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## 物理代写|结构力学代写Structural Mechanics代考|The spectral decomposition

The spectral decomposition. If the eigenvalues and eigenvectors are known, we can express the original tensor in terms of those objects in the following manner
$$\mathbf{T}=\sum_{i=1}^{3} \mu_{i} \mathbf{n}{i} \otimes \mathbf{n}{i}$$
Note that we need to suspend the summation convention because of the number of times that the index $i$ appears in the expression. This form of expression of the tensor $\mathbf{T}$ is called the spectral decomposition of the tensor. How do we know that the tensor $\mathbf{T}$ is equivalent to its spectral decomposition? As we indicated earlier, the operation of a second-order tensor is completely defined by its operation on three independent vectors. Let us assume that the eigenvectors $\left{\mathbf{n}{1}, \mathbf{n}{2}, \mathbf{n}{3}\right}$ are orthogonal (which means that any eigenvectors associated with repeated eigenvalues were orthogonalized). Let us examine how the tensor and its spectral decomposition operate on $\mathbf{n}{j}$
$$\mathbf{T n}{j}=\sum{i=1}^{3} \mu_{i}\left[\mathbf{n}{i} \otimes \mathbf{n}{i}\right] \mathbf{n}{j}=\sum{i=1}^{3} \mu_{i}\left(\mathbf{n}{j} \cdot \mathbf{n}{i}\right) \mathbf{n}{i}=\sum{i=1}^{3} \mu_{i} \delta_{i j} \mathbf{n}{i}=\mu{j} \mathbf{n}_{j}$$
Thus, we have concluded that both tensors operate the same way on the three eigenvectors. Therefore, the spectral representation must be equivalent to the original tensor. A corollary of the preceding construction is that any two tensors with exactly the same eigenvalues and eigenvectors are equivalent.

## 物理代写|结构力学代写Structural Mechanics代考|Vector and Tensor Calculus

A field is a function of position defined on a particular region. In our study of mechanics we shall have need of scalar, vector, and tensor fields, in which the output of the function is a scalar, vector, or tensor, respectively. For problems defined on a region of three-dimensional space, the input is the position vector x. A function defined on a three-dimensional domain, then, is a function of three independent variables (the components $x_{1}, x_{2}$, and $x_{3}$ of the position vector $\mathbf{x}$ ). In certain specialized theories (e.g., beam theory, plate theory, and plane stress) position will be described by one or two independent variables.

A field theory is a physical theory built within the framework of fields. The primary advantage of using field theories to describe physical phenomena is that the tools of differential and integral calculus are available to carry out the analysis. For example, we can appeal to concepts like infinitesimal neighborhoods and limits. And we can compute rates of change by differentiation and accumulations and averages by integration.

Figure 15 shows the simplest possible manifestation of a field: a scalar function of a scalar variable, $g(x)$. A scalar field can, of course, be represented as a graph with $x$ as the abscissa and $g(x)$ as the ordinate. For each value of position $x$ the function produces as output $g(x)$. The derivative of the function is defined through the limiting process as

$$\frac{d g}{d x} \equiv \lim _{\Delta x \rightarrow 0}\left(\frac{g(x+\Delta x)-g(x)}{\Delta x}\right) \equiv g^{\prime}(x)$$
The derivative has the familiar geometrical interpretation of the slope of the curve at a point and gives the rate of change of $g$ with respect to change in position $x$. Many of the graphical constructs that serve so well for scalar functions of scalar variables do not generalize well to vector and tensor fields. However, the concept of the derivative as the limit of the ratio of flux, $g(x+\Delta x)-g(x)$ in the present case, to size of the region, $\Delta x$ in the present case, will generalize for all cases.

## 物理代写|结构力学代写Structural Mechanics代考| The spectral decomposition

$$\mathbf{T}=\sum_{i=1}^{3} \mu_{i} \mathbf{n} i \otimes \mathbf{n} i$$

$$\mathbf{T} \mathbf{n} j=\sum i=1^{3} \mu_{i}[\mathbf{n} i \otimes \mathbf{n} i] \mathbf{n} j=\sum i=1^{3} \mu_{i}(\mathbf{n} j \cdot \mathbf{n} i) \mathbf{n} i=\sum i=1^{3} \mu_{i} \delta_{i j} \mathbf{n} i=\mu j \mathbf{n}{j}$$ 因此，我们得出结论，两个张量在三个特征向量上以相同的方式工作。因此，光谱表示必须等价于原始张量。上述构造的推论是， 任何两个具有完全相同特征值和特征向量的张量都是等价的。

