{\partial}_t u)(t)$ satisfies (\[weak\_sol\_def\]) with $\phi=\phi_{p,q}$, where $p$ and $q$ are arbitrary. By (\[id-conv\]) and (\[u-strong\]) we have $u(0)=u_0$. The compatibility condition follows from (\[nm-comp\]), (\[ut-conv\]) and (\[v\_conv-b\]).

We conclude the proof of the existence of weak solutions by showing that any function $\phi$ in ${{\cal L}}_T$ can be approximated by a sequence of functions of the form (\[phi-pq\]), we refer to [@ChuRyz2011] for a similar argument in the case of zero in-plane deformations.

This solution satisfies the energy [*inequality*]{}: $$\begin{gathered}
{{\cal E}}(v(t), u(t), u_t(t))+\nu \int_0^t ||{{\nabla}}v||^2_{{\cal O}}d\tau + {{\gamma}}\int_0^t ||w_t(\tau)||^2_{H_\alpha} d\tau \\ \le {{\cal E}}(v_0, u_0, u_1)
+\int_0^t(G_f(\tau),  v)_{{\cal O}}d\tau +\int_0^t (G_{sh}(\tau),u_t)_{{\Omega}}d\tau\end{gathered}$$ for almost all $t>0$, where the energy functional ${{\cal E}}$ is defined by .

[*Step 3. Existence of weak solutions in the case ${{\varrho}}=0$.*]{} We fix some ${\bar{u}}_1$ from $L_2({{\Omega}})$ and consider the corresponding solution $(v_{{\varrho}}(t); u_{{\varrho}}(t))$ with ${{\varrho}}>0$. As above, Proposition \[pr:Korn\] and relation imply the following a priori estimate: $$\begin{gathered}
\label{a-pri-ro}
 {\rm ess}\!\!\sup_{t\in [0,T]}\left[\|v_{{{\varrho}}}(t)\|_{{\cal O}}^2 +
 \|M^{1/2}_{{\alpha}}{\partial}_t w_{{{\varrho}}}(t)\|_{{\Omega}}^2
 +\|{\Delta}w_{{\varrho}}(t) \|^2_{{\Omega}}+ \|{\bar{u}}_{{\varrho}}(t)\|_{1,{{\Omega}}}^2\right]  \\
 +\int_0^T\|\nabla v_{{{\varrho}}}(t)\|_{{\Omega}}^2 dt +
  \int_0^T \|{\partial}_t {\bar{u}}_{{{\varrho}}}(t)\|_{[H_*^{1/2}({{\Omega}})]^2}^2 dt
 \le C_T
 \end{gathered}$$ for any interval $[0,T]$, where the constant $C_T$ does not depend ${{\varrho}}\in (0,1)$. This implies that the family $\{(v_{{{\varrho}}}; u_{{\varrho}}; {\partial}_t u_{{\varrho}})\}$ 