Ceramic Materials, Fracture Mechanics & Semiconductor Wafer Reliability
Griffith Brittle Fracture & Weibull Ceramic Volume Scaling
Calculate Griffith Critical Brittle Fracture Stress SQRT((2*E*gamma_s)/(PI()*a)), Weibull Weakest-Link Effective-Volume Knockdown Factor (V_coupon / V_part)^(1/m), Full-Component Characteristic Strength, and Design Safety Factor.
Account for microscopic Griffith flaws and Weibull volume-size knockdown in structural ceramics!
Why can a tiny silicon-nitride flexural test bar survive 180 MPa while a 64x-larger ceramic turbocharger rotor made from the EXACT same powder batch shatters at 90 MPa? A.A. Griffith’s thermodynamic brittle-fracture criterion shows that fracture initiates at the single worst microscopic pore or machining scratch of half-length a: sigma_Griffith (MPa) = SQRT((2 * E_GPa * gamma_s * 1000) / (PI() * a_um))! Because larger stressed volumes V_part have a higher probability of containing a larger tail-distribution flaw, Weibull’s Weakest-Link Theory scales characteristic strength downward by (V_coupon / V_part)^(1 / m), where Weibull Modulus m (typically 5–15 for technical ceramics vs 50–100 for ductile steels) governs defect scatter!
Verify on a Technical Ceramic Component—Youngs_Modulus E = 300 GPa (B2), Fracture_Surface_Energy gamma_s = 5.0 J/m^2 (B3), Critical_Coupon_Flaw a = 30.0 um (B4 -> Coupon Griffith Strength = SQRT((2 * 300 * 5 * 1000) / (PI() * 30)) = 178.41 MPa in E2), Coupon_Effective_Volume = 100 mm^3 (B5), Full_Part_Effective_Volume = 6,400 mm^3 (64x larger, B6), Weibull_Modulus m = 6.0 (B7), and Applied_Tensile_Stress = 70.0 MPa (B8): Weibull Volume Knockdown Factor is (100 / 6,400)^(1 / 6) = (1 / 64)^(1 / 6) = 0.5000 (E3); Full-Part Characteristic Strength drops to 178.41 * 0.5000 = 89.21 MPa (E4); and Component Factor of Safety is 89.21 / 70.0 = 1.27 (E5)!
| Ceramic Fracture Metric (E=300 GPa, gamma=5 J/m^2, a=30 um, V2/V1=64, m=6) | Formula | Verified Output |
|---|---|---|
| Coupon Griffith Strength (MPa) | SQRT((2 * E_GPa * gamma * 1000)/(PI()*a_um)) | 178.41 MPa (89.21 @ a=120 um) |
| Weibull Volume Scaling Factor | (V_coupon / V_part)^(1 / m) | 0.5000 (1.0000 @ V_part=100) |
| Full-Part Strength (MPa) | Coupon_Strength * Weibull_Factor | 89.21 MPa |
| Component Safety Factor (x) | Full_Part_Strength / Applied_MPa | 1.27x |
Griffith-Weibull Size-Scaled Ceramic Component Strength Formula
Validate strictly positive E_GPa, gamma_s, Flaw_a_um, Coupon_Vol, Part_Vol, Weibull_Modulus m, and Applied_Stress!
=LET(e_gp, B2, gm_s, B3, fl_u, B4, vl_1, B5, vl_2, B6, wb_m, B7, ap_s, B8, IF(OR(e_gp <= 0, gm_s <= 0, fl_u <= 0, vl_1 <= 0, vl_2 <= 0, wb_m <= 0, ap_s <= 0), "Invalid Ceramic Fracture Input", LET(sg_c, SQRT((2 * e_gp * gm_s * 1000) / (PI() * fl_u)), wf_v, (vl_1 / vl_2) ^ (1 / wb_m), ROUND(sg_c * wf_v, 2))))4x Larger Machining Flaw (a = 120 um) & Equal-Volume Checks
Quadruple Coupon_Flaw_Size B4 from 30 um to 120 um: Griffith Strength E2 halves by 1/SQRT(4) to 89.21 MPa! Or restore B4 = 30 um and set Part_Volume B6 = 100 mm^3: Weibull Factor E3 equals 1.0000!
=IF(ISTEXT(E4), E4, ROUND((B5 / B6) ^ (1 / B7), 4))Try the example
Use the original values or change them to test your understanding.