Mohr-Coulomb Faulting Simulator (2D)

Faulting Regime

Stress State

Max principal stress σ₁ 150 MPa

Min principal stress σ₃ 40 MPa

Cohesion C₀ 20 MPa

Friction angle φ 30°

Fracture plane angle θ (from σ₁) 45°

STABLE — circle below envelope

Normal stress σₙ

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Shear stress τ

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Envelope τ at σₙ

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Coulomb-optimal θ

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How it works

On any plane inclined at angle θ to the maximum principal stress σ₁, the normal and shear stresses are the Mohr-circle equations (σ₁, σ₃ compressive, σ₁ ≥ σ₃):

σn = (σ1+σ3)/2 + (σ1-σ3)/2 · cos(2θ)
τ  = (σ1-σ3)/2 · sin(2θ)

Every possible plane traces the full Mohr circle: center (σ1+σ3)/2, radius (σ1-σ3)/2. The rock fails the instant that circle touches or crosses the linear Mohr-Coulomb failure envelope, set by cohesion C₀ and internal friction angle φ:

τ_failure = C0 + σn · tan(φ)

Geometrically the circle is tangent to the envelope exactly when the perpendicular distance from the circle's center to the envelope line equals the circle's radius:

(σ1-σ3)/2  ==  C0·cos(φ) + (σ1+σ3)/2 · sin(φ)   ← tangency
(σ1-σ3)/2  >   C0·cos(φ) + (σ1+σ3)/2 · sin(φ)   ← failure (circle crosses the line)

Solving for the tangent point on the circle gives the Coulomb-optimal fracture angle, the plane that fails first as stress increases:

θ_opt = 45° + φ/2   (measured from σ1)

Real crustal rock typically has φ ≈ 25–35°, so shear fractures form in conjugate pairs about 32–52° from σ1 — not at 45°, the classic textbook departure from simple maximum-shear intuition.

Regime — Anderson's theory of faulting: which principal axis is vertical decides the fault style. σ1 vertical → normal faults. σ1 horizontal, σ3 vertical → reverse/thrust faults. σ1 and σ3 both horizontal → strike-slip faults. The cross-section below reorients to match.
θ slider — rotates the candidate fracture plane through the stress field; the moving point on the Mohr circle and the fracture line in the rock block move together.
Snap to optimal — jumps straight to θ_opt, the angle real rock actually breaks along, found by maximizing τ − σn·tanφ over θ (equivalently, the point where the circle is tangent to the envelope).

σ₁ (max compression) σ₃ (min compression) candidate fracture plane failure envelope Mohr circle