An empirical engineering failure criterion for soils and rocks that represents shear strength as a linear envelope: shear stress at failure τf = c + σ'·tan(φ), where c is cohesion, φ is the internal friction angle, and σ' is the effective normal stress; failure is predicted when the stress state reaches this envelope under the assumed stress path and drainage conditions.

Principle

Shear failure is predicted when the mobilized shear stress reaches a linear function of effective normal stress; the apparent cohesion and internal friction angle parameterize strength as an approximately linear relationship within the tested stress range.

Demonstration

Illustrative scenario → A saturated sand sample is consolidated and sheared in a drained triaxial test. Recognition: the experimenter plots Mohr circles for successive confining stresses and fits a straight line tangent to the outer envelope. Action: extract c and φ from the tangent line and use them to evaluate a slope’s factor of safety by comparing driving and resisting shear along a potential failure surface under drained conditions. Consequence: if the calculated shear equals the envelope value, the criterion indicates instability for the assumed drainage and stress‑path conditions.

Misapplication

Treating c and φ as immutable material constants valid for all stress paths, strain levels, rates, microstructures or scales. The semantic error is assuming parameters obtained from one test type (e.g., drained triaxial) apply unchanged to different field stress paths, undrained rapid loading, cemented or highly anisotropic materials.

Consequence

When parameterized to matching stress‑path and drainage conditions, the criterion provides a simple basis for preliminary design of slopes, foundations and retaining structures; when misapplied it can under‑ or over‑predict strength, producing unsafe designs or excessive conservatism because it neglects nonlinearity, strain softening/hardening, anisotropy, rate effects and scale dependence.

Reversal

The linear Mohr–Coulomb form is inadequate where strength depends strongly on confinement (nonlinear envelope), the intermediate principal stress is influential, fabric anisotropy or cementation dominates, or rapid undrained pore‑pressure response occurs; in those cases alternative models (e.g., nonlinear frictional models, Hoek–Brown for rock masses, or undrained strength concepts) are more appropriate.

Boundary

Clearly within: cohesionless and weakly cohesive geomaterials under typical engineering confinements where laboratory tests yield an approximately linear relation between shear strength and effective normal stress. Boundary case: lightly cemented or aged soils whose apparent cohesion varies with strain and history—applicability depends on matching test and field conditions. Clearly outside: highly fractured rock masses whose failure is controlled by discontinuity networks and scale effects requiring rock‑mass specific criteria.

Semantic Tension

Simplicity and analytical tractability (a two‑parameter linear envelope) ↔ fidelity to complex material behaviour (nonlinearity, anisotropy, path‑ and rate‑dependence); engineers must trade ease of use against the risk of misrepresenting actual strength mechanisms.

Synthesis

Mohr–Coulomb is a practical, parameterized approximation that captures emergent shear resistance from interparticle friction and short‑range cohesion; its usefulness depends on matching parameters and stress‑path assumptions to the problem and explicitly accounting for its limits (nonlinearity, drainage state, fabric and scale effects).