Constitutive Models
A constitutive model relates stress to strain. Implementing one correctly is a different problem from writing down its yield surface. These articles work through the parts that decide whether an implementation converges: return mapping at corners and apexes, consistent tangent operators, and the parameters that quietly control the answer.
7 articles
The consistent tangent: 7 iterations against 887
Newton converges quadratically only with the exact Jacobian. On one elastoplastic bar the correct tangent needs 7 iterations where the elastic one needs 887.
Hoek-Brown tunnel: the closed-form ground reaction curve
A circular tunnel in Hoek-Brown rock, derived in closed form from radial equilibrium and solved in KATAI 2D: plastic radius, wall closure, every number given.
What a Mohr circle actually tells you
The Mohr circle turns a stress state into geometry. The angles on it are doubled, the envelope touches it once, and both facts are derived here from scratch.
Drucker-Prager: four ways to match Mohr-Coulomb
A cone fitted to a hexagon is right in some directions and wrong in others. Matched at triaxial compression it overstates extension strength by exactly 40%.
Dilatancy angle: what the choice of psi actually costs
Associated flow is convenient and wrong for soil. On one bearing capacity problem it inflates the collapse load from 200.8 to 301.4 kPa — a third more.
Mohr-Coulomb return mapping: the corners and the apex
The single-surface return is three lines of algebra and it produces an inadmissible state. Worked in full, with the principal ordering violated by 37.494 kPa.
The K0 procedure: initial stresses before anything moves
A model starting from the wrong horizontal stress is wrong before the first load. Jaky, elasticity and overconsolidation give 0.429, 0.470 and 0.980 here.
Terms defined
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