【摘要】：格林函數(shù)法是求解數(shù)學(xué)物理方程的一種常用方法,在固體力學(xué)的理論和應(yīng)用研究中起著重要作用。格林函數(shù)的描述對(duì)象是點(diǎn)源產(chǎn)生的場(chǎng),它是許多理論和數(shù)值計(jì)算的基礎(chǔ),如邊界元法等。采用疊加原理,格林函數(shù)可以用于對(duì)任意載荷作用下的復(fù)雜工程問題進(jìn)行求解。此外它們還可以被用來對(duì)工程中常見的裂縫,損傷和雜質(zhì)問題進(jìn)行解析求解。 近幾年來正在高速發(fā)展的功能材料,實(shí)際使用中一般需要與其他材料結(jié)合在一起,由此產(chǎn)生界面效應(yīng)和多物理場(chǎng)耦合,無論從結(jié)構(gòu)的安全性,還是從功能元件的可靠性考慮,都必須作多場(chǎng)耦合的分析和評(píng)價(jià)。流固耦合及其界面問題是這些工程實(shí)際問題的重要組成部分。橫觀各向同性焦熱電陶瓷是工程中常見的材料,這種材料經(jīng)常與勻質(zhì)介電流體配合使用,本文以這兩種材料作為流固耦合模型的對(duì)應(yīng)兩相,采用格林函數(shù)分析方法,研究點(diǎn)源作用下兩相體的多物理場(chǎng)耦合關(guān)系。力電耦合是焦熱電陶瓷材料的固有特性,此外該材料也對(duì)溫度相當(dāng)敏感,鑒于焦熱電陶瓷材料制品實(shí)際工作環(huán)境的復(fù)雜性,在研究焦熱電陶瓷材料的流固兩相界面效應(yīng)時(shí),應(yīng)主要著眼于兩相體內(nèi)部熱-電-彈性場(chǎng)的變化。 本文首先介紹了橫觀各向同性焦熱電材料在考慮力-熱-電耦合效應(yīng)下的三維本構(gòu)方程、平衡方程和熱傳導(dǎo)方程。利用微分算子理論和Almansi理論,得到了用單調(diào)和函數(shù)表示的三維穩(wěn)態(tài)簡(jiǎn)潔通解。對(duì)于勻質(zhì)介電流體,給出了用調(diào)和函數(shù)表示的電位和溫度增量的通解。這些調(diào)和函數(shù)形式表示的穩(wěn)態(tài)通解為后續(xù)的兩相體多場(chǎng)耦合研究奠定了基礎(chǔ)。 后面章節(jié)分別對(duì)點(diǎn)熱源作用在焦熱電陶瓷材料流固兩相體固體內(nèi)部和流體內(nèi)部,點(diǎn)電荷、點(diǎn)力作用于焦熱電陶瓷材料流固兩相體固體內(nèi)部,點(diǎn)電荷作用于兩相體中流體內(nèi)部的情況,運(yùn)用格林函數(shù)分析方法,引入多個(gè)含待定常數(shù)的調(diào)和函數(shù),考慮相應(yīng)的邊界條件、連續(xù)性條件和平衡條件來確定待定常數(shù),求解擴(kuò)展的Mindlin和擴(kuò)展的Lorentz解。各章最后通過數(shù)值算例分析,給出了各點(diǎn)源作用形式下,耦合場(chǎng)無量綱溫度增量、應(yīng)力分量和電位移分量的等值線圖,獲得了一些有工程應(yīng)用價(jià)值的結(jié)論,比如指導(dǎo)俘能器和傳感器的設(shè)計(jì)。格林函數(shù)的分析方法,為考慮力-熱-電耦合效應(yīng)的流固耦合問題進(jìn)行理論分析和數(shù)值計(jì)算提供了一種可以選擇的平臺(tái),所構(gòu)造的基本解用初等函數(shù)表示,形式簡(jiǎn)潔,便于在理論和工程中推廣應(yīng)用。
[Abstract]:The Green's function method is a common method for solving mathematical and physical equations, which plays an important role in the theoretical and applied research of solid mechanics. The description object of Green's function is the field produced by point source, which is the basis of many theories and numerical calculations, such as boundary element method. Using the superposition principle, Green's function can be used to solve complex engineering problems under arbitrary loads. In addition, they can be used to solve the problems of cracks, damage and impurity in engineering. In recent years, functional materials, which are developing at a high speed in practice, usually need to be combined with other materials in practical use, resulting in interface effects and multi-physical field coupling, both in terms of structural safety and reliability of functional components. The analysis and evaluation of multi-field coupling must be made. The fluid-solid coupling and its interface problems are important components of these engineering practical problems. Transversely isotropic pyroelectric ceramics are common materials in engineering. These materials are often used in conjunction with homogeneous dielectric fluids. In this paper, the two materials are used as the corresponding two phases of the fluid-solid coupling model, and the Green's function analysis method is used. The multi-physical field coupling relation of two-phase body under the action of point source is studied. Electromechanical coupling is an inherent characteristic of pyroelectric ceramic materials. In addition, the material is also sensitive to temperature. In view of the complexity of the actual working environment of pyroelectric ceramic materials, the fluid-solid two-phase interface effect of pyroelectric ceramic materials is studied. The change of thermoelectric-elastic field in the two-phase body should be focused on. In this paper, the three dimensional constitutive equations, equilibrium equations and heat conduction equations of transversely isotropic pyroelectric materials are first introduced. By using the theory of differential operator and Almansi theory, a simple and simple solution of three-dimensional steady state is obtained, which is expressed by monotone sum function. For homogeneous dielectric fluid, the general solution of potential and temperature increment expressed by harmonic function is given. The steady-state general solutions of these harmonic function forms lay a foundation for the further study of two-phase multifield coupling. In the following chapters, the point heat source acting on the solid of the fluid-solid two-phase body and the fluid, the point charge and the point force acting on the solid of the fluid-solid two-phase body of the pyroelectric ceramic material, The point charge acts on the fluid in a two-phase body. By using Green's function analysis method, several harmonic functions with undetermined constants are introduced, and the corresponding boundary conditions, continuity conditions and equilibrium conditions are considered to determine the undetermined constants. The extended Mindlin and the extended Lorentz solution are solved. At the end of each chapter, through numerical example analysis, the isoline diagrams of dimensionless temperature increment, stress component and electric displacement component of coupling field under the action of point source are given, and some valuable conclusions of engineering application are obtained. Such as guiding the design of energy capture and sensors. The analysis method of Green's function provides an alternative platform for the theoretical analysis and numerical calculation of fluid-solid coupling problem considering the force thermoelectric coupling effect. The basic solution is expressed by elementary function and the form is simple. It is easy to be popularized and applied in theory and engineering.
【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O302;TQ174.1

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