires and hydraulics. An extensive testing progra has been conducted at the University of Bristol to investigate the influence of specien size on the open hole tensile strength of coposites and the echaniss behind variations [2]. t has been found that delaination and splitting play a significant role in deterining the strength of a lainate. The traditional virtual crack closure technique (VCCT) [3] based on linear elastic fracture echanics has been successfully used in the prediction of delaination growth [4]. However the use of this technique requires that an initial delainated area ust be predefined and only self-siilar delaination growth can be dealt with satisfactorily. To overcoe the liitation associated with VCCT, an interface eleent has been developed in LS-DYNA. The interface eleents are located between adjacent lainae of a lainated coposite structure to siulate both initiation of delaination and non-self-siilar growth of delaination cracks without specifying an initial delaination. A bilinear softening cohesivedecohesive constitutive law which relates the interfacial traction coponents to the displaceent coponents has been ipleented. The failure of the interface eleent under ixed ode condition is defined by a power law using experientally deterined fracture energies without the need for further curve fitting paraeters. A siple new state variable is introduced to trace the daage accuulated at the interface. Since coposite delaination processes often involve strong nonlinear events such as snap-back and snap through, convergence is usually very difficult to achieve if the siulations are perfored using an iplicit code without coplex path following algoriths [5]. The use of an explicit solver such as LS-DYNA overcoes any of these difficulties. This paper describes the forulation of the interface eleent, its application in odels of the experiental prograe and results obtained which show good correlation to location, extent and failure stress of delaination and splitting which occurs.

3 5 th European LS-DYNA Users Conference Coposites CONSTTUTVE EQUATONS The stress singularities predicted by FE at a crack tip in the linear elasticity solution cannot be reconciled with any realistic local rupture process. The softening type of cohesive zone odel presented here is intended to represent the degradation of the aterial ahead of the crack tip. t captures strength-based bond weakening, and fracture-based bond rupture. The softening constitutive law is ipleented within an interfacial surface aterial consisting of a pair of bonded surfaces S + and S - connected by continuous distributed nonlinear springs (see Figure 1). The bilinear interface forulation adopted in this paper for ixed-ode softening law can be illustrated in a single three-diensional ap by representing open ode (Mode ) on the 0 - σ - δ noral plane, and shear ode (ode ) on the 0 - σ - δ shear plane, as shown in Figure 2. The triangles 0 - T - δ and 0 S - δ are the bi-linear response in pure open ode and in pure shear ode respectively. Any point on the 0 - δ noral - δ shear plane represents a ixed-ode relative displaceent. The total ixed-ode relative displaceent is defined as: 2 2 δ = δ + δ (1) where δ = ( δ 1,0) is the noral opening relative displaceent, 2 2 δ = δ 2 + δ 3 the resultant shear relative displaceent. δ 1, δ 2 and δ 3 are relative displaceent coponents for a pair of interface connecting points in a local orthogonal coordinate syste. δ 1 is the coponent noral to interface idplane (see Figure 1). The daage initiation under a ulti-axial stress state adopted is: 2 2 ( σ,0) σ + = 1 (2) σ σ where σ is the noral interlainar tensile stress, σ the shear stress resultant of the interface, σ the interlainar tensile strength, and σ the interlainar shear strength. We assue that the noral interlainar copressive stress does not affect to the onset of delaination. The daage initiation locus represented e by the relative displaceent corresponding to softening onset, δ, can be deterined by this condition (see figure 2). The failure criterion of interface eleent under ixed ode condition used is defined by the power law proposed by Whitcob [6]: α α G G + = 1 (3) GC Gc where α (1.0 ~ 2.0) is an epirical paraeter derived fro ixed-ode tests, G C and G C are critical energy release rates for ode (open) and ode (shear) respectively. The fully debonded locus represented by the relative displaceent corresponding to the coplete interface failure, δ, can be deterined by this condition (see figure 2).

