PSI - Issue 35

Domen Šeruga et al. / Procedia Structural Integrity 35 (2022) 150–158 Sˇ eruga et al. / Structural Integrity Procedia 00 (2021) 000–000

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in the case of extreme events is therefore very important. There are several studies regarding the behaviour of pipes and pipe bends during operation. Some recent studies include the plastic response of initially deformed thin-walled pressurised pipe bends under in-plane opening bending moment (Roy et al. (2021)). Li et al. (2020) used a finite element method to determine limit load solutions for pipe bends with 180 ◦ bending angle under in-plane bending moment. Shim et al. (2016) experimentally verified numerical results of double-stage forming using the analytically calculated pre-bending radius in roll bending of pipes. Ancellotti et al. (2019) inspected the role of the fabrication load history on the mechanical response of complicated spatial tubular structures. Lu et al. (2020) presented a detailed three-dimensional finite element model for simulating flexible pipes subjected to combined axisymmetric and one fully reversed bending load cycle. In this paper, the structural behaviour of a pipe bend under variable thermomechanical conditions is analysed using the Prandtl operator approach. The major advantage of the Prandtl operator approach is the calculation speed and a small number of material parameters needed to carry out the simulations. The approach has been recently implemented into an Abaqus UMAT routine to enable simulations of cyclically loaded mechanical components with temperature-dependent isotropic elastoplastic material properties and multilinear kinematic hardening. The results of the simulation in this paper have been compared against the well-known Besseling model for cyclic elastoplastic solutions with multilinear kinematic hardening to investigate the validity of the approach (Besseling (1958)). For the both simulations, accumulated fatigue damage has been calculated to analyse the influence of the di ff erences in the stress-strain simulation. An extensive overview of other approaches to simulate the stress-strain response of thermomechanically loaded mechanical components can be found in e.g. Nagode et al. (2021). Moreover, a complete presentation of the closed-form solution for temperature-dependent elastoplastic problems using the Prandtl operator approach can be also found in Nagode et al. (2021). Here, initial and final equations are provided to give an insight into the simulation procedure.

Nomenclature

Kronecker delta

δ i j

ε (1) i j ε (2) i j

strain tensor in the last equilibrium step

strain tensor in the current step

strain-tensor increment

∆ ε i j σ (1) i j σ (2) i j ∆ σ i j T (2)

stress tensor in the last equilibrium step

stress tensor in the current step

stress-tensor increment

temperature in the current step

λ ∗ T (2) Lame´ first constant µ ∗ T (2) Lame´ second constant

2. Method

Thermomechanical loads are applied to the observed mechanical component in steps. The aim of the Prandtl operator approach is then to determine the stress tensor in an arbitrary integration point of an arbitrary finite element of the analysed mechanical component in the current step, denoted by index (2), as σ (2) i j = σ (1) i j + ∆ σ i j , (1) if the stress tensor σ (1) i j and the strain tensor ε (1) i j in the last equilibrium step, denoted by index (1), including the strain-tensor increment ∆ ε i j , are known, ε (2) i j = ε (1) i j + ∆ ε i j . (2)