Analytical and experimental investigation of tensile properties of cross-ply and angle-ply GFRP composite laminates

3.1 Netting analysis

Netting analysis is an empirical composite laminate analysis method. It assumes that only lamina fiber direction provides stiffness or strength, i.e., no contribution from off-axis plies. The composite laminate tensile mechanical properties can be derived as follows:

where t₀ is the 0° ply total thickness and t_t is the laminate total thickness.

However, because there are no 0° plies in the [+67.5₄, -67.5₄]_s laminate, the conventional netting analysis is unable to predict any of the tensile mechanical properties for this specific laminate.

3.2 Laminate mixture rule

The laminate mixture rule is also an empirical composite laminate analysis method. It assumes an empirical contribution from off-axis plies, e.g., 10% off-axis contribution (Hart-Smith rule). The composite laminate tensile mechanical properties can be derived as follows:

where t₀ is the 0° ply total thickness, t_θ is the θ° ply total thickness and t_t is the laminate total thickness.

3.3 Classical laminate theory (CLT)

CLT is a linear elastic ply constitutive relations method at the macroscopic level. It works for plane stress cases to derive in-plane laminate mechanical properties. Tensile tests are considered to be a linear-elastic stress-strain analysis load configuration, which can be numerically investigated using CLT. The latter temperature thermoelastic analysis is not considered for this case. CLT uses the maximum strain failure criterion, which accounts for:

• Ply transverse, shear and fiber failure modes in tension or compression

• Partial ply failure with transverse and shear property degradation

• Last ply failure at fiber failure

Generally, CLT is applicable to symmetric balanced laminates with multiple orthotropic or isotropic plies such as the unidirectional glass used in this work. CLT also accounts for different material properties, thicknesses and fiber angles. In a composite laminate, stresses are generally different on plies with different orientation (as in this work) or different material properties. Consequently, some plies are bound to fail first before others, which is generally referred to as first-ply failure [9]. The tensile properties are estimated using the following equation:

where the N_i vector represents the loading intensities, and

where t is the laminate thickness and t_k is the kth laminate thickness.

Because the laminates used in this work are symmetric and only longitudinal loading is considered, Eq. (7) reduces to:

which gives

The rest of the ply failure calculations are carried out using one of the many freely available CLT software packages. These programs initially perform laminate analysis for given loading (axial in this case), and then they obtain stresses and strains in each layer in 1-2 ply axes, which are checked against the failure criteria for each mode (1-1 fiber failure, 2-2 transverse failure and 1-2 shear matrix failure). If a ply fails in matrix-dominated mode (i.e., 2-2 or 1-2), then the code degrades layer stiffness properties (E₂ and G₁₂) and it repeats analysis. If, however, a ply has failed in fiber-dominated mode, i.e., 1-1, then the code assumes final failure and the code stops analysis.

3.4 Experimental results

Figures 1 and 2 show the tensile test results of three specimens, each of both laminates considered for this work. The achieved stress and strain values for each specimen are very consistent with each other, i.e., with little scatter. The obtained ultimate values are summarized in Tables 1 and 2 for [90₄, 0₄]_s and [+67.5₄, -67.5₄]_s laminates, respectively. From Figure 1, it can be seen that the slope was linear up to around 150 MPa, when a visible change in slope occurred, with the rest of the curve showing linear variation to final failure. Rusnáková et al. [10] used similar glass fibers (Fiberglas 92145) and reported lower ultimate failure stresses and strains for 0/90 cross-ply laminates compared to the findings of this work. They reported a maximum failure stress of 371 MPa and a maximum failure strain of 0.0548. However, it should be noted that the matrix system they used is different from the one used for this work. Moreover, Figure 2 shows that the strength of [+67.5₄, -67.5₄]_s laminates is considerably weaker compared to that of [90₄, 0₄]_s laminates due to mainly the matrix taking the tensile load and not much being carried by the fibers. It is also noted that the stress/strain curve starts to change its slope just after the 30-MPa mark, indicating the onset of matrix failure.

Figure 1

Stress vs. strain curves for [90₄, 0₄]_s specimens.

Figure 2

Stress vs. strain curves for [+67.5₄, -67.5₄]_s specimens.

3.5 Comparison between all methods

Tables 2 and 3 and Figures 3 and 4 summarize the results of all methods used in this work and how they compare to experimental data. The conventional netting analysis is unable to estimate the [+67.5₄, -67.5₄]_s laminate properties because this method does not account for any off-axis plies.

Figure 3

Summary of [+90₄, -90₄]_s comparison results.

Figure 4

Summary of [+67.5₄, -67.5₄]_s comparison results.

For the [90₄, 0₄]_s laminate, it can be seen from the summary of results that for the ultimate failure stress, the 10% rule gave the lowest error (-22%) compared to experimental data, with the netting analysis underestimating the ultimate tensile strength by 35% and the CLT overestimating it by 44%. In terms of Young’s modulus, the netting analysis gave the lowest error (11%), whereas the 10% rule and CLT gave comparable overestimates of around 20% compared to the experimental data. However, for ultimate failure strain predictions, the CLT produced the lowest error of 30%, whereas both the netting analysis and the 10% rule underestimated the ultimate failure strain by around 50%.

For the [+67.5₄, -67.5₄]_s laminate, it can be seen from the summary of results in Table 3 that for ultimate failure stress the 10% rule gave the lowest error (-26%) compared to experimental data, with the CLT underestimating it by almost three times. In terms of Young’s modulus, the CLT underestimated it by 74%, and the 10% rule underestimated it by more than 100%. However, for ultimate failure strain predictions, the CLT produced the lowest error of -37%, whereas the CLT overestimated the ultimate failure strain by around 45%.