OBSERVATION AND PREDICTION - Introduction to Soft Soil Geotechnique: Content, Context and Application - page 144

formation of a shear band and its inclination is far from obvious, neither in
numerical simulation nor in physical reality.
Teunissen showed an elegant and holistic way to understand this characteristic
problem in geotechnical engineering. The maximum stress ratio (maximum
strength) follows from the Mohr-Coulomb model (cohesion is disregarded for
convenience)

: 7

1 sin

8
9

v

5
6

(9.12)

1 sin

h

max

h are principal stresses. Obviously, the maximum
stress ratio (9.12) depends on the friction angle

Here, the stresses v and

.
Whether this expression gives the stress ratio for ongoing deformations (residual
strength) depends on the formulation of the plastic deformations when the failure is
reached.
In order to understand the ambiguity in localisation, a biaxial test where a shear
plane has developed with angle

and not on the dilation angle

to the horizontal axis (see Fig 9.6a) will be
analysed. The equilibrium conditions (in terms of forces) are expressed by

b tan T cos N sin 0 (9.13a)

h

b T sin N cos 0 (9.13b)

v

If R = T/N expresses the shear ratio, equations (9.13) results in 45

2

cos sin R sin tan

v

(9.14)

2

cos sin R cos tan( atan R )

h

According to Coulomb, the lowest ratio of (9.14) is a measure for the strength of
the material, referred to as critical strength, and it occurs at a specific value of

cr ,

namely

R cr = cot(2 cr ) or cr = 45 = + ½ atan R cr (9.15a)

and it can be found that (9.14) for this value leads to the critical strength 45

: 7

2

cos sin R sin 2

8
9

v

5
6

cr cr cr cr

tan (9.15b)

cr

2

cos sin R cos

h cr cr cr cr

cr

If the same material is tested for large ongoing deformations, the ultimate shear
stress ratio for a Mohr-Coulomb model can be expressed by the ratio R res ,
according to (Davis), valid for non-associative behaviour

45

Elaboration, see Application 9.5