Evaluating failure modes through energy dissipation mechanisms in hybrid composites under bending loads

Evaluating failure modes through energy dissipation mechanisms in hybrid composites under bending loads J.M. Parente a , L.M. Ferreira b,c,* , P.N.B. Reis d a C-MAST, Center for Mechanical and Aerospace Science and Technologies, Universidade da Beira Interior, Rua Marquˆ es d’Avila e Bolama 6201001 Covilh˜ a, Portugal b Grupo de Elasticidad y Resistencia de Materiales, Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla, Camino Descubrimientos, S/N, Sevilla 41092, Spain c Escuela Polit´ ecnica Superior, Universidad de Sevilla, C/ Virgen de ´ Africa, 7, Sevilla 41011, Spain d Department of Mechanical Engineering, CEMMPRE, ARISE, University of Coimbra 3030-788 Coimbra, Portugal ARTICLE INFO Keywords: Polymer matrix composites (PMCs) Hybridization effect Bending response Damage mechanisms Numerical analysis Energy balance ABSTRACT Understanding the bending behaviour of composite materials is essential for effective design, particularly with the increasing use of complex components with multiple bends. Carbon fibres are often preferred in such applications due to their superior tensile properties; however, given their limited compressive performance, hybridization with glass fibres is commonly used. In this context, this study analyses the energy contributions of intralaminar and interlaminar damage mechanisms leading to failure in hybrid carbon/glass fabric-reinforced laminates under bending loads. Different hybridization ratios and configurations, specifically the positioning of glass and carbon fabric reinforcements relative to the load application, are evaluated experimentally and numerically. The experimental results show that the bending performance of hybrid laminates falls between those of non-hybrid carbon (8C) and glass (8G) laminates, with a clear dependence on the hybridisation ratio. When glass fibres are positioned in the compression region, the hybrid laminates exhibit slightly higher force and displacement values. Notably, the 3G/5C configuration (glass on the compression side) achieves a force and a displacement of 255.1 N and 4.23 mm, respectively, representing increases of approximately 5.9% and 13.1% compared to the 5C/3G configuration, which reaches 240.9 N and 3.74 mm. Numerical models show a good agreement with the experimental data, with force errors predominantly within ±5.3% and displacement errors within ±6.8%. Non-hybrid configurations demonstrate a more predictable damage progression, whereas hybrid laminates introduce variability due to differences in fiber type and placement, influencing overall energy dissipation and structural performance. Additionally, the energy analysis reveals that intralaminar damage is the dominant energy dissipation mechanism, followed by delamination and friction. * Corresponding author at: Grupo de Elasticidad y Resistencia de Materiales, Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla, Camino Descubrimientos, S/N, Sevilla 41092, Spain. E-mail address: [email protected] (L.M. Ferreira). Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech https://doi.org/10.1016/j.engfracmech.2025.110855 Received 26 October 2024; Received in revised form 20 December 2024; Accepted 20 January 2025 Engineering Fracture Mechanics 316 (2025) 110855 Available online 21 January 2025 0013-7944/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). 1. Introduction There are increasing challenges for engineers to use lighter and more efficient materials without compromising their mechanical performance. It is in this context that composite materials emerge as the a promising substitute for traditional metal-based materials [1,2]. In addition to a polymer-based matrix, they can use various fibres as reinforcing elements, taking advantage of their intrinsic properties [3,4]. In the case of reinforcement, the aim is to provide strength and stiffness to withstand the applied load [5], while the matrix acts as a stress distributor between the reinforcing fibres, as well as protecting the reinforcing fibres from external factors (i.e. moisture, corrosion, chemicals, etc.) [6,7]. From this perspective, carbon fibres are preferred for many engineering applications, primarily due to their excellent tensile properties. However, their poor performance under compressive loads represents a significant limitation, reducing their broader application. For example, this drawback is particularly evident in carbon fibre-reinforced composites used in structural elements subjected to bending loads, a loading