Computational Geomechanics - Manuel Pastor

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K Subscript f Baseline slash n Over upper D plus left-parenthesis upper K Subscript f Baseline slash n right-parenthesis EndFraction equals 0.973"/> Schematic illustration of zones of sufficient accuracy for various approximations.

      We note that, for instance, fully undrained behavior is applicable when Π1 < 10−2 and when Π1 > 102, the drainage is so free that fully drained condition can be safely assumed.

      To apply this table in practical cases, some numerical values are necessary. Consider, for instance, the problem of the earthquake response of a dam in which the typical length is characterized by the height L = 50 m, subject to an earthquake in which the important frequencies lie in the range

0.05 seconds less-than upper T less-than 5 seconds

      Thus, with the wave speed taken as

upper V Subscript s Baseline equals 1000 normal m slash normal s

      we have

ModifyingAbove upper T With ampersand c period circ semicolon equals StartFraction 2 upper L Over upper V Subscript s Baseline EndFraction equals 0.1 seconds

      the parameter Π2 is, therefore, in the range 3.9 × 10−3 < Π2 < 39

StartLayout 1st Row 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 equals pi squared left-parenthesis StartFraction 0.1 Over Number 0 period 0 5 tilde 5 EndFraction right-parenthesis squared almost-equals 39 tilde 0.0039 EndLayout

      and Π1 is dependent on the permeability k with the range defined by

0.97 k Superscript prime Baseline less-than normal upper Pi 1 less-than 97 k prime StartLayout 1st Row normal upper Pi 1 equals StartFraction 2 Over 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 k prime pi Over beta g upper T EndFraction StartFraction 1 Over pi squared EndFraction left-parenthesis StartFraction upper T Over ModifyingAbove upper T With ampersand c period circ semicolon EndFraction right-parenthesis squared equals StartFraction 2 pi Over beta g EndFraction StartFraction 1 Over normal upper Pi 2 upper T EndFraction k Superscript prime Baseline almost-equals StartFraction 2 pi Over beta g EndFraction left-parenthesis StartFraction 1 Over 39 times 0.05 tilde 0.0039 times 5 EndFraction right-parenthesis k Superscript prime Baseline almost-equals left-parenthesis 0.97 comma tilde comma 97 right-parenthesis k prime EndLayout

      According to Figure 2.2, we can, with reasonable confidence:

      1 assume fully undrained behavior when Π1 = 97k′ < 10−2 or the permeability k′ < 10−4 m/s.

      2 We can assume u–p approximation as being valid when k′ < 10−3 m/s to reproduce the complete frequency range. However, when k′ < 10−1 m/s, periods of less than 0.5 seconds are still well modeled.

      We shall, therefore, typically use the u–p formulation appropriately in what follows reserving the full form for explicit transients where shocks and very high frequency are involved.

      2.3.1 Why Is Inclusion of Partial Saturation Required in Practical Analysis?

      In the previous, fully saturated, analysis, we have considered both the water pore pressure and the solid displacement as problem variables. In the general case of nonlinear nature, which is characteristic of the problems of soil mechanics, both the effective stresses and pressures will have to be determined incrementally as the solution process (or computation) progresses step by step. In many soils, we shall encounter a process of “densification” implied in the constitutive soil behavior. This means that the history of straining (associated generally with shear strain) induces the solid matrix to contract (or the material to densify). Such densification usually will cause the pore pressure to increase, leading finally to a decrease of contact stresses in the soil particles to near‐zero values when complete liquefaction occurs. Indeed, generally, failure will occur prior to the liquefaction limit. However, the reverse may occur where the soil “dilation” during the deformation history is imposed. This will imply the development of negative pressures which may reach substantial magnitudes. Such negative pressures cannot exist in reality without the presence of separation surfaces in the fluid which is contained in the pores, and consequent capillary effects. Voids will therefore open up during the process in the fluid which is essentially incapable of sustaining tension. This opening of voids will probably occur when zero pressure (or corresponding vapor pressure of water) is reached. Alternatively, air will come out of the solution – or indeed ingress from the free water surface if this is open to the atmosphere. The pressure will not then be vapor pressure but simply atmospheric.

      The presence of negative water pressures will, of course, increase the strength of the soil and thus have a beneficial effect. This is particularly true above the

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