# Lifetime Prediction of Single Crystal Nickel-Based Superalloys
## 1 Introduction  
Single crystal nickel-based superalloys are widely preferred in turbine blade components due to their high temperature resistance and mechanical properties [1, 2]. Turbine blades are subjected to both severe HCF and LCF loading conditions on top of high operating temperatures, resulting in both fatigue (HCF and LCF) and creep as the main mechanisms of failure [3, 4]. Although the presence of creep-fatigue interaction has a negative influence on component life, fatigue fracture is considered to be the main concern of failure in such applications [1, 5]. High rotational speed and the fast transient startups of the turbine result in LCF loadings [3]. High-temperature conditions caused by the heated gas inside cause additional creep damage along with LCF loadings [4]. As a result, nickel-based single crystals are often subjected to severe LCF and TMF [6, 7, 8]. Understanding the creep-fatigue interaction and LCF fatigue failure of these components is crucial for proper life prediction of such components. Turbine blades are subjected to vibration loads due to the gas flow within the engine [3, 9], resulting in severe HCF loading conditions in addition to LCF and TMF. Therefore, special attention is paid to HCF lifetime assessment of single crystal superalloys since the reliability and security of aircraft engines are significantly related to the HCF properties of such alloys. In the case of single crystals, crack initiation points and growth directions are highly dependent on the cutting plane and loading types [10; 11], unlike most polycrystalline materials. Depending on the crystal structure and operating temperature, plasticity is observed through active slip planes and directions. Although nominal stress is in the elastic region for HCF loadings, plasticity is observed in stress concentration zones as geometric irregularities and micro-cracks. Hence, considering active slip systems is essential to the lifetime assessment of such components. Fatigue life prediction of single crystal materials is widely investigated from many different aspects. Experimental observations conducted by [12] suggest that for multiaxial cases, fatigue crack initiation and growth occur along dominant slip planes. Based on this phenomenon, many lifetime prediction formulations have been developed with critical plane formulations on probabilistic approaches and modifications to isotropic fatigue formulations to account for multiaxial anisotropy [2; 9; 13; 14], in which resolved shear stresses on the critical planes are chosen as the main concern for failure since it is proven that (being controlled by shear strains) fatigue cracks initiate along crystallographic directions [12]. Once initiated, crack growth is also controlled by the maximum shear strain direction. However, the maximum stress direction of these mechanisms may differ from one another depending on the crystallographic orientations. Fatigue life prediction methods based on continuum damage mechanics evaluate the damage parameter, \(D\), cycle by cycle, with crack nucleation occurring when the damage reaches a critical level. Early approaches, such as the Palmgren-Miner rule, model damage as a linear function of lifetime, later extended to nonlinear behaviors using models such as Double Linear Damage Rule (DLDR), Damage Curve Approach (DCA), and Double Damage Curve Approach (DDCA) [15]. These methods are commonly paired with stress- or strain-based approaches for lifetime prediction under variable amplitude loading. [PERSON] and [PERSON] [16] introduced a nonlinear damage evolution equation incorporating multiaxial loading via equivalent stress amplitudes. Due to its mathematical formulation, the Chaboche model can be readily implemented in computational tools, facilitating its application in complex fatigue analyses.