Here $\mathrm{H}$ represents sources and sinks of $u$, and ${D u}/{D t}$ is the material time derivative of $u$. The time derivatives of the unknowns are not present in the `PETSc` template but, after time-discretisation, they produce terms that are combinations of fluxes and flux history terms (which combine with $\boldsymbol{\sigma}$ to contribute to $F$) and forces (which combine with $\mathbf{h}$ to contribute to $f$). The explicit time / position dependence in $\sigma$ is to highlight potential changes to boundary conditions or constitutive properties.
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