Mohr–Coulomb shear strength envelope: identify the correct statements Regarding the Mohr–Coulomb failure criterion for soils and rocks, which of the following statements hold true?

Introduction / Context:
The Mohr–Coulomb criterion underpins most introductory geotechnical analyses. It describes shear strength as a function of normal stress, cohesion, and internal friction angle, and it connects principal stresses at failure with a straight-line envelope in τ–σ space for idealized soils.

Given Data / Assumptions:

• Isotropic, homogeneous material idealization for the envelope.
• Effective stress interpretation for drained conditions (or total stress for undrained clays as appropriate).
• Plane strain approximations common in retaining walls and slopes.

Concept / Approach:

The classical form is τf = c + σ′ tan φ. Coulomb’s insight was the linear relation; Mohr’s circle construction connects principal stresses at failure to a tangent straight line at angle φ. Real geomaterials may deviate slightly from perfect linearity (curved envelopes at very low or very high stresses), motivating generalised forms that still reference Mohr circles.

Step-by-Step Solution:

Verification / Alternative check:

Laboratory direct shear and triaxial test data typically produce near-linear envelopes over common stress ranges, validating the criterion.

Why Other Options Are Wrong:

Each statement (a)–(d) correctly characterises a facet of Mohr–Coulomb; hence “All of the above” is the only fully correct choice.

Common Pitfalls:

Confusing total and effective stress; assuming linearity at extreme stresses without verification; neglecting scale and fabric effects.

Final Answer:

All of the above.