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下面介绍变形体力学的要点
前面提到了变形体力学涉及到三个方面
力的平衡、变形的描述、材料的行为
要引入三大类变量和三大类方程
怎么来引入呢
这里面的核心就是研究对象不是刚体，而是变形体
那么变形体的力学行为首先需要实验确定
如图所示，我们首先设计一个试样
我们大多数实验使用的试样都是等截面的
有些是圆形的、有些是长方形的
同时这个试样还有一定的长度
我们放到拉伸机上进行拉伸试验
可以得到拉伸的曲线
那我们在进行材料的拉伸试验时都有试验规范
我们假定
同一种材料我们考虑三个单拉实验
第一个我们作为基准实验，叫做实验1
假定它的横截面积是A1
长度是L1
两端的拉力是F1
这个作为基准实验
可以得到一个拉伸实验的曲线，叫做实验1的曲线
当然实验中还是小变形
实验曲线是力和拉伸长度也就是拉伸位移的实验曲线
它是线性的，小变形情况
再设计一个实验2
这个实验2在刚才基准实验的基础上
把横截面的面积增加一倍
长度不变
拉伸的从零拉到最大的力也不变
同样的就力从零拉到F2和位移做一个实验曲线
我们把这个曲线叫做实验2的实验曲线
我们再设计一个实验三
和基准实验1相比
我们把长度增加一倍
面积保持不变
拉伸的最大的力也保持不变
这样我们得到拉伸的力和拉伸长度的曲线关系
我们还可以设计很多不同横截面不同长度的实验
得到很多条实验曲线
那我们下面要思考一个问题
同一种材料
通过刚才的三个实验得到了三个实验曲线
有没有可能我们反映材料力学行为就是一条曲线
它不随着我们实验中设计的试样横截面积的大小改变
以及长度的改变
而改变实验性质
有没有可能呢
我们把实验曲线
稍微做一个数据处理
把拉伸的力除以对应的试样的横截面积
把拉伸的伸长量除以试样的原长
我们定义这两个参量
一个叫σ，也就是应力
另外一个叫相对伸长量，也叫应变
我们把三个实验的实验曲线重新绘制
我们发现三个实验的曲线完全变成了一条曲线
这条实验曲线的比例因子
就是E，这就是我们说的一维的胡克定理
这个E就叫弹性模量，也就是材料的常数
这三个实验说明
只要我们定义了应力和应变
不管试样的几何形状比如横截面面积或试样长度如何变化
我们获取的材料的行为就是唯一的
也就是说用弹性模量可以唯一的描述材料的行为
也就是说可以消除材料试样的几何因素的影响
由此，我们定义力学的参量
从刚才可以看出来
我们需要定义应力和应变
实验中我们可以看到试样右端有伸长量，也就是位移
所以我们把位移、应力和应变
作为描述变形体力学变量的基本变量
刚才的是一维拉伸的情况
也叫做1D问题比较简单
对于二维和三维问题
定义的三大类变量还是这三大类（位移、应力、应变）
但是具体的描述方式会更复杂一些
在我们定义了三大类变量之后
同样也要描述试样的平衡关系
由于试样是均匀的
试样的平衡关系可以直接写出来
对于每一个截面
应力都满足F/A
同样这个物理方程对于每一个位置都是满足的
我们把任意一点的位移除以原长
定义出几何方程
当然现在看这个方程还是比较简单的线性方程
实际上对于二维和三维的情况会更加复杂
主要的方程是微分方程
那就涉及到用三大类方程求解三大类变量
如果要用微分方程求解是有相当大难度的
-Finite element, infinite capabilities
--Video
-1.1 Classification of mechanics：particle、rigid body、deformed body mechanics
--1.1 Test
-1.2 Main points for deformed body mechanics
--1.2 Test
-1.3 Methods to solve differential equation solving method
--1.3 Test
-1.4 Function approximation
--1.4 Test
-1.5 Function approximation defined on complex domains
--1.5 Test
-1.6 The core of finite element: subdomain function approximation for complex domains
--1.6 Test
-1.7 History and software of FEM development
--1.7 Test
-Discussion
-Homework
-2.1 Principles of mechanic analysis of springs
--2.1 Test
-2.2 Comparison between spring element and bar element
--2.2 Test
-2.3 Coordinate transformation of bar element
--2.3 Test
-2.4 An example of a four-bar structure
--2.4 Test
-2.5 ANSYS case analysis of four-bar structure
--ANSYS
-Discussion
-3.1 Mechanical description and basic assumptions for deformed body
--3.1 Test
-3.2 Index notation
--3.2 Test
-3.3 Thoughts on three major variables and three major equations
--3.3 Test
-3.4 Test
-3.4 Construction of equilibrium Equation of Plane Problem
-3.5 Test
-3.5 Construction of strain-displacement relations for plane problems
-3.6 Test
-3.6 Construction of constitutive relations for plane problems
-3.7 Test
-3.7 Two kinds of boundary conditions
- Discussion
-- Discussion
-4.1 Test
-4.1 Discussion of several special cases
-4.2 Test
-4.2 A complete solution of a simple bar under uniaxial tension based on elastic mechanics
-4.3 Test
-4.3 The description and solution of plane beam under pure bending
-4.4 Test
-4.4 Complete description of 3D elastic problem
-4.5 Test
-4.5 Description and understanding of tensor
-Discussion
-5.1 Test
-5.1Main method classification and trial function method for solving deformed body mechanics equation
-5.2 Test
-5.2 Trial function method for solving pure bending beam: residual value method
-5.3 Test
-5.3How to reduce the order of the derivative of trial function
-5.4 Test
-5.4 The principle of virtual work for solving plane bending beam
-5.5 Test
-5.5 The variational basis of the principle of minimum potential energy for solving the plane bending
-5.6 Test
-5.6 The general energy principle of elastic problem
-Discussion
-6.1Test
-6.1 Classic method and finite element method based on trial function
-6.2 Test
