For many decades the strong non-linearity of the shear strength of intact rock and of non-planar fractures has been well documented, at least when considering wide variations in effective normal stress. Yet USA university researchers working in areas such as geothermal energy and CO2 sequestration, and petroleum service company personell working on so-called but actually common 'unconventional' (i.e. fractured) reservoirs are almost collectively showing linear 'Byerlee-type' friction coefficients, without effective normal stress dependence. We have relatively simple descriptions of the non-linearity. It is time that the improved accuracy of sub-surface predictions and modelling obtainable with the acceptance of non-linearity is applied, also in the USA and in the Middle East.
Linear approximation is reasonable in the case of small variation in stress level. You can represent nonlinear strength envelope with multiple linear segments without losing much accuracy.. I am not an expert in abovementioned fields but if high stress is involved linear approximation seems reasonable? Nonlinearity is much more pronounced in the low stress range.
The same question was asked by prof. Maksimovic (unfortunately he passed away few years ago) regarding nonlinear shear strength envelope for slope stability analysis
Zoran! Thanks for the reply. I agree with you that for the fractures, non-linearity is more marked in the low effective stress range. Nevertheless when injecting and changing the effective stress considerably, 'meeting' the strength envelope at different 'times' (also depending on different fracture set characteristics) surely has to be important. As for the intact rock, which may be weak in the sedimentary provinces, the non-linearity is very marked due to critical state behaviour. According to extensive analyses by others (Singh et al. 2011, Shen et al. 2018) the strength envelope approaches horizontal when the critical confining pressure reaches the level of UCS. So the maximum shear strength 'only' reaches to half the effective normal stress. (Barton, 1976). This incidentally can be used to explain why the world's highest mountain ranges reach 'only' to 8-9km - they are shear-strength limited, and maybe sub-brittle-ductile transition as well.
For what is worth: we are aiming to introduce non-linear envelopes (HB) in the next edition of Eurocode EC7. This is one aspect of the current draft that has gathered a very large consensus.
That is (partially) good to hear Marcos! Thanks for the EC7 update. Unfortunately there are different points of view that then enter the shear strength 'arena'. One is that, if we are talking rock masses and not intact rock, then there are some of us (and countless progressive failure cases of slopes) that suggest we cannot add c and sigma n Tan phi - linear MC or non linear H-B. If - or because - there are intact bridges that may need to fail first, then these rougher fractures, then suitably oriented joints, then perhaps filled discontinuities, there may therefore be many components ('crack, 'crunch', scrape', 'swoosh' has been suggested!) and at least c 'then' sigma n tan phi will be more representative. (If only joints were involved HB would not be needed, and the slope might have failed before it was formed - so to speak.). Non-'ideally' oriented joints will need some of the above components to be mobilized.) Concerning the intact rock we now have a more scientifically sound non-linear criterion than HB (Shen et al. 2018) - it has more curvature - and this may be important in the case of weaker rocks - even in civil engineerng.
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