{"title":"周期性三维小尺寸固体波传播分析的新计算框架","authors":"Andrea Francesco Russillo, Giuseppe Failla","doi":"10.1016/j.ijengsci.2025.104350","DOIUrl":null,"url":null,"abstract":"<div><div>Analysing elastic wave propagation in periodic small-size structures plays an important role in the design of many micro- and nano-engineering devices. However, as ad hoc size-dependent continuum theories are required to capture size effects, pertinent computational tools shall be developed to characterize the wave propagation properties. In this context, this paper introduces an original computational framework to build the dispersion diagram of periodic 3D small-size solids of arbitrary shape, as modelled by the well-established Eringen’s nonlocal integral theory. The framework makes use of a suitable periodic Bloch ansatz to represent the response variables involved in the weak formulation of the integro-differential free-vibration equilibrium equations of the unit cell. Building on the periodicity of the Bloch ansatz and introducing an appropriate change of variables, it is shown that the integral coupling the response at a given point of the unit cell to the responses at all points of the solid can be reverted to the summation of integrals defined on the domain of the unit cell only. This remarkable result paves the way to solve the wave propagation problem by a finite element formulation of the free-vibration equilibrium equations of the unit cell, which involves a standard mass matrix, a local stiffness matrix and a nonlocal stiffness matrix, with the latter being expressed by the infinite summation of nonlocal matrices accounting for the nonlocal interactions between the unit cell and the surrounding cells of the solid. In fact, the summation can be truncated to a finite order depending on the nonlocal horizon of the kernel function selected for the nonlocal integral model and the dispersion diagram can be obtained from a linear eigenvalue problem, derived enforcing the Bloch conditions in the finite element free-vibration equilibrium equations of the unit cell. Numerical applications substantiate correctness and accuracy of the proposed framework, which enables a consistent application of the Eringen’s nonlocal integral theory to study wave propagation in periodic 3D small-size structures of arbitrary shape, for the first time to the best of authors’ knowledge.</div></div>","PeriodicalId":14053,"journal":{"name":"International Journal of Engineering Science","volume":"217 ","pages":"Article 104350"},"PeriodicalIF":5.7000,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A novel computational framework for wave propagation analysis of periodic 3D small-size solids\",\"authors\":\"Andrea Francesco Russillo, Giuseppe Failla\",\"doi\":\"10.1016/j.ijengsci.2025.104350\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Analysing elastic wave propagation in periodic small-size structures plays an important role in the design of many micro- and nano-engineering devices. However, as ad hoc size-dependent continuum theories are required to capture size effects, pertinent computational tools shall be developed to characterize the wave propagation properties. In this context, this paper introduces an original computational framework to build the dispersion diagram of periodic 3D small-size solids of arbitrary shape, as modelled by the well-established Eringen’s nonlocal integral theory. The framework makes use of a suitable periodic Bloch ansatz to represent the response variables involved in the weak formulation of the integro-differential free-vibration equilibrium equations of the unit cell. Building on the periodicity of the Bloch ansatz and introducing an appropriate change of variables, it is shown that the integral coupling the response at a given point of the unit cell to the responses at all points of the solid can be reverted to the summation of integrals defined on the domain of the unit cell only. This remarkable result paves the way to solve the wave propagation problem by a finite element formulation of the free-vibration equilibrium equations of the unit cell, which involves a standard mass matrix, a local stiffness matrix and a nonlocal stiffness matrix, with the latter being expressed by the infinite summation of nonlocal matrices accounting for the nonlocal interactions between the unit cell and the surrounding cells of the solid. In fact, the summation can be truncated to a finite order depending on the nonlocal horizon of the kernel function selected for the nonlocal integral model and the dispersion diagram can be obtained from a linear eigenvalue problem, derived enforcing the Bloch conditions in the finite element free-vibration equilibrium equations of the unit cell. Numerical applications substantiate correctness and accuracy of the proposed framework, which enables a consistent application of the Eringen’s nonlocal integral theory to study wave propagation in periodic 3D small-size structures of arbitrary shape, for the first time to the best of authors’ knowledge.</div></div>\",\"PeriodicalId\":14053,\"journal\":{\"name\":\"International Journal of Engineering Science\",\"volume\":\"217 \",\"pages\":\"Article 104350\"},\"PeriodicalIF\":5.7000,\"publicationDate\":\"2025-07-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Engineering Science\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0020722525001375\",\"RegionNum\":1,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Engineering Science","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020722525001375","RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
A novel computational framework for wave propagation analysis of periodic 3D small-size solids
Analysing elastic wave propagation in periodic small-size structures plays an important role in the design of many micro- and nano-engineering devices. However, as ad hoc size-dependent continuum theories are required to capture size effects, pertinent computational tools shall be developed to characterize the wave propagation properties. In this context, this paper introduces an original computational framework to build the dispersion diagram of periodic 3D small-size solids of arbitrary shape, as modelled by the well-established Eringen’s nonlocal integral theory. The framework makes use of a suitable periodic Bloch ansatz to represent the response variables involved in the weak formulation of the integro-differential free-vibration equilibrium equations of the unit cell. Building on the periodicity of the Bloch ansatz and introducing an appropriate change of variables, it is shown that the integral coupling the response at a given point of the unit cell to the responses at all points of the solid can be reverted to the summation of integrals defined on the domain of the unit cell only. This remarkable result paves the way to solve the wave propagation problem by a finite element formulation of the free-vibration equilibrium equations of the unit cell, which involves a standard mass matrix, a local stiffness matrix and a nonlocal stiffness matrix, with the latter being expressed by the infinite summation of nonlocal matrices accounting for the nonlocal interactions between the unit cell and the surrounding cells of the solid. In fact, the summation can be truncated to a finite order depending on the nonlocal horizon of the kernel function selected for the nonlocal integral model and the dispersion diagram can be obtained from a linear eigenvalue problem, derived enforcing the Bloch conditions in the finite element free-vibration equilibrium equations of the unit cell. Numerical applications substantiate correctness and accuracy of the proposed framework, which enables a consistent application of the Eringen’s nonlocal integral theory to study wave propagation in periodic 3D small-size structures of arbitrary shape, for the first time to the best of authors’ knowledge.
期刊介绍:
The International Journal of Engineering Science is not limited to a specific aspect of science and engineering but is instead devoted to a wide range of subfields in the engineering sciences. While it encourages a broad spectrum of contribution in the engineering sciences, its core interest lies in issues concerning material modeling and response. Articles of interdisciplinary nature are particularly welcome.
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