= Mean stress and strain boundary identities
{title2=$V\overline\sigma_{ij}=\int_Sx_it_j\,dS+\int_V\rho x_i(b_j-a_j)\,dV$}
Apply the <divergence theorem> to $\partial_k(x_i\sigma_{jk})$ and use momentum balance $\sigma_{jk,k}=\rho(a_j-b_j)$ and stress symmetry to obtain the displayed mean <stress> identity. Applying the same theorem to the <infinitesimal strain tensor> gives $V\overline e_{ij}=\frac12\int_S(u_in_j+u_jn_i)\,dS$. The acceleration term cannot be dropped during a transient.
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