Abstract
Stress, displacement, and pore pressure of a partially sealed spherical cavity in viscoelastic soil condition have been obtained in Laplace transform domain. Solutions of axisymmetric surface load and fluid pressure are derived.
1. Introduction
Biot [1, 2] presented the propagation theory of elastic waves and the general solutions for fluid-saturated porous viscoelastic medium. Akkas and Zakout [3] discussed the solution for the transient response for an axisymmetric and nontorsional load of an infinite, isotropic, elastic medium containing a spherical cavity with and without thin elastic shell embedment. In this paper, considering a viscoelastic model presented by Eringen [4], the transient response of a spherical cavity with a partially sealed shell embedded in viscoelastic soil is investigated. The solutions of stresses, displacements and pore pressure induced by axisymmetric surface load and fluid pressure are derived in Laplace transform domain. Durbin’s [5] inverse Laplace transform is used to analyze the influence of partial permeable property of boundary and relative rigidity of shell and soil on the transient response of the spherical cavity. The solutions of permeable and impermeable boundary without shell are considered as two extreme cases.
2. Basic Equations and Solutions
In infinite viscoelastic saturated soil, a thin elastic shell shown in Figure 1 with inner radius , outer radius , and thickness , has been bored. are the spherical coordinates, where and are the meridional and circumferential angles, respectively, , , are nonvanishing components of stress tensor in case of an axisymmetric nontorsional load, that is, independent of and acting on the shell surface.
In spherical coordinate system , the equilibrium equation for soil mass is where and are radial displacement of soil skelton and displacement of pore fluid with respect to soil skelton, respectively; , the density of soil; and are densities of fluid and soil grains respectively; is porosity.
The pore fluid equilibrium equation is given by where is excess pore pressure; is the fluid viscosity, and is the intrinsic permeability of soil.
Soil is not an ideal medium. Due to overcoming the interior friction of soil, a part of energy of the propagation wave is changed into heat energy during the propagation. This property is known as damping of material. Assuming that the viscoelastic property of soil may be simulated by Kelvin-Voigt model. Following Eringen [4], the stress-strain relationship is expressed as where and , dilations of solid and fluid, respectively; and are Lame constants of the bulk material; and are the dilatant and shear constant of the viscoelastic soil; and are the compressibility parameters of the two phase medium, , and , for a material with incompressible constituents.
Substituting (3), (4), and (5) into (1) and (2), the governing equations of the transient response of a spherical cavity in viscoelastic solid condition can be reduced as Now, displacements and are assumed to be of the forms for solving (6) and (7). Consider Then, Similarly, Substituting (8), (9), (11), and (12), into (6), we obtain Integrating w.r.t. , we have where .
Substituting (8) to (12), in (7), we obtain Integrating w.r.t. , we have
Taking Laplace transform in (14) yields
Taking Laplace transform in (14) yields
In a great range of vibration frequencies, viscoelastic damp coefficient of rock and soft soil may be assumed as a constant. The dimensionless damp coefficient is considered as Also, are nondimensional Lame constant, compressibility parameter, fluid density, and permeability coefficient of soil respectively. is the radius of spherical shell, .
Using (19) and (20) in (17) yields Similarly, using (19) and (20) in (18) yields Solving (21) and (22) yields Equation (24) can be written as where and are the complex wave number of two dilation waves, that is, with
The general solutions of and in (25) are
Considering the limitation property of radial displacement when , that is, In (28), we have, , Constants , , , and in (30) and (31) are linearly dependent and may be related by using (25) to obtain where .
