Chapter 6 Fatigue Life Calculation

Transcription

1 Chapter 6 Fatigue Life Calculation 6.1Strain Life Equations with Mean Stress The compressive mean normal stresses are beneficial, and tensile mean normal stresses are detrimental to fatigue life is revealed from all experimental data. However, it actually depends on the damage mechanism in the material; a shear parameter sensitive material would not necessarily have a longer fatigue life if the compressive stress is not aligned with the shear plane. This has been observed under the condition when the fatigue behavior falls in the high cycle fatigue regime where elastic strain is dominant [41]. In conjunction with the local strain life approach, many mean stress correction models have been proposed to quantify the effect of mean stresses on fatigue behavior. The modified Morrow equation (Morrow, 1968) and the Smith Watson Topper model (Smith et al., 1970) are commonly used and described in the following sections [60] Strain-Based Fatigue Analysis and Design: Stress-based fatigue life prediction for the fatigue analysis of components works well for situations in which only elastic stresses and strains are present. Generally most components may appear to have nominally cyclic elastic stresses. The notches, welds, or other stress concentrations present in the component may result in local cyclic plastic deformation. Under these conditions, another approach that uses the local strains as the governing fatigue parameter (the local strain life method) was developed in the late 1960s. The local strain life method is based on the assumption that the life spent on crack nucleation and small crack growth of a notched component can be similar to that for polished laboratory specimen under the same cyclic distortion at the point where crack initiation occurs. The fatigue life at a point in a cyclically loaded component can be determined using local strain life method if the relationship between the localized strain in the specimen and fatigue life is known. This strain life relationship is typically represented as a curve of strain versus fatigue life. The curve generated by conducting strain-controlled axial fatigue tests on smooth, polished specimens of the material. The material at stress concentrations and notches in a component may be under cyclic plastic 60

2 deformation even when the bulk of the component behaves elastically during cyclic loading therefore strain-controlled axial fatigue testing is recommended [37]. During early design stage of the component the local strain life method can be used. Fatigue life estimates can be made for various prospective design geometries and manufacturing processes before any actual component is produced if the material properties are available. Identifying and rejecting unsatisfactory designs early in the design process will result in a reduction in the number of design iterations. The total design cycle time is reduced and the product come to market quickly. If the load history is uneven or random and the mean stress and the load sequence effects are thought to be of importance, the local strain life approach is preferred. This method also provides a rational approach to differentiate the high-cycle fatigue and the low-cycle fatigue regimes and to include the local notch plasticity and mean stress effect on fatigue life. In the local strain life approach the material data used are only related to the laboratory specimen. The fabrication effects in the actual component, such as surface roughness/finish, residual stress, and material properties variations due to cold working and welding, may not appropriately be taken into account. [4] Morrow Morrow presented his mean stress correction model in the stress-life equation. By proposing t h a t the mean stress effect is negligible in the LCF regime. Stress life can be modeled by the Morrow equation for its noticeable effect in the HCF regime, the strain life equation (so-called the modified Morrow equation) is then modified as follows: εa = 2 2 (6.1) where This equation has been extensively used for steels and used with considerable success in the HCF regime. Walcher, Gray, and Manson (1979) have noted that for other materials, such as Ti-6Al-4V, σ f is too high a value for the mean stress correction, and an intermediate value of kmσ f is introduced. Thus, a generic formula was proposed: εa = 2 2..(6.2) 61

3 6.2.2 Smith, Watson, and Topper Smith, Watson, and Topper (1970) developed another mean stress correction model, by postulating the fatigue damage in a cycle is determined by the product of σmaxεa, where σmax is the maximum stress. They stated that σaεa for a fully reversed test is equal to σmax εa for a mean stress test. Vogel expressed this concept in following form: σmax εa = σa,rev εa,rev for σmax > 0..(6.3) The value of εa,rev should be obtained from the fully reversed, constant amplitude strain life curve and the value of σarev, from the cyclic stress strain curve. The SWT parameter predicts no fatigue damage if the maximum tensile stress becomes zero and negative. The solutions to Equation (6.3) can be obtained by using the Newton Raphson iterative procedure. For a special case of Equation (6.3), where a material satisfies the compatibility condition among fatigue and strain properties (i.e., n = b/c and K = σ f/(ε f) n ), the maximum tensile stress for fully-reversed loading is then given by σmax = σa = 2.(6.4) by multiplying the fully reversed, constant amplitude strain life equation, the SWT mean stress correction formula becomes σmax εa = 2 2 σmax > 0. (6.5) Equation (6.5) has been the widely adopted SWT equation that has been successfully applied to grey cast iron (Fash & Socie, 1982), hardened carbon steels (Koh & Stephens, 1991), micro alloyed steels (Forsetti & Blasarin, 1988), and precipitationhardened aluminum alloys in the 2000 and 7000 series (Dowling, 2009). It is considered that during cyclic loading the material follows the cyclic stress strain curve for the initial phase and the hysteresis stress strain behavior for the subsequent loading reversals. The strain by and stresses are calculated for the entire block. The stress strain curve up to point 1 is calculated using a cyclic stress strain curve equation (The Ramberg Osgood equation): (6.6) From then on all incremental reversals with respect to a reference turning point are calculated based on Masing s model: 2.(6.7) 62

