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Ceramics & Glass formula solvers

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Effective Volume in Three-Point Bending

Veff=V2 (m+1)2V_{\text{eff}} = \frac{V}{2\,(m+1)^2}

Ceramics & GlassOnly a sliver of a bend bar is meaningfully stressed: the stress falls to zero at the neutral axis and at both supports, so most of the bar is a spectator. The effective volume is the volume of uniformly stressed material that would carry the same risk of failure, and it is what belongs in every Weibull scaling calculation.

Median Rank Failure Probability

Pf,i=i−0.3N+0.4P_{f,i} = \frac{i - 0.3}{N + 0.4}

Ceramics & GlassBefore a set of broken specimens can be plotted, each one needs a failure probability attached to it. Bernard and Bos-Levenbach's 1953 approximation to the median rank is the estimator almost everyone uses: sort the strengths, and the i-th of N gets (i − 0.3)/(N + 0.4).

Modulus of Rupture, Four-Point Bending

σ=3PL4bd2\sigma = \frac{3PL}{4bd^2}

Ceramics & GlassHow ceramic strength is actually measured: a rectangular bar on two supports, loaded by two inner rollers at the quarter points, broken, and the peak load converted to a stress by beam theory. Nearly every strength figure on a ceramic datasheet came from this test.

Ryshkewitch-Duckworth Porosity-Strength Relation

σ=σ0 e−bP\sigma = \sigma_0 \, e^{-bP}

Ceramics & GlassStrength falls exponentially with porosity, and much faster than the missing cross-section alone would explain. Ryshkewitch measured it on alumina and zirconia in 1953 and Duckworth gave it its usual form in the same volume of the Journal of the American Ceramic Society; it has held up across sixty years of sintered materials.

Thermal Shock Resistance Parameter R

R=σf (1−ν)E αR = \frac{\sigma_f\,(1-\nu)}{E\,\alpha}

Ceramics & GlassKingery's 1955 ranking parameter for how much of a temperature drop a brittle material can take. It has the units of a temperature difference and it is tempting to read it as one — which is the mistake this page exists to prevent.

Thermal Shock Resistance Parameter R′

R′=k σf (1−ν)E αR' = \frac{k\,\sigma_f\,(1-\nu)}{E\,\alpha}

Ceramics & GlassKingery's R with the thermal conductivity multiplied in, for the far more common case of a merely brisk temperature change rather than an instantaneous quench. It has the units of power per unit length, and it is the parameter that finally explains why glass cracks in the sink and alumina does not.

Vogel-Fulcher-Tammann Viscosity

log⁡10η=A+BT−T0\log_{10}\eta = A + \frac{B}{T - T_0}

Ceramics & GlassGlass viscosity spans roughly fifteen orders of magnitude between the melt and the annealed solid, and no Arrhenius line goes near it. The Vogel-Fulcher-Tammann form — published independently by Vogel in 1921, Fulcher in 1925 and Tammann and Hesse in 1926 — bends the line by putting a divergence temperature in the denominator, and it fits.

Weibull Modulus from Two Points

m=ln⁡ ⁣[ln⁡(1/P1)ln⁡(1/P2)]ln⁡(σ1/σ2)m = \frac{\ln\!\left[ \dfrac{\ln(1/P_1)}{\ln(1/P_2)} \right]}{\ln(\sigma_1 / \sigma_2)}

Ceramics & GlassThe slope of a Weibull plot taken between two points on it. Plot ln[ln(1/P_s)] against ln σ and the two-parameter Weibull becomes a straight line whose slope is m; this is that slope, worked from any two points that lie on it.

Weibull Strength Size Scaling

σ1σ2=(V2V1) ⁣1/m\frac{\sigma_1}{\sigma_2} = \left( \frac{V_2}{V_1} \right)^{\! 1/m}

Ceramics & GlassTake one ceramic, make two parts of different size out of it, and the bigger one is weaker. Not weaker in total load — weaker in STRESS, in megapascals, before any design decision is made. This is the most surprising and most consequential result in brittle materials, and it falls straight out of the weakest-link argument.

Weibull Survival Probability

Ps=exp⁡ ⁣[−(σσ0) ⁣m]P_s = \exp\!\left[ -\left( \frac{\sigma}{\sigma_0} \right)^{\! m} \right]

Ceramics & GlassThe chance a brittle part survives a stress, given the two numbers a Weibull fit produces: a characteristic strength and a modulus describing the scatter. It is the ceramic engineer's replacement for a single strength value, because a ceramic does not have one.

Weibull Survival with Volume Scaling

Ps=exp⁡ ⁣[−VV0(σσ0) ⁣m]P_s = \exp\!\left[ -\frac{V}{V_0} \left( \frac{\sigma}{\sigma_0} \right)^{\! m} \right]

Ceramics & GlassThe survival probability with the specimen's size written into it. This is the form Weibull actually derived in 1939 from the weakest-link argument, and it is the one that explains why a laboratory bend bar and the component it was meant to qualify do not have the same strength.