Influence of the 90/0 ply thickness ratio on the stresses associated with the edge effect phenomenon

Influence of the 90/0 ply thickness ratio on the stresses associated with the edge effect phenomenon S. S´ anchez-Carmona * , A. Barroso , E. Correa Grupo de Elasticidad y Resistencia de Materiales, Departamento de Mec´ anica de Medios Continuos y Teoría de Estructuras, Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla 41092 Sevilla, Spain ARTICLE INFO Keywords: Scale effect Free edge Carbon and glass fibres Cross-ply laminate ABSTRACT The scale effect phenomenon has shed light on the thermo-mechanical behaviour of ultra-thin plies. These laminas in cross-ply laminates (under certain 90/0 thickness ratios) have given rise to a relevant edge effect phenomenon, causing significant through-the-thickness stress in the weakest ply along its free edge under both thermal and (0-longitudinal tension) mechanical loading. Thus, a biaxial stress state must be considered to analyse what happens along the free edges from experimental samples to structural components, such as drill holes. A parametric numerical analysis is performed taking three different features into account: the thickness of the ply blocks, the cross-ply stacking sequences and the type of fibre, either carbon or glass. The numerical predictions are in accordance with experimental results, which are obtained under a thermal cooldown. The biaxial stress state predictions could be used in future numerical procedures including the presence of components’ free edges. 1. Introduction The possibility of manufacturing ultra-thin plies industrially leads to performing several research works to improve the knowledge concerning the phenomenon known as the scale effect, which has been studied since the appearance of composite materials by Parvizi et al [1], Flaggs and Kural [2] and Pagano et al [3], among others. The interest of the scientific community in ultra-thin plies in the last decades has shed light on their behaviour under different thermal and mechanical conditions [4,5,6,7]. Some years ago, París et al [8] gave a physically based explanation of the scale effect phenomenon. In this way, these authors started to study these ultra-thin plies under different loading conditions, finding some non-conventional damages on the ultra-thin 90◦ply blocks of cross-ply laminates which appeared longitudinal to the fibres of the 0◦ply blocks along the samples’ free edges, even just after the cooling of the curing process [9]. The presence of free edges, from experimental specimens (for characterisation and research programmes) to structural components (holes, edge finishing of workpieces and joints, etc.), gives importance to the edge effect phenomenon, since an accurate knowledge of the stress state is required. Due to this fact, several studies have been performed concerning this phenomenon, such as Kassapoglou and Lagace [10], Becker [11], Pagano et al[3], Lorriot et al [12] and Mittelstedt and coworkers ([13,14]), among others. It is worth highlighting that Hajikazemi and Van Paepegem [15] showed the importance of the through-the-thickness stresses that appear when thin plies are considered. Taking this idea into account, S´ anchez-Carmona et al [16] profoundly studied the edge effect phenomenon for the case of cross-ply laminates, which takes special relevance when ultra-thin plies are involved in the 90◦ply block. The high constraints of the surrounding 0◦plies to the 90◦ultra-thin ones led to important intralaminar throughthe-thickness stresses in the 90◦ply block along the parts’ free edges. The relevance of this stress component implies that a biaxial stress state appears in the 90◦ply block just after the curing process (due to the thermal decrement associated with the cooling process) and when tested under uniaxial tension, a fact recently studied by the authors (S´ anchezCarmona et al [17]). In this way, this work aims to establish the edge-effect consequences associated with the use of different t 90 /t 0 ratios in cross-ply laminates to analyse from thick to ultra-thin ply blocks. A parametric numerical analysis is performed based on different thicknesses of 0◦and 90◦ply blocks, different cross-ply configurations, and