Critical state strength of soil - The Mechanics of Soils and Foundations - page 123

Figure 9.4 Critical states of soils.

and Bransby (1978) and byMuir Wood (1991). From Figs. 9.1 and 9.3 the critical state
is the state reached after strains of at least 10% and is associated with turbulent flow.
The relationships between the shear stress, the normal effective stress and the voids
ratio of soils at the critical states are illustrated in Fig. 9.4.
Figure 9.4(a) and (b) shows the critical state line (CSL). This shows that, at the
critical state, there is a unique relationship between the shear stress, the normal effective
stress and the voids ratio. Figure 9.4(c) is the same as Fig. 9.4(b) but with the normal
stress on a logarithmic scale. Also shown on Fig. 9.4(c) is the one-dimensional normal
compression line from Fig. 8.10(b).
The critical state line is given by

τ f = σ f tan φ c

(9.2)

σ f

e f = e − C c log (9.3)

where the subscripts f denote that the stresses and the voids ratio are those at failure
at the critical state. It is essential to recognise that Eqs. 9.2 and 9.3 contain effective,
not total stresses.
In Fig. 9.4(c) the normal compression and critical state lines are parallel and both
have the same gradient, C c . The parameter e defines the position of the critical state
line in the same way that e 0 defines the position of the normal compression line.
Equation (9.2) is the friction failure criterion discussed in Sec. 3.3 and

φ c is the critical
friction angle. The critical state line shown in Fig. 9.4(c) is directly above the critical
overconsolidation line shown in Fig. 8.7. (The height of the critical state line above the
critical overconsolidation line is τ f given by Eq. (9.2).) Later, in Chapter 11, we will