Throw a vessel on a Ukrainian kick wheel, dry it, and fire it in a wood-fired updraft kiln. Every number comes from a model: clay yield stress, shell stresses on the wall, moisture diffusion, a kiln energy balance and heat conduction with the quartz inversion at 573 C.
Drag the round handles sideways to reshape the radius profile r(z); drag the square handle up or down to change the height. The outline is coloured by the local safety factor of the wet wall (green = safe, red = about to fail).
Wet clay shrinks while water leaves the gaps between the particles; once they touch (critical moisture) shrinkage stops. The thick base dries slower than the thin wall, which is where drying cracks start.
Decoration is visual only. Ornaments are rendered in Blender as generic folk-style motifs in the spirit of the living Ukrainian traditions; they are not copies of any master's design.
Question: for a wall of a given thickness, what is the fastest steady heating rate that keeps the tensile stress (thermal gradient plus the sudden quartz expansion) below the strength of the ware? The model heats a bone-dry wall from 20 C to 680 C at a constant rate and records the worst strength/stress ratio.
Each check recomputes a result by a different route (a second formula, a closed-form solution or a conserved quantity) and compares it with the live simulation.
Throwing clay behaves like a Bingham solid: it holds its shape until the shear stress exceeds a yield stress tau, then flows. Tau falls roughly exponentially with water content; this lab uses tau = 6 kPa * exp(-(w-0.22)/0.045) (an assumed fit: real clays differ by a factor of 2 or more). A Tresca criterion says the wall fails when the largest difference between principal stresses exceeds 2 tau.
The wall is treated as a thin axisymmetric shell loaded by its own weight (membrane theory). The meridional force and the hoop force follow from vertical and normal equilibrium:
N_phi = -Np sqrt(1+r'^2) Np = rho g (integral of th r dz above z) / r N_theta = r (rho g th r' - Np r'') / sqrt(1+r'^2) sigma_theta,rot = rho w^2 r^2 (centrifugal, ring)Compression along the wall is also compared with elastic shell buckling, 0.2 x 0.605 E t / R (0.2 is a knock-down for imperfections; E is assumed to be 250 tau). A wide pot at high speed fails by centrifugal flaring long before a narrow one, and a tall thin cylinder buckles. A real potter's hands add support, friction and wobble that are not modelled.
The clay volume is the area of the wall section revolved around the axis: V = pi |closed integral of r^2 dz|. As a check Pappus' theorem gives V = 2 pi r_centroid A from the shoelace area and centroid.
Water moves through the wall by diffusion (liquid water above the critical moisture, vapour below it) and leaves at the surface at a rate J = k_m rho_vs(T) (a_s - RH). While the clay stays saturated its volume falls by exactly the volume of water lost, which fixes the linear drying shrinkage from the critical moisture: about 4.5-8 % for the bodies here. Differential shrinkage between the surface and the core, and between the thin wall and thick base, builds stress that relaxes with an assumed 2 h Maxwell time. The crack index is that stress divided by an assumed moisture-dependent strength; it shows trends (cover it, thin the base) but is not a calibrated failure prediction.
One lumped energy balance: C dT/dt = eta m_dot LHV - m_gas c_p (T - Ta) - G (T - Ta). Wood has LHV about 14.5 MJ/kg and needs about 5 kg of air per kg. The damper sets the draught, the draught sets the air factor lambda; lambda < 1 starves the fire (smoke, a reducing atmosphere that turns the clay black), lambda large wastes heat up the chimney. Opening the door floods the chamber with room air and shocks the ware.
The wall is a 1-D slab heated on both faces (radiation + convection). The effective heat capacity includes the dehydroxylation endotherm near 530 C and, in damp ware, the latent heat of water at 100 C. Thermal expansion plus the 0.45 % linear jump of quartz at 573 C (times the quartz fraction) form an eigenstrain field; for a free plate the stress is sigma = -E/(1-nu) (eps*(x) - mean - slope x). For a steady ramp beta the centre lags the surface by beta t^2 / (8 alpha), which the sweep compares with the solver. Strength rises only once the body sinters (assumed Arrhenius rate, ~0.5 after an hour at 950 C), so the weakest moment is just after dehydroxylation, around 573 C.
Total linear shrinkage is 1 - (1 - s_dry)(1 - s_fire): about 10 % for the default red clay and 7-13 % across the bodies here. To make a 20 cm jug, throw it about 22 cm tall.