different fibre systems. Once this wide range of possibilities is calculated, an accurate prediction of the biaxial stress state along the 90◦ply block’s free edge is obtained, * Corresponding author. E-mail address: [email protected] (S. S´ anchez-Carmona). Contents lists available at ScienceDirect Composites Part A journal homepage: www.elsevier.com/locate/compositesa https://doi.org/10.1016/j.compositesa.2025.108961 Received 6 February 2025; Received in revised form 16 April 2025; Accepted 17 April 2025 Composites: Part A 195 (2025) 108961 Available online 20 April 2025 1359-835X/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC license ( http://creativecommons.org/licenses/bync/4.0/ ). providing a far better approximation of the actual stress state that can be calculated by simply using the Classical Laminate Theory. These numerical predictions are corroborated experimentally for the thermal case (just after the curing process) for both fibre systems. Finally, these estimations may be used to predict the biaxial stress state that occur along components’ free edges for future numerical studies performed by the scientific community. 2. Definition of the case studies The laminate scheme under study is defined in Fig. 1, where the 0◦fibre ply block is aligned to the x-axis and the 90◦fibre ply block is aligned to the y-axis. The free edges are those marked in green which correspond to the planes y=0 and y=w. The edge effect phenomenon in the cross-ply configuration is associated with two property mismatches, the different thermal expansion coefficients and elastic properties, which can be studied separately. On the one hand, the curing process entails that the cross-ply laminate suffers a thermal decrement during the cooling stage, which provokes free-edge stresses due to the different coefficients of thermal expansion (CTE) of the 90◦and 0◦ply blocks. On the other hand, a mechanical load leads to a nominal σ xx in the laminate and also generates free-edge stresses due to the differences in the elastic modulus and the Poisson ratios between the 0◦and 90◦ply blocks. Both problems provoke through-the-thickness stresses, σ zz , in the 90◦ply block. Hence, a biaxial stress state is present along the laminate’s free-edges (S´ anchez-Carmona et al [17]), the analysis of which is the main motivation of this work. As mentioned above, the previous work of these authors [16] showed the relevance of the edge effect phenomenon in a cross-ply laminate when an ultra-thin 90◦ply block is used together with thick 0◦ply blocks. A more in-depth study about the different ply constraints between 0◦and 90◦ply blocks is performed in this work. The case studies which are going to be analysed are based on a parametric analysis to consider the effect of three parameters, summarised in Table 1: the ply thicknesses 0◦and 90◦, the ply locations giving rise to different lay-ups, and the type of material used, varying the type of fibre while keeping the same thermoset matrix. First, a range of different thicknesses, from ultra-thin to ultra-thick, are used for each ply block, 0◦and 90◦. The selected thicknesses are 50, 65, 100, 170, 340, 510 and 680 µm. In Table 1, a parameter, n, varying from 1 to 7, is employed to represent each thickness in sets of stacking sequences with the same thickness of the 0◦ply blocks. Second, two different cross-ply laminates are selected as the extreme cases, [0/90/0] and [0/90/0/90/0] s . The first laminate has all the 0◦and 90◦plies concentrated in one ply block on one side with respect to the central plane of the laminate, named as the regular cross-ply, whereas the second cross-ply configuration has the 0◦and 90◦plies alternatively distributed. Third, the influence of two different fibres on the edge effect phenomenon is assessed. T700S Carbon and 2026 E-glass fibres are considered using the same epoxy resin, TP-402. Any combination of these three different characteristics is evaluated to provide the biaxial stress state that occurs along the 90-ply free edges of these material systems. 