PSI - Issue 61

102 Aliyye Kara et al. / Procedia Structural Integrity 61 (2024) 98–107 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 5 inherent numerical discretization. Theoretically, if both σ ( σ a ) and (−) were known in closed-form by analytical formulas, the transformed CDF ε ( ε ) could be computed in closed-form, either. Once the strain amplitude CDF has been obtained, its corresponding PDF can finally be calculated, by definition, upon differentiation as: ( ) = ( ) (7) This calculation is done numerically if the analytical expression of ε ( ε ) is not available. 4. Time-domain simulations Here, the method explained in Section 3.1 is benchmarked against time-domain simulations carried out by the finite element method. The main goal of time-domain simulations was to understand how material plasticity changes the probability density function of local stress and strain amplitudes in response to stress-time realizations with a certain variance. The procedure is explained in the following sections. 4.1. Finite element model The notched specimen, already used in Rognon et al. (2011, 2014), was considered to simulate the elasto-plastic material response. Figure 2(a)-(b) represents the notch geometry and the corresponding finite element model of the specimen. Axisymmetric finite elements and a model symmetry along the X horizontal axis were exploited to shorten the computational time. After a mesh convergence analysis, the final mesh consisted of 1712 quadratic, eight-node quadrilateral elements and 5353 nodes.

b.

Axis of axisymmetry

Axis of symmetry

Distance 0 3.5

Fig. 2. (a) Notch details (b) Mesh model of the notch specimen (c) Multiaxial stress-state at notch.

A linear elastic static analysis was run to calculate the stress distribution at the notch and the stress concentration factor K t =σ max /σ g , where σ g is the remote nominal stress at the gross section (applied at the upper specimen's edge) and is the maximum principal stress at the notch along the Y axial direction. The calculated stress concentration factor is 7.81. Fig. 2(c) displays the trend of normal stress components (radial σ x , hoop σ z , axial σ y ) at the notch section for an applied remote stress of σ g = 1 MPa. Figure 2(c) points out that, for this particular specimen geometry, the state of stress at the notch section is multiaxial (biaxial), even if the applied remote stress is uniaxial.

4.2. Generation of random stress-time histories

The finite element model in Fig. 2 is loaded by a random stress-time history σ g ( t ) applied at the upper edge of the specimen. The applied random stress-time histories are zero-mean stationary and Gaussian, with a narrow-band PSD with a central frequency of 20 Hz (125. 7 rad/sec) and a highest-to-lowest frequency ratio of 1.222, as shown in Fig.