Fatigue Behavior under Constant Load of CFRP Sheets Compared to Their Performance under Static Load

3.1 Analysis of large deflections in orthotropic thin plates

It is assumed that the principal axes of the circle's diameter, which is subjected to the applied load, align parallel with the principal directions of the orthotropic material. When an orthotropic circular plate with radius (a) and a uniform applied load (q) is set up with a = b, it is expressed by Eq. (1) [19].

$W=\frac{q}{64 D_1}\left(a^2-r^2\right)^2$                (1)

where,

$r=\sqrt{x^2+y^2}$             (2)

$D_1=\frac{1}{8}\left(3 D_x+2 H+3 D_y\right)$                 (3)

$H=D_{x y}+2 G_{x_y}$              (4)

$D_{x y}=\frac{t^3 E_x^{\prime} v_y}{12\left(1-v_x v_y\right)}=\frac{t^3 E_y^{\prime} v_x}{12\left(1-v_x v_y\right)}$                (5)

$G_{x_y}=\frac{r^3 \mathrm{G}}{12}$                (6)

$D_x=\frac{t^3 E_x^{\prime}}{12\left(1-v_x v_y\right)}$               (7)

$D_y=\frac{t^3 E_y^{\prime}}{12\left(1-v_x v_y\right)}$              (8)

The stresses that were observed in the center of the composite plate when r = 0, regardless of r and θ, can be articulated as follows:

$\sigma_r=\sigma_\theta=\frac{3 q}{4}\left[\frac{a}{h}\right]^2$                (9)

The tensile strain at the center of the plate during significant deflections of the composite material may be determined by study [20].

$\varepsilon_r=\varepsilon_\theta=0.462 \frac{w^2}{a^2}$              (10)

The tangential strain (ε_t) and effective strain (ε_eff) experienced by the composite plate can be determined using the following equations [21]:

$\varepsilon_t=-\left[\varepsilon_r+\varepsilon_\theta\right]$             (11)

$\epsilon_{e_{f f}}=\sqrt{\frac{2}{3}\left(\varepsilon_r^2+\varepsilon_\theta^2+\varepsilon_t^2\right)}$                  (12)

3.2 Mechanical considerations of fatigue fracture

An analysis of stress and strains under cyclic loading is essential for engineering applications. In certain practical applications, the material functions under constant maximum and minimum stress levels. This is referred to as constant amplitude stressing. The mean stress, σ_m, is the average of the maximum σ_max and minimum σ_min stress values, and the algebraic difference is the difference between the maximum and minimum stress values, ∆σ = σ_max - σ_min. The half range is referred to as stress amplitude. The following are the mathematical expressions [22]:

$\sigma_m=\frac{\sigma_{\max }+\sigma_{\min }}{2}$               (13)

$\sigma_a=\frac{\sigma_{\max }-\sigma_{\min }}{2}$                  (14)

The stress ratio R is defined as the ratio of minimum stress to maximal stress and is:  

$R=\frac{\sigma_{\min }}{\sigma_{\max }}$               (15)

where, R=1 denotes the static tensile load.

3.3 Experimental findings comparing the two tests and discussion

Two composite carbon fiber samples, each with a uniform thickness of 0.25 mm, were selected for the experimental test. Pressures ranging from 1 bar to the value leading to the complete collapse of both samples were applied to the 150 mm diagonal cross-sectional area. Using a disc displacement gauge set at r = 0, the maximum deflection values of the samples were recorded. The samples were firmly placed on the test cylinder, and the experimental test was initiated using the specially designed testing mechanism.

In both static loading and fatigue testing under constant load, the deformation of the samples resulted in convexity as the applied loads increased. This deformation exhibited linear behavior, with deflection values in the fatigue test under constant load exceeding those recorded in the static loading test. The fatigue test mechanism stabilized maximum deflection values after several cycles of constant loading, explaining this disparity.

In the static loading test, when a load of 6 × 10⁵ N/m² was applied, the deformation increased steadily until a cracking sound was detected, indicating microstructural damage in the sample. As the pressure increased further, reaching 6.2 × 10⁵ N/m², the deformation briefly escalated, leading to instantaneous failure characterized by an explosive rupture of the circular plate's diameter.

