Living Wood Joint

What is modelled

Wood holds water in its cell walls and changes size with it: across the grain it swells by a few tenths of a percent for each percent of moisture gained, and along the grain hardly at all. Moisture moves slowly by diffusion, so a thick piece lags behind the air for weeks. A joint cut with no clearance between pieces of different moisture therefore changes its fit while the pieces dry or wet towards the moisture set by the humidity of the room: a green mortise piece shrinks around a dry tenon and grips it, a wet tenon shrinks away from a dry mortise and goes loose.

Equations

• Equilibrium moisture from the Hailwood-Horrobin sorption curve in the relative humidity.
• Diffusion in a slab of half-thickness a: average moisture Me + (M0 − Me) (8/π2) Σ exp(−n2π2 Fo / 4) / n2, Fo = D t / a2, n odd.
• Size change Δs / s = β (M − M0) with β_tangential = 0.0030 and β_radial = 0.0016 per percent below the fibre saturation point.
• Pressure p = E δ / s (capped at the crushing strength), grip force μ p A.

Checks run in the page

• A finite-difference diffusion run against the cosine series.
• The sorption curve against the tabulated 12 % at 20 C and 65 %.
• A hundred thousand random clearances against the normal-distribution estimate.
• Moisture conservation in a sealed board.

Honest simplifications

The embedment stiffness, friction coefficient, shrinkage coefficients and the effective diffusivity (which also stands in for the resistance of the surface) are typical values of a hardwood, not data for one species, the joint is treated as a flat contact without creep or crushing history, and the pieces are straight and defect-free.

• Press Wet tenon, dry mortise and watch the grip fall away.
• Press Grain turned 90 degrees and compare the final grip.
• Press Dry season to see the effect of low humidity.