Real shear failure envelope is not linear as in MC criterion, but more hyperbolic. This means that in low stress level MC cohesion is lower than usual, and friction angle is higher. Check out nonlinear (3 parameters) hyperbolic envelope by Professor Milan Maksimovic, whose parameters can be obtained from conventional direct shear or triaxial test.

       The failure criterion (envelope) of soils over the whole range of stresses (from small to large stresses) stresses is a topic that has attracted many researchers. The failure envelopes of many geomaterials (soils and rocks) are curved especially in the range of small normal stresses and this fact has been demonstrated experimentally in the laboratory for many soils (Penman 1953; Holtz and Gibbs 1956; Ponce and Bell 1971; Atkinson and Farrar 1985; Maksimovic 1989). It is common practice to fit a linear Mohr-Coulomb criterion to experimental data, due to its ease of manipulation, but as was mentioned by Stephane Bonelli this yields an erroneously high cohesion at zero normal stress.

Maksimovic (1989), as substantiated by Srbulov (1997), had demonstrated that the failure envelope for sands (granular materials) over a very wide range of normal stress is best represented by a hyperbolic law

τ = σ tan φ ------------------- (1)

where φ is the secant friction angle that varies hyperbolically with the normal stress σ viz:

φ = φ∞ + Δφ/(1+ σ/σN) -------------------(2)

φ∞ is the limiting friction angle at σ = ∞, Δφ is the change in friction angle from σ = 0 to σ = ∞, σN is the normal stress value at which φ has changed by Δφ/2 from its value at σ = 0.

However, for small to medium range of normal stresses, the failure envelope of cohesive and granular soils can be represented accurately by the power-type law

τ = (a + bσ)n ----------------------------------(3)

Baker (2004) had revealed how the parameters a, b, n could be determined from triaxial test data and also proved that this criterion fits quality test data for Dune sand, St. Bees sandstone, soft Singapore marine clay and overconsolidated London clay. For drained tests on sands and normally consolidated clays ‘a’ is very near zero. This type of criterion (suitably modified) had been utilized by Li (2007) in slope stability analysis. The advantage of Eq. 3 is that even when test data for σ greater than or equal to 69 KN/m2 are available it could extrapolate the true cohesion ‘an’ fairly accurately.

Anyaegbunam (2014) has given a detailed investigation of the properties of Eq.3 and provided a Qbasic4.5 program for evaluating a, b and n from triaxial test data.

References

Anyaegbunam, A.J. (2015).“Nonlinear power-type failure laws for geomaterials:

synthesis from triaxial data, properties and applications.” International                 Journal of Geomechanics, American Society of Civil Engineers, 15(1), Feb 2015 http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000348. 

Atkinson, J. H., and Farrar, D. M. (1985). “Stress path tests to measure soil strength parameters for shallow landslips.” Proc., 11th Int. Conf. on Soil         Mech. and Found.Eng.., San Francisco, Vol. 2, 983–986.

Baker, R. (2004). “Non-linear strength envelopes based on triaxial test data.”         J. Geotech. Geoenv. Eng., 130(5), 498-506

Holtz, W. G., and Gibbs, H. J. (1956). “Shear strength of pervious gravelly             soils.” J. Soil. Mech. Found. Div., 82 (SM 1), 1-22.

Maksimovic, M. (1989). “Nonlinear failure envelope for soils.” J. Geotech.               Geoenv. Eng.,115(4), 581-586.

Penman, A. (1953). “ Shear characteristics of saturated silts measured in              triaxial compression.” Géotechnique, 3(8), 312-328.

Ponce, V. M., and Bell, J. M. (1971). “Shear strength of sand at extremely low pressures.” J. Soil Mech. Found. Div., 97(SM4), 625-637.

Srbulov, M. (1997). “On the influence of soil strength brittleness and                 nonlinearity on slope stability.” Comput. Geotech., 20(1), 95-104.

Interesting discussion. If you measure failure stresses over a range of low stresses (e.g. 5 - 50kPA), I would think you could fit a linear Coulomb Criterion without major problem. This will not work over a wide range of stresses, of course. I would expect the hyperbolic trend having the most significant impact at near zero or minimal negative confining stress, or not?

We measure frictional parameters of analog model materials like sand and other granular materials under very low stresses (500Pa - 10kPa) and can always approximate the failure envelope with a linear Coulomb Criterion and we will get a very low Cohesion value as well.

However, we use a spezialised ring shear tester for powder testing which enable measurements under very low stresses. I would expect that some of your observations may be caused by the restrictions of the triaxial testing setup not designed to measure under very low confining pressure.

See below references for some measurement results under low stresses in granular materials.

Cheers, Jürgen

Lohrmann, J., Kukowski, N., Adam, J., and Oncken, O., 2003, The impact of analogue material properties on the geometry, kinematics, and dynamics of convergent sand wedges: Journal of Structural Geology, v. 25, no. 10, p. 1691-1711.

Panien, M., Schreurs, G., and Pfiffner, A., 2006, Mechanical behaviour of granular materials used in analogue modelling: insights from grain characterisation, ring-shear test and analogue experiments: Journal of Structural Geology, v. 28, p. 1710-1724.

Hello Jurgen Adam, Though a linear Mohr-Coulomb (M-C) envelope can be fitted to test data for low normal stresses, another M-C envelope will be needed for higher normal stresses thereby requiring bilinear or even trilinear envelopes to model a single material for a finite stress range. Difficulties can be encountered in using multiple envelopes in applications such as limit analysis and elastoplastic deformation computations in addition to the corner problem that occurs when a failure law ignores the intermediate principal stress. You have given two reference articles. That is good. Could you please provide about two data sets for test results obtained in the small and medium stress range obtained using  the ring shear tester. It will help me to demonstrate that a single power-type envelope can model the data very well in lieu of the bilinear M-C envelope as you suggested.   Thank you.