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The aim of this paper is to prove the existence of almost global weaksolutions for the unsteady nonlinear elastodynamics system in dimension $d=2$or $3$, for a range of strain energy density functions satisfying some givenassumptions. These assumptions are satisfied by the main strain energiesgenerally considered. The domain is assumed to be bounded, and mixed boundaryconditions are considered. Our approach is based on a nonlinear parabolicregularization technique, involving the $p$-Laplace operator. First we provethe existence of a local-in-time solution for the regularized system, by afixed point technique. Next, using an energy estimate, we show that if the dataare small enough, bounded by $\varepsilon >0$, then the maximal time ofexistence does not depend on the parabolic regularization parameter, and thebehavior of the lifespan $T$ is $\gtrsim \log (1/\varepsilon)$, defining whatwe call here {\it almost global existence}. The solution is thus obtained bypassing this parameter to zero. The key point of our proof is due to recentnonlinear Korn's inequalities proven by Ciarlet \& Mardare in$\mathrm{W}^{1,p}$ spaces, for $p>2$.

Almost global existence of weak solutions for the nonlinear elastodynamics system for a class of strain energies