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<p>Shells, curved slender structural surfaces, are essential to the function of many biological and engineered systems. Yet despite the predictive success of numerical models, there lacks an understanding of how their stiffness changes with wrinkles or folds. Here, it is shown that, of the six loads generated by pulling, shearing, bending and twisting, a shell resists exactly three loads and complies with the other three, provided the shell is simply connected, meaning it has no holes and no handles. Thus, precluding tears and stitches, wrinkles and folds redistribute stiffness and compliance among modes of deformation but cannot stiffen modes without softening others. This balance law between stiffness and compliance provides a fundamental heuristic for the design and optimization of advanced material&#xa0;surfaces across scales.</p>

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A topological counting rule for shells

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月14日

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  • Hussein Nassar

摘要

Shells, curved slender structural surfaces, are essential to the function of many biological and engineered systems. Yet despite the predictive success of numerical models, there lacks an understanding of how their stiffness changes with wrinkles or folds. Here, it is shown that, of the six loads generated by pulling, shearing, bending and twisting, a shell resists exactly three loads and complies with the other three, provided the shell is simply connected, meaning it has no holes and no handles. Thus, precluding tears and stitches, wrinkles and folds redistribute stiffness and compliance among modes of deformation but cannot stiffen modes without softening others. This balance law between stiffness and compliance provides a fundamental heuristic for the design and optimization of advanced material surfaces across scales.