The main challenge when analyzing the delamination is how it develops across the laminate. It is difficult to track and also establish methods for that. The first step is the knowledge of the fracture modes. Once this is known, it is possible to define the proper tests to calculate important parameters as fracture toughness G and, mainly, the crack length. This is article proposes a brief review about those methods.

Fracture modes

The delamination has three fracture modes, the tensile, the shear and the tear ones. The first occurs when the loads are vertically oriented with respect to the plies. The shear is defined by the longitudinal loads, while the tear is based in transversal loads applies to the crack front. These two modes are rather similar with respect to the results on the laminate, which is the the shear of the crack front. However, the tear fracture mode has no standard for testing, because it would be a quite difficult arrange to emulate those conditions. Hence, considering that and the similarities with the shear mode, the tear one is usually assumed as shear mode. The fracture modes of tensile, shear and tear have fracture toughness referred as G_IC, G_IIC and G_IIIC, respectively. The combination of these defines the fracture toughness of the laminate G_C. G_IIC and G_IIIC are quite different since the first is caused by the pulling of the laminate, while the shear fracture toughness is caused by longitudinal loads. When G_IIC and G_IIIC are compared, the results are quite similar, because both are based on the shear of the crack front. Therefore, when accounting G_C, its value depends on how much the tensile and shear loads are applied on the laminate.

Mode 1: Double cantilever beam test

C = δ/P → C = 2a³/(3E_f1I) → G_I = (P²/2w)(dC/da) = P²a²/(wE_f1I)

The evaluation of the tensile fracture mode is performed by the double cantilever beam test. This is based in a plate at a cantilever in an extremity, while the other is under pulling forces that tend to separate the laminate. The test simulates a incrusted plate exposed to a load P at the tip that results in a crack displacement δ. Regarding composite materials, this load is applied also at the region of the crack. More precisely, at its starting crack. Actually, this term refers to a laminate that is suffering delamination in some plastic layers. These are characterized by the poor adhesion with the epoxy matrix. Since the test measures the load P and the displacement δ, it is possible to calculate the compliance C.

However, for this test, C has a different definition. It is based in the crack length a, the flexural Young’s modulus at longitudinal direction E_f1 and the moment of inertia I. In other words, this formulation is based on the beam theory. The reason is that, this test is based in cantilever beam. The problem of this test is not how to estimate E_f1. Actually, it is E_f1 itself. This occurs because the test is done in bending, which has a loading condition that result in tension and compression stresses, while the shear ones are quite absent for in-plane loading. Then, E_f1 is not the in-plane Young’s modulus for the most general cases. Its value can be lower than the in-plane Young’s modulus, thus using E_f1 can result in significant errors.

The strain energy release rate formula is defined by substituting and deriving the compliance C with respect to the crack length. Hence, the final form has two problems, the already mentioned E_f1 and a. The crack length is very time consuming to be calculated. However, there are three possible solutions for those issues. These are relationships between C and a that can be used for perfect cantilever beam cases. They are proposed by standards since the real applications are based in bonded structures made from composites, instead of incrusted ones. This means that, the fixtures have significant degree of flexibility, while the cantilever is rigid. Hence, the fact that this test assumes incrusted structures reduces the precision of the results. The flexibility in real joints works as a lower bound of the compliance.

G_I = 3Pδ/(2b(a + |∆|)) , C ∝ (a + |∆|)³

G_I = nPδ/2ba , C ∝ aⁿ

G_I = 3P²C^2/3/(2A∙b∙h) , C ∝ (a/h)³

The method to work out this issue are proposed by standards. These are the modified beam theory (MDT), the compliance calibration (CC) and the modified compliance calibration (MCC). Basically, those tests deliver very similar results. In any case, those are based in the fracture toughness test in order to measure the compliance C. This is the parameter used to calculate the crack length a. Once many samples are measured, it is possible to correlate C and a in a plot in order to observe a trend. From this step, one of those methods are applied. Considering the choose of MDT, each C-a point in the plot is connected by a straigth line. When this is extended until the ordinate axis (a-axis), it is possible to measure the variation of the crack length, which is given by ∆. MDT also proposes a correlation between ∆ and a in order to account the imperfection of the incrust in the cantilever. In other words, it is a solution to consider the flexibility of real cases. Therefore, C, ∆ and a are obtained and used to calculate the strain energy release rate G_I.

The arrange of the double cantilever test can be of two types, the piano hinges or the loading block. These refers to type of fixtures used to incrust the beam. Its selection depends on the level of loading. Usually, higher loads require the loading block arrange. Both arranges deliver the same results. These are plots of P and δ, the load and the displacement between the two plies, respectively. This graph illustrates that, the peak of load is observed just at the beginning. At this point, it is possible to spot the initial crack length a0. This region is also characterized by the transition from a linear to a non-linear relation between P and δ and, mainly, the point of visual onset. This is the point that, if the structure was observed in a microscope, then it would be possible to observe the crack tip. Hence, if it is possible to observe the crack advancing, this can be considered as the point of visual onset. Therefore, the peak of the load basically corresponds to the crack propagation.

