Computational Geomechanics. Manuel Pastor

      or

      (1.12b)

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

      (1.13a)

      or

      (1.13b)

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

      or

      we can write

      (1.14a)

      or simply

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

      (1.14b)

and

      For isotropic materials, we note that,

      (1.15a)

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

(1.15b)

      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)

      or

      (1.16b)

      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 K_s of the grains is much larger than that of the whole material. In such a case, we can write, of course

      (1.17a)

      or

      (1.17b)

      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 K_T/K_s 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, S_w that for water and S_a 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)

      Clearly,