We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of $\mathrm{\Gamma }$-convergence. Hereby, we generalize the results of the purely elastic setting [M. G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by $\mathrm{\Gamma }$-convergence, Calc. Var. Partial Differential Equations 18 (2003) 287–305] to a framework of free discontinuity problems. The effective one-dimensional model features a classical elastic bending–torsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a discontinuity of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [M. Friedrich, L. Kreutz and K. Zemas, Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces, preprint (2021), arXiv:2107.10808].

我们通过以下方式推导出具有材料空隙的三维棒的降维极限：$\mathrm{\gamma }$-收敛。在此，我们概括了纯弹性设置的结果 [MG Mora 和 S. Müller，不可延伸杆的非线性弯曲扭转理论的推导：$\mathrm{\gamma }$-收敛，计算。变种。偏微分方程 18 (2003) 287–305] 到自由不连续问题的框架。有效的一维模型具有经典的弹性弯曲扭转能量，但也考虑了限制杆可以分解成几段或折叠的可能性。后一种现象的发生可能是由于极限中空隙的持续存在，或者由于它们塌陷成极限变形或其导数的不连续性。证明的主要成分是在曲率消失正则化下不同域中的新颖刚度估计，在[M. Friedrich、L. Kreutz 和 K. Zemas，可变域中的几何刚度和自由表面弹性材料线性化模型的推导，预印本 (2021)，arXiv:2107.10808]。