ClAssier Chapter o The Basic solution of The Temperature stress Problems
1 Elasticity
弹单性生力学 第六章温均本解
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The Basic Soltion o he l ura Stress Problems Chapter 6 The Basic Solution of Temperature Stress Problems 86-1 The basic concept of temperature field and heat conduction 86-2 The differential equation of heat conduction 86-3 The boundary conditions of temperature filed 86-4 Solve plane problem of temperature stresses by displacement 86-5 The introducing of potential function of displacement 86-6 The plane problems of thermal stresses in axisymmetric temperature field
3 Chapter 6 The Basic Solution of Temperature Stress Problems §6-4 Solve plane problem of temperature stresses by displacement §6-3 The boundary conditions of temperature filed §6-2 The differential equation of heat conduction §6-1 The basic concept of temperature field and heat conduction §6-5 The introducing of potential function of displacement §6-6 The plane problems of thermal stresses in axisymmetric temperature field
温动向边签解法 第六章温度应力问题的基本解法 §6-1温度场和热传导的基本概念 §6-2热传导微分方程 □§6-3温度场的边界条件 §6-4按位移求解温度应力的平面问题 §6-5位移势函数的引用 §6-6轴对称温度场平面热应力问题
4 第六章 温度应力问题的基本解法 §6-4 按位移求解温度应力的平面问题 §6-3 温度场的边界条件 §6-2 热传导微分方程 §6-1 温度场和热传导的基本概念 §6-5 位移势函数的引用 §6-6 轴对称温度场平面热应力问题
The Basic Soltion he demerara es Problems Chapter f the basic solution of he temperature Stress Problems When the temperature of a elastic body changes, its volume will expand or contract. If the expansion or contraction cant happen freely due to the external restrictions or internal deformation compatibility demands, additional stresses will be produced in the structure. These stresses produced by temperature change are called thermal stresses, or temperature stresses Neglecting the effects of the temperature change on the material performance, to solve the temperature stresses, we need two aspects of calculation: (1) Solve the temperature field of the elastic body by the initial conditions and boundary conditions, according to heat conduction equations And the difference between the former temperature field and the later temperature field is the temperature change of the elastic body. (2) Solve the temperature stresses of the elastic body according to the basic equations of the elastic mechanics. This chapter will present these two aspects of calculation simply 5
5 When the temperature of a elastic body changes, its volume will expand or contract. If the expansion or contraction can’t happen freely due to the external restrictions or internal deformation compatibility demands, additional stresses will be produced in the structure. These stresses produced by temperature change are called thermal stresses, or temperature stresses. Neglecting the effects of the temperature change on the material performance, to solve the temperature stresses, we need two aspects of calculation: (1) Solve the temperature field of the elastic body by the initial conditions and boundary conditions, according to heat conduction equations. And the difference between the former temperature field and the later temperature field is the temperature change of the elastic body. (2) Solve the temperature stresses of the elastic body according to the basic equations of the elastic mechanics. This chapter will present these two aspects of calculation simply
温动向边签解法 第六章温应力题的基本解法 当弹性体的温度变化时,其体积将趋于膨胀和收缩,若 外部的约束或内部的变形协调要求而使膨胀或收缩不能自由 发生时,结构中就会出现附加的应力。这种因温度变化而引 起的应力称为热应力,或温度应力。 忽略变温对材料性能的影响,为了求得温度应力,需要 进行两方面的计算:(1)由问题的初始条件、边界条件, 按热传导方程求解弹性体的温度场,而前后两个温度场之差 就是弹性体的变温。(2)按热弹性力学的基本方程求解弹 性体的温度应力。本章将对这两方面的计算进行简单的介绍
