grains.simulation.hallpetch_constants

grains.simulation.hallpetch_constants(sigma_y, d)

Determines the two Hall-Petch constants. Given available measurements for the grains sizes and the yield stresses, the two constants in the Hall-Petch formula are computed.

Parameters

• sigma_y (list of floats) – Yield stresses.

• d (list of floats) – Diameters of the grains.

Returns

• sigma_0 (float) – Starting stress for dislocation movement (material constant).

• k (float) – Strengthening coefficient (material constant).

Notes

If two data points are given in the inputs (corresponding to two measurements), the output parameters have unique values:

k = (sigma_y[0] - sigma_y[1]) / (1/sqrt(d[0]) - 1/sqrt(d[1])) sigma_0 = sigma_y[0] - k/sqrt(d[0])

If there are more than two measurements, the resulting linear system is overdetermined. In both cases, the outputs are determined using least squares fitting.