CHAPTER 6. DISCUSSION

In this chapter, the experimental results obtained in this thesis have been analysed and discussed with a global perspective. Three aspects of the fracture and fatigue behaviour have been considered: static residual strength, infinite fatigue life and finite fatigue life.

6.1 Residual strength diagrams

Kitagawa-Takahashi diagram can be extended to the static failure occurring at one cycle. In this failure diagram for static loading, the limit in the MSC regime is taken as the tensile strength, σ_t, and the prediction for LC regime is calculated with the fracture toughness, K_IC, being the allowable nominal stress given by:

$$\sigma =\frac{K_{Ic}}{\sqrt{\pi a}}\mathrm{ (6-1)}$$

El Haddad model is also suitable for this static case, introducing the characteristic length, $I_0^{\mathrm{s}}$, for the static response:

$${\sigma }_{Hd}=\frac{K_{IC}}{\sqrt{\pi (a+I_0^s)}}\mathrm{ (6-2)}$$

where:

$$I_0^s=\frac{1}{\pi }{\left(\frac{K_{IC}}{{\sigma }_t}\right)}^2\mathrm{ (6-3)}$$

Figure 6-1 shows the residual strength curves, in continuous line, and the modified El Haddad model, in dashed line, of SLS PA-12 at 0° and at 90° orientations and of IM PA-12.

Experimental data from tensile tests and fracture tests presented in sections 5.5 and 5.6, respectively, were also included. The former was assigned a defect of equal size to the surface roughness shown in section 5.2. The diagrams were completed with the results obtained from monotonic loading tests carried out in specimens with small and medium-size crack lengths in the range between 0.2 and 3 mm for SLS at 0° orientation and for IM PA-12.

To get an easier visual comparison of the materials, Figure 6-2 displays the three residual strength curves for static loading all together. SLS PA-12 in the MSC regime shows higher allowable stresses, due to their higher tensile strengths, but the three materials present similar behaviours at LC regime. Differences in l_o^s for each material are also distinguished.

The characteristic static lengths $I_0^{\mathrm{s}}$ are shown in Table 6-1. As it can be seen, both SLS PA-12 orientations have a similar $I_0^{\mathrm{s}}$ value, which is clearly smaller than that of IM PA-12.

A normalized diagram may help to unify all the results corresponding to different materials in just one curve. Figure 6-3 contains the same information as the three curves presented in Figure 6-1 but, in this case, the Y-axis was normalized by the tensile stress of each material and the X-axis was also normalized by the $I_0^{\mathrm{s}}$ values contained in Table 6-1.

Figure 6-3 displays the prediction capabilities of the empirical models as the modified Kitagawa-Takahashi and El Haddad regarding the static strength of the PA-12 manufactured by different techniques. The diagram is a prediction of the frontier between the integrity and the failure regimes and, consequently, the modified Kitagawa-Takahashi is too conservative. The modified El Haddad model provides better predictions as the experimental data tend to fall on the model curves.

6.2 Infinite fatigue lifetime

The Kitagawa Takahashi diagrams including the predictions of El Haddad model for infinite fatigue lifetime have been constructed from the values of fatigue limit Δσ_fl at 10⁶ cycles and crack propagation threshold $\mathrm{\Delta }{\sqrt{G}}_{th}$. As the stress range for infinite fatigue lifetime is calculated from the threshold stress intensity factor, ΔK_th (eq. (2-4)), all the data expressed in terms of the control parameter $\mathrm{\Delta }\sqrt{G}$ were transformed using the following expression:

$$\mathrm{\Delta }K=\mathrm{\Delta }\sqrt{G}\sqrt{\frac{2E}{B\frac{dg}{d\alpha }}f\left(\alpha \right)}\mathrm{ (6-4)}$$

Figure 6-4 displays the Kitagawa-Takahashi diagrams of SLS and IM PA-12 specimens. Experimental data from $\mathrm{\Delta }\sqrt{\mathrm{G}}$-decreasing Fatigue Crack Propagation tests (FCP Results) have been depicted with filled orange circles, and the results employed for the calculation of the fatigue limit have been also drawn, the run-outs as diamonds and the failures as crosses. Despite the fatigue life tests were carried out in plain specimens, a defect size was assigned equal to the surface roughness measured and displayed in section 5.2. For SLS PA-12 at 0° orientation (Figure 6-4.a) and for the IM PA-12 (Figure 6-4.c), fatigue tests on specimens with small cracks were included.

A characteristic length, I₀, can be obtained from the intersection of MSCand LC estimations (equation (2-6)) in the Kitagawa-Takahashi diagrams. This length can be considered as the limiting dimension of a subcritical defect, above which the long crack ΔK_th becomes the control parameter. Table 6-2 shows the values of I₀ for the materials under study. While there are no differences between the characteristic lengths of SLS PA-12 at 0° and 90° orientations, the value of IM PA-12 is much smaller. Comparing the static characteristic length, ${\mathit{I}}_0^s$, (Table 6-1) and I₀ (Table 6-2), the latter is always smaller, ranging the difference from 25% in SLS PA-12 up to 300% in IM PA-12.

Figure 6-5 displays the Kitagawa-Takahashi diagrams and El Haddad model. As observed, the behaviour of the three materials differ, being IM PA-12 the most resistant in the MSC regime.

This could be accounted for the high number of defects in form of pores and unmolten particles present in SLS PA-12. On the other hand, in the LC regime, the trend is the opposite, as both SLS orientations undergo a better performance than IM PA-12. This may be linked to microstructural features. The spherulite size of SLS PA-12 is 4 times bigger than that of IM PA-12 (Table 5-3), and this could be the reason behind the higher long crack threshold values of the former.

