The effect of stress singularities in the failure of composite off‐axis specimens subjected to tension

Polymer Composites, 2025; 0:1–14 https://doi.org/10.1002/pc.70319 1 Polymer Composites RESEARCH ARTICLE OPEN ACCESS The Effect of Stress Singularities in the Failure of Composite OffAxis Specimens Subjected to Tension J.C.Marín | A.Barroso Elasticity and Strength of Materials Group, School of Engineering, University of Seville, Seville,Spain Correspondence: A. Barroso ([email protected]) Received: 6 May 2025 | Revised: 31 July 2025 | Accepted: 4 August 2025 Funding: This work was supported by Ministerio de Ciencia, Innovación y Universidades; (PID2021126279OBI00) State Plan 2021–2023: Knowledge Generation Projects. Keywords: composites| failure| finite element| offaxis tensile test| stress singularities ABSTRACT This work is motivated by the observation of premature failures in offaxis tension test specimens for the characterization of intralaminar shear strength in unidirectional composite materials. These failures have been mainly observed in the neighborhood of the corners formed by the test coupon with the tab used to place the specimen inside the jaws of the testing machine. A study of the singular stress state in these critical points and its influence on the geometric and mechanical parameters that characterize the problem has been carried out. The parametric study carried out has allowed for the obtaining of variation plots of the parameters that define the singular stress state with the geometric parameters of the problem. These results have been checked against experimental results, observing that the 3D model can be a more precise representation of the singular stress state in the corner of the offaxis specimen. 1 | Introduction Shear behavior in composite materials is a research topic of interest, as shown by the works concerning this issue in recent years [1–7]. In [1] the failure strain was measured for unidirectional laminates with different orientations in the range 0°–90° obtaining a high dispersion for almost all orientations. Improvements on the composite strength was achieved in [3] by adding short fibers into the long continuous fiber composite. In [4] the shear damage was studied in needled C/SiC composites. High nonlinear behavior in shear was addressed in [5] for woven composites at high temperature. Different procedures were used in [6] (offaxis) and [7] (modified threerail shear and [+45/−45]ns) to analyze the shear behaviour of composite in thermoset and thermoplastic composites respectively. In particular, shear characterization in composite materials is still a matter of research, despite the numerous proposals and publications during the last decades. In a recent workshop on the topic [2] the different testing alternatives for the shear strength characterization, their advantages and drawbacks were summarized, among them the offaxis tension test appears as one of the simplest proposals. The offaxis test is often used for the shear characterization of composite materials [5, 6, 8–10], as well as for the analysis of the mechanical behavior of unidirectional laminas with different fiber orientations under static load [1, 11–14] and fatigue load [15–18]. In [8–10] different test methods are used to characterize the shear response of the composites, for example, ofaxis, [+45/−45]ns, tensioncompression biaxial test. The role of endtab shape was addressed in [11], nonlinearities in [12], woven composites in [13] and also 3D printed composites in [14]. Under fatiogie loadings, temperature effects were studied in [15], different stress ratios, were analyzed in [16] and the role of the test frequency was studied in [17]. Nevertheless, there are some wellknown issues associated to the practical performance of the test [19, 20], due to the coupling between the normal and shear stress components. To solve these problems, different alternatives This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. © 2025 The Author(s). Polymer Composites published by Wiley Periodicals LLC on behalf of Society of Plastics Engineers. 