A new version of Hill’s lemma for Cosserat continuum

Abstract

A new version of Hill’s lemma is presented for micro- to macrohomogenization modeling of heterogeneous Cosserat continuum in the frame of average-field theory. From the presented Hill’s lemma, new forms of rotational displacement and surface couple boundary conditions on the representative volume element are extracted and checked.

Appendix A: Verification of Hill’s lemma given by Eq. (47)

With the use of Eqs. (1), (26), and Gauss theorem, the first term at the right-hand side of Hill’s lemma given by Eq. (47) can be deduced as

$$\begin{aligned}&\frac{1}{V}\int \limits _S {(u_i -\bar{{u}}_{i,j} x_j )\left( n_k \sigma _{ki} -n_k \bar{{\sigma }}_{ki} \right) \hbox {d}S} \nonumber \\&\quad =\frac{1}{V}\int \limits _S {\left( {n_k \sigma _{ki} u_i +\bar{{\sigma }}_{ki} \bar{{u}}_{i,j} n_k x_j -\bar{{u}}_{i,j} n_k \sigma _{ki} x_j -\bar{{\sigma }}_{ki} n_k u_i } \right) \hbox {d}S} \nonumber \\&\quad =\frac{1}{V}\int \limits _S {n_k \sigma _{ki} u_i \hbox {d}S} +\bar{{\sigma }}_{ki} \bar{{u}}_{i,j} \frac{1}{V}\int \limits _S {n_k x_j \hbox {d}S} -\bar{{u}}_{i,j} \frac{1}{V}\int \limits _S {n_k \sigma _{ki} x_j \hbox {d}S} -\bar{{\sigma }}_{ij} \frac{1}{V}\int \limits _S {n_i u_j \hbox {d}S} \nonumber \\&\quad =\frac{1}{V}\int \limits _V {\frac{\partial (\sigma _{ki} u_i )}{\partial x_k }\hbox {d}V} +\bar{{\sigma }}_{ki} \bar{{u}}_{i,j} \frac{1}{V}\int \limits _V {\frac{\partial x_k }{\partial x_j }\hbox {d}V} -\bar{{u}}_{i,j} \frac{1}{V}\int \limits _V {\frac{\partial (\sigma _{kj} x_i )}{\partial x_k }\hbox {d}V} -\bar{{\sigma }}_{ij} \frac{1}{V}\int \limits _V {u_{j,i} \hbox {d}V} \nonumber \\&\quad =\overline{\sigma _{ij} u_{j,i} } +\bar{{\sigma }}_{ij} \bar{{u}}_{j,i} -\bar{{\sigma }}_{ij} \bar{{u}}_{j,i} -\bar{{\sigma }}_{ij} \bar{{u}}_{j,i} =\overline{\sigma _{ij} u_{j,i} } -\bar{{\sigma }}_{ij} \bar{{u}}_{j,i} \end{aligned}$$

(A.1)

With the use of Eqs. (20), (37), and Gauss theorem, the second term at the right-hand side of Hill’s lemma given by Eq. (47) can be deduced as

$$\begin{aligned}&\frac{1}{V}\int \limits _S {(\omega _i -\bar{{\omega }}_{i,j} x_j )(n_k {\mu }^{{\prime }{\prime }}_{ki} -n_k \bar{{\mu }}^{{\prime }{\prime }}_{ki} )\hbox {d}S} \nonumber \\&\quad =\frac{1}{V}\int \limits _S {\left( {\omega _i n_k {\mu }^{{\prime }{\prime }}_{ki} -\omega _i n_k \bar{{\mu }} ^{{\prime }{\prime }}_{ki} -n_k {\mu }^{{\prime }{\prime }}_{ki} \bar{{\omega }}_{i,j} x_j +n_k \bar{{\mu }}^{{\prime }{\prime }}_{ki} \bar{{\omega }}_{i,j} x_j } \right) \hbox {d}S} \nonumber \\&\quad =\frac{1}{V}\int \limits _S {n_k {\mu }^{{\prime }{\prime }}_{ki} \omega _i \hbox {d}S} -\bar{{\mu }}^{{\prime }{\prime }}_{ki} \frac{1}{V}\int \limits _S {n_k \omega _i \hbox {d}S} -\bar{{\omega }}_{i,j} \frac{1}{V}\int \limits _S {n_k {\mu }^{{\prime }{\prime }}_{ki} x_j \hbox {d}S} +\bar{{\mu }}^{{\prime }{\prime }}_{ki} \bar{{\omega }}_{i,j} \frac{1}{V}\int \limits _S {n_k x_j \hbox {d}S} \nonumber \\&\quad =\frac{1}{V}\int \limits _V {\frac{\partial ({\mu }^{{\prime }{\prime }}_{ki} \omega _i )}{\partial x_k }\hbox {d}V} -\bar{{\mu }}^{{\prime }{\prime }}_{ki} \bar{{\kappa }}_{ik} -\bar{{\omega }}_{i,j} \frac{1}{V}\int \limits _V {\frac{\partial ({\mu }^{{\prime }{\prime }}_{ki} x_j )}{\partial x_k }\hbox {d}V} +\bar{{\mu }}^{{\prime }{\prime }}_{ki} \bar{{\omega }}_{i,j} \frac{1}{V}\int \limits _S {\frac{\partial x_j }{\partial x_k }\hbox {d}S} \nonumber \\&\quad =\overline{{\mu }^{{\prime }{\prime }}_{ij} \kappa _{ij} } -\bar{{\mu }}^{{\prime }{\prime }}_{ij} \bar{{\kappa }}_{ij} -\bar{{\mu }}^{{\prime }{\prime }}_{ij} \bar{{\kappa }}_{ij} +\bar{{\mu }}^{{\prime }{\prime }}_{ij} \bar{{\kappa }}_{ij} =\overline{{\mu }^{{\prime }{\prime }}_{ij} \kappa _{ij} } -\bar{{\mu }}^{{\prime }{\prime }}_{ij} \bar{{\kappa }}_{ij} \end{aligned}$$

(A.2)

From Eqs. (A.1)–(A.2), we can obtain Eq. (47), i.e.,

$$\begin{aligned}&\overline{\sigma _{ij} u_{j,i} } -\bar{{\sigma }}_{ij} \bar{{u}}_{j,i} +\overline{{\mu }^{{\prime }{\prime }}_{ij} \kappa _{ij} } -\bar{{\mu }}^{{\prime }{\prime }}_{ij} \bar{{\kappa }}_{ij} \nonumber \\&\quad =\frac{1}{V}\int \limits _S {(u_i -\bar{{u}}_{i,j} x_j )(n_k \sigma _{ki} -n_k \bar{{\sigma }}_{ki} )\hbox {d}S} +\frac{1}{V}\int \limits _S {(\omega _i -\bar{{\omega }}_{i,j} x_j )(n_k {\mu }^{{\prime }{\prime }}_{ki} -n_k \bar{{\mu }}^{{\prime }{\prime }}_{ki} )\hbox {d}S} \end{aligned}$$

(47)