-6.2 Natural discretization and approximated discretization in finite element method
-6.3 Test
-6.3 Basic steps in the finite element method
-6.4 Test
-6.4 Comparison of classic method and finite element method
-Discussion
-7.1 Test
-7.1 Construction and MATLAB programming of bar element in local coordinate system
-7.2 Test
-7.2 Construction and MATLAB programming of plane pure bending beam element in local coordinate syste
-7.3 Construction of three-dimensional beam element in local coordinate system
-7.4 Test
-7.4 Beam element coordinate transformation
-7.5 Test
-7.5 Treatment of distributed force
-7.6 Case Analysis and MATLAB programming of portal frame structure
-7.7 ANSYS case analysis of portal frame structure
-8.1 Test
-8.1 Two-dimensional 3-node triangular element and MATLAB programming
-8.2 Test
-8.2 Two-dimensional 4-node rectangular element and MATLAB programming
-8.3 Test
-8.3 Axisymmetric element
-8.4 Test
-8.4 Treatment of distributed force
-8.5 MATLAB programming of 2D plane rectangular thin plate
-8.6 Finite element GUI operation and command flow of a plane rectangular thin plate on ANSYS softwar
-Discussion
-9.1 Three-dimensional 4-node tetrahedral element and MATLAB programming
-9.2 Three-dimensional 8-node hexahedral element and MATLAB programming
-9.3 Principle of the isoparametric element
-9.4Test
-9.4Numerical integration
-9.5 MATLAB programming for typical 2D problems
-9.6 ANSYS analysis case of typical 3Dl problem
-Discussion
-10.1Test
-10.1Node number and storage bandwidth
-10.2Test
-10.2 Properties of shape function matrix and stiffness matrix
-10.3Test
-10.3 Treatment of boundary conditions and calculation of reaction forces
-10.4Test
-10.4 Requirements for construction and convergence of displacement function
-10.5Test
-10.5C0 element and C1 element
-10.6 Test
-10.6 Patch test of element
-10.7 Test
-10.7 Accuracy and property of numerical solutions of finite element analysis
-10.8Test
-10.8 Error and average processing of element stress calculation result
-10.9 Test
-10.9 Error control and the accuracy improving method of h method and p method
-Discussion
-11.1 Test
-11.1 1D high-order element
-11.2 Test
-11.2 2D high-order element
-11.3 Test
-11.3 3D high-order element
-11.4 Test
-11.4 Bending plate element based on thin plate theory
-11.5 Test
-11.5 Sub-structure and super-element
-12.1Test
-12.1 Finite element analysis for structural vibration: basic principle
-12.2 Test
-12.2 Case of finite element analysis for structural vibration
-12.3 Test
-12.3 Finite element analysis for elastic-plastic problems: basic principle
-12.4 Test
-12.4 Finite element analysis for elastic-plastic problems: solving non-linear equations
-Discussion
-13.1 Test
-13.1 Finite element analysis for heat transfer: basic principle
-13.2 Test
-13.2 Case of finite element analysis for heat transfer
-13.3 Test
-13.3 Finite element analysis for thermal stress problems: basic principle
-13.4 Test
-13.4 Finite element analysis for thermal stress problems: solving non-linear equation
-Discussion
-2D problem: finite element analysis of a 2D perforated plate
-3D problem: meshing control of a flower-shaped chuck
-Modal analysis of vibration: Modal analysis of a cable-stayed bridge
-Elastic-plastic analysis: elastic-plastic analysis of a thick-walled cylinder under internal pressur
-Heat transfer analysis: transient problem of temperature field during steel cylinder cooling process
-Thermal stress analysis: temperature and assembly stress analysis of truss structure
-Probability of structure: Probabilistic design analysis of large hydraulic press frame
-Modeling and application of methods: Modeling and analysis of p-type elements for plane problem