2Polymer Composites, 2025 have been historically proposed, from the use of correction factors [21, 22] to modify the values directly obtained from the test, to other proposals which modify the test configuration oriented to have a deformed shape of the sample close to the ideal configuration [21, 23, 24]. In this sense, the use of oblique tabs at the ends of the sample [23] has been shown to be an adequate solution for samples with fiber orientation below 20° [25]. The use of tabs, bonded at the ends of the sample, is a common practice. According to the scheme in Figure1, conventional tab configuration is a straight (ϕ = 90°) or oblique (ϕ ≠ 90°) as proposed, for example, by Sun and Chung [23]. The machine jaws create approximately a certain prescribed displacement condition along the line between the tab and the sample [22, 26] (line AB in Figure1). The abrupt change in materials properties and geometry at these points (A and B) makes the stress state to be singular [27], giving rise to unbounded stresses under linear elastic behavior. These singular stress values could originate premature failures of the sample in the neighborhood of these points, introducing some uncertainties and doubts about the real strength values of the material, which are based on the assumption of nominal stress values appearing at the central part of the specimen. Although, at both points (A and B) the stress state is singular, due to the heterogeneous nature of the material, only failures starting at corner A could lead to a catastrophic failure of the specimen. In fact, a failure initiated in corner A would originate damage in the matrix, which would run parallel to the fiber direction along the whole specimen width. On the contrary, a potential failure initiated in corner B would need the failure of the adjacent fibers to progress into the specimen, requiring much higher stress values than in the former case (corner A) which only implies matrix failure. These high stress values are not reachable in this test configuration. Therefore, in the present work, we will focus on the stress state of corner A and its variation with the geometric parameters that define the test configuration (fiber angle θ, tab angle ϕ, and length to width ratio of the sample L/w). For that end, an own semianalytical code has been used to determine the stress singularity orders (δ) which will be defined in Section2; numerical tools (finite element models) have been used to determine the associated generalized stress intensity factors (K), which will be also defined in Section2. Different plots have been obtained showing the influence that the different geometrical parameters (θ, ϕ, L/w) have on the stress singularity parameters (δ, K). Experiments have been carried out under different configurations (straight and oblique tabs), different fiber orientations, and different L/w ratios; a clear correlation has been observed between the stress singularity parameters and the observed failures in the tests. The material used for the present work is a graphiteepoxy composite AS4/8552 from HEXCEL Composites, with mechanical properties: E11 = 125.159 GPa, E22 = 8.112 GPa, G12 = 4.28 GPa, ν12 = 0.3 [1, 25]. In the framework of accredited laboratories performing mechanical testing, the results of the present work might be of help for identifying potential problems in premature failures of certain test specimens. 2 | SemiAnalytical Study of the Stress Singularities 2.1 | Stress Singularity Parameters Taking a polar coordinate system (r, φ) with its origin at corner A, the asymptotic stress representation can be written, in the majority of cases, in variable separation form, as: where Kk are the Generalized Stress Intensity Factors (GSIFs), δk (with 0 < δk < 1) are the orders of stress singularity, fαβ(k)(φ) are the characteristic angular functions, and Lo is a characteristic length of the problem (in our case, Lo = 1 mm). For the determination of the orders of stress singularities, an own developed code has been used, which implements the Stroh formalism of anisotropic elasticity in a semianalytic way (everything is analytic except the search of roots of a characteristic equation, which is carried out numerically). All details of the implementation and the limitations of this code are detailed in [28, 29]. Although in the present case the corner is just made of two materials with perfect adhesion between them, the code, in general, allows any number of material wedges at the corner, different boundary conditions, different constitutive laws, and different contact conditions between the material wedges (typically, perfect adhesion, frictionless or friction contact). (1) 𝜎 𝛼𝛽(r,𝜑)= ∑ k Kk r −𝛿 k L −𝛿k o f(k) 𝛼𝛽 (𝜑)(𝛼,𝛽=r,𝜑 ) Summary • Singularity stress parameters obtained for the offaxis tension test. • Parametric numerical study varying the test configuration. • Comparison of results using 2D and 3D numerical models. • Experimental tests carried out to check numerical predictions. FIGURE 1 | Sample geometry and material orientation at the specimen end.   15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 3 2.2 | Variation of the Order of Stress Singularity The corner configuration (Corner A) has only one singular term in Equation(1) (only one term with 0 < δk < 1) and no subindex for δ will be used from now on. Figure2 shows the values of the order of stress singularity for four different fiber orientations (θ = 5°, 10°, 15° and 20°) and tab angles ranging from ϕ = 20° to the straight tab configuration ϕ = 90°. As a general trend, for all fiber angles (θ) considered in Figure2, δ increases with ϕ, being maximum for the straight tab configuration ϕ = 90°, which supports that the oblique tab configuration is more favorable for strength determination than the conventional one (ϕ = 90°). For 40° ≤ ϕ ≤ 90°, the order of stress singularity δ increases with the fiber angle θ, whereas for ϕ < 40°, the combination of θ and ϕ determines the optimal configuration of the test, which consists of having the tab angle ϕ equal to the isodisplacement line of the ideal configuration of the “offaxis” problem[23]. According to [25] an analytic expression for ϕ is available in terms of θ and the mechanical properties of the material (μ = G12/E11, ν12, and the ratio E11/E22). The order of stress singularity (δ) and the theoretical tab angles (ϕ) corresponding to each fiber orientation (θ) are: (θ = 5°, ϕ = 29.6°, δ = 0.04594 / θ = 10°, ϕ = 23.1°, δ = 0.05733 / θ = 15°, ϕ = 24.2°, δ = 0.07176 / θ = 20°, ϕ = 27.5°, δ = 0.09158). Although the values are low in all cases, the order of stress singularity δ decreases with the fiber orientation angle θ. It is also important to notice that, for all analyzed cases (θ = 5°, 10°, 15° and 20°), there always exists a tab angle value ϕ for which the order of stress singularity vanishes at corner A. With this fact in mind, the test configuration with θ = 10° seems to be the most favorable one, as it is the test configuration in which this angle (ϕ≈21.8°), which makes δ = 0, is closer to the theoretical one obtained from Equation(2) (ϕ = 23.1°) which makes the tab line coincide with the longitudinal isodisplacement line in the test. 2.3 | Influence on the Stress State With the Variation of G12 It is important to remark that the results in Figure2 have been obtained using G12 = 4.28 GPa, which would correspond to the tangent initial value in the shear stress (σ12) vs. shear strain (γ12) plot. Nevertheless, it is well known that the nonlinear character under shear of these materials exists. This nonlinear behavior makes G12 significantly decrease with the load level [1], which suggests using the secant modulus of G12 if a linear analysis is considered at the instant of failure. According to previous experimental testing with this material [1] this secant modulus would be around 2 GPa (approximately a 46% of the initial considered value). Figure 3 shows an analogous plot to that previously shown in Figure2 for the four fiber orientations (θ = 5°, 10°, 15° and 20°), but now considering a G12 value corresponding to the secant modulus value just before failure (2 GPa). In general terms, for all fiber orientations included in Figure3, variations of δ vs ϕ follow the same trend as that previously observed in Figure2, but with higher values. For ϕ ≥ 40°, the order of stress singularity δ increases with the value of the fiber angle θ, while for values ϕ < 40°, it is necessary to consider the relationship between the fiber angle θ and the tab angle ϕ (Equation2). The δ values associated with the theoretical tab angle ϕ associated with each fiber orientation θ, with G12 = 2 GPa are: (θ = 5°, ϕ = 29.6°, δ = 0.18146 / θ = 10°, ϕ = 23.1°, δ = 0.22591 / θ = 15°, ϕ = 24.2°, δ = 0.23209 / θ = 20°, ϕ = 27.5°, δ = 0.23415). It is important to notice that the theoretical tab angles are the same as those considered in the previous case, as they have been calculated using the initial shear modulus value G12 of the material. The test configuration is defined (before the test is done) with the nominal properties of the material. Now, the order of stress singularity considerably increases for all fiber orientations, being now the lowest for θ = 5° and the highest for θ = 20° with intermediate values for 10° and 15°. (2) 𝜙 =arctan ⎡ ⎢ ⎢ ⎢ ⎣ 1 tan2𝜃− � 2𝜈12 −1 𝜇 � +E11 E22 tan2𝜃 −�2+2𝜈12 −1 𝜇�1 tan 𝜃+�2E11 E22 +2𝜈12 −1 𝜇�tan 𝜃 ⎤ ⎥ ⎥ ⎥ ⎦ FIGURE 2 | Order of stress singularity as a function of the tab angle (ϕ) and fiber orientations (θ). 