Responses of the multilayered transversely isotropic medium subjected to dynamic rectangular loadings

• Published in: Archive of applied mechanics (1991) Vol. 92; no. 1; pp. 73 - 98
• Main Authors: Qiao, Hong, Zhang, Pengchong, Du, Xianting, Long, Peiheng
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2022; Springer Nature B.V
• Subjects: Cartesian coordinates; Classical Mechanics; Differential equations; Domains; Elastodynamics; Engineering; Equilibrium equations; Integral transforms; Isotropic media; Material properties; Matrices (mathematics); Ordinary differential equations; Original; Soil layers; Theoretical and Applied Mechanics; Thickness; Elastodynamic responses; Mixed dual vector technique; Multilayered transversely isotropic medium; Time-harmonic rectangular loadings; Precise integration approach
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-021-02042-9

The elastodynamic responses of the multilayered transversely isotropic medium subjected to uniform pressures occupying a rectangular flexible domain are precisely investigated by virtue of the precise integration approach (PIA) and the mixed dual vector technique. The dynamic rectangular distributed loadings are prescribed either at the external surface or within the stratified soil. The axes of material symmetry for different strata are assumed to be normal to the interfaces of adjacent layers and thus parallel with each other. As for the introduced technology, there is no limit of the thickness and number of soil layers to be discussed. The classical two-dimensional Fourier integral transformation is applied to convert the governing partial differential equilibrium equations corresponding to elastic displacements in the Cartesian coordinate system, and a set of easily solved second-order ordinary differential equations are acquired. Aided by the mixed dual vectors, the second-order ordinary differential equations are further reduced into first-order ordinary differential matrix equation and its general solution is analytically expressed as the exponential matrix. The PIA is adopted to estimate the changing patterns of displacement components by means of solving the exponential matrix in the Fourier-transformed domain. As a highly precise methodology, the PIA is advantageous to obtain the dynamic solutions with any desired accuracy. The physical-domain counterpart is then derived with the aid of the double Fourier inverse transform. Finally, comparisons with existing numerical results are provided to validate the high accuracy of the employed scheme. Selected numerical examples are presented to portray the dependence of the dynamic responses of the multilayered media on material properties, thicknesses, frequencies of excitation and dimensions of the loading size.