Predicting failure load of a non-crimp fabric composite by means of a 3D finite element model including progressive damage

1 Predicting failure load of a non-crimp fabric composite by means of a 3D finite element model including progressive damage Authors: Luis Miguel Ferreira a,b (corresponding author), Enrique Graciani a, Federico París a Affiliation: a. Grupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla. b. Abu Dhabi Men’s College, Higher Colleges of Technology. Address: a. Grupo de Elasticidad y Resistencia de Materiales. Escuela Técnica Superior de Ingeniería. Camino de los Descubrimientos s/n. 41092 Sevilla, España. Phone: +34 954 487 300. Fax: +34 954 461 637. b. Abu Dhabi Men’s College, 25035 Abu Dhabi, United Arab Emirates. Email: [email protected] (Luis Miguel Ferreira), [email protected] (Enrique Graciani), [email protected] (Federico París). Abstract A mesoscopic scale 3D finite element model of its representative unit cell is used to study the progressive damage of a [0,90]n non-crimp fabric laminate under compressive loading. The tows of the unit cell have been generated with a straight finite element mesh, and the out-of- plane fibre crimp has been incorporated into the model by defining the mechanical properties of each element according to the actual direction of the fibres. The material properties degradation (MPDG) method has been used to study the damage evolution. Non-interactive criteria (Maximum Stress and Maximum Strain), and interactive criteria (Hashin and Puck), associated with failure modes, have been employed to determine the onset of the material degradation at the fibre tows. The progressive damage throughout the mesoscopic unit cell, from the load at which damage is initiated, until the load at which the failure of the laminate is predicted, has been analysed. The mechanism responsible for the failure of the laminate has also been identified. The numerical predictions of the failure stress and failure strain, for the considered failure criteria, are discussed and compared with experimental data obtained from direct compression tests on biaxial cross-ply NCF laminates. A satisfactory agreement between the numerical and experimental failure stress, failure strain as well as the compressive stress-strain curves has been obtained for the MPDG method when using Maximum Stress, Hashin’s or Puck’s failure criteria. Keywords Composites; Non-crimp fabric (NCF); Finite element model (FEM); Compressive behaviour; Progressive damage; Failure mechanism. This is an Accepted Manuscript of an article published by Elsevier in Composite Structures on October 2019, available at: https://doi.org/10.1016/j.compstruct.2019.111115 Copyright Elsevier 2019. En idUS: Licencia Creative Commons CC BY-NC-ND 2 1. Introduction The attractive cost-performance relationship of non-crimp fabric (NCF) composites led to a substantial increase in their use in the past decade, particularly in the aerospace and automotive industries. The adoption rate of NCF composites has been increasing because they combine the benefits of automated manufacturing, easy handling and lower cost than the conventional woven fabric composites, with the superior mechanical properties of the unidirectional pre-impregnated tapes composites (UDPT). NCF composites present a complex mesoscopic 3D structure that includes, for example, fibre tows in different in-plane orientations and with different cross-sections, resin rich areas, stitching yarns and fibre crimp due to the nesting of the tows or to the presence of the stitching yarns [1-5]. To illustrate these heterogeneities in the NCF structure at the mesoscopic scale, a typical NCF composite cross-section is shown in Fig. 1. Fig. 1. Mesoscopic scale structure [1]. Numerical simulation, particularly finite element (FE) modelling, is often used to evaluate the mechanical performance of NCF composites prior to experimental characterisation. However, generating FE models that replicate the complex 3D structure of these materials requires establishing an important trade-off between accuracy and computational cost of the solution. Different approaches to model the mesoscopic 3D architecture of these materials can be found in literature. González et al. [6] developed a 3D meso-scale