  We devise and evaluate numerically Hybrid High-Order (HHO) methods for
hyperelastic materials undergoing finite deformations. The HHO methods use as
discrete unknowns piecewise polynomials of order $k\ge1$ on the mesh skeleton,
together with cell-based polynomials that can be eliminated locally by static
condensation. The discrete problem is written as the minimization of the broken
nonlinear elastic energy where a local reconstruction of the displacement
gradient is used. Two HHO methods are considered: a stabilized method where the
gradient is reconstructed as a tensor-valued polynomial of order $k$ and a
stabilization is added to the discrete energy functional, and an unstabilized
method which reconstructs a stable higher-order gradient and circumvents the
need for stabilization. Both methods satisfy the principle of virtual work
locally with equilibrated tractions. We present a numerical study of both HHO
methods on test cases with known solution and on more challenging
three-dimensional test cases including finite deformations with strong shear
layers and cavitating voids. We assess the computational efficiency of both
methods, and we compare our results to those obtained with an industrial
software using conforming finite elements and to results from the literature.
Both methods exhibit robust behavior in the quasi-incompressible regime.
