Many geomaterials, such as shale, and biomaterials, including human bone and cartilage, are anisotropic. Understanding the dynamic responses of such anisotropic porous media is essential in field operations and laboratory measurements. This paper presents analytical solutions to the transversely isotropic poroelastodynamics Mandel's problem, using the Fourier transform method and Cardano formula to solve coupled six-order partial differential equations. Potential applications of the solutions include explaining coupled fluid-solid dynamics, validating numerical algorithms, and interpreting dynamic responses of anisotropic porous media. As an example, we apply the solutions to simulate a fluid-saturated transversely isotropic Trafalgar shale specimen subjected to harmonic excitation. The simulation shows that pore pressure builds up as frequency increases. The buildup occurs at a lower frequency for a lower horizontal permeability. The amount of pore pressure buildup due to the Mandel-Cryer effect is sensitive to material anisotropy. The porous material's effective stiffness reaches its periodic minimum and maximum at the resonant and anti-resonant frequencies controlled by mechanical anisotropy and pore fluid.

中文翻译：

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横向各向同性多孔弹性动力学曼德尔问题的基本解

许多地质材料（例如页岩）和生物材料（包括人骨和软骨）都是各向异性的。了解这种各向异性多孔介质的动态响应对于现场操作和实验室测量至关重要。本文提出了横向各向同性多孔弹性动力学曼德尔问题的解析解，使用傅立叶变换方法和卡尔达诺公式求解耦合的六阶偏微分方程。这些解决方案的潜在应用包括解释耦合流固动力学、验证数值算法以及解释各向异性多孔介质的动态响应。例如，我们应用这些解决方案来模拟受谐波激发的流体饱和横向各向同性 Trafalgar 页岩标本。模拟表明，孔隙压力随着频率的增加而增加。对于较低的水平渗透率，堆积发生在较低的频率。由 Mandel-Cryer 效应引起的孔隙压力增加量对材料各向异性很敏感。多孔材料的有效刚度在由机械各向异性和孔隙流体控制的共振和反共振频率下达到其周期性最小值和最大值。