FE model of a [0,90]2 NCF laminate with out-of-plane fibre crimped tows to study the apparent in-plane stiffness properties. Marklund and Asp [7] generated a 3D FE model with straight rectangular crosssection fibre tows in their study of combined micro-meso methodology for transverse matrix failure prediction and a similar approach was used by Mattsson et al. [8, 9] to study the performance of NCF laminate with different stacking sequences. Lomov et al. [10], in their study of meso-FE modelling of textile composites refer to the geometric modeller software, WiseTex, developed by Lomov and Verpoest [11], as a tool to create NCF composite geometries. This approach was used in [12-14] to generate the meso-scale structure of the NCF laminates and study their mechanical performance. Another geometric textile modeller software, Texgen, that resulted from the work of Robitaille et al. [15, 16] was also used by Serra Topal et al. [17] to generate a 3D non-crimp woven composite to study the last-stage fatigue damage. Even though these studies use different approaches to generate 3D FE models to evaluate the mechanical response of NCF composites under different types of loads, creating these 3D FE models efficiently is still a challenge. A progressive damage model requires a stress analysis to obtain the stress distribution across the model, a failure criterion to estimate the initiation of damage, and a damage evolution law 3 to control the stiffness reduction of the damaged material. Currently, many damage models for composite materials can be found in literature. However, predicting their failure load still remains a challenge [18, 19]. Since the concept of damage was firstly presented by Kachanov [20] to study the creep rupture of metals, several other continuum damage mechanics approaches have been proposed to model damage progression [21-23]. Matzenmiller et al. [24] proposed the first damage model for fibre-reinforced composites in which the onset of damage was defined by Hashin’s failure criterion [25] and the damage evolution was controlled by damage coefficients that reduce the stiffness of the material. Since then, other authors have also contributed to extend the number of approaches available to study damage progression in composite materials [26-29]. Various examples can be found in literature regarding the FE implementation of progressive damage models to composites [30-35]. In this study ANSYS® 18 [36] is employed to perform the numerical analyses, which allow us to choose between two distinct damage evolution methods: the material property degradation (MPDG) and the continuum damage mechanics (CDM). The MPDG method assumes that stiffness immediately reduces once the onset of material is reached and that the reduction is based on explicit values defined by damage coefficients which can take values from 0 (undamaged status) to 1 (completely damaged status). This method is a generalization of the concept of continuum damage evolution for composite laminates that was proposed by Camanho et al. [26]. In the CDM method, the damage coefficients increase progressively based on the energy amounts dissipated for the different damage modes (fibre tension, fibre compression, matrix tension and matrix compression). However, to be able to implement the CDM method, the material viscous damping coefficients and the energy dissipated per unit area need to be defined. Since these material properties were not available for the studied material, the MPDG method has been used in the progression damage model of the [0,90]n NCF laminate. This paper is strongly related to previously published work [6, 37-39]. It follows the modelling approach presented in [37], which makes use of the rotation of the elements’ coordinate system to model the fibre crimp in the geometrically straight 3D FE meshes. The efficiency/effectiveness of this approach was demonstrated in [6]. The qualitative mechanism responsible for the failure of NCF laminates under compressive loading was presented in [38], followed by a preliminary study to assess the quantitative value of the failure load using the Maximum Strain criterion and the MPDG method [39]. The current paper describes in greater detail the work initiated in [39] and extends the analysis