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An exact closed-form solution of three-dimensional transient Green's functions in a constrained transversely isotropic half-space
International Journal of Engineering Science ( IF 6.6 ) Pub Date : 2020-11-17 , DOI: 10.1016/j.ijengsci.2020.103394
Mehdi Raoofian-Naeeni , Morteza Eskandari-Ghadi , Ernian Pan

In this paper, we derive an exact closed-form solution of three-dimensional transient Green's functions (GFs) at the free surface of a constrained transversely isotropic (TI) half-space due to an arbitrarily oriented surface point force of Heaviside time variation. The solution clearly delineates the distinct features of body waves (P-, SV- and SH- waves) as well as Rayleigh surface wave in displacement components and demonstrates the travel time of each wave amplitude in terms of inequalities among elastic coefficients. It is further shown that owing to material anisotropy, the radial and angular displacements due to horizontal point force may contain a special disturbance, which attenuates proportionally to the inverse square of epicentral distance from the source whilst varies linearly with time. This novel feature is attributed to the peculiarities of shear wave splitting, a consequence of polarization, with different velocities, of the SV- and SH- waves in TI materials, and as such, does not exist in isotropic materials. It is also worthwhile to mention that for this constrained anisotropy, the Rayleigh wave function is degenerated into a new elegant formulation whose real root specifies the Rayleigh wave velocity in terms of elastic coefficients. Moreover, the transient GFs are computed for the entire range of the elastic coefficients including also the case when the Rayleigh function has complex conjugate roots. For validation, the new GFs derived in this paper are suitably degenerated into the corresponding solutions for the isotropic (Poisson's) material and are compared with the existing solutions. The effect of Poisson's ratio on the isotropic response functions is graphically illustrated and so is the effect of material anisotropy for different synthetic TI materials.



中文翻译:

受约束的横向各向同性半空间中三维瞬态格林函数的精确闭式解

在本文中,由于Heaviside时间变化的任意定向表面点力,我们在约束的横向各向同性(TI)半空间的自由表面上导出了三维瞬态格林函数(GFs)的精确封闭形式解。该解决方案清楚地描述了体波的独特特征(P-SV-SH-波)以及位移分量中的瑞利面波,并根据弹性系数之间的不等式展示了每个波幅的传播时间。进一步表明,由于材料的各向异性,水平点力引起的径向和角位移可能包含特殊的扰动,该扰动与距震源的震中距离的平方成反比,而随时间线性变化。这种新颖的特征归因于SV-SH的剪切波分裂的特殊性,这是极化的结果,具有不同的速度极化-TI材料中的波,因此在各向同性材料中不存在。还值得一提的是,对于这种受约束的各向异性,瑞利波函数退化为一个新的优雅公式,其真实根以弹性系数来指定瑞利波的速度。此外,对于弹性系数的整个范围计算瞬态GFs,还包括瑞利函数具有复共轭根的情况。为了进行验证,将本文中衍生的新GF适当地退化为各向同性(泊松)材料的相应解决方案,并将其与现有解决方案进行比较。图解说明了泊松比对各向同性响应函数的影响,以及不同合成TI材料的材料各向异性的影响。

更新日期:2020-11-17
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