When the crack length begins to increase, the displacement δ follows. Conversely, the load P decreases with respect to a. This occurs as observed on the definition of G, that when the laminate is under propagation, it is equal to the fracture toughness (crack resistance). This parameter is fairly constant with respect to the crack length. The reason is due to the phenomena that occurs with fibers and non-linear effects. The first, states that fibers are linked to the surface, while the non-linear effects result in some increasing of G_IC during propagation. Those effects justify the decrease of P as the crack propagates. Interestingly, the propagation from initial to final crack lengths exhibits a parabolic pattern. If the fracture toughness was not constant with respect to a, this curve would be a constant value.

Another graph that can be built is G_IC-a one. It indicates the fracture toughness variation during the test. It illustrates how constant is the fracture toughness along the propagation. In addition, it better illustrates the different points at the initial points of the propagation. These are three, the deviation from the linearity, the visual onset and the 5% offset. The first is where the first crack is about to start, then when it happens this becomes a visual onset. Considering again the P-δ graph, if it is drawn a line that goes until the point of 5% offset, this will have a slope 5% lower than the initial line. This also means that, the stiffness of the material reduces by 5%. Hence, while the crack propagates, δ increases, P and the stiffness decrease. These two graphs represent different methods to track the crack propagation.

Mode 2: End notch flexure test (ENF)

C = (3a² + 2L³)/(8E_f1∙b∙h) ; G_II = 9Pδ/4b∙a[1+(a/L)³]

C = (6L+3a-L³/a₂)/(8E_f1∙b∙h) ; G_II = 9P²∙a²(C-C_shear)/4b∙L³[1+1-5(a/L)³]

The end notch flexure test is based in a three point bending arrange. The plate is displaced over two fixtures and a load P is applied at the middle of it. The dimensions of the plate are given by L, a, b and 2h, which represent the half length, the width, the crack length and the total thickness. As in the previous test, this one also require a previous calculation of the compliance C in order to evaluate the crack length. The standard also states propert formulas for compliance C and the strain energy release rate G. The main problem of this type of test is its stability. The three point bending arrange is unstable along the test due to the following relation, a ≤ L³/√3. For this reason, this test do not register the fracture toughness with respect to the crack length. It only delivers the value of G_IIC at the crack initiation. Since the three point bending also result in some shear stress, it can be accounted in the G calculation by the compliance C_shear.

Mixed mode 1-2: Mixed-mode bending (ASTM D 6671)

The mixed mode bending test is a sort of combination between the double cantilever and the end notch flexure test. The specimen is similar to the ones from those tests. However, the arrange at which it is submitted is a three point bending one with a piano hinge as one of the fixtures. In addition, there is an optional position for an extra weight. As a result, the test device combine an incrusted plate with three point bending, which allows to obtain combined results from double cantilever and the end notch flexure test. The standard proposes that, these have to be summed in order to obtain the final fracture toughness.

G_I = [12P²(3c – L)²/(16b²h³L²E_f1)]∙(a+χh)²

G_II = [9P²(c + L)²/(16b²h³L²E_f1)]∙(a+0.42χh)²

G = G_I + G_II

(G_II/G) = G_II/(G_I + G_II)

χ ≡ √[E₁₁/11G₁₃]{3-2(Γ/(1+Γ))²} ; Γ = 1.18((E₁₁E₂₂)^1/2/G₁₃)

Actually, this sum is a function of the percentage of G_II. However, these are respective from G_I and G_II obtained from the mixed mode 1-2 test. Hence, the formulas for the fracture toughness G_I and G_II are quite different from the previous tests. Therefore, it is introduced the parameter Γ and χ to account the material properties at directions 1 and 2. In addition, G_I and G_II formulas must be updated when the equipment is fitted with an extra weight.

G_I = [(12P²(3c – L)+Pg(3Cg – L))²/(16b²h³L²E_f1)]∙(a+χh)²

G_II = [(9P²(c + L)+Pg(Cg + L))²/(16b²h³L²E_f1)]∙(a+0.42χh)²

The results are usually given in plot that relate the overall fracture toughness with the percentage of it with respect to the mode 2. Usually, for general cases of composite materials, G_IC and G_IIC represent the lowest and the highest values that can be obtained. Hence, the mixed mode test allows to evaluate the intermediate cases, but it is still difficult to track the crack propagation for the mode 2 cases.

References

• This article is based in the notes taken during the Composite Materials course attended by the author in Muner Advanced Automotive Engineering Master Degree.