Developed at Cambridge University, the MCC model is an elastoplastic model tailored specifically for clays. It features an elliptical yield surface in space (where p′p prime is mean effective stress and is deviatoric stress):
This rule dictates how the yield surface changes in size, shape, or position as plastic deformation progresses.
To construct a valid plasticity model for geomechanics, four fundamental mathematical components must be defined. Yield Criteria (
The direction of plastic strain increment is given by: [ d\varepsilon_ij^p = d\lambda \frac\partial g\partial \sigma'_ij ] fundamentals of plasticity in geomechanics pdf
| | Real-World Application | | --- | --- | | Mohr-Coulomb yield criterion | Bearing capacity of shallow footings (Terzaghi’s method is derived from plasticity). | | Critical State & Hardening | Settlement analysis of soft clay under embankment loading. | | Non-associated flow rule | Prediction of lateral earth pressure on retaining walls. | | Softening & Strain localization | Stability of slopes in stiff, overconsolidated clays. | | Drucker-Prager cone | 3D numerical modeling of tunnels in rock. |
Calculating ultimate bearing capacity without over-simplifying soil failure geometry.
Natural soil deposits are formed via vertical sedimentation over millennia, making them inherently stiffer in the vertical direction than the horizontal. Advanced constitutive models distort the yield surface to account for this directional dependency. Developed at Cambridge University, the MCC model is
Beyond theory, these plasticity models are implemented to solve real-world engineering problems:
Explicit integration. Easy to implement but requires incredibly small time steps to prevent numerical drift.
is the second invariant of the deviatoric stress tensor, and Yield Criteria ( The direction of plastic strain
A hexagonal cone shape in principal stress space, ideal for defining failure.
dϵijp=dλ𝜕g𝜕σijd epsilon sub i j end-sub to the p-th power equals d lambda the fraction with numerator partial g and denominator partial sigma sub i j end-sub end-fraction is a non-negative scalar called the plastic multiplier.