We propose a method for deriving equivalent one-dimensional models for slender non-linear structures. The approach is designed to be broadly applicable, and can handle in principle finite strains, finite rotations, arbitrary cross-sections shapes, inhomogeneous elastic properties across the cross-section, arbitrary elastic constitutive laws (possibly with low symmetry) and arbitrary distributions of pre-strain, including finite pre-strain. It is based on a kinematic parameterization of the actual configuration that makes use of a center-line, a frame of directors, and local degrees of freedom capturing the detailed shape of cross-sections. A relaxation method is applied that holds the framed center-line fixed while relaxing the local degrees of freedom; it is asymptotically valid when the macroscopic strain and the properties of the rod vary slowly in the longitudinal direction. The outcome is a one-dimensional strain energy depending on the apparent stretching, bending and twisting strain of the framed center-line; the dependence on the strain gradients is also captured, yielding an equivalent rod model that is asymptotically exact to higher order. The method is presented in a fully non-linear setting and it is verified against linear and weakly non-linear solutions available from the literature.

我们提出了一种用于导出细长非线性结构的等效一维模型的方法。该方法被设计为广泛适用，并且原则上可以处理有限应变，有限旋转，任意横截面形状，横截面的不均匀弹性特性，任意弹性本构定律（可能具有低对称性）和预应力的任意分布-应变，包括有限的预应变。它基于实际配置的运动学参数化，该参数化利用了中心线，指向矢的框架以及捕获截面详细形状的局部自由度。采用一种松弛方法，在保持局部自由度的同时，保持框架中心线固定。当棒的宏观应变和特性在纵向方向上缓慢变化时，这是渐近有效的。结果是一维应变能，取决于框架中心线的表观拉伸，弯曲和扭曲应变。还捕获了对应变梯度的依赖性，从而得出了一个渐近精确到更高阶的等效杆模型。该方法在完全非线性的环境中提出，并针对文献中的线性和弱非线性解决方案进行了验证。产生一个渐近精确到更高阶的等效杆模型。该方法在完全非线性的环境中提出，并针对文献中的线性和弱非线性解决方案进行了验证。产生一个渐近精确到更高阶的等效杆模型。该方法在完全非线性的环境中提出，并针对文献中的线性和弱非线性解决方案进行了验证。