  A stress equilibration procedure for hyperelastic material models is proposed
andanalyzed in this paper. Based on the displacement-pressure approximation
computed with a stable finite element pair, it constructs, in a
vertex-patch-wise manner, an $H(div)$-conforming approximation to the first
Piola-Kirchhoff stress. This is done in such a way that its associated Cauchy
stress is weakly symmetric in the sense that its anti-symmetric part is zero
tested against continuous piecewise linear functions. Our main result is the
identification of the subspace of test functions perpendicular to the range of
the local equilibration system on each patch which turn out to be rigid body
modes associated with the current configuration. Momentum balance properties
are investigated analytically and numerically and the resulting stress
reconstruction is shown to provide improved results for surface traction forces
by computational experiments.
