Dynamic response analysis of fluid-saturated porous rectangular plates

1 Introduction

Since Biot (1956) [1], [2] put forward the basic equations describing the dynamic characteristics of saturated porous media, the theory of porous media has become the foundation for studying the dynamic characteristics and performing dynamic analysis of saturated porous media, and has been widely used in different engineering of a variety of fields.

To date, research on saturated porous media in the fields of geotechnical engineering, earthquake engineering and geophysical has mainly focused on geotechnical materials whose geometric characteristics are associated with the half-space domain or an infinite horizontal layer, including theoretical analysis and numerical simulation of dynamic responses [3], [4], [5], [6], [7], [8], [9], [10] and wave propagation characteristics [11], [12], [13], [14]. However, few studies have been focused on flexible structures, such as porous media beams, plates and shell structures. On the one hand, the behaviour of flexible porous structures is an important issue in biomechanics [15], [16], such as the mechanical analysis of cartilage tissue and the stems of plants. On the other hand, porous materials such as polyurethane foams and fibrous materials have been widely used in the automotive and aerospace industries for damping and sound absorption [17], [18]. Therefore, it is necessary to further study the static and dynamic mechanical behaviours of flexible porous structures.

In terms of fluid-saturated porous plate structures, considering the lateral diffusion of fluid, Taber [19] established the solid-phase and fluid-phase governing equation of the isotropic fluid-saturated porous media based on Biot’s model to analyse the dynamic bending of simply supported rectangular porous plates using the Laplace transformation and the perturbation method. Leclaire et al. [20] analysed the transverse vibration problem of rectangular thin plates for four-edge-clamped porous media using the Rayleigh–Ritz method, which takes the effect of fluid viscosity on energy dissipation into account. Based on the research of Theodorakopoulos and Beskos [21] on the bending vibration of porous elastic plates and Biot’s theory, Anke and Heinz [22] established the dynamic mathematical model of saturated porous elastic Mindlin plates and gave the principle of virtual work using the deflection, angle and pore stress as basic unknown quantities. Based on the classical theory of homogeneous plates and Biot’s stress–strain relations in an isotropic porous medium with a uniform porosity, Feng-xi and Xiao-lin [23] researched the dynamic bending mathematical model of saturated porous elastic plates, and the influence of porosity, tortuosity and permeability on the resonances was studied to determine the condition of maximum damping considering these parameters. Nagler and Schanz [24], using the series approximation method, obtained another plate shear deformation theory of porous material plates. Many studies on the mechanical response of fluid-saturated porous plates used various simplified plate theories, such as classical thin plate theory and Reissner–Mindlin thick plate theory. Rezaei and Saidi [25] presented an exact solution for the free vibration analysis of porous rectangular plates under undrained conditions, and the results show that the effect of coupled fluid–solid deformation may not be disregarded. However, other studies have shown that the mechanical quantity cannot be a polynomial of coordinate variables in the thickness direction. In various simplified theories, incompatibility among the fundamental equations can be found, that is, some mechanical quantities can meet only some and not all of the basic equations, and the error will increase greatly with increasing thickness [26].

Based on Biot’s model of porous media and three-dimensional elastic theory, the dynamic governing equation of fluid-saturated porous rectangular plates is established in this paper. The free vibration and forced vibration responses of simply supported fluid-saturated porous rectangular plates are researched. The influence of the surface infiltration conditions, porosity, pore fluid permeability coefficient and loading frequency parameters on the free vibration frequency, the solid skeleton stress, the pore fluid pressure, the solid skeleton displacement and pore fluid displacement of plates are analysed via numerical examples.

2 The governing equation of fluid-saturated porous rectangular plates

A rectangular plate occupying the region [0, L₁] × [0, L₂] × [−H/2, H/2] in the unstressed reference configuration is described in rectangular Cartesian coordinates x_i (i = 1, 2, 3). Based on Biot’s model of porous media, the basic equations of homogeneous saturated porous media are as follows.

The constitutive equations for a fluid-saturated porous media are

The strain tensor for infinitesimal deformations is related to the displacements u_i by

The equilibrium equations in the absence of a body force are

where σ_ij and p are the total stress components and pore fluid pressure (i, j = 1, 2, 3), respectively. u_i and w_i are the displacement components of the solid skeleton and pore fluid, that is e = u_i,i, and ζ = −w_i,i. λ and μ are the Lame constants and ε_ij is the strain vector of the solid skeleton. b is a parameter accounting for internal friction due to the relative motion between the solid and the pore fluid and b = η/k_f, where k_f is the dynamic permeability and η is the viscosity of the fluid. α and M are the Biot parameters considering compressibility of the two-phase material, and α = 1 − K/K_s, 1/M = (α−n)/K_s + n/K_f, where K, K_s and K_f are the bulk modulus of the solid skeleton, solid particles and pore fluid, respectively, while n is the soil porosity. ρ is the total density of the saturated soil and can be represented by ρ = (1 − n)ρ_s + nρ_f, where ρ_s and ρ_f are the densities of the solid phase and liquid phase, respectively. m is a parametric representation related to the mass density of the pore fluid and pore geometry features m = ρ_f/n.

