Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

中文翻译：

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不可压缩弹性弦的渐近变分分析

从不可压缩性限制下的三维非线性弹性出发，我们推导出简化模型来捕捉弦对外力的响应行为。我们对收缩截面极限中的约束能量泛函的Γ收敛分析产生了明确的一维极限能量。后者取决于所施加的力的比例。局部体积保持的效果通过横截面变量的约束最小化或在允许变形的类别中反映在它们的能量密度中。有趣的是，所有缩放机制都允许压缩和/或拉伸字符串。Γ-limit 证明的主要困难是建立适应由不可压缩性施加的非线性微分约束的恢复序列。为此，我们借助为 3d-1d 降维问题量身定制的内部扰动参数来修改无约束情况下的经典结构。