mode that is gaining importance with the development of increasingly complex components with multiple bends, designed to minimize welding and assembly requirements. In this context, there is interest in combining various types of fibres (hybridization) to improve the mechanical performance of composite laminates. For example, hybridization can increase damage tolerance by combining fibres that promote strength with others that have higher ductility and, consequently, better impact performance [8–11], or simply to ensure a wider range of applications, where fibres with higher mechanical performance can be combined with more economical ones to achieve a balance between mechanical properties and cost [10,12]. These advantages translate into significant advancements across various sectors [13–16]. In automotive applications, for example, hybrid composites can enable the production of lighter and stronger components such as bumpers and body panels [17,18]. In terms of marine industry, they can contribute to more durable and cost-effective boat hulls [19–21], while in the aerospace sector, they bring advantages in the Nomenclature Abbreviation Description VG fGlass fibre volumetric fraction VC fCarbon fibre volumetric fraction VmMatrix volumetric fraction tThickness of the laminate ρ Density E1+,−Tensile/compressive Young’s modulus along fibre direction 1 E2+,−Tensile/compressive Young’s modulus along fibre direction 2 G12 In-plane shear modulus ν 12 In-plane Poisson’s ratio X1+,−Tensile/compressive strength along direction 1 X2+,−Tensile/compressive strength along direction 2 S12 In-plane shear strength G1,2 fIntralaminar fracture toughness along direction 1 and 2 dmax 12 Maximum shear damage  σ y0Initial effective shear yield stress CCoefficient in hardening equation pPower term in hardening equation knElastic normal interlaminar stiffness ksElastic shear interlaminar stiffness ktElastic tangential interlaminar stiffness τ 0 nMaximum normal contact stress τ 0 sMaximum 1 st shear contact stress τ 0 tMaximum 2 nd shear contact stress GIc Interlaminar normal fracture toughness GIIc Interlaminar 1 st shear fracture toughness GIIIc Interlaminar 2 nd shear fracture toughness η Benzeggagh-Kenane exponent CdDilatational wave speed Lmin Smallest characteristic length of the element ΔtTime increment FE Finite Element 3PB 3-Point Bending CDM Continuum Damage Mechanics SbCB Surface-based Cohesive Behaviour J.M. Parente et al. Engineering Fracture Mechanics 316 (2025) 110855 2 manufacture of lightweight and high-strength aircraft components, such as wings and fuselage panels [22–24]. Furthermore, hybrid composites also have several advantageous applications in the renewable energy sector, such as wind turbine blades to increase their improved strength-to-weight ratio and, consequently, increase energy capture efficiency [25–27]. However, to integrate this type of material into industrial applications, requires the evaluation of its static, fatigue, and viscoelastic performance. This data is crucial for the design of safe, reliable, and efficient composite structures. Bending tests are widely considered as an essential method to evaluate the mechanical properties of composite materials. These tests provide key parameters such as bending modulus, bending strength, and fracture toughness, which are used to predict the material behaviour under service conditions. Moreover, the anisotropic and heterogeneous nature of composite materials leads to the development of various failure mechanisms, such as tensile and compressive fibre fracture, enclosed or side-opened delaminations (interlaminar shear), and localized crushing under the crosshead, which can be studied through bending tests [28–31]. Numerical simulations, can provide an excellent tool to improve the understanding of the material’s response, thereby contributing to the development of safer, more efficient, and innovative composite structures. Therefore, numerous studies in the literature have employed Finite Element (FE) models to investigate the bending behaviour of hybrid composites. For example, Dong et al. [32–37] focused on modelling the bending response of hybrid composites reinforced with carbon and glass fibres. They concluded that the optimal ratio of glass to carbon fibre is 0.125 [34] and that glass and carbon fibre configuration influences the failure behaviour of the laminates [33]. These authors also found that placing glass fibres on the compressive side can improve the bending properties of the hybrid composite compared to an all-carbon composite, while their placement on the