feat(fem): second moment of area of a hollow circular (tube) section#442
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Add beam::hollow_circular_second_moment_of_area(outer_diameter, inner_diameter), the bending second moment of a tube/pipe cross-section: I = pi * (D^4 - d^4) / 64 (m^4) the annulus = outer disc minus inner disc -- the I of structural tubing, far stiffer per unit weight than a solid bar since the removed core carries little bending stress. Guard: D > 0, 0 <= d < D, finite. Analytic test hollow_circular_second_moment_of_area_is_pi_d4_minus_d4_over_64: (a) worked D=0.1, d=0.05 -> I ~= 4.6019e-6 m^4; (b) threads circular_second_moment_of_area (#419) (non-tautological): the annulus is I(D) - I(d); (c) solid limit: a zero bore is a solid circle; (d) monotonicity: a larger bore removes more material so I drops; (e) 0-sentinel guards for D<=0, d>=D, non-finite, negative bore. Re-exported from lib.rs alphabetically in the beam block. valenx-fem 265 lib tests (was 264), cargo clippy --all-targets -D warnings clean.
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… set (#451) Adds the shear modulus (modulus of rigidity) G = E/(2(1+ν)) (Pa) as a public closed-form primitive in beam.rs — the second Lamé constant μ, relating a shear stress to the shear strain it produces. Together with the bulk modulus K (#440) and the Lamé first parameter λ (#442) it completes the isotropic elastic-constant set: λ = K − 2G/3, and E = 9KG/(3K+G). Named shear_modulus_from_youngs to avoid colliding with the crate's existing private material-based shear_modulus(&FemMaterial) helper; the new fn is the closed-form (E, ν) → f64 public companion. Returns 0 for non-physical input (E ≤ 0 or non-finite, ν outside (−1, 0.5)); G > 0 throughout that range, so the 0 sentinel is unambiguous. Analytic test (5 checks): worked anchor (E=200e9, ν=0.3 → G≈7.6923e10); two non-tautological cross-checks threading the prior constants — the Lamé identity G = 1.5(K − λ) and the interconversion E = 9KG/(3K+G); the ν=0 limit G = E/2; and non-physical guards → 0. valenx-fem 269 lib tests (was 268), clippy clean. Research-grade closed-form, not a substitute for a full constitutive model.
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…amily (#453) Adds the P-wave (longitudinal / constrained) modulus M = E(1−ν)/((1+ν)(1−2ν)) (Pa) as a public closed-form primitive in beam.rs — the uniaxial-strain stiffness (stress to compress along one axis while laterally constrained, as in an oedometer / 1-D consolidation test), which also sets the compressional wave speed v_p = √(M/ρ). It is the capstone tying the whole isotropic elastic-constant family together: M = K + 4G/3 = λ + 2G, where K = bulk_modulus (#440), G = shear_modulus_from_youngs (#444), λ = lames_first_parameter (#442). At ν=0 it reduces to E. Returns 0 for non-physical input (E ≤ 0 or non-finite, ν outside (−1, 0.5)); M > 0 throughout that range, so the 0 sentinel is unambiguous. Analytic test (5 checks): worked anchor (E=200e9, ν=0.3 → M≈2.6923e11); two exact non-tautological identities threading the prior constants — M = K + 4G/3 and M = λ + 2G (the latter ties all four together); the ν=0 limit M = E; and non-physical guards → 0. valenx-fem 270 lib tests (was 269), clippy clean. Research-grade closed-form, not a substitute for a full constitutive model.
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What
Adds
beam::hollow_circular_second_moment_of_area(outer_diameter, inner_diameter)— the bending second moment of a tube/pipe cross-section:// I = pi * (D^4 - d^4) / 64 (m^4)Why
The annulus = outer disc minus inner disc — the
Iof structural tubing, far stiffer per unit weight than a solid bar since the removed core carries little bending stress. Guard:D > 0,0 <= d < D, finite. Extends the FEM section-property library from solid sections to hollow tubing.Test
hollow_circular_second_moment_of_area_is_pi_d4_minus_d4_over_64:D=0.1, d=0.05 -> I ~= 4.6019e-6 m^4;circular_second_moment_of_area(feat(cfd): vorticity standard deviation (variance identity) #419) (non-tautological) — the annulus isI(D) - I(d);Idrops;D<=0,d>=D, non-finite, negative bore.Re-exported from
lib.rsalphabetically in the beam block.valenx-fem265 lib tests (was 264);cargo clippy -p valenx-fem --all-targets -- -D warningsclean. Research-grade section-property primitive.