Fig. 10

Selected macroscopic moduli depending on the size \(l_{micro}\)of square-shaped RVEs with circular inclusions with volume fraction \(f_0 = \pi /16\). The chemical parameter of the matrix and the inclusions are chosen to \(A_{mat} = A_{incl} = 10\,\hbox {N/mm}^{2}\). The mobility parameters of the matrix and the inclusions are \(M_{mat}=0.1~\hbox {mm}^{4}\)/(Ns) and \(M_{incl} = 0.0001~\hbox {mm}^{4}\)/(Ns) , respectively. a We observe that for vanishing size of the RVE \(l_{micro} \rightarrow 0\), the macroscopic chemical parameter \(\partial ^2_{{\overline{s}}{\overline{s}}}{\overline{\pi }}\) converges to the homogeneous solution \(\langle \partial ^2_{ss} \hat{\psi }\rangle = \frac{1}{|{\mathcal {B}}|} \int _{{\mathcal {B}}} \partial _{ss} \hat{\psi }\, \mathrm {d}V\). b Furthermore, for vanishing size of the RVE, the macroscopic mobility parameter lies firmly within the Voigt and Reuss bound according to (42)