DIRECT NUMERICAL SIMULATION OF DAMAGE PROGRESSION IN LAMINATED COMPOSITE PLATES USING MULTI-SCALE MODELLING

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1 DIRECT NUMERICAL SIMULATION OF DAMAGE PROGRESSION IN LAMINATED COMPOSITE PLATES USING MULTI-SCALE MODELLING DIRECT NUMERICAL SIMULATION OF DAMAGE PROGRESSION IN LAMINATED COMPOSITE PLATES USING MULTI-SCALE MODELLING N.K. Karna *, H.J. Kang, K.J. Park, K.M. Na, C.H. Chung, I.H. Choi 2, M.K. Ki 2 and S.J. Shin Departent of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Republic of Korea 2 Korea Aerospace Research Institute, Daejeon, Republic of Korea * Corresponding author (karna.nitesh@gail.co) Keywords: Multi-Scale Modelling, Parallel Processing, Progressive Failure Analysis (PFA), Direct Nuerical Siulation (DNS), Eleent Failure Method (EFM) Abstract: The specific characteristics of coposite aterial, such as light-weight, high strength, corrosive resistance, ake it one of the ost appealing aterials to be used for various applications. At the sae tie, understanding their failure behaviour due to the coposition of two or ore constituents poses enorous challenge. One of such studies is the daage progress in the coposite lainates. Therefore, this paper focuses on the developent of a unique coputational odel capable of deterining the ultiate strength of open-hole lainated coposite plate under tension. Direct nuerical siulation (DNS) of coposites was developed and integrated into the in-house developed finite eleent (FE) code DIAMOND/IPSAP and the progressive failure analysis (PFA) was carried out using eleent failure ethod (EFM) as the coputational schee and icroechanics based failure criteria. 2 Background: 2. Direct Nuerical Siulation (DNS) Several researches based on the acroscopic approach have been perfored during the recent years. However, understanding and predicting the echanis of the failure initiation turned out to be indirect since the failure of coposite aterials initiates fro the icroscopic level and then propagates to the acroscopic level []. Also the inforation like the influence of an individual fibre on the echanical behaviour will disappear in the acro-scale odelling. Therefore, there exists necessity to realize icro-scale odelling to copletely understand the different aspect of coposites. In order to achieve such goal, this paper introduces DNS of the coposites using the clustered cell odelling. DNS of the coposites can be perfored by using either a unit cell or a clustered cell odel. The cluster cell odel has significant advantage over the unit cell odel. For exaple, the unit cell odel approach has the liitations of predicting the local behaviour of the constituents of coposite aterials which is not the case with the clustered cell odel approach. As noted in Ki et al. [], by using the clustered cell odel the icroscopic level stress distribution as well as failure initiation can be investigated conveniently. Also its siulation can be carried out without any assuption of local bending or local boundary conditions. It was also successfully deonstrated that the effect of nonunifor arrangeents of constituents fro defect or irregularity can be taken into account with ease. However, this will be ore liited to achieve by using the unit cell and acroscopic approach. A typical figure depicting the unit cell as well as clustered cell odel is shown in Fig.. This paper discusses both the unit cell approach and the clustered cell odelling approach. 2.2 Microechanics of Failure (MMF) Due to the aterial and geoetric inhoogeneity arising fro the inclusion of the fibres in the fibre reinforced coposites, non-unifor icro-stresses at the constituent level is developed by external echanical and theral loadings. For each fibre, atrix or fibre-atrix interface region, ply failure can have different failure echaniss. Therefore, proper failure criteria should be used to distinguish where failure initiates. Due to the lack of inforation about the acro-stress/strain and icro-stress/strain