by comparing different noninteractive (Maximum Stress and Maximum Strain) and interactive failure criteria (Hashin [25] and Puck [40]). The description of the material properties, mesoscopic approach and the boundary conditions employed are presented in sections 2.1, 2.2 and 2.3, respectively. The progressive damage model used in the study is detailed in section 2.4. The progressive damage throughout the representative unit cell (RUC), from the load at which damage is initiated until the load at which the failure of the NCF laminate is predicted, is presented in section 3.1. The mechanisms responsible for the failure of the laminate are discussed in section 3.2. In section 3.3, the numerical results are validated against experimental results obtained from direct compression tests on biaxial cross-ply NCF laminates with similar average stiffness properties to those used in the numerical models [41]. Finally, the conclusions are presented in section 4. 4 2. Numerical Modelling The following subsections are dedicated to describing the material properties of the constituents, the mesoscopic modelling approach, the boundary conditions and the progressive damage model used in the generation of the FE model. 2.1. Material properties of the constituents Fibre tows can be considered, at the mesoscopic scale, as a homogeneous transversely isotropic material made from parallel fibres uniformly distributed and impregnated by an isotropic resin. Since no experimental measurements have been performed to evaluate the elastic constants of the fibre tows, the corresponding stiffness matrix in the FE model has been calculated using micro-meso homogenisation, based on Chamis’ semi-empirical model [42]. This micromechanical model has been used in several studies of meso-FE modelling for textile composites as it gives the formulation for all independent elastic constants [43-45]: 𝐸11 =𝑉𝑓𝐸11 𝑓+(1−𝑉𝑓)𝐸𝑚 ; 𝐸22 =𝐸33 =𝐸𝑚 1−√𝑉𝑓(1−𝐸𝑚 𝐸22 𝑓) (1) 𝐺12 =𝐺13 =𝐺𝑚 1−√𝑉𝑓(1−𝐺𝑚 𝐺12 𝑓) ; 𝐺23 =𝐺𝑚 1−√𝑉𝑓(1−𝐺𝑚 𝐺23 𝑓) (2) 𝜈12 =𝜈13 =𝑉𝑓𝜈12 𝑓+(1−𝑉𝑓)𝜈𝑚 ; 𝜈23 =𝐸22 2𝐺23 −1 (3) where 𝑉𝑓 is the fibre volume fraction of the tows and E , G and 𝜈 are elastic moduli, shear moduli and Poisson’s ratios, respectively. Superscript 𝑓 is used for the fibre properties while superscript 𝑚 is used for the matrix properties. Subscript 1 designates the longitudinal fibre direction while 2 and 3 designate the transverse fibre directions in the material (3 being the through-thickness direction of the laminate). The constituents of the biaxial cross-ply NCF composites, tested under direct compression for the FALCOM project and presented in Table 1, have been considered in the estimation of the elastic properties of the fibre tows [41]. The epoxy resin - Hexply® M36 [46] is characterized by, 𝐸𝑚= 3.5 GPa, 𝜈𝑚= 0.42 and 𝐺𝑚= 1.2 GPa. The carbon fibre Tenax® HTS 5631 12K [47] is characterized by, 𝐸11 𝑓= 237 GPa, 𝐸22 𝑓= 13 GPa, 𝜈12 𝑓= 0.2, 𝐺12 𝑓= 18 GPa, 𝐺23 𝑓= 6 GPa. Notice that the elastic properties of the carbon fibres have been extracted from the manufacturer’s datasheet and those which were not available have been obtained from literature, using averaged values derived from experimental measurements [42, 47-52]. Parameter Description/Value Representative unit cell size [x] 2.6 mm ´ [y] 2.6 mm ´ [z] 0.96 mm Tows Carbon fibre - Tenax® HTS 5631 12K Matrix Epoxy resin - Hexply® M36 5 Table 1. [0,90]n NCF laminate parameters. Fig. 2. Shear stress-strain behaviour of the fibre tows. The behaviour of the fibre tows under shear loads in the 12-plane and 13-plane, is controlled by the resin, whose behaviour is nonlinear [2, 5, 53]. This nonlinear behaviour can be appreciated in Fig. 2, in which the shear stress-strain experimental curve obtained from a unidirectional laminate T300/914 is represented [2, 5, 53]. In previous studies [37, 38], the bilinear behaviour was included in the RUC to account for the shear nonlinear behaviour of the fibre tows. Notwithstanding, to be able to carry out a progressive damage study with ANSYS® 18 [36], the fibre tows have been modelled with a linear elastic shear behaviour in all planes, as both material behaviour models cannot be added simultaneously to the FE model. The fibre tows have been defined with an averaged fibre volume fraction of 60% taking into consideration the values observed in [54]. The estimated stiffness properties of the fibre tows are presented in Table 2. 