Characterization and simulation methods for acoustoelastic effect in the ribbed plate structure

doi:10.21203/rs.3.rs-2085919/v1

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Characterization and simulation methods for acoustoelastic effect in the ribbed plate structure

https://doi.org/10.21203/rs.3.rs-2085919/v1

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The acoustoelastic effect in complex structures is gradually focused by scholars. However, there are no valid methods to characterize and simulate the acoustoelastic effects in complex structures such as ribbed plate structures. Therefore, this paper proposes methods to characterize the acoustoelastic effects in such structures respectively based on wave propagation delay and eigenfrequency, and then develops corresponding simulation methods for acoustoelastic effects of the representative cross-sections of the ribbed plate structure. Finally, the acoustoelastic effects in cross-sections of ribbed plate structures with different fillet radii are characterized based on the Degree of Acoustoelastic Effects (DAEE) defined by wave propagation delay and eigenfrequency respectively, and the related laws of DAEE are summarized.

Ribbed Plate Structure

Acoustoelastic Effect

Simulation model

Time Delay

Eigenfrequency

The acoustoelastic effect refers that the velocity of waves in solid materials is related to the stress state of this material [1]. Acoustoelastic effects in simple structures have been studied adequately by many scholars with analytical [2] experimental [3] [4] and simulation [5] approaches. A few of research about the acoustoelastic effect of waves in complex structures has been done [6–12]. For example, Mase [6] and Leisk [7] studied the acoustoelastic effect in fiber reinforced composites, while Manchem [8] and Minicucci [9] applied the acoustoelastic effect to measure residual stresses in railroad rails and acoustoelastic coefficients in railway wheels, respectively.

However, few papers have been reported with respect to the acoustoelastic effect in geometrically complex structures such as ribbed plates. Because of the complex geometric structure of ribbed plates compared to the simple structure such as plates, the stress state of this structure is more complex when it is subjected to external loads, leading to more complex acoustoelastic effects of waves in this structure. Apparently, acoustoelastic effects of waves in complex structures are difficult to be characterized by analytical methods, therefore, it is vital to adopt suitable indices to characterize the acoustoelastic effects of waves in the ribbed plates and to develop a simulation model for solving the acoustoelastic effects of waves in this structure.

The ribbed plate structure in this paper refers to the specific structure formed by a piece of metal plate through mechanical milling and chemical etching process, as shown in Fig. 1, which is widely used in engineering structures such as rocket propellant tanks, large oil storage tanks, etc. Due to the limitations of this forming process, there is always a fillet between the rib and the plate and with a radius comparable to the thickness of the rib.

2.1 Characterization of acoustoelastic effects in simple waveguide structures

As described in the introduction, the acoustoelastic effect refers to the effect of stress on the velocity of waves in solid materials. When the direction of wave propagation in solid materials is parallel to the direction of stress, the analytical relationship between the velocity of longitudinal and transverse waves and the stress [2, 3] are respectively shown as Eqs. (1) and (2). When the propagation direction of the wave in solid material is perpendicular to the direction of stress, the analytical relationship between the velocity of longitudinal and transverse waves and stress are respectively shown as Eqs. (3) and (4).

$${\rho }_{0}{C}_{Lx}^{2}=\lambda +2\mu -\frac{\sigma }{3{k}_{0}}\left[2l+\lambda +\frac{\lambda +\mu }{\mu }\left(4m+4\lambda +10\mu \right)\right]$$

$${\rho }_{0}{C}_{Tx}^{2}=\mu -\frac{\sigma }{3{k}_{0}}\left[m+\frac{\lambda n}{4\mu }+\lambda +2\mu \right]$$

$${\rho }_{0}{C}_{Ly}^{2}=\lambda +2\mu -\frac{\sigma }{3{k}_{0}}\left[2l-\frac{2\lambda }{\mu }\left(m+\lambda +2\mu \right)\right]$$

$${\rho }_{0}{C}_{Ty}^{2}=\mu -\frac{\sigma }{3{k}_{0}}\left[m-\frac{\lambda +\mu }{2\mu }-2\lambda \right]$$

where, ${\rho }_{0}$ is the density of the material, ${C}_{L}$ and ${C}_{T}$ are the velocity of longitudinal and transverse waves respectively, the subscripts x and y mean that the direction of wave propagation are parallel and perpendicular to the stress respectively, $\sigma$ refers to the axial stress, ${k}_{0}$ represents the bulk modulus of solid materials, $\lambda$ and $\mu$are Lame constants, $l$, $m$ and $n$ are third-order elastic constants.

It follows that the acoustoelastic effect in a simple structure subjected to a simple external load can be directly characterized by an analytical expression.

2.2 Characterization of acoustoelastic effects in complex waveguide structures

For a complex waveguide structure under external loads, on the one hand the irregularity of its geometry inevitably leads to a non-uniform internal stress, and the wave propagation direction and stress in this structure are usually neither parallel nor perpendicular on the other. As a result, it is difficult to characterize the relationship between wave velocity and stress in this structure by analytical expressions. Consequently, other suitable parameters are put forward to characterize this acoustoelastic effect necessarily.

