\eta_k \partial_k^x p
or
(y_k + \eta_k) \partial_k^x p
Micro displacements.
\bm{u}^1 = \bm{\chi}^{ij}\, e_{ij}^x(\bm{u}^0)
\widetilde\pi^P\,p
\sum_{ij} \int_0^t {\mathrm{d} \over \mathrm{d} t} \widetilde\pi^{ij}(t-s)\, {\mathrm{d} \over \mathrm{d} s} e_{ij}(\bm{u}(s))\,ds + \int_0^t {\mathrm{d} \over \mathrm{d} t}\widetilde\pi^P(t-s)\,p(s)\,ds
Macro-induced pressure.
\partial_j^x p\,(y_j - y_j^c)
\sum_{ij} \bm{\omega}^{ij}\, e_{ij}(\bm{u})
Notes
\sum_{ij} \left[ \int_0^t \bm{\omega}^{ij}(t-s) {\mathrm{d} \over \mathrm{d} s} e_{ij}(\bm{u}(s))\,ds\right] + \int_0^t \widetilde{\bm{\omega}}^P(t-s)\,p(s)\,ds
Macro-induced displacements.
e_{ij}^x(\bm{u})\,(y_j - y_j^c)
\int_0^t f(t-s) p(s) ds
Notes
\int_0^t f^{ij}(t-s) p_{ij}(s) ds
Notes
Notes
note that
\widetilde{\pi}^P
is in corrs_pressure -> from time correctors only ‘u’, ‘dp’ are needed.