2 relation, prediction of the constituent failure will be a difficulty [2]. The relation is generally expressed in ters of the stress aplification factor, as in Equations ()-(2). M σ = M M 2 M 3 M 4 σ = M σ σ + A σ T () t = M t σ + A t T (2) M 2 M 22 M 32 M 42 M 3 M 23 M 33 M 43 M 4 M 24 M 34 M 44 M 55 M 65 M 56 M 66 ] [ A σ = [A A 2 A 3 A 4 ] T The stress aplification factor was calculated using the direct finite eleent analysis with the unit cell odel. Failure criterion can be expressed in ters of the icro level strength, as in Equations (3)-(5) -Fibre failure criterion C f < σ x < T f (3) -Matrix failure criterion ( σ VM σcr ) VM 2 + ( I I cr) = (4) σ cr VM = T ( α2 +α )/2 α+, cr α I = T 2 +α, α = C α 2 /T -Fibre-Matrix Interface failure criterion ( t n ) 2 + ( t s ) 2 = (5) Y n Y s σ VM is von Mises equivalent stress. I is voluetric stress invariant. T f, C f, T, C, Y n and Y s are icro level strengths. Micro level strength was obtained fro the acro level strength of a ply using the proper failure criteria and a detailed distribution of icro stresses. (i) T f = ax (M σ,f X T + A σ,f T) (6) (i) C f = ax ( M σ,f (i) (i) X C + A σ,f T) (7) Figure 3 is a flow chart to obtain the atrix icro strength. Equivalence stress, σ eq is, σ eq = (α )I + (α ) 2 I ασ VM 2α p = σ eq /T (8) Stress aplification factor can be obtained using DNS(Direct Nuerical Siulation). In DNS, RVE(Representative Volue Eleent) odel is used. To represent the icro level behaviour, periodic boundary condition is applied on RVE odel. Periodic boundary condition can be expressed as in Equation (9). u +j i u j j i = c i (9) where, u i j indicates the displaceent along the i- direction of one node located at the boundary face whose noral vector is along the j-direction. 2.3 Eleent Failure Method (EFM) Eleent-failure algorith, proposed by Beissel et al. [3] is a basic concept which deals with the analysis of dynaic crack propagation. In this technique, the eleents through which crack propagates, loses the ability to sustain deviatoric and tensile voluetric stresses; i.e. they are failed as shown in Fig 2 [3]. These failed eleents are not reoved fro the esh, but rather a set of external nodal forces are applied on the failed eleents. No requireents of reeshing or the definition of new contact surfaces as well as the allowance of crack propagation in any direction are cited as its ain advantage [3]. In this paper, EFM as ipleented by Tay et al. [4] is used as a coputational platfor for the progression of daage throughout the finite eleent (FE) odel with the icro-echanics based failure criterion as shown in Fig. 4 instead of the ore traditional aterial properties/stiffness degradation ethod (MPDM). Before beginning the EFM procedure, linear elastic analysis is conducted on the FE odel to find the initiation of daage using proper failure criteria. Whenever a failed eleent is detected the EFM is eployed to siulate the progression of daage in the odel. Since, the basic idea of EFM is to odify the nodal force of a failed eleent to represent the state of daage in the coposite aterial [4], the external forces will be applied until the net nodal forces of adjacent eleents approaches zero. The external forces to be applied are unknown and are calculated through the iterative procedure which is ipleented into the present IPSAP FE-code. Since, the icroechanics based failure criteria distinguishes between the atrix and fibre failure ode, a special care is taken while applying the external force when either of the fails. When the atrix only fails, an external force will be applied in the transverse direction to the fibre and its properties 2

3 DIRECT NUMERICAL SIMULATION OF DAMAGE PROGRESSION IN LAMINATED COMPOSITE PLATES USING MULTI-SCALE MODELLING be degraded. Siilar is the case with the occurrence of failure in the fibre, where the properties of fibre are degraded. If both of the fails copletely, the odel cannot sustain any load and the external force is applied in both fibre and atrix direction until the assigned convergence criteria is satisfied. If there is no failed eleent detected, the applied load is increased by sall increent at each step. The ain advantage of using EFM over MPDM is that it assures coputational convergence and stability as the stiffness atrix is not altered in EFM. This also results in saving the coputational tie and effort [4]. In this paper, the prediction of daage progression in lainated coposite plate is conducted within the icroechanical FE odel where fibres and atrices are odelled explicitly. 3 Model Description: The tensile loaded carbon fibre reinforced plastic (CFRP) with centrally located hole is odelled in DIAMOND. The diension of the lainate with open hole is taken fro the study done by Arunkuar and Przekop [5]. The diensions of the specien is with centrally located hole of diaeter 2.7. The aterial used in this paper is different fro the one used by Arunkuar because of the unavailability of icroechanical properties of IM7/8552. The coposite plate is therefore ade of IM7/855-7 with the three different stacking sequence. The thickness of the lainate with their respective lay-up configuration is also shown in Table. The aterial properties are shown in Table. The properties of the atrix, fibre as well as the laina are obtained fro WWFE-II (PART A) [6]. The finite eleent odel for coposite plate with hole is prepared using DIAMOND as shown in Fig 6. The odel consists of two diensional shell eleents and is relatively fine eshed around the hole with 4,92 as the total nuber of eleents. The left end of the odel is copletely fixed and a tensile load is applied to the right end. The applied load is increased at each step of the iteration until the lainate fails copletely. The unit cell odel with three diensional solid eleents is created using the recently ipleented direct nuerical siulation generator (DNSG) in DIAMOND as shown in Fig 4. The icro strength and stress aplification factor data are calculated using the unit cell odel and transferred to IPSAP to perfor the progressive failure analysis using EFM. The stiffness values for fibre can also be calculated by using the rule of ixture as in Equation () [7]. E E f E 2 f 2 f f f E E f E 2 2 f G G f G 2 2 () 4 Results and Discussion The results obtained are plotted using the present DIAMOND as the post processor. The failure distribution in atrix and fibre for each layer are shown in Fig (7) and (8), respectively. The first ply failure load and the last ply failure load are predicted for each lay-up configuration and are shown in Table 3. Since, there are not any experiental data available for the aterial used in this paper, it will be uneasy to verify the results obtained iediately. However, the results can be obtained by using acro level strength failure criteria like axiu stress failure criterion with EFM and are copared with the one obtained fro MSC Nastran using one of the available acro level failure criteria like Hill failure criterion. 5 Conclusion and Further Study The PFA with EFM as the coputational schee is developed in IPSAP and the results obtained are presented in this paper. The results obtained fro the developed algorith will be verified experientally in the future. Also, the effect of rando fibre arrangeent in coposite structure can be taken into account through the developed DNSG. 6 Acknowledgeent The research was conducted in co-operation with Korea Aerospace Research Institute (KARI) under the project The Developent of Analysis Progra and Technology for Coposite Material. References [] S.J. Ki, C.S. Lee, H.J. Yeo, J.H. Ki and J.Y. Cho Direct nuerical siulation of coposite 3

