Evaluation of failure criteria for residual tensile strength of composite laminates with circular holes

2.1. Point stress criterion (PSC)

For an infinitely anisotropic plate with a circular hole of radius $R$ subjected to uniform stress $\sigma$ in the $Y$ direction at infinity, this criterion assumes that the laminate fails when the stress at a characteristic distance from the hole edge reaches the ultimate strength of the unnotched laminated plate:

where ${\sigma }_y\left(x,0\right)$ is the distribution of the $Y$-direction stress on the smallest cross-section of the hole; $d_0$ is the characteristic distance, which is determined experimentally.

The laminate is assumed to be equivalent to a homogeneous orthotropic plate. For an orthotropic infinite plate with a circular hole subjected to uniaxial tension, the elastic stress distribution along the axis perpendicular to the loading direction at the edge of the hole can be derived from the Lekhnitskii formula [15], i.e.:

The stress concentration factor at the edge of the hole in the infinitely wide laminated plate is calculated by the formula [16], i.e.:

where, $A_{ij}$ is the in-plane stiffness coefficient of the laminate. For an orthogonal anisotropic infinite plate with a circular hole under uniaxial tension, the failure strength can be obtained by substituting the stress distribution around the hole into Eq. (1):

The critical failure strength of an infinitely wide plate is ${\sigma }_c^{\infty }=\eta {\sigma }_c$, ${\sigma }_c$ is the failure strength of a finite width plate corrected by a finite width correction factor. When the ratio of defect length to the plate width is not greater than 1/3, for a laminated plate with a central circular hole of radius $c$, we have:

where, $W$ is the width of the plate.

2.2. Average stress criterion (PSC)

This criterion assumes that the laminated plate fails when the average stress within a certain characteristic distance from the hole edge reaches the ultimate strength of a notch-free laminated plate, i.e.:

where, $a_0$ – characteristic distance determined by experiments. Substitute Eq. (2) into Eq. (6) to obtain the formula for calculating the strength of destruction. For an orthotropic infinitely plate with a central circular hole under uniaxial tension, the formula for calculating the strength can be obtained by substituting Eq. (2) into Eq. (6), i.e.:

where, ${\zeta }_2=R/(R+a_0)$.

2.3. Stress field intensity (SFI)

The stress field intensity method assumes that the failure of a component with stress concentration depends not only on the peak stress but also on the stress-strain field caused by the stress concentration zone. This method takes into account the influence of stress gradients in the damage zone at the root of the notch, and considers the combined contribution of stress at different material points to the failure of the notch root by allocating weight functions to the stress vectors of all material points in the field strength area. The formula for calculating the failure strength of laminated plates with circular holes based on this method is given by:

where $d_s$ – characteristic distance determined by experiments; $\lambda =R/(R+d_s)\text{;}$$C=2/(W-2R)$.

3.1. Test specimens

The test specimens are smooth composite laminates made of medium modulus carbon fiber reinforced M21C epoxy resin unidirectional tapes with a nominal thickness of 0.187 mm per layer. The material properties of the specimens are shown in Table 1.

The specimens were manufactured using automatic fiber placement and autoclave curing auxiliary process. Two configurations were used: without notch and with a centrally circular hole, as shown in Fig. 1.

The specimens contain 4 typical ply configurations, as shown in Table 2.

According to the different configurations and layups, the test specimens were divided into 28 groups with a total of 144 specimens, as shown in Table 3.

Fig. 1Specimen with hole

3.2. Test results

The far-field failure strain of the intact specimen reached around 16000 με, while the far-field failure strain of the specimen with a hole decreased to between 5000-8000 με. The experimental failure loads are shown in Table 4, where $e$ is the ratio of the hole diameter to the plate width, $F_{max}$ is the average breaking load, ${\sigma }_c$ is average value of failure stress, CV is the coefficient of variation of failure load.

The experimental results showed low dispersion, with a maximum coefficient of variation of 5.38 %, indicating the reliability of the results. For the four types of layups, the strength of specimens with holes of diameter 6.35 mm was reduced by 44.3 %, 51.3 %, 48.0 %, and 39.4 %, respectively, compared to the intact specimens. It can be seen that the greater the proportion of 0° plies, the greater the reduction in strength of specimens with holes. With the constant ratio of hole diameter to plate width, the tensile strength decreased with the increasing hole diameter.

3.3. Evaluation of failure criteria

Based on the obtained experimental data, the estimation models of PSC, ASC, and SFI were validated. The three failure criteria all require obtaining the corresponding characteristic distances through experiments. Taking the average value of the characteristic distances for different hole diameters, the relative average error (RAE) can be defined as the maximum relative error between the obtained characteristic distance values and the average value of the characteristic distance for the same type of specimen. RAE can be used to evaluate the effectiveness of the estimation models.

The characteristic distances obtained by calculation are shown in Table 5. It can be seen that the characteristic distances obtained for different hole diameters and diameter-width ratios are different. Generally, when the diameter-width ratio is constant and the diameter is less than 25.4 mm, the larger the hole diameter, the larger the characteristic distance. When the hole diameter is constant, the difference in characteristic distances is relatively small, but still shows a slightly decreasing trend with increasing diameter-width ratio.

The average characteristic distance and relative error are shown in Table 6, where $\overline{d_0}$ is the average characteristic distance of PSC, $\overline{a_0}$ is the average characteristic distance of ASC, $\overline{d_s}$ is the average characteristic distance of SFI, RAE is the relative average error. It can be seen that the prediction accuracy of the average stress criterion is better than that of the point stress criterion and the stress field intensity method.

In fact, by substituting the average value of the characteristic distance into the corresponding residual strength calculation formula, the errors of the three criteria relative to the test results can be obtained. The maximum errors for each type of laminate were listed in Table 7. It can be seen that in any case, the average stress criterion is the better criterion, while for the A and D layers, the accuracy obtained by the stress field strength criterion has made it inapplicable. For the D layer, which is dominated by the 45-degree layer, all three criteria showed relatively large errors.