Legacy · Mechanical validation · §8.3 / Fig. 6c

Hollow cylinder (Fig. 6c)

Separable thermal field — §8.3 / Figure 6(c). Uses the same $\theta(t)$ (and $\lambda$ ) as the solid cylinder.

Mechanical material: Itumbiara — $\nu=0.20$ ; $\alpha_T=10\times10^{-6}\,^{\circ}\mathrm{C}^{-1}$ . Diffusivity: $h^{2}=0.10\,\mathrm{m}^{2}/\mathrm{d}$ (do not confuse with $a_T$ ). Idea: $T=\theta(t)\,h(\mathbf{x})$ ⇒ $\sigma_{ij}=g_{ij}(\mathbf{x})\,S(t)$ .

Control points (all Fig. 6 cases): ref_fig6_pontos_controle.csv.

Schematic

Data sheet

Item	Value
Geometry	Long hollow cylinder: inner radius $a=0.15\,\mathrm{m}$ , outer $b=1\,\mathrm{m}$
Readout cut	$r/b=0.23$ (i.e. $r=0.23\,\mathrm{m}$ )
Boundary	$\sigma_r(a)=\sigma_r(b)=0$ ; free ends
$h(r)$	$\ln(b/r)/\ln(b/a)$ (steady state)
$\theta(t)$	same as the solid cylinder ( $\lambda=h^{2}\mu_1^{2}/b^{2}$ )
Control point $g_\theta$	$g_\theta(0.23)\approx -0.35311$ in the CSV (numerical integration of the map); the closed form T&G (`hollow_cylinder_log_closed_form`) gives $\approx -0.35286$ — difference ~0.07%, typically from radial discretisation

Input files

ref_cilindro_oco_theta.csv
Material: Itumbiara

Output figures

Maps · Contours

Output files

Time mesh and tolerance

Same Fig. 6 mesh: age_grid(3, 123, 240). Expected tolerance ~0.2% on the temporal incremental response (see solid cylinder).