On the Reconstruction of a Two-Dimensional Density of a Functionally Graded Elastic Plate

IF 0.6 4区 工程技术 Q4 MECHANICS Mechanics of Solids Pub Date : 2024-11-01 DOI:10.1134/S0025654424602532
V. V. Dudarev, R. M. Mnukhin
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Abstract

In this work, based on the general formulation of the problem of steady-state vibrations of an inhomogeneous elastic isotropic body, a direct problem of planar vibrations of a rectangular plate within the framework of a plane stress state is formulated. The left side of the plate is rigidly fixed, vibrations are forced by tensile load applied at the right side. The properties of the functionally graded material are described by two-dimensional laws of change in Young’s modulus, Poisson’s ratio and density. For generality of consideration, a dimensionless formulation of the problem is given. The solution to the direct problem of determining the displacement field was obtained using the finite element method. The effect of material characteristics on the displacement field and the value of the first resonant frequency are shown. An analysis of the obtained results was carried out. The inverse problem of determining the law of density from data on the values of the displacement field components at a fixed frequency is considered. To reduce the error in calculating derivatives of table functions of two variables, an approach based on spline approximation and a locally weighted regression algorithm is proposed. Reconstruction examples of different laws are presented to demonstrate the possibility of using this approach.

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论功能梯度弹性板二维密度的重构
在这项工作中,基于非均质弹性各向同性体稳态振动问题的一般表述,在平面应力状态的框架内直接提出了矩形板的平面振动问题。板的左侧是刚性固定的,振动是由施加在右侧的拉伸载荷迫使产生的。功能分级材料的特性由杨氏模量、泊松比和密度的二维变化规律描述。为了考虑问题的普遍性,给出了问题的无量纲表述。使用有限元法获得了确定位移场的直接问题的解决方案。图中显示了材料特性对位移场和第一共振频率值的影响。对所得结果进行了分析。考虑了根据固定频率下的位移场分量值数据确定密度定律的逆问题。为了减少计算两变量表函数导数时的误差,提出了一种基于样条近似和局部加权回归算法的方法。本文介绍了不同规律的重建实例,以证明使用这种方法的可能性。
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来源期刊
Mechanics of Solids
Mechanics of Solids 医学-力学
CiteScore
1.20
自引率
42.90%
发文量
112
审稿时长
6-12 weeks
期刊介绍: Mechanics of Solids publishes articles in the general areas of dynamics of particles and rigid bodies and the mechanics of deformable solids. The journal has a goal of being a comprehensive record of up-to-the-minute research results. The journal coverage is vibration of discrete and continuous systems; stability and optimization of mechanical systems; automatic control theory; dynamics of multiple body systems; elasticity, viscoelasticity and plasticity; mechanics of composite materials; theory of structures and structural stability; wave propagation and impact of solids; fracture mechanics; micromechanics of solids; mechanics of granular and geological materials; structure-fluid interaction; mechanical behavior of materials; gyroscopes and navigation systems; and nanomechanics. Most of the articles in the journal are theoretical and analytical. They present a blend of basic mechanics theory with analysis of contemporary technological problems.
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