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Soil Mechanics / Critical-state soil behaviour

The critical state

The unique end-point of shearing, and the critical-state line.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

  • A geotechnical reviewer must check laboratory interpretation, classification conventions and the limits of drained, undrained and one-dimensional idealisations.
  • Stress-path and critical-state illustrations simplify stress anisotropy, total vs excess pressure, loading history, drainage, structured soils, strain softening and cyclic behaviour. Full specialist review remains necessary; selected corrections do not certify the models.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

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Why this matters

So far the course has treated compression (Module 10) and strength (Module 11) as separate topics. Critical-state soil mechanics shows they are two faces of one framework. The idealised framework describes continued shearing towards a constant-volume, constant-stress condition. Its applicability depends on the soil and loading path; structure, particle crushing and strain localisation can complicate this response.

What you should already know

  • Void ratio and the e–log σ′ compression line (Module 10)
  • Shear strength, φ′ and the q–p′ invariants (Modules 9, 11)
  • Dilation and the peak/critical-state distinction (Module 11)

At the critical state a soil shears continuously at constant volume and constant stress — it has stopped changing. Dense soils reach it after dilating; loose soils after contracting; but both arrive at the same state for a given stress. Because the void ratio no longer changes there, the critical state links a soil's volume to the stress at which it can flow — the bridge between compression and strength.

Plotted with the invariants of Module 9, the critical states of a soil fall on a single critical-state line (CSL):

  • in q–p′ space (deviatoric vs mean effective stress) the CSL is a straight line through the origin, $q = M p′$;
  • in e–ln p′ space (void ratio vs mean effective stress) it is a line parallel to the normal compression line.

The slope MM in q–p′ is the strength at the critical state, and it is fixed by the critical-state friction angle.

Critical-state line and stress ratio
q=Mp′,M=6sin⁡φcs′3−sin⁡φcs′q = M p', \qquad M = \dfrac{6\sin\varphi'_{cs}}{3 - \sin\varphi'_{cs}}

What it calculates: the deviatoric stress at the critical state, and its link to the friction angle

q
deviatoric stress at critical state (kPa)
p′
mean effective stress (kPa)
M
critical-state stress ratio (slope of the CSL)
φcs′\varphi'_{cs}
critical-state friction angle (degrees)

In plain terms: M and φ′cs carry the same information (triaxial compression). φ′cs = 30° gives M = 1.2; a stronger soil has a larger M and a steeper CSL.

Try it

Critical-state explorer

The critical-state line q = Mp′ and a constant-confining-stress drained path. Their intersection describes critical-state strength for this model; a dense or overconsolidated soil may reach peak strength earlier.

30 °
100 kPa
Stress ratio M
1.200
p′ at failure
167 kPa
q at failure
200 kPa
Axial σ′₁f = σ′₃ + q
300 kPa

What the plot shows

  • Red line = critical-state line q = Mp′ (slope M from φ′cs).
  • Orange path = drained triaxial compression (slope 3 in q–p′).
  • The intersection agrees with Mohr–Coulomb using c′ = 0 and the same critical-state friction angle; it does not predict a separate peak strength.
p′ (kPa)qCSL q = Mp′p′₀failure

φ′cs = 30° gives M = 1.20. The drained path meets the critical-state line at p′ = 167, q = 200 kPa, i.e. σ′₁ = 300 kPa — identical to the Mohr–Coulomb failure stress.

Worked example

Failure state of a drained triaxial test in q–p′ space

Given

  • Critical-state friction angle φ′cs = 30° (so M = 1.2)
  • Drained triaxial compression at cell pressure σ′3 = 100 kPa

Find

The mean and deviatoric stress at failure, and the axial stress σ′1f.

Assumptions

  • Drained test: σ′3 constant; the drained path in q–p′ has slope 3.

    Summary

    • At the critical state a soil shears at constant volume and constant stress, independent of its start.
    • The critical states lie on the critical-state line: q = Mp′ in stress space, parallel to the NCL in e–ln p′.
    • M = 6 sinφ′cs/(3 − sinφ′cs) links the CSL slope to the friction angle (M = 1.2 at φ′cs = 30°).
    • Critical-state and Mohr–Coulomb give the same failure strength; critical state adds the volume dimension.