tensile side has a negative effect [33,36]. Jiang et al. [38] carried out an experimental and numerical study to investigate the interactive failure of a hybrid composite comprising carbon, Kevlar®, and glass fibres. They concluded that the hybrid carbon/Kevlar® and carbon/glass/Kevlar® composites exhibited superior energy absorption capabilities compared to carbon fibre and Kevlar® only composites. Furthermore, the authors reported that the mechanical properties can be improved through the interplay between brittle and ductile failure mechanisms. Burgani et al. analysed the bending failure of a carbon fibre-reinforced composite by comparing experimental and numerical results [39]. They found that the Hoffman and Tsai-Wu criteria were the most accurate in predicting the onset of bending failure. Additionally, the study revealed that the type of carbon fibre and resin used could influence the composite’s failure mechanisms. Massarwa et al. [40] employed FE models to study the failure behaviour of a hybrid composite composed of glass and carbon fibres. They observed that the positioning of the carbon layers relative to the neutral axis significantly influenced the bending modulus. Furthermore, damage initiation occurred in the compressive outer layers and then propagated throughout the specimen. Regarding the energy dissipation capabilities of composite materials, the studies conducted by Jiang et al. [41], and Jiang and Ren [42], demonstrated the importance of engineered interfaces in enhancing the performance of CFRP composites. Through the incorporation of microand nanofillers and the application of techniques such as CNT-modified resin pre-coating, these interfaces can be tailored to promote specific failure mechanisms that maximize energy absorption. From the available literature, it is possible to conclude that most studies focus on the numerical replication of experimental data and, in some cases, on analysing the morphology of the resulting damage. While these studies have significantly advanced the understanding of the overall behaviour of hybrid fibre-reinforced composites, some aspects remain underexplored, particularly regarding the characterization of the initiation and progression of damage. Moreover, to the best of the authors’ knowledge, no study has systematically quantified the energetic contributions of damage mechanisms at the intraand interlaminar level, specifically in hybrid fabric-reinforced composites. To address these gaps, the present study investigates the bending response of hybrid fabric-reinforced composite laminates with various hybridization ratios and fibre positioning, focusing on numerically predicting the energy dissipation contributions of fibre failure and matrix cracking (intralaminar damage), delamination (interlaminar damage), and friction. For this purpose, and considering that FE modelling provides a robust tool for analysing the complex interactions between different fabric reinforcement types and their arrangement, 3D FE models were developed using ABAQUS®/Explicit [43]. These 3D FE models integrate a Continuum Damage Mechanics (CDM) model for simulating intralaminar damage in fabric-reinforced composites and a Surface-based Cohesive Behaviour (SbCB) model for interlaminar damage. It is important to acknowledge that while 3D FE models have proven to be highly effective in characterising the mechanical response of composite laminates [44], there are still some challenges. These include, for example, the sensitivity of the simulation results to material property variations and FE discretization, and the complexity of accurately modelling post-failure behaviour. In this way, to ensure the reliability of the simulations, the developed 3D FE models were validated against the experimental force–displacement curves and the post-failure damage mechanisms observed in both hybrid and non-hybrid composite laminates. A total of eight configurations were analysed, including two non-hybrid configurations (a carbon and a glass fabric-reinforced composite laminate) and six hybrid configurations, providing a robust assessment of the predictive accuracy of the 3D FE models. Fig. 1. Lay-up configurations of the hybrid and non-hybrid composite laminates employed. J.M. Parente et al. Engineering Fracture Mechanics 316 (2025) 110855 3 A detailed understanding of these mechanisms is fundamental to optimizing the design of hybrid composites for improved mechanical performance and reliability. The numerical approach and results presented in this study aim to contribute to the optimization of hybridization ratios and fibre arrangements for specific applications, reducing dependency on extensive experimental testing. 