  This paper presents an efficient algorithm for the approximation of the
rank-one convex hull in the context of nonlinear solid mechanics. It is based
on hierarchical rank-one sequences and simultaneously provides first and second
derivative information essential for the calculation of mechanical stresses and
the computational minimization of discretized energies. For materials, whose
microstructure can be well approximated in terms of laminates and where each
laminate stage achieves energetic optimality with respect to the current stage,
the approximate envelope coincides with the rank-one convex envelope. Although
the proposed method provides only an upper bound for the rank-one convex hull,
a careful examination of the resulting constraints shows a decent applicability
in mechanical problems. Various aspects of the algorithm are discussed,
including the restoration of rotational invariance, microstructure
reconstruction, comparisons with other semi-convexification algorithms, and
mesh independency. Overall, this paper demonstrates the efficiency of the
algorithm for both, well-established mathematical benchmark problems as well as
nonconvex isotropic finite-strain continuum damage models in two and three
dimensions. Thereby, for the first time, a feasible concurrent numerical
relaxation is established for an incremental, dissipative large-strain model
with relevant applications in engineering problems.