𝐸11 𝐸22 =𝐸33 𝜈12 =𝜈13 𝜈23 𝐺12 =𝐺13 𝐺23 147 GPa 8.1 GPa 0.29 0.26 4.43 GPa 3.2 GPa Table 2. Estimated stiffness properties of the fibre tows. The strength properties of the fibre tows have been obtained from averaged strength values of carbon fibre unidirectional laminates with a fibre volume fraction of 60% [55, 56]. The values employed are presented in Table 3, where 𝑆𝑖𝑗 𝑙 are the strengths of the fibre tows with indices 𝑖,𝑗= 1, 2, 3 corresponding to the aforementioned orthogonal directions, and 𝑙= +, −,𝑠 standing for the fibre tows tensile, compressive and shear strength, respectively. The strain limits used in the Maximum Strain criterion have been calculated dividing the strength limits by the appropriate stiffness coefficients, i.e., assuming a linear relationship [55, 57]. 6 𝑆11 + 𝑆11 − 𝑆22 +=𝑆33 + 𝑆22 −=𝑆33 − 𝑆12 𝑠=𝑆13 𝑠=𝑆23 𝑠 2100 MPa 1500 MPa 55 MPa 200 MPa 67 MPa Table 3. Strength properties of the fibre tows. 2.2. Mesoscopic approach of the NCF laminate A mesoscopic scale FE model of the RUC of a [0,90]n NCF laminate has been modelled using 3D 8 node (three degrees of freedom at each node) solid element SOLID185 from ANSYS® 18 FE code [36]. This element enables the definition of an element’s coordinate system with an orientation not imposed by the node locations, thus allowing us to define the orthotropic material directions appropriately at each point. Each lamina of the unit cell is composed of two halves of a fibre tow which are separated by a resin rich area. The stacking sequence and the fibre tows orientation and shape are represented in Fig. 3. Although the fibre tows have been generated using a straight 3D FE mesh, an out- of-plane fibre crimp has been modelled following the approach presented in [37]. The fibre crimp has been defined with the same maximum angle (𝛽 = 9°) and transversal to the fibre direction in all fibre tows. The fibre crimp orientation, upwards or downwards, has been made dependent on the lamina stack position in the laminate. To ensure the appropriate modelling of the fibre crimp and to improve the accuracy of the numerical solution, the FE mesh has been refined inside and around the fibre crimp locations. Notice that the maximum fibre crimp angle that has been employed in the FE model was defined based on previous studies [6, 58], in which (using an identical RUC) a satisfactory agreement has been found between the apparent in-plane stiffness properties and the experimental evidence for angles ranging from 9º to 12º. Fig. 3. 3D FE model of the RUC of the [0,90]n NCF laminate (fibre tows only). 7 Fig. 4. (a) One-quarter of the 3D FE model of the RUC of the [0,90]n NCF laminate with geometric parameters. (b) rotation of the element’s coordinate system. Taking advantage of the symmetry of the problem, only one-quarter of the RUC has been modelled in this study, see Fig. 4(a). The RUC has been defined parametrically with the geometric parameters being the length of the tow not affected by the fibre crimp (a=2.06 mm), the thickness of each lamina (t=0.24 mm), and the width of the gap left between two adjacent tows of each lamina (g=0.3 mm), see Fig. 4(a). The values used in the dimensioning of the RUC (a, t and g) were estimated from the microstructure characterization performed in [48] and the total size of the RUC is shown in Table 1. In accordance with the global coordinate system xyz, the plane of the laminate is the xy-plane, and z is the through-thickness direction while the local coordinate system 123 is oriented considering the actual direction of the fibres, direction 1 in Fig. 4(b). As an indication, the quarter of the RUC of the [0,90]n NCF laminate shown in Fig. 4a) has a total of 36952 elements and 150121 nodes. 2.3. Boundary conditions The boundary conditions applied to the faces of the FE model serve to impose a compressive stress state and to guarantee the displacements compatibility in the limits of the model with the surrounding structure. Symmetry conditions have been assumed at the faces parallel to the xz-plane and at one of the faces parallel to yz-plane, while at the opposite face, parallel to the yz-plane, a constant displacement in the compression load direction has been applied to all nodes. In this way, the compression load is given by the sum of the reaction forces of all nodes at 𝑥= 0. The boundary conditions at 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 at both faces were coupled to assure that the extension in the through-thickness direction is constant throughout the entire model. 