2. Materials and Methods A two-component epoxy resin, SR8100 and SD8824 supplied by Sicomin, was used as the matrix, while reinforcements involved combinations of woven bidirectional carbon fabric 195 T (196 g/m 2 ) and woven bidirectional glass fabric 1195P (195 g/m 2 ), both supplied by Rebelco. According to the supplier, the thicknesses of the individual carbon and glass fabric reinforcements are 0.25 mm and 0.21 mm, respectively. A total of eight lay-up configurations were produced with the following combinations: “1G/7C”, “7C/1G”, “2G/6C”, “6C/2C”, “3G/5C”, “5C/3G”, “8C”, and “8G”, as illustrated in Fig. 1. The numbers represent the quantities of layers, while the letters represent the type of reinforcement, i.e. “C” for carbon fibres and “G” for glass fibres. It should be noted that these antisymmetric configurations were used to study the damage mechanisms and their propagation until final failure, aiming to optimize the bending response of hybrid composites involving glass and carbon fibres. The resin and hardener were mixed using a mechanical stirrer at 300 rpm for five minutes, followed by the removal of any air bubbles in a vacuum chamber. This mixture was then used to fabricate the composite laminates using hand lay-up technique. The laminates were placed in vacuum bags and compressed in a hydraulic press with a load of 2.5 kN for 24 hours to ensure a constant fibre volume fraction and uniform thickness. During the initial 30 minutes, the vacuum bag was connected to a vacuum pump to remove any air bubbles that may have been introduced during the manufacturing process. Finally, the laminates underwent post-curing at 40◦C for four hours. The corresponding fibre volume fractions, determined by the burn-off test in accordance with ASTM D2584-11 standard [45], are shown in Table 1. The fibre volume fractions for the carbon and glass fabric reinforcements are represented as VC f and VG f, respectively, whereas the matrix volume fraction is denoted as Vm. Specimens with dimensions of 60 ×10 ×t mm 3 were prepared for the static tests, where “t” represents the laminate thickness: 1.4 mm for the “8C” configuration, 1.52 mm for “8G”, 1.50 mm for “1G/7C” and “7C/1G”, 1.49 mm for “2G/6C” and “6C/2G”, and 1.49 mm for “3G/5C” and “5C/3G”. These specimens were then subjected to 3-point bending tests (3PB) in accordance with the ISO 178–2019 standard [46]. The tests were carried out using a Shimadzu AG-X universal testing machine equipped with a 10 kN load cell. The displacement rate was set at 2 mm/min, and the span between the supports was 35 mm, ensuring compliance with the standard for all laminate configurations. For each condition, eight specimens were tested to ensure statistical significance and the accuracy of the test results. The experimental setup used for the 3PB tests is shown in Fig. 2. 3. Experimental results The experimental static bending response of both hybrid and non-hybrid composite laminates, represented by force–displacement curves for all studied configurations is shown in Fig. 3. These representative curves were selected because they closely reflect the average response of the eight tests. Moreover, it should be noted that the experimental bending results presented a low variation, with the standard deviation for the average force ranging from 2% to 15%, and for the average displacement, ranging from 2.5% to 12%. A quasi-brittle behaviour is observed across all composites, characterized by a nearly linear increase in force with displacement up to a peak value, after which the force drops abruptly. It is noticed that some of the curves exhibit a noticeable zigzag pattern in the region of maximum force, attributed to the sequential failure of different fibres. This behaviour is particularly noticeable in hybrid composites with carbon fibres on the compression side (“7C/1G”, “6C/2G”, and “5C/3G”) because the high compressive stress concentration in the crosshead load contact region, combined with the low compressive strength of the fibres, promotes fibre breakage leading to the zigzag pattern observed [47–49]. Notably, the highest force was achieved by the non-hybrid carbon fabric-reinforced laminates “8C”, while the non-hybrid glass fabric-reinforced laminates “8G” exhibited the lowest. This trend is reversed when considering displacement at maximum load. The hybrid composite laminates, showing a strong dependence on the hybridization ratio, fall between these force and displacement values. In more detail, the