An experimental analysis of the end-notched flexure composite laminates beams with elastic couplings

Fiber-reinforced plastic (FRP) laminates, besides steel, are widely used in many industry sectors. The main advantages of polymeric composites are high strength and corrosion resistance. The most common type of failure in FRP laminates is delamination which is the major weakness of advanced composite structures. Another problem which can have a considerable effect on delamination resistance may be an elastic coupling phenomena which may be present when laminates are manufactured with specific configuration of layers. The fracture toughness (\(G_{\mathrm{C}}\)) in the form of the critical strain energy release rate (c-SERR) values can be determined for three basic modes including the mode I (tensile opening), the mode II (in-plane shear) and the mode III (transverse shear). Among them, the mode II delamination is one of the common damages in the industrial structures because of applying flexural loads to the parts. There are various standardized test methodologies available to determine the mode II fracture toughness of fiber-reinforced polymer composites, namely the end-notched flexure (ENF) test [1], the end-loaded split (ELS) [2] test and the mix-mode bending (MMB) test [3]. All these methods are widely used by the scientific community and the industry to obtain the fracture properties of materials. The advantages and limitations of each configuration were reviewed in [4] where it was found that one of the main problems in the mode II testing is poor comparability among the existing test methods. The causes of such discrepancies in results are diverse, ranging from the friction [5, 6] and large deflections involved, to the complex damage mechanism occurring at the crack tip. Nevertheless, the end-notched flexure is among others the most widely used technique to study the mode II fracture owing to only requiring a common three-point bending fixture. Therefore, many researchers have examined the mode II delamination of laminated composites by using the ENF test [7‐16].

The respective ASTM Standards cover the measurements of the \(G_{\mathrm{IIC}}\) only of unidirectional (UD) laminates. However, most applications in composite structures involve multidirectional (MD) laminates, in which delamination tends to occur between layers of different orientations. Moreover, MD laminates usually present elastic couplings which can be the source of significant errors in the \(G_{\mathrm{IIC}}\) measurements [17]. Couplings lead to a non-uniform distribution of the energy release rate along delamination front [18] resulting in skewed and curved crack fronts when propagating an initially straight crack in fracture test and possibly also mix-mode effects. This may in turn lead to an error in the evaluation of the fracture toughness based on analysis which assumes a uniform crack extension across the specimen width and assumption of fracture mode from global considerations. Davidson et al. [19] presented a three-dimensional finite element results for the energy release rate (ERR) distribution in symmetric and antisymmetric ENF specimens. It has been shown that the difference between the maximum and minimum ERRs along the delamination front is a function of the specimen’s \(D_{{c}}\) ratio. It has also been shown that the asymmetry in the energy release rate distribution about the center of the specimen’s width is a function of the \(B_{{t}}\) ratio. For the ENF loading, the peak energy release rates occurred at one or both edges of specimen. Moreover, larger bending–twisting coupling resulted in larger asymmetries in the energy release rate, whereas larger longitudinal–transverse bending coupling resulted in larger peak values. Similar conclusions were drawn by Samborski [20]. He performed a numerical study concerned the mode II critical strain energy release rate distribution along delamination front in mechanically coupled laminated composites subjected to the standardized end-notched flexure test described in the ASTM D7905 Standard. Finite element results indicated that the assumptions of the uniform distribution of the mode II SERR along delamination front typical for unidirectional laminates were no longer valid for the general ply-angle laminates. The strongest deviations of the \(G_{\mathrm{II}}\) distributions were observed for the laminated beam models exhibiting simultaneous bending–extension/extension–twisting/shearing–bending coupling. Apart from numerical simulations, many researchers conducted experimental determination of the mode II interlaminar fracture of multidirectional composite laminates. Ozdil et al. [21] prepared the ENF tests on unidirectional and angle-ply E-glass/polyester specimens and revealed that the fracture toughness increased with increased angle \(\theta \) at the \(\pm \theta \) interface. This is attributed to the fracture work associated with debonding of transversely oriented fiber bundles in the quasi-unidirectional plies. Moreover, post-fracture examination of the specimens revealed deviations of the crack front form its initial straight shape in all specimens as a result of elastic coupling effects in the laminate beams. Pereira et al. [22] prepared experimental study on the mode II fracture toughness of carbon/epoxy multidirectional laminates with delamination interfaces \(0^{\circ }//0^{\circ }\), \(\theta //\theta \) and \(0^{\circ }//\theta \). Experimental results obtained for the ENF test revealed that \(G_{\mathrm{IIC}}\) values increased with the ply angle \(\theta \) for both \(\theta //-\theta \) and \(0//\theta \) specimens. Similar conclusions were drawn in [23] where glass/epoxy multidirectional laminates were subjected to the end-notched flexure test.