The normalized diagram of Figure 6-6 displays the complete behaviour of the PA-12 manufactured by different techniques. The stress range has been normalized by the corresponding fatigue limit values at 10⁶ cycles for each material and the crack length by their characteristic length.

El Haddad model predictions are more accurate than those of the Kitagawa-Takahashi, especially for SLS PA-12. Nevertheless, for IM PA-12, El Haddad model does not seem to describe the PSC regime as some failures fall within the integrity zone.

6.3 Finite Fatigue Lifetime

There are several authors (Larsen et al. [180], Ciavarella et al. [181]) who have included in the Kitagawa-Takahashi diagrams the residual strength curves and information about the finite fatigue lifetime. Figure 6-7, Figure 6-8 and Figure 6-9 display the Kitagawa-Takahashi diagrams (continuous line) and El Haddad model predictions (dashed line) for static and infinite fatigue lifetime of SLS PA-12 at 0° and 90°orientations, and of IM PA-12, respectively.

Red colour is used to represent the static boundary conditions and the infinite fatigue lifetime boundary below which no crack propagates is displayed in green. Between these two limits, finite fatigue life can be predicted by means of a combination of the lifetime fatigue curves and the crack growth propagation curves. In fact, in the LC regime, the Fatigue Crack Propagation curves obtained from $\mathrm{\Delta }\sqrt{G}$-increasing tests (FCP Results) have been plotted in blue dotted lines.

With the aim of comparing the finite fatigue lifetime band, defined as tensile strength to fatigue limit ratio in MSC zone or fracture toughness to long crack threshold in LC regime, Figure 6-10 displays all together the modified Kitagawa-Takahashi diagrams of SLS PA-12 at both orientations and of IM PA-12. In metals, the tensile strength to fatigue limit ratio tends to remain roughly constant, independently of the microstructure [182]. This is not the case for these materials. While this ratio holds constant independently of the orientation in SLS PA-12, it is reduced for IM PA-12. This supports the fact that the band size between these two limiting conditions in the MSC regime seems to be related to the defects population, which are more numerous in SLS PA-12.

On the oher hand, in metals, the fracture toughness and the threshold values are closely related to microstructure [182]. In this study, in the LC regime, the fracture toughnes to long crack threshold values ratio of IM PA-12 was much higher than that of SLS PA-12, being hardly differences between 0° and 90° orientations in the former.

The fracture toughness is dependent on the spherulite size, being higher as the spherulite size is smaller, especially appreciated when the fracture values are expressed in terms of the J-integral (Figure 5-24). Nevertheless, although the threshold propagation values are lower in IM PA-12 and this could be linked to the smaller spherulitic dimension, this topic still needs further investigation.

Ciavarella et al. [181] extended the Kitagawa-Takahashi diagrams for finite life predictions by analogy in the construction of the originals. Therefore, for finite life N, the stress range for each crack length is obtained as the minimum of the stress range resulting from the Basquin law and the integrated form of the Paris law as follows:

$$\mathrm{\Delta }\sigma (N,a)=min\left(\genfrac{}{}{0ex}{}{{\left(\frac{B_B}{N}\right)}^{\frac{1}{m_B}},}{{\left(\left(\frac{n-2}{2}\right)A{\pi }^{\frac{n}{2}}N\right)}^{-\frac{1}{n}}{a}^{\frac{1}{n}\frac{1}{2}}}\right)\mathrm{ (6-5)}$$

In this case, the key issue is the transition from the Basquin dominated regime to the Paris dominated zone, which for a specific number of cycles, N, it is obtained by equating the stress range obtained from the Basquin law and the integrated form of the Paris law. In terms of crack size, I_t, it is given by [181]:

$$I_t\left(N\right)={\left(\left(\frac{n-2}{2}\right)A{\pi }^{\frac{n}{2}}{B}_B^{\frac{n}{m_B}}{N}^{m_B-\frac{n}{m_B}}\right)}^{\frac{2}{2-n}}\mathrm{ (6-6)}$$

As a way of example, Figure 6-11 shows the fatigue maps including finite life predictions for fatigue lifetimes of 10.000 cycles and 100.000 cycles of SLS PA-12 at 0° and 90°orientations and of IM PA-12. Normalized coordinates have been used, that is, the Y-coordinates have been divided by σ_t, and the X-coordinates by $I_0^{\mathrm{s}}$, I_t(N) or I₀ for static, finite life for a given number of cycles N and infinite life cases, respectively. Little differences were observed among the different materials, only that the region of possible crack propagation is smaller for IM PA-12.

6.4 On the physical meaning of I₀

Especially in metals, many authors have tried to give a physical meaning to the characteristic lengths. When the values of I₀ are small, some authors have linked them to microstructural characteristics. El Haddad related them with the crystallographic grain size in steels and titanium alloys [127] and Ciavarella, also in steels, with the distance from the surface to the first microstructural barrier which the crack could found [181]. In other occasions, as in non-metals (building materials, composites or some polymers) or in materials with no microstructure because they are amorphous, the characteristic length can be associated with the size of the damage zone. In this case, the characteristic length can range from only some hundreds of microns to millimetres, being mostly at least one order of magnitude larger than the microstructural features as there can be many factors involved in the creation and growth of the damage.

For the materials under study, the characteristic lengths could be associated neither with any microstructure feature (spherulite size, skin layer dimension, surface roughness, etc) nor with damage zone size. Therefore, further research should be done in this topic to elucidate if the values of I₀ obtained could be related to material properties or failure processes.

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