highest average bending force was obtained for the configuration “8C” (366.6±27.7 N) and the lowest for “8G” (219.6±5.5 N), while the displacement at maximum force shows an inverse behaviour (2.58±0.2 mm and 5.96±0.18 mm, respectively). These differences are approximately 1.67 times lower for the bending force and 2.31 times higher for the displacement. Regarding the hybrid laminates, the effect of the glass fabric reinforcement position on the bending response should be emphasized, because when they are in the compression region of the specimen lead to slightly higher results in terms of force and displacement. Moreover, it can be observed that regardless of their position, a higher number of glass fabric reinforcement layers decreases the Table 1 Fibre volume fraction for the different laminates obtained by the burn-off test. Laminates VG f(%)VC f(%)Vm(%) 3G/5C 16 38.07 45.93 2G/6C 10.28 44.03 45.69 1G/7C 5.16 51.62 43.22 8C −56.40 43.60 8G 55.91 −44.09 J.M. Parente et al. Engineering Fracture Mechanics 316 (2025) 110855 4 Fig. 2. Experimental setup for the 3PB tests according to ISO 178-2019 [46]. Fig. 3. Force-displacement curves for the fabric-reinforced composite laminates with: a) glass fabric-reinforcement placed in the compressive region; b) glass fabric-reinforcement placed in the tensile region. Fig. 4. Damage mechanisms observed in fabric-reinforced laminates “8C”, “8G”, “3G/5C” and “5C/3G”. J.M. Parente et al. Engineering Fracture Mechanics 316 (2025) 110855 5 bending force but increases the bending displacement. To clarify the previously mentioned observations, Fig. 4 provides a summary of the typical damage mechanisms observed in the “8C”, “8G”, “3G/5C” and “5C/3G” configurations, with the hybrid arrangement being representative example of the others. Regarding the non-hybrid laminates, the “8C” configuration presents fibre breakage and delaminations on the compression side, while in the “8G” configuration shows fibre breakage on the tensile side, due to higher bending capacity compared to the carbon laminate, along with delaminations in the crosshead load contact region caused by the stress concentrations in that area. In the case of “8G”, a more gradual failure progression was observed. These damage mechanisms are consistent with the literature [48,50–54], and are related to the intrinsic properties of the fibres. It should be noted that carbon fibres are four times stiffer than glass fibres and have a tensile strength 20%–50% greater [55–57], the glass fibres show higher strains (around 650%) than carbon fibres and, consequently, higher toughness [58–60]. On the other hand, the compressive strength of the carbon fibre/carbon fibre composites is between 30% to 50% of the tensile strength [61], which means that carbon fibres are much more sensitive to compression loads than glass fibres [36]. For the hybrid configurations “3G/5C” and “5C/3G”, the damage mechanisms are similar to those observed in the non-hybrid configurations. For instance, when carbon fibres are placed on the compression side, many of them break, and delaminations form around them, while the glass fibres on the tensile side break punctually. This progression of fibre failure in the compression region contributes to the zigzag pattern observed in the force–displacement curves for the carbon/glass configurations [47–49]. On the other hand, when the glass fibres are placed on the compression side, the damage is more extensive. In addition to delaminations, it includes breaking of the carbon fibres in the tensile region and glass fibres in the compression region. This more severe damage is responsible for the abrupt drop in the force–displacement curves after the peak load is reached. Further analysis of the damage evolution and its correlation with the force–displacement curves is provided in the following sections. 4. Numerical models 4.1. Damage models Under bending loading, composite materials experience tensile and compressive stresses that can lead to intralaminar damage, typically manifesting as matrix cracking and fibre breakage. Additionally, interlaminar shear stresses, particularly concentrated near the loading point and supports, can cause separation between the laminate layers, known as delamination. These damage mechanisms contribute to a reduction in the material’s stiffness and load-bearing capacity, making it essential to develop FE models capable of predicting the material’s behaviour. To simulate the damage mechanisms that occur in the fabric-reinforced composite laminates under 3PB bending loading, two damage models were used: a