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.

翻译：本文提出了一种高效算法，用于非线性固体力学中秩一凸包络的近似计算。该方法基于分层秩一序列，同时提供计算机械应力和离散能量最小化所需的一阶及二阶导数信息。对于微结构可通过层合板良好近似、且每个层合阶段均能实现当前阶段能量最优的材料，该近似包络与秩一凸包络完全一致。尽管所提方法仅提供秩一凸包络的上界，但对所得约束条件的仔细分析表明其在力学问题中具有良好的适用性。本文详细讨论了算法的多个方面，包括旋转不变性恢复、微结构重构、与其他半凸化算法的比较以及网格无关性验证。整体而言，本文通过经典数学基准问题以及二维和三维非凸各向同性有限应变连续损伤模型，验证了算法的高效性。由此首次为具有工程应用价值的增量式耗散大应变模型建立了可行的并行数值松弛框架。