Skip to main content

Table 1 Diffusion equation forms for coupled temperature displacement, gradient-damage, and phase-field models

From: A simple and unified implementation of phase field and gradient damage models

Name

Field variable

“Diffusion” coefficient \({\varvec{\varGamma }}\)

“Flux” term \({\varvec{\nabla \psi }}\)

“Source” term \({{\varvec{h}}}_{{\varvec{s}}}\)

Temperature

T

\(\kappa \)

\(\nabla T\)

\(\dot{q}\)

Gradient-damage

D

\(l_n ^{2 (1)}\)

\(\nabla \bar{D}\)

\(\bar{D} -D\)

Phase-field

d

\(\mathcal{G}_c l_d ^{(1)}\)

\(\nabla d\)

\(\left( {\mathcal{G}_c /l_d } \right) d-2\mathcal{H}\left( {1-d} \right) \)

  1. \(\kappa \) Thermal conductivity, \(h_s \) heat source, \(l_n \) characteristic length for gradient damage model, \(\mathcal{G}_{\mathrm{c}} \) fracture energy, \(\mathcal{H}\) elastic energy density, \(l_d \) characteristic length for phase field-model