Mohr-Coulomb Failure Condition and the Direct Shear Test Revisited

An alternative critical plane orientation is proposed in the Mohr-Coulomb failure criterion for soils with an extreme property. Parameter identification from the direct shear test is extended to incude the lateral normal stress.


Introduction
The Coulomb failure condition is defined by the equation f c = + -= t s j tan 0 (1) where t and s are the shear and normal traction components respectively on the critical plane in the material, c is the apparent cohesion and j is the angle of shearing resistance (internal friction).The usual sign convention is used for the normal stress s, compression is negative.In the classical Mohr-Coulomb formulation, the critical plane normal is inclined by the angle a p j = -4 2 from the s 1 direction to the s 3 direction.Ordered principal stresses s s s plane touches the envelope (1) as shown in Fig. 1.Stresses s cx , s cz and t c are implied in the coordinate frame associated with the critical plane.The Mohr-Coulomb condition is natural but the assumed orientation of the critical plane in fact lacks a rigorous substantiation.Other orientations could be assumed.A rational modification of the Mohr-Coulomb condition can be obtained when the critical plane orientation is not a priori restrained.Instead, it can be determined so that f attains its maximum on the critical plane.The resulting criterion should be more severe than the classical one.

Mohr-Coulomb criterion based on an extreme property
Direct notation is used in the development, and a general triaxial stress is assumed for full generality.Stress tensor s is assumed to have principal stresses s i with direction vectors n i .The unknown critical plane normal is denoted n.The normal and tangential traction components on the plane are The extreme of f is sought when n is subject to variation with subsidiary condition n n × =1.Lagrange multiplier l is introduced and the extended criterion The equation is contractively multiplied by n and the resulting scalar equation is used to eliminate l from Eq. 3. As- is obtained for the unknown n.The equation can be rewritten in a comprehensive form when tensor r is introduced The eigenvectors of r deliver extremes of f.Eigenvectors of r and s are the same, however, so these extremes are minima (t = 0) of f.In order to find the other extremes, all variables are decomposed in terms of the eigenvalues s i and principal vectors n i of s: and with the Coulomb condition (1) applied on the plane of the maximum shear stress The latter condition represents the third option for the critical plane orienation.For plane stress conditions s 2 0 = the graphic representation of all three yield locuses is in Fig. 2. The modified yield locus is the most severe, as expected.
Intersections of the modified (A) and classical (M) yield locuses with the rendulic plane s s whereas for the original Mohr-Coulomb Positive signs pertain to the lower branches of the yield locus intersections with the rendulic plane.
The three options for the critical plane orientation distinguish three slightly different material models of the Mohr--Coulomb type.The practical value of these modifications can be assessed in connection with the solutions of actual problems.The problem tackled below is the parameter identification in the direct shear test.

Evaluation of the direct shear test
Most applications of constitutive equations include a) the parameter calibration and b) solution of the actual task analytically or numerically.Let us assume first that the triaxial test is used in the first step.Tests provide points in the rendulic plane and parameters c and tan j are selected to best fit the points.Other procedures are available for identifying of the parameters using, for instance, the modified and alternate Mohr-Coulomb diagrams as recommended in [1] and [4].Different parameter values are obtained for the three versions of the yield locus.The calibrated locus remains nearly the same for all versions, however.Application of the three versions in any actual problem solution does not thus make any difference in the results, in spite of the difference in the parameter values.
Differences might occur when direct shear apparatus is used in the first step, see Fig. 4. The failure plane orientation is imposed by the test arrangement in this case.Strictly speak- orientation of the plane follows from the postulated condition that the Mohr circle in the s s1 3

Fig. 1 :
Fig. 1: Inclination a of the critical plane in the classical Mohr--Coulomb yield condition

1 2 =
are shown in Fig. 3. Functions f A and f M are important for parameter calibration of the model by the triaxial test.The modified Coulomb (A) condition intersection with the rendulic plane is: