{"title":"解决功能分级材料复杂形状板材几何非线性弯曲问题的数值和分析方法","authors":"S. M. Sklepus","doi":"10.1007/s11223-023-00583-8","DOIUrl":null,"url":null,"abstract":"<p>This study proposes a new numerical-and-analytical method for solving geometrically nonlinear problems of bending of complex-shaped plates made of functionally graded materials developed. The problem was formulated within the framework of a refined higher-order theory considering the quadratic law of distribution of transverse tangential stresses along the plate thickness. To linearize the nonlinear problem, we used the method of continuous continuation in the parameter associated with the external load. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear plate bending (displacements, strains, and stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta–Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometrically linear deformation. The right-hand sides of the differential equations, at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme, were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution as a formula. This solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. Test problems are solved for a homogeneous rigidly fixed and functionally graded hinged square plate subjected to a uniformly distributed load of varying intensity. The results for deflections and stresses obtained by the developed method are compared with the solutions obtained by radial basis functions. The problem of bending of a functionally graded plate of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress-strain state is investigated.</p>","PeriodicalId":22007,"journal":{"name":"Strength of Materials","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Numerical-and-Analytical Method for Solving Geometrically Nonlinear Bending Problems of Complex-Shaped Plates from Functionally Graded Materials\",\"authors\":\"S. M. 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The Cauchy problem was solved by the Runge-Kutta–Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometrically linear deformation. The right-hand sides of the differential equations, at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme, were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution as a formula. This solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. Test problems are solved for a homogeneous rigidly fixed and functionally graded hinged square plate subjected to a uniformly distributed load of varying intensity. The results for deflections and stresses obtained by the developed method are compared with the solutions obtained by radial basis functions. The problem of bending of a functionally graded plate of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress-strain state is investigated.</p>\",\"PeriodicalId\":22007,\"journal\":{\"name\":\"Strength of Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-12-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Strength of Materials\",\"FirstCategoryId\":\"88\",\"ListUrlMain\":\"https://doi.org/10.1007/s11223-023-00583-8\",\"RegionNum\":4,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATERIALS SCIENCE, CHARACTERIZATION & TESTING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Strength of Materials","FirstCategoryId":"88","ListUrlMain":"https://doi.org/10.1007/s11223-023-00583-8","RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATERIALS SCIENCE, CHARACTERIZATION & TESTING","Score":null,"Total":0}
引用次数: 0
摘要
本研究提出了一种新的数值-分析方法,用于解决由功能分级材料制成的复杂形状板材弯曲的几何非线性问题。该问题是在考虑到横向切向应力沿板厚分布的二次定律的精炼高阶理论框架内提出的。为了使非线性问题线性化,我们使用了与外部载荷相关的参数连续延续法。为了对线性化问题进行变分计算,我们构建了一个拉格朗日函数,该函数定义于运动学上可能的位移速度。为了找到非线性板弯曲问题的主要未知数(位移、应变和应力),提出了常微分方程系统的 Cauchy 问题。Cauchy 问题采用自动选择步长的 Runge-Kutta-Merson 方法求解。初始条件是从几何线性变形问题的解法中找到的。在与 Runge-Kutta-Merson 方案相对应的载荷参数固定值下,微分方程的右边是从拉格朗日函数的变分问题求解中得到的。变分问题是通过里兹法结合 R 函数法求解的。后者使得以公式形式给出近似解成为可能。这种解结构完全满足边界条件的全部(一般结构)或部分(部分结构)。对承受不同强度的均匀分布载荷的同质刚性固定和功能分级铰链方板的测试问题进行了求解。将所开发方法得到的挠度和应力结果与径向基函数求解结果进行了比较。解决了形状复杂的功能梯度板的弯曲问题。研究了材料的梯度特性和几何形状对应力-应变状态的影响。
Numerical-and-Analytical Method for Solving Geometrically Nonlinear Bending Problems of Complex-Shaped Plates from Functionally Graded Materials
This study proposes a new numerical-and-analytical method for solving geometrically nonlinear problems of bending of complex-shaped plates made of functionally graded materials developed. The problem was formulated within the framework of a refined higher-order theory considering the quadratic law of distribution of transverse tangential stresses along the plate thickness. To linearize the nonlinear problem, we used the method of continuous continuation in the parameter associated with the external load. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear plate bending (displacements, strains, and stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta–Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometrically linear deformation. The right-hand sides of the differential equations, at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme, were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution as a formula. This solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. Test problems are solved for a homogeneous rigidly fixed and functionally graded hinged square plate subjected to a uniformly distributed load of varying intensity. The results for deflections and stresses obtained by the developed method are compared with the solutions obtained by radial basis functions. The problem of bending of a functionally graded plate of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress-strain state is investigated.
期刊介绍:
Strength of Materials focuses on the strength of materials and structural components subjected to different types of force and thermal loadings, the limiting strength criteria of structures, and the theory of strength of structures. Consideration is given to actual operating conditions, problems of crack resistance and theories of failure, the theory of oscillations of real mechanical systems, and calculations of the stress-strain state of structural components.