Fatigue life prediction under mixed-mode loading using equivalent stress intensity factor models

Damage tolerance principles are widely used to assess the structural integrity and failure of engineering components. The advances in numerical simulation techniques facilitate the prediction of fatigue life of engineering component, which is essential in damage tolerance design. For the components under mixed mode (I/II) loading, the fatigue crack growth and life is predicted by using a modified form of Paris' law along with the equivalent stress intensity factor ( ) eq K  . Numerous eq K  models are available for correlating the equivalent stress intensity factor range and the fatigue crack growth rate. The knowledge of proper eq K  model is essential for the accurate fatigue life estimation. In this work, the authors numerically assess the performance of eq K  models in mixed mode fatigue life prediction with the help of published experimental mixed mode data.


Introduction
Engineering components and parts are generally assumed to be safe, but they contain randomly oriented microcracks, voids, inclusions and flaws, and that may grow under cyclic loading. For the cracks subjected to repeated or cyclic loading, fatigue crack growth analysis based on fracture mechanics principles plays an important role in damage tolerance assessments and in the evaluation of residual fatigue life. Structural components having complex geometries require the use of numerical simulation techniques for the fatigue crack growth (FCG) rate and fatigue life prediction. Numerical techniques such as boundary element and finite element methods are generally used for this purpose. For the ease of analysis, most of the engineering components containing cracks are considered to be either under mode I or mode II loading and are assumed to be under static loading. Numerous crack growth laws are available for the fatigue life correlation for cracks subjected to only mode I loading. For cracks under opening mode/mode I loading, the power law proposed by Paris [1] has received wide acceptance. Under mixedmode (I/II) loading conditions, the crack propagation is in non-self-similar manner, i.e., the crack deviates from its original crack path and crack growth rate is significantly affected by the mixed mode loading condition. So in mixed mode loading, the crack path as well as the crack growth rate has to be determined. For cracks subjected to mixed mode loading, Tanaka [2] proposed the modified form of Paris law, a relationship between the fatigue crack growth rate and equivalent stress intensity factor (equivalent SIF) range, which is given by where C and m are material constants. Equation (1) represents the linear relation between on logarithmic scale. The material constants C and m can be obtained from mode I fatigue tests [3]. This section briefly describes the numerical procedure for simulation of mixed mode (I/II) fatigue crack growth. Fig.1 shows different steps to carryout fatigue crack growth numerical simulation under mixed mode loading. An incremental crack tip extension method is used in present study, in which the crack tip is advanced to a new tip position after each step. At each step, SIFs I K and II K are calculated. At the end of each step, the fatigue crack growth direction, as well as fatigue life, is calculated using the SIFs. The new crack tip coordinates are being calculated and the crack is advanced to the new position. The crack step increments considered are relatively small (w:r:t the initial crack length), for obtaining a smooth crack path. The finite element numerical simulations were carried out using the crack propagation simulator FRANC2D, which is a finite element based crack propagation simulator. The necessary scripts for the post-processing were developed in MATLAB.

Fatigue crack path prediction
Accurate estimation of fatigue crack path is essesntial for the fatigue life pediction. Erdogan and Sih [4] proposed maximum tangential stress criterion (MTS), which is given as Solving the above equation for the propagation angle   I  I  c  II  II  II   I  I  c  II  II  II where positive c  is the crack growth direction from the initial crack axis in anticlockwise direction.

Fatigue life prediction
The fatigue crack propagation prediction using the equivalent SIF eq K  is the most widely used approach for components under mode I/II cyclic loading. Using a modified form of Paris law, Tanaka [2] proposed a power law for the correlation of fatigue crack growth rate and equivalent SIF eq K  ( under mixed mode I/II), which is given as From Eq. (5), for a crack increment, the fatigue life cycles can be determined as Some of the commonly used eq K  models along with the proposed authors are summarized in Table 1.
Richard et al. [8]   The equivalent SIF models listed in Table 1 were developed to establish a linear relationship between the experimental da dN and equivalent SIF eq K  on a logarithmic scale. A performance evaluation of the equivalent SIF models (in predicting the fatigue life) is carried out, and they are discussed in the following section.

Performance assessment of Equivalent SIF models
This section discusses the finite element simulations performed for the present investigation. The numerical simulation is based on the experimental studies using mixed mode I/II CTS specimen by Ma et al. [9] The geometry and boundary conditions of the mixed mode CTS specimen used for the present investigation is given in Fig. 2(a). The loading direction is making an angle of 60 0 with crack axis. The experimental specimen made of steel is used by Ma et al. [9] in his experimental studies. For applying mixed mode loading using the uni-axial testing machine, Richard's loading device [10] is used. Fig. 2  The material used for the experiment, material properties and test conditions given by Ma et al. [9] along with material properties and other constants used for present analysis are given in table 2. An incremental crack tip extension method is followed in the present numerical investigation, and the step crack increment 0.5 mm a   is used. For modeling the material response, plane stress condition is assumed.  Using the crack kink direction and the constant crack increment, the new location of the crack tip is calculated, and the crack is advanced to the new position. Fig. 4 shows the crack path obtained from the present numerical simulations. The experimental crack path was not available for the present problem considered. Fig. 5 shows experimental fatigue life curves by Ma et al. [9] and a six-degree polynomial fit on experimental data. It can be noted from the goodness of fit of Fig. 5 that the polynomial fit represents the experimental data accurately.  The numerically estimated fatigue life data using different eq K  models are plotted in Fig. 6. The polynomial fit of the experimental fatigue life is also plotted in Fig. 6 for comparison. For better understanding, the close-up views of Fig. 6 near the origin and at the end of the graph is shown in Fig. 7. The percentage relative error in the predicted fatigue life is calculated using the fitted data as reference and is plotted in Fig. 8. In the present numerical investigation, the percentage relative error is calculated by using the fitted polynomial as reference. Fig. 5 also shows the goodness of fit.
It can be seen from Fig. 8 that at the initial stages all the eq K  models presented in Table 1  A detailed investigation of the present study using a variety of benchmark examples along with some recommendation for the reduction of initial errors is available in [11,12].

Conclusions
Performance analysis of various eq K  models in predicting the fatigue life is carried out in this study. The present study is conducted using the limited available experimental results; the present study provides certain useful observations. The results of the present investigation indicate that fatigue life predicted using the eq K  models based on Irwin [5] and Hussain et al. [7] models are closer to the experimental fatigue life used in the present numerical investigation. Among the above two, fatigue life prediction using Irwin's [5] model is more close to the experimental fatigue life. One of the models proposed by Tanaka [2] is also predicting fatigue crack growth life close to the experimental data. The remaining three models unsed for the analysis provides more conservative solution, among them the model proposed by Richard et al. [8] is the most conservative model. It is observed in the present investigation that the when the mode mixity is less, most of the models are predicting fatigue life close to the experimental results. However, when there exists a higher mode mixity, there is a significant deviation among the fatigue life prediction by different models and the models proposed by Irwin [5], and Hussain et al. [7] are found to be performing well. The present investigation offers some useful insights into the prediction of mixed mode fatigue crack growth life, which is essential from the damage tolerance point of view.