8 2.4. Progressive damage modelling Progressive damage has been modelled at the mesoscopic level using the built-in MPDG method, available in ANSYS®18 [36]. This damage evolution method requires the definition of a damage initiation criteria to determine the onset of the material damage, and a damage evolution law that controls the stiffness reduction of the material. Since there are no well-established specific damage initiation criteria for NCF composites, and given that at this scale fibre tows can be considered to behave as UD composite laminas, the most commonly used failure criteria for laminates [57] have been applied to the fibre tows to predict the onset of the material damage and its progression until the compressive failure of the NCF laminate. Non-interactive criteria, Maximum Stress and Maximum Strain, and interactive criteria, Hashin [25] and Puck [40], have been employed to determine the onset of the material damage of the fibre tows under the compressive loading (see Appendix A). For the sake of simplification and knowing beforehand that the material degradation of the resin rich areas will have a very small impact on the overall stiffness reduction of the NCF laminate, the Maximum Stress criterion has been employed for all the resin rich areas. Notice that all the required tensile and compressive strengths for the resin are available in the manufacturer’s datasheet [46]. According to the MPDG method, after reaching the onset of the material damage defined by each criterion, the stiffness reduction occurs immediately, and each damage mode (fibre tension, fibre compression, matrix tension and matrix compression) is assumed to be uncoupled. The amount of stiffness reduction is based on damage coefficients specified in the damage evolution law. These coefficients can vary from 0 to 1, where 0 corresponds to no damage and 1 to the complete loss of stiffness in the affected damage mode. The Hooke’s law for a damaged orthotropic material is given by (4), where Sij are the components of the compliance matrix of the undamaged orthotropic material, 𝝈 is the nominal Cauchy stress tensor, and 𝜺 is the elastic strain tensor [11]. 𝜺= ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑆11 (1−𝑑𝑓)𝑆12 𝑆13 0 0 0 𝑆21 𝑆22 (1 −𝑑𝑚)𝑆23 0 0 0 𝑆31 𝑆32 𝑆33 (1 −𝑑𝑚)0 0 0 0 0 0 𝑆44 (1 −𝑑𝑠)0 0 0 0 0 0 𝑆55 (1 −𝑑𝑠)0 000 0 0 𝑆66 (1 −𝑑𝑠) ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ 𝝈 (4) The damage coefficients for calculating the damaged elasticity matrix are determined by, 9 𝑑𝑓={𝑑𝑓 +,𝑖𝑓 𝐼𝑓+≥1 𝑑𝑓 −,𝑖𝑓 𝐼𝑓−≥1 𝑑𝑚={𝑑𝑚 +,𝑖𝑓 𝐼𝑚 +≥1 𝑑𝑚 −,𝑖𝑓 𝐼𝑚 −≥1 𝑑𝑠= 1 −(1−𝑑𝑓 +)(1−𝑑𝑓 −)(1−𝑑𝑚 +)(1 −𝑑𝑚 −) (5) where, 𝑑𝑠 is the shear damage coefficient and 𝐼𝑓+,𝐼𝑓−,𝐼𝑚 + and 𝐼𝑚 − are respectively the fibre tension, fibre compression, matrix tension and matrix compression damage initiation indexes calculated from the effective Cauchy stress measured in the undamaged domain. Each criterion computes the damage initiation indexes differently [25, 40], hence different progressive damage results are obtained with the Hashin’s, Puck’s, Maximum Stress and Maximum Strain failure criteria. As mentioned before, there are no well-stablished failure criteria for a fibre tow of an NCF laminate, so several criteria that were initially developed for a prepreg lamina were object of analysis in this study: Maximum Stress, Maximum Strain, Puck and Hashin. Therefore, four different progressive damage FE models have been generated. In all of them, the matrix has been modelled as an isotropic material and the onset of failure has been estimated with the Maximum Stress criterion. Otherwise, the fibre tows have been modelled as a transversely isotropic material and a different failure criterion to estimate the onset of damage has been used in each of the four models. Notice also that each of these failure criteria is in fact a close set of four different failure criteria associated