  In this work we propose and analyze a novel Hybrid High-Order discretization
of a class of (linear and) nonlinear elasticity models in the small deformation
regime which are of common use in solid mechanics. The proposed method is valid
in two and three space dimensions, it supports general meshes including
polyhedral elements and nonmatching interfaces, enables arbitrary approximation
order, and the resolution cost can be reduced by statically condensing a large
subset of the unknowns for linearized versions of the problem. Additionally,
the method satisfies a local principle of virtual work inside each mesh
element, with interface tractions that obey the law of action and reaction. A
complete analysis covering very general stress-strain laws is carried out, and
optimal error estimates are proved. Extensive numerical validation on model
test problems is also provided on two types of nonlinear models.