3. Numerical modelling The numerical model is defined as indicated in Fig. 2 using a 2D geometry (element type PLANE 182) under the assumption of Generalized Plane Strain, ui=ui(y,z)for i=x,y,z, in ANSYS® software. The results obtained from the Finite Element Model (FEM) were processed using the MATLAB® software. Each ply block has a particular thickness, t 0 for the 0◦and t 90 for the 90◦ply blocks, leading to a total of 49 combinations of 50, 65, 100, 170, 340, 510 and 680 µm (as shown in Table 1) for each specific case of stacking sequence and fibre system, giving rise to a total of 196 cases studied. The model is implemented as a quarter of the 2D geometry with a width (w) of 5 mm, considering symmetry along the middle plane of the laminate through the thickness and the width, as indicated by the red dash lines in Fig. 2. The smallest elements (10 -6 mm length) are located in the 90◦ply block close to the 0/90 interface. As detailed in Section 2, the model is solved for the thermal and mechanical problems independently; hence, a total of 392 numerical cases are performed in this parametric analysis. The thermal case is simulated using a thermal decrement of −110 ◦C, representing the cooling down process from cure temperature (135 ◦C) to room temperature (25 ◦C). The mechanical problem is associated with a longitudinal displacement (u x ) applied in the elastic centre of the problem (where tensile force and bending moments are uncoupled) as it does not coincide with the geometrical centre because only the upper half part of the problem is modelled. This prescribed displacement, u x , gives rise to a longitudinal strain ( ε x ) that generates in the 90◦ply block a nominal σ xx =38 MPa for the carbon/epoxy laminate and σ xx =46 MPa for the glass/ epoxy one, which coincides with the experimentally measured transverse strength (Y T ) of each material system. This value, predicted by the Classical Laminate Theory, has been verified in areas far away from the free edge in the numerical model. It is worth highlighting that a stress singularity exists due to the presence of the bimaterial corner at the 0/90 free-edge interface. The present authors performed a numerical analysis to remove the influence of the weak stress singularity through a local geometrical modification of the corner geometry, giving rise to a length affected by the singular stress field below 10 -2 mm [16] unlike the extent of the edge effect phenomenon, which is of the order of 1 or 2 thicknesses across the width w. As a last comment, let’s take into account that this distance is of the same order of magnitude as the fibre diameter itself, and the homogenised calculated stresses may not fully capture the real heterogeneous nature of the material. Hence, the free-edge consequences shown below are exclusively due to the edge effect phenomenon without being affected by the stress singularity at the 0/90 interfaces. Taking the carbon/epoxy [0/90/0/90/0] s laminate (t 0 =680 µm and t 90 =50 µm) into consideration, Fig. 3 shows the σ zz contour plots along the 90◦ply blocks, showing the through-the-width depth of the edge effect phenomenon (purple frame) and remarking two different details at different scales; first, the bi-material corner at the 0/90 interface where the singular stress field arises (magenta frame); and, second, the zone Fig. 1. Scheme of the material system under study. S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 2 considered for the calculation of the average value of the edge-effect stresses provoked is inside an orange frame. The thermo-mechanical properties of both prepreg grammages needed for these models are indicated in Table 2, taken from S´ anchezCarmona et al [18] for the carbon/epoxy material and measured experimentally for the glass/epoxy tape. The out-of-plane properties (zaxis) are estimated from the in-plane properties. E 33 and α 3 are taken equal to E 22 and α 2 , respectively. Regarding ν 23 , an increasing percentage of its corresponding ν 12 is assumed following the literature ([19,20]). A particular consideration has been made regarding G 23 due to the difficulties in obtaining this elastic property experimentally. These authors are experienced in the complexities of characterising shear properties [21]; hence, its assumption is performed from the literature. Some publications obtain this value by assuming the transversely isotropic behaviour whereas