Similarly, during the fatigue test under constant load, an audible fracture indication was observed at the same applied load of 6 × 10⁵ N/m², specifically during the crack initiation stage. As the applied load increased to 6.5 × 10⁵ N/m², the internal structural cracking sounds became more pronounced, signaling the transition to the crack propagation stage, where microcracks expanded and became more visible within the material's microstructure. The applied stress was gradually raised to a total load of 6.8 × 10⁵ N/m², at which point the cracking noise intensified significantly, accompanied by a noticeable increase in maximum deflection. After 15 cycles of cyclic loading, the specimen suffered complete and explosive failure, similar to the behavior observed under static loading conditions. However, the fragmentation of the plate was more pronounced, with smaller scattered sections, compared to its state under static loading. This observation suggests a gradual structural failure during cyclic loading, commonly referred to as fatigue failure.

A summary of the key mechanisms influencing CFRP panel performance under different loading conditions (static and fatigue) can be outlined as follows:

•The maximum deflection values under static loading were lower than those recorded in the fatigue test. This is attributed to the single application of force in static loading, whereas the fatigue test involves multiple loading cycles, allowing deflection to accumulate over time.

•The duration for crack propagation and damage manifestation in fatigue testing was longer due to the cyclic nature of loading. The brief intervals between cycles allowed damage to accumulate gradually, unlike static loading, where failure occurred more abruptly.

•The appearance of the failed CFRP sample varied between the two loading conditions. After fatigue failure, the sample exhibited a more fragmented and crumbled state compared to its failure under static loading.

Figure 6 illustrates the sample during testing, while Figure 7 presents the two samples after failure. Image (a) depicts the sample post-failure under static loading, whereas Image (b) shows the sample after failure under fatigue loading. Table 3 compares the maximum deflection values for both tests, and Figure 8 provides a graphical comparison of the maximum deflection values.

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Figure 6. CFRP specimen under 5 × 10⁵ N/m² distributed compression test

Table 3. Experimental aberration results for static load and fatigue under constant load conditions

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Figure 7. The scattered section of the sample was subjected to static loading compared to that under cyclic loading

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Figure 8. The maximum deflection resulting from static load and fatigue under constant load

3.4 SEM microstructure analysis and Vickers hardness testing

The microstructure of the CFRP sample was examined microscopically after exposure to low-cycle stresses. Figure 9 presents the microscopic analysis of the sample before failure, while Figure 10 depicts the sample after failure, consisting of two images:

•Image (a) illustrates the initial deterioration of the matrix and the formation of microcracks. This phase involves the development and expansion of fractures in regions with a high concentration of voids, as the applied loads increase.

•Image (b) shows the critical stress concentration at the fracture tips, reaching its peak at 6.8 × 10⁵ N/m², ultimately leading to the specimen's failure.

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Figure 9. SEM image for CFRP specimen before testing

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Figure 10. Images of the microstructure of a CFRP sample subsequent to fatigue testing under constant load

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Figure 11. Microscopic views of the CFRP specimen utilized for Vickers hardness testing

A Vickers hardness test was conducted on the specimen before the experimental procedure, using an applied load of 500 g for 15 seconds, with three readings taken from different locations on the sample. The test yielded an average hardness of 111.33 H.V. Figure 11 (a) illustrates the hardness test conducted before the experimental procedure.

Following the experimental test, another Vickers hardness test was performed under the same conditions to assess the hardness of the failed specimen. The hardness measurement after failure was 28.22 H.V., as shown in Figure 11 (b). This significant reduction in hardness indicates material degradation and a deterioration of the internal structure due to the applied loads.

Figure 11 provides a microscopic image demonstrating the impact of the Vickers hardness test tool on the sample:

•Image (a) represents the sample condition prior to testing.

•Image (b) displays the sample condition after failure, highlighting the structural changes caused by the applied stresses.

Scanning electron microscope analysis is essential for identifying the stages of structural failure in specimens subjected to different stresses and loads; the material hardness test before and after the experimental procedure is also considered one of the most important evidence for determining these stages.

3.5 Theoretical vs. Experimental outcomes

The experimental results of the sample deformation stages were compared with the theoretical results, focusing on the maximum deflection under an ascending load gradient. This study utilized Tables 4, 5, and 6 to calculate the maximum deflection, stresses, and strains experienced by the sample due to both theoretically and experimentally applied loads. Figures 12, 13, and 14 present graphical representations of the results derived from these tables.