15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4Polymer Composites, 2025 According to the previous authors' experience [25], it would be advisable to choose, in the case of using oblique tabs, tab angles lower than those obtained using the nominal G12 value, as this would lead to lower values of the order of stress singularity. As a final remark, it is important to stress out that the semianalytical tool used for the determination of the order of stress singularities is conceived for 2.5D problems, under generalized plane strain configurations (ui = ui(x1, x2), i = 1, 2, 3). The evaluation of the order of stress singularities of the real 3D problem will be addressed in the following section “Numerical Study” together with the evaluation of the Generalized Stress Intensity Factors. 3 | Numerical Study For the evaluation of the Generalized Stress Intensity Factors (GSIFs), 2D and 3D numerical models have been prepared, the latter ones including the tabs to take into account the real 3D nature of the local problem at the considered sample corner. The 2D finite element model was performed with the use of the commercial software ANSYS [30] using shell elements (SHELL63) with 30,729 nodes. The mesh was conveniently refined in the vicinity of the corner (corner A). The distance between the first two nodes, at corner A, along the horizontal axis being 1.32 10−3 mm. The material properties taken into account are those of a graphiteepoxy composite (AS4/8552): E11 = 125.159 GPa, E22 = 8.112 GPa, G12 = 4.28 GPa, ν12 = 0.3. The 3D finite element model was carried out also using ANSYS [30] but now with 3D elements (SOLID185) and 40,963 nodes. The mesh was also conveniently refined toward the corner point, taking care to keep the same order of magnitude for the nodal distance (element size) of the elements adjacent to the corner, 6 10−3 mm. Only half of the problem has been modeled due to the symmetry with respect to the horizontal midplane of the sample. The material is the same as in the 2D case (AS4/8552), the complete set of 3D mechanical properties being: E11 = 125.159 GPa, E22 = E33 = 8.112 GPa, G12 = G13 = 4.28 GPa, G23 = 3.571 GPa, ν12 = ν13 = 0.3, ν23 = 0.07. The material used for the tabs is a woven fabric glassepoxy composite with mechanical properties: E11 = E22 = 25.2 GPa, E33 = 17.6 GPa, G12 = G13 = 4.4 GPa, G23 = 2 GPa, ν12 = ν13 = 0.07, ν23 = 0.3. 3.1 | Evaluation of Generalized Stress Intensity Factors The GSIFs define the stress state in the neighborhood of a singular point. In the present work, the stress state has been evaluated along the horizontal axis passing through the corner (see Figure 1), and the product K·fαβ (φ = 0), see Equation(1), has been evaluated and defined as Kαβ. Notice that, with such definition, Kαβ depends on the stress component being used for its evaluation. To allow the different configurations to be comparable, the results have been made dimensionless, dividing by the longitudinal nominal stress σn applied to each configuration. 3.2 | Results of the 2D Model Figure4a–c (for K11), (for K22), and (for K12) show the GSIF values with variations of the specimen ratio (L/w) for the fiber orientation θ = 5° and for both the initial tangent value of G12 = 4.28 GPa (continuous lines) and the secant one G12sec = 2.0 GPa (dashed lines). For all stress components and all considered ratios, GSIF values take higher values for the secant shear modulus (G12sec) than those obtained with the initial tangent shear modulus (G12). As can be observed from Figure4a–c, the trends are almost the same for each stress component, although with different values. For the problem under analysis, the stress component that controls the stress state in terms of failure is the shear stress σ12; FIGURE 3 | Order of stress singularity as a function of the tab angle (ϕ) and fiber orientations (θ) (G12 = 2 GPa).         15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 5 therefore, in what follows, only results of K12 from this stress component will be shown. It can be also observed in Figure4 that for the fiber orientation θ = 5° there only exists influence of the L/w ratio for tab angles ϕ > 30°. For those cases, the GSIFs are lower the higher the L/w ratio is. For tab angles ϕ < 30°, where oblique tab configurations typically are, there is no significant influence at all in the GSIF values with the L/w ratio. Figure 5a–c show the GSIF values vs L/w specimen ratio for fiber angles (θ) of 10°, 15°, and 20° respectively, for both the initial tangent value of G12 = 4.28 GPa (continuous lines) and the secant one G12sec = 2.0 GPa (dashed lines). Similarly to what happened for θ = 5°, for the other considered fiber orientations (θ = 10°, 15° and 20°), the L/w ratio does not play a role for tab angles ϕ < 30°, whereas for tab angles ϕ > 30°, GSIF values decrease with higher L/w ratio values. For fiber orientations θ = 10° (Figure 5a), in a similar way as occurred in the case for θ = 5°, GSIF values are, in