Modeling of environmentally assisted fatigue crack growth behavior

1 Introduction

The effect of environment on stress corrosion cracking of metals was documented more than a century ago, when cavalry stored rifle cartridges made of brass under saddles (Brown, 1972). Brass, which was cold drawn during manufacturing, as such contains residual stresses, when exposed to horse urine containing ammonia vapor showed signs of metal cracking. The combination of material-environmental-stress systems determines the extent of the effects of environment on crack propagation in metals. Even in ambient air, the detrimental effect of environment on the fatigue strength of metals was first suggested by Haigh (1917) and experimentally reported by Gough and Sopwith (1932a,b) on copper and brass. Wadsworth (1961) showed a comparison of strain life curve in air and vacuum for carbon steel. In vacuum, the materials exhibited longer fatigue life compared to air. Even in air, Bennet (1964) showed that the fatigue life of aluminum improved in dry air compared to humid air. This indicates that the level of humidity in air should also be considered as an environmental parameter. Thus, fatigue limits, stress life, strain life, and fatigue crack growth (FCG) behavior obtained at ambient air needs to be regarded as corrosion/environment-affected fatigue data. Many of the exciting data strongly demonstrate that environment strongly enhances FCG behavior (Gangloff 1990; Garud, 1991; Lee, Vasudevan, & Glinka, 2009; Wei & Gangloff, 1989). This is termed as environmentally assisted cracking (EAC).

2 Environmentally assisted cracking

To date, the evidence indicates that adsorption-based mechanisms, involving weakening of interatomic bonds, dislocation emission, or decohesion, facilitate EAC behavior in many systems (Gangloff 1990; Garud, 1991; Lee et al., 2009; Wei & Gangloff, 1989). These mechanisms are generally believed to oxide products or cause hydrogen embrittlement leading to reduced fatigue strength. In this paper, it is not our intention to describe all these possible mechanisms of EAC but rather to consider environment as a parameter in modeling of environmentally assisted crack growth rate behavior. To consider environment as a parameter, we need to separate environment from mechanical load on FCG behavior. Thus, it is critical to establish a reference or so-called pure fatigue behavior.

Guadladt and Petit (1991), Henaff, Marchal, and Petit (1995), and Petit, Henaff, and Sarrazin-Baudoux (2000, 2003) considered reference FCG behavior at high vacuum in terms of effective stress intensity factor (SIF), ΔK_eff, which utilized crack closure assumption near the threshold region. They proposed that crack-tip embrittlement is due to hydrogen released by the dissociation of previously absorbed water vapor molecules and which are dragged into the cyclically strained material. In the continuous crack growth region, or otherwise considered as the Paris region, they suggested that the crack growth enhancement is induced by adsorption of water vapor molecules that reduce the surface energy. The contribution of environment is evaluated by comparing FCG rates in ambient air versus inert environment. Sadananda and Vasudevan (2001, 2003) and Vasudevan and Sadananda (2009) analyzed FCG data in terms of two-parameter driving force, ΔK and K_max, and characterized FCG behavior using trajectory map concept. They suggested that two parameters, ΔK and K_max, influence FCG rate. When FCG data at constant crack growth rate is transferred on to ΔK versus K_max plot, they form L-shaped curves. Each L-shaped curve corresponds to a constant crack growth rate. The asymptotic values for these L-shaped curves are used to create a trajectory map. It is suggested that for pure fatigue (vacuum), the trajectory shows a slope of 45° and any deviation from this is considered to be the effect of environment. Thus, studying the role of environment on fatigue requires reference data from a “good” vacuum. It can be said that using only environmental data to make life prediction can lead to erroneous interpretation as the data have effects from both mechanical load and environment.

3 Experimental observations

3.1 Macro observations

One of the standard methods to evaluate material resistance to cracking in a given environment is through the slow strain rate test (SSRT). As described in the ASTM standard (G-129, 2006), in a SSRT, the material undergoes monotonic elongation in a corrosive environment. The selection of elongation rate, otherwise called the strain rate, is important and depends on the material/environment system. A very high strain rate will fail to capture the environmental influence, while a very low strain rate would result in unreasonably long testing time. The strain rate that captures the influence of environment within a reasonable testing time is usually found experimentally through trial and error and is called critical strain rate. For most common materials like steel, aluminum, and other alloys, these critical strain rates can be found in the literature (Bradford & Lee, 1978; Lee, Kim, Jeong, & Kim, 2012; Parkins, 1993).

SSRT was performed on aluminum 7075 T651 in laboratory air and in corrosive environment. The chemical composition is provided in Table 1. Dog-bone-shaped specimens were machined such that the specimens are straining in the short-transverse direction. The straining direction is specifically chosen so as to enhance the effect of environment on fracture strain. The setup of the test is shown in Figure 1, where only half of the gauge length of the specimen is immersed into corrosive solution. The corrosive solution is 3.5% NaCl with a pH of 2.5 (acidic). The specimen was strained to failure at a nominal strain rate of 8×10^-7 s^-1. Figure 2 shows stress strain response in air and in corrosive solution. The elongation to failure in air was 16.9%, while in corrosive solution, it was only 3.5%, indicating a 75% decrease in ductility of the material.