to the four different damage modes (fibre tension, fibre compression, matrix tension and matrix compression). The damage coefficients values employed in this study for the fibre tows and for the resin rich areas (fibre tension, 𝑑𝑓 +, fibre compression, 𝑑𝑓 −, matrix tension, 𝑑𝑚 +, and matrix compression, 𝑑𝑚 −) are presented in Table 4. A parametric study has been carried out to select the damage coefficients which produce results that better adjust to the experimental evidence. In the parametric study, different damage coefficient values, ranging from 0 to 0.4, have been considered for both the fibre tows and for the resin rich areas. The results have shown that for the resin rich areas, changing the matrix damage coefficient values, 𝑑𝑚 + and 𝑑𝑚 −, within this range, adds no relevant effect on the stiffness degradation of the NCF laminate. However, for the fibre tows, the sensitivity to changes in the coefficients values has shown to be much higher. The matrix damage coefficients, 𝑑𝑚 + and 𝑑𝑚 −, control the amount of the initial stiffness reduction of the damaged elements. At this stage, the fibre damage coefficients are still not initiated (𝑑𝑓 +=𝑑𝑓 −= 0). The amount of the second and last stiffness reductions in the completely damaged elements are controlled by all the damage coefficient values, 𝑑𝑚 +, 𝑑𝑚 −, 𝑑𝑓 + 𝑎𝑛𝑑 𝑑𝑓 −. In this way, it is clear that the damage coefficients used in the fibre tows can control both the nonlinear behaviour and the failure load of the NCF laminate. The relation between the damage coefficient values and the stiffness reductions is addressed with greater detail in the next section. 16 Fig. 10. Damage development analysis for an element with maximum misorientation in a 0° fibre tow for all the failure criteria. As can be seen, results shown in Fig. 10 are in a fairly good agreement with the non-linear shear behaviour of the fibre tows shown in Fig. 2. Notice that, material degradation is used to model the non-linear shear behaviour of the fibre tows since, as already mentioned, ANSYS® 18 [36] does not allow us to use two non-linear material models at the same time. However, the existence of different failure indexes and damage coefficients associated to different failure modes has made possible to obtain good agreement with experimental results using the progressive damage to model both the failure and the non-linear shear response of the material. 3.3. Failure load predictions comparison and validation of the numerical results. The numerical predictions of failure stress and failure strain, for each failure criteria, are discussed in this section. The numerical results are also validated against the experimental data obtained from direct compression tests on biaxial cross-ply NCF laminates with similar average stiffness properties to those used in the numerical models [41]. The results presented in Table 5 and Fig. 6, make evident the similarity in the failure stress and failure strain values obtained for the Hashin’s, Puck’s and the Maximum Stress failure criteria. The predicted failure stress and failure strain are almost identical for all the aforementioned criteria, with values between 645-650 MPa and 1.06-1.07%, respectively. The experimental mean values for the failure stress and failure strain are only 4% and 1% higher than the numerically predicted results, respectively. The Maximum Strain criterion predicted approximately a failure stress and failure strain values 10-14% lower than the all other criteria. This difference is justified by the fact that the strain limits have been calculated dividing the strength limits by the initial stiffness coefficients. The compressive elastic modulus is another relevant parameter for the validation of the numerical models. The stiffness of the [0,90]n NCF laminate has been calculated for a strain of 𝜀11 ≅0.01% (no damage initiation). The numerical and the experimental mean values of the compressive elastic modulus are in satisfactory agreement as is possible to appreciate from the stress-strain curves represented in Fig. 6. The numerical and experimental mean values obtained are, 𝐸𝑥 𝑛𝑢𝑚= 65.5 GPa and 𝐸𝑥 𝑒𝑥𝑝= 67.4 GPa, respectively. 4. Conclusions A mesoscopic 3D RUC has been used to study the progressive damage of a [0,90]n NCF laminate under in-plane compressive loading. The RUC has been defined with an out-of-plane fibre crimp with a maximum misorientation angle of β % = % 9° in the 0° fibre tows. The material properties degradation method has been used in this study and non-interactive criteria (Maximum Stress and Maximum Strain), and interactive criteria (Hashin and Puck), associated with failure modes, have been employed to determine the onset of the material damage of the fibre tows. The Maximum Stress criterion has been used for all the resin rich areas. The following stages have been identified in the analysis of the progressive damage of the NCF laminate when submitted to the compressive loading. Firstly, the presence of high localized shear stresses, 𝜎13, and shear strains, 𝜀13, in the crimp location of the 0° fibre tows, 17 particularly in elements with higher misorientation angle, are responsible for the damage initiation, i.e., they exceed the respective allowable that has been defined in the numerical model. Secondly, once the onset of the material damage is reached, the affected elements become damaged and suffer an instant stiffness reduction based on the values defined for the damage coefficients. Thirdly, damage propagates to the surrounding elements of the fibre crimp location and completely damaged (all damage coefficients are initiated) elements also arise, suffering then a second stiffness reduction. Finally, both damaged and completely damaged elements spread rapidly, mainly throughout the 0° tows and the resin reach areas below the crimp location. These events motivate the sudden loss of the NCF laminate stiffness, and thus failure of the NCF laminate is predicted, see Fig. 5. The numerical results obtained with the MPDG method in which the onset of the material damage was determined by Hashin’s, Puck’s and Maximum Stress criteria presented a satisfactory agreement with the experimental data. On the other hand, the MPDG method combined with the Maximum Strain criterion predicted values outside the range of the experimented results. A satisfactory agreement between the numerically calculated compressive elastic modulus of the [0,90]n NCF laminate and the experimental mean values has been found. The results have also shown that the shear modulus of the tows does not substantially affect the compressive elastic modulus of the NCF laminate. The scope of the study can be widened in the future to include also the Continuum Damage Mechanism (CDM) method available in ANSYS® 18 [36] which uses the Hashin’s failure criterion to predict the onset of the material damage. Appendix A The expressions employed to compute damage initiation indexes in ANSYS® 18 [36] for the non-interactive criteria (Maximum Stress and Maximum Strain) and the interactive criteria (Hashin and Puck), are summarized in Table 6 and Table 7, respectively. Maximum Stress Failure Criterion Maximum Strain Failure Criterion Fibre Tension: 𝜎11 > 0 𝐼𝑓+=𝜎11 𝑆11 + 𝐼𝑓+=𝜀11𝐸11 𝑆11 + Fibre Compression 𝜎11 < 0 𝐼 𝑓 −=𝜎11 𝑆11 − 𝐼 𝑓 −=𝜀11𝐸11 𝑆11 − Matrix Tension 𝜎22,𝜎33 > 0 𝐼 𝑚 +=𝜎22 𝑆22 + , 𝐼 𝑚 +=𝜎33 𝑆33 + 𝐼 𝑚 +=𝜀22𝐸22 𝑆22 + , 𝐼 𝑚 +=𝜀33𝐸33 𝑆33 + Matrix Compression 𝜎22,𝜎33 < 0 𝐼 𝑚 −=𝜎22 𝑆22 − , 𝐼 𝑚 −=𝜎33 𝑆33 − 𝐼 𝑚 −=𝜀22𝐸22 𝑆22 − , 𝐼 𝑚 −=𝜀33𝐸33 𝑆33 − Table 6. Non-interactive failure criteria associated with failure modes. 18 Hashin’s Failure Criterion [25, 36] Fibre Tension 𝜎11 ≥0 𝐼𝑓+=(𝜎11 𝑆11 +)2 +𝜎12 2+𝜎13 2 (𝑆12 𝑠)2 Fibre Compression 𝜎11 < 0 𝐼𝑓−=(𝜎11 𝑆11 −)2 Matrix Tension 𝜎22 +𝜎33 > 0 𝐼𝑚 +=(𝜎22 +𝜎33)2 (𝑆11 +)2+𝜎23 2−𝜎22𝜎33 (𝑆23 𝑠)2+𝜎12 2+𝜎13 2 (𝑆12 𝑠)2 Matrix Compression 𝜎22 +𝜎33 < 0 𝐼𝑚 −=1 𝑆22 −[(𝑆22 − 2𝑆23 𝑠)2−1](𝜎22 +𝜎33) + (𝜎22 +𝜎33 2𝑆23 𝑠)2 +𝜎23 2−𝜎22𝜎33 (𝑆23 𝑠)2+𝜎12 2+𝜎13 2 (𝑆12 𝑠)2 Puck’s Failure Criterion [36, 60] Fibre Tension 𝜎11 ≥0 𝐼𝑓+=𝜎11 𝑆11 + Fibre Compression 𝜎11 < 0 𝐼𝑓−=𝜎11 𝑆11 − Inter-fibre Tension [Matrix Tension] 𝜎𝑛≥0 𝐼𝑚 += ⎷ √ √ √ (1 𝑆22 +−𝐾+)2𝜎𝑛2+(𝜏𝑛𝑡 𝑆23 𝑠)2 +(𝜏𝑛1 𝑆12 𝑠)2 +𝐾+𝜎𝑛 𝐾+= ⎩ ⎪ ⎨ ⎪ ⎧ 1 𝜏𝑛𝑡 2+𝜏𝑛1 2(𝑃23 + 𝑆23 𝑠𝜏𝑛𝑡 2+𝑃13 + 𝑆12 𝑠𝜏𝑛1 2) if 𝜏𝑛𝑡 2+𝜏𝑛1 2> 0 0 if 𝜏𝑛𝑡 2+𝜏𝑛1 2= 0 Inter-fibre Compression [Matrix Compression] 𝜎𝑛< 0 𝐼𝑚 −=√(𝜏𝑛𝑡 𝑆23 𝑠)2 +(𝜏𝑛1 𝑆12 𝑠)2 + (𝐾−𝜎𝑛)2+𝐾−𝜎𝑛 𝐾−= ⎩ ⎪ ⎨ ⎪ ⎧ 1 𝜏𝑛𝑡 2+𝜏𝑛1 2(𝑃23 − 𝑆23 𝑠𝜏𝑛𝑡 2+𝑃13 − 𝑆12 𝑠𝜏𝑛1 2) if 𝜏𝑛𝑡 2+𝜏𝑛1 2> 0 0 if 𝜏𝑛𝑡 2+𝜏𝑛1 2= 0 Table 7. 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