Different steels were used to measure typical inclusion size to be used to build the FEA model. The steels used in this metallographic study are presented in Table 1 for reference purpose. All these steels are commonly used in general industrial applications. Steel A has been subjected to rare earth inclusion shape control practice and presents a microstructure with fine and rounded inclusions as shown in Figure 1. Steels B to J are spring steels and present typical elongated manganese sulfide inclusions as seen on Figure 2.

Metallographic was used to characterize the inclusions. Polished and lightly etched specimens were used. A Clemex image analyzer linked to a Nikon Ephiphot optical microscope has been used to evaluate the morphology and distribution of inclusions. Inclusions in the transverse direction were considered due to their elongated shape. The stress concentration factors at inclusions have been calculated using the following formula [1] :

K t = 1 + 2 c ρ (1)

K_t = stress concentration factor

c = semi-major axis of the inclusion

ρ = radius of the notch

For simplicity, this equation may be written as:

K t = 1 + 2 L t (2)

L = total length of the inclusion

t = thickness of the inclusion

The results are shown in Table 2.

Based on results of micro structural analysis, it was decided to simulate two extreme inclusion morphologies, namely, the round and the elongated shapes. For round a void, the radius was set at 1 µm (refer to Figure 4). For elongated voids, two round voids were combined with their centers set 22 µm apart, thus creating a 24-µm stringer (refer to Figure 5). This void shape resembles the elongated manganese sulfide (MnS) stringers that are commonly observed in steels. This assessment is based on quantitative metallography.

The stress concentration factor is estimated as follows:

K t = σ max σ ∞ (3)

where σ max is the maximum stress calculated with FEA and σ ∞ is the remote stress.

Analytical expressions can be found for the stress concentration factor for the two types of void and, therefore, may be compared to FEA with no-crack configurations. As mentioned earlier, a round void behaves as a circular hole in an infinite medium, for which K_t = 3 [2] . This study found a convergent solution with K_t = 3.03 (a difference of only 1%). For the elongated void shown in Figure 5, the FEA result is K_t = 8.26. Rice [3] proposed the following equation for the stress concentration factor of a flat type void:

K t = 2.43 c ρ (4)

which gives K_t = 8.42 with c = 12 µm and ρ = 1 µm as in Figure 5. Here again the discrepancy between FEA and analytical results is very small (a difference of about only 2%, refer to Table 3).

Simulation of inclusions poses some questions. First, it is well known that typical inclusions in steel are considered weakly bonded to the matrix [4] while, in FEA models, the displacement at an interface is fully compatible between the two materials. Similar questions have been dealt with in FEA simulation of graphite nodules in cast iron [5] . In this particular case, the nodules have been considered to behave like voids because of the weak bond and the low stiffness of the graphite nodules. In this study, inclusions were considered as voids in the matrix. Attributing an extremely low stiffness value to an element of material data is a way to introduce a void in the model where otherwise it would be a steel element, thus allowing the mesh to represent various configurations.

Stress intensity factors were determined using the quadratic element mid-side node re-localized at the quarter position. Such crack elements were located along the path of the crack. The only effective one, in any simulation, is the one located at the crack tip. The stress intensity factor was calculated from [6] , assuming that the state of stress is plane strain:

K = E 4 ( 1 − ν 2 ) ( 2 π L ) ( 4 u B − u C ) (5)

where u_B is the crack opening displacement at the quarter node, u_C is the crack opening displacement at the corner of the element (position C), E is the Young’s Modulus, ν the Poisson’s ratio and L is the element size.

The stress intensity form factor, defined as:

F ( a ) = K I σ ∞ π a (6)

with K_I the SIF in mode I, as calculated from Equation (5), σ ∞ , the remote stress and a, the crack length, was calculated from FEA results.

The stress intensity form factor for round and elongated voids were calculated using Equation (6). The results are presented in Table 4.

With the result of Table 4, we may then plot the graph of the geometrical correction curve against crack length. The results are presented in Figure 6.

As it can be seen from this figure, the F curve exhibits a sudden exponential drop at the first three steps of propagation, after which it stabilizes as the crack behaviour becomes only dependent on the crack length. One particularity of this curve is the relatively high value of F for very short cracks.

For very short cracks, FEA yielded SIF significantly higher than that predicted from LEFM [7] [8] [9] . For example, a typical LEFM calculation [9] gave a form

factor F(a, K_t) of 3.3 at a 0.023 µm crack emerging from a 2 µm diameter round void while the FEA result was 27. For crack lengths above 0.2 µm, however, both methods led to practically the same results. The same observation holds for elongated voids (with K_t = 8.34) for which a form factor of 28 was obtained. Therefore, it seems that, for a very short crack, the radius of the void appears to be the controlling SIF parameter, whereas, beyond 1 µm of propagation, this phenomenon disappears and the crack tip singularity combined with the stress concentration factor takes over.

In view of this unexpectedly steep F function seen at very short cracks, SIF were alternatively estimated from the Hellen-Parks [10] virtual crack extension method. With this method, the J-Integral is first estimated and hence the SIF is derived. These numerical SIF values, either based on the crack tip opening displacement formula of Leslie-Banks [6] or on the potential energy difference of Hellen-Parks, differed by only 8%, thus providing the same estimation whatever the calculation technique.