Table 1

Chemical composition of Al 7075-T651 (max wt.%).

Si	0.40	Fe	0.50
Cu	1.2–2.0	Mn	0.3
Mg	2.1–2.9	Others, total	0.15
Zn	5.1–6.1	Others, each	0.5
Ti	0.20	Balance	Aluminum
Cr	0.18–0.28

Figure 1:

Schematic of test setup for slow strain rate test.

Figure 2:

Stress-strain curve from a SSRT on Al 7075-T651 in air and in corrosive solution.

3.2 Micro observations

Pippan, Zelger, Gach, Bichler, and Weinhandl (2010) carried out extensive experimental work to understand the effect of environment at low and intermittent crack propagation rate. They suggested that the crack propagation mechanism is the formation of new surface due to crack-tip blunting during loading and resharpening during unloading. This process of crack-tip blunting is affected by the material/environment system. The authors performed experiments on steel and aluminum in 3.5% NaCl and in vacuum (10^-7 torr) and compared the results with those of tests in air. After the tests, the fracture surface was mapped to reproduce the shape of crack tip through stereo-photogrammetric analyses using stereo scanning electron microscope. They reported the precise measurements of crack-tip opening displacement (CTOD), crack extension per cycle (da/dN), and crack tip blunting angle (CTBA) (2θ) (see insert in Figure 3). The result shows that in corrosive environment, the crack tip is sharper than in vacuum; this is evident from the measured CTBA, which is highest for vacuum. The reported crack extension shows accelerated crack growth in corrosive environment compared to vacuum. The trend was reported to be similar for both austenitic 220 steel and 7020 aluminum alloy, as they are presented in Table 2. Table 2 clearly indicates that at the same CTOD, the actual crack growth rate da/dN is a function of CTBA, which in turn depends on environment.

Figure 3:

Schematic of a cycle showing unloading part (1–2) and loading part (2–3), with the reference SIF, K_ref as 2*.

Table 2

CTOD, CTBA, and crack growth rate (da/dN) in vacuum, air, and 3.5% NaCl for austenitic 220 steel and 7020 Al alloy (Pippan et al., 2010).

Material	Environment	CTODa (μm)	CTBA (2θ)	da/dN (μm/c)
Austenitic 220 steel	Vacuum	4.6	~105°	0.5
	Air	4.3	~90°	1.0
	3.5% NaCl	4.7	~60°	1.2
7020 Al alloy	Vacuum	5.0	~130°	0.3
	Air	4.3	~90°	0.7
	3.5% NaCl	4.7	~25°	1.7

^aCTOD was measured 7.5 μm behind a crack tip.

4 FCG model

A FCG model is presented here, which incorporates the influence of environment through CTBA. Following the two-parameter approach (Vasudevan & Sadananda, 1996), incorporating residual stress similar to the Willenborg model (Willenborg, Engle, & Wood, 1971) and small time scale model (Lu & Liu, 2010), a new FCG model is developed. The model predicts FCG rate by integrating the crack extension within a cycle. The model takes into account the crack extension per cycle through reversed plasticity in front of the crack tip, which is influenced by the material-environment system through CTBA.

The model assumes that (1) crack extends only during the loading part of the cycle, from 2 to 3 in Figure 3; (2) the crack does not extend during the unloading part of the cycle, from 1 to 2 in Figure 3; and (3) even during loading, crack extension occurs only after crossing a reference stress point, K_ref, from 2* to 3 in Figure 3. This reference stress identifies the crossing of the forward plastic zone and reversed plastic zone. The forward plastic zone is created during the loading part of the cycle, while the reversed plastic zone is created during the unloading part of the cycle. Rice (1967) suggested that the reversed plastic zone may be estimated using Eq. (1), while the forward plastic zone may be estimated by Eq. (2), as suggested by Dugdale (1961).

and

where σ_y is the yield strength. Equating Eq. (1) and Eq. (2), the reference stress point may be evaluated for R≥0 as in Eq. (3).

The forward plastic zone and reversed plastic zones are considered to develop Eq. (3) and is valid for stress ratios greater than or equal to zero. To include negative stress ratios, material strain hardening and changes in material properties for monotonic loading versus cyclic loading have to be taken into account. A general form of Eq. (3) is given by Kujawski and Lin (2008) in the form of Eq. (4). This equation considers an asymptotic value for very large negative stress ratios, as suggested by Lang (2000).

The crack extension is calculated by integrating between K_ref and K_max, i.e. from 2* to 3 in Figure 3, while considering the CTBA, 2θ. The final form after integration is presented here as Eq. (5), which represents the stable crack growth rate.

To account for the intermittent crack growth rate near threshold and at very high crack growth region, additional nondimensional terms are introduced. Thus, Eq. (5) takes the final form as Eq. (6).

