PSI - Issue 83

288 Teresa Morgado et al. / Procedia Structural Integrity 83 (2026) 286–294 r and the horizontal axis (xx), k is a constant, and ௜௝ is a dimensionless function dependent on the angle . For the higher-order terms, ௠ represents the amplitude, while ௜௝ ሺ௠ሻ is a dimensionless function of of the m terms. ௜௝ ሺ , ሻ ൌ √ଶ ௞ గ௥ ௜௝ ሺ ሻ ൅ ∑ ௠ ೘మ ௜௝ ሺ௠ሻ ሺ ሻ ஶ௠ୀ଴ (1) It is important to note that the solution of equation 1 for each type of geometry contains a principal term that is proportional to 1/√ . Therefore, as r approaches zero, the value of the principal component tends to infinity (Anderson, 2017). Each loading case produces a singularity of the type 1/√ at the crack tip. However, the constants k and ௜௝ depend on the loading mode. Substituting k for ஼ ൌ݇ √2 , critical stress intensity factor, the stress field at the crack tip is given by equation 2 for each propagation mode (Anderson, 2017). ௥՜଴ ௜௝ሺூ,ூூ,ூூூ ሻ ൌ ௄ ಴ √ଶగ௥ ௜௝ሺூ,ூூ,ூூூሻ ሺ ሻ (2) 2.1. J-Integral The J-Integral (JIT) concept was first introduced by Rice and Rosengren (1968) and aims to determine and describe the stress and strain fields in the region surrounding the crack tip, enabling the modelling of a material’s elastoplastic behaviour through a nonlinear elastic approximation. JIT quantifies the rate of energy deformation release per unit area of the newly created fracture surface. It is a quantity directly associated with the material’s intrinsic resistance to crack propagation and the energy balance during fracture propagation. Equation 3 presents the J-Integral formulation, where the integration contour, denoted by Γ , is an arbitrary curve around the crack tip in a counterclockwise direction, starting at the bottom face and ending at the top face of the crack tip; ω is the deformation energy density at points on the contour, u is the displacement in the x direction and ds is the length of the increment along the contour Γ (Anderson, 2017). ௜ is the tensile vector at the contour points calculated by equation 4, where, ௜௝ is the stress tensor, ௝ is the director cosine of the vector, and a is the crack length. ൌ׬ ሺ߱ ݕ െ ௜ డ డ ௨ ௫ ௜ ሻ ௰ ௪ (3) ௜ ൌܽ ௜௝ ௝ (4) JIT and FIT are two approaches to evaluate the stress state at the crack tip and can be related by equations 5 and 6, assuming, respectively, the stress and deformation plane states. Where K is the SIF in mode I (Anderson, 2017), E is the Young modulus, and is Poisson’s ratio. ܫ ൌ ௄ ಺ మ ா (5) ܫ ൌ ሺଵି ఔሻ మ ா ூ ଶ (6) 2.2. Crack Propagation There are two different SIFs, the ௧௛ , which defines the stress state at the threshold of crack propagation, and the ௖ , which occurs when the crack achieves a critical toughness and results in unstable fracture of the material. Crack propagation occurs whenever the value of SIF range, ∆ , exceeds the value of ∆ ௧௛ , stress intensity factor range at the propagation threshold at the crack tip, characteristic of the material and calculated experimentally. Crack propagation by fatigue relates the crack growth rate to the energy release rate (ASM, 1997). Fatigue life can be studied through crack growth rate per number of cycles, da/dN, versus stress intensity factor range, ∆ K. The Paris Law (equation 7) describes fatigue behaviour of the material where C and m are empirical material constants, related