Three dimensional finite element study of the behaviour and failure mechanism of non-crimp fabric composites under in-plane compression

1 Three dimensional finite element study of the behaviour and failure mechanism of non-crimp fabric composites under in-plane compression Authors: Luis Miguel Ferreira 1 (corresponding author), Enrique Graciani, Federico París Affiliation (All Authors): Grupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingenieros, Universidad de Sevilla. Address: Grupo de Elasticidad y Resistencia de Materiales. Escuela Técnica Superior de Ingenieros. Camino de los Descubrimientos s/n. 41092 Sevilla, España Phone: (+34) 954 487 300 Fax: (+34) 954 461 637 Email: [email protected] (Luis Miguel Ferreira), [email protected] (Enrique Graciani), [email protected] (Federico París). Abstract The compressive behaviour and the mechanism responsible for failure of a [0,90]n noncrimp fabric (NCF) laminate are studied using a 3D finite element (FE) model of the representative unit cell at the mesoscopic scale. The tows of the unit cell were generated using a straight FE mesh taking into account the waviness of the fibres with the definition of the mechanical properties of each element according to the actual direction of the fibres. A parametric study has been carried out to evaluate the influence of the non-linear behaviour of the tows and of the fibre crimp on the compressive failure mechanism of the laminate. The numerical predictions are discussed and compared with experimental data. The results lead to think that the mechanism of failure of a [0,90]n NCF laminate under a pure compressive load is controlled by the shear strains that appear in the crimp part of the 0º tows. It is also found that the non-linear behaviour of the tows and the fibre crimp substantially contribute to the development of the potential failure initiation mechanism. A satisfactory agreement between the numerical and experimental compressive stress-strain curves is obtained for the highest fibre crimp angles considered. Keywords 1 Higher Colleges of Technology, Abu Dhabi Men's College P.O. Box 25035, Abu Dhabi, United Arab Emirates 2 Composites; Non-crimp fabric (NCF); Finite element model (FEM); Compressive behaviour; Failure Mechanism. 1. Introduction The need to combine the excellent in-plane behaviour of the composites made from unidirectional layers of pre-impregnated tapes (UDPT) with the lower cost and better out- of-plane behaviour of the woven fabric composites led to the appearance of the non-crimp fabric (NCF) composites [1]. These composites are made from dry preforms which are produced in a continuous process by sequentially laying groups of fibre tows, or tows for short, at a specific orientation, sequentially on a conveyor system. Once the tows are laid at all the orientations required, they are held in position by a stitching yarn creating, in theory, a fabric without fibre crimp. In practice, however, some geometrical features are introduced at the manufacturing stage of the preforms and others later on at the moulding stage, such as resin pockets between the tows and some mesoscopic fibre crimp, (Fig. 1 [2]); the latter has particular interest for this study since it is strongly related to the compressive performance of NCF composites [3-5]. Fig. 1. Cross-section of a typical NCF composite. Due to the complex internal structure of the NCF composites, performing representative studies using Finite Element (FE) models becomes a complicated task. Some representative studies have been performed using a 2D approach [3, 6, 7]. However, the complex mesoscopic architecture of these materials may not be completely represented by a 2D FE model [5], since it is highly irregular along all directions of the material. Moreover, this approach does not allow the stresses and strains at the 3D mesomechanical level to be calculated. These limitations motivated the development of parametric mesoscopic 3D FE models capable of reproducing the complex 3D structure of a NCF laminate. The study presented in this paper, of the compressive behaviour and failure mechanism of NCF composites, has been achieved using 3D FE models of the representative unit cell of a [0,90]n NCF laminate, represented in Section 2.1. The stiffness properties of the constituents and the boundary conditions imposed are described in Sections 2.2 and 2.3, respectively. The compressive behaviour results, which are presented in Section 3, emphasize the shear stresses/strains components associated with the mechanism of failure and how