1. Introduction
With the desire for high strength-to-weight ratio materials in aerospace, automotive, medical, and energy sectors, carbon-fiber-reinforced polymer composites (CFRP) have been attracting growing attention as a lightweight and strong material. Extensive composite material characterization ensures safety compliance and provides critical design data in the listed industries. Modeling the elastic behavior of orthotropic materials with transverse isotropy usually requires the characterization of five independent elastic constants, namely,
and
(or
v23), which represent Young’s modulus in the fiber direction, Young’s modulus perpendicular to the fiber direction, the in-plane shear modulus, the in-plane Poisson’s ratio, the out-of-plane shear modulus, and the out-of-plane Poisson’s ratio, respectively [
1]. However, there are few significant mathematical relations for material strength resulting in numerous resource-intensive destructive tests, often providing single failure mode results. Accordingly, it is advantageous to study composite behavior under combined loading, and their dominating failure modes.
A relatively inexpensive and straightforward test for composites is the ASTM D7264, utilized for obtaining flexural properties of polymer-reinforced composites. Though seldom used for design, flexural tests are often used for quality control in composites. Flexural testing requires a simple rectangular sample and uses a three-point bending setup to measure the flexural response of specimens. The dominating flexural failure modes are compression at the top ply (either fiber microbuckling or ply-level buckling) and tension at the bottom-most ply [
2]. The flexural strength of different CFRP systems often surpasses their compressive strength (
). Flexural strength lies between the fiber direction compressive and tensile strengths in the CFRP systems investigated here [
3,
4,
5].
Through finite-element analysis (FEA), this study investigates stress interactions occurring in unidirectional, 20 ply CFRP samples loaded through a three-point flexural test, and predicts the flexural strength of each system with reasonable accuracy. Three CFRP systems of interest are investigated where their flexural strengths vary from their compressive strength, an average of the tensile and comprehensive strengths, to their tensile strength. Different failure criteria were evaluated with respect to their ability to predict the flexural strength of the three CFRP systems.
3. Results
Experimental and FEA strengths for three systems are summarized in
Table 4. The FEA results were compared with experimental values obtained from datasheets for tensile, compressive, and flexural strengths. The Tsai–Wu failure criterion was utilized to determine stress at the onset of flexural failure. As mentioned earlier, all numerically obtained strengths were conservative due to nonprogressive damage analysis. The flexural specimen likely did not completely fail even if the Tsai–Wu index reached a value of 1 for a single finite element.
Tsai–Wu failure indices through the thickness the three systems are plotted in
Figure 6. Failure occurs for a Tsai–Wu index above 1. An interesting finding was the failure of the Hexcel system in tension at the bottom ply, as opposed to the compressive failure experienced by the other two CFRPs.
Stresses across the thickness near the loading location are plotted and shown in
Figure 7. Only stresses for the Toray system were plotted, as all analyzed systems showed similar trends. Shear stresses
and
were not plotted, as their values were near zero throughout the thickness, having negligible contribution to laminate failure.
4. Discussion
A key trend in all three systems was that their flexural strength was between their tensile and compressive strengths. This implies that flexural failure occurs because of combined modes instead of a single failure mode.
Table 5 summarizes the maximal FEA compressive stresses at the top ply under the loading nose, compared with the reported limits, i.e., compressive strengths.
The Tsai–Wu criterion showed the closest results to experimental data. With compressive interactions occurring at the top ply between
and
due to Poisson’s effect, the effective stress limit was expanded [
1]. Terms from the Tsai–Wu failure criterion are further explored in
Table 6 for each analyzed system. Terms in bold font are those contributing to the enhanced load-bearing capacity of the samples.
The Hexcel system, which fails in a tensile combined mode, had the highest reported compressive strength across the considered composite systems. Daniel and Ishai showed that, for similar stress interactions in combined loading, compressive load-bearing capabilities are much better than the tensile capabilities for a similar AS4 system, which could explain the tensile failure obtained by the FEA with the Hexcel AS4 system [
1].
Lastly, the flexural finite-element model underpredicted flexural strength across all systems. This could be due to static structural analysis and no progressive damage being considered. The model effectively predicts a first ply failure (FPF), while an ultimate ply failure (UPF) prediction would likely return more accurate results. Nonetheless, the FPF returned values with reasonable accuracy at a lower computational cost to a UPF model.
Error in the prediction of failure consistently increases as flexural strength approaches its tensile strength. This could stem from an increased need for a UPF model, as brittleness could be less pronounced from a tensile failure due to crack propagation. As the system progresses through cracking, large rotations may occur, which in turn can induce significant membrane resultant forces. Equation (1) may not be a valued estimation in such a case, and a more complex theory of beams would be needed (e.g., geometrically exact beam theory) [
17]. Lastly, both Hexcel and Solvay systems did not have centralized test data containing all strengths necessary for utilizing the Tsai–Wu criterion and reported flexural strength. Though the resin contents were nearly identical, a possible source of error stems from using mixed test data.
Though the Tsai–Wu failure criterion was chosen due to its fully interactive nature, other failure criteria were also explored. Numerical trends were similar for all systems, and thus only the data for the Toray system are summarized in
Table 7. The Tsai–Wu index is provided, and safety factors for other criteria and their respective predicted flexural strength are included.
The Tsai–Wu failure criterion predicted the strength most accurately from all failure criteria. Hoffman also predicted good results since it is fully interactive, with minor differences from the Tsai–Wu criterion. Partially interactive criterion Hashin had among the lowest predicted accuracy values, likely because of its lack of interaction between the fiber and transverse direction stresses. Noninteractive criteria (max stress, max strain) also underpredicted the flexural strength. Max strain exhibited marginally better performance, as its formulation allows for some Poisson effects in combined loading.
Tsai–Wu and Hoffman predicted failure at the bottom for the Hexcel system, while the other systems predicted a compressive failure occurring in the top ply. This likely stemmed from the lack of interaction and combined loading strength expansion.