general, higher for G12sec than for the initial nominal G12 value. Nevertheless, this trend changes for fiber orientations θ = 15° and 20°. For θ = 15° (Figure 5b) GSIF values are higher with G12sec when the tab angle ϕ < 60°, for higher tab angles ϕ > 60° GSIF values are almost identical or slightly lower. For θ = 20° (Figure 5c), GSIF values are higher with G12sec when the tab angle ϕ < 40°, for higher tab angles ϕ > 40°, GSIF values are lower with G12sec than those obtained considering the initial G12 value. Figure6a–c show the GSIF values for L/w, ratios of 10, 15 and 20 respectively, and different fiber orientations θ (5°, 10°, 15° and 20°). GSIF values in these figures have been evaluated using the secant shear modulus (G12sec) which was previously set as the most representative value at the instant of failure, at corner A. Remember that the Figures show dimensionless GSIF values (standardized with the nominal longitudinal normal stress σn). In Figure6a, with a L/w ratio equal to 10, the maximum GSIF values, for all tab angles ϕ are associated to the fiber orientation θ = 10°. For θ = 5° values are slightly lower than for the case of 10°, and for θ = 15° and 20°, GSIF values are significantly lower than for θ = 10°, the minimum values being obtained for this last fiber orientation θ = 20°. Figure6b,c, for L/w ratios 15 and 20, respectively, show that a different trend can be observed for tab angles below and above ϕ = 40°. Thus, for ϕ > 40° GSIF values decrease with increasing values of fiber orientation θ, being minimum at θ = 20° and maximum at θ = 5°. Thus, for ϕ < 40° GSIF values increase with the following fiber orientation order (20°, 15°, 5°, 10°) being minimum at θ = 20° and maximum at θ = 10°. As previously mentioned, when calculating the order of stress singularity, results interpretation for oblique tab configurations FIGURE 4 | Values of (a) K11/σn, (b) K22/σn and (c) K12/σn, vs ratio L/w for θ = 5°, and G12 and G12sec. 15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6Polymer Composites, 2025 (typically ϕ < 30°) requires consideration of the relationship between ϕ and θ (Equation 2), whose values were summarized previously. In Table1, GSIF values corresponding to these combinations of fiber angle (θ) and tab angle (ϕ) for the different L/w ratio values are shown. GSIF values in Table1 are lower with increasing values of the fiber orientation angle θ, being minimum for θ = 20° and maximum for θ = 5°. It is remarkable that GSIF values slightly decrease when increasing L/w ratio, thus corroborating the lack of influence of the L/w ratio in these oblique tab configurations. It is also important to notice that in these oblique tab configurations, negative deviations in the tab angle value ϕ would lead to lower GSIF values, and therefore to more favorable strength configurations. 3.3 | Results of the 3D Model 3.3.1 | Order of Stress Singularity In general terms, there exist no analytical or semianalytical procedures to obtain the orders of stress singularities in 3D corners with anisotropic materials; therefore, the results shown in this section correspond to those values obtained directly by means of a least squares procedure using the numerical results of the stresses at the corner neighborhood. In Figure7, the orders of stress singularity for corner A are shown for the 4 considered fiber orientations (θ = 5°, 10°, 15° and 20°), and for G12sec (2 GPa). For all considered fiber orientations θ, the order of stress singularity δ increases up to ϕ = 30° to 40°, with slight variations up to ϕ = 90°. In any case, the values of the 3D orders of stress singularities are higher than those of the 2D case. As mentioned previously (Section2) the interpretation of results for the oblique tab configurations (typically ϕ < 30°) depends on the relationship between θ and ϕ (Equation2). The maximum value of δ corresponds to the fiber orientation θ = 10°, although for θ = 5° the value is only slightly lower. For the other two fiber orientations θ = 15° and 20°, the values of δ are lower. The values of δ for the combinations of fiber orientation θ and tab angle ϕ are: (θ = 5°, ϕ = 29.6°, δ = 0.40076 / θ = 10°, ϕ = 23.1°, δ = 0.40218 / θ = 15°, ϕ = 24.2°, δ = 0.35394 / θ = 20°, ϕ = 27.5°, δ = 0.31000). 