Computational Geomechanics. Manuel Pastor

Computational Geomechanics

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

StartLayout 1st Row sigma double-prime Subscript x Baseline equals sigma prime Subscript x Baseline equals sigma Subscript x Baseline plus p 2nd Row sigma prime Subscript x Baseline equals upper D epsilon Subscript x Baseline left-parenthesis epsilon Subscript y Baseline equals epsilon Subscript z Baseline equals 0 right-parenthesis 3rd Row upper D equals StartFraction upper E left-parenthesis 1 minus nu right-parenthesis Over left-parenthesis 1 plus nu right-parenthesis left-parenthesis 1 minus 2 nu right-parenthesis EndFraction EndLayout

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

epsilon Superscript 0 Baseline equals 0 comma upper S 0 equals 0 comma b equals 0 comma u 1 equals u comma u 2 equals u 3 equals 0

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

StartFraction partial-differential sigma Over partial-differential x EndFraction minus rho StartFraction partial-differential squared u Over partial-differential t squared EndFraction minus rho Subscript f Baseline StartFraction partial-differential w Over partial-differential t EndFraction equals 0

In place of (2.13):

minus StartFraction partial-differential p Over partial-differential x EndFraction minus StartFraction w Over k EndFraction minus rho Subscript f Baseline StartFraction partial-differential squared u Over partial-differential t squared EndFraction minus StartFraction rho Subscript f Baseline Over n EndFraction StartFraction partial-differential w Over partial-differential t EndFraction equals 0

and in place of (2.16):

StartFraction partial-differential w Over partial-differential x EndFraction plus StartFraction partial-differential epsilon Over partial-differential t EndFraction plus StartFraction 1 Over upper Q EndFraction StartFraction partial-differential p Over partial-differential t EndFraction equals 0

with

sigma equals italic upper D epsilon minus p

Taking K_s → ∞

StartFraction 1 Over upper Q EndFraction equals StartFraction n Over upper K Subscript f Baseline EndFraction

For a periodic applied surface load

q equals q overbar normal e Superscript italic i w t

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

p equals p overbar e Superscript italic i w t Baseline e t c period

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:

normal upper Pi 1 equals StartFraction k prime upper V Subscript c Superscript 2 Baseline Over italic g beta omega upper L squared EndFraction equals StartFraction 2 Over italic beta pi EndFraction StartFraction k prime Over g EndFraction StartFraction upper T Over ModifyingAbove upper T With ampersand c period circ semicolon squared EndFraction equals StartFraction 2 italic k rho Over pi EndFraction StartFraction upper T Over ModifyingAbove upper T With ampersand c period circ semicolon squared EndFraction

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

In the above

ModifyingAbove upper T With ampersand c period circ semicolon equals StartFraction 2 upper L Over upper V Subscript c Baseline EndFraction

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

upper V Subscript c Baseline equals StartRoot StartFraction upper D plus left-parenthesis upper K Subscript f Baseline slash n right-parenthesis Over rho EndFraction EndRoot

is the speed of sound, ModifyingAbove upper T With ampersand c period circ semicolon is the natural vibration period and T is the period of excitation.

The second nondimensional parameter is defined as

Schematic illustration of the soil column depicting the variation of pore pressure with depth for various values of pi 1 and pi 2.

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.

normal upper Pi 2 equals pi squared left-parenthesis StartFraction ModifyingAbove upper T With ampersand c period circ semicolon Over upper T EndFraction right-parenthesis squared

In the study, the following values were assumed:

beta identical-to rho Subscript f Baseline slash rho equals 0.333 comma n left-parenthesis porosity right-parenthesis equals 0.333 comma

and

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