As shown in Fig. 2, due to the acoustoelastic effect, the wave velocity at various points inside solid materials consequentially changes when the solid materials are subjected to external loads. When an elastic wave propagates in this solid material, the change of wave velocity directly leads to the variation of the time for the elastic wave to travel a fixed distance in a fixed direction, resulting in a time delay. It follows that this time delay reflects the cumulative effect of the variety of wave velocity at each point in the structure along a fixed direction and distance. Therefore, the time delay can be used as a parameter to indirectly reflect the acoustoelastic effect of complex waveguide structures.

Furthermore, the dispersion characteristics of guided waves in waveguide structures are also a matter of concern. It has been shown that the dispersion curves (phase velocity versus frequency curves) of guided waves in waveguide structures under stress will be shifted [4], as shown in Fig. 3 for the dispersion curves of A0 mode Lamb waves in plates subjected to a series of unidirectional stresses. However, the dispersion curves of guided waves in complex waveguide structures are difficult to obtain, so it is impossible to characterize the acoustoelastic effect of guided waves in complex structures by the degree of shift of dispersion curves.

However, the change in phase velocity under stress is related to the variety of the wave number, which in turn can be reflected by the eigenfrequency of the waveguide structure. Therefore, the acoustoelastic effect can be characterized by the change in the eigenfrequency of the waveguide structure [5].

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.

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

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 l₀, 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.

The results of the time delay study and the eigenfrequency study have been obtained after the simulations. All simulations were run on a workstation with Windows 10, equipped with a 24-core, 48-thread Intel Xeon W3265M processor and 224GB RAM.

4.1. Acoustoelastic effects characterized by time delay

Taking the results in the cross-section with R/H = 0 as an example, the time curves of the X-component u of the displacement field obtained at the receiving point under different external loads are shown in Fig. 7. Through analyzing the results around the fifth wave crest of these curves, it can be found that the acoustoelastic effect characterized by the time delay as an indicator under different pre-deformation shows an approximately linear characteristic.

For quantitatively characterizing the acoustoelastic effects in cross-sections under loads, the values of the time delays must be found accurately. As an example, the time delay in a section with R/H = 0 at the pre-deformation l₀ = 0.5 mm can be obtained by taking the difference between the corresponding moments of points 0 and 5 as shown in Fig. 7. The time delays in different sections under loads can also be obtained by the same method, and the results are shown in Table 1 and plotted in Fig. 8.

Table 1

Time delays in cross-sections with different fillet radii under loads
$\varDelta t$	l₀ = 0.1 mm	l₀ = 0.2 mm	l₀ = 0.3 mm	l₀ = 0.4 mm	l₀ = 0.5 mm	l₀ = 0.6 mm	l₀ = 0.7 mm	l₀ = 0.8 mm	l₀ = 0.9 mm	l₀ = 1.0 mm
R/H = 0	0.009	0.018	0.027	0.036	0.045	0.054	0.063	0.072	0.081	0.090
R/H = 0.2	0.011	0.022	0.033	0.044	0.055	0.066	0.077	0.088	0.099	0.110
R/H = 0.4	0.009	0.018	0.027	0.037	0.046	0.055	0.065	0.074	0.083	0.093
R/H = 0.6	0.008	0.015	0.023	0.030	0.038	0.046	0.053	0.061	0.069	0.076
R/H = 0.8	0.008	0.016	0.023	0.031	0.039	0.046	0.054	0.062	0.070	0.077
R/H = 1.0	0.010	0.019	0.028	0.038	0.047	0.057	0.066	0.075	0.085	0.094

From the results shown in Fig. 8, the time delay in every section increases linearly with the increase of the prestress characterized by the predeformation. The rate of time delay and predeformation, the slope of the line in the figure, can be defined as the degree of acoustoelastic effects (DAEE), which can be used to characterize the sensitivity of time delay to stress in cross-sections with different corner radii. All values of DAEE of cross-sections were calculated, as shown in Table 2.

Table 2

The values of DAEE of cross-sections
R/H	0	0.2	0.4	0.6	0.8	1.0
DAEE (µs mm^-1)	0.090	0.110	0.093	0.076	0.073	0.094

It follows that the time delay in the cross-section with R/H = 0.2 is most sensitive to stress, the DAEE values in the cross-sections with R/H = 0, 0.4, 1.0 are approximated, while the time delays in the cross-sections with R/H = 0.6 and 0.8 are least sensitive to stress. Apparently, there is no linear law between DAEE value and the fillet radius of the cross-section.

4.2. Acoustoelastic effects characterized by the variety of eigenfrequency

As described in Section 2.2, the acoustoelastic effect in complex waveguide structures can be characterized by the variation of the eigenfrequencies, in another word, the eigenfrequencies of the structure will change when external loads are applied. The simulation results take 10 eigenfrequencies near 200 kHz and derive the variation of these eigenfrequencies under a series of predeformations. The eigenfrequency values of the cross-section with R/H = 1.0 under loads are plotted as a line graph, as shown in Fig. 9.