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Figure 12. Experimental deflection under static load vs theoretical deflection of CFRP material

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Figure 13. Relationship between strain and stress based on experimental results of maximum deflection of CFRP

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Figure 14. The relationship between the applied load and strain based on the theoretical findings of CFRP

The experimental findings identified four key phases in relation to the theoretical maximum deflection outcomes:

Initial Phase:

A distributed load of 1 × 10⁵ N/m² caused the sample to adopt a convex shape.

The difference between the theoretical and experimental deflection values can be attributed to the application of significant deviation theory, which does not fully capture the slight variations observed in the experimental data.

Second and Third Phases:

The theoretical and experimental values closely aligned, with only minor discrepancies observed.

These phases demonstrated a strong correlation between predicted and actual deflections.

Fourth Phase:

At 6.2 × 10⁵ N/m², the difference between the theoretical and experimental deflection values became more pronounced.

The experimental data recorded the complete collapse of the sample, whereas the theoretical model continued to provide readings with consistent differences.

This discrepancy arises because the theoretical model does not account for sudden failure, whereas the experimental results reflect the actual structural breakdown of the sample.

Considering the inherent uncertainty in material properties due to industrial defects, the experimental results demonstrated strong convergence with theoretical predictions. This alignment allows for reliable scientific conclusions, as highlighted in the comparison table provided below.

Table 4. Deflections of CFRP material: Experimental vs. theoretical under static load

Table 5. Maximum stresses and strains for CFRP center experimental deflection

Table 6. Maximum stresses and strains for CFRP center theoretical deflection

Table 5 presents the relationship between stress and strain based on the maximum experimental deflection, as illustrated in Figure 13. This relationship remains linear from the initial loading phase until it begins to deviate from linearity during the early stages of laminate matrix damage, occurring under a load of 4 × 10⁵ N/m². The deflection continues to increase until the damage reaches its peak, culminating in structural failure at the maximum load, where the maximum deflection values correspond to the stresses that lead to the complete collapse of the specimen.

Table 6 presents the theoretical strain values corresponding to stresses at the maximum theoretical deflection, with Figure 14 illustrating their trajectory. The agreement between the experimental and theoretical results demonstrates a high level of convergence, with a strain progression pattern in response to the applied loads that closely matches the experimental data.

In fatigue testing under constant load, the maximum deflection values require a longer duration to reach stabilization. The deflection measurements reflect the sample deformation caused by the cyclic loading technique, in which each load is applied in repeated cycles. This approach extends the stabilization period of deflections and results in higher deflection values compared to static loading conditions.

All variations observed in the first cycle correspond to those recorded under static loading, where each pressure value is applied in a single batch. The theoretical predictions regarding maximum deflection closely align with the experimental findings.

Figure 15 illustrates the relationship between the number of cycles and the maximum deflection values for each applied load. The black marker at the end of each deflection line represents the stability stage of the maximum deflection value, which is reached after several cycles, based on both the experimental test results and the detailed theoretical calculations.

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Figure 15. The relationship between the number of cycles and the maximum deflection values during fatigue testing under constant load for CFRP

Table 7 presents the values of maximum stress ($\sigma \max$), minimum stress ($\sigma \mathrm{min}$), and mean stress ($\sigma \mathrm{m}$), along with the algebraic difference $(\Delta \sigma)$, stress amplitude $(\sigma \alpha)$, and stress ratio (R) for loads ranging from 1 to $6.8 \times 10^5 \mathrm{~N} / \mathrm{m}^2$. The stress ratio values exhibited an upward trend, aligning with the gradual increase in applied loads until the sample reached failure.

Figure 16 illustrates the relationship between the number of cycles and the stress values corresponding to each applied load. The graphs indicate that:

•As the loads increase, the stress values also increase.

•Higher loads result in higher stress values.

•The ratios between increasing stress values gradually decrease until they converge at peak loads.

This trend suggests a reduction in panel stiffness, attributed to the deterioration of the microstructure and the propagation of cracks throughout the sample. Ultimately, the sample fails under an applied load of 6.8 × 10⁵ N/m² during cycle 14.

Table 7. Results of testing fatigue under constant load for all applied load levels for CFRP specimen

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Figure 16. The relationship between the stress values and the number of cycles for each load applied to the CFRP sample