3.3.2 | Influence of the GSIF With the L/w Ratio Figure8a–d show the standardized GSIF values (K12/σn) vs. the tab angle ϕ and different L/w ratios for θ = 5°, 10°, 15°, and 20° respectively, for the 3D model using the secant shear modulus G12sec (2 GPa). In these figures, to allow a comparison, previous results of the corresponding 2D model (those in Figures4c and 5a–c) have FIGURE 5 | Values of K12/σn vs ratio L/w for (a) θ = 10°, (b) θ = 15°, and (c) θ = 20°, and G12 and G12sec. (a) (b) 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 K12/n (degree) = 10º r10 G12r15 G12r20 G12 r10 G12sec r15 G12sec r20 G12sec 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 K 12 / n (degree) = 15º r10 G12r15 G12r20 G12 r10 G12sec r15 G12sec r20 G12sec (c) 0 0,005 0,01 0,015 0,02 0,025 0,03 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 K 12 / n (degree) = 20º r10 G12r15 G12r20 G12 r10 G12sec r15 G12sec r20 G12sec 15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 7 also been included (dashed lines). As it also occurred in 2D models, 3D results show influence with L/w ratio (for the considered fiber orientations, θ = 5°, 10°, 15° and 20°) for tab angles ϕ > 30°. For a θ = 5° fiber orientation and ϕ > 25° (see Figure8a–d), the GSIF values obtained by the 3D model are lower than the values of the 2D model. On the contrary, for tab angles ϕ < 25°, the GSIF values of the 3D model are somewhat higher than those evaluated with the 2D model. For fiber orientations θ = 10°, 15°, and 20°, and tab angles ϕ > 20° (see figures Figure8a–d), the GSIF values obtained using the 3D model are lower than those of the 2D model. In the case of tab angles ϕ < 20°, the GSIF values of the 3D model are similar to those evaluated with the 2D model. 3.3.3 | GSIF Values With the Fiber Orientation θ Figure9a–c show the GSIF values, obtained with the 3D model, for L/w ratio of 10, 15, and 20, respectively, and different values of the fiber orientation θ = 5°, 10°, 15°, and 20°. FIGURE 6 | GSIF values for different fiber orientations θ, with (a) L/w = 10, (b) L/w = 15, and (c) L/w = 20. (a) (b) 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 K 12 / n (degree) ratio 10 G12=2 GPa 2D     0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 K 12 / n (degree) ratio 15 G12=2 GPa 2D     (c) 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 K 12 / n (degree) ratio 20 G12=2 GPa 2D     TABLE 1 | Theoretical tab angle ϕ and associated GSIF values for different fiber orientations θ and L/w ratios. θ (°) ϕ theoretical (°) K12/σnK12/σnK12/σn L/w = 10 L/w = 15 L/w = 20 529.6 0.02997 0.02938 0.02862 10 23.1 0.02972 0.02906 0.02860 15 24.2 0.02603 0.02584 0.02433 20 27.5 0.02333 0.02304 0.02300 FIGURE 7 | Order of stress singularities vs. the tab angle (ϕ) for different fiber orientations (θ) (G12 = 2 GPa).           15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8Polymer Composites, 2025 Results of the 3D model show that, for specimen ratios 10 and 15 (Figure9a,b), the most unfavorable fiber orientation is 5° for the whole tab angle range ϕ, the GSIF values being lower as the fiber orientation θ gets higher. For the specimen ratio 20 (Figure9c) the trend is almost the same, except for the tab angle range 20° < ϕ < 25° where the GSIF values are slightly higher for θ = 10° than for θ = 5°. As previously mentioned for the 2D model, the interpretation of the results for the oblique tab configurations (typically ϕ < 30°) requires taking into account the relationship between θ and ϕ (Equation 2). In Table2, GSIF values corresponding to these fiber and tab angle combinations are summarized (for the different specimen ratios). As shown in Table 2, GSIF values are lower for higher fiber orientations θ, being minimum for θ = 20° and maximum for θ = 5°. It is remarkable that the GSIF values almost remain the same for the different specimen ratios L/w, corroborating the low influence of this geometrical parameter in oblique tab configurations. Once again, it is noticeable that in these oblique tab configurations, negative deviations of the tab angle ϕ would lead to lower GSIF values, and then, to more favorable configurations from a resistant point of view. 4 | Experimental Study As it could be deduced from Figure8, for all fiber orientations (5°, 10°, 15° and 20°), there only exists some influence of the ratio L/w for tab angles ϕ > 30°. In those cases, the higher the ratio L/w is, the lower the GSIF values are. To verify this finding experimentally, offaxis tension tests have been carried out with fiber orientation θ = 10° and straight tabs (ϕ = 90°). Laminates with 6 unidirectional plies (approximate thickness of 1.9 mm) and fiber orientation of 10° of the material system (AS4/8552) were prepared and cured in an autoclave using a vacuum bag. Glass fiber tabs were bonded at the specimen ends, with L/w ratios of 10, 15, and 20. The tensile tests have been carried out in an Instron 4483 universal testing machine, with a 150 KN load cell and a 50 mm length extensometer. The room temperature for the