Mechanical modeling of damage accumulation and life evaluation for stress corrosion cracking

3.1 Stress-enhanced corrosion damage evolution

We propose here a phenomenological evolution model as follows instead of deriving it from Eq. (1):

where c₀ is the proportional coefficient of natural corrosion (without any loading), and c₁σ^m expresses the enhancing effect of applied stress on corrosion, m is the enhancing exponent, and β is the corrosion damage exponent. The reasons for assuming evolution function as shown in Eq. (2) are as follows: (i) natural corrosion damage accumulation exists too even if there is no applied loading, (ii) an exponent function can express the enhancing effect of stress more extensively, and (iii) 1-D indicates the effective section [S_ef=(1-D)S₀] in material where further damage accumulates, so the damage evolution rate should be dependent on 1-D rather than D.

The time needed for damage that accumulated to a certain value D_e due to corrosion can be estimated from Eq. (2) as

where D₀ is the initial damage that can be always set as zero for metals. For natural corrosion, the final damage D_e can be accumulated near to 1, and at last the surface layer will fall off. That is, uniform weight or thickness loss is the result of natural corrosion, and no macrocracking will happen. Denoting the referenced thickness of surface layer as h (but it is not necessary to be specified in detail) and the life corresponding to D_e=1 as t_e0, the thickness loss rate of natural corrosion can be derived as

whereas, for the case with applied stress, the thickness loss rate becomes

where s_σ is the enhancing coefficient of stress. If s_σ is very small, the stress-enhancing effect can be neglected when the applied stress is not extremely large, so the thickness loss rate is almost the same as that of natural corrosion. However, if s_σ is relatively large, the loss rate may increase with the applied stress. The enhancing coefficient s_σ can be determined simply by corrosion tests with the applied stress smaller than the threshold based on Eq. (5). If the applied stress is large (larger than the threshold stress), the static fatigue damage accumulation mechanism will be activated when corrosion damage is accumulated to a transitional damage value D_s, and further damage evolution is no longer dominated by Eq. (2). The time needed for damage accumulated to D_s by corrosion damage accumulation only can be derived as

or rearranging it in a convenient form as

We call t_in the incubation life because it is the life before the static fatigue mechanism is active. From Eq. (6), it can be seen that the incubation life is almost a constant for not extremely high stress levels if the stress-enhancing effect on corrosion damage accumulation is weak. Although when the effect is strong, the incubation life may decrease with the increase of applied stress.

3.2 Corrosion-enhanced static fatigue damage evolution

After the damage has been accumulated to the transitional damage D_s, further damage accumulation must take corrosion-enhanced static fatigue damage into account.

3.3 Total SCC life

The total SCC life t_SCC is the sum of incubation and static fatigue lives

Usually, the corrosion-enhanced static fatigue mechanism becomes active at early stage if the applied stress is larger than the threshold. That is, usually t_f>>t_in, so t_SCC≈t_f. However, if the applied stress is extremely large, such as that at a crack front, the static fatigue life may become very short, even shorter than the incubation life, which is nearly a constant as explained before. For such a case, the incubation life of internal layers cannot be neglected and may be even the main part of total SCC life.

4.1 SCC S-t curve with a stress threshold

Figure 2 shows the comparison of Eq. (8) to experimental results (Wu et al., 2004) for Al-Li alloys. The experiments were conducted in a simulated seawater environment. Lives were counted by the times to failure of smooth specimens, not by the time that an SCC crack has been observed. It can be seen that Eq. (8) can well characterize the whole S-t curve of SCC. In Figure 2, various transitional damages have been tried in the computation of Eq. (8). It is found that the effects of C_f and D_s on SCC S-t curve are coupled. By adjusting C_f, which should be determined by experiments at last, any transitional damage can be used in calculation. This fact means that it might be not necessary to specify the detailed transitional damage, and one can always set it as zero if the proportional life coefficient is determined by fitting Eq. (8) with experiments. On the contrary, the concept of transitional damage is still necessary, otherwise the reasons for the static fatigue mechanism becoming active and incubation behavior cannot be explained.

Figure 2:

Comparisons of life evaluation formula with experiments.

