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所在平台: Udemy |
课程主页: https://www.udemy.com/course/theory-of-elasticityadvanced-solid-mechanics/
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课程名称：弹性理论/先进固体力学 概述：本课程介绍弹性理论的基本概念和方法，研究材料在外力作用下的变形和应力。课程目标是使学生能够理解力、应力、应变和胡克定律在二维和三维中的定义和符号；应用不同坐标系统下应力和应变分量的变换；识别主应力和主应变、应力和应变不变量、应变能和叠加原理；利用艾里应力函数法解决二维笛卡尔坐标及极坐标下的问题；通过应力函数、能量和数值方法分析不同截面形状的杆件扭转。课堂还将介绍解决弹性问题的一些实验技术和分析工具，如肥皂膜和普朗德特膜类比。学生需要将课堂学习的概念和方法运用到作业和设计项目中。本课程还将为更高级的固体力学课程（如塑性、断裂力学和有限元分析）做好准备，适合已经完成工程力学、数学和物理课程的学生。本课程将通过在线视频讲座进行授课。 主要内容包括： - 介绍：力量和应力的定义和符号、应力和应变分量、广义胡克定律、三方向的应力-应变关系、二维和三维的平面应力与平面应变、平衡方程与兼容性方程、倾斜平面上的应力分量、坐标系统变化下的应力分量变换。 - 主应力和主平面：应力不变量、均值和偏差应力、单位体积应变能、单位体积畸变应变能、八面体剪应力、线元的应变。主应变、应变不变量、体积应变、叠加原理、对偶定理。 - 笛卡尔坐标下的二维问题：通过多项式求解、圣维南原理、解的唯一性、基于艾里应力函数的应力分量。应用于悬臂梁、简支梁和固定梁的简单负载。 - 极坐标下的二维问题：应力-应变分量、平衡方程、兼容性方程、应用艾里应变函数解决关于对称轴的应力分布、孔对拉伸板应力分布的影响、载荷在半无限直边界上的点产生的应力、直径加载下圆盘的应力。 - 扭转：各种形状杆的扭转、应用于圆形和椭圆形杆的应力函数法、矩形杆的扭转、通过能量法解决扭转问题、运用肥皂膜解决扭转问题、普朗德特膜类比。利用瑞利-里茨法和有限差分法解决矩形杆的扭转问题。 本课程为希望深入了解弹性理论及其应用的学生提供了良好的基础。
This course introduces the basic concepts and methods of elasticity theory, which is the study of how materials deform and stress under external forces. The course objectives are to enable the students to:Understand the definitions and notations of forces, stresses, strains, and Hooke's law in two and three dimensions.Apply the transformation of stress and strain components under different coordinate systems.Identify the principal stresses and strains, stress and strain invariants, strain energy, and superposition principle.Solve two-dimensional problems in Cartesian and polar coordinates using Airy's stress function method.Analyze torsion of bars with different cross-sectional shapes using stress function, energy, and numerical methods.The course will also expose the students to some experimental techniques and analytical tools for solving elasticity problems, such as soap films and Prandtl's membrane analogy. The course will require the students to apply the concepts and methods learned in class to solve homework assignments and design projects. The course will also prepare the students for more advanced courses in solid mechanics, such as plasticity, fracture mechanics, and finite element analysis. The course is suitable for students who have completed courses in engineering mechanics, mathematics, and physics. The course will be taught through online video lectures.The topics covered in this course are:Introduction: Definition and notation for forces and stresses, components of stress and strain, Generalized Hooke's law, Stress-strain relations in three directions, Plane stress and plane strain, Equations of equilibrium and compatibility in two and three dimensions, Stress components on an oblique plane, Transformation of stress components under change of co-ordinate system.Principal stresses and principal planes: Stress invariants, Mean and Deviator stress, Strain energy per unit volume, Distortion strain energy per unit volume, Octahedral shear stress, Strain of a line element. Principal strains, Strain invariants, Volume strain, Principle of superposition, reciprocal theorem.Two dimensional problems in Cartesian co-ordinates: Solution by polynomials, St. Venant's Principle, Uniqueness of solution, Stress components in terms of Airy's stress function. Applications to Cantilever, simply supported and fixed beams with simple loading.Two dimensional problems in Polar co-ordinates: Stress-strain components, Equilibrium equations, Compatibility equations, Applications using Airy's strain functions in polar co-ordinates for stress distributions symmetric about an axis, Effect of hole on stress distribution in a plate in tension, Stress due to load at a point on a semi-infinite straight boundary, Stresses in a circular disc under diametrical loading.Torsion: Torsion of various shapes of bars, Stress function method of solution applied to circular and elliptical bars, Torsion of rectangular bars, Solution of Torsional problems by energy method, use of soap films in solving torsion problems, Prandtl's membrane analogy. Solution of torsion of rectangular bars by (i) Raleigh Ritz method and (ii) Finite difference method.
