CavityGrowthDamageRate

Source: models/solid_mechanics/traction_separation_law/CavityGrowthDamageRate.py

The part of \(\dot f\) driven by growth of the existing cavities,

\[ \dot{f}_\text{growth} = \frac{\sqrt{\pi}\, N^{3/2}}{2 h \sqrt{f}} \dot{V}. \]

This is the geometric conversion from a single-cavity volume rate to an area-fraction rate, obtained from \(f = \pi a^2 N\) and \(V = \tfrac{4}{3}\pi a^3 h\) by eliminating \(a\). Feeding it the diffusive rate \(\dot V = 8\pi\mathcal{D}\langle T_n\rangle / q(f^*)\) recovers the Sham-Needleman growth term \(4\pi^{3/2}\mathcal{D} N^{3/2} \langle T_n \rangle / (h \sqrt{f} q(f^*))\) exactly.

Keeping the conversion separate from the growth law is what lets a second growth mechanism be summed onto \(\dot V\) (with a ScalarLinearCombination) without restating this geometry.

Note the physical \(f\) appears here, not the effective \(f^*\): the coalescence map accelerates the softening functions only, so that the identity \([\![u]\!]_n^v = NV - N_0V_0\) keeps holding.

Inputs

cavity_fraction — input · Scalar · required

Physical cavitated area fraction

cavity_density — input · Scalar · required

Cavity number density per unit boundary area

cavity_volume_rate — input · Scalar · required

Rate of change of a single cavity volume

Outputs

cavity_fraction_growth_rate — output · Scalar · required

Area fraction rate contributed by growth of existing cavities

Parameters

cavity_shape_factor — parameter · Scalar · required

Cavity shape factor h(psi); h is about 0.61 at a tip angle of 75 degrees