, can be obtained from boundary conditions. Now, The Laplace transformed solution of radial displacement , that is, is given by Similarly, the Laplace transform solution of is : Next, by (5), Taking Laplace transform and using (34) and (35), we have Next, by (3), Taking Laplace transform of both sides Similarly, by (4), (35), and (37), we have Then
3. Solution of Shell Embedment and Axisymmetric Loading
Dynamic loads applied on the surface of shell considered herein are an axially symmetric radial traction and axially symmetric fluid pressure with the step style shown in Figure 2, where is the nondimensional step load time , is actual step load time; , the nondimensional time; is actual time; is maximum of the step load. In the domain of Laplace transform, the load can be expressed as
Here, the case of a thin, elastic shell embedded in infinite viscoelastic saturated soil subjected to axisymmetric surface load and fluid pressure is considered. The equation of motion of this shell under nontorsional axisymmetric loading is where ;   and are the dilatational wave velocity and plate velocity, respectively; , are the modulus and Poission’s rate of shell, respectively; is the net outward radial pressure. For a thin shell, the thick can be omitted without significant error. The interface shell and soil can be defined as . The stress and displacement condition at the interface is expressed as where is the radial stress applied at the inner surface of the shall; is the stress exerted by the soil on the shell and can be given by (39).
In practical situation, the condition is frequently found in two extreme cases; permeable and impermeable.
The partial permeable flow boundary condition is where is a dimensionless permeability parameter that defines the flow capacity of the shell. The parameter depends on the relative permeability of the shell and soil as well as the geometry of the shell, that is,(1)when the spherical shell is impermeable, that is, , tends to zero and(2)when the shell is permeable, that is, , tends to infinity.
Substituting (34), (39), (44) into (43) and (37) into (46), we obtain where
Under the fluid pressure on shell surface, the displacement and stress components are continuous at the kinematic interface between the spherical shell and soil. In this case, flow boundary conditions are
Substituting (34), (39), (49), (50) into (43) and (37) into (51) yields or Using inverse Laplace transform and numerical computation, the final solutionin time domain can be obtained after determining and .
4. Results and Discussion
In this paper, we will discuss the influences of partial permeable property of boundary and relative rigidity of shell and soil (defined as ) on the transient response of the spherical cavity. The numerical results are presented for the material and geometric parameters which are listed in Table 1.
4.1. Solutions Corresponding to Fluid Pressure
The histories of dimensionless radial displacement under fluid pressure are shown in Figure 2 when parameters . It is noted that at a certain time instant as shown in Figure 2, there exists maximum displacement at the interface of shell and soil. With the increase of time, radial displacement decreased vibrationally and finally to an asymptotic value of zero. Radial displacement decreased obviously with increasing relative rigidity, and increased with increasing of parameter (Figure 3).
The excess pore pressures induced by fluid pressure are shown in Figure 4. However, the influence of parameter (Figure 4) is significant. When the shell boundary became almost impermeable , almost no excess pore pressure existed, whereas with the increasing of time, the excess pore pressure at the interface equaled the fluid pressure when the shell boundary became almost permeable .
4.2. Solutions Corresponding to Radial Load
The histories of dimensionless radial displacement at the interface of shell and soil induced by axially symmetric radial surface load are shown in Figure 5 when the parameter . With the increase of the dimensionless time ), radial displacement increases to maximum value, then decreases and is noted once again. Eventually, it tends to an asymptotic value. When relative rigidity , the shell is complete flexible, there is the maximum radial displacement at the interface of shell and soil. The value of radial displacement decreases with the increase of relative rigidity. The influence of permeability parameter on radial displacement is indicated in Figure 6. It can be seen that the influence of parameter on radial displacement induced by axisymmetric radial surface load is not remarkable.
The histories of dimensionless pore pressure are shown in Figure 7 at the interface of shell and soil for the parameter . Pore pressure is zero at and increases rapidly with time in the interval and reaches to its peak value nearly at . Thereafter, it decreases with time and reaches to its maximum suction values. With increasing time the values of suction decreases and pore pressure is noted once again. On the other hand, the pore pressure decreases with the increase of parameter (Figure 8). As a result, both the relative rigidity and parameter have great influence on the pore pressure under the condition of axisymmetric radial surface load.
5. Conclusions
An extensive parameters study conducted to investigate the influence of therelative rigidity of shell and soil and permeability parameter , showed that permeability parameter depends on the relative permeability of the liner and soil as well as the geometry of the liner. Relative rigidity and parameter have significant influences on the transient response of spherical cavity with a shell embedded in viscoelastic saturated soil. The solutions under permeable and impermeable boundary conditions are only two extreme cases. Thus partially sealed boundary condition and the relative rigidity of shell and soil in the designing and computation of spherical shell in viscoelastic saturated medium are remarkable.