三维非线性弹性力学高阶杆模型的渐近推导
Asymptotic derivation of high-order rod models from non-linear 3D elasticity
摘要：
提出了一种细长非线性结构等效一维模型的推导方法。该方法被设计为具有广泛的适用性，并且原则上可以处理有限应变、有限旋转、任意横截面形状、横截面上的非均匀弹性性质、任意弹性本构关系（可能具有低对称性）和任意预应变分布，包括有限预应变。它是基于实际配置的运动学参数化，利用中心线，一个框架的董事，和当地的自由度捕捉详细形状的横截面。一个松弛的方法，保持固定的框架中心线，同时放松当地的自由度，它是渐近有效的宏观应变和性能的杆在纵向方向上缓慢变化时。结果是一个一维的应变能取决于明显的拉伸，弯曲和扭转应变的框架中心线;应变梯度的依赖性也被捕获，产生一个等效杆模型，是渐近精确到高阶。该方法是在一个完全非线性设置，它是验证对线性和弱非线性的解决方案，可从文献中。
英文摘要：
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.
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