Coupling Effects of Initial Stresses and Nonlinear Elasticity on the Propagation Characteristics of Elastic Waves
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摘要: 初始应力场与介质的非线性特性会影响弹性波在固体介质中的传播行为，准确表征受初始应力介质中弹性波的传播规律对于应力无损检测、结构健康监测及地球物理勘探等具有重要的意义. 目前研究对初始应力相关的几何非线性(有限变形)与物理非线性如何影响弹性波的传播还缺乏清晰认识. 本研究基于声弹性理论，建立了耦合有限的初始变形与物理非线性的弹性波传播理论框架，并通过摄动近似方法获得体波相速度的近似解析解，为快速计算受初始应力介质中弹性波的传播提供有效手段. 此外，通过求解平面波的特征方程并进行对比分析，系统揭示了初应力场与非线性弹性耦合效应作用下的弹性波传播规律. 研究结果表明，初始应力对弹性波传播的影响是几何非线性效应与物理非线性效应竞争的结果. 在拉伸初始应力作用下，几何非线性会使弹性波速度提高，而物理非线性会使弹性波速度降低. 物理非线性效应使得初始应力产生更显著的体波相速度改变及横波各向异性，在7075-T651铝合金材料中，横波各向异性程度AS可达到2%~3%.
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关键词:
- 初始应力 /
- 声弹性理论 /
- 弹性波传播 /
- 几何非线性与物理非线性 /
- 摄动理论
Abstract: The propagation of elastic waves in solids is influenced by initial stresses and material nonlinearity. Accurately characterizing the propagation of elastic waves in initially stressed media is crucial for stress nondestructive testing, structural health monitoring, and geophysical exploration. However, the distinct roles of stress-related geometric nonlinearity and material nonlinearity remain unclear. A theoretical framework based on acoustoelasticity and coupling finite initial deformation with material nonlinearity was developed. Approximate analytical solutions for the phase velocities of body waves were derived with the perturbation theory, to give an efficient approach for the rapid calculation of elastic wave propagation in initial stress media. Furthermore, the characteristic equation was solved for plane waves, the coupled effects of initial stress and nonlinear elasticity were systematically analyzed. The results show that, the effects of initial stresses stem from the competition between geometric and physical nonlinearity. Under tensile initial stresses, geometric nonlinearity will increase wave speeds, while physical nonlinearity will decrease them. Physical nonlinearity induces more pronounced changes in phase velocity and velocity anisotropy of shear waves. For the 7075-T651 aluminum alloy, the shear wave anisotropy reaches 2%~3%. -
图 1 自然构形、中间构形及当前构形的变形示意图
Figure 1. Schematic of the deformation among the natural, intermediate, and current states
图 2 物理非线性条件下P波、S1波和S2波相速度的摄动近似解和数值精确解的比较
Figure 2. Comparisons of the phase velocities of P-waves, S1-waves and S2-waves through perturbation approximations and exactly numerical solutions with physical nonlinearity
图 3 几何非线性条件下P波、S1波和S2波相速度的摄动近似解和数值精确解的比较
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 3. Comparisons of the phase velocities of P-waves, S1-waves and S2-waves through perturbation approximations and exactly numerical solutions with geometrically nonlinearity
图 4 受单轴初应力介质几何非线性及物理非线性条件下P波的相速度相对变化、极化矢量及极化各向异性
Figure 4. Relative variations of phase velocities, polarization vectors and polarization anisotropy of P-waves under geometric nonlinearity or physical nonlinearity in uniaxially stressed media
图 5 受单轴初应力介质几何非线性及物理非线性条件下S1波和S2波的相速度相对变化和横波相速度各向异性
Figure 5. Relative variations of phase velocities of S1-waves and S2-waves as well as the shear-wave phase velocity anisotropy under geometric nonlinearity or physical nonlinearity in uniaxially stressed media
图 6 受单轴初应力介质几何非线性及物理非线性条件下S1波和S2波的极化矢量与横波极化各向异性
Figure 6. Polarization vectors and polarization anisotropy of S1 and S2 waves under geometric nonlinearity or physical nonlinearity in uniaxially stressed media
图 7 受双轴初应力介质几何非线性及物理非线性条件下P波的相速度、极化矢量及极化各向异性
Figure 7. Relative variations of phase velocities, polarization vector and polarization anisotropy of P-wave under geometric nonlinearity or physical nonlinearity in biaxially stressed media
图 8 受双轴初应力介质几何非线性及物理非线性条件下S1波和S2波的相速度和横波相速度各向异性
Figure 8. Relative variations of phase velocities of S1-waves and S2-waves as well as the shear-wave phase velocity anisotropy under geometric nonlinearity or physical nonlinearity in biaxially stressed media
图 9 受双轴初应力介质几何非线性及物理非线性条件下S1波和S2波的极化矢量与横波极化各向异性
Figure 9. Polarization vectors and polarization anisotropy of S1 and S2 waves under geometric nonlinearity or physical nonlinearity in biaxially stressed media
图 10 受三轴初应力介质几何非线性及物理非线性条件下P波的相速度、极化矢量及极化各向异性
Figure 10. Relative variations of phase velocities, polarization vector and polarization anisotropy of P-wave under geometric nonlinearity or physical nonlinearity in triaxially stressed media
图 11 受三轴初应力介质几何非线性及物理非线性条件下S1波和S2波相速度和横波相速度各向异性
Figure 11. Relative variations of phase velocities of S1-waves and S2-waves as well as the shear-wave phase velocity anisotropy under geometric nonlinearity or physical nonlinearity in triaxially stressed media
图 12 受三轴初应力介质几何非线性及物理非线性条件下S1波和S2波极化矢量与横波极化各向异性
Figure 12. Polarization vectors and polarization anisotropy of S1 and S2 waves under geometric nonlinearity or physical nonlinearity in triaxially stressed media
表 1 7075-T651铝合金弹性常数
Table 1. Elastic constants of the 7075-T651 aluminum alloy
λ/GPa μ/GPa A/GPa B/GPa C/GPa 54.9 26.5 -351.2 -149.4 -102.8 
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