Direct measurement of the moment-curvature response of sheets or wires up to high curvatures can aid in modeling creasing, pleating, and other forming operations. We consider theoretical issues related to the geometry of pure bending. We present a linkage design that, for homogeneous deformation of the sample, results in an exact pure bending state up to arbitrarily high curvature, by combining a cochleoidal trajectory with an angle-doubling mechanism. The full mechanism is single degree of freedom and position controlled, a desirable feature for measurement of soft materials. We also present an optimal approximation to the cochleoid using a more easily implemented circular trajectory, and compare this with an existing commercial system for fabric testing. We quantify the error of the approximate test by calculating the deformation of an Euler elastica, a structure with linear moment-curvature response. While the circular mechanism can approximate the exact boundary conditions quite well up to moderately large curvatures, the resulting curvature of the test sample still deviates significantly from homogeneous response. We also briefly discuss expectations of localization behavior in pure bending tests.

中文翻译：

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板材纯弯曲的精确和近似机制

直接测量片材或线材在高曲率下的弯矩-曲率响应有助于对折痕、打褶和其他成型操作进行建模。我们考虑与纯弯曲几何相关的理论问题。我们提出了一种连杆设计，对于样品的均匀变形，通过将耳蜗轨迹与倍角机制相结合，可以在任意高曲率下产生精确的纯弯曲状态。完整的机制是单自由度和位置控制的，这是测量软材料的理想特性。我们还使用更容易实现的圆形轨迹呈现了对耳蜗的最佳近似，并将其与现有的用于织物测试的商业系统进行比较。我们通过计算 Euler elastica（一种具有线性弯矩-曲率响应的结构）的变形来量化近似测试的误差。虽然圆形机制可以很好地近似精确的边界条件，直到适度大的曲率，测试样品的结果曲率仍然显着偏离均匀响应。我们还简要讨论了纯弯曲测试中定位行为的预期。