Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in Reissner’s Approximation

IF 0.9 4区地球科学 Q4 GEOCHEMISTRY & GEOPHYSICS Izvestiya, Physics of the Solid Earth Pub Date : 2023-08-06 DOI:10.1134/S1069351323040122

A. P. Trubitsyn, V. P. Trubitsyn

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Abstract—Elastic bending stresses and deformations in the lithosphere are usually calculated based on the Kirchhoff–Love theory for thin plates. The criterion for its applicability is the smallness of the ratio of plate thickness to plate length. In oceanic plates, due to the buoyancy force of the mantle, the main deformations are not uniformly distributed along the plate but concentrate in the vicinity of the subduction zone. Therefore, the effective length of the bending part of the plate is a few fractions of its actual length, and the plate thinness criterion is partially violated. In this paper, we analyze the possibility of applying thick plate bending equations. The existing variational theories of 3D bending of thick plates are substantially more complicated than the Kirchhoff–Love theory, as they involve solving three differential equations instead of one, and have limited application due to their complexity. Since geophysical applications frequently use 2D models, in this paper we analyze in detail the potential and accuracy of the thick plate bending theory for 2D models. After the conversion to the 2D plane strain and plane stress approximation, the original 3D Reissner thick plate bending equations are written out in the form similar to the Kirchhoff equations with additive corrections and are supplemented with the explicit formulas for longitudinal displacement. The comparison of the analytical solutions of the 2D Reissner equations with the exact solutions shows that the 2D approximation only provides a correction for the plate deflection function. However, this correction refines the Kirchhoff–Love theory by almost an order of magnitude. At the same time, the solution of the equations in this case turns out to be almost as simple as the solution of the thin plate equations.

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二维模型薄板弹性弯曲理论在Reissner近似中的修正

摘要岩石圈的弹性弯曲应力和变形通常是基于Kirchhoff-Love薄板理论计算的。其适用性的标准是板厚与板长之比较小。在大洋板块中，由于地幔的浮力作用，主要变形并非沿板块均匀分布，而是集中在俯冲带附近。因此，板的弯曲部分的有效长度是其实际长度的一小部分，部分违反了板的厚度准则。本文分析了应用厚板弯曲方程的可能性。现有的厚板三维弯曲变分理论比Kirchhoff-Love理论复杂得多，因为它们需要求解三个微分方程而不是一个，并且由于其复杂性而限制了应用。由于地球物理应用经常使用二维模型，本文详细分析了厚板弯曲理论在二维模型中的潜力和准确性。将原始的三维Reissner厚板弯曲方程转换为二维平面应变和平面应力近似后，以加性修正的Kirchhoff方程的形式写成，并补充了纵向位移的显式公式。二维Reissner方程解析解与精确解的比较表明，二维近似只对板挠度函数提供了修正。然而，这一修正几乎将Kirchhoff-Love理论改进了一个数量级。同时，这种情况下方程的解几乎和薄板方程的解一样简单。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Izvestiya, Physics of the Solid Earth 地学-地球化学与地球物理

CiteScore

1.60

自引率

30.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Izvestiya, Physics of the Solid Earth is an international peer reviewed journal that publishes results of original theoretical and experimental research in relevant areas of the physics of the Earth''s interior and applied geophysics. The journal welcomes manuscripts from all countries in the English or Russian language.