Legacy · Mechanical validation · §9.2

Restrained slab (foundation)

Slab $L=3\,\mathrm{m}$ on ground, Hill-type generation, restraint $\varepsilon_x=\varepsilon_y=0$ (Zhu). Comparison of the relaxation coefficient × incremental algorithm. Only case with CSV overlay on the portal.

Article: Part I · DOI 10.70271/rirc.v1n2a007 · §9.2

Material: Itumbiara

Reference figure

Data sheet

Item	Value
Type	Slab on ground (foundation); restraint $\varepsilon_x=\varepsilon_y=0$ ; free face $\sigma_z=0$
Thickness	$L=3\,\mathrm{m}$
Reported point	Centre ( $x=1.5\,\mathrm{m}$ )
Initial cond.	$T_0=T_{\mathrm{amb}}=20\,^{\circ}\mathrm{C}$
1D conduction	$k=7.5\,\mathrm{kJ/(m\cdot h\cdot^{\circ}C)}$ , $\rho=2400\,\mathrm{kg/m^{3}}$ , $c=0.75\,\mathrm{kJ/(kg\cdot^{\circ}C)}$ → $h^{2}=k/(\rho c)=0.10\,\mathrm{m}^{2}/\mathrm{d}$
Convection	$h_c=43.2\,\mathrm{kJ/(m^{2}\cdot h\cdot^{\circ}C)}$ (faces exposed to air, in the one-dimensional model of the article)
Generation (Hill)	$\alpha(t)=\exp\bigl[-(\tau/t)^{\beta}\bigr]$ ; $T^{Q}(t)=\Delta T_{\mathrm{adi}}\,\alpha(t)$ with $\Delta T_{\mathrm{adi}}=50\,^{\circ}\mathrm{C}$ , $\tau=36\,\mathrm{h}$ , $\beta=2$ (time in the Hill law is maturity / equivalent time when $E_a/R\neq 0$ )
Maturity (Arrhenius)	$E_a/R=4000\,\mathrm{K}$ , $T_{\mathrm{ref}}=20\,^{\circ}\mathrm{C}$ — enters generation and $f_t(t_{\mathrm{eq}})$ ; not $E(z)$ nor $J(t,z)$
Dilatation	$\alpha_T=10\times10^{-6}\,^{\circ}\mathrm{C}^{-1}$
Strength $f_t$	$f_t=f_{t,\infty}\,t_{\mathrm{eq}}^{n}/(a^{n}+t_{\mathrm{eq}}^{n})$ with $f_{t,\infty}=4.5\,\mathrm{MPa}$ , $a=2\,\mathrm{d}$ , $n=1.2$
Peak at centre (CSV)	$68.4\,^{\circ}\mathrm{C}$ at $2.51\,\mathrm{d}$ ( $t_{\mathrm{eq}}\approx 10.51\,\mathrm{d}$ )

$\tau$ and $\beta$ enter the degree-of-hydration law $\alpha(t)$ , not the dilatation $\alpha_T$ .

Chronological age × equivalent age

In the mechanical algorithm of this case (Zhu and incremental), $E$ and $J$ depend on chronological age $z=t$ (process time from casting). Arrhenius maturity $t_{\mathrm{eq}}$ governs Hill kinetics and the $f_t$ curve; it does not replace $z$ in $E(z)$ nor in $J(t,z)$ . Replacing $z$ by $t_{\mathrm{eq}}$ in the material shifts the compression peak (~−6.5 MPa → ~−7.3 MPa) and does not reproduce the CSV.

Canonical thermal history

To validate the viscoelastic solver, use the published history:

ref_laje_fixa_T_centro_media.csv

The article describes the thermal model in §9.2 (Hill + Arrhenius + convection). The slab_temperature function in the irc-creep package belongs to the free slab (§8.3.5: different thickness, different generation) and does not regenerate this CSV. Anyone who rebuilds $T$ with only symmetric convection on both faces of an isolated slab will get faster cooling (~32 °C at 20 d) than the CSV (~37.4 °C); the published history already incorporates coupling to the ground from the original simulation. The IRC mechanical reference always starts from the $T$ CSV.

Input files

ref_laje_fixa_T_centro_media.csv — $T$ at centre and mean
Material: Itumbiara

Output files

File	Content
ref_laje_fixa_centro.csv	Full (includes EVT, $f_t$ , $\sigma_{\mathrm{ef}}$ …)
ref_laje_fixa_centro_slim.csv	Slim for overlay

Columns of the full CSV

Column	Meaning
`tempo[h]`	process time from casting
`idade[d]`	chronological age (= time in days on this mesh)
`T_centro[C]`	temperature at the centre
`teq[d]`	equivalent time (Arrhenius maturity)
`f_t[MPa]`	tensile strength $f_t(t_{\mathrm{eq}})$
`sigma_Zhu_K[MPa]`	stress via relaxation coefficient $K=R/E$ (Zhu)
`sigma_incremental[MPa]`	stress from the $\eta_n$ , $q_n$ algorithm (IRC reference)
`dT_solic[C]`	thermal drop of the criterion: $(T_{\max}-T)_+$
`EVT[C]`	capacity in temperature: $f_t J(1-\nu)/(K_R\alpha_T)$ ; `nan` until the thermal peak (~2.5 d) — the tension check applies only on cooling
`sigma_ef[MPa]`	“classical” effective stress (effective modulus / strain capacity), §9.2
`sigma_ef_star_multi[MPa]`	multi-age $\sigma_{\mathrm{ef}}^{}$ from $t^{}$
`sigma_ef_star_1J[MPa]`	form with a single $J(t,t^{*})$

To validate the viscoelastic solver, idade[d], T_centro[C], sigma_Zhu_K and sigma_incremental suffice. Columns EVT / sigma_ef* are verification criteria of Section 9.2, not input to the incremental algorithm.

Control points (centre)

Check	Value (CSV)
Thermal peak	$68.4\,^{\circ}\mathrm{C}$ at $2.51\,\mathrm{d}$
Maximum compression (incremental)	$-6.54\,\mathrm{MPa}$ around $2.1\,\mathrm{d}$
Tension at $80\,\mathrm{d}$	$5.47\,\mathrm{MPa}$ (above $f_t\approx 4.48\,\mathrm{MPa}$ )

Overlay — compare your CSV

The chart loads the IRC reference (sigma_incremental vs age). Subscribers upload their own CSV (columns idade[d] and sigma_incremental[MPa], or sigma_user[MPa]). Parsing is in the browser; the file is not sent to the server.