  We consider hyperelastic problems and their numerical solution using a
conforming finite element discretization and iterative linearization
algorithms. For these problems, we present equilibrated, weakly symmetric,
$H(\rm{div)}$-conforming stress tensor reconstructions, obtained from local
problems on patches around vertices using the Arnold--Falk--Winther finite
element spaces. We distinguish two stress reconstructions, one for the discrete
stress and one representing the linearization error. The reconstructions are
independent of the mechanical behavior law. Based on these stress tensor
reconstructions, we derive an a posteriori error estimate distinguishing the
discretization, linearization, and quadrature error estimates, and propose an
adaptive algorithm balancing these different error sources. We prove the
efficiency of the estimate, and confirm it on a numerical test with analytical
solution for the linear elasticity problem. We then apply the adaptive
algorithm to a more application-oriented test, considering the Hencky--Mises
and an isotropic damage models.