the non-linear behaviour of the tows and the fibre crimp affect the compressive performance of the NCF laminate. In Section 4, the numerical predictions are discussed 3 and compared with the experimental evidence obtained from the work carried out on the FALCOM Project (Failure Performance and Processing Prediction for Enhanced Design with Non-Crimp Fabric Composites). The numerical analysis has been carried out using a FE model with straight mesh, following the modelling approach presented in [8]. In Section 5, the results obtained with the FE models with a straight mesh are compared with those obtained with the Classical Modelling Approach (using a curved mesh) to show the better performance of the FE models with a straight mesh. Finally, the conclusions are set out in Section 6. 2. Mesoscopic approach of the NCF laminate Before presenting the mesoscopic scale 3D FE models developed for this study, it is necessary to define the concept of the representative unit cell (RUC): a repeatable cell which allows any part of the NCF laminate to be generated just by assembling multiple RUC’s. This concept is illustrated in Fig. 2 using a [0,90]2 NCF laminate. Fig. 2. Process used to obtain the representative unit cell of a [0,90]2 NCF laminate. At the mesoscopic level, an NCF laminate can be thought of as a stack of layers of fibre tows (in different orientations) with homogeneous transversely isotropic behaviour, the isotropic plane normal to the fibres being surrounded by an homogeneous isotropic matrix. 2.1. Numerical models A mesoscopic scale 3D FE model of the RUC of a [0,90]n NCF laminate with fibre crimp, due to nesting of the tows, was generated using structural solid element SOLID45 from ANSYS® FE code [9]. The unit cell is composed of 4 laminas (0º, 90º, 0º, 90º) and each lamina contains two half-tows with a resin pocket between them, see Fig. 3(a) and Fig. 3(b). Notice that the resin has been removed to better understand the tows’ shape and orientation. Some of the geometrical parameters, which appear in Fig. 3, will be defined later. 4 Fig. 3. 3D FE model of the representative unit cell without resin. (a) Single NCF lamina, (b) [0,90]n NCF laminate. The curvature of the tows has not been included in the geometry of the model, however the waviness of the fibres has been taken into consideration, see Fig. 4(a), following the modelling approach presented in [8]. This approach obviously allows a straight FE mesh, as represented in Fig. 4(b), to be used. An isolated element with the appropriate orientation of the fibres in shown in Fig. 4(c). Notice that a is the angle between the fibre direction 1 and the ideal longitudinal direction of the tow (x in the 0º tows and y in the 90º tows) and will be referred to as the misorientation angle from now on. is an angular parameter that corresponds to the maximum misorientation angle and thus represents the maximum fibre crimp of the tow, that is: . A maximum fibre crimp angle of 9º was chosen for the initial studies, based on the fact that previous results showed a satisfactory agreement between the apparent in-plane stiffness properties and the experimental evidence for angles in the range of 9º to 12º [4, 10]. In what follows, the fibre crimp was parametrically varied (in a range of 0º to 12º) in order to reveal the influence of this geometrical heterogeneity over the compressive behaviour and failure mechanism of [0,90]n NCF laminates. The geometrical parameters employed are shown in Fig. 3(a), where a represents the length of the tow not affected by the fibre crimp, t the thickness of each lamina and g the width of the gap between two adjacent tows for each lamina. The values of these parameters have been estimated from the results of the microstructure characterization measured in [11] having taken, , and The value of these parameters and the maximum fibre crimp angle are identical for all the tows. An average fibre volume fraction for each lamina of has been obtained from the observed average fibre volume fraction of the tows [12] and the relative dimensions of the cross-sections of the tow and of the lamina. b ( ) max ba = 2.06 mma= 0.24 mmt= 0.30 mm.g= 60% l f V= 69% t f V= 5 Fig. 4. Schematic representation of the approach used for modelling the waviness of the fibres in 3D FE models [8]: (a) NCF composite with fibre crimp, (b) detailed view of the FE mesh in the fibre crimped area, (c) isolated straight element. Taking advantage of the symmetry of the problem to improve the computing efficiency, only one quarter of the RUC has been employed in this study, see Fig. 5. In accordance with the global coordinate system xyz represented in Fig. 5, the plane of the laminate is the xy-plane and z is the through-thickness direction. The local coordinate system 123 corresponds to the actual direction of the fibres. Fig. 5. 