Although the ENF test is a widely used technique to study the mode II fracture, it suffers from a limited number of reliable and precise data about crack propagation and process zone phenomena. Actually, the difficulties in monitoring crack propagation and the fiber bridging phenomena often limit the mode II test to the measurement of initiation toughness values. Moreover, the lack of crack opening and the particularly large fracture process zone (FPZ) associated with the mode II delamination pose additional problem in defining the crack tip position during the test. Therefore, a so-called the compliance-based beam method (CBBM) [24, 25] has been proposed to overcome those difficulties and to take into account the effect of the FPZ on the \(G_{\mathrm{IIC}}\) values. The main idea of this method was to use the beam theory to obtain the equation that relates the current compliance and the crack length. This equation can be used to estimate the equivalent crack length as a function of current compliance. The standardized test methods based on visual crack length monitoring show differences in determination of the fracture toughness because they mainly depend on the measurements of the crack length and a well-defined crack tip does not exist when testing laminates under the mode II loadings. Therefore, they are not reliable for measuring the fracture toughness. On the contrary, methods that are not based on a direct measurement of the crack length such as the CBBM, results in small differences between the different test typologies and materials. Even though the CBBM is based on linear elastic fracture mechanics (LEFM) it rely on calculating an equivalent crack length that somehow takes into account the FPZ length. This is why the CBBM results show better agreement with an experiment.

In practice, determination of the interlaminar fracture resistance may be a challenge when elastic couplings occurred in laminate. Therefore, in this situation the question of author of this paper is, whether the standardized test methods of determination of the mode II fracture toughness—elaborated for the unidirectional (UD) composites with the respective data reduction schemes are still valid.

In the present work an experimental determination of the mode II fracture toughness of carbon/epoxy composite laminates with elastic couplings was prepared. Critical strain energy release rates were obtained both for the classical data reduction schemes and for the method based on equivalent crack concept (compliance-based beam method). Recognition of delamination initiation was supported by the acoustic emission technique. Delamination surface of coupled laminates after the ENF tests were investigated by using SEM analysis. Novelty in the present article is an experimental evaluation of the influence of elastic couplings phenomena on delamination process and fracture toughness of the carbon/epoxy composite laminates subjected to the mode II end-notched standardized test.

The response of laminate beams subjected to the ENF test may be examined by using the laminate constitutive equations according to the classical laminate theory (CLT)where N and M are the force and moment resultants, \(\varepsilon ^{0}\) and \(\varkappa ^{0}\) are the mid-plane strains and curvatures and A, B and D are the extensional, the coupling and the bending stiffness matrices, respectively, and are defined as follows:Here, i, \(j = 1\ldots 6\), \(z_{k}\) is the distance from laminate mid-plane to the top of the kth ply, \(z_{k}^{{c}} \) is the kth ply center of gravity location parameter counting from the neutral plane, \(t_{k}\) is the kth ply thickness and \(\bar{Q}_{ij}\) stands for the transformed reduced stiffness matrix.

Laminate behavior under the mode II loading is dependent on the components of each of the coupling B and the bending D stiffness matrices. Namely, for the bending–twisting (BT) coupled laminate with [\(\theta /0/\theta /\theta /0/-\theta /0/-\theta /-\theta /-\theta /-\theta /0/-\theta /\theta /0/0/\theta /\theta \)] ply sequence the forms of each of the B and the D matrices are as follows:whereas, for the bending–extension (BE) laminate with [\(\theta /-\theta /0/-\theta /0/\theta /90/\theta /-\theta \)] stacking sequence, the coupling and the bending stiffness matrices were taken from:Application of bending moment \(M_{{x}}\) to the BT laminate produced twisting curvature around its central axis which was caused by nonzero \(D_{{16}}\) term. In addition, the coupling matrix B was uncoupled which meant absence of couplings between the bending moment and the mid-plane normal strains as well as between the twisting moment and the mid-plane shear strains. Considering the BE laminate, a \(D_{16}\) component of the bending stiffness matrix was equal zero which caused absence of twisting curvatures. Moreover, the sub-laminates might experienced a mid-plane strain deformation in opposite directions caused by opposite signs of the \(B_{11}\) term for the laminate arms above and below the mid-plane.