Continuum Damage Mechanics (CDM) model and a Surface-based Cohesive Behaviour (SbCB) model, which account for intralaminar damage and interlaminar damage, respectively. To evaluate damage evolution at the intralaminar level in the tested laminates, which were reinforced with carbon and/or glass fabrics, the constitutive damage model for fabric-reinforced composites available in ABAQUS®/Explicit was used [43]. This model, implemented via a VUMAT subroutine and accessed by defining a user-defined material with a string “ABQ_PLY_FABRIC”, is compatible exclusively with plane-stress elements and considers each layer a homogeneous orthotropic elastic material. It accounts for stiffness degradation resulting from fibre failure, matrix cracking, and plastic deformation under shear loading. This CDM model applies the maximum stress failure criterion to identify the onset of fibre damage and uses a damage evolution approach, governed by fracture energies, to control the reduction in stiffness. Table 2 presents the VUMAT subroutine inputs for the carbon and glass fabricreinforced composite layers used in this study. The subscripts “+” and “-” denote the tensile and compressive loading modes, respectively, while “1” and “2” refer to the principal directions of the layer. The elastic and strength properties, as well as the fracture energies of the non-hybrid composite laminates, were initially estimated from the experimental data. These values were then refined through a parametric study to achieve a good correlation with the experimental results, ensuring an accurate representation of the laminate behaviour. The shear plasticity data was obtained from [62–64]. The bond between the layers was simulated using the SbCB model. This approach is suitable for interfaces with negligible thickness and provides capabilities similar to the cohesive elements. The cohesive behaviour was defined as a surface interaction property, Table 2 Intralaminar properties of the carbon and glass fabric-reinforced composite layers. Property Symbol Units Carbon Fabric Glass Fabric Value Value Density ρ kg/m 3 1900 1600 Stiffness properties E1+,−=E2+,−GPa 40 9.5 G12 GPa 6 3 ν 12 −0.14 0.11 Strength properties X1+,−=X2+,−MPa 640 320 SMPa 120 40 Fracture energy G1,2 fN/mm 15,000 25,000 Shear plasticity dmax 12 −1 1  σ y0MPa 55 25 C−800 800 p−0.552 0.552 J.M. Parente et al. Engineering Fracture Mechanics 316 (2025) 110855 6 governed by a traction-separation constitutive model. Delamination onset was determined using the output damage variable “CSQUADSCRT”, which corresponds to the stress-based damage initiation criterion for cohesive surfaces in general contact within ABAQUS® FE code [43]. Once delamination initiates (i.e., when CSQUADSCRT=1), its progression until complete debonding, was captured using the output damage variable “CSMDG” for cohesive surfaces. This variable is controlled by the power law represented in Eq. (1), where complete delamination is predicted when CSMDG reaches the value of one. (GI GIc) η +(GII GIIc) η +(GIII GIIIc) η =1 (1) The material interlaminar properties used to define the SbCB model for this study are detailed in Table 3. These properties were obtained from the literature and subsequently adjusted to meet the specific requirements of the laminate configurations under analysis. The stiffness values were obtained from [65], while the damage initiation parameters were taken from [66]. Damage evolution energies, which define the material’s resistance to progressive delamination, were obtained from [67] and [68]. Finally, the interaction parameter was defined from [63]. The intralaminar and interlaminar damage models used in this study are thoroughly explained in [43,69,70]. These models have been applied by the authors in several previous studies [63,64,71,72] and have demonstrated reliable performance in predicting damage behaviour in fabric-reinforced composite laminates. 