As shown in Fig. 9, the values of all the eigenfrequencies of the cross-section with R/H = 1.0 decrease linearly with the increase of predeformation, but the degree of linear decrease varies for different eigenfrequency. As a result, the Degree of Acoustoelastic Effects (DAEE) based on the characteristic frequencies can be defined as the change of eigenfrequencies per unit predeformation increment, i.e., the slope of line in the Fig. 9, to characterize the degree of eigenfrequencies affected by stresses in the cross-sections. The values of DAEE at these 10 eigenfrequencies of cross-sections are calculated separately, as shown in Table 3, in Hz/mm.

Table 3

The values of DAEE at these 10 eigenfrequencies of cross-sections
R/H	1	2	3	4	5	6	7	8	9	10
0.0	8403.1	7882.1	5357.7	6156.5	4438.2	9329.6	2462.1	807.1	10557.9	3055.3
0.2	-101.6	-366.0	-79.5	-157.1	-627.3	-159.9	-375.7	-484.2	-218.6	-233.0
0.4	-114.7	-283.0	-80.5	-191.2	-587.9	-188.4	-390.4	-456.0	-222.1	-271.1
0.6	-141.2	-258.2	-100.2	-234.6	-563.4	-177.8	-338.6	-492.7	-225.7	-292.5
0.8	-85.6	-180.1	-136.9	-333.5	-209.8	-530.1	-159.4	-274.4	-568.3	-228.3
1.0	-98.7	-213.4	-161.0	-448.9	-137.1	-501.4	-156.5	-229.5	-628.3	-221.2

It follows that the values of eigenfrequency of the section with R/H = 0 increase with the increase of the predeformation value, while the eigenfrequency values of the remaining sections vary reversely. The eigenfrequency 9 of the sections with R/H = 0, 0.8, 1.0 are the most sensitive to stress, while the eigenfrequency 5 of the other sections are the most affected by stress.

In this paper, in view of the lack of characterization and simulation methods for the acoustoelastic effect in complex structures such as ribbed plate with fillets, the characterization methods for the acoustoelastic effect based on wave propagation time delay and eigenfrequency are proposed respectively, and a simulation method for the acoustoelastic effect in this structure is developed with the cross-section representing the ribbed plate structure. Finally, the degree of acoustoelastic effect (DAEE) is defined respectively based on time delay and eigenfrequency, characterizing the acoustoelastic effects in cross-sections of the ribbed plate structures with different fillet radii, whose laws can be summarized as follows.

(1) When the wave propagation time delay is used to characterize the acoustoelastic effect in the ribbed plate structure, the time delays of wave propagation in cross-sections of the ribbed plate structures with different fillet radii increase linearly with the increase of predeformation. the DAEE value in the cross-section with R/H = 0.2 is the largest which means that the time delay in this structure is the most affected by the stress, and the DAEE values in the cross-sections with R/H = 0, 0.4, and 1.0 are the second, while the time delays in cross-sections with R/H = 0.6 and 0.8 are the least sensitive to the stress. In addition, there is no linear law between the DAEE values and the radius of the fillet in the cross-section of the ribbed plate structure.

(2) When the acoustoelastic effect in the ribbed plate structure is characterized based on the eigenfrequency, the eigenfrequencies of cross-sections of the ribbed plate structures with different fillet radii vary linearly with the predeformation, among which eigenfrequencies of the cross-section of the ribbed plate structure with R/H = 0 increase linearly with the increase of predeformation, while those of other sections are just the opposite. And the of the cross-sections of the ribbed plate structures with R/H = 0, 0.8 and 1.0 are the most affected by the stress, which means that the absolute values of DAEE at eigenfrequency 9 of these cross-sections are the largest respectively, while the eigenfrequency 5 of the other sections are the most affected by stress.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability

All data generated or analyzed during this study are included in this published article.

Funding

This study was supported by the National Natural Science Foundation of China (Grant no. 51975581).

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No competing interests reported.

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Characterization and simulation methods for acoustoelastic effect in the ribbed plate structure

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Abstract

Figures

1. Introduction

2. Characterization Of Acoustoelastic Effects

2.1 Characterization of acoustoelastic effects in simple waveguide structures

2.2 Characterization of acoustoelastic effects in complex waveguide structures

3. Simulation Model Of Acoustoelastic Effect In The Ribbed Plate Structure

3.1. Geometric model

3.2. Physical field settings

3.3. Meshing

3.4. Settings for time delay study

3.5. Settings for eigenfrequency study

4. Results Of Simulation

4.1. Acoustoelastic effects characterized by time delay

4.2. Acoustoelastic effects characterized by the variety of eigenfrequency

5. Summary And Conclusions

Declarations

References

Additional Declarations

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