To examine possibility of non-uniform crack extension the non-dimensional parameters \(D_{{c}}\) and \(B_{{t}}\) (Eqs. 7, 8) [19] were calculated for the laminates with elastic couplings (BT and BE laminates) as well as for the laminates with [\(0^{\circ } _{{18}}/\theta //\theta /0^{\circ }_{18}\)] and [\(0^{\circ }_{16}/\theta //-\theta /0^{\circ }_{16}\)] stacking sequences in which delamination interfaces were respectively: \(\theta //\theta \) and \(\theta //-\theta \) where \(\theta \) set angle was \(\{30^{\circ }, 45^{\circ }, 60^{\circ } \}\) . All calculations were based on the measured quasi-unidirectional ply properties (Table 1) of carbon/epoxy composite and were done by using MATLAB software environment. Obtained results of calculations were summarized in Table 2. Values of the \(D_{{c}}\) parameter were constant for the specimens with interfaces \(\theta // \theta \) and \(\theta //-\theta \) and were equal 0.0048. For the bending–twisting laminate value of this parameter reached 0.2384. Nevertheless, for all these layups, however, \(D_{{c}}\) was below the upper limit of 0.25 as suggested by Davidson et. al in [26]. The bending–extension laminate exhibited the greatest value of \(D_{{c}}\) equal 0.4089 indicating tendency for non-uniform crack extension. The skewness parameter \(B_{{t}}\) reached the greatest value for the BT specimen (\(B_{{t}}=0.2363\)), which indicating in this case, that crack might propagated in a skewed manner. Although, for the multidirectional laminates with \(\theta //\theta \) interface the non-dimensional parameter \(B_{{t}}\) was non-zero, its influence on delamination process was insignificant. For the remaining layups the \(B_{{t}}\) values were equal zero.

Figures 4 and 5 present results of calculations of the mode II fracture toughness of multidirectional laminates with different ply orientation angles at delamination plane. For specimens with \(\theta //\theta \) interface the critical strain energy release rate increased with growth of \(\theta \) angle. Namely, the c-SERR values obtained by using the corrected beam theory were respectively 0.686 N/mm for specimen with \(30^{\circ }//30^{\circ }\) interface, 0.872 N/mm for \(45^{\circ }//45^{\circ }\) interface and 1.254 N/mm for \(60^{\circ }//60^{\circ }\) interface. Results obtained by using the direct beam theory and the corrected beam theory were on similar level. The greatest discrepancies between values of the \(G_{\mathrm{IIC}}\) calculated by using those methods were obtained for the \(60^{\circ }//60^{\circ }\) specimen. Opposite situation was observed for the \(\theta //-\theta \) composite laminates. For those laminates, the critical strain energy release rate decreased with growth of \(\theta \) angle reaching maximum value equal 1.021 N/mm for specimen with \(30^{\circ }//-30^{\circ }\) delamination interface and minimum value \(G_{\mathrm{IIC}}=0.465\,\hbox {N}/\hbox {mm}\) for \(60^{\circ }//-60^{\circ }\) specimen. Laminate with interface \(45^{\circ }//-45^{\circ }\) exhibited value of the c-SERR equal 0.582 N/mm. Moreover, in the case of specimen with \(\theta //-\theta \) interfaces results obtained by using different standardized methods were also on similar level. Nevertheless, fracture toughness calculated by using the compliance-based beam method for all multidirectional specimens gave greater values than results obtained by using the classical data reduction schemes. Globally, those differences could be explained by the effect of the fracture process zone. In fact, the CBT method does not account for this damaged region which influences the compliance. Direct beam theory is expected to give more precise results in this point of view because it includes P and \(\delta \) in its equation instead of load alone. On the other hand, the compliance-based beam method captures the effect of FPZ and thus yields much more accurate toughness values.

