Powder Metallurgy, Metal Injection Molding (MIM) & Technical Ceramics, Sintering Shrinkage
Sintering Linear Shrinkage & Die Tooling Oversize Formula
Calculate Volumetric Sintering Shrinkage (%), Isotropic Linear Shrinkage 1 - (rho_g/rho_s)^(1/3) (%), Tooling Oversize Scale Factor, and Green Die Cavity Dimension (mm).
Multiplying a 32 mm finished MIM gear by (1 + 0.1394) cuts the mold cavity 0.72 mm UNDERSIZE—because tooling scale is 1 / (1 - S_L), NOT 1 + S_L!
In **Metal Injection Molding (`MIM`)**, **Binder-Jet `3D` Printing**, **WC-Co Hardmetal Pressing**, and **Alumina/Zirconia Sintering**, a porous **De-bound Brown/Green Compact of Density `rho_g`** (`g/cm³`) densifies at `1,250–1,600°C` by grain-boundary/volume diffusion to **Final Sintered Density `rho_s`** (`g/cm³`) at constant metal/ceramic mass (`m_g = m_s -> V_s / V_g = rho_g / rho_s`)! Hence **Volumetric Shrinkage `Delta_V / V_0 = (1 - rho_g / rho_s) * 100%`** maps via exact cube-root geometry to **Isotropic Linear Shrinkage `S_L = 1 - (rho_g / rho_s)^(1/3)`** (defined relative to *green* tooling length `L_0`: `S_L = (L_0 - L_f) / L_0`)! Consequently, to hit **Target Finished Dimension `L_f`** (`mm`), the EDM toolmaker MUST machine the cavity to **Oversize Scale Factor `OSF = L_0 / L_f = 1 / (1 - S_L) = (rho_s / rho_g)^(1/3)`**—NEVER `1 + S_L`!
Verify on a `17-4PH` Stainless-Steel `MIM` Surgical Jaw—De-bound_Green_Density `rho_g` = 4.92 g/cm³ (`63.7%` solids loading, B2), Vacuum-Sintered_Density `rho_s` = 7.72 g/cm³ (`98.3%` theoretical, B3), and Target_Finished_Length `L_f` = 32.000 mm (B4): Volumetric Shrinkage equals `36.27%` (E2), Linear Shrinkage `S_L` equals `13.94%` (E3), True Tooling Oversize Factor equals `1.1620` (E4, vs naive `1.1394`!), and Required Mold Cavity Length equals **`37.185` mm** (E5—whereas naive `32 * 1.1394 = 36.461` mm scraps an expensive `$25,000` H13 mold cavity `0.724` mm undersize!)!
| MIM vs Die-Pressed PM Sintering KPI (Target L_f = 32.000 mm) | Formula | Verified Output |
|---|---|---|
| Volumetric Shrinkage (%) | (1 - (Green_Rho / Sintered_Rho)) * 100 | 36.27% MIM (8.05% uniaxial PM) |
| Isotropic Linear Shrinkage SL (%) | (1 - ((Green_Rho / Sintered_Rho) ^ (1/3))) * 100 | 13.94% MIM (2.76% uniaxial PM) |
| Exact Tooling Oversize Factor (OSF) | 1 / ((Green_Rho / Sintered_Rho) ^ (1/3)) | 1.1620 (vs wrong 1+SL = 1.1394!) |
| Required Mold / Die Cavity L0 (mm) | Target_Dim_mm / ((Green_Rho / Sintered_Rho) ^ (1/3)) | 37.185 mm MIM (32.908 mm PM) |
Mass-Conserved Cube-Root Sintering Shrinkage & Tooling Cavity Scale Formula
Validate `Green_Density rho_g > 0.5` g/cm³, `Sintered_Density rho_s > rho_g` up to `22.5` g/cm³ (tungsten/platinum), realistic packing ratio `rho_g / rho_s >= 0.35`, and `Target_Dimension L_f > 0.1` mm!
=LET(rg_d, B2, rs_d, B3, lf_m, B4, lr_t, (rg_d / rs_d) ^ (1 / 3), IF(OR(rg_d <= 0.5, rs_d <= rg_d, rs_d > 22.5, (rg_d / rs_d) < 0.35, lf_m <= 0.1), "Invalid Sintering Input", ROUND(lf_m / lr_t, 3)))Uniaxial Die-Compacted PM Steel Gear (`rho_g = 6.85 -> rho_s = 7.45` g/cm³) Low-Shrinkage Check
Compare against conventional `600`-MPa uniaxially die-pressed `FN-0205` PM steel (`B2 = 6.85`, `B3 = 7.45` g/cm³): Volumetric Shrinkage drops to `8.05%` (E2), Linear Shrinkage to `2.76%` (E3), Oversize Factor to `1.0284` (E4), and Die Cavity Size to `32.908` mm (E5)!
=IF(ISTEXT(E5), E5, ROUND(1 / ((B2 / B3) ^ (1 / 3)), 4))Try the example
Use the original values or change them to test your understanding.