Starting from the theory of elastic plates, we derive a non-linear one-dimensional model for elastic ribbons with thickness $t$, width $a$ and length $\ell$, assuming $t\ll a\ll \ell$. It takes the form of a rod model with a specific non-linear constitutive law accounting for both the stretching and the bending of the ribbon mid-surface. The model is asymptotically correct and can handle finite rotations. Two popular theories can be recovered as limiting cases, namely Kirchhoff’s rod model for small bending and twisting strains, $\left|{\kappa }_i\right|\ll t∕a^2$, and Sadowsky’s inextensible ribbon model for $\left|{\kappa }_i\right|\gg t∕a^2$; we point out that Sadowsky’s inextensible model may be a poor approximation even for ribbons having a very thin cross-section (say, with $t∕a$ as small as $0.02$). By way of illustration, the one-dimensional model is applied (i) to the lateral-torsional instability of a ribbon, showing good agreement with both experiments and finite-element shell simulations, and (ii) to the stability of a twisted ribbon subjected to a tensile force. The non-convexity of the one-dimensional model is discussed; it is addressed by a convexification argument.

从弹性板理论出发，我们推导了具有厚度的弹性带的非线性一维模型 $Ť$， 宽度 $\mathrm{一种}$ 和长度 $\ell$， 假设 $Ť\ll \mathrm{一种}\ll \ell$。它采用具有特定非线性本构律的棒模型的形式，该模型考虑了碳带中间表面的拉伸和弯曲。该模型是渐近正确的，并且可以处理有限的旋转。作为限制情况，可以找到两种流行的理论，即基尔霍夫的弯曲和扭转应变小的杆模型，$\left|{\kappa }_{\mathrm{一世}}\right|\ll Ť∕{\mathrm{一种}}^{\mathrm{2个}}$和Sadowsky的不可扩展的功能区模型 $\left|{\kappa }_{\mathrm{一世}}\right|\gg Ť∕{\mathrm{一种}}^{\mathrm{2个}}$; 我们指出，即使对于横截面非常薄的色带，Sadowsky的不可扩展模型也可能是一个差的近似值（例如，$Ť∕\mathrm{一种}$ 一样小 $0。02$）。通过举例说明，将一维模型应用于（i）薄带的横向扭转不稳定性，与实验和有限元壳模拟都显示出良好的一致性，并且（ii）经受扭曲的薄带的稳定性拉伸力。讨论了一维模型的非凸性；它通过凸化论证得到解决。