Measurement of in-plane fibre misalignment results in a Gaussian distribution of the untransformed fibre orientation and after the transformation of the fibre misalignment angles . The distribution range of for GFRP is from −4° to +5°. Since 90% of all values are between −2.75° and +2.75°, a mean misalignment angle of is valid for GFRP. For CFRP, the distribution range is smaller (−2° to +3°) with 92% of all values lying between −2° and +2°. Hence, a mean misalignment angle of is present in the CFRP specimens. The median misalignment angle as well as the standard deviation of the misalignment for CFRP is smaller compared to GFRP.
4.2. Compression Tests and Comparison with Analytical Models
The UD compressive strength as experimentally determined is
for GFRP and
for CFRP (refer to
Table 4). Compressive strength of CFRP is approx. 50% higher than that of GFRP. The higher strength for CFRP can be attributed to the higher strength of carbon fibres in comparison to glass fibres.
Figure 3 shows photos of a representative CFRP and GFRP specimen after final failure. Final failure occurs in the form of a kink-band that is visible for both materials. The CFRP specimens exhibit a slightly higher amount of delaminations and fibre breakage next to the kink-band, which indicates the higher load at failure that results in more severe visible damage.
The prediction accuracy for UD compressive strength of the analytical models is compared with the compression test results. With the determined plastic shear parameters (
Table 3) and the material properties of fibre and matrix (refer to
Section 3.1), the strength is calculated for different fibre misalignments
by using the various models presented briefly in
Section 1.
The predicted strength values of the different models are summarised in
Table 5 and compared with the experimental results. A fibre misalignment angle of 3° is selected for this comparison because it reflects the measured misalignment in the specimens well. Best prediction accuracy is achieved with the kinking model from Budiansky [
19,
20], especially for CFRP.
The
shear model from Rosen [
2] (refer to Equation (
1)) highly overestimates the compressive strength with a predicted strength of approximately
for both GFRP and CFRP in the shear mode. For the extension mode, the values are even higher. The model is not able to differentiate between different fibre types. This was expected because only the matrix shear modulus and the fibre volume fraction are used as input parameters in this approach to predict the strength of a material with a very complex damage process under the given load case and is in accordance with previous results from other authors [
32].
For the
fibre kinking model from Budiansky [
19,
20], the compressive strength is calculated with Equation (
2). The model predictions in comparison with the experimental results are shown in
Figure 4 for GFRP and CFRP. For small misalignment angles, the model highly over-predicts the compressive strength. The predicted compressive strength of
for CFRP with a fibre misalignment angle of 3° correlates well with the experimental results, with the predicted value being within the standard deviation but below the mean value. As the fibre type is regarded only indirectly via the results from the shear test in the model parameters, the difference between predicted strength and failure strength is higher for GFRP. For a misalignment angle of 4°, which is higher than the measured misalignment in the GFRP specimens, the predicted value of
is within the standard deviation of the experimentally determined strength. The better agreement of predicted values with experimental results compared to the literature [
16,
17,
32] can be explained with the fibre volume content
that is lower in our specimens. With a decreasing fibre volume content, plastic kinking is facilitated and the matrix properties become more and more relevant. Therefore, the fibre kinking model, which is mainly based on the matrix shear behaviour, is more accurate for lower
.
In the
fibre microbuckling model from Berbinau et al. [
16], the fibre type determines the fibre cross section area
and second modulus of inertia
I in Equation (
4). For the calculation of
and
, the measured mean fibre radius is used. The amplitude value of the unstressed fibre
is calculated with Equation (
10). It is reported that the wavelength
equals the kink band width [
39] and for
a value of
is reasonable [
9,
13,
14,
16,
17], thus this approach is used here as well:
A graphic representation of
is plotted over the compressive stress and an asymptotic increase of the fibre amplitude predicts a compressive strength of approximately
for CFRP. Therefore, this model overestimates the compressive strength, which is critical for conservative design of composite parts and in contrast to what is previously reported for this model [
32]. The predicted strength for our material is lower compared to the values predicted for CFRP with a higher volume fraction [
17]. Hence, the general influence of fibre content is represented qualitatively correct by the model, but the predicted value has a large error compared to the experiments with specimens that have a lower
. Decreasing
leads to a higher decrease of
than predicted by the model.
For GFRP, the microbuckling model in the current form is not applicable because the graphic representation of
over the compressive stress results in an asymptotic decrease. With the slope of
tending to zero instead of to infinite as expected, the compressive strength cannot be read at the
x-axis and an adaption of the microbuckling model is necessary to be applicable for GFRP. This is related to the fact that the larger fibre diameter of the glass fibres that determines the moment of inertia
I and the cross section area
, in combination with the lower fibre Young’s modulus
, in comparison to CFRP, leads to a negative denominator in Equation (
4) and thus a decrease of
over
. In other words, the term
has a higher value than the term
for glass fibres. Consequently, the fibre type and diameter
, are important factors and should be considered for compressive strength prediction. When writing Equation (
4) with the introduced abbreviations in the form
, for an asymptotic increase,
must be valid, which is not the case for GFRP. This can be avoided, when the absolute value of the term in the denominator is used, leading to the adapted Equation (
11). This equation with the absolute values for the denominator is applicable for both CFRP and GFRP:
The graphic representation of
versus compressive stress
for GFRP is shown in
Figure 5 for both the original microbuckling model from Equation (
4) in
Figure 5a and for the adapted model from Equation (
11) in
Figure 5b. Curves are plotted for misalignment angles between 1° and 5°, but the influence of fibre misalignment of the graphic representation of
and thus the predicted strength is negligible.
