Coupled wave propagation in meso-scale heterogeneous medium
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摘要: 非均匀介质在自然界中十分常见,针对细观非均匀介质的波动力学行为和非均匀性描述的研究具有重要意义并充满挑战。建立了反映细观非均匀材料压剪耦合特性的一般压剪耦合本构关系,提出了描述材料非均匀性的耦合系数,并建立了广义波动方程。广义波动方程数值分析表明,耦合系数的正负、取值和组合与应力/应变张量共同影响耦合波动传播过程。作为算例,给出了一阶近似的压剪耦合参数确定的本构关系以及3个压剪耦合特征波速的表达式,并利用有限差分法得到了耦合压缩波和剪切波的传播过程。研究了4个非均匀性耦合系数对应力状态、耦合波速和波传播过程的影响。耦合压缩波速反映了剪切对压缩的耦合效应和体积压实效应2种机制的竞争,耦合剪切波速反映了压缩对剪切的耦合效应和介质持续畸变带来的剪切弱化效应2种机制的竞争。这些机制可通过压剪耦合参数的不同组合来实现。应用真三轴实验系统测量了花岗岩、由砂浆制成的模型材料、具有粗骨料的水泥砂浆制成的材料3种非均匀介质在不同压剪应力下的纵波波速。结果表明,体积压实效应普遍存在,而非均匀程度越高,材料伸缩的同时完成切向的畸变导致压缩波的速度显著降低,剪切对纵波波速的影响越占据主导。理论计算结果与实验结果整体趋势基本一致。本研究可为非均匀材料的波速和动态力学性能研究提供物理机制方面的解释。Abstract: Heterogeneous media are very common in nature. Due to the complex internal structure, the heterogeneous compressive shear coupled stress field is inside heterogeneous media, which leads to a mutual influence of compression and shear waves. The study of wave mechanics behavior and description of heterogeneity in heterogeneous media is of great significance and full of challenges. This article establishes a general constitutive relationship that reflects the compression shear coupling characteristics of heterogeneous materials, proposes coupling coefficients to describe material heterogeneity, combines momentum conservation law to establish a generalized wave equation, and provides a general method for solving the generalized wave equation. As an example, expressions for the three characteristic wave velocities of compression shear coupling under the first-order compression shear coupling constitutive relationship are provided, and the finite difference method is employed to obtain the propagation process of coupled compression and shear waves. The effects of four heterogeneous coupling coefficients on stress state, coupled wave velocity, and wave propagation process are studied. The positive and negative values of coupling parameters and their combinations reflect the structural characteristics of heterogeneous media and also determine the properties of compression shear coupling waves. For heterogeneous media with high-pressure effects, shear dilation effects, and shear weakening effects, the coupled compression wave velocity is lower than the elastic compression wave velocity corresponding to uniform media, and the coupled shear wave velocity is higher than the elastic shear wave velocity. The effect of shear on compression delays the propagation of compressive stress, while compression promotes the propagation of shear. Coupled compression wave velocity is the result of the competition between the coupling effect of shear on compression and the volume compaction effect. Coupled shear wave velocity is the result of the competition between the coupling effect of compression on shear and the shear weakening effect caused by continuous distortion of the medium. These mechanisms could be achieved through different combinations of compression shear coupling parameters. A true triaxial experimental testing system was used to measure the longitudinal wave velocity of granite, model materials made of mortar, and materials made of cement mortar with coarse aggregates under different compressive and shear stresses. The results indicate that for heterogeneous media, the longitudinal wave velocity decreases with the increase of static water pressure and equivalent shear stress, and the shear expansion and shear weakening effects dominate. The experimental results and theoretical results have the same trend. The conclusion of this study is expected to provide a physical mechanism explanation for the phenomenon of the variation of wave velocity with stress state in heterogeneous materials.
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表 1 3个试样的参数
Table 1. Parameters for three specimens
试样材料 x方向长度/mm y方向长度/mm z方向长度/mm 质量/g 密度/(g·cm−3) 杨氏模量/GPa 泊松比 花岗岩 50.10±0.34 50.14±0.34 50.18±0.34 321 2.64 70.0 0.20 MMM 49.70±0.48 50.02±0.34 49.64±0.37 263 2.12 16.5 0.15 MMMA 50.12±0.34 50.20±0.34 50.20±0.34 292 2.31 45.0 0.25 表 2 试样的围压条件
Table 2. Confining pressure conditions of the specimen
序号 试样材料 压应力/MPa 静水压力/MPa 等效剪应力/MPa x方向 y方向 z方向 1 花岗岩 16 16 16 16.00 0 2 5 16 5 8.67 6.35 3 15 5 20 13.33 7.64 4 20 5 20 15.00 8.66 5 20 12 8 13.33 6.11 6 MMM 5 5 5 5.00 0 7 16 5 16 12.33 6.35 8 5 16 5 8.67 6.35 9 16 16 16 16.00 0 10 MMMA 5 5 5 5.00 0 11 16 16 16 16.00 0 12 16 5 16 12.33 6.35 13 5 16 5 8.67 6.35 14 20 12 8 13.33 6.11 -
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