With the life evaluation formula Eq. (8), experiments needed to obtain the SCC properties of material can be reduced greatly. However, applying Eq. (8) to evaluate SCC life implies that the incubation life can be neglected in total SCC life. This is true only when the static fatigue life is relatively long (i.e. the stress is not extremely high). For relatively high stress, lives obtained by experiments should be compared to Eq. (8) by reducing a certain value as t_SCC-t_in. By fitting experimental t_SCC-t_in with theoretical S-t curve, t_in can also be determined as a constant by the test-and-error method. For example, for test results of 2091-T8X shown in Figure 2, if the incubation life is assumed as t_in=5 h, then the experimental t_SCC-t_in results can fit theoretical curve of Eq. (8) better as shown in Figure 3. In other words, the incubation life of this material under simulated seawater can be determined as 5 h.

Figure 3:

Determination of incubation life.

4.2 SCC S-t curve without obvious stress threshold

Figure 4 compared Eq. (11) to Kumar’s experimental results (Kumar et al., 2015) for super 304H under 45% MgCl₂ boiling at 155°C. Experimental lives were also counted by the times to failure of smooth specimens (i.e. lives for general corrosion). The enhancing effect of stress on corrosion has been neglected, and other coefficients used in calculating lives have been shown in the Figure 4. It can be seen that the life formula Eq. (11) can well estimate SCC life. It is noted that, to determine the property coefficients, some techniques may be required. The following two methods can be considered: (i) determining static fatigue coefficients through lives at high stress levels (quite larger than the nominal threshold but not extremely large) by Eq. (8) at first, because corrosion damage accumulation is relatively small at these cases, and then determining corrosion coefficients by fitting Eq. (11) to lives at lower stresses (near or below the nominal threshold), and (ii) determining corrosion coefficients through lives at low stress levels (below the nominal threshold) by Eq. (3) at first, where D_e should be taken as D_e=1-σ/σ_SCCf because static fatigue will lead to rapid failure after this damage value, and then determining static fatigue coefficients by fitting Eq. (11) to lives at larger stresses.

Figure 4:

Example of coupled corrosion and static fatigue.

5.1 Is there an SCC threshold?

Because the static fatigue mechanism can always be activated by corrosion damage accumulation even for very low stresses, there may be no SCC threshold theoretically. However, if the proportional coefficient of corrosion damage is small, the SCC S-t curve would show threshold behavior phenomenally. Figure 5 illustrates the effect of the proportional coefficient of corrosion damage on S-t curve. Only the proportional coefficient of corrosion damage has been adjusted, and other parameters are just the same as shown in Figure 4. It can be seen that, when the proportional coefficient is small enough, the threshold behavior seems possible, whereas, if it is large, there would be no threshold behavior. However, even for the small proportional coefficient, the S-t curve will be curved down at longer time. Therefore, the SCC threshold is only a concept depending on the tested life range, and it does not mean that the life can become infinite (but can be long enough).

Figure 5:

Shapes of SCC S-t curve.

5.2 Equivalent stress for SCC under complicated stress state

An engineering structure usually would be subjected to multiaxial stresses rather than a uniaxial loading. How to apply uniaxial SCC test results or evaluation models to complicated stress states is of great interest in application. Obviously, if an equivalent stress can be introduced, the life evaluation for a complicated stress state can be carried out in the same way. The problem is: what should an equivalent stress be in SCC problems? At first, in the stage of stress-enhanced corrosion, from the view of Gibbs free energy in which strain energy is involved, the effect of complicated stress state must be reflected by the strain energy density. Therefore, by the equivalence of strain energy under complicated stress state to that under uniaxial state, the definition of equivalent stress can be derived. That is, let

where E is the Young’s modulus and σ_eq denotes the equivalent stress for multiaxial stresses σ_ij. Assuming that the material is isotropic linear elastic, from Eq. (13), one gets

where ν is the Poisson’s ratio and I₁ and I₂ are the first and second stress invariants, that is,

On the contrary, in the stage of static fatigue, we have found that the strain-energy-equivalent stress can be used to estimate static fatigue life by the comparison of test results under multiaxial and uniaxial stress states (Cai & Xu, 2016). Therefore, Eq. (14) can always be used as the equivalent stress to evaluate SCC life under multiaxial stress states. However, the critical damage D_C should be determined by the maximum principle stress σ₁ as

because the instantaneous fracture condition is dominated by the quasi-brittle fracture criterion σ₁/(1-D_C)=σ_b. In other words, damage accumulation is dominated by the equivalent stress, but the final fracture condition is dominated by the maximum principle stress.

