research background

    In the aerospace, automotive and other transportation industries pursuing the wave of "lightweight + high strength", carbon fiber reinforced polymer (CFRP) laminates become the core material, but the service is prone to buckling under bidirectional compression load, especially the failure mechanism of laminates with strong coupling of bending and torsion is more complex.

    The extensive application of CFRP laminates in primary load-bearing structures has made multi-axis load stability a critical challenge in engineering design. There is a lack of biaxial compression test data, and the buckling behavior of coupled laminates under bending-torsion loading remains unpredictable. Additionally, the applicability of cross-shaped specimens under compression-compression loading has yet to be conclusively determined.

    In 2024, the research team led by M.C. Serna Moreno at Spain's University of Castilla-La Mancha published a study in *Composites Part B*. The experiment focused on [±45] S CFRP laminates with significant bending-torsion coupling characteristics. The cross-shaped test specimens adopted boundary conditions similar to those of fixed square plates in their central testing zones, with boundary constraint control achieved by adjusting the thickness ratio between the test arms and the central region (H/h=4). A biaxial buckling clamp was used to suppress arm instability during testing. The LaVision Strain-Master stereoscopic digital image correlation (DIC) system combined with strain gauges simultaneously captured three-dimensional deflection surface data and local strain information in the central region. The strength reduction method was applied to estimate the critical stress under biaxial compression. This study was the first to experimentally validate the effectiveness of cross-shaped specimens for biaxial compression buckling analysis of CFRP laminates, successfully capturing the 3D buckling patterns of anisotropic laminates. The findings provide crucial experimental evidence for structural design under complex stress conditions.

Materials and Tests

1.Materials and Sample Design

    Test materials: The experiment employed ∓45° symmetrically layered CFRP laminates (pre-impregnated M21E/34%/UD268/IMA-12K), with a single-layer thickness of 0.25 mm. The in-plane principal direction elastic and strength parameters are shown in Table 1.

         Table 1: Average Elastic Properties of the Principal Material Direction in the Plane

    The cross-shaped specimen features a thickness ratio of H/h=4 between the arm and central region (arm [±45] S layer, 4 mm thick; central [±45] S layer, 1 mm thick). Double fillet transitions are applied to the arm with reduced length to prevent instability, and a 3 mm-thick quasi-isotropic glass/epoxy end plate (Ee≈6.76 GPa) is bonded, with 45° chamfering at the inner corner to prevent compression. The specimen is hand-laid into a 300 mm× 300 mm slab, which is then CNC-milled after curing.

Figure 1: Geometric shape and dimensions of the cross-shaped specimen

2. Experimental Equipment and Testing Methods

    Loading equipment: A multi-axis electromechanical testing machine with four sets of synchronized actuators applying equal biaxial compression loads (loading rate 20N/s). Equipped with a cross-shaped anti-buckling fixture to restrict the out-of-plane displacement of the specimen arm, ensuring buckling occurs exclusively in the central test area.

    Observation technology: The LaVision stereo DIC system (3D full-field displacement/strain measurement) combined with strain gauges was employed to simultaneously capture deformation data in the central region, achieving a resolution of 3.45μm per pixel.

Figure 2 (a) Disassembled components of the biaxial buckling clamp [24,25]: blue represents the cross-shaped plate, yellow the L-shaped support, and purple its base; (b) Experimental setup for compression-compression (CC) testing.

3.results and discussion

3.1  stress-strain response

    The failure mode of all specimens is matrix cracking and buckling twisted zone of parallel fibers, which is consistent in morphology, but the stress-strain parameters are discrete due to manufacturing difference and loading asymmetry.

    [∓45] The layered laminates exhibit differential strain measurements in the x and y directions due to local shear strain variations, but the shear effects of adjacent layers compensate for each other, resulting in zero global shear strain. Therefore, the average value of both measurements is used for analysis. The stress-strain curve demonstrates that the apparent biaxial stiffness during the linear phase aligns with the classical laminates theory (CLPT) estimates. When stress reaches the critical value, strain bifurcation occurs on both surfaces of the specimen, indicating the onset of instability, and the response mode shifts from biaxial compression to bending-torsion moment-driven.

    The arm stress did not reach the pseudo-ductility deformation threshold, with no geometric changes or fiber orientation adjustments observed post-test, confirming that the nonlinear response in the central region remains unaffected by the arm. In an ideal scenario, deflection should not occur before buckling, but deflection appeared during the initial loading phase. This phenomenon is likely attributed to initial geometric defects, asymmetric loading, and variations in actuator displacement. Future research should focus on optimizing displacement control strategies.