Values of the critical strain energy release rate obtained for specimens with the bending-twisting and the bending–extension couplings were presented in Fig. 6. Comparing the results obtained by using the standardized data reduction methods the greatest values of the c-SERR were obtained for the BE specimen and were equal 0.746 N/mm (for the CBT method) and 0.735 (for the DBT method). For the BT laminate the fracture toughness values were equal 0.582 N/mm and 0.594 N/mm for the corrected beam theory and the direct beam theory, respectively. In this case, results obtained by using the compliance-based beam method were much more greater and reached maximum equal 1.052 N/mm for the bending–extension laminate and minimum equal 0.934 N/mm for the bending–twisting laminate. Considerable discrepancies between results obtained from the classical methods and the method based on equivalent crack concept might be caused by larger fracture process zone which resulted from the mode II circumstances and presence of elastic couplings. Moreover, non-zero components in the matrices the B and D might influenced arising shearing deformation and twisting curvatures at delamination front which might caused errors in the mode II fracture toughness measurements, even if accurate observations of the crack positions were achieved. Influence of elastic couplings on behavior of the bending–extension laminate subjected to the ENF test was presented in Fig. 7 where transverse shearing of sub-laminates was observed. Although this phenomena was observable only for large deflection it could influence on the fracture toughness at delamination initiation.

Figure 8 shows the comparison of scanning electron microscope (SEM) photographs of fracture surfaces after the ENF tests for the bending–extension and the bending–twisting laminates. The investigation showed the mode II-dominated shear fracture for all tested laminates. In addition, for both specimens delamination surfaces exhibited a very rough fracture plane with the characteristic hackles attributed to the mode II delamination which were created by coalescence of numerous, small, tensile microcracks that formed between the fibers under the shear loading. The hackles were quite varied over the fracture surfaces on the BE and BT specimens which could be caused by the actual percentage of the local mode I versus mode II or mode III load conditions, the amount of resin between the fibers and the orientation of those fibers. Smooth fracture surface observed on the BE laminate delamination plane was dominated by matrix failure. For the specimen with the bending–twisting coupling some evidence of the mode I tension fracture could be seen. It might be caused by \(D_{{16}}\) component of the bending stiffness matrix which produced twisting curvature around central axis and—in consequence tensile load contribution. On the other hand, non-zero components in the coupling stiffness matrix for the bending–extension laminate caused shearing deformation which was shown in Fig. 7. Moreover, the hackles observed on delamination plane of the BE specimen were perpendicular to the fibers and not to the crack propagation direction. This would suggest a presence of the local mode III fracture component. The strain energy release rates obtained during the experiments were slightly greater for the bending–extension specimen than for the bending–twisting laminate. It could be explained by contribution of the mode III that correlated with greater fracture toughness energy and showed material’s ability to absorb more energy under the mode II loading. Although some fibers breakage were observed, its contribution to the total fracture work was deemed minor. Moreover, for all coupled specimens an interlaminar delamination was observed which was desirable in the experimental point of view.

Experimental ENF tests were prepared according to the ASTM D7905 Standard on multidirectional laminates with different delamination interfaces and laminates exhibiting the bending–extension and the bending–twisting elastic couplings. Critical strain energy release rates were calculated by using standardized data reduction methods and the compliance-based beam method. Results showed that the fracture toughness values obtained for the classical data reduction schemes were lower than for the CBBM method. It might be explained by effect of the fracture process zone. In fact, even if accurate observations of the crack position were achieved the corrected beam theory and the direct beam theory fail to take into account the FPZ. Moreover, elastic coupling phenomena can introduce errors in the mode II strain energy release rate determination. The non-zero terms in the coupling stiffness matrix caused transverse shearing of the BE laminate. In addition, elastic coupling phenomena influenced on curved delamination front which was subsequent source of errors during experimental determination of the critical strain energy release rate. Fractographic analysis prepared after the ENF tests on coupled laminates confirmed contribution of the opening mode I and the transverse mode III components during delamination process.

The acoustic emission method turned out to be a very helpful technique in recognizing delamination initiation. The AE point corresponding to the first sharp growth of cumulative count was taken into account as delamination onset. Moreover, the AE investigations revealed that delamination initiated before the maximum point on the load–displacement plot without any observable crack propagation. Thus, effect of the fracture process zone was confirmed.

To sum up, applicability of the end-notched flexure test to carbon/epoxy composite laminates exhibiting elastic couplings can be prepared in the case of measurements of the total fracture toughness. The acoustic emission technique and additional data reduction method allowed to precise determination of the total delamination resistance of laminates subjected to the bending load. On the other hand, percentage contribution of the opening mode I and transverse shear mode III in the ENF test should be recognized by using an additional method, i.e., numerical analysis, when pure mode II is determined.