J2 bar

Verification methodWARNING BY DESIGN · 68 nodes / 140 elements

A monotone uniaxial elastoplastic case checks stress, equivalent plastic strain, equilibrium residual, and load-step sensitivity.

01 Model and scope

A monotone bar load drives elastic response to yield and then plastic flow. Stress and accumulated equivalent plastic strain can be compared with the one-dimensional return mapping; changing load increments reveals whether the result depends materially on discretization of the path.

Route assumptions and calculation checks

A numerical pass on the frozen fixture does not make it general material qualification.
Small displacements and monotonic loading do not cover contact or damage evolution.
The engineering-profile WARNING is part of the result and must remain visible.

Useful foundations and prerequisites

Units, signs and error measures Interpolation, mapping and assembly

02 Essential equations and symbols

Applied axial stress = 3.0\times{}10⁸ Pa

Definitions and conventions

$\sigma{}$: Axial stress.
$\sigma{}eq$: Equivalent von Mises stress.
$\alpha{}$: Accumulated equivalent plastic strain.
$r$: Global equilibrium residual.

01Monotone loading

02Return mapping

03Stress + plastic strain

04WARNING stays visible

J2 bar · method mapA monotone path tests the material point response and global bar result while keeping the deliberate WARNING visible.

3 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Separate the elastic predictor from plastic correction

Under monotone uniaxial tension, the trial stress is elastic until the yield stress is reached. Thereafter the isotropic-hardening branch couples total strain to elastic strain and accumulated plastic strain. This one-dimensional oracle complements, but does not replace, the three-dimensional material-point tests.

\sigma^{tr}=E(\varepsilon-\alpha_n),\qquad f^{tr}=|\sigma^{tr}|-(\sigma_{y0}+H\alpha_n)

2 · Compare the post-yield closed form

For monotone uniaxial loading after yield, the scalar hardening law provides a direct check of stress and accumulated equivalent plastic strain. Compare the expected constitutive response with the axial force equilibrium and the mean integration-point history.

\sigma=\sigma_{y0}+H\alpha,\qquad\varepsilon=\frac{\sigma}{E}+\alpha\quad(\text{monotone post-yield branch})

3 · Preserve the warning and increment sensitivity

The named 140-element bar reports 3.0×10⁸ Pa applied axial stress, mean equivalent plastic strain 0.05, and a relative stress error near 9.93×10⁻¹⁶; its engineering-profile verdict is deliberately WARNING. A nearly exact fixture value does not certify general plasticity or erase that warning.

\sigma_{applied}=3.0\times10^8\,\mathrm{Pa},\qquad\bar\alpha=0.05,\qquad e_\sigma\approx9.93\times10^{-16}

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

WARNING BY DESIGN · 68 nodes / 140 elements

Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.

J2 benchmark report Generated result summary Material status

06 Sources and related sheets

J2 benchmark report Generated result summary Material status

Continue the reasoning

TET4 force, pressure, and affine patch Beam and discrete oscillator MITC shell benchmarks External solver comparison Recorded structured-TET4 workloads

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