Computational Geomechanics. Manuel Pastor

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Computational Geomechanics - Manuel Pastor

Computational Geomechanics - Manuel Pastor

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double-prime Baseline equals normal d sigma Subscript italic i j Baseline plus italic alpha delta Subscript italic i j Baseline normal d p equals upper D Subscript italic ijkl Baseline left-parenthesis normal d epsilon Subscript italic k l Baseline minus normal d epsilon Subscript italic k l Superscript 0 Baseline right-parenthesis"/>

      or

      (1.12b)normal d sigma Superscript double-prime Baseline equals normal d sigma plus alpha bold m normal d p equals bold upper D left-parenthesis normal d bold epsilon minus normal d bold epsilon 0 right-parenthesis

      where a new “effective” stress, σ″, is defined. In the above

      (1.13a)italic alpha delta Subscript italic i j Baseline equals delta Subscript italic i j Baseline minus upper D Subscript italic ijkl Baseline delta Subscript italic k l Baseline StartFraction 1 Over 3 upper K Subscript s Baseline EndFraction

      or

      (1.13b)alpha bold m equals bold m minus bold upper D m StartFraction 1 Over 3 upper K Subscript s Baseline EndFraction

      and the new form eliminates the need for separate determination of the volumetric strain component. Noting that, in three dimensions,

delta Subscript italic i j Baseline delta Subscript italic i j Baseline equals 3

      or

bold m Superscript normal upper T Baseline bold m equals 3

      we can write

      (1.14a)alpha bold m Superscript normal upper T Baseline bold m equals bold m Superscript normal upper T Baseline bold m minus bold m Superscript normal upper T Baseline bold upper D m StartFraction 1 Over 3 upper K Subscript s Baseline EndFraction

      or simply

alpha equals 1 minus StartFraction bold m Superscript upper T Baseline bold upper D m Over upper K Subscript s Baseline EndFraction

      Alternatively, in tensorial form, the same result is obtained as

      (1.14b)italic alpha delta Subscript italic i j Baseline delta Subscript italic i j Baseline equals delta Subscript italic i j Baseline delta Subscript italic i j Baseline minus delta Subscript i j Baseline upper D Subscript italic ijkl Baseline delta Subscript italic k l Baseline StartFraction 1 Over 3 upper K Subscript s Baseline EndFraction

alpha equals 1 minus StartFraction delta Subscript italic i j Baseline upper D Subscript italic ijkl Baseline delta Subscript italic k l Baseline Over 9 upper K Subscript s Baseline EndFraction

      For isotropic materials, we note that,

      (1.15a)StartFraction bold m Superscript normal upper T Baseline bold upper D m Over 9 EndFraction equals StartFraction delta Subscript italic i j Baseline upper D Subscript italic ijkl Baseline delta Subscript italic k l Baseline Over 9 EndFraction equals StartFraction delta Subscript italic i j Baseline left-parenthesis italic lamda delta Subscript italic i j Baseline delta Subscript italic k l Baseline plus mu left-parenthesis delta Subscript italic i k Baseline delta Subscript italic j l Baseline plus delta Subscript italic i l Baseline delta Subscript italic j k Baseline right-parenthesis right-parenthesis delta Subscript italic k l Baseline Over 9 EndFraction equals StartFraction 9 lamda plus 6 mu Over 9 EndFraction equals upper K Subscript upper T

      which is the tangential bulk modulus of an isotropic elastic material with λ and μ being the Lamé’s constants. Thus we can write

      The reader should note that in (1.12), we have written the definition of the effective stress increment which can, of course, be used in a non‐incremental state as

      (1.16a)sigma double-prime Subscript italic i j Baseline equals sigma Subscript italic i j Baseline plus italic alpha delta Subscript italic i j Baseline p

      or

      (1.16b)bold sigma double-prime equals bold sigma plus alpha bold m p

      assuming that all the stresses and pore pressure started from a zero initial state (for example, material exposed to air is taken as under zero pressure). The above definition corresponds to that of the effective stress used by Biot (1941) but is somewhat more simply derived. In the above, α is a factor that becomes close to unity when the bulk modulus Ks of the grains is much larger than that of the whole material. In such a case, we can write, of course

      (1.17a)sigma double-prime Subscript italic i j Baseline equals sigma prime Subscript italic i j Baseline identical-to sigma Subscript italic i j Baseline plus delta Subscript italic i j Baseline p

      or

      (1.17b)bold sigma double-prime equals bold sigma prime identical-to bold sigma plus bold m p

      recovering the common definition used by many in soil mechanics and introduced by Terzaghi (1936). Now, however, the meaning of α is no longer associated with an effective area.

      It should have been noted that in some materials such as rocks or concrete, it is possible for the ratio KT/Ks to be as large as 1/3 with α = 2/3 being a fairly common value for determination of deformation.

      We note that in the preceding discussion, the only assumption made, which can be questioned, is that of neglecting the local damage due to differing materials in the soil matrix. We have also implicitly assumed that the fluid flow is such that it does not separate the contacts of the soil grains. This assumption is not totally correct in soil liquefaction or flow in the soil‐shearing layer during localization; therefore, it is not clear if Terzaghi’s definition of effective stress still applies when the soil is liquefied.

      1.3.3 Effective Stress in the Presence of Two (or More) Pore Fluids – Partially Saturated Media

      The interstitial space, or the pores, may, in a practical situation, be filled with two or more fluids. We shall, in this section, consider only two fluids with the degree of saturation by each fluid being defined by the proportion of the total pore volume n (porosity) occupied by each fluid. In the context of soil behavior discussed in this book, the fluids will invariably be water and air, respectively. Thus, we shall refer to only two saturation degrees, Sw that for water and Sa that for air, but the discussion will be valid for any two fluids.

      It is clear that if both fluids fill the pores completely, we shall always have

      (1.18)upper S Subscript w Baseline plus upper S Subscript a Baseline equals 1

      Clearly,

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