alt="Schematic illustration of various idealized structures of fluid-saturated porous solids."/>
Figure 1.3 Various idealized structures of fluid-saturated porous solids: (a) a granular material; (b) a perforated solid with interconnecting voids.
Using the concept of effective stress, which we shall discuss in detail in the next section, it is possible to reduce the soil mechanics problem to that of the behavior of a single phase once all the pore pressures are known. Then we can again use the simple, single‐phase analysis approaches. Indeed, on occasion, the limit load procedures are again possible. One such case is that occurring under long‐term load conditions in the material of appreciable permeability when a steady‐state drainage pattern has been established and the pore pressures are independent of the material deformation and can be determined by uncoupled calculations.
Such drained behavior, however, seldom occurs even in problems that we may be tempted to consider as static due to the slow movement of the pore fluid and, theoretically, the infinite time required to reach this asymptotic behavior. In very finely grained materials such as silts or clays, this situation may never be established even as an approximation.
Thus, in a general situation, the complete solution of the problem of solid material deformation coupled to a transient fluid flow needs to be solved generally. Here no shortcuts are possible and full coupled analyses of equations which we shall introduce in Chapter 2 become necessary.
We have not mentioned so far the notion of the so‐called undrained behavior, which is frequently assumed for rapidly loaded soil. Indeed, if all fluid motion is prevented, by zero permeability being implied or by extreme speed of the loading phenomena, the pressures developed in the fluid will be linked in a unique manner to deformation of the solid material and a single‐phase behavior can again be specified. While the artifice of simple undrained behavior is occasionally useful in static studies, it is not applicable to dynamic phenomena such as those which occur in earthquakes as the pressures developed will, in general, be linked again to the straining (or loading) history and this must always be taken into account. Although in early attempts to deal with earthquake analyses and to predict the damage and response, such undrained analyses were invariably used, adding generally a linearization of the total behavior and a heuristic assumption linking the pressure development with cycles of loading and the behavior predictions were poor. Indeed, comparisons with centrifuge experiments confirmed the inability of such methods to predict either the pressure development or deformations (VELACS – Arulanandan and Scott 1993). For this reason, we believe that the only realistic type of analysis is of the type indicated in this book. This was demonstrated in the same VELACS tests to which we shall frequently refer in Chapter 7.
At this point, perhaps it is useful to interject an observation about the possible experimental approaches. The question which could be addressed is whether a scale model study can be made relatively inexpensively in place of elaborate computation. A typical civil engineer may well consider here the analogy with hydraulic models used to solve such problems as spillway flow patterns where the cost of a small‐scale model is frequently small compared to equivalent calculations.
Unfortunately, many factors conspire to deny in geomechanics a readily accessible model study. Scale models placed on shaking tables cannot adequately model the main force acting on the soil structure, i.e. that of gravity, though, of course, the dynamic forces are reproducible and scalable.
To remedy this defect, centrifuge models have been introduced and, here, though, at considerable cost, gravity effects can be well modeled. With suitable fluids substituting water, it is indeed also possible to reproduce the seepage timescale and the centrifuge undoubtedly provides a powerful tool for modeling earthquake and consolidation problems in fully saturated materials. Unfortunately, even here a barrier 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 σij using indicial notation, these are determined by summing the appropriate forces in the i‐direction on the projection, or cuts, dxj (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 δij 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
