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explicit: hyperelastic materials on the shared finite-strain material point
Neo-Hookean, Mooney–Rivlin and Yeoh materials in explicit dynamics, in total form from the deformation gradient (objective by construction): the one-point hex8 takes its mean F over the undeformed element and the stress τ/J; tet4 takes the isochoric stress of its own F with the average nodal pressure of the decoupled volumetric energy (Bonet–Burton at finite strain). Each element's frequency bound follows the tangent's largest normal stiffness. Small-displacement steps and thermal strain of hyperelastic elements are refused by name. Verified by uniaxial strain to λ = 1.5 of all three laws on hex8 and tet4 against the material point (to rounding), and the oracle case explicit-neo-hookean-implicit: a quasi-static explicit pull against the implicit nonlinear solver (lateral contraction to 0.09%, stress to 0.03%). Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
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jyvikoandClaude Opus 5.5 authored and Irakli Kourtis committed
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# Neo-Hookean cube: explicit against implicit (quasi-static)
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A unit cube of 2 × 2 × 2 cells of tet4 (six per cell) of a compressible neo-Hookean material
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(E = 10, ν = 0.3, ρ = 1), quarter symmetric (symmetry planes x = 0, y = 0, z = 0), pulled on
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x = 1 to λ = 1.5 in uniaxial stress. The same model is solved by the implicit nonlinear solver
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and by explicit dynamics with the pull applied over 100 time units (a few hundred wave transits):
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the lateral contraction and the axial stress must agree, both coming from the shared
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finite-strain material point.
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{
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"title": "Neo-Hookean cube stretched to λ = 1.5: explicit dynamics, loaded slowly, against the implicit nonlinear solver",
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"kind": "theory",
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"source": "Uniaxial stress of a compressible neo-Hookean solid (Holzapfel, Nonlinear Solid Mechanics, Wiley 2000, §6.5): a homogeneous state that a quasi-static explicit run must share with the implicit Newton solution on the same material point; the implicit solver is itself checked against the closed form by nl-neo-hookean-uniaxial",
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"quantities": [
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{"id": "lateral", "quantity": "lateral displacement of the corner (1, 1, 1) along y, explicit over implicit",
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"measure": {"command": "solve.probe", "params": {"step": "Dynamic", "at": [[1, 1, 1]]}, "pointer": "/nodes/0/displacement/1",
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"over": {"command": "solve.probe", "params": {"step": "Static", "at": [[1, 1, 1]]}, "pointer": "/nodes/0/displacement/1"}},
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"value": 1.0, "tolerance": {"relative": 0.005},
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"reason": "the same material point and a homogeneous state; the explicit run is pulled over a few hundred wave transits, and what is left of its waves is a fraction of a percent"},
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{"id": "stress", "quantity": "σ_xx of element 1, explicit, over σ_xx at the node at (0, 0, 0), implicit",
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"measure": {"command": "solve.probe", "params": {"step": "Dynamic", "elements": [1]}, "pointer": "/elements/0/stress/0",
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"over": {"command": "solve.probe", "params": {"step": "Static", "at": [[0, 0, 0]]}, "pointer": "/nodes/0/stress/0"}},
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"value": 1.0, "tolerance": {"relative": 0.005},
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"reason": "as for the displacement"}
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]
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}
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{
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"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
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"values": {
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"lateral": {
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"band": 1e-9,
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"value": 0.9990781754549147
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},
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"stress": {
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"band": 1.0002554925493198e-9,
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"value": 1.0002554925493197
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}
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}
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}
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@@ -0,0 +1,245 @@
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[
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{
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"command": "mesh.block",
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"params": {
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"body": "Cube",
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"kind": "tet4",
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"corners": [
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[
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0,
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0,
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0
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],
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[
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1,
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0,
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0
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],
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[
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1,
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1,
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0
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],
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[
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0,
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1,
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0
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],
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[
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0,
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0,
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1
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],
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[
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1,
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0,
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1
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],
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[
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1,
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1,
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1
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],
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[
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0,
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1,
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1
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]
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],
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"divisions": [
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2,
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2,
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2
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],
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"faces": {
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"6": "x0",
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"3": "y0",
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"1": "z0",
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"4": "x1"
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}
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}
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},
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{
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"command": "model.set.create",
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"params": {
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"name": "all",
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"geometry": [
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{
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"body": "Cube"
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}
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]
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}
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},
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{
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"command": "model.set.create",
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"params": {
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"name": "x0",
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"geometry": [
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{
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"body": "Cube",
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"face": "x0"
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}
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]
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}
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},
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{
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"command": "model.set.create",
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"params": {
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"name": "y0",
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"geometry": [
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{
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"body": "Cube",
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"face": "y0"
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}
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]
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}
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},
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{
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"command": "model.set.create",
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"params": {
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"name": "z0",
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"geometry": [
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{
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"body": "Cube",
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"face": "z0"
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}
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]
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}
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},
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{
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"command": "model.set.create",
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"params": {
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"name": "x1",
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"geometry": [
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{
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"body": "Cube",
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"face": "x1"
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}
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]
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}
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},
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{
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"command": "model.material.create",
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"params": {
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"name": "Rubber",
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"elastic": {
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"type": "isotropic",
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"young": 10.0,
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"poisson": 0.3
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},
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"density": 1.0,
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"hyperelastic": {
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"type": "neo_hookean"
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}
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}
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},
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{
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"command": "model.section.create",
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"params": {
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"name": "cube",
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"set": "all",
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"material": "Rubber",
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"kind": {
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"type": "solid"
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}
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}
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},
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{
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"command": "model.boundary_condition.create",
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"params": {
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"name": "sx",
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"type": "symmetry",
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"target": {
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"set": "x0"
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},
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"plane": "x"
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}
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},
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{
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"command": "model.boundary_condition.create",
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"params": {
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"name": "sy",
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"type": "symmetry",
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"target": {
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"set": "y0"
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},
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"plane": "y"
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}
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},
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{
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"command": "model.boundary_condition.create",
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"params": {
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"name": "sz",
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"type": "symmetry",
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"target": {
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"set": "z0"
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},
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"plane": "z"
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}
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},
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{
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"command": "model.boundary_condition.create",
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"params": {
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"name": "pull",
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"type": "displacement",
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"target": {
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"set": "x1"
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},
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"values": [
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0.5,
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null,
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null,
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null,
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null,
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null
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]
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}
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},
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{
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"command": "model.step.create",
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"params": {
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"name": "Static",
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"procedure": {
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"type": "nonlinear_static"
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},
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"boundary_conditions": [
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"sx",
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"sy",
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"sz",
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"pull"
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],
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"loads": []
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}
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},
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{
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"command": "model.step.create",
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"params": {
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"name": "Dynamic",
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"procedure": {
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"type": "explicit_dynamic",
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"end_time": 100.0
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},
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"boundary_conditions": [
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"sx",
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"sy",
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"sz",
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"pull"
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],
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"loads": []
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}
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},
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{
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"command": "solve.nonlinear",
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"params": {
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"step": "Static"
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},
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"id": "implicit"
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},
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{
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"command": "solve.explicit",
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"params": {
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"step": "Dynamic"
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},
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"id": "explicit"
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}
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]
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