4 Coposites 5 th European LS-DYNA Users Conference A siple new state (or history) variable d is introduced to track the extent of daage accuulated at the interface: e δ δ d( δ ) = e δ (4) δ 0 ts initial value is d = 0 and will reain zero until the daage initiation condition (2) is et. t is a onotonically increasing value which starts to grow when δ > and reaches the failure value 1 when δ. e δ δ The state variable d is defined in such a way as to assure that transition between load steps is sooth, even for the case of significant ode ratio change. To coplete the constitutive law, unloading is siply assued to return linearly back to zero stress state at the origin as shown in Figure 2. FNTE ELEMENT ANALYSES AND COMPARSON WTH TESTS t is well known that the strength of coposites is related to the size of the specien tested. To study the size effects, a series of scaled speciens anufactured fro Carbon-Fibre epoxy were tested and analyzed. A syetric quasi-isotropic lay-up with ply orientations of (45 /90 /-45 /0 ) s was used. Lainate thickness was increased by increasing the thickness of each of the individual ply blocks. The geoetry and diensions are given in Figure 3. The rectangular lainates with a centrally located circular hole subjected to tension were odelled with eight noded fully integrated brick eleents using LS-DYNA software. To reduce the odel size, only the gage length was odelled. Since the layup is syetric, it was only necessary to odel half the thickness. Plane syetric boundary condition on the id-plane of the lainates were applied. Duplicate nodes are placed on either side of any adjacent ply interfaces where delaination is expected, and these are connected together with interface eleents to odel delaination between plies [7]. A siilar ethod is used to odel splitting which occurs between fibres within a ply by eans of interface eleents which are located in the sites where the splits were found to occur fro the tested saples [2], see Figures 4 and 5. The properties used for the interface eleents were taken to be G C =0.2N/, G C =1.0N/, α=1.0, σ =90MPa. σ =60MPa, The aterial properties of individual layers of M7/8552 carbon/epoxy on the principal aterial coordinates are taken to be E1=161GPa, E2=E3=11.38GPa, G12=G13=5.17GPa, G23=3.98GPa, ν12=ν13=0.32, ν23=0.436 [8]. The theral expansion coefficients are α11=0.0, α22=α33=3x10-5 C -1. To include the theral residual stress fro the fabrication of the lainate in the analysis, theral load (teperature drop, -160 o C) was applied first prior to echanical extension. A relatively fine esh was used adjacent to the central hole of the saple, with larger eleents away fro the notch to reduce the odel size. Figure 6 shows a reduced density in-plane esh. The coarse in-plane esh shown is denoted as esh density 1x1. Finer eshes are denoted as esh 2x2, and esh 3x3 respectively. A esh sensitivity analysis through the thickness showed that using one eleent per ply in the through thickness direction is adequate (Figure 7). Figure 7 shows typical load-extension curves predicted using three different esh densities for the specien with hole diaeter d=3.175 and lainate thickness t=4. The delaination and splitting process for this specien is also illustrated in Figure 8. All the featured loading stages (arked A to G in

