We present a global-in-time variational approach to dynamics of elastic bodies. We introduce a family of parameter-dependent functionals defined over entire trajectories and we prove that they admit minimisers which are solutions of the corresponding Euler-Lagrange problem. By passing to the limit in the parameter, we prove that those minimisers converge to the solutions of a generalized hyperbolic PDE describing the dynamics of hyperelastic materials. Then, small deformations are taken into account. We rescale the family of functionals with respect to a deformation parameter and we prove that they Gamma-converge to the family of linearized functionals as this parameter converges to 0. This is a joint work with Ulisse Stefanelli and Martin Kružík.

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