Analytical Solution for 2D and 3D Lamb Problem in Saturated Soil Incorporating Effects of Compressibility of Solid and Pore Fluid

In this paper, to avoid the complexity, a simple and efficient analytical solution is derived for both 2D and 3D Lamb problems, respectively, in saturated soil under harmonic excitations. Unlike the existing solutions, the proposed solutions for both 2D and 3D Lamb problems in saturated soil under harmonic excitation are capable of well revealing the effect of compressibility of both liquid phase and solid phase on the ground displacements. By applying Fourier transforms and Hankel transforms on the governing equations of wave propagation in saturated soil, wave equations are transformed to ordinary differential equations. Combining the boundary conditions and draining conditions on the ground surface, the displacement solutions on the surface of saturated porous soil due to line and point harmonic excitations are derived, respectively. Then, the solutions in frequency domain are obtained by inverse integral transforms. In the meanwhile, for the sake of discussion without losing its generality, the nondimensional solutions for three-dimensional Lamb problem are derived. The effectiveness and accuracy of the proposed solutions are demonstrated by employing three different approaches. Finally, parametric studies are conducted to investigate the effects of the governing parameters (i.e., exciting frequency, bulk modulus of soil matrix, and bulk modulus of pore fluid) on variation of non-dimensional displacement with the increasing distance away from the excitation source. The results indicate that, in contrast to the effect of the compressibility of soil matrix, the exciting frequency as well as the compressibility of the pore fluid play significant role in affecting the variation of displacement on ground surface subjected to excitations, which particularly highlights that the compressibility of the pore fluid should be carefully considered for evaluating the ground movements.


INTRODUCTION
The classical problem of solutions for elastic waves generated by external forces in isotropic elastic half-space with plane surfaces was presented, (Lamb, 1904) [1], which has been called Lamb's problem. Since then, this problem has been widely used in various fields, such as foundation engineering (Liu, G. Y. et al. 2007) [2] earthquake engineering, layered elastic half-space, general anisotropy. Also, the Lamb's problem has been effectively extended to different areas, including contact mechanics and ultrasonic nondestructive evaluation (Han, J. H. et al. 2015) [3].
By means of Fourier transforms with respect to space and time variables, integral form expressions were obtained by Lamb. For the integral expressions, Cagniard (1938) [4] stated that a suitable distortion of the integration contour can lead to an exact closed expression. The Cagniard's method was then simplified (de Hoop, 1960) [5]. The foregoing technique for integration has become known as the Cagniard-de Hoop method. By applying the integral transform method and the inverse transformation technique, exact solution for a two-dimensional Lamb's problem due to a strip impulse loading in elastic media was presented   [6]. The details for solving Lamb's problem using finite element method were explored (Kravtsov, 2011) [7]. The efficiency and accuracy of integration operation were studied (Guenfoud, 2009;Arcos, 2013) [8,9].
For anisotropic elastic solids, Kraut (1960) [10] solved the Lamb problem in a transversely isotropic solid when the surface of the half-space is parallel or perpendicular to the symmetry axis of material using Cagniard-de Hoop method (Kraut, E. A. et al. 1963) [10]. Burridge (1970) [11] applied Cagniard-de Hoop method to an anisotropic solid in the most general class. Hereafter, the Lamb problem was studied by means of different methods, such as self-similar boundary-initial value problem method (e.g. Willis (1973), Willis (1975) , Wang (1996)) [12][13][14].
For saturated porous media, Lamb problem due to line source and point source was solved   [15,16] using potential decomposition and Laplace-Fourier transforms. The complicated displacement solution was presented by means of Cagniard-de Hoop method. Since the item denoting couple effects between soil phase and fluid phase in the governing equations cannot be measured easily and the fluid viscosity is neglected with a result that the solution could not be used widely. Using Helmholtz decomposition method, Lamb problem in half-space due to harmonic loading was studied and the solution with integral form was obtained (Philippacopoulos, 1988) [17]. But Sharma (1992) [18] pointed out that the boundary conditions used by Philippacopoulos were not correct. Again the problem was solved using the foregoing method by Sharma. Lamb problem in three-dimensional nonaxisymmetrical space in transversely isotropic saturated porous media was studied by Huang et al. (2004) [19] using Helmholtz decomposition and integral transforms method. Transient Lamb problem in cylindrical coordinates in transversely isotropic saturated porous media was calculated using Laplace-Fourier-Hankel transforms (Wang, 2011) [20]. However, the compressibility of fluid phase and solid phase is not taken into account in the above existing studies. In particular, expressions of the solutions to the Lamb problem in saturated soil, such as transversely isotropic saturated porous soil, are often significantly complicated. It is well known that the compressibility of materials is a basic property. The compressibility of fluid and (or) solid is considered in some engineering problems, such as analytical solutions for damage initiation induced by pore pressure variation (Mattos, H. S. D. C. et al. 2018) [21], analysis of beams on layered poroelastic soils (Ai, Z. Y. et al. 2016) [22], and optimal control problem for flows (Tamellini, M, et al. 2018) [23]. Likewise, it is of great importance for considering the compressibility of fluid phase and solid phase in the Lamb problem.
To address the above issues, in this paper, a simple and efficient analytical solution is derived for the Lamb problems in saturated soil under harmonic excitation, where the compressibility of liquid phase and solid phase is considered. Considering a large number of plane strain problems in engineering, not only the solutions of 3D case but also those of 2D case are presented, in order to give all aspects of the problem. By applying Fourier transforms and Hankel transforms on the governing equations of wave propagation in saturated soil, the displacement solutions on the surface of saturated porous soil due to line and point harmonic excitations are derived, respectively. Furthermore, the solutions in frequency domain are obtained by inverse integral transforms. The effectiveness and accuracy of the proposed solutions is demonstrated by employing three different approaches. In the meanwhile, for the sake of discussion without losing its generality, the non-dimensional solutions for three-dimensional Lamb problem are derived. Finally, parametric studies are conducted to investigate the effects of the governing parameters (i.e., exciting frequency, bulk modulus of soil matrix, and bulk modulus of pore fluid) on variation of non-dimensional displacement with the increasing distance away from the excitation source. The results and discussion are presented in details.
These equations are the governing equations for steady state problems in compressible saturated porous media.

