According to the analysis in Section 2.2, the time delay of wave propagation and the variation of the eigenfrequency can be used as quantitative indexes to characterize the acoustoelastic effect, respectively. For solving these indexes, the corresponding simulation models of the acoustoelastic effect must be developed.

## 3.1. Geometric model

The 3D unit model of the ribbed plate is shown in Fig. 4, which has all the structural features of the ribbed plate structure. However, it is obviously unadvisable to use this model as the geometric model for simulation, because the simulation of a complex 3D model requires a huge number of meshes and computational degrees of freedom, which occupies huge computational resources and consumes a lot of computational time. Therefore, the problem needs to be further simplified.

Since the A and B boundaries of the structure maintain a constant cross-section along its normal direction within a certain distance, it is advisable to use the cross-section model shown in Fig. 5 as the study object. The study hence can be simplified from a three-dimensional problem to a two-dimensional problem by studying the wave propagation characteristics in the cross-section to reflect that in the three-dimensional structure. In order to study the difference of the acoustoelastic effect in the ribbed plate structure with different fillet radii, the simulation model must include a series of cross-sections with different fillet radii. The relevant geometrical parameters of the cross-section model are as follows: plate length *l* = 125 mm, plate thickness *h* = 3.3 mm, rib width *w* = 6.5 mm, rib thickness H = 10.7 mm, and R/H, the ratios of fillet radius to rib thickness are shown in Fig. 5.

## 3.2. Physical field settings

Firstly, since all problems under study involve only elastic waves in solids, solid mechanics must be selected as the physical field for simulation. Secondly, structural steel is applied to the materials of all sections, and the Murnaghan model of the hyperelastic material is used for all domains. Thirdly, how to completely constrain the model and reasonably apply external loads become the next simulation procedures. the model is done as follows, the edges on both sides of cross-section of the ribbed plate are set as spring bases, whose elasticity type uses material data, material is the same as the domain material and the length is \(10\times l\). The external loads are applied by setting different degrees of pre-deformation of the spring bases along the normal direction of the edges. Finally, there is the problem of excitation application and signal reception. As shown in Fig. 5, Load excitation points (A1-A6) and signal reception points (B1-B6) are set at the center of the plates on both sides of the rib. A point load excitation is applied to the load excitation point along the X direction with the value \({F}_{0}\times \text{H}\left(\text{t}\right)\), where F0 = 2.0 N and H(t) is the Hanning window function defined by Eq. (5). And the receiving point takes u, X component of the displacement field, as the receiving parameter.

$$H\left(t\right)=\left\{\begin{array}{c}0 , t<N/{f}_{c}\\ \left[1-\text{cos}\left(2\pi {f}_{c}t/N\right)\right]\text{sin}\left(2\pi {f}_{c}t\right), t\ge N/{f}_{c}\end{array}\right.$$

5

Where, *f**c* is the center frequency, which is taken as 200 kHz, and *N* is the number of periods with the value 5.

## 3.3. Meshing

The cross-section is divided into four domains as shown in Fig. 6. For domains 1, 2 and 3, divide the structural mesh with 5 nodes uniformly distributed on edges *ab* and *ij*, 60 nodes uniformly distributed on edges *bc* and *hi*, 6 nodes uniformly distributed on edge *ef* and 10*(1-R/H) nodes uniformly distributed on edge *de*. The free triangle mesh is divided for domain 4, and the maximum cell growth rate is 1.05, and all meshes are shown in Fig. 6.

## 3.4. Settings for time delay study

For solving the time delays of the wave propagation in cross-sections under external loads, the stress states of cross-sections under static external loads must be found first, and then transient simulation is carried out on this basis. Therefore, two study steps, steady-state study step and transient study step, are supposed to be added successively in one study. In the steady-state study step, the physical field configuration needs to be modified by disabling the point load. As far as the transient solver, the step size used in the transient default solver is set to be exact, and the calculation time is 0 µs to70 µs with a step size of 0.001 µs.

In order to study the acoustoelastic effect in cross-section under different loads, a parametric scan is added with the parameter *l**0*, whose value varies from 0mm to 1mm with 0.1mm as an increment. The differences of the time delay in cross-sections under external loads can be found by comparing the time domain signals of the received points in the cross-sections with different fillet radii under different loads.

## 3.5. Settings for eigenfrequency study

For solving the eigenfrequencies of the cross-sections under external loads, the stress states of cross-sections under static external load must be found first, and then the eigenfrequency study is carried out on this basis. Therefore, two study steps, steady state study step and eigenfrequency study step are supposed to be added successively in one study. In the eigenfrequency study step, the reference value of eigenfrequency search is 200 kHz, and the number of solved eigenfrequencies is 10. Furthermore, for comparing the differences between the eigenfrequency of cross-sections under external loads and that without loads, a parametric scan must be added to the study, with the same settings as in Section 3.4.