Residual Stress Analysis in Additive Manufacturing
Theory: shrinking layers and restraint
Overview
A metal-3D-printed part warped when removed from the baseplate. Can this be predicted?
Yes. In metal additive manufacturing, a small laser-melted region cools and solidifies rapidly, then the next layer is stacked on top. Each new layer tries to shrink but is restrained by the layers beneath, leaving tensile stress. That stress is held while attached to the baseplate; on removal the part warps and some of it is released. Resolving melting and cooling in detail is costly, so in practice the inherent strain method — giving each layer the strain it tries to shrink by — is often used to estimate overall warpage quickly.
Size of the inherent strain
For 316L (modulus 190 GPa, expansion 17×10⁻⁶ /K, as-built yield 450 MPa), the stress equivalent to thermal shrinkage from near melting to room temperature is about ten times yield. Layers therefore yield, and the remaining inherent strain is about the yield strain (about 0.24%).
Stress and warp after release
Subtracting from the restrained stress $\sigma^*$ the linear distribution that cancels net force $N$ and moment $M$ gives the post-release stress and curvature $\kappa$; $\delta$ is the mid-span warp over length $L$.
A method born of welding know-how
The inherent strain concept was developed in the 1970s by Japanese welding researchers to predict welding distortion. In welding, melted metal shrinks as it cools and, restrained by its surroundings, produces distortion and residual stress — exactly the mechanism of metal additive manufacturing, which can be seen as stacking a vast number of tiny welds. As metal 3D printers entered industrial use in the 2010s, the inherent strain method was adopted in commercial software to predict whole-part warpage in minutes.
Worked example: a build on a baseplate
In the simplest form, a uniform inherent strain (the yield strain) was applied to all deposited layers on a baseplate, and stress and warp over 100 mm were computed elastically after unbolting. For a 10 mm build with varying plate thickness:
| Plate thickness | Top of build | Build bottom | Plate bottom | Warp (100 mm) |
|---|---|---|---|---|
| 5 mm | −146 MPa | +246 MPa | −2 MPa | 0.26 mm |
| 10 mm | −109 MPa | +222 MPa | +111 MPa | 0.22 mm |
| 20 mm | +2 MPa | +198 MPa | +149 MPa | 0.13 mm |
Warp on a 10 mm plate for different build heights: 0.21 mm at 2 mm, 0.26 mm at 5 mm, 0.22 mm at 10 mm and 0.13 mm at 20 mm.
I expected taller builds to warp more, but it peaks partway and then decreases.
Like a bimetal, bending peaks when the flexural stiffness of the shrinking layers and the restraining plate are balanced; as the build grows taller, it stiffens and bends less. But this is the simplest model, applying the same inherent strain to all layers at once. In reality, sequential layer stacking, plasticity, support structures and the order of cutting from the plate (e.g. by wire EDM) change the stress distribution. In practice, calibrate the inherent strain on test coupons' warpage, then solve the whole part by finite elements.
Analysis workflow
- Build simple coupons (e.g. cantilever shapes) and measure warp on cutting.
- Calibrate inherent strain magnitudes (per direction) to match coupon warp.
- Build up the whole part layer by layer in finite elements to predict warp and stress.
- Compensate the geometry by the inverse of predicted distortion.
- Compare stress-relief heat treatment before versus after removal from the plate.
“It cracked when supports came off”
A part with a thin overhang was cut from the baseplate by wire EDM straight after the build, and it cracked partway through the cut. Large tensile residual stress had built up at the overhang root; as cutting removed restraint, stress concentrated there. Tracing the cutting sequence with the inherent strain method reproduced when stress peaked. Switching to stress-relief heat treatment while still attached to the plate eliminated the cracking.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Elastic use of full thermal strain | Stress ten times yield | Plasticity or inherent strain |
| Uncalibrated inherent strain | Warp magnitude off | Calibrate on coupons |
| Ignoring cutting sequence | Cracks and distortion missed | Simulate the cutting steps |
| Omitting supports | Restraint changes distortion | Model supports |
| Assuming taller means more warp | Wrong design decisions | Consider height-to-plate ratio |
I'd like to try related calculations.
Check thermal-stress basics with the thermal stress calculator and thermal expansion calculator. Related analyses include thermal residual stress, welding distortion, CTE-graded design and additive manufacturing thermal analysis.
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