Baseline equals sigma prime Subscript italic i j Baseline minus delta Subscript italic i j Baseline italic n p"/>
or if the vectorial notation is used, we have
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
where nw is now called the effective area coefficient and is such that
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 us now consider the effect of the simultaneous application of a total external hydrostatic stress and a pore pressure change, both equal to Δp, to any porous material. The above requirement can be written in tensorial notation as requiring that the total stress increment is defined as
(1.8a)
or, using the vector notation
(1.8b)
In the above, the negative sign is introduced since “pressures” are generally defined as being positive in compression, while it is convenient to define stress components as positive in tension.
It is evident that for the loading mentioned, only a very uniform and small volumetric strain will occur in the skeleton and the material will not suffer any damage provided that the grains of the solid are all made of identical material. This is simply because all parts of the porous medium solid component will be subjected to identical compressive stress.
However, if the microstructure of the porous medium is composed of different materials, it appears possible that nonuniform, localized stresses, can occur and that local grain damage may be suffered. Experiments performed on many soils and rocks and rock‐like materials show, however, that such effects are insignificant. Thus, in general, the grains and, hence, the total material will be in a state of pure volumetric strain
(1.9)
where Ks is the average material bulk modulus of the solid components of the skeleton. Alternatively, adopting a vectorial notation for strain in a manner involved in (1.1)
(1.10a)
where ε is the vector defining the strains in the manner corresponding to that of stress increment definition. Again, assuming that the material is isotropic, we shall have
(1.10b)
Those not familiar with soil mechanics may find the following hypothetical experiment illustrative. A block of porous, sponge‐like rubber is immersed in a fluid to which an increase in pressure of Δp is applied as shown in Figure 1.4. If the pores are connected to the fluid, the volumetric strain
