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Table 1 Summary of homogenization equations

From: A thermo-elastoplastic self-consistent homogenization method for inter-granular plasticity with application to thermal ratcheting of TATB

Effective propertyEquationEquation number
Elastic compliance\(\overline{\mathbb {M}}^e = \langle ( \mathbb {M}^e + \mathbb {R}^e){:} \hat{\overline{\mathbb {B}}} \rangle \)(65)
Plastic compliance\(\overline{\mathbb {M}}^p = \langle \mathbb {R}^p{:}\overline{\mathbb {B}} \rangle \)(60)
Elastic strain rate\(\dot{\varvec{E}}^{e} = \overline{\mathbb {M}}^e {:} \dot{\varvec{\Sigma }}\)(64)
Thermal strain rate\(\dot{\varvec{E}}^{th} = \langle \mathbb {M}^e {:}\hat{\overline{\varvec{b}}}+ \dot{\overline{\varvec{\varepsilon }}}^{th} \rangle + \langle \mathbb {R}^e{:}\hat{\overline{\varvec{b}}}\rangle \)(66)
Plastic strain rate\(\dot{\varvec{E}}^p = \overline{\mathbb {M}}^p {:}\varvec{\Sigma } + \dot{\varvec{E}}^p_0\)(59)
Thermoplastic strain rate\( \dot{\varvec{E}}^p_0 = \langle \mathbb {R}^p{:}\overline{\varvec{b}}\rangle \)(59)