forked from 1b-agent/fieldcraft-upstream
jyvikoandClaude Opus 5.5
7a69fe5e2f
Four cases, each against its closed form: uniaxial tension of a J2 cube to a logarithmic strain of 0.2 (Kirchhoff stress exact; 0.13 % from the explicit solver's hypoelastic Cauchy stress, the elastic volume change between the two laws), the Bauschinger effect of linear kinematic hardening on a load path back to zero strain (exact), Hill's elastic-perfectly plastic thick cylinder (bore displacement at a plastic radius of 1.5a within 0.009 %, collapse pressure within 2e-8; the pending case is now active) and a cantilever's plastic collapse, converged 3.1 % above beam theory's hinge, the bound a continuum exceeds. docs/fem.md, the literature, ROADMAP (M15), parity and the oracle page follow. Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
17 lines
1.1 KiB
Markdown
17 lines
1.1 KiB
Markdown
# Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit)
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The cube of `j2-uniaxial-explicit` solved by the implicit nonlinear solver: one fully integrated
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hex8, E = 200000, ν = 0.3, σ_y = 250, H = 2000 (isotropic), held by symmetry on three faces and
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stretched on the fourth to λ = e^0.2 in a nonlinear static step, lateral faces free.
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Simo's exponential return map integrates this proportional path exactly in the logarithmic
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strain, so the Kirchhoff stress is the small-strain curve τ = σ_y + E_t (ε − σ_y/E) = 643.564 at
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ε = 0.2 and the equivalent plastic strain (τ − σ_y)/H = 0.196782. The Cauchy stress is
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σ = τ/J with Hencky's volume law ln J = τ/(3K): 642.737.
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Cross-check with the explicit solver: `j2-uniaxial-explicit` reaches 643.57 for the Cauchy stress
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with the same plastic strain. Its hypoelastic update takes the logarithmic strain rate against
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the Cauchy stress, while the implicit model is hyperelastic in the Kirchhoff stress: the two
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laws differ by the factor J = 1.0013 (the elastic volume change), which is the 0.13 % between
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them; both meet the closed form of their own law.
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