Each new layer solidifies hot, bonds to the already-cooled layers beneath it, then contracts as it cools — but the stiff material below restrains that contraction. This is the temperature-gradient mechanism (TGM) described by Mercelis & Kruth (2006): it leaves the most recently built layers in tension, with a compensating compressive zone lower in the stack and a second tensile peak right where the part is still bonded to the rigid build plate — the classic "M-shaped" residual-stress profile measured in laser powder bed fusion (LPBF) parts.
Layer eigenstrain: ε0(z) = α·ΔT_eff·c_cool · shape(z) ΔT_eff = T_process − T_preheat
While plate-bonded: σ(z) ≈ E·(−ε0(z)) (fully restrained)
After release, a free beam must satisfy:
∫σ dA = 0 → ε_uniform = mean(ε0)
∫σ·z dA = 0 → κ = ∫ε0(z)·z dA / I (bending curvature)
σ(z) = E·[ε_uniform + κz − ε0(z)] (clipped at yield, ±950 MPa)
Sagitta (end deflection): δ ≈ κ·L² / 8
- Layer count sets the through-height resolution of the eigenstrain profile — more, thinner layers approach the continuous TGM stress curve.
- Deposition temperature sets ΔT_eff and therefore the magnitude of the eigenstrain — hotter melt pools mean more shrinkage fighting the solid structure below.
- Preheat raises the baseline temperature the layer cools to, shrinking ΔT_eff — this is exactly why real LPBF/EBM machines heat their build plates to cut warping.
- Cooling rate scales the retained eigenstrain c_cool — a faster quench leaves less time for stress relaxation, steepening the thermal gradient and locking in more shrinkage strain; a slower cool lets more of it relax away.
- Part length sets the beam's moment arm — the same stress profile bends a longer part further, exactly as κ·L² predicts.
- Material properties used: Ti-6Al-4V, E = 114 GPa, α = 8.6×10⁻⁶ /K, yield ≈ 950 MPa.
Bending is visually exaggerated ×6 for legibility — real LPBF cantilevers typically warp by tenths of a millimetre to a few millimetres, which is exactly why parts are stress-relief annealed before cutting them off the plate.