Refined Analysis of Shear Stress Distribution in Tapered Rods Accounting for Gradient Effects

IF 0.8 Q2 MATHEMATICS Lobachevskii Journal of Mathematics Pub Date : 2024-08-28 DOI:10.1134/s1995080224602522
A. V. Volkov, K. S. Golubkin, Y. O. Solyaev
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Abstract

In this article, we propose to apply the strain gradient elasticity theory for the refined stress analysis in the tapered rods with variable cross section. Corresponding statement of the higher-order boundary value problem with extended set of boundary conditions is derived based on the variational approach. Considering an example for the cylindrical/conical rod loaded by self-equilibrated body and end forces, we provide the comparison between the stress distributions that can be obtained within the classical 3D elasticity, classical 1D rod theory and the established 1D rod theory with the strain gradient effects. It is shown that the last one allows to obtain the smoothed solution for the shear stresses that can be fitted to 3D elasticity solution under appropriate choice of additional length scale parameter of gradient theory. In contrast, solution of classical rod theory cannot be fitted exactly to 3D elasticity solution and contains unavoidable discontinuities of shear stresses.

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考虑梯度效应的锥形杆剪应力分布精细分析
摘要 本文提出应用应变梯度弹性理论对变截面锥形杆进行精细应力分析。基于变分法,推导出了具有扩展边界条件集的高阶边界值问题的相应表述。考虑到圆柱形/圆锥形杆在自平衡体力和端力作用下的载荷实例,我们对经典三维弹性理论、经典一维杆理论和具有应变梯度效应的既定一维杆理论所能获得的应力分布进行了比较。结果表明,在适当选择梯度理论的附加长度尺度参数的情况下,最后一种方法可以获得与三维弹性解相匹配的剪应力平滑解。与此相反,经典杆理论的解无法与三维弹性解精确拟合,并且包含不可避免的剪应力不连续性。
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来源期刊
CiteScore
1.50
自引率
42.90%
发文量
127
期刊介绍: Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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