tests has been 22°C, the relative humidity 50%, and the crosshead displacement velocity 1 mm/ min. The offaxis test has no specific international standard, but as it basically consists of a tensile test, ASTM D 3039 [31] applies. Figure10a–d (a) shows the test configuration, (b–d) show the samples after failure for specimen ratios (L/w) 10, 15, and 20 respectively. In Figure10b, 6 out of 7 specimens of L/w ratio equal to 10, failed in corner A under the influence of the singularity stress FIGURE 8 | Variation of K12/σn with L/w for (a) θ = 5°, (b) θ = 10°, (c) θ = 15° and (d) θ = 20°, models 2D and 3D.             15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 9 state, and only one sample (R105 in Figure10b) failed far from the corner. This implies a failure probability affected by the corner stress state of 85.71%. With the specimen ratio L/w = 15, 4 out of 6 samples failed at corner A, and only two samples (R1514 and R1516 in Figure10c) failed far from the tabs. This means a 66.67% probability of failure affected by the stress singularity. In Figure10d, with the specimen ratio L/w = 20, only 1 out of 6 samples failed at corner A (R203 in Figure10d) affected by the singularity stress field, while 5 samples failed far from the tabs. In this case the failure probability at the corner is just 16.67%. Although using a reduced number of samples, it is reasonably clear that for a fiber orientation θ = 10° and straight tabs (ϕ = 90°) an increase in the specimen ratio L/w implies a decrease in the severity of the singular stress state at corner A. These results corroborate the predictions of 2D and 3D models (Figure8b) which showed that GSIF decrease with the increase in the L/w ratio. Regarding the influence of the fiber angle θ, the predictions of the 2D and 3D models, for ratio L/w = 10 (Figures6a and 9a) show some discrepancies. While the 2D model predicts that θ = 10° would be the most unfavorable fiber orientation (giving rise to higher GSIF values), 3D models predict that the most unfavorable fiber orientation would be θ = 5°. To contrast the results of the numerical models, off axis tension tests have been carried out with fiber orientations θ = 5°, 10°, 15° and 20° and oblique tabs. For that end, four laminates of (AS4/8552) with 4 plies (approximate thickness of 0.85 mm) and fiber orientations 5°, 10°, 15° and 20° were prepared in autoclave with vacuum bag. Glass fiber tabs were bonded at the sample ends with tab angles of ϕ = 29°, 23,°, 24° and 27° for the fiber orientations θ = 5°, 10,°, 15° and 20° respectively, which correspond to those theoretical angles derived previously, all specimens with L/w = 10. The tests have been carried out in an Instron 4483 universal testing machine, with a 150 KN load cell, and a 50 mm length extensometer. The room temperature for the tests has been 22°C, the relative humidity 50%, and the crosshead displacement velocity 1 mm/min. Figure11a–d show the samples after failure for the fiber orientations θ = 5°, 10°, 15°, and 20° respectively. In Figure 11a, all samples with fiber orientation θ = 5° failed at corner A affected by the stress singularity field, two of them having additional failure locations at the interior of the tab. In Figure11b, with θ = 10°, only 1 out of 7 tested samples failed at corner A under the influence of the stress singularity field, the other 6 failing far from the tabs. From Figure 11c, with θ = 15°, no sample failed at corner A under the influence of the singularity stress field, all samples failing far from the tabs. Failures were mainly observed in a lateral area, near the sample end where the combined action of σ22 and σ12 controls the stress state at failure, as previously observed by the Finite Element analysis in [25]. It can be observed in Figure11d, that for the fiber orientation θ = 20° no sample failed at corner A affected by the singular stress state. Similarly, as in the case of θ = 15°, failure mainly occurs at the lateral side near the tab by a combination of the stress components σ22 and σ12. FIGURE 9 | GSIF vs tab angle ϕ for different fiber orientations θ and ratio L/w (a) 10, (b) 15, and (c) 20. TABLE 2 | Theoretical tab angles ϕ and GSIF values for the fiber orientations θ and ratios L/w. θ (°) ϕ theoretical (°) K12/σnK12/σnK12/σn L/w = 10 L/w = 15 L/w = 20 529.6 0.02873 0.02846 0.02826 10 23.1 0.02600 0.02627 0.02669 15 24.2 0.02191 0.02308 0.02267 20 27.5 0.01845 0.01875 0.01892 15480569, 0, Downloaded from https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.70319 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [09/09/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License