Tutorial: single-element plasticity

The smallest possible demonstration of a nonlinear constitutive law. One 1×1×1 m hexahedral element is stretched past its yield point under von Mises (J2) plasticity, and the reaction force plateaus at the yield stress — the material yields instead of carrying more load. Every number below comes from a real run against Kratos 10.4 with the ConstitutiveLawsApplication.

The kratos://examples/plasticity-cube resource is this exact case with its files and result baked in, and notebooks/materials.ipynb walks through it as a live MCP client (alongside the material- and solver-preset tools).

1. Generate the mesh

A single hexahedron is create_box_mesh with one division per axis — 8 nodes, 1 element, with xmin/xmax/ymin/… face sub-model-parts:

mdpa_create_structured_mesh({
  "path": "/tmp/plast/mesh.mdpa", "kind": "box",
  "size": [1.0, 1.0, 1.0], "divisions": [1, 1, 1]
})
→ { "num_nodes": 8, "num_elements": 1 }

2. Build the material from a preset

create_materials accepts a preset — a ready-made constitutive law plus default variables. small_strain_plasticity_von_mises_3d fills the von Mises plasticity law (E = 210 GPa, yield stress = 250 MPa, perfect plasticity):

create_materials({ "output_file": "/tmp/plast/Materials.json", "materials": [
  { "model_part_name": "Structure.domain",
    "preset": "small_strain_plasticity_von_mises_3d" }
]})

List everything available with list_material_presets; discover a process' parameters with kratos_get_process_defaults.

3. The ProjectParameters

Plasticity is nonlinear, so the solver needs analysis_type: "non_linear" (a Newton–Raphson loop with a residual convergence criterion) — this is the one example whose ProjectParameters is hand-authored rather than rendered from the linear structural_static template. Symmetry (roller) supports on the xmin/ymin/zmin faces and a prescribed x-displacement ramped on xmax:

"processes": { "constraints_process_list": [
  { "...": "fix ux=0 on xmin, uy=0 on ymin, uz=0 on zmin (rollers)" },
  { "python_module": "assign_vector_variable_process", "Parameters": {
      "model_part_name": "Structure.xmax", "variable_name": "DISPLACEMENT",
      "value": ["0.001*t", null, null], "constrained": [true, false, false] } }
]}

4. Run

run_simulation({ "case_dir": "/tmp/plast", "wait_seconds": 60 })
→ { "state": "succeeded", "elapsed_seconds": 1.0 }

Five steps ramp the imposed x-displacement from 0.001 m to 0.005 m.

5. The elastic → plastic transition

Read the reaction force in x on the fixed xmin face at each step:

step 1  strain 0.001  |Fx| = 2.10e8 N   ← elastic: E·strain·A = 210 MPa
step 2  strain 0.002  |Fx| = 2.50e8 N   ← yielded
step 3  strain 0.003  |Fx| = 2.50e8 N   ← plateau
step 4  strain 0.004  |Fx| = 2.50e8 N
step 5  strain 0.005  |Fx| = 2.50e8 N

Below the yield strain (σ_y/E = 250e6/210e9 = 0.00119) the response is linear elastic and |Fx| = E·strain·A exactly. Once past yield the reaction plateaus at |Fx| = σ_y·A = 250 MPa × 1 m² = 2.5e8 N no matter how much further you stretch it. A LinearElastic3DLaw would keep climbing; the plateau is the nonlinearity.

Variations

• Other yield surfaces — the preset uses Von Mises; the constitutive-laws catalog (kratos_list_constitutive_laws) also has Drucker–Prager, Mohr–Coulomb and Tresca variants.
• Damage instead of plasticity — swap to the small_strain_damage_von_mises_3d preset for stiffness degradation.
• Hardening — set HARDENING_CURVE to a hardening law so the reaction keeps rising (gently) after yield instead of plateauing flat.