With Equation (
11), a compressive strength of
is predicted for GFRP (refer to
Figure 5). This is higher than the predicted strength for CFRP, which results from the increased stability of thicker fibres against microbuckling in the model but does not represent realistic behaviour. When using comparable specimen geometry, CFRP achieves higher compressive strength than GFRP, as is also the case in the experiments. Regarding the predicted strength values, the adapted microbuckling model significantly overestimates the compressive strength. The prediction error is even larger for GFRP due to the fact that a higher predicted strength coincides with lower measured strength when compared to CFRP. The suggested adaption allows application of the microbuckling model, although originally derived for CFRP. However, a higher predicted strength for GFRP than for CFRP is not reasonable.
The microbuckling model uses the shear modulus of the composite
for predicting the microbuckling behaviour. For thicker fibres and lower fibre volume fractions, it can be argued that the local microbuckling of a fibre depends more on the shear modulus of the surrounding matrix than on that of the composite due to the larger inter-fibre distance. This could be the case for the GFRP used in this study, which exhibits a significantly lower fibre volume fraction (
) compared to the CFRP prepreg system against which the microbuckling model was verified [
16,
17]. When using the matrix shear modulus
instead of
in Equation (
11), a compressive strength of
is predicted for GFRP, which is lower than the experimentally determined strength but far more realistic. For CFRP, a compressive strength of
is predicted when using the matrix shear modulus instead of the composite shear modulus in Equation (
11). This underestimation of strength agrees better with the general behaviour of the microbuckling model as described in literature that led to the introduction of the combined modes model [
32].
The
combined modes model by Jumahat [
32] consists of a microbuckling part and a kinking part, as described by Equation (
6). For calculating the ratio of compressive strength attributed to fibre microbuckling, the adapted Equation (
11) is used, so that the combined model is also applicable for GFRP. The fibre kinking ratio is calculated with Equation (
7), with the parameters determined in the tensile tests for the respective material. Results from the combined modes model for different fibre misalignment angles
in comparison with the experimental results (mean value and standard deviation) are shown in
Figure 6 for CFRP and in
Figure 7 for GFRP. Realistic misalignment angles between 1° and 5° are chosen to analyse a certain range of fibre misalignment with the model.
Since the microbuckling model already overestimates the compressive strength of the specimens, the combined modes model by Jumahat et al. [
32] does so as well. The predicted UD compressive strength of
for CFRP with an initial fibre misalignment of 3°, which is the local out-of-plane misalignment measured in our specimens, is lower compared to predicted strength values reported for prepreg-CFRP with a higher
[
32], but significantly higher than measured specimen strength. Therefore, the model is able to represent the general influence of a lower fibre volume fraction with its prediction, but leads to a overestimation of compressive strength for lower
. For GFRP, the combined model also predicts higher compressive strength values for the investigated range of misalignment than measured in the experiments. For a misalignment angle of 3°, the predicted value is
. This is unrealistically higher that the value predicted for CFRP and results from the higher microbuckling ratio compared to CFRP because the microbuckling model predicts higher strength for GFRP.
If the matrix shear modulus instead of the composite modulus is used to calculate the microbuckling ratio, the prediction accuracy for GFRP is quite good. The use of the matrix shear modulus is motivated by the lower fibre volume fraction that results in more matrix dominated microbuckling of the fibres. For small misalignment angles, the predicted strength is within the standard deviation of the test results. For larger misalignment angles, the strength is slightly underestimated, which is less critical for conservative design. For CFRP, usage of is not meaningful because the compressive strength is highly underestimated.
When comparing the different analytical models for predicting the compressive strength of FRP (refer to
Table 5), the fibre kinking model achieves the best results in comparison to the experiments. This is unexpected because it is in contrast to previous investigations [
16,
17,
32]. Probable reasons for this deviation are the material and the manufacturing process. In the other investigations, a CFRP prepreg material with a fibre volume fraction of approximately
was used that was autoclave cured. In our study, we used a non-crimp fabric and prepared specimens via a VARTM process, resulting in a lower fibre volume fraction (
). Both the infusion process and the achieved fibre volume fraction are typical for many applications of FRP such as wind turbine blades or sporting goods and thus of relevance for an accurate prediction of compressive properties. As expected, the lower fibre volume fraction results in a lower compressive strength compared to the values in literature [
16,
17,
32]. The trend of decreasing strength with lower
is represented by the kinking model and the combined modes model, although the latter highly overestimates the strength for lower
. It can be concluded that the matrix properties become more important with decreasing fibre content and that fibre kinking is the dominant failure mechanisms in that case. This is more pronounced for GFRP than for CFRP, where usage of matrix instead of composite shear properties leads to accurate prediction of compressive strength in the combined modes model considering both microbuckling and kinking.
It has to be noted that, in our experiments, the fibres exhibit circular cross sections and such a shape is used for calculation of fibre cross section area
and moment of inertia
I. However, in some composite parts, the fibre cross-section is of a kidney shape, which influences the mechanical properties and failure behaviour under compressive loading [
40]. This should be considered, when applying the analytical models to predict the UD compressive strength of laminates with kidney-shaped fibres (e.g., by different equations for calculating
and
I in the microbuckling and combined modes model).