4 structures. Journal of Coposite Materials, Vol. 36, No. 24, pp , 22. [2] S.K. Ha, K.K. Jin and Y.C. Huang Micro-echanics of failure (MMF) for continuous fibre reinforced coposites. Journal of Coposite Materials, Vol. 42, No. 8, pp , 28. [3] S.R. Beissel, G.R. Johnson and C.H. Popelar An eleent-failure algorith for dynaic crack propagation in general directions. Engineering Fracture Mechanics, Vol. 6, Issues 3-4, pp , 998. [4] T.E. Tay, G. Liu, V.B.C. Tan, X.S. Sun, and D.C. Pha Progressive failure analysis of coposites. Journal of Coposite Materials, Vol. 42, No. 8, pp , 28. [5] S. Arunkuar, and A. Przekop Predicting failure progression and failure loads in coposite open-hole tension coupons. NASA/CR-2-267, 2. [6] A.S. Kaddour and M. J. Hinton Input data for test cases used in bencharking triaxial failure theories of coposites. Journal of Coposite Materials, 22. [7] K.K. Jin, Y.C. Huang, Y.H. Lee and S.K. Ha Distribution of icro stresses and interfacial tractions in unidirectional coposites. Journal of Coposite Materials, Vol. 42, No. 8, pp , 28. Unit Cell Approach Hoogeneous Effect Hoogenization Method Assuptions are required Cluster Cell Approach Larger scale than unit cell Large DOF and detailed behaviour Inhoogeneous Effect Microscopic Failure Fig.. Description of the clustered and unit cell approach. Fig. 3: Process of obtaining atrix s icro strength. Fig 2: Crack path odel of the eleent-failure algorith. Fig 4: Unit cell odel generated in DIAMOND. 4

5 DIRECT NUMERICAL SIMULATION OF DAMAGE PROGRESSION IN LAMINATED COMPOSITE PLATES USING MULTI-SCALE MODELLING Start Input data: Macro strength Find icro strength using unit cell odel Output data: Micro strength Obtain data fro file directly/ fro DNS Find stress aplificati on factor FE Analysis and Get icro stress for each eleent Perfor Static Analysis Apply MMF failure criteria to obtain failed eleents Increase the load Eleent fails? NO YES Find nodal forces of failed eleents Proceed to fail additional eleents until the desired nuber is reached Apply external nodal forces at nodes Solve for updated displaceents and nodal forces Continue iterations YES Check Convergence NO Fig. 5: Ipleentation of the present EFM with MMF. EOD (EFM) 5

6 Fig 6: Finite eleent odel of lainated coposite plate created in DIAMOND. (a) (b) (c) 6

7 DIRECT NUMERICAL SIMULATION OF DAMAGE PROGRESSION IN LAMINATED COMPOSITE PLATES USING MULTI-SCALE MODELLING (d) Fig 7: Matrix Failure Distribution in open hole lainate at (a) layer - 45, (b) layer 2-9, (c) layer and (d) layer 4 - where red coloured eleents are failed. (a) (b) (c) 7

8 (d) Fig 8: Fibre Failure Distribution in open hole lainate at (a) layer - 45, (b) layer 2-9, (c) layer and (d) layer 4 -. Table : Stacking sequence of the present odel Stacking Sequence Thickness () [45/9/-45/9] s.962 [45 2/9 2/-45 2/9 2] s.9688 [45 4/9 4/-45 4/9 4] s Table 2: Material properties of IM7/855-7 Matrix E 4.8E+9 Pa G.48E+9 Pa.38 Table 3: First ply failure (FPF) load and last ply failure (LPF) load for each configuration Stacking Sequence FPF (N) LPF (N) [45/9/-45/9] s 7,8 2, [45 2/9 2/-45 2/9 2] s 5,6 4,2 [45 4/9 4/-45 4/9 4] s 34,2 8,26 Stiffness Fibre IM7 E f 2.76E+ Pa E f22.9e+ Pa G f2 2.7E+ Pa f2.2 G f23 7.E+9 Laina IM7/855-7 E.65E+ Pa E E+9 Pa G 2 = G 3 5.6E+9 Pa G E+9 Pa 2.34 fibre volue fraction.6 Strength X 2.56E+9 Pa X'.59E+9 Pa Y 7.3E+7 Pa Y'.85E+8 Pa S 9.E+7 Pa 8