Consolidation of viscoelastic soil by vertical drains incorporating fractional‐derivative model and time‐dependent loading

Abstract

Summary This paper presents a general solution to the consolidation system of viscoelastic soil by vertical drains incorporating a fractional‐derivative model and arbitrary time‐dependent loading. The fractional‐derivative Merchant model is introduced to describe the viscoelastic behavior of saturated soil around the vertical drains. Based on this model, the governing partial differential equation of a consolidation system incorporating vertical and horizontal drainage is obtained for the equal strain condition. Then, a general solution to the consolidation system involving arbitrary time‐dependent loading is derived using the Laplace transform technique and eigenfunction expansion method. Further, two comparisons are presented to verify the exactness of the proposed solution, and the consolidation behavior involving four time‐dependent loadings is illustrated and discussed.

A One-Dimensional Fractional-Derivative Viscoelastic Model for the Aquitard Consolidation of an Aquifer System

“…As a result, a decrease in the fractional order increases the vertical effective stress and accelerates the dissipation of the pore-water pressure further. This phenomenon is the same as that found in the consolidation of the vertical drain system [43]. Figure 4 illustrates the results of the settlement of the aquitard and the degree of consolidation under different values of the modulus ratio (i.e., κ).…”

“…A larger fractional order leads to a smaller settlement at the initial time, while it results in a greater settlement at a later stage. This phenomenon is identical to that of the soil deformation behavior of the soft soil subjected to external loading and that of the vertical drain consolidation system [43]. As shown in Figure 3(b) with different fractional orders, the calculated results of the degree of consolidation also appear as a set of curves intersecting with each other.…”

“…It has a property wherein its constitutive equation has continuity from the ideal solid state to the ideal fluid state. Based on this model, the vertical strain of the aquitard can be expressed in a convolution form as [40,43]…”

“…where κ = E s0 /E s1 stands for the modulus ratio of the Hookean springs; λ = η/E s1 is defined as the creep time of the fractional-derivative Voigt body with the viscosity coefficient η; α is the fractional order of the fractional-derivative element, and it is commonly considered to be a dimensional coefficient concerning the memory of the real materials; ⊗ is called the convolution of two functions; ℒ ⋅ represents the Laplace transform of a real function and its inverse operator is denoted as ℒ −1 ⋅ ; and s is a complex variable, respectively. Moreover, it has been found [43] that…”

“…For the consolidation problem of soft clay, Wang et al [42] extended the one-dimensional consolidation theory using a two-element fractional-derivative model. Huang and Li [43] investigated the consolidation process of the vertical drain system installed in soft clay using a fractional-derivative Merchant model. For the aquifer system, however, there are few studies on the consolidation process by considering a fractional-derivative model.…”