others directly take a percentage of the value of G 23 with respect to G 12 , even normally taking a slight reduction ([20,22]). To avoid any influence of G 23 in this numerical analysis, two extreme values are chosen in this parametric model for the glass fibre case. The lowest value selected is G 23 =3200 MPa, representing a minimum reduction of G 12 , and the highest value used is G 23 =4181 MPa, calculated under the assumption of transversely isotropic behaviour. Fig. 4 shows the negligible influence of this elastic property on the average mechanical solution. It is worth noting that the extreme value due to the stress singularity increases for the first case considered although the average values are quite similar. Both results can be considered equal as the results affected by the stress singularity will be removed to just analyse the effect of the nominal edge effect phenomenon. The other three stress components analysed show the same fact. Thus, G 23 is selected as a slight reduction of G 12 for both fibres cases in this work (although only the glass fibre is shown here), taking similar percentages of G 23 with respect to G 12 than those used in Kaddour et al [20]. The next two sections present the thickness-based parametric analysis of the presented problem, showing the dimensionless in-plane ( σ xx ) and out-of-plane ( σ zz ) stresses, i.e. each stress is divided by Y T (due to the biaxial stress state associated with the edge-effect consequences, on average along the 90◦ply block thickness) versus the thicknesses of the 0◦and 90◦ply blocks (t 0 , t 90 ) for both thermal and mechanical problems (from Fig. 8 to Fig. 11). These four 3D charts show the thicknesses of each ply block, t 0 and t 90 , in the horizontal axes, and the dimensionless stress analysed in the vertical axis. Furthermore, a grey constant plane of dimensionless stress equal to 1 is drawn in each plot to help the visualization of the dimensionless stresses that are above the Y T value. The first section shows the dependence on the stacking sequence for each fibre type and the second section details the effect of the fibre type on each stacking sequence. Regarding the average edge-effect stress obtained from each study case analysed, which is defined as the selected stress versus the material transverse strength, some considerations are marked: •These average values are calculated by discarding the values associated with the stress singularity field, i.e. removing those values that are within 10 -2 mm from each 0/90 free-edge interface according to S´ anchez-Carmona et al [16]. •Most curves are quite flat along the 90◦thickness considered; nevertheless, there are some cases whose distributions require a clarification. In particular, and with respect to the [0/90/0] configuration, the edge effect trends are not so uniform for some specific 90/0 thickness ratios. Fig. 5 shows the thermal dimensionless in-plane stress, σ xx , versus the 90◦ply block thickness for the t 0 =340 µ m, varying t 90 from 50 to 680 µ m (CP-50 up to CP-680, respectively). A monotonically increasing trend of the edge-effect stress with respect to the thermal σ xx component is observed especially for CP-680 (bottom curve). The minimum, average and maximum values from this trend are marked, taken as the maximum value that remains after removing those values associated with the stress singularity field. Although this average value, 0.596, is not as representative as can be the one selected for instance for CP-50, it is an intermediate value that can be seen as a reference. However, all the cases that present this specificity have a dimensionless stress less than 1; hence, those values after the residual thermal relaxation considered show a lower relevance. This change from the flat trend occurs because the resin of thick 90◦ply blocks tends to decrease these blocks volume with a minor effect from the constraint of the 0◦ply blocks. Table 1 Parametric analysis matrix, varying the 0◦and 90◦ply block thicknesses from 50 to 680 µm (only the extreme thicknesses case for the 0◦ply block, n =1 and 7, are shown). Carbon fibre – Epoxy resin Glass fibre – Epoxy resin   [0/90/ 0] [0/90/0/ 90/0] s [0/90/ 0] [0/90/0/ 90/0] s Combination of thicknesses (in µm) for 0◦and 90◦ply blocks n =1 (50 µm) [50/ 50/50] [50/50/ 50/50/ 50] s [50/ 50/50] [50/50/ 50/50/ 50] s [50/ 65/50] [50/65/ 50/65/ 50] s [50/ 65/50] [50/65/ 50/65/ 50] s [50/ 100/ 50] [50/100/ 50/100/ 50] s [50/ 100/ 50] [50/100/ 50/100/ 50] s [50/ 170/ 50] [50/170/ 50/170/ 50] s [50/ 170/ 50] [50/170/ 50/170/ 50] s [50/ 340/ 50] [50/340/ 50/340/ 50] s [50/ 340/ 50] [50/340/ 50/340/ 50] s [50/ 510/ 50] [50/510/ 50/510/ 50] s [50/ 510/ 50] [50/510/ 50/510/ 50] s [50/ 680/ 50] [50/680/ 50/680/ 50] s [50/ 680/ 50] [50/680/ 50/680/ 50] s …From n =2 (65 µm) to n =6 (510 µm) … n =7 (680 µm) [680/ 50/ 680] [680/50/ 680/50/ 680] s [680/ 50/ 680] [680/50/ 680/50/ 680] s [680/ 65/ 680] [680/65/ 680/65/ 680] s [680/ 65/ 680] [680/65/ 680/65/ 680] s [680/ 100/ 680] [680/ 100/ 680/ 100/ 680] s [680/ 100/ 680] [680/ 100/ 680/ 100/ 680] s [680/ 170/ 680] [680/ 170/ 680/ 170/ 680] s [680/ 170/ 680] [680/ 170/ 680/ 170/ 680] s [680/ 340/ 680] [680/ 340/ 680/ 340/ 680] s [680/ 340/ 680] [680/ 340/ 680/ 340/ 680] s [680/ 510/ 680] [680/ 510/ 680/ 510/ 680] s [680/ 510/ 680] [680/ 510/ 680/ 510/ 680] s [680/ 680/ 680] [680/ 680/ 680/ 680/ 680] s [680/ 680/ 680] [680/ 680/ 680/ 680/ 680] s S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 3 •The constraint of thick 0◦ply blocks leads to edge-effect stresses that are not uniformly distributed along the 90◦ply block thickness for the distributed cross-ply configuration, as shown in Fig. 6. In the distributed stacking sequence, the inner 90◦ply block is more restrained by the central 0◦ply block (located in the plane of symmetry) than by the intermediate 0◦ply block, which is also constraining the outer 90◦ply block. In other words, the 0/90 interfaces close to the intermediate 0◦ply block are less restrained because this 0◦ply block is surrounded by two different 90◦ply blocks. This can be illustrated graphically by employing the scheme of Fig. 7, where the dimensionless stresses for the CP-680 from Fig. 6 are shown in each 90◦ply block. Fig. 2. FEM mesoscopic model of the material systems under study. Fig. 3. σ zz contour plots (MPa) for the carbon/epoxy [0/90/0/90/0] s laminate (t 0 =680 µm and t 90 =50 µm), highlighting the singular stress field at the 0/90 interface and the 90◦-ply zone taken for the calculation of the edge-effect stress average value. Table 2 Thermomechanical properties of unidirectional carbon/epoxy and glass/epoxy tape prepreg. Carbon/Epoxy Glass/Epoxy E 11 (MPa) 113,910 43,210 E 22 (MPa) 7810 11,540 E 33 (MPa) 7810 11,540 ν 12 0.321 0.269 ν 13 0.321 0.269 ν 23 0.4 0.38 G 12 (MPa) 3254 3380 G 13 (MPa) 3254 3380 G 23 (MPa) 2350 3200 α 1 (◦C −1 ) 6.16e-6 1.13e-5 α 2 (◦C −1 ) 3.03e-5 3.54e-5 α 3 (◦C −1 ) 3.03e-5 3.54e-5 S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 4 3.1. On the stacking sequence All the results obtained for the comparison of the two cross-ply configurations considered are shown below. Fig. 8 shows the results associated with the 90◦carbon fibre blocks (both regular cross-ply and distributed) whereas Fig. 9 presents the same collection of results for the E-glass fibre case. A similar trend occurs for each dimensionless stress analysed in Fig. 8 and Fig. 9. The 90◦ply block in the regular cross-ply configuration is more constrained by the surrounding 0◦ply blocks than those 90◦ply blocks in the distributed lay-up. Thus, the regular cross-ply configuration shows greater edge-effect stresses, except for the thermal in-plane stress. This particular case, shown in Fig. 8.(b) for carbon fibre and Fig. 9.(b) for glass fibre, has a lower edge-effect stress than the distributed configuration, except for the extreme constraint cases of thick 0◦ply blocks against thin 90◦ply blocks. This different behaviour comes from the non-uniform edge effect consequences expounded in the first part of section 3, specifically in Fig. 8. Nevertheless, the greater values obtained in each chart of Fig. 8 and Fig. 9 are not so different for the two cross-ply configurations, except for the thermal through-the-thickness stress that has values around 20 % higher for all the thicknesses ratios. It is important to remark that the in-plane mechanical stresses σ xx are always greater than their corresponding Y T value because the external mechanical load applied to the laminate is Y T ; the σ xx difference with Y T is therefore the edge effect component. 