PSI - Issue 45

Xianwen Hu et al. / Procedia Structural Integrity 45 (2023) 20–27 Hu, X., Liang, P., Ng, C.T., and Kotousov, A. / Structural Integrity Procedia 00 (2019) 000 – 000

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3. Numerical simulation 3.1. Modelling of material nonlinearity

The microstructural damage of the aluminium plate was considered as weak material nonlinearities by incorporating Murnaghan’s strain energy equation (Hu et al. 2022; Yang, Ng & Kotousov 2019). . The constitutive equations were modified through the subroutine VUMAT (user-defined material for explicit dynamic). The deformation gradient ( F ) can be expressed as F = ∂x / ∂ X , where X is defined as the coordinate in the reference configuration, and x is defined as the coordinate in the current configuration. The Green-Lagrange strain tensor is defined as E = (F T F-I) / 2, where I is defined as the identity tensor. According to Yang, Ng and Kotousov (2019) , Murnaghan’s strain energy function can be written as (5)

= 1 2 +2 12 + 1 3 +2 13 − 2 2 − 2 1 2 + 3 1 = 2 = 1 2 12 − 2 3 = = − 1

where and l, m, n are the third-order elastic constants, and the strain invariants ( i 1 , i 2 , i 3 ) can be expressed as follows: (6)

Also, in the isotropic media, the second Piola-Kirchhoff stress can be written as T= ∂W(E) / ∂E. The relationship between the second Piola-Kirchhoff stress and Cauchy stress can thus be given by the following:

(7)

where J -1 is defined as the Jacobian determinant, and J = det( F ) (Zhu, Ng & Kotousov 2022).

3.2. 3D FEM This section presents numerical simulations for the edge wave propagation in metallic plates. A 3D FEM was developed by ABAQUS CAE 2017. As shown in Figure 3, an isotropic aluminum plate (500 × 300 × 16mm) was modelled. The symmetric boundary condition was applied to the left plate edge to reduce the computational cost for the numerical analysis. Table 1 and Table 2 show the linear and nonlinear material properties of the test plate, respectively. To generate edge waves, out-of-plane displacement was applied to a 5 mm radius half-circle (the excitation area) at the top left of the plate (Hughes et al. 2021b). The excitation signal was a 120 kHz Hanning windowed tone burst with 10 cycles, and the corresponding FTV is 3.87. The maximum peak amplitudes were 210 μm. A measurement point was defined at center line of the top plate edge as shown in the figure. The distance between the center of the excitation area and the measurement point was 150 mm.