3D FE model and dimensions of a quarter of the [0,90]n NCF laminate developed. Different mesh discretizations were used in the study to assess its influence on the results. The use of coarser meshes led to the appearance of intermittent evolutions of stresses and strains along the part where the tows present the fibre crimp. As there are no stress 6 intensification factors, the mesh was progressively refined until these evolutions became smoother. As it is possible to appreciate in Fig. 5, particular attention was given to crimped part of the tows where a finer mesh was employed. This area corresponds to a potential failure initiation location when looking for a mechanism of failure. 2.2. Mechanical properties of the constituents At the mesoscopic scale, the stiffness properties required in order to carry out the FE analysis are those of the resin pockets and of the tows. Since the tows are constituted by a large number of parallel fibres, uniformly distributed and impregnated by a resin, their behaviour can be considered transversely isotropic (with elastic constants: ). The FE model has been generated according to the NCF composites tested within the FALCOM Project [13-15]. However, as no experimental measurements were performed to estimate the stiffness properties of the tows, the rule of mixtures has been employed as an averaging technique to estimate their values as a function of the elastic properties of the fibre and of the resin, see Table 1. 60% 165.5 11.43 0.30 0.42 4.03 4.03 Table 1. Stiffness properties of the tows obtained using the rule of mixtures. The resin employed has the following elastic properties: an elastic modulus , a Poisson’s ratio and a corresponding shear modulus Carbon fibres, being a transversely isotropic material, exhibit the following relationships: . According to the rule of mixtures, the values of the elastic constants in the planes transverse to the fibre axis, are not required to estimate the apparent stiffness properties of the tows neither in the longitudinal plane nor in the transverse planes. Therefore, the carbon fibre has been characterized by a longitudinal elastic modulus , a Poisson’s ratio and a shear modulus . The tows were modelled with linear shear behaviour in the transverse plane (23-plane) and with non-linear shear behaviour in the longitudinal and transverse through-thickness planes (12-plane and 13-plane, respectively), because in these latter planes the behaviour of the tows under shear loads is affected by the resin, whose behaviour is also non-linear [3]. This non-linearity in the mechanical behaviour of the tows was introduced into the FE model by a bilinear approximation of the shear stress-strain curve obtained by Ditcher [16]. The change of slope was defined to occur at a value of 60 MPa of stress and 1.4% of strain, see Table 2. 11 12 12 22 23 23 , , , , and tttt t t EGE G nn t f V ( ) 11 GPa t E ( ) 22 GPa t E 12 t n 23 t n ( ) 12 GPa t G ( ) 23 GPa t G 3.50 GPa m E= 0.42 m n = 1.23 GPa. m G= 22 33 12 13 12 13 , , ffffff EE GG nn === ( ) 22 23 23 , and , ff f EG n 11 237 GPa f E= 12 0.25 f n = 12 94.8 GPa f G= ( ) 12 s * 7 Slope I Slope II 60 4.03 0.75 Table 2. , slope values I and II used to define the non-linear shear behaviour of the tows. Several studies were also performed in order to assess the influence of the behaviour of the neat resin within the resin pockets (linear elastic and non-linear elastic) on the compressive response of the NCF laminate. No significant changes in the results were appreciated. Therefore, in order to facilitate the convergence of the solutions, the behaviour of the resin within the resin pockets has been defined as linear. 