Figure 3: Stress-strain response of [∓45] S configuration (laminated plate) under compression-compression (CC) test: (a) Local strain measured by strain gauge arrays at the bottom layer; (b) Global biaxial strain on both sides of the central region, measured by digital image correlation (DIC) technology and strain gauges.

Figure 4: (a) Variation of central stress with maximum deflection in the biaxial loading zone; (b) Variation of applied stress (Equation (3)) on the specimen arm with average x-and y-direction actuator displacements

3.2  Evaluation of buckling modes at bifurcation initiation

    Using 3D digital image correlation (DIC) technology, we successfully captured the 3D deflection surface and 2D projection of the small central region, achieving the first visual observation of the buckling mode of anisotropic laminated plate cross-shaped specimens. The experimentally measured 2D projections of the buckling modes showed high qualitative consistency with the numerical simulation results, both exhibiting characteristic elliptical shapes, which confirmed the reliability of the numerical model.

Figure 5: Out-of-plane displacements of [∓45] S-shaped laminates under biaxial loading, (a) 3D digital image correlation (DIC) deflection surface; (b) 2D DIC projection of the deflection surface; (c) 2D projection of numerically simulated buckling modes; (d) 2D projection of analytical deflection surface results.

    By analyzing the ratio of the long to short axis wavelengths of the ellipse, the proportional coefficient between torsional and bending curvatures was calculated, quantifying the strong torsional-bending coupling characteristics of the [±45]S laminate. Through derivation using the CLPT moment stiffness matrix, it was demonstrated that torsional moments and bending moments also satisfy this proportional relationship, revealing the intrinsic mechanical coupling mechanism under biaxial compression. The buckling mode orientation analysis indicates that the ellipse's long axis aligns with the outer fiber direction. The outer ply simultaneously bears both maximum normal stress and shear stress, where their combined effect induces localized deformation in the matrix direction, establishing it as the critical region for buckling initiation—a finding consistent with the ultimate failure mode.

Figure 6: (a) Experimental measurements of the normalized deflection w/d along the long axis direction of the deflection surface, and (b) along the short axis direction, as a function of the coordinate ρ; representative schematics for wavelengths 2A and 2B are labeled.

Conclusion：

1. This study investigates the [±45]S CFRP laminated plate with strong flexural-torsional coupling characteristics, demonstrating the applicability of cross-shaped specimens with biaxial buckling arresters in coupon-scale biaxial compression stability analysis. By optimizing the thickness ratio (H/h=4) between the specimen arms and central region, the boundary conditions of the central region were approximated as four-sided fixed support. The results confirm that the nonlinear response of the central region is independent of the arms and unaffected by the pseudo-ductility effect of the arms.

2. Utilizing 3D DIC technology, we have for the first time observed the 3D deflection surface in the central region of an anisotropic laminated plate cross-section, with its buckling mode qualitatively consistent with numerical simulation results. The geometric characteristics of the buckling mode quantified the coupling relationship between bending and torsion, where the ratio coefficients between torsional curvature and bending curvature, as well as between torsional moment and bending moment, were both 0.81. This provides critical evidence for analyzing the mechanical behavior of laminated plates under biaxial compression.

3. The critical stress of biaxial compression is estimated by the strength reduction method of the metal plate, which is about 123.74 MPa, and it is highly consistent with the experimental value (about 120 MPa). It proves that the method can provide the reference value of critical stress effectively under the assumption of simplified material behavior, but the accuracy depends on the reasonable determination of the ultimate stress.

4. The cross-shaped specimen offers distinct advantages: load application is distributed away from the central test region, effectively reducing stress concentration and clamping damage risks. Moreover, its thickness ratio can be adjusted to flexibly simulate various boundary conditions. However, experimental results exhibit some variability due to manufacturing geometric variations, imperfect boundary conditions, and misalignment during loading. Future research should focus on optimizing specimen fabrication precision and refining loading control strategies.

5. The experimental methodology and data analysis framework developed in this study provide a viable approach for investigating the biaxial compression stability of small-sized composite laminates. The findings offer technical support for buckling-resistant design of lightweight structures in transportation, aerospace, and related fields.

Title：M.C. Serna Moreno, S. Horta Muñoz,Experimental evaluation of the use of cruciform specimens for biaxial stability analysis,Composites Part B: Engineering,Volume 286,2024,111764

Original link：https://doi.org/10.1016/j.compositesb.2024.111764