6 当弹性体的温度变化时，其体积将趋于膨胀和收缩，若 外部的约束或内部的变形协调要求而使膨胀或收缩不能自由 发生时，结构中就会出现附加的应力。这种因温度变化而引 起的应力称为热应力，或温度应力。 忽略变温对材料性能的影响，为了求得温度应力，需要 进行两方面的计算：（1）由问题的初始条件、边界条件， 按热传导方程求解弹性体的温度场，而前后两个温度场之差 就是弹性体的变温。（2）按热弹性力学的基本方程求解弹 性体的温度应力。本章将对这两方面的计算进行简单的介绍
86-1 The Basic Concept of Temperature Field And Heat Conduction 1. The temperature field The total of the temperature at all the points in a 0 Unstable temperature filed or nonsteady temperature field: The temperature clastic body at a certain moment, denoted by t in the temperature field changes with time i.e. T-T(x, y, 2, t) Stable temperature filed or steady temperature field The temperature in the temperature field is only the function of positional coordinates i.e. TT(x,y, Plane temperature field: The temperature in temperature field only changes ith two positional coordinates i.e. T=T O x,y,t) 7
7 §6-1 The Basic Concept of Temperature Field And Heat Conduction 1.The temperature field: The total of the temperature at all the points in a elastic body at a certain moment, denoted by T. Unstable temperature filed or nonsteady temperature field: The temperature in the temperature field changes with time. i.e. T=T（x，y，z，t） Stable temperature filed or steady temperature field: The temperature in the temperature field is only the function of positional coordinates. i.e. T=T（x，y，z） Plane temperature field: The temperature in temperature field only changes with two positional coordinates. i.e. T=T（x，y，t）
温动向边签解法 §6-1温度场和热传导的基本概念 1.温度场:在任一瞬时,弹性体内所有各点的温度值的总体。用 7表示。 不稳定温度场或非定常温度场:温度场的温度随时间而变化。 即77(x,y,z,t) 稳定温度场或定常温度场:温度场的温度只是位置坐标的函数。 即=(x,y,z) 平面温度场:温度场的温度只随平面内的两个位置坐标而变。 即7=7(x,J,t)
8 §6-1 温度场和热传导的基本概念 1.温度场：在任一瞬时，弹性体内所有各点的温度值的总体。用 T表示。 不稳定温度场或非定常温度场：温度场的温度随时间而变化。 即 T=T（x，y，z，t） 稳定温度场或定常温度场：温度场的温度只是位置坐标的函数。 即 T=T（x，y，z） 平面温度场：温度场的温度只随平面内的两个位置坐标而变。 即 T=T（x，y，t）
The Basic Soltion he demerara es Problems 2. Isothermal surface: The surface that connects all the points with the same temperature in the temperature field at a +2△T certain moment. apparently, the temperature doesnt changes along the T+△T isothermal surface; The changing rate is T-△T the largest along the normal direction of the isothermal surface 3.Temperature gradient: The vector that points to the direction in which temperature increase along the normal direction of the isothermal surface It is denoted by t and its value is denoted by ar an, where n is the normal direction of the isothermal surface. The components of temperature gradient at each coordinate are 9
9 2.Isothermal surface: The surface that connects all the points with the same temperature in the temperature field at a certain moment. Apparently, the temperature doesn’t changes along the isothermal surface; The changing rate is the largest along the normal direction of the isothermal surface. T+2△T T+△T T T-△T o x y 3.Temperature gradient:The vector that points to the direction in which temperature increase along the normal direction of the isothermal surface. It is denoted by △T, and its value is denoted by , where n is the normal direction of the isothermal surface. The components of temperature gradient at each coordinate are n T
温动向边签解法 2.等温面:在任一瞬时,连接温度 场内温度相同各点的曲面。显然, w2△T沿着等温面,温度不变;沿着等温 T+△T 面的法线方向,温度的变化率最大 3.温度梯度:沿等温面的法线方向,指向温度增大方向的矢 T 量。用△T表示,其大小用动表示。其中n为等温面的法线方 向。温度梯度在各坐标轴的分量为 10
10 2.等温面：在任一瞬时，连接温度 场内温度相同各点的曲面。显然， 沿着等温面，温度不变；沿着等温 面的法线方向，温度的变化率最大。 T+2△T T+△T T T-△T o x y n T 3.温度梯度：沿等温面的法线方向，指向温度增大方向的矢 量。用△T表示，其大小用 表示。其中n为等温面的法线方 向。温度梯度在各坐标轴的分量为