5 5 th European LS-DYNA Users Conference Coposites Figure 7) are included. The two 45 splits which are tangent to the hole boundary within the 45 surface ply occured at very low load level (stage A). At stage C, the delaination between 45 /90 started to develop fro the splits, and the 90, - 45 and 0 splits within their corresponding plies were all fored close to the hole boundary. At stage D, the -45 and 90 splits had fored copletely across the specien width, the triangular areas within the 45,/90 and 90 /-45 interfaces were delainated, and the -45 /0 interface was partially delainated in one side of the specien. The split crack front in the 0 ply is acopaning the -45 /0 delaination. The non-syetric delaination phenoenon between the -45 and 0 plies was also observed in the tests. The peak load before significant delaination daage prediction (loading point C) for all speciens showed good correlation with the test failure load (first load peak before 5% load drop), with exception for the sall thin saple (d=3.175, t=1), see Figure 9. Further investigation shows that the failure echanis for this saple is due to fibre failure before delaination, which has not yet included in the current finite eleent odel. SUMMARY AND CONCLUSONS An interface eleent with a bi-linear softening law for ixed ode delaination was proposed and ipleented in LS-DYNA. The constitutive equation proposed uses a new single daage variable to track the extent of daage at the interface under general loading conditions. This odel has been applied to a series of scaled saples. The finite eleent odel proposed can capture the iportant features of the progressive delaination processes involved and good agreeent has been obtained for delaination strength predictions. ACKNOWLEDGEMENTS This work was supported by the UK Engineering and Physical Sciences Research Council, Ministry of Defence and Airbus UK. REFERENCES 1. Wisno M.R., Size effects in the testing of fibre-coposite aterials, Coposite Sciences and Technology, Vol. 59, N0. 13, , Green B., Wisno M.R., Hallett S.R., Tensile scaling effects in notched coposites, 2 nd nternational conference on coposites testing and odel identification, Paper No. 19, Septeber 2004, University of Bristol, UK. 3. Rybricki E.F., Kanninen, M.F., A finite eleent calculation of stress intensity factors by a odified crack closure integral, Engineering Fracture Mechanics, Vol. 9, , Krueger R., O brien T.K., A shell/3d odeling technique for the analysis of delainated coposite lainates, Coposites: Part A, Vol. 32, 25-44, Alfano G., Crisfield M.A., Finite eleent interface odels for the delaination analysis of lainated coposites: echanics and coputational issues, nternational Journal for Nuerical Methods in Engineering, Vol. 50(7), , Whitcob J.D., Analysis of instability-related growth of a through-width delaination, NASA TM 86301, Wisno M.R., Chang F.K., Modelling of splitting and delaination in notched cross-ply lainates. Coposites Science and Technology, Vol. 60, , O brien T.K., Krueger R., Analysis of ninety degree flexure tests for characterization of coposite transverse tensile strength, NASA.TM ARL-TR-2568, October 2001.

6 Coposites 5 th European LS-DYNA Users Conference Figure 1. Scheatic of interface eleent S + δ 1,σ 1 δ 3,σ 3 S - δ 2,σ 2 Figure 2. nterfacial bilinear ixed ode softening law σ σ σ S Shear ode Unloading T 0 e δ δ δ δ shear δ δ δ Fully debond locus δ δ noral Daage initiation locus

7 5 th European LS-DYNA Users Conference Coposites Figure 3. specien geoetry [2] Lainate thickness, t () Hole diaeter, d () = Configuration tested Figure 4. Siplified geoetry of the saple odelled Lines show the locations of potential splits introduced in the FE odel F F Not to scale The sall degenerated areas are neglected to avoid sall degenerated eleent

8 Coposites 5 th European LS-DYNA Users Conference Figure 5. A tested saple showing typical failure ode [2] Figure 6. Saple finite eleent esh (in-plane esh, esh density 1x1)

9 5 th European LS-DYNA Users Conference Coposites Figure 7. Saple load-extension curve showing the esh convergence 400 age stress level, F/LW, (MPa) 200 3x3 in-plane esh with one eleent throug 2x2 in-plane esh with 4 eleent through 2x2 in-plane esh with one eleent throug A B C D E F Figure 8. Typical delaination and splitting developing progress fro Loading stage in Fig. 7 (Extension strain %) Stress level (MPa) Location of interlainar interface 45 /90 90 / /0 Location of splitting within plies All layers (superiposed) A (0.25) 141 B (0.35) 204 C (0.40) 229 D (0.45) 189 E (0.60) 230 F (0.65) 183 G (0.84) 262

10 Coposites 5 th European LS-DYNA Users Conference Figure 9. Coparison of failure stress level Failure stress (MPa) d=3.175 t=1 d=3.175 t=2 d=3.175 t=4 d=6.35 t=2 d=6.35 t=4 d=12.7 t=4 Specien diension ()