不可压缩非线性弹性固体分岔引起的薄壁圆柱颈缩
Necking of thin-walled cylinders via bifurcation of incompressible nonlinear elastic solids
AI总结：
本文通过不可压缩非线性弹性固体的二维分岔理论，建模了软聚丙烯薄壁圆柱在准静态拉伸下的多重颈缩及高阶模态现象，并给出了压缩起皱与Euler屈曲的闭式解，解释了实验观察。
摘要：
在准静态单轴拉伸下，实验观察到由软聚丙烯制成的延性薄壁圆柱管发生颈缩局部化。颈缩在管上多个位置成核并扩展至整体，涉及高阶模态，显示出三叶形和四叶形（但很少出现五叶形）的横截面形状。在其他延性材料的薄壁圆柱中，准静态加载下未发现如此复杂的颈缩发生与扩展的证据。为了模拟这一现象以及所有其他可能的分岔，引入了一种二维公式，其中仅考虑管的平均表面，这与著名的Flügge对轴向压缩圆柱壳的处理方法相平行。该处理方法被扩展以包括拉伸和一类广泛的材料非线性超弹性本构定律，同时假设材料是不可压缩的。该理论框架产生了许多新结果，不仅针对拉伸轴向力（其中对颈缩进行了建模，并且作为特例，经典的Considère公式被证明代表极薄管的极限），也针对压缩力，提供了起皱的闭式公式（表明直接应用Flügge方程可能是不正确的）和Euler屈曲的闭式公式。结果表明，J2塑性变形理论（通过非线性弹性模拟材料塑性分支的最简单本构假设）能够捕捉多重颈缩和高阶模态的出现，从而解释了实验现象。所呈现的结果对于多种应用都很重要，从航空航天和汽车工程到血管力学生物学，在这些领域中薄壁管（例如动脉、导管或支架）不仅在压缩下而且在拉伸下都可能变得不稳定。
英文摘要：
Necking localization under quasi-static uniaxial tension is experimentally observed in ductile thin-walled cylindrical tubes, made of soft polypropylene. Necking nucleates at multiple locations along the tube and spreads throughout, involving higher-order modes, evidencing trefoil and fourthfoiled (but rarely even fifth-foiled) shaped cross-sections. No evidence of such a complicated necking occurrence and growth was found in other ductile materials for thin-walled cylinders under quasi-static loading. With the aim of modelling this phenomenon, as well as all other possible bifurcations, a twodimensional formulation is introduced, in which only the mean surface of the tube is considered, paralleling the celebrated Flügge treatment of axially-compressed cylindrical shells. This treatment is extended to include tension and a broad class of nonlinear-hyperelastic constitutive law for the material, which is also assumed to be incompressible. The theoretical framework leads to a number of new results, not only for tensile axial force (where necking is modelled and, as a particular case, the classic Considère formula is shown to represent the limit of very thin tubes), but also for compressive force, providing closed-form formulae for wrinkling (showing that a direct application of the Flügge equation can be incorrect) and for Euler buckling. It is shown that the J2-deformation theory of plasticity (the simplest constitutive assumption to mimic through nonlinear elasticity the plastic branch of a material) captures multiple necking and occurrence of higher-order modes, so that experiments are explained. The presented results are important for several applications, ranging from aerospace and automotive engineering to the vascular mechanobiology, where a thin-walled tube (for instance an artery, or a catheter, or a stent) may become unstable not only in compression, but also in tension.
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