K_σ = ∫ (∇N_a · σ ∇N_b) I dV for a stress state at the stiffness points (solids, plane and
axisymmetric elements with the hoop term), for linear buckling. Checked: uᵀK_σu = σ V for a
uniform stress on a distorted hex20 and a plane strain quad8, rigid translations in its null space.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Temperatures now strain the material in nonlinear steps instead of being
refused. At finite strain the deformation gradient splits multiplicatively,
F = F_m F_theta with F_theta = diag(1 + alpha_i dT) at each point: the material
sees F_m and the loops run on the thermally expanded volume J_theta dV0 (mixed
elements project det F_m), so the tangent stays consistent. At small strain the
split is additive, and elements on their linear stiffness take the equivalent
nodal forces as a dead load. The step's temperatures are its temperature loads,
or the steady conduction of its temperature conditions and heat loads (one-way
coupling, as the linear coupled step); both at once are refused. They ramp with
the load factor from each material's reference temperature. Johnson-Cook
plasticity softens at the point's temperature. Verified: free expansion and a
fully constrained neo-Hookean block at 20 % thermal stretch, full and hybrid
(exact); a bar held at its ends heated past yield (exact); the linear limit and
the conducted temperatures against the linear solvers; tangents by central
differences with a temperature field.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
Two more large-strain options on the isotropic constants, decoupled like
neo-Hookean with the volume kappa/2 (J - 1)^2: Mooney-Rivlin with C10 = mu/2 - C01
(0 <= C01 <= mu/2) and Yeoh with C10 = mu/2 and the higher terms C20, C30, so the
small-strain material is the elastic one everywhere. MaterialModel::Polynomial
gives the isochoric stress and tangent from Holzapfel's fictitious quantities
(eqs. 6.168, 6.193), consistent by central differences, equal to neo-Hookean
with C10 alone; hybrid elements take it. Abaqus writes *HYPERELASTIC,
MOONEY-RIVLIN and YEOH; LS-DYNA keeps refusing hyperelastic materials in
nonlinear steps by name. Verified in uniaxial and equibiaxial tension and pure
shear against the compressible closed form (exact).
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
The radial return of the explicit update is now J2::radial_return, called by
both solvers, with the consistent tangent J2::tangent (Simo & Hughes 1998,
Box 3.2). The new elastoplastic module gives the implicit material point:
Simo's multiplicative plasticity with the exponential return map in the
principal logarithmic strains (CMAME 99, 1992) and its spatial tangent through
the Daleckii-Krein divided differences of the logarithm, and the additive
radial return of small strain with kinematic hardening. MaterialModel::Plastic
carries it, decoupled (deviatoric return, Hencky volume) so hybrid elements
take it. The element loops of finite.rs gain small-strain kinematics for
inelastic materials and pass history through the mixed path. Every tangent is
checked against central differences, element by element, from a plastic
history.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
The material point interface returns the Kirchhoff stress and the spatial
tangent of its Lie derivative from the deformation gradient, with per-point
history committed by the solver: St Venant–Kirchhoff (pushed forward from
S = D E) and the decoupled compressible neo-Hookean with its isochoric and
volumetric parts. Total Lagrangian solids (tet4, tet10, wedge6, hex8, hex20
full and reduced) and plane strain and axisymmetric elements give internal
forces and the consistent tangent with its geometric stiffness; hybrid
solids use the Simo–Taylor–Pister three-field formulation on the existing
pressure fields. Incompatible modes and plane stress are refused by name.
Follower pressure on faces and 2D edges with its load stiffness. Every
tangent is checked against central differences of its forces.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