3.2. On the fibre type The results obtained comparing the two fibre types are shown in this section. Fig. 10 shows the results associated with [0/90/0] regular laminates whereas Fig. 11 presents the results for the distributed crossply configuration. Although the edge-effect stresses are lower for the distributed crossply laminates, the trends of the edge-effect consequences for the two types of fibre are almost identical for the two laminates studied, as deduce from the comparison of Fig. 10 and Fig. 11. Fig. 10.(a) and Fig. 11.(a) show a relevant edge effect phenomenon when there are thick 0◦ply blocks against ultra-thin 90◦ply blocks for the carbon/epoxy case. In contrast, this phenomenon is not so pronounced for the glass/epoxy material, which presents a moderate increase for thick 0◦ply blocks versus ultra-thin 90◦ply blocks. Since a certain relaxation of the residual thermal stresses has to be taken into account after the cooling process, as experimentally confirmed by the authors [9], the 90◦ply block would finally bear stresses lower than Y T for glass fibre whereas these stresses may be equal to or higher than Y T for carbon fibre. Concerning the dimensionless thermal in-plane stress ( σ xx / Y T ), Fig. 10.(b) shows almost identical surfaces for both types of fibre whereas some differences can be observed in Fig. 11.(b). In the case of the carbon/epoxy material, the surface tendency changes for thick 90◦ ply blocks against ultra-thin 0◦ply blocks, being greater than for the glass case. It should be noted that the edge-effect stresses are so low in these constraint cases that the difference between the two materials may be considered negligible. These thermal differences are related to the different coefficients of thermal expansion (CTE) experienced by these materials in the Fig. 4. Free-edge consequences concerning mechanical out-of-plane stresses for the glass/epoxy material system: (a) G 23 =3200 MPa, and (b) G 23 =4181 MPa. Fig. 5. Free-edge consequences concerning the thermal in-plane stresses ( σ xx ) for t 0 =340 µm (varying t 90 ), highlighting the minimum, the average and the maximum values for the thickest 90◦ply block thickness (t 90 =680 µm, CP-680). S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 5 longitudinal direction to the fibre ( α 1 ), while α 2 and α 3 values are almost identical for both bi-material systems. Mechanical σ zz / Y T , Fig. 10.(c) and Fig. 11.(c), show the same trend for each fibre type used. Whereas the out-of-plane stress is quite similar for all 90/0 thicknesses ratios in glass fibre, the carbon fibre shows important increments. The lower the t 90 /t 0 ratio is, the higher the edgeeffect stress is. Therefore, the extreme case of a thick 0◦ply block and an ultra-thin 90◦ply block leads to relevant mechanical stresses in the thickness direction ( σ zz ) in the 90◦ply block (~80 %Y T ). A similar trend can be seen for the in-plane stress, Fig. 10.(d) and Fig. 11.(d), although the increase is less pronounced for lower t 90 /t 0 ratios. As commented in the first part of Section 3, all results are above Y T because a displacement is prescribed in the FEM model that is associated with an in-plane transverse normal stress in the 90◦ply block of Y T . In the extreme lowest t 90 /t 0 case, there is an increase of ~ 40 % along the free edges of the 90◦ply block. These mechanical trends can be explained by the fact that the E 11 /E 22 ratio is much higher for carbon fibre (=14.6) than for glass fibre (=3.7). 