Combining Eqs. (1)–(5), the governing equations of the dynamic response of the saturated media can be obtained as

Considering the simply supported boundary conditions, that is

3 The exact solution for fluid-saturated porous rectangular plates

A solution for the displacement field in the fluid-saturated porous rectangular plate is sought in the form.

where ω denotes the angular frequency. Assume that ξ₁ = m₁π/L₁, ξ₂ = n₁π/L₂ and that m₁ and n₁ are positive integers. The chosen displacement field described by Eq. (9) satisfies the homogeneous boundary conditions described by Eq. (8) at simply supported boundaries. Substituting Eqs. (3) and (9) into the constitutive equation of Eq. (1), the stresses can be given in terms of displacements as

where a prime denotes differentiation with respect to x₃. Substituting Eq. (9) into Eqs. (6) and (7), the following coupled system of second-order ordinary differential equations can be obtained:

The power series method is used to obtain the solution for Eq. (11). According to a general method, we assume a power series solution for the displacement functions U_i(x₃) and W_i(x₃) as

Substituting Eq. (12) into Eq. (11) yields the following coupled recurrence algebraic relations:

Clearly, the recurrence relations described in Eq. (13) are evaluated successively for β = 0, 1, … , to obtain U˜i(β) and W˜i(β) In terms of the eight arbitrary constants U˜1(0), W˜1(0), W˜2(0), U˜3(0), W˜3(0), U˜1(1), U˜2(1), and U˜3(1) and the angular frequency ω as

Here, Aij(β)(ω) and Bij(β)(ω) are known polynomials in ω and they are determined by the recurrence formula in Eq. (13).

Substituting Eq. (14) into Eq. (12) gives the rewritten displacement expression as

where Aij(x3)=∑β=0∞Aij(β)x3β, and Bij(x3)=∑β=0∞Bij(β)x3β. The degree of each of the polynomials increases as more terms are retained in the series expansion in Eq. (12).

Substituting the expressions of displacement in Eq. (15) into the stress components in Eq. (10) represented by displacement yields the following expression of stress:

where S = [σ₃₃σ₁₃σ₂₃p]^T, U˜=[U˜1(0)W˜1(0)U˜2(0)W˜2(0)U˜3(0)W˜3(0)U˜1(1)U˜3(1)]T, and the matrix of coefficients Q_ij is a 4 × 8 matrix for each i, j given in Appendix A.

At this point, a series solution of the displacement and stress components are obtained.

4 Results and discussion

4.3 Harmonic vibration

The response of the fluid-saturated porous rectangular plates under the dynamic load is analysed with the harmonic load q = 1000 pa. The results for the forced vibration are plotted in Figure 1 with the harmonic load and forcing frequencies ω‾ = 5, 10, 20 and 50. The solid skeleton stress, the pore fluid pressure, the solid skeleton displacement and the pore fluid displacement of the fluid-saturated porous rectangular plates decrease with increasing forcing frequencies whether the upper and lower surfaces are permeable or completely impermeable, but the transverse shear stress σ₂₃ and the transverse normal stress σ₃₃ of the solid skeleton are influenced only slightly. Due to the strong transverse pressure gradients that occur on the beam sections and are undertaken by both the solid skeleton and pore fluid as the beam bends, the transverse bending deformation causes compression in the upper beam for x₃ < h/2 and tension in the lower beam for x₃ > h/2, which is similar to the stress distribution in a single-phase continuous elastic beam. The relationship between the total pressure and pore pressure leads to positive pore pressures in the compression zone of the upper beam and negative pore pressures in the tension zone of the lower beam, resulting in pore suction, as shown in Figure 1.

Figure 1:

The variations in the solid skeleton stress, pore fluid pressure, solid skeleton displacement and pore fluid displacement in the thickness direction with ω‾.

(1), (3), (5), (7), (9), (11), (13) and (15) are under freely permeable conditions, and (2), (4), (6), (8), (10), (12), (14) and (16) are under completely impermeable conditions.

Considering different values of porosity, the variations in the solid skeleton stress, pore fluid pressure and displacements of the solid skeleton and fluid in the thickness direction are shown in Figure 2 with a frequency of ω‾ = 10. The results indicate that porosity affects the solid skeleton stress, pore fluid pressure and solid skeleton and fluid displacements, although the upper and lower surface infiltration conditions have a more significant effect.

Figure 2:

The variations in the solid skeleton stress, pore fluid pressure, solid skeleton displacement and pore fluid displacement in the thickness direction with n.

(1), (3), (5), (7), (9), (11), (13) and (15) are under freely permeable conditions, and (2), (4), (6), (8), (10), (12), (14) and (16) are under completely impermeable conditions.

To analyse the influence of the permeability coefficient on the dynamic responses of plates, Figure 3 shows the variations in the solid skeleton stress, pore fluid pressure and solid skeleton and fluid displacements in the thickness direction. The results indicate that the permeability coefficient has an impact on the solid skeleton stress, pore fluid pressure and solid skeleton and fluid displacements. Moreover, these factors can be affected significantly by the surface infiltration conditions.

Figure 3:

The variations in the solid skeleton stress, pore fluid pressure, solid skeleton displacement and pore fluid displacement in the thickness direction with k_f.

(1), (3), (5), (7), (9), (11), (13) and (15) are under freely permeable conditions, and (2), (4), (6), (8), (10), (12), (14) and (16) are under completely impermeable conditions.

5 Conclusions

A series solution is presented for the dynamic responses of a simply supported fluid-saturated porous rectangular plate. Considering the compressibility of solid particles and fluid and the viscosity of pore fluid, the dynamic responses of simply supported fluid-saturated porous rectangular plates and the influence of the surface infiltration conditions, porosity and pore fluid permeability coefficient on the free vibration frequency of porous plates are discussed. Parametric studies indicate that the effect of coupling between a solid and fluid is important for increasing the frequency and must be considered in the case of dynamic responses.