With the equivalent stress shown in Eq. (14), the threshold condition under multiaxial stress state can be simply expressed as

For example, for pure shear stress state, it leads to

which means

Therefore, the shear stress threshold τ_SCC is not an independent property and can be determined from the axial stress threshold. For a combined axial and shear stress state, Eq. (17) leads to

With the use of Eq. (18b), it can be rearranged as

That is, the elliptical formula in fact has a theoretical foundation.

5.3 SCC crack growth rate curve

SCC crack growth rate curve is a very important property in SCC life evaluation of engineering structures (Kalnaus et al., 2011; Ramamurthy & Atrens, 2013). It is well known that the curve would show a plateau region at which the growth rate is kept constant, which is independent of the magnitude of the applied stress intensity factor (SIF). Why is there a plateau region and what affects the constant growth rate? From the model introduced above, these questions can be understood clearly.

Considering the SCC crack growth as a fatigue process of material within small lengths at crack front step-by-step as shown in Figure 6, denoting the SCC life of first length as t_1f, the growth rate can be expressed as

Figure 6:

Considering crack growth as a process of small lengths fatigued step-by-step.

where t_1f is composed of incubation and static fatigue lives as shown in Eq. (12), and the small length L_f must be taken properly as will be explained later. It should be noted that SCC crack propagation is in fact a localized corrosion phenomenon at crack front, and thereby, the SCC life of crack front may be different from that for overall/general corrosion. Here, we just use the general SCC life approximately for small length L_f to investigate the general features of SCC crack propagation. Due to the singular stress at crack front, the effective steady stress at crack front is very large, so the static fatigue life may become very short, even much shorter than the incubation life. For such a case, the growth rate mainly depends on incubation life only, i.e.

If the stress-enhancing effect on corrosion is weak, incubation life t_in is nearly a constant, so the growth rate becomes nearly a constant too. This is the reason why there is a plateau region. To explain the general features of SCC crack growth rate curve comprehensively, we try to calculate it from Eq. (21). It should be noted that a crack front is always under multiaxial stress state. Therefore, to consider the SCC life of crack front, an equivalent stress should be introduced at first. According to Eq. (14), from the singular stress field at a mode I crack tip (σx=σy=KI/2πr), it can be derived as

where K_I is the mode I SIF and F is a material constant related to stress state as

Although the equivalent stress is also singular, it does not lead to instantaneous fracture immediately. To calculate the SCC life of crack front, average equivalent stress within the length L_f might be a reasonable choice because the fatigue of the whole region within L_f should be considered:

The SCC stress threshold condition can then be expressed as

On the contrary, the threshold condition of crack propagation is

where K_ISCC is the SCC threshold of crack growth. It should be noted that K_ISCC is a macroproperty measured by macrocrack propagation. For small or short cracks, similar to that of threshold SIF range for common fatigue crack propagation (Tanaka et al., 1981), threshold SIF might be dependent on the crack size. These two equations expressed the same threshold condition of crack front. Substituting Eq. (27) into Eq. (26) yields

That is, to consider the SCC life of crack front, the length cannot be taken arbitrary, and it should be taken as a constant as shown above. With such a specified length, the average equivalent stress at crack front can be calculated in detail. The incubation and static fatigue lives can now be calculated from Eqs. (6b) and (8) as

The crack growth rate can be calculated from Eq. (21). To illustrate the general features of the crack growth rate curve, assuming incubation life formula as (1+4.8×10^-61σ²⁰)t_in=800 s and static fatigue life formula as σ^2.8t_f=3.9×10⁷I(σ), ζ=8, the normalized growth rate curve calculated from Eq. (21) is shown in Figure 7. Obviously, it shows just the typical features of SCC crack growth rate curves. The lower part shows that the growth rate increases with applied SIF as the static fatigue life of crack front decreases, whereas the upper part beyond the plateau is induced by an extremely high average stress level, which makes the incubation life decreasing rapidly too even if the stress-enhancing coefficient is small. This example shows that the general features of SCC crack growth behavior can be well explained by the proposed life evaluation model. However, it must be noted that it is still difficult to obtain the exact propagation curve because the SCC life of small region L_f may be different from that obtained by an overall evaluation model, which is developed for relatively large specimens. In other words, the above discussion is helpful only for understanding the general features of the SCC crack propagation curve.

Figure 7:

Calculated propagation curve.