Table 1 Summary of homogenization equations

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)