3.3. Polynomial fit of the edge-effect stresses The results presented in the previous sections (Fig. 8 to Fig. 11) may be useful for future stress predictions needed by designers and researchers. In order to facilitate the use of these results, a 4th-order polynomial function, eq. (1), is obtained using the least squares method and is proposed for each stress analysed ( σ ii, i =x,z) to give rise to the stresses that appear in the weakest lamina, the 90◦ply block, along the cross-ply laminates’ free edges for these two material systems. These predictions could be obtained for any bi-material case. σ ii YT (t0,t90) =p40t04+p31t03t90 +p22t02t902+p13t0t903+p04t904 +p30t03+p21t02t90 +p12t0t902+p03t903+p20t02 +p11t0t90 +p02t902+p10t0+p01t90 +p00 (1) The values of the polynomial coefficients required for eq. (1) for each different configuration (24 in total) and their fit statistics are detailed in Annex 1. An example of one of the most non-planar surfaces is shown in Fig. 12, specifically the carbon [0/90/0] case. It is worth pointing out that the polynomial that the polynomial fit of this case is in perfect agreement with the edge-effect stresses predicted by FEM, as shown by its fit statistics, the sum of error squares and its R-squared (see Annex 1). 4. Edge-effect stresses as a function of the 90/0 thickness ratio The results obtained so far remark the influence of the thicknesses of the 0◦and 90◦ply blocks on the edge-effect stresses. In an attempt to undertake a deeper analysis, the stress distributions are plotted directly versus the 90/0 thickness ratio for each cross-ply configuration, Fig. 13. From these results, it can be seen that there is a clear and straightforward relationship between the edge effect stresses and the thickness ratio. Specifically, t 90 /t 0 ratio is calculated as 1/2 for the regular configuration and 2/3 for the distributed case. Once each dimensionless stress is charted versus the 90/0 thickness ratio, it can be seen that there is a rational function that follows the expression: Fig. 6. Free-edge consequences concerning thermal out-of-plane stresses ( σ zz ) for distributed cross-ply stacking sequence using t 0 =680 µm and varying the thickness of the 90◦ply block from 50 to 680 µm: (a) inner 90◦ply, and (b) outer 90◦ply. Fig. 7. Scheme of the distribution of the free-edge consequences concerning thermal out-of-plane stresses ( σ zz ) for the distributed cross-ply configuration with t 0 = t 90 =680 µm. S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 6 Fig. 8. Comparison of the thickness effect on the average edge-effect stresses (dimensionless values) for the carbon/epoxy cross-ply configurations: (a) thermal σ zz , (b) thermal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx . Fig. 9. Comparison of the thickness effect on the average edge-effect stresses (dimensionless values) for the glass/epoxy cross-ply configurations: (a) thermal σ zz , (b) thermal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx . S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 7 Fig. 10. Comparison of the thickness effect on the average edge-effect stresses (dimensionless values) for the regular cross-ply configuration: (a) thermal σ zz , (b) thermal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx .Fig. 11. Comparison of the thickness effect on the average edge-effect stresses (dimensionless values) for the distributed cross-ply configuration: (a) thermal σ zz , (b) thermal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx . S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 8 σ ii YT (t90/t0) = k a+t90 t0 +b(2) where a, b and k are the fitting parameters adjusted from the FEM values obtained. The parameters values for each case are detailed in Annex 2. Eq. (2) sheds light on the influence of the thickness’s relation between 0◦and 90◦plies, whereas eq. (1) allows researchers to have a good prediction of the edge effect stresses as the 3D stress surfaces (from Fig. 8 to Fig. 11) are fitted. It is worth pointing out that only the stress values for the outer 90◦ ply block are shown in Fig. 13, as those for the inner ply block are quite similar. Fig. 12. Comparison of the average thermal σ zz along the 90◦ply block of the carbon/epoxy regular cross-ply laminates’ free edges between the FEM prediction and the polynomial fit. Fig. 13. σ ii/Y T (i =x,z) vs. t 90 /t 0 (predicted −FEM, and adjusted −Fit) of each case (carbon/glass fibre, regular/distributed cross-ply configuration) for: (a) thermal and (b) mechanical out-of-plane stress, σ zz , and (c) thermal and (d) mechanical in-plane stress, σ xx . Fig. 14. Thermal out-of-plane stress, σ zz , (divided by Y T ) along the free edge of a [0/90/0] laminate made of carbon/epoxy material, for t 0 =680 µm and t 90 = 50, 65, 100 and 510 µm. S. S´ anchez-Carmona et al. Composites Part A 195 (2025) 108961 9