This study proposes a time-domain spectral finite element (SFE) method for simulating the second harmonic generation (SHG) of nonlinear guided wave due to material, geometric and contact nonlinearities in beams. The time-domain SFE method is developed based on the Mindlin–Hermann rod and Timoshenko beam theory. The material and geometric nonlinearities are modeled by adapting the constitutive relation between stress and strain using a second-order approximation. The contact nonlinearity induced by breathing crack is simulated by bilinear crack mechanism. The material and geometric nonlinearities of the SFE model are validated analytically and the contact nonlinearity is verified numerically using three-dimensional (3D) finite element (FE) simulation. There is good agreement between the analytical, numerical and SFE results, demonstrating the accuracy of the proposed method. Numerical case studies are conducted to investigate the influence of number of cycles and amplitude of the excitation signal on the SHG and its performance in damage detection. The results show that the amplitude of the SHG increases with the numbers of cycles and amplitude of the excitation signal. The amplitudes of the SHG due to material and geometric nonlinearities are also compared with the contact nonlinearity when a breathing crack exists in the beam. It shows that the material and geometric nonlinearities have much less contribution to the SHG than the contact nonlinearity. In addition, the SHG can accurately determine the crack location without using the reference data. Overall, the findings of this study help further advance the use of SHG for damage detection.

中文翻译：

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梁中材料、几何和接触非线性引起的导波二次谐波生成建模的时域谱有限元方法

本研究提出了一种时域谱有限元 (SFE) 方法，用于模拟由于梁中的材料、几何和接触非线性而产生的非线性导波的二次谐波 (SHG)。时域 SFE 方法是基于 Mindlin-Hermann 杆和 Timoshenko 梁理论开发的。通过使用二阶近似调整应力和应变之间的本构关系来模拟材料和几何非线性。采用双线性裂纹机理模拟呼吸裂纹引起的接触非线性。对 SFE 模型的材料和几何非线性进行分析验证，并使用三维 (3D) 有限元 (FE) 仿真对接触非线性进行数值验证。解析、数值和 SFE 结果之间有很好的一致性，证明了所提出方法的准确性。进行了数值案例研究，以研究激励信号的周期数和幅度对 SHG 及其在损伤检测中的性能的影响。结果表明，SHG的幅度随着激励信号的周期数和幅度的增加而增加。当梁中存在呼吸裂纹时，还将由材料和几何非线性引起的 SHG 的幅度与接触非线性进行比较。它表明材料和几何非线性对 SHG 的贡献远小于接触非线性。此外，SHG 可以在不使用参考数据的情况下准确确定裂纹位置。总体而言，本研究的结果有助于进一步推进使用 SHG 进行损伤检测。进行了数值案例研究，以研究激励信号的周期数和幅度对 SHG 及其在损伤检测中的性能的影响。结果表明，SHG的幅度随着激励信号的周期数和幅度的增加而增加。当梁中存在呼吸裂纹时，还将由材料和几何非线性引起的 SHG 的幅度与接触非线性进行比较。它表明材料和几何非线性对 SHG 的贡献远小于接触非线性。此外，SHG 可以在不使用参考数据的情况下准确确定裂纹位置。总体而言，本研究的结果有助于进一步推进使用 SHG 进行损伤检测。进行了数值案例研究，以研究激励信号的周期数和幅度对 SHG 及其在损伤检测中的性能的影响。结果表明，SHG的幅度随着激励信号的周期数和幅度的增加而增加。当梁中存在呼吸裂纹时，还将由材料和几何非线性引起的 SHG 的幅度与接触非线性进行比较。它表明材料和几何非线性对 SHG 的贡献远小于接触非线性。此外，SHG 可以在不使用参考数据的情况下准确确定裂纹位置。总体而言，本研究的结果有助于进一步推进使用 SHG 进行损伤检测。