PSI - Issue 81

Oleh Yasniy et al. / Procedia Structural Integrity 81 (2026) 244–250

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the crack network, the growth of these cracks will continue. Therefore, an important task is to predict the evolution of the surface crack network caused by thermal fatigue, which forms at the early stages of operation and largely determines the service life of the structural element. The rate of microcrack growth depends on material properties, thermomechanical operating conditions, the mutual arrangement of microcracks and the distances between them, and has a statistical nature. Both experimental studies and numerical simulations have been carried out to assess the growth of crack networks (Haddar & Fissolo, 2005; Haddar et al., 2005). It has been found that the reduction in crack growth rate at a certain distance from the surface, as well as the degree of this reduction, depends on the amplitude and frequency of temperature fluctuations and on the thermal conductivity coefficient (Kasahara et al., 2002; Taheri, 2007). The behavior of materials under thermal fatigue is strongly influenced by various types of uncertainties. Current industrial design standards account for these uncertainties by applying empirical safety factors, which makes the design overly conservative. Such approaches do not allow for a quantitative assessment of risks associated with design decisions. Therefore, it is important to introduce probabilistic models as an alternative to existing standards. Despite a number of studies devoted to the development of probabilistic approaches for modeling multiple surface cracking of structural elements under thermal fatigue, this problem still requires further investigation and improvement, particularly in the analysis of specific structural components depending on their operating conditions. In addition, crack initiation and propagation remain insufficiently studied. In most cases, the influence of temperature and mechanical factors is not taken into account. The proposed approaches to thermal fatigue assessment are predominantly deterministic and therefore do not allow for consideration of the probabilistic nature of crack initiation and their subsequent development. In addition, the effect of thermocyclic loading of 21CrMoV5-7 steel on changes in hardness and microhardness in the surface layers was investigated (Yasniy et al., 2009). It was found that with an increasing number of thermal cycles, the microhardness of the surface layer increases. The static fracture process of the material is considered with account for deformation processes. Under thermocyclic loading, surface defects accumulate and the mismatch of deformation mechanisms at different scale levels increases. This leads to active formation of mesobands and promotes the redistribution of normal and shear strains, which increases the plasticity of the material but localizes stresses. In view of the above, investigating the evolution of a thermo-fatigue crack network requires an accurate evaluation of the stress distribution under biaxial loading conditions. Therefore, in this work, the initiation of thermo-fatigue cracks is modeled using the Monte Carlo method, explicitly taking into account the stress state at potential crack initiation points. This approach makes it possible to incorporate the inherently random nature of defect formation and local stress fluctuations into the analysis. The subsequent growth of fatigue cracks is simulated numerically using the boundary element method, which provides high accuracy in describing stress concentration and crack-tip fields. The proposed methodology thus combines probabilistic modeling of crack nucleation with a deterministic simulation of crack propagation, enabling a more realistic and comprehensive description of the thermo-fatigue damage process. This integrated framework allows for improved prediction of crack network development and enhances the assessment of structural durability under complex thermomechanical loading. 2. Numerical modeling of thermo-fatigue processes 2.1. Crack initiation A model for multiple crack initiation and their subsequent growth under thermo-fatigue conditions was developed based on the Monte Carlo method. A plate with dimensions of 5x5 mm (Kamaya & Taheri, 2008), made of 21CrMoV5-7 steel, was considered as the computational domain. It was assumed that a new crack could nucleate only at specific predefined locations, referred to as crack initiation nuclei. A total of 20 × 20 = 400 such nuclei were uniformly distributed over the surface of the plate, formi ng a regular grid. Consequently, the distance between neighboring nuclei was 0.25 mm in both the horizontal and vertical directions (Fig. 1). This discretization scheme allows the stochastic nature of crack initiation to be represented in a controlled and physically meaningful way, while ensuring a uniform spatial resolution for probabilistic analysis. Each nucleus serves as a potential site for crack formation if the local stress state and damage criteria are satisfied. Damage was accumulated independently at each initiation nucleus. The rate of damage accumulation per loading cycle, / dD dN , was determined according to an experimentally obtained relationship (Yasniy et al., 2009):

dD dN

0.016 6.48 10 a  

(1)

,



where a  is the stress amplitude (in MPa) calculated at each nucleus. As the governing stress parameter a  , the Huber – Mises stress was adopted, evaluated for a plane stress state. This formulation allows the progressive degradation of material at each potential crack initiation site to be quantified as a function of the local stress level. By using the von Mises (Huber – Mises) equivalent stress, the model accounts for the combined effect of normal and shear stress components in a physically consistent manner. As the number of loading cycles increases, the accumulated damage at a nucleus grows, and when a critical damage level is reached, a crack is assumed to initiate at that location.