PSI - Issue 74

Jiří Vala et al. / Procedia Structural Integrity 74 (2025) 91–98 J.Vala & V.Koza´k / Structural Integrity Procedia 00 (2025) 000–000

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namely to the mesh modifications and refinements on active parts of Λ , handling various crack tip phenomena, crack opening and closing, etc., cf. Belytchko et al. (2009), Wang et al. (2024) and Koza´k and Vala (2025).

Fig. 2. Elongation region between the a priori crack and the creep region, magnification 150 × (left-hand-side figure). Proportional elongation for J ˙equal to a) 47 and b) 52 N / mm / h obtained via XFEM (right-hand-side graph). The illustrative example on Fig. 2 shows the results of the simulation process; for more detail see Koza´k and Vala (2024). For practical modelling of crack propagation respecting the microscopic processes in front of the crack, the use of the cohesion approach has been applied. The propagation of a macroscopic crack during the creep of austenitic steel AISI 304L is the focus of both theoretical and experimental investigation. Experiments were carried out at the temperature 973 K on bodies with and without a priori cracks in order gather input data (creep proportionality constant for power law is 1 . 36 · 10 − 20 MPa − 7 / s; creep stress sensitivity parameter is 7; Young modulus is 1 . 45 · 10 5 MPa). It is demonstrated that a large region of stationary creep follows an unstable primary stage area. This model can be applied to many di ff erent types of materials and faults, due to the simple description between stress and displacement in front of the crack. Model for the exponential traction-separation law was used, with an initial value of J ˙ (or a similar integral), an estimate of the initial value of J ˙ for the given material can be found in Brust and Atluri (1986), for the details of numerical treatment cf. Chen et al. (2024). It is assumed that this value will not change during the creep violation. From the mathematical point of view, still other generalization for the derivation of Fig. 2 is needed: instead of a linear viscous element β C in Fig. 1 a power-law creep term, coming from the Norton law, is needed, whereas a second elastic element α C is not implemented for simplicity. This is the source of still other nonlinearity in (4), (5), etc., but this can be still handled, including all XFEM computations, as a quasilinear problem. 5. Conclusions In this contribution we have highlighted the analysis of the SLS viscoelastic model, enlarged by two types of damage. Notice that some similar (or even more simple) problems, not involved in the preceding considerations literally, can be handled in an analogous way, e. g. the case with the missing second elastic element α C in Fig. 1, which leads to the analysis of the classical Kelvin viscoelastic model by Vala and Koza´k (2020), with the possibility of some modifications, as that presented in our illustrative example. The need of a related class of models, referred as generalized Kelvin chains, comes from Zhang et al. (2020) and Trcala et al. (2024a): the scheme by Fig. 1 is replaced of a serial composition of a finite number of parallel Kelvin models and one purely elastic component; the method of possible implementation of smeared damage is described in Trcala et al. (2024b). Fortunately, from the both mathematical and computational points of view, similar results as in our model problem can be derived also in such a case. Still more generalizations, mentioned in two last paragraphs of Introduction , enable us to come to more correct both physical and mathematical formulations, but with numerous unclosed theoretical questions and missing reasonable computational tools. A lot of research challenges in the analysis of elastic, plastic, viscous, etc. materials with damage is open. The authors and their collaborators at Brno University of Technology are now interested in the detailed analysis of energy dissipation on contacts / impacts of multiple deformable bodies with potential damage, following Neˇmec et al. (2022), with additional problems of searching for contact candidates using selected results from the graph