4 Fig.6.1 shows SWT Parameter versus Reversal to Failure for Three materials (a) C 45 (b) Al2024 T4 and (c) Al7075T6 The next step is to extract the hysteresis loops and identify the maximum and minimum stress and strain points for each loop so these data could be entered into a damage calculation. Please note that the three-point rain flow cycle counting technique can be used to check the identified hysteresis loops. Using this information, the Smith Watson Topper and the modified Morrow formulas are used to evaluate the number of block cycles under which this component will survive. An individual damage number is calculated for each counted cycle by employing the linear damage accumulation rule, which is also known as the Palmgren Miner linear damage rule (Palmgren, 1924; Miner, 1946). The total damage D is defined as:, D = (6.8) Where, ni = the number of applied cycles to a constant stress amplitude Kn = the total number of the stress blocks Nf,i = the so-called fatigue life as the number of cycles to failure calculated from either the modified Morrow equation or the SWT mode The number of repeats of the given load time history (the number of blocks) can be estimated by assuming failure occurs when D = 1. 63

5 6.3 Notch Analysis The numerical analysis method to estimate the local stress strain response at a notch root -a stress concentration site, based on the virtual stress time history from a linear elastic finite element analysis is called Notch analysis. The virtual stress (σe) can also be obtained by the product of the elastic stress concentration factor (Kt) and nominal stress (S) for a component with a well-defined notch geometry and configuration. Many efforts are made for the development of notch analyses for expeditious stress strain calculations. Among these, Neuber s rule (Neuber,1961) and Molsky Glinka s energy density method (Molsky & Glinka, 1981) have been widely used and will be discussed in the following sections [95]. 6.4 Modified Neuber Rule The local strain life approach is not taking in to account the cyclic behavior of the materials. Smith, Watson and Topper proposed to modify the Neuber rule by replacing Kt with the fatigue notch factor (Kf). Considering these conflicting results, we endorse that the modified Neuber rule be used for local notch stress strain estimates and fatigue life predictions. The notched component analysis will experience a problem with the use of Kf factor in Neuber s rule where the notch geometry and configuration is complex because the nominal stress S and the Kt factor are difficult to measure. However, the product of KtS (the pseudo stress) is obtained from a linear elastic finite element analysis. The factor KtS needs to be converted to KfS in the modified Neuber rule. The Kt Kf relationship can only be obtained experimentally. In the past empirical average stress models (Peterson, 1969; Neuber, 1946; Heywood, 1962) have been developed to estimate the Kt Kf factor with the reference to a notch radius (r) and the ultimate tensile strength St,u. Moreover, the following expression for the Kt/Kf ratio was developed by Siebel and Stieler (1966): 1 (6.9) Equation (6.9) is the generalized formula for the Kt/Kf ratio for various materials with different yield strengths. 64

6 The modified Neuber model can be used with the help of the unique relationship between the Kt/Kf ratio and G. The Kt/Kf ratio can be obtained from Equation (6.9) for the given G and material 1yield strength. And the pseudo stress at a notch (σe = KtS1) or the pseudo stress range at a notch (Δσe = KtΔS) can be obtained from the elastic finite element analysis [49] If Kt/Kf = ASS, The modified Neuber equation can be rewritten as follows for normally elastic behavior: + (6.10) + 2Δ 2. (6.11) The Newton Raphson iteration technique can be used to solve above equations containing the stress gradient effect for the local stress or stress range. 65