The delamination is the main problem in race car chassis. Even though it is pretty well known, its analysis is quite difficult. The reason is that, this is a problem that occurs inside the laminate and when it comes out, it is in its last stages. In addition, the methods for crack tracking are different from the usual stress intensity factor (SIF). Actually, the fracture mechanics of composite materials is more based in the energy approach to fracture. This article proposes a brief comment about the fracture mechanics of a composite material under delamination.

Fracture mechanics

The delamination is an internal problem of composite materials, it is hard to track it even with ultrasonic transducer. Hence, the resistance to crack propagation is very difficult to be defined. Its analysis is called, fracture mechanics. The objective is to understand how a crack propagates. The trigger is already known, it can be due to manufacturing process or impacts, then the structure fails by fatigue along the life cycle. This is the critical point since the stress at the interface have an oscillatory behaviour. In addition, the stress parameter loses its meaning, because it is theoretically infinite. The fracture mechanics propose the stress intensity factor (SIF) in order to track the delamination. However, this one is not advisable since it is based on the stress. Hence, it is used a more general parameter, the strain energy release rate G. This approach is also difficult since it is required the identification of the fracture toughness (crack resistance). This parameter defines the delamination extension. Then, it is possible to evaluate the mixed mode of loading, which depends on how G and fracture toughness vary with respect to the mode of loading. Therefore, the solution to analyze the crack propagation is by the stress release rate G, but it is necessary a detailed approach for that. This is called, energy approach to fracture.

Energy approach to fracture

For the delamination analysis, it is applied the theory of Griffth-Orowan-Irwin. This is based in the energy balance of a block of material which is exposed to a load P and a deformation ∆. The block has thickness B and an internal crack. This is linear, symmetric and pass-through. In other words, its side lengths are equal and it extends through the entire thickness. Then the length is 2a, while each side is given by a. The Griffth-Orowan-Irwin theory states that, the total energy of the system is given by the sum of the potential energy (∏), the kinect energy (KE) and the energy to create a new surface (WS). In addition, it also states that, the time derivative of the total energy of the system is zero.

E” = ∏” + W_S” + K_E“

d/dt = (∂A/∂t)∙(∂A/∂A) = A”(∂/∂A)

Assuming that the system is steady, the objective now is to convert the time derivative based system into an area derivative based system. Once the system is steady, the kinect energy is null (KE” = 0). Hence, the system is now relying on ∏” and WS”. The potential energy summed with the energy to create new cracks is defined as the Young’s modulus of the system for an increase of the crack surface. This is a parameter that must be controlled, thus it must be analyzed the variation on ∏” and WS”. The change in potential energy is necessary to absorb the energy when the crack surface increase. In other words, ∏” reduces while new crack surfaces are created. The general definition of the potential energy is the elastic energy of the material, which is a sort of internal energy, and the work produced due to external forces. Understanding that the system is a material block exposed to an internal crack, this definition is modified.

∏ = U – L

∏ = ∏₀ – (πσ²a/E)

W_S = 2∙A∙γ_S = 4∙a∙B∙γ_S

The potential energy becomes the initial potential energy (∏₀) subtracted by the energy due to the presence of a defect. This is given by W_S, which is twice the value of the crack surface times the surface tension. Since the thickness is B and the length of the crack is 2a, the crack energy is again multiplied by 2. The surface tension parameter γ_S is necessary, because to break a drop, it is necessary some work.

∂E/∂A = ∂∏/∂A + ∂W_S/∂A = 0

G = – d∏/dA = πσ²a/E

R = dWS/dA = G_C = 2γ_S

G = (1/B)∙(dU/da)_P = – (1/B)∙(dU/da)_∆

The next step is to convert the total energy of the system into a crack length derivative based system. The derivative of the potential energy with respect to the area is called, crack driving force (G). Actualy, this parameter is not exactly a force, instead, it is a specific energy since its unit is Joule per squared meters (J/m²). Since these are is given by the thickness times the crack length, the first can be out of the derivative because B is constant. Hence, the crack driving force, or strain energy release rate of the crack, is obtained as a function of the crack length. The product of B and da is the increment of the crack and U is the strain energy of the body. The resistance is given by R. If it is substituted the value of WS inside it, the resistance becomes twice the surface tension γ_S. Then, it becomes G_C = 2γ_S, because it is dealing with two surfaces. It is possible to notice that, G can assume two alternative forms. One assumes a constant load P and a variable deformation ∆. The other form assumes that, the load P is the variable parameter, while the deformation ∆ is constant. This is an important step of the energy approach to fracture, because it allows to elaborate tests with more parameters.

dU = Pd∆/2

dL = Pd∆

d∏ = dU – dL = – pd∆ = – dU

G = – d∏/Bda = dU/Bda = Pd∆/2Bda

In the first form it is considered a material block with a crack inside and exposed to constant load P. If when the block is exposed to P the crack reaches the conditions to increase in size, this occurs with no load variation. Hence, applying the potential energy definition allows to calculate the work. This is given by P∙d∆ and is the area inside the graph. The elastic energy follows the same principle. At the beginning, the crack was “a” at a load P. The following conditions is a crack defined by “a + da”. The area inside this variation is the elastic energy into the material. Therefore, the variation in the potential energy is just the negative in sign of the elastic internal energy due to the crack. This principle can be used to update the strain energy release rate definition G. Its final form is a function of the crack increase (L = P∙d∆) with the crack length “a”, because P and B are constant parameters. The first form is called, Load Control form.

dU = ∆dP/2

∆L = 0

d∏ = dU – dL = dU

G = – d∏/Bda = – dU/Bda = – ∆dP/2Bda

The second form proposes the same material block, but this is tightly fixed so that the material do not perform any work. This can be assumed that, the material block is not able to have a crack increase with respect to its size. Hence, the crack increase dL is zero. Then, the definition of the potential energy becomes equal to the elastic internal energy. In this mode, when the conditions for the crack elongation is reached, the load decreases. Hence, the strain energy release rate formula can be updated. Now its final form is a function of the load P and the crack length da, because ∆ and B are now the constant parameters. It is possible to notice that, as the crack length increases, the load and the strain energy decrease, which is the reason why their derivatives are negative. This form is called, Displacement Control form.

G = – d∏/Bda = dU/Bda = Pd∆/2Bda → ∆ = C∙P → G = (P²/2B)(dC/da) → Load Control Form

G = – d∏/Bda = dU/Bda = Pd∆/2Bda → ∆ = C∙P → G = (∆/2B)(d∆C^-1/da) = (∆²/2B)(1/C²)(dC/da) → Displacement Control Form

Actually, there is no significative difference between these methods. The only difference is that, G is given by a derivative form of ∆ and P for load and displacement control forms, respectively. The last step of this process is to make those forms more convenient. Taking advantage of the linear relation between displacement and load, it is possible to define ∆ = C∙P. Where C is the compliance of the system. Interestingly, substituting ∆ into the load and displacement control forms it is obtained the same formulas for both cases. At these ones, the compliance varies with respect to the crack length. In other words, the strain energy release rate G is given by a derivative of the crack length. The reason of this approach is due to C and a parameters are easily measured experimentally. If the applied load P and the open along the load line are measured, thus C can be calculated. Therefore, the strain energy release rate calculation is independent of the forcing mode. Actually, it is defined by the change of the compliance with respect to the crack length. Those methods are very useful for experimental analysis.

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.