Subject Area
Mechanical Engineering
Abstract
In this dissertation research, new non-classical models for Kirchhoff and Mindlin plates are developed and applied to study band gaps for flexural wave propagation in composite plate structures.
In Chapter 2, a new non-classical model for a Kirchhoff plate resting on an elastic foundation is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved.
In Chapter 3, a new non-classical model for circular Kirchhoff plates subjected to axisymmetric loading is presented based on the same modified couple stress theory and surface elasticity theory but using cylindrical polar coordinates. The new non-classical plate model includes the circular plate models considering the microstructure influence or the surface energy effect alone as special cases and recovers the classical elasticity-based Kirchhoff plate model when both the microstructure and surface energy effects are suppressed. To demonstrate the new model, the static bending problem of a clamped solid circular Kirchhoff plate subjected to a uniform normal load is analytically solved.
In Chapter 4, a new non-classical model for a Mindlin plate resting on an elastic foundation is developed in a general form using the modified couple stress theory, the surface elasticity theory and the two-parameter Winkler–Pasternak foundation model, which are the same as those employed in Chapter 2. It includes all five kinematic variables possible for a Mindlin plate and treats the microstructure, surface energy and foundation effects in a unified manner. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all neglected. In addition, the new model includes the Mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases, and it degenerates to the Timoshenko beam model including the microstructure effect. To illustrate the new Mindlin plate model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived.
In Chapter 5, a new non-classical model for circular Mindlin plates is furnished using the modified couple stress theory, the surface elasticity theory, Hamilton’s principle, and cylindrical polar coordinates, as was done in Chapter 3. The non-classical model includes the circular plate models considering the microstructure influence only and the surface energy effect alone as special cases, and it recovers the classical elasticity-based circular Mindlin plate model when both the microstructure and surface energy effects are not considered. To illustrate the new model, the static bending problem of a clamped circular Mindlin plate under a uniform normal load is analytically solved.
In Chapter 6, a new model for determining band gaps for flexural elastic wave propagation in a periodic composite plate structure with square inclusions is developed by directly using the non-classical model for Kirchhoff plates presented in Chapter 2. The band gaps predicted by the newly developed model depend on the microstructure and surface elasticity of each constituent material, the elastic foundation moduli, the unit cell size, and the volume fraction of the inclusion phase. To quantitatively illustrate the effects of these factors, a parametric study is conducted.
In Chapter 7, a new model for predicting band gaps for flexural elastic wave propagation in a periodic composite plate structure with square or cruciform inclusions is provided by using the non-classical model for Mindlin plates proposed in Chapter 4. The band gaps predicted by the new model depend on the microstructure and surface elasticity of each consitituent material, the unit cell size, and the volume fraction. To quantitatively illustrate the effects of these factors, a parametric study is conducted for periodic composite plate structures containing square and cruciform inclusions.
Degree Date
Fall 12-15-2018
Document Type
Dissertation
Degree Name
Ph.D.
Department
Mechanical Engineering
Advisor
Xin-Lin Gao
Format
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License
Recommended Citation
Zhang, Gongye, "Non-classical Plate Models Incorporating Microstructure and Surface Energy Effects: Their Variational Formulations and Applications" (2018). Mechanical Engineering Research Theses and Dissertations. 12.
https://scholar.smu.edu/engineering_mechanical_etds/12
