Peak states - The Mechanics of Soils and Foundations - page 152

dilating but, for an instant it is straining at constant volume. At C at the critical state
the stress ratio is again equal to tan φ c and the angle of dilation is zero; the sample

is straining at constant stress and at constant volume. From A to C the sample dilates.
The maximum rate of dilation given by the largest value of

ψ

ψ occurs at the point P

where the stress ratio is a maximum.
Figure 10.12(b) shows a frictional block on a plane. The forces on the block are
T and N , the angle of friction is

and the slope angle is i . The mechanics of the
sliding block are similar to the mechanics of the shear sample in Fig. 10.12(a) so the
relationships between

µ

σ in Fig. 10.12(a) will be analogous to the relationships
between T and N in Fig. 10.12(b), the angle of dilation

τ and

ψ is analogous to the slope

φ .
From Fig. 10.12(b), resolving vertically and horizontally and after some algebra

angle i and µ is analogous to

T
N =

µ +

tan( i ) (10.18)

Following the analogy between the shear test sample and the sliding block, the soil
behaviour can be represented by

τ
σ =

φ c + ψ

tan( ) (10.19)

Equation (10.19) corresponds with the observations from Fig. 10.13. At the points A
and C τ / σ φ c and at the point P both τ / σ and ψ = 0 and the stress ratio is = tan

τ / σ ψ have their maximum values. In fact Eq. (10.19) relates stress ratio to angle of

dilation throughout the whole test from O to the ultimate critical state at C . This

is the essence of the stress-dilatancy theory (Taylor, 1948).
Figure 10.14 shows a stress path O-A-P-C for the shear test illustrated in Fig. 10.13.
The peak stress ratio, and indeed any stress ratio is the sum of a component due to
friction and a component due to dilation. The stress path represents the changes of
stress throughout the test in which the normal effective stress was constant. The stress
ratio

ψ

τ / σ is given by Eq. (10.19) at all stages of the test: at A and C the stress ratio

τ / σ φ c because τ / σ < φ c ; for the

= tan ψ = 0; for the path O-A ψ< 0 and tan

Figure 10.14 Peak strength of dilating soil.