Layer-by-Layer Residual Stress & Warping in Metal 3D Printing

Print Process

Peak melt-pool temperature 900°C

Build-plate preheat 20°C

Build height 25 mm

Part length 80 mm

Live Readouts

Build progress

0 / 32 layers

Peak stress

0 MPa

End deflection

0.00 mm

Residual stress (Ti-6Al-4V, released state)

compressivetensile

How it works

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 · 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

Peak temperature sets ΔT_eff and therefore the magnitude of the eigenstrain — hotter melt pools and faster cooling 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.
Build height & part length set the beam's cross-section and 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.

Building…

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