Computational Geomechanics. Manuel Pastor

is reached which appears to be insurmountable. As we shall see later under conditions when two fluids, such as air and water, for instance, fill the pores, capillary effects occur and these are extremely important. So far, no significant success has been achieved in modeling these and, hence, studies of structures with free (phreatic) water surface are excluded. This, of course, eliminates the possible practical applications of the centrifuge for dams and embankments in what otherwise is a useful experimental procedure.

1.3 Concepts of Effective Stress in Saturated or Partially Saturated Media

      1.3.1 A Single Fluid Present in the Pores – Historical Note

The essential concepts defining the stresses which control the strength and constitutive behavior of a porous material with internal pore pressure of fluid appear to have been defined, at least qualitatively two centuries ago. The work of Lyell (1871), Boussinesq (1876), and Reynolds (1886) was here of considerable note for problems of soils. Later, similar concepts were used to define the behavior of concrete in dams (Levy 1895 and Fillunger 1913a, 1913b, 1915) and indeed for other soil or rock structures. In all of these approaches, the concept of division of the total stress between the part carried by the solid skeleton and the fluid pressure is introduced and the assumption made that the strength and deformation of the skeleton is its intrinsic property and not dependent on the fluid pressure.

      If we thus define the total stress σ by its components σ_ij using indicial notation, these are determined by summing the appropriate forces in the i‐direction on the projection, or cuts, dx_j (or dx, dy, and dz in conventional notation). The surfaces of cuts are shown for two kinds of porous material structure in Figure 1.3 and include the total area of the porous skeleton.

      In the context of the finite element computation, we shall frequently use a vectorial notation for stresses, writing

      (1.1a)

      or

      (1.1b)

      This notation reduces the components to six rather than nine and has some computational merit.

      Now if the stress in the solid skeleton is defined as the effective stress σ′ again over the whole cross sectional area, then the hydrostatic stress due to the pore pressure, p acting, only on the pore area should be

      (1.2)

      where n is the porosity and δ_ij is the Kronecker delta. The negative sign is introduced as it is a general convention to take tensile components of stress as positive.

      The above, plausible, argument leads to the following relation between total and effective stress with total stress

(1.3)

      or if the vectorial notation is used, we have

(1.4)

      where m is a vector written as

      (1.5)

The above arguments do not stand the test of experiment as it would appear that, with values of porosity n with a magnitude of 0.1–0.2, it would be possible to damage a specimen of a porous material (such as concrete, for instance) by subjecting it to external and internal pressures simultaneously. Further, it would appear from Equation (1.3) that the strength of the material would be always influenced by the pressure p.

      Fillunger introduced the concepts implicit in (1.3) in 1913 but despite conducting experiments in 1915 on the tensile strength of concrete subject to water pressure in the pores, which gave the correct answers, he was not willing to depart from the simple statements made above.

It was the work of Terzaghi and Rendulic (1934) and by Terzaghi (1936) which finally modified the definition of effective stress to

(1.6)

      where n_w is now called the effective area coefficient and is such that

(1.7)

      Much further experimentation on such porous solids as the concrete had to be performed before the above statement was generally accepted. Here the work of Leliavsky (1947), McHenry (1948), and Serafim (1954, 1964) made important contributions by experiments and arguments showing that it is more rational to take sections for determining the pore water effect through arbitrary surfaces with minimum contact points.

      Bishop (1959) and Skempton (1960) analyzed the historical perspective and, more recently, de Boer (1996) and de Boer et al. (1996) addressed the same problem showing how an acrimonious debate between Fillunger and Terzaghi terminated in the tragic suicide of the former in 1937.

Zienkiewicz (1947, 1963) found that interpretation of the various experiments was not always convincing. However, the work of Biot (1941, 1955, 1956a, 1956b, 1962) and Biot and Willis (1957) clarified many concepts in the interpretation of effective stress and indeed of the coupled fluid and solid interaction. In the following section, we shall present a somewhat different argument leading to Equations (1.6) and (1.7).

If the quantity σ′ of (1.3) and (1.4) is interpreted as the volume‐averaged solid stress (1 − n) t _s used in the mixture theory (partial stress), see Gray et al. (2009), then we recover the stress split introduced in Biot (1955). There the fluid pressure, as opposed to the effective stress concept, is weighted by the porosity. Biot (1955) declares that “the remaining components of the stress tensor are the forces applied to that portion of the cube faces occupied by the solid.” In this book, we use the much more common concept of effective stress.

      1.3.2 An Alternative Approach to Effective Stress

      Let