forked from bai/curriculum-project-hub
feat(examples): migrate TH-141 sample to declarative layout (WU-6)
Real-content fixture for the end-to-end pipeline. TH-141 (39 parts: 22 segment, 15 lemma, 2 example) migrated from the prototype's typst `#let parts` manifest to ADR-0008: declarative manifest.toml (project+info+ordered parts+targets) + per-element element.toml (kind + scalars; examples carry `source`). Content .typ files copied byte-identical (no math corruption); per-element main.typ + meta.toml dropped (wiring is now generated). Part order matches source exactly; 5 lemmas have no proof.typ (optional); no cross-file imports / paralearn refs / figs. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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kind = "lemma"
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设固体体积 $V$、表面积 $A$,应变 $epsilon$ 为均匀小量。体相单位体积的自由能 $f_b = Y epsilon^2 \/ 2$,表面单位面积的自由能 $gamma(epsilon)$,总自由能
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$ F = V f_b + A gamma(epsilon) . $
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对均匀拉伸 $V epsilon$ 求导得单位横截面承担的应力
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$ sigma_("tot") = (partial F) / (partial (V epsilon)) = Y epsilon + A / V dot (dif gamma) / (dif epsilon) , $
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即 @总应力。
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对横截面被均匀拉伸的固体,单位横截面承担的总应力为
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$ sigma_("tot") = Y epsilon + A / V dot (dif gamma) / (dif epsilon) , $ <总应力>
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其中 $Y$ 为杨氏模量、$epsilon$ 为应变、$A \/ V$ 为表面积与体积之比。第一项为体相弹性贡献,第二项为表面贡献。
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