Computational Geomechanics. Manuel Pastor

of the equation is possible. We consider thus that the only physical variation is in the vertical, x₁, direction (x₁ = x) and then we have

      where D is called the one‐dimensional constrained modulus, E is Young’s modulus and ν is Poisson’s ratio of the linear elastic soil matrix, also

      The differential equations are, in place of (2.11):

      In place of (2.13):

      and in place of (2.16):

      with

      Taking K_s → ∞

      For a periodic applied surface load

      a periodic solution arises after the dissipation of the initial transient in the form

      and a system of ordinary linear differential equations is obtained in the frequency domain which can be readily solved by standard procedure.

The boundary conditions imposed are as shown in Figure 2.1. Thus, at x = L, u = 0, w = 0, and at x = 0, σ_x = q, p = 0.

      In Figure 2.1 (taken from Zienkiewicz et al. (1980)), we show a comparison of various numerical results obtained by various approximations:

      1 exact solution (Biot’s, labelled B)

      2 the u−p equation approximation (labelled Z)

      3 the undrained assumption (w = 0) and

      4 the consolidation equation obtained by omitting all acceleration terms (labelled C).

      The reader will note that the results are plotted against two nondimensional coefficients:

      where k′ and k are the two definitions of permeability discussed earlier.

      In the above

      where L is a typical length such as the length of the one‐dimensional soil column under consideration, g is the gravitational acceleration,

      is the speed of sound, is the natural vibration period and T is the period of excitation.

      The second nondimensional parameter is defined as

Figure 2.1 The soil column – variation of pore pressure with depth for various values of π₁ and π₂ –––– B (Biot theory) – – – – Z (u–p approximation theory) ––C (Consolidation theory) (Solution (C) is independent of π₂).

      Source: Reproduced from Zienkiewicz et al. (1980) by permission of the Institution of Civil Engineers.

      (a) π₂ ≤ 10⁻³. (b) π₂ = 10⁻². (c) π₂ = 10⁻¹. (d) π₂ = 10⁰. (e) π₂ = 10¹. (f) π₁ = 10⁻¹ π₂ = 10². Reproduced from Zienkiewicz (1980) by permission of the Institution of Civil Engineers.

      In the study, the following values were assumed:

      and