Metallurgy & Heat Treatment formula solvers

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C H J L M N P S T V

Chvorinov's Rule

t_s = B \left( \dfrac{V}{A} \right)^{2}

Metallurgy & Heat TreatmentA casting freezes in a time proportional to the square of its volume-to-surface-area ratio. Chvorinov published it in 1940, and it is the whole basis of riser design: make the riser's modulus larger than the casting's and the riser will still be liquid when the casting needs feeding.

Considère Criterion and Uniform Elongation

\varepsilon_u = n, \qquad e_u = e^{\,n} - 1

Metallurgy & Heat TreatmentConsidère's 1885 argument: a tensile bar necks the moment the metal stops hardening faster than the section shrinks. Put a Hollomon curve into that condition and the answer is startlingly simple — the true strain at the onset of necking equals the strain-hardening exponent. This page turns that n into the uniform elongation a tensile certificate reports.

Hall–Petch Relation

\sigma_y = \sigma_0 + k_y \, d^{-1/2}

Metallurgy & Heat TreatmentFiner grains make a stronger metal, and the strength climbs as the inverse square root of the grain diameter. Hall measured it on mild steel in 1951 and Petch on cleavage in 1953, and it is the one strengthening mechanism that costs nothing in toughness — every other route to strength trades ductility away.

Hollomon Flow Curve

\sigma = K \, \varepsilon^{\,n}

Metallurgy & Heat TreatmentThe power law that describes how a metal hardens as it deforms: true stress rises as true plastic strain raised to a small exponent. Hollomon published it in 1945, and the exponent n is the single number that tells a press shop whether a sheet will stretch or split.

JMAK (Avrami) Transformed Fraction

X = 1 - \exp\!\left( -k \, t^{\,n} \right)

Metallurgy & Heat TreatmentThe S-shaped curve every isothermal transformation follows: slow while nuclei are still appearing, fast while they grow into free space, slow again as they run into each other. Avrami wrote it in 1939, Johnson and Mehl the same year, and it is the arithmetic behind every TTT diagram and every recrystallisation anneal.

Larson–Miller Parameter

P = T \left( C + \log_{10} t \right)

Metallurgy & Heat TreatmentThe time–temperature trade that lets a 1000-hour rupture test at 700 °C stand in for a 100,000-hour life at 600 °C. Larson and Miller published it in 1952, and every creep-rupture master curve since has been plotted against it. Two warnings live on this page and both matter: the temperature is ABSOLUTE, and the logarithm is of a time in HOURS.

Lever Rule

W_\alpha = \dfrac{C_0 - C_\beta}{C_\alpha - C_\beta}

Metallurgy & Heat TreatmentHow much of each phase is present in a two-phase field, read straight off a tie line. It is nothing more than a mass balance — the solute has to be somewhere — and that is exactly why it is exact, and exactly why it knows nothing whatever about how long anything takes.

Monkman–Grant Relation

t_r \, \dot{\varepsilon}_s^{\,m} = C_{MG}

Metallurgy & Heat TreatmentThe empirical observation that a metal creeping twice as fast lasts about half as long, whatever the stress and temperature that made it creep. Monkman and Grant found it in 1956 across a wide range of alloys, and it is the reason a short test that never reaches rupture can still predict one.

Norton Creep Law

\dot{\varepsilon} = A \, \sigma^{\,n} \exp\!\left( -\dfrac{Q}{RT} \right)

Metallurgy & Heat TreatmentSteady-state creep, written as a power law in stress multiplied by an Arrhenius term in temperature. Norton put the stress dependence down in 1929 and the Arrhenius factor came from Dorn's work, and between them the two exponents identify WHICH creep mechanism is running — which is the real reason to fit them.

Parabolic Grain Growth Law

D^{\,n} - D_0^{\,n} = K \, t

Metallurgy & Heat TreatmentOnce recrystallisation is finished, the new grains start consuming each other to reduce total boundary area, and the mean size grows as a power of time. Beck and co-workers fitted it in 1948. It is why an over-long soak undoes exactly the strength that Hall–Petch said the fine grain had bought.

Scheil–Gulliver Segregation

C_S = k \, C_0 \left( 1 - f_S \right)^{k-1}

Metallurgy & Heat TreatmentWhat actually happens when an alloy freezes at a real rate: the first solid is lean in solute, the rejected solute piles up in the shrinking liquid, and every later layer of solid is richer than the one before. Gulliver set out the argument in 1913 and Scheil put it in this form in 1942, and it is why castings are cored and why homogenisation anneals exist.

True Stress from Engineering Stress

\sigma_t = \sigma_e \, (1 + e), \qquad \varepsilon_t = \ln(1 + e)

Metallurgy & Heat TreatmentA tensile machine divides load by the ORIGINAL area, because that is the only area it knows. True stress divides by the area the bar actually has at that instant, which is smaller — so true stress is always the larger number, and the gap grows with every percent of strain. Every plasticity relation on this site wants the true values.

Vickers Hardness

HV = \dfrac{2F \sin(136^\circ/2)}{d^{2}} = \dfrac{1.8544\,F}{d^{2}}

Metallurgy & Heat TreatmentPress a square-based diamond pyramid into the metal, measure the two diagonals of the mark it leaves, and divide the load by the SURFACE AREA of that mark. The 1.8544 is pure geometry from the indenter's 136° face angle, and because the pyramid is self-similar, the number is independent of the load — which no other common hardness scale can claim.