4.2. Finite element model Three-dimensional FE models replicating the experimental 3PB tests described in Section 2 were developed using ABAQUS® FE code [73]. Accordingly, the eight different laminate lay-up configurations illustrated in Fig. 1 were generated with the corresponding geometric parameters. Notice that the laminate thickness, represented with the parameter “t” in Fig. 2, was defined based on the specific dimensions of each lay-up configuration. The laminate layers were simulated using 8-node continuum shell elements (SC8R), while the crosshead and supports were modelled with 4-node discrete rigid elements (R3D4). Given the negligible thickness of the layer’s interfaces, cohesive surfaces were employed to model the bonding, eliminating the need for element definition. It should be noted that in SbCB models, the cohesive zone is represented as a zero-thickness interface, capturing the mechanical behaviour of bonded surfaces without explicitly modelling the material’s finite thickness. The 3D FE mesh discretization employing quadrilateral-shaped elements is represented in Fig. 5. Mesh refinement was applied in the contact regions with the crosshead and supports, where elements were assigned a characteristic length of 0.1 mm. This refinement ensures an accurate representation of stress concentrations and gradients near the loading point and supports, leading to a more precise prediction of material behaviour under bending load. To replicate the experimental setup, fixed boundary conditions were applied to both supports and a ramped displacement, u y , was imposed on the crosshead. To improve computational efficiency, only half of the geometry was modelled, taking advantage of the system’s symmetry. In this way, symmetry boundary conditions were imposed along the zy-plane. For reference, the developed 3D FE models are composed of a total 35,464 elements and 64,660 nodes. The penalty enforcement contact method was used to model the surface-to-surface interaction between the crosshead and the laminate. This method was also applied at the layers’ interfaces, to account for friction that may develop once delamination occurs. A friction coefficient of 0.3 was assigned to the crosshead-composite and supports-composite interactions, while a coefficient of 0.5 was used for the layers’ interfaces [64,72,74]. 4.3. Quasi-static analyses Efficient quasi-static solutions can be achieved using the explicit dynamics solver in ABAQUS® [75]. This approach is welldocumented in the literature [67,76–78] and it involves increasing the loading rate and employing mass scaling. To keep the inertia forces negligible throughout the analysis and ensure the accuracy and reliability of the numerical predictions, several general recommendations were implemented. Firstly, the loading velocity was limited to approximately 1.2 m/s, which is less than 1% of the material’s dilatational wave speed, Cd, which can be calculated for a linear elastic material as, Cd= E ρ √(2) where, E is the elastic modulus and ρ the material density. Notice that, according to the literature, the dilatation wave speed Cd for Table 3 Interlaminar properties. Property Symbol Units Value Stiffness kn=ks=ktN/mm 3 10 6 Damage initiation τ 0 nMPa 50 τ 0 s= τ 0 tMPa 42 Damage evolution energies GIc J/m 2 0.5 GIIc =GIIIc J/m 2 2.0 Interaction parameter η −1.45 J.M. Parente et al. Engineering Fracture Mechanics 316 (2025) 110855 7 composite materials can range from 2500–6500 m/s depending on the specific composition of the laminate [79–82]. Secondly, the stability limit in the explicit dynamics’ procedure, Δt, should follow Eq. (3), where Lmin represents the smallest characteristic element length. Δt≤Lmin Cd (3) Based on the range of Cd values for composite materials and knowing that the developed 3D FE models have an Lmin of 0.1 mm, the stable time increment Δt was maintained below 1.5 ×10 –8 s throughout the quasi-static analysis. Lastly, to minimize stress wave propagation throughout the model, a smooth ramp-up of the loading velocity from zero was applied. To evaluate if the simulations provide an appropriate quasi-static response, an energy balance was performed for all laminate configurations. As an example, Fig. 6 shows the energy balance predictions for configuration “8C”, where is excluded the contribution of the rigid bodies (crosshead and supports). Furthermore, the results are presented for a 3D FE model with fixed mass scaling, applied at the beginning of the step with a factor of four, and without mass scaling. Notice that it is recommended that the kinetic energy of the deforming material (output variable ALLKE) does not exceed 1%–5% of the internal energy (output variable ALLIE) for most of the quasi-static analysis [75]. The results demonstrate that during the quasi-static analysis, the kinetic energy remains negligible, regardless of whether the model’s density is scaled by a factor of four. Across all configurations, the ALLKE/ALLIE ratio consistently stayed below 5% after approximately 1.5 ms, which corresponds to 0.2 mm of crosshead displacement. It is important to note that achieving such balance in Fig. 5. 