Failure envelope behavior: Depending on the material properties and stress range, the shear strength (failure) envelope may exhibit which characteristics?

Introduction / Context:
The Mohr–Coulomb model often uses a straight-line failure envelope (τ = c + σ′ tan φ). However, real geomaterials can show curvature at low or high stresses, and the intercepts depend on bonding, suction, or cementation. Understanding these possibilities is important when extrapolating lab data across stress ranges.

Given Data / Assumptions:

• Material may be purely frictional, cohesive-frictional, or bonded.
• Effective stress framework applies for drained conditions.
• Observed envelope may vary with fabric and confining stress.

Concept / Approach:

(a) Many soils and rocks show curved envelopes (e.g., non-linear Hoek–Brown or parabolic low-stress behavior); linearization is a convenience over limited ranges. (b) For clean sands with zero cohesion, the envelope passes through the origin. (c) Presence of apparent or true cohesion yields a positive τ-intercept when projected to σ′ = 0.

Step-by-Step Solution:

Verification / Alternative check:

Laboratory triaxial envelopes often curve at low confining stress and may straighten at higher stress; direct shear on sands gives near-origin passage.

Why Other Options Are Wrong:

Since all three statements are valid, “All the above” must be chosen; “None” contradicts established observations.

Common Pitfalls:

Assuming strict linearity for all soils; ignoring suction-induced cohesion in partially saturated soils.

Final Answer:

All the above