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物理代写|结构力学代写Structural Mechanics代考|CE262 The Deformation Map

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物理代写|结构力学代写Structural Mechanics代考|The Deformation Map

The description of the geometry of deformation must begin with a description of the body in question. For our purposes it is sufficient to imagine a continuous, solid body located in three-dimensional space. We must be able to completely characterize the geometry of the body in some configuration in order to make any headway in describing the geometry of deformation. We will call the known geometry the reference configuration. The reference configuration is often taken to be the unstressed and unstrained configuration of the body, although such a restriction is not necessary. Our primary assumption about the initial geometry is that in this configuration we know the position of every point in the body. A second, equally crucial, assumption is that the body is continuous (as opposed to, say, a collection of discrete particles). The assumption of continuity will allow us to use the tools of differential geometry.

Our prototypical body $\mathscr{B}$ is illustrated in Fig. 28. The initial geometry has two basic features: the domain, which is everything inside the body, and the boundary, which is the surface of the body. In the reference configuration $\mathscr{B}$ $\left{z_{1}, z_{2}, z_{3}\right}$ relative to the origin of the coordinate system. The vector pointing from the origin of coordinates to the point $\mathscr{P}$ is called the position vector $\mathbf{z}$.

物理代写|结构力学代写Structural Mechanics代考|The Stretch of a Curve

Our imaginary curve is a good starting point for the definition of strain in a three-dimensional body because we can examine the change in length of this line under the action of the deformation map. From elementary considerations we already know what strain means for the stretching of a line. The arbitrariness of the choice of our curve will allow us to generalize our concept of strain to three dimensions. Let us examine the change in length of the curve $\mathrm{C}$ between two points.

Consider two points on the curve $\mathcal{C}$, one described by the position vector $\mathbf{z}(s)$ and the other by the position vector $\mathbf{z}(s+\Delta s)$, as shown in Fig. 30. The vector connecting the first point to the second is $\Delta \mathbf{z} \equiv \mathbf{z}(s+\Delta s)-\mathbf{z}(s)$, and the length of this vector measures the straight-line distance between the two points. The two points are mapped to the positions $\mathbf{x}(s)$ and $\mathbf{x}(s+\Delta s)$, respectively, in the deformed configuration. The vector connecting the two points in the deformed configuration is $\Delta \mathbf{x} \equiv \mathbf{x}(s+\Delta s)-\mathbf{x}(s)$, and the length of this vector measures the straight-line distance between the two points. In the limit as $\Delta s \rightarrow 0$, the straight-line distance between two points and the distance measured along the arc become equal. Hence, in the limit, the lengths of the vectors $\Delta \mathbf{z}$ and $\Delta \mathbf{x}$ are appropriate measures of the lengths of the respective curves.
In the limit as $\Delta s \rightarrow 0$, the length of the vector $\mathbf{z}(s+\Delta s)-\mathbf{z}(s)$ approaches zero, but the ratio of the length of the vector to the length of the arc approaches unity. Taking the limit of this ratio as $\Delta s \rightarrow 0$, we obtain the expression for the tangent vector to the curve
$$\lim _{\Delta s \rightarrow 0} \frac{\mathbf{z}(s+\Delta s)-\mathbf{z}(s)}{\Delta s}=\frac{d \mathbf{z}}{d s}$$
Thus, the derivative of a position vector along a curve is always tangent to the curve. If it is normalized with respect to the measure of distance along the curve, then it is always a unit vector because as $\Delta s \rightarrow 0$, the secant line length approaches the arc length, i.e., $\Delta s \rightarrow|\mathbf{z}(s+\Delta s)-\mathbf{z}(s)|$.

物理代写|结构力学代写STRUCTURAL MECHANICS代考|THE DEFORMATION MAP

asopposedto, say, acollectionofdiscreteparticles.连续性假设将允许我们使用微分几何工具。 坐标系的原点。从坐标原点指向该点的向量 $\mathscr{P}$ 称为位置向量z.

物理代写|结构力学代写STRUCTURAL MECHANICS代考|THE STRETCH OF A CURVE

$\Delta \mathrm{x} \equiv \mathbf{x}(s+\Delta s)-\mathbf{x}(s)$ ，并且这个向量的长度测量了两点之间的直线距离。在极限为 $\Delta s \rightarrow 0$ ，两点之间的直线距离与沿圆弧测量的距离相等。因此，在极限 情况下，向量的长度 $\Delta \mathrm{z}$ 和 $\Delta \mathrm{x}$ 是相应曲线长度的适当度量。

$$\lim _{\Delta s \rightarrow 0} \frac{\mathbf{z}(s+\Delta s)-\mathbf{z}(s)}{\Delta s}=\frac{d \mathbf{z}}{d s}$$

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。