3D FE model developed to analyse the different laminate configurations under 3PB loading. Fig. 6. Kinetic energy (ALLKE) and internal energy (ALLIE) energy with and without mass scaling, throughout the quasi-static analysis for laminate configuration “8C”. J.M. Parente et al. Engineering Fracture Mechanics 316 (2025) 110855 8 the early stages of the analysis is generally not possible because the laminate, as the only deformable body, moves before experiencing significant deformation. Additionally, the use of mass scaling improved the computational time of the solutions by about 50%. Larger scale factors were also considered; however, they led to an increase in inertia forces, and without any significant improvement in computational efficiency. Overall, these results confirm that the system presents no significant dynamic effects. 5. Numerical results 5.1. Numerical-Experimental correlation The experimental force–displacement curves of all the laminate configurations presented in Fig. 3, were used to validate the numerical predictions obtained with the 3D FE models. As an example, the numerical predictions and experimental results for the nonhybrid configurations “8C” and “8G”, and the hybrid configuration “5C/3G” are presented in Fig. 7. Points A to D correspond to the damage stages illustrated in Fig. 8, while points E and F represent those shown in Fig. 9. A more detailed analysis for all configurations, supported by quantitative values, is provided in Table 4. The results show that the numerical models effectively predict the mechanical response of the hybrid and non-hybrid laminate configurations under 3PB loading. The numerical and experimental force–displacement curves present a good correlation, with forces peaking at similar displacements. In the post-peak region, the experimental data exhibits more variability compared to the smoother numerical predictions, which are related to the complex damage mechanisms in laminates after yielding. When analysing the peak force results for all configurations in Table 4, it can be concluded that the numerical models tend to slightly overestimate the experimental mean values, with errors ranging from +1.1% to +15.9%. However, the numerical predictions show percentage errors that fall within the standard deviation observed in the experimental tests. The largest discrepancy is observed in the “7C/1G” configuration, while “8G” shows an excellent agreement between numerical and experimental data. Notably, “7C/1G” also has the highest variability in the experimental results, which is evidenced by the greatest standard deviation in peak force (39.8 N). Displacement at peak force predictions shows more variation compared to force, with several configurations underestimating displacement, particularly in “8G” (−4.7%), “2G/6C” (−6.8%), and “3G/5C” (−17.5%). Conversely, “7C/1G” overestimates displacement by +8.7%. 5.2. Damage evolution The implemented damage models allow the numerical prediction of the intralaminar and interlaminar damage mechanisms of the fabric-reinforced laminates under the 3PB loading, as discussed in Section 4.1. The predicted damage at two different damage stages, peak force and delamination onset, for the non-hybrid configurations “8C” and “8G”, and hybrid configurations “5C/3G” and “3G/5C”, are presented in Figs. 8 and 9, respectively. These stages are identified in the numerical force–displacement curves shown in Fig. 7. The intralaminar damage is represented as tensile and compressive damage along the fibre directions, with output variables ranging from 0.5 to 1, while delamination is captured by the output variable “CSDMG” for complete debonding between the layers, that is when CSDMG=1. It should be noted that although only two of the six hybrid composite laminates were selected for detailed analysis, the damage evolution predictions for the remaining four laminates are in agreement with the experimental results. The selection of representative results was made to provide a focused and clear comparison, ensuring that key trends were effectively highlighted without compromising the overall consistency of the findings. In all the hybrid and non-hybrid configurations, intralaminar damage is the first damage mode to be predicted. This damage is confined to the regions aligned with the crosshead axis, primarily in the upper and bottom layers where tensile and compressive stresses are the highest. It is subsequently followed by intralaminar damage (delamination), which develops in the areas with the most Fig. 7. Numerical predictions and experimental force–displacement curves for laminate configurations “8C”, “8G” and “5C/3G”. J.M. Parente et al. 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