General Solutions for Displacement
To solve the Lamb problem, the general solutions for the governing equations will be derived by means of Fourier transforms. For two-dimensional problems, assuming that the functions in Eq. (4) are independent of coordinate y, Fourier transforms of the function f (x, z) with respect to coordinate x can be defined as: Applying Fourier transforms defined above to Eq. (4) leads to: Substituting Eq. (9) into Eq. (7), the following equation can be obtained: From Eq. (1), equations can be given as: Again, applying Fourier transforms defined in Eq. (4) to Eq. (12) gives: Subtracting α•Eq. (13b) from Eq. (13a) leads to: Then substituting Eq. (14) into Eq. (13a) gives another ordinary differential equation: According to Eq. (15), z u  can be solved as follows: where A 3 is the arbitrary function with respect to p. Substituting Eq. (9) and Eq. (16) Considering e plying the Fourier transforms to both sides of these equations, the following equations can be obtained: Then substituting Eq. (7), Eq. (9) The following parameters are introduced for simplification.

 
Thus, the above equations are the general solutions for governing equations for wave propagation in the porous saturated media in two-dimensional space.

Boundary Conditions and Special Solutions
The arbitrary functions, such as 1 * A , 2 * A and 3 * A , in the Eq. (21) should be determined by boundary conditions in Lamb problem. Then special solutions, namely Lamb problem's solutions, will be presented. Assuming the ground surface is permeable, Fig. 1 shows a schematic diagram for two-dimensional Lamb problem. The vertical concentrated harmonic force is acting on the surface at origin point. The boundary condition can be given as follows:

Figure 1 Sketch of 2-D Lamb problem
Applying Fourier transforms defined in Eq. (4) to constitutive equation Eq. (2) gives: Substituting Eq. (21) into Eq. (23) and in turn into Eq. (22) yields: Solution of Eq. (24) gives the following expressions: Substituting Eq. (25) into Eq. (21), the displacement solutions in transformed domain for two-dimensional Lamb problem can be given by Applying inverse Fourier transforms to Eq.(26), the displacement solutions in frequency domain can be expressed as:

THREE-DIMENSIONAL LAMB PROBLEM 3.1 General Solutions for Displacement
The n-th order Hankel transforms of F (r, z) with respect to r is defined as: where   n J pr is the Bessel function of the first kind of order n.
Utilizing symmetrical cylindrical coordinate in threedimensional space, and applying Hankel transforms instead of Fourier transforms to Eq. (4), the displacement solutions can be obtained as follows: A are arbitrary functions with respect to p, which can be determined by the boundary conditions of special problem. Fig. 2 shows a schematic diagram for three-dimensional Lamb problem, where the ground surface is assumed to be permeable. The harmonic force, namely i 0 e t P   , is acting on the ground surface at origin point along vertical direction. Assuming the force is acting within a circle plane with radius of r 0.

Boundary Conditions and Special Solutions
The vertical stress due to the force can be written as: Finally, applying inverse Hankel transforms to Eq. (34), the displacement solutions in frequency domain can be obtained as:

PARAMETRIC STUDY FOR SOLID SKELETON DISPLA-CEMENT IN 3D PROBLEM 5.1 Non-Dimensional Governing Equations and Displacement Solutions
Prior to parametric study, the solid skeleton displacement solutions for three-dimension against distance to origin are firstly discussed in detail herewith. For the sake of discussion without losing its generality, the non-dimensional governing equations are given firstly.
Taking the non-dimensional parameters as shown below:  . It is observed that, for real part of ground displacements of solid skeleton in both horizontal and vertical directions, the effect of non-dimensional excitation frequency *  on the variation of non-dimensional displacement against the distance from excitation source is insignificant. In contrast, for imaginary part of horizontal displacements, an increase in *  leads to a significant increase in the amplitude of horizontal displacement of solid skeleton, but this effect would decrease gradually with the increasing distance away from the excitation source. It is found that the maximum imaginary amplitude occurs when frequency *  is taken the value 0.06. Also, the effect of *  on the amplitude of vertical displacement of solid skeleton is relatively insignificant.  Fig. 7 present the displacements of solid skeleton along horizontal and vertical direction, respectively, with the increasing distance from the excitation source corresponding to the parameter  varying from 0.502 to 0.902. Note that, based on the definition of parameter  , the greater the parameter  is, the larger the bulk modulus of soil matrix is if the bulk modulus of soil skeleton keeps unchanged.
In this study, the other parameters (except  ) in Tab. 2 remain constant. As observed in Fig. 6 and Fig. 7, it can be found that the variation of the displacements of solid skeleton along both horizontal and vertical direction is insensitive to the change in the value of  (i.e., bulk modulus of soil matrix). Fig. 8 and Fig. 9 present the displacements of solid skeleton along horizontal and vertical direction, respecti-