The model has three parameters, threshold SIF, ΔK_TH, CTBA, 2θ, and fracture toughness range, ΔK_C.

5 Model predictions

5.1 Model predictions for air

Using the analytical solution, a model was created using MatLab to predict crack growth curves. Figure 4 shows crack growth curves predicted by the model for Ti-6Al-4V (Yuen et al., 1974) for stress ratios ranging from -5 to 0.8. The model shows fairly good correlation with experimental data. The CTBA, 2θ, increases with increasing stress ratio, with an attendant decrease in the threshold stress intensity range. The fracture toughness range, ΔK_C, was evaluated from fracture toughness of the material as ΔK_C=K_IC (1-R). In this model, the threshold intensity range, ΔK_TH, and the fracture toughness values, K_IC, are determined from experimental data, while CTBA is a material/environmental parameter. The model predicted similar results for aluminum, steel, and their alloys (Kirby & Beevers, 1973; Pao, Gao, & Wei, 1985; Salvik, 1993; Tesch, Pippan, & Doker, 2007; Yuen at al., 1974).

Figure 4:

FCG data of Ti-6Al-4V for stress ratio between -5 and 0.8. Open symbols indicate experimental data, while solid symbols represents model predictions (Yuen, Hopkins, Levernt, & Rau, 1974).

5.2 Model predictions for vacuum

Figure 5 shows experimental data for three stress ratios for Al 7075-T7531 (Kirby & Beevers, 1973) in air (closed symbols) and in vacuum (open symbols). Figure 6 shows the model predictions by solid lines; the results indicate that the model captures the influence of stress ratio in air (Figure 6A) and also the absence of its influence in vacuum (Figure 6B). It can be noted that the CTBA, 2θ, in air varies from 82° to 87° with the stress ratio while there is negligible variation in vacuum.

Figure 5:

FCG rate experimental data for Al 7075-T351 (Kirby & Beevers, 1973).

Figure 6:

Model predictions for experimental data in Figure 5 for (A) air and (B) vacuum (Kirby & Beevers, 1973).

5.3 Model predictions for 3.5% NaCl environment

It is widely agreed that FCG rate behavior at the threshold region is predominantly influenced by corrosive environment in comparison to high crack growth rate region. This is due to the fact that at threshold, the environment has ample time to weaken the material compared to damage caused due to mechanical loading. Thus, the two mechanisms influences the FCG curves, where near threshold, the damage is predominantly due to electro-chemical processes, while in the Paris region, mechanical damage is the dominating the FCG mechanism. This is reflected by the model predictions for FCG curve for AA2090-T81 (Salvik, 1993) in 3.5% NaCl (Figure 7A). The model distinguishes the CTBA for both mechanisms along with the threshold stress intensity range, ΔK_TH, and critical fracture toughness range, ΔK_C, while this dual mechanism does not appear to operate in the model predictions for vacuum (Figure 7B).

Figure 7:

FCG rate experimental data (Salvik, 1993) and model predictions for AA2090-T81 in (A) 3.5% NaCl and (B) vacuum.

5.4 Model predictions for water vapor pressure

FCG rate in particular for Al alloys is influenced by the level humidity in air. Pao et al. (1985) performed FCG experiments at different water vapor pressure. The results showed that as water vapor pressure increased, the threshold SIF decreased and the FCG rate increased for a given ΔK. The model prediction shows good agreement with the experimental data, as it is indicated in Figure 8A. The model also predicts the decrease in CTBA as water vapor pressure increases (Figure 8B). That is, the model suggests that the crack tip is sharper in high humid conditions compared to low humidity. On the other hand, ΔK_C values are approximately independent on humidity (Figure 8C).

Figure 8:

FCG experimental data (Pao et al., 1985) and model predictions for Al 7075-T651 for R=0.1, (B) CTBA for various water vapor pressures, (C) fracture toughness for various vapor pressure.

5.5 Model predictions for frequency effect on FCG behavior

The influence of aggressive environment on the FCG rate is governed by test frequency. Gingell and King (1997) performed FCG experiments at different frequencies while exposing the specimen to corrosive environment. The model predictions show a good agreement with experimental data (Figure 9), with higher frequency showing the largest angle.

Figure 9:

FCG experimental data model predictions for Al 7150-T651 for various frequencies in corrosive environment (Gingell & King, 1997).

6 Conclusions

An experimentally based FCG model is proposed that includes the role of environment. This model has three parameters: fracture toughness range (ΔK_C), threshold stress intensity range (ΔK_TH), and CTBA (2θ). ΔK_TH is dependent on stress ratio and environment, while ΔK_C is independent of environment and CTBA is associated with material/environment system. The model predictions for air showed the effect of stress ratio on FCG curves. FCG predictions for AA2090-T81 in 3.5% NaCl showed the presence of two mechanisms, which depend on the ΔK range applied. Varying water vapor pressure, the model successfully predicted the blunting of the crack tip with decreased humidity. Thus, the proposed model predicts the FCG rate at different material/environment systems through the CTBA parameter.