端部有导向约束的压杆Euler屈曲公式的非线性修正
Nonlinear Correction to the Euler Buckling Formula for Compressed Cylinders with Guided-Guided End Conditions
摘要：
著名的欧拉屈曲公式给出了半径为$B$，长度为$L$的细长圆柱屈曲的临界载荷$N$为\[ N / (π^3 B^2) = (E/4)(B/L)^2, \]，其中$E$为杨氏模量。它的推导依赖于线性弹性适用于这个问题的假设，以及细长$(B/L)$是一个无穷小的量。在这里，我们提出以下问题：当保留直到$(B/L)^4$的项时，这个方程右边的第一个非线性校正是什么？为了回答这个问题，我们专门为三阶弹性理论的增量非线性弹性的均匀压缩厚的可压缩圆柱润滑端的精确解。特别是，我们强调的方式二阶和三阶常数-包括泊松比-都出现在系数的$(B/L)^4$。
英文摘要：
Euler's celebrated buckling formula gives the critical load $N$ for the buckling of a slender cylindrical column with radius $B$ and length $L$ as \[ N / (π^3 B^2) = (E/4)(B/L)^2, \] where $E$ is Young's modulus. Its derivation relies on the assumptions that linear elasticity applies to this problem, and that the slenderness $(B/L)$ is an infinitesimal quantity. Here we ask the following question: What is the first nonlinear correction in the right hand-side of this equation when terms up to $(B/L)^4$ are kept? To answer this question, we specialize the exact solution of incremental non-linear elasticity for the homogeneous compression of a thick compressible cylinder with lubricated ends to the theory of third-order elasticity. In particular, we highlight the way second- and third-order constants ---including Poisson's ratio--- all appear in the coefficient of $(B/L)^4$.
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