2.3. Boundary conditions The boundary conditions applied to the faces of the model, see Table 3, serve to impose a compressive stress state and to guarantee the displacements compatibility between the limits of the model and the rest of the structure. With reference to the global coordinate system xyz, represented in Fig. 5, symmetry conditions have been assumed in the faces parallel to the xz-plane and one of the faces parallel to the yz-plane. On the opposite face, parallel to the yz-plane, a constant displacement (k) in the compression load direction has been applied to all nodes, and to ensure that they all displace the same, all nodes have been coupled. The compression load, therefore, is the result of the sum of the reaction forces of all nodes at . The boundary conditions on the top and bottom faces (parallel to the xy-plane) must guarantee the compatibility of the unit cell under consideration with the adjacent unit cells. Thus, the displacements of the nodes on both faces are coupled to ensure that the extension in the through-thickness direction is constant throughout the whole model. Notice that with the boundary conditions employed, the NCF laminate has an infinite length along the y-axis. Model faces Boundary conditions Schematic representation Table 3. Boundary conditions imposed. ( ) 12 MPa s * ( ) ,, 12 13 GPa tI tI GG= ( ) ,, 12 13 GPa tII tII GG= 12 s * 0x= 0x= , 0| 0| 0 x xy xz ukk ss => = = ( ) 2 atg x++ = 0| 0| 0 xxyxz u ss === 0y= 0| 0| 0 yxyyz u ss === ( ) 2 atg y++ = 0| 0| 0 yxyyz u ss === 0 | 4zzt== || tbtbtbd xxyyzzz uuuuuuu==-= 8 3. Compressive behaviour and failure mechanism This section focuses on studying the compressive behaviour and failure mechanism of a [0,90]n NCF laminate by means of FE analysis. The study was performed with the command option NLGEOM, ON [9], so that it includes large displacements, large rotations and large strain effects. The equilibrium is satisfied in the deformed shape and the load is applied in a linear and progressive form until instability occurs. It is clear that, due to the boundary conditions and to the load imposed, the normal stress component in the 0º tows will present larger values than any other stress component in any other part of the unit cell (90º tows or resin). However, given the presence of the fibre crimp, it is expected that the 0º tows will suffer a shear instability phenomenon, known as mesobuckling, that causes localized shear strains [3, 5]. In the present study (and with reference to the aforementioned), the evolutions of these shear strains and shear stresses stood out when looking for a mechanism of failure. 3.1. Stresses and strains in the potential failure location Since the fibre crimp has been modelled in the through-thickness direction, the most relevant shear stress/strain components to be analysed when looking for the mechanism of failure are and (results expressed in the global axis of the problem). The distribution of and throughout the unit cell with for a compressive load of 0.5 GPa, can be observed in Fig. 6(a) and Fig. 6(b), respectively. Note that Fig. 6 displays the solution results as continuous contours across element boundaries for the selected nodes and elements. These contours are determined by linear interpolations within each element from the nodal values, which are averaged at a node whenever two or more elements connect to the same node. Fig. 6. Numerical results throughout the unit cell for a compressive load of 0.5GPa. (a) Shear stresses (b) Shear strains . It can be observed that in the left-hand part of the figures, where there are neither matrix elements nor fibre crimp, no significant changes in and appear. The greatest stress and strain gradients arise in the part where the 0º tows are affected by the fibre crimp and xz s xz g xz s xz g 9º b = xz s xz g xz s xz g 9 where the packages of resin appear between each pair of 90º tows. The results also show that both highly positive and highly negative shear strains are placed very close together and are located at the part of the 0º tow where the misorientation angle is greatest. This mechanism can be clearly observed in Fig. 7 where detailed views of the undeformed and deformed shapes are represented in the area where the fibres are crimped. The same scale has been employed for both figures. Fig. 7. Undeformed and deformed shapes in the area under analysis with location of nodes N1 and N2. It is also possible to appreciate the distribution of shear strain along two consecutive vertical rows of elements in the 0º tows, one of them with a positive value of and the other with a negative value of (greater in absolute value than the previous one). This evolution of throughout the xy-plane along the actual direction of the fibres in the middle of a 0º tow, is shown in Fig. 8 as direction A. Notice that several levels of the applied compressive load are represented and that “Load 100%” corresponds to the load at which the numerical solution stops converging (0.58 GPa). For load values higher than “Load 100%”, the elements in the crimped area of the 0º tows (aligned with the load direction), with higher values of , became exceedingly distorted and the solution was unable to converge. Notice that the load at which the solution stops converging has nothing to do with the failure of the laminate, rather it depends on a great number of parameters, such as the quality of the mesh, the element type, time control of the solution and the selected equation solver. xz g xz g xz g a 16 independently of the crimp angle considered. These evidences can be appreciated in Fig. 14, where the numerical results for both methods (for fibre crimp angles of 9º and 15º) and the experimental data are presented. Fig. 14. Numerical results obtained with RUC’s with the fibre crimp modelled with the Classical Modelling Approach and with a Straight FE Mesh and the experimental data. Apart from the observable influence of the fibre crimp angle on the convergence of the solution, several studies have also been performed to analyse how parameters, such as: the FE mesh discretization (coarser and finer 3D FE meshes); the element type (SOLID45 and SOLID185); the time control of the solution (different time step sizes) and the selected equation solver (sparse direct, pre-condition CG), could affect the convergence [9]. It has been found that the load at which the numerical solution stops converging, for any of the fibre crimp angles employed, is strongly dependant on these parameters. The results presented were obtain after a process of optimization of the aforementioned parameters so that the lack of convergence of the solution occurred at the highest load possible. 6. Conclusions The behaviour of a [0,90]n NCF laminate under an in-plane compressive load has been studied using 3D FE models with fibre crimped tows. It can be observed from the results that the potential mechanism of failure of a [0,90]n NCF laminate that has been identified under this type of load is controlled by the shear strains that appear in the crimp part of the 0º tows. Maximum positive and maximum negative shear strains, on the xz-plane, are seen to be located very close together, in the area where the 0º tow stops being in contact with the adjacent 90º tow and reaches the maximum misorientation angle. These results are in agreement with studies, using a 2D approach, which were performed to investigate 17 the compressive strength of NCF laminates. However, this 2D approach did not allow the stresses and strains at the 3D mesomechanical level to be calculated. This limitation motivated the development of the parametric 3D FE models capable of reproducing the complex 3D mesoscopic structure of a NCF laminate. The influence of the non-linear shear behaviour of the tows on the compressive failure mechanism has also been analysed. The results show that including the non-linear behaviour in the tows causes an increase in the shear stresses/strains particularly in the crimped area of the 0º tows (aligned with the load direction) hence contributing to the development of the potential failure mechanism. It has also been demonstrated that the size of the jump of the shear strains is directly related to the fibre crimp angle: the higher the fibre crimp angles, the higher the shear strains. To validate the results obtained with the unit cell, the numerically predicted compressive stress-strain curves were compared with the experimental measurements carried out on the FALCOM Project. A fairly good prediction of the numerical compressive stress-strain curve for the NCF laminate has been obtained for fibre crimp angles in a range of 12º to 18º. In particular, the solution that best adjusted to the mean value of all tested specimens was obtained with a 15º but convergence issues appeared for lower strains. In this way, it is not possible to verify if the solution would continue adjusting for higher strains when the accumulation of damage started to affect the global behaviour of the specimens nor it is possible predict failure. To validate the straight FE mesh modelling approach the results were compared with the results obtained with equivalent RUC’s in which the fibre crimp was modelled according to the Classical Modelling Approach. Similar solutions were obtained with both methods but the Classical Modelling Approach presented more convergence issues in the nonlinear analyses than the straight FE mesh approach. The parametric 3D models of the RUC of a [0,90]n NCF laminate that have been generated were employed in a progressive damage analysis [18] and a satisfactory agreement with experimental data has been obtained. 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