Computational Geomechanics. Manuel Pastor

Computational Geomechanics

Computational Geomechanics - Manuel Pastor

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so‐called phreatic line. Usually, one is tempted to assume simply a zero pressure throughout that zone but for non‐cohesive materials, this means almost instantaneous failure under any dynamic load. The presence of negative pressure in the pores assures some cohesion (of the same kind which allows castles to be built on the beach provided that the sand is damp). This cohesion is essential to assure the structural integrity of many embankments and dams.

      2.3.2 The Modification of Equations Necessary for Partially Saturated Conditions

      The necessary modification of Equations (2.20) and (2.21) will be derived below, noting that generally we shall consider partial saturation only in the slower phenomena for which u–p approximation is permissible.

      Before proceeding, we must note that the effective stress definition is modified and the effective pressure now becomes (viz Section 1.3.3)

      with the effective stress still defined by (2.1).

      neglecting the weight of air. The correction is obviously small and its effect insignificant.

      However, (2.21) will now appear in a modified form which we shall derive here.

      First, the water momentum equilibrium, Equation (2.13), will be considered. We note that its form remains unchanged but with the variable p being replaced by pw. We thus have

      (2.26a)minus p Subscript w comma i Baseline minus upper R Subscript i Baseline minus rho Subscript f Baseline ModifyingAbove u With two-dots Subscript i Baseline plus rho Subscript f Baseline b Subscript i Baseline equals 0

      (2.26b)minus italic nabla p Subscript w Baseline minus bold-italic upper R minus rho Subscript f Baseline ModifyingAbove bold-italic u With two-dots plus rho Subscript f Baseline bold-italic b equals 0

      As before, we have neglected the relative acceleration of the fluid to the solid.

      Equation (2.14), defining the permeabilities, remains unchanged as

      (2.27a)k Subscript italic i j Baseline upper R Subscript j Baseline equals w Subscript i

      (2.27b)bold k upper R equals bold w

      However, in general, only scalar, i.e. isotropic, permeability will be used here

      (2.28a)k Subscript italic i j Baseline equals k delta Subscript italic i j

      (2.28b)bold k equals k bold upper I

      where I is the identity matrix. The value of k is, however, dependent strongly on Sw and we note that:

      (2.29)k equals k left-parenthesis upper S Subscript w Baseline right-parenthesis

      Such typical dependence is again shown in Figure 1.6.

      Finally, the conservation Equation (2.16) has to be restructured, though the reader will recognize similarities.

      The mass balance will once again consider the divergence of fluid flow wi,i to be augmented by terms previously derived (and some additional ones). These are

      1 Increased pore volume due to change of strain assuming no change of saturation: δijdεij = dεii

      2 An additional volume stored by compression of the fluid due to fluid pressure increase: nSwdpw/Kf

      3 Change of volume of the solid phase due to fluid pressure increase: (1 − n)χwdpw/Ks

      4 Change of volume of solid phase due to change of intergranular contact stress: −KT/Ks(dεii + χwdpw/Ks)

      5 And a new term taking into account the change of saturation: ndSw

      Adding to the above, as in Section 2.2, the terms involving density changes, on thermal expansion, the conservation equation now becomes:

      (2.30a)StartLayout 1st Row w Subscript i comma i Baseline plus alpha ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i plus upper S Subscript w Baseline StartFraction n Over upper K Subscript f Baseline EndFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript w plus StartFraction alpha minus n Over upper K Subscript s Baseline EndFraction chi Subscript w Baseline ModifyingAbove p With ampersand c period dotab semicolon Subscript w plus n ModifyingAbove upper S With ampersand c period dotab semicolon Subscript w plus italic n upper S Subscript w Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript w Baseline Over rho Subscript w Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 2nd Row identical-to w Subscript i comma i Baseline plus alpha ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i Baseline plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript w Baseline Over upper Q Superscript asterisk Baseline EndFraction plus italic n upper S Subscript w Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript w Baseline Over rho Subscript w Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0 EndLayout

      or

      (2.30b)italic nabla Superscript upper T Baseline bold-italic w plus alpha bold-italic m Superscript upper T Baseline ModifyingAbove bold-italic epsilon With ampersand c period dotab semicolon plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript w Baseline Over upper Q Superscript asterisk Baseline EndFraction plus italic n upper S Subscript w Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript w Baseline Over rho Subscript w Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0

      (2.30c)StartFraction <hr><noindex><a href=Скачать книгу