Basic Classification of Fatigue Analysis: Stress Based and Strain Based Fatigue Analysis: 12

1. INTRODUCTION AND LITERATURE REVIEW

1.3. Basics of Metal Fatigue Analysis

1.3.2. Basic Classification of Fatigue Analysis: Stress Based and Strain Based Fatigue Analysis: 12

Based on consideration of either stress or strain values towards fatigue life estimation, fatigue analysis theories can be broadly classified into either stress-life approach or strain- life approach. The stress⎼life, linear relationship (Figure 13), firstly introduced by Basquin [41] is

∆𝜎 2⁄ = 𝜎_𝑓^′(2𝑁_𝑖)^𝑏 (1) where ∆𝜎 is alternating stress range, 𝜎_𝑓^′ is fatigue strength coefficient, 𝑏 is fatigue strength exponent and 𝑁_𝑖 is fatigue initiation life.

This stress-based fatigue life relationship is more appropriate in cases with low load levels where stresses and strains are linearly dependent. At high load levels i.e., in the low cycle fatigue regime, this relationship is observed to be conservative. In low cycle fatigue (LCF) regime, material behavior can be best modelled with strain-controlled conditions and the fatigue damage is dependent upon the resultant strain conditions [42]. Consequently, strain-based fatigue life relationship has been proposed later, especially for the LCF life calculation.

The plastic strain-life relationship proposed by Coffin [43] and Manson [44] is

∆𝜀_𝑝⁄ = 𝜀2 _𝑓^′(2𝑁_𝑖)^𝑐 (2) Further, combining both elastic and plastic strains, finally the total strain-life relationship [45] (as shown in Figure 14) is

∆𝜀 2⁄ = (𝜎_𝑓^′⁄ )(2𝑁𝐸 _𝑖)^𝑏+ 𝜀_𝑓^′(2𝑁_𝑖)^𝑐 (3) where ∆𝜀 is alternating total strain range, ∆𝜀_𝑝 is alternating plastic strain range, 𝜀_𝑓^′ is fatigue ductility coefficient, 𝑐 is fatigue ductility exponent, 𝐸 is elastic modulus, 𝑁_𝑖 is fatigue initiation life, 𝜎_𝑓^′ is fatigue strength coefficient, 𝑏 is fatigue strength exponent.

At large strain amplitudes, the strain-life curve coincides with the plastic line and at low strain amplitudes, the curve coincides with the elastic line. Thus, this relationship can be used for both LCF and high cycle fatigue (HCF) life calculations.

Figure 13: Stress-life relationship

Figure 14: Total strain-life relationship

As compared to the stress-based life calculations, strain-based life calculations are observed to predict comparatively more accurate results, especially for low life results, where the component stresses are observed to be above or near yield limit. The above equations hold good for cases where the load cycle is fully reversible and needs to be modified in order to consider the mean stress effect. Fretting fatigue is a special fatigue damage phenomenon of the contacting bodies, as a result of cyclic stresses on the contacting bodies and reduces the fatigue life extensively compared with non-fretting fatigue.

1.4. Basics of Fretting

Most general definition of fretting is “damage at the junction of contacting bodies due to presence of oscillating force, which generates relative displacement between the contacting bodies”. Resultant loads during the fretting phenomenon produce a similar effect on the material as a sharp, dynamic loading notch [36]. It is observed that the predicted life correlation is improved by a factor of 2, after considering fretting fatigue corrections [46]. This observation signifies the effect of consideration of fretting related in contact problems.

Generally, the contact surface consists of two regions i.e.,

1. Stick region: is the region of the contact surface where relative displacement is zero/negligible.

2. Stick-slip region occurs near the edges of the contact, where comparatively higher relative displacement occurs. In the stick-slip region, relative displacement is observed between the contacting bodies, along with high contact stresses, causing the surface degradation i.e., fretting.

It is observed that the transition from fretting to reciprocating sliding is usually in the range till 300 𝜇𝑚, regardless of the normal load [5]. Further, based on the amount of the resultant slip at the contact junction, fretting damage is classified into two categories:

1. Fretting wear: It occurs in the gross-slip region of contact, typically when the slip amplitude is in the range of 20-300 𝜇𝑚 . It results in the surface damage; however, the crack formation is very limited or absent.

2. Fretting fatigue: It occurs at low slip amplitude (generally below 20 𝜇𝑚) and the development of crack is dominant. Fretting fatigue is usually dominant at the mixed stick-slip region of the contact.

Figure 15: Illustration of fretting damage (Reproduced based on [52])

The combined effect of normal and bulk loads at the contact interface, causes the loss of mechanical energy in form of frictional energy due to friction and energy dissipation process. The area under the hysteresis curve between the transverse load (Q) and relative slip ( 𝛿), as shown in Figure 15, corresponds to the frictional energy generated during the resultant oscillatory movements at the contact surface. As the resultant frictional energy

exceeds the threshold strain energy limit, the dislocation of shear slip bands occurs, and crack starts to nucleate. Further, based on the relative slip magnitude, either fretting wear failure or fretting fatigue failure is observed at the contact surface.

As shown in Figure 16, it can be observed that fretting fatigue life decreases with increase in relative displacement value till certain threshold slip amplitude. Beyond this limit value, fretting fatigue life is observed to start increasing again. With the initial increase in relative slip displacement, the resultant surface traction also increases, resulting in the slip bands forming at the critical locations and thus forming the micro-cracks. However, with the higher relative displacement, the formed micro-cracks get shredded off and no further crack propagation can occur. Instead, the wear effect becomes more dominant beyond the critical value of the relative slip magnitude.