## 物理代写结构力学代写Structural Mechanics代考| Vector and Tensor Calculus

$$\frac{d g}{d x} \equiv \lim _{\Delta x \rightarrow 0}\left(\frac{g(x+\Delta x)-g(x)}{\Delta x}\right) \equiv g^{\prime}(x)$$

## MATLAB代写

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。

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## 物理代写|结构力学代写Structural Mechanics代考|Tensor invariants

Tensor invariants. In subsequent chapters we will have occasions to wonder whether there are properties of the tensor components that do not depend upon the choice of basis. These properties will be called tensor invariants. The identities of Eqn. (43) will be useful in proving the invariance of these properties. The argument will go something like this: Let $f\left(T_{i j}\right)$ be a function of the components of the tensor $\mathbf{T}$. Under a change of basis, we can write this function in the form $f\left(Q_{i k} Q_{j i} T_{k 1}\right)$. If the function has the property that
$$f\left(Q_{i k} Q_{j l} T_{k l}\right)=f\left(T_{i j}\right)$$
then the function $f$ is a tensor invariant. Since it does not depend upon the coordinate system, we can say that it is an intrinsic function of the tensor $T$, and write $f(\mathrm{~T})$. Three fundamental tensor invariants are given by
$$f_{1}(\mathbf{T}) \equiv T_{i i} \quad f_{2}(\mathbf{T}) \equiv T_{i j} T_{j i} \quad f_{3}(\mathbf{T}) \equiv T_{i j} T_{j k} T_{k i}$$

## 物理代写|结构力学代写Structural Mechanics代考|Eigenvalues and eigenvectors of symmetric tensors

Eigenvalues and eigenvectors of symmetric tensors. A tensor has properties independent of any basis used to characterize its components. As we have just seen, the components themselves have mysterious properties called invariants that are independent of the basis that defines them. It seems reasonable to expect that we might be able to find a representation of a tensor that is canonical. Indeed, this canonical form is the spectral representation of the tensor that can be built from its eigenvalues and eigenvectors. In this section we shall build the mathematics behind the spectral representation of tensors.

Recall that the action of a tensor is to stretch and rotate a vector. Let us consider a symmetric tensor $\mathbf{T}$ acting on a unit vector $\mathbf{n}^{\dagger}$ If the action of the tensor is simply to stretch the vector but not to rotate it then we can express it as
$$\mathbf{T n}=\mu \mathbf{n}$$
where $\mu$ is the amount of the stretch. This equation, by itself, begs the question of existence of such a vector $\mathbf{n}$. Is there any vector that has the special property that action by $\mathrm{T}$ is identical to multiplication by a scalar? Is it possible that more than one vector has this property?

Equation (47) is called an eigenvalue problem. Eigenvalue problems show up all over the place in mathematical physics and engineering. The tensor in three dimensional space is a great context in which to explore the eigenvalue problem because the computations are quite manageable (as opposed to, say, solving the vibration eigenvalue problem of structural dynamics on a structure with a million degrees of freedom).

## 物理代写|结构力学代写Structural Mechanics代考| Tensor invariants

（43）的晅等式侍有助于证明这些性质的不变性。争论将变成䢒样: 让 $f\left(T_{i j}\right)$ 是张量分量的函数 $\mathbf{T}$. 在基数栾化的情况下，我们可 以将此函数编写为 $f\left(Q_{i k} Q_{j i} T_{k 1}\right)$. 如果函数具有以下属性:
$$f\left(Q_{i k} Q_{j l} T_{k l}\right)=f\left(T_{i j}\right)$$

$$f_{1}(\mathbf{T}) \equiv T_{i i} \quad f_{2}(\mathbf{T}) \equiv T_{i j} T_{j i} \quad f_{3}(\mathbf{T}) \equiv T_{i j} T_{j k} T_{k i}$$

## 物理代写|结构力学代写Structural Mechanics代考| Eigenvalues and eigenvectors of symmetric tensors

$$\mathbf{T n}=\mu \mathbf{n}$$

## MATLAB代写

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。

Posted on Categories:Structural Mechanics, 物理代写, 结构力学

## avatest™帮您通过考试

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## 物理代写|结构力学代写Structural Mechanics代考|The Geometry of Three-dimensional Space

We live in three-dimensional space, and all physical objects that we are familiar with have a three-dimensional nature to their geometry. In addition to solid bodies, there are basically three primitive geometric objects in three-dimensional space: the point, the curve, and the surface. Figure 1 illustrates these objects by taking a slice through the three-dimensional solid body $\mathscr{B}$ (a cube, in this case). A point describes position in space, and has no dimension or size. The point $\mathscr{P}$ in the figure is an example. The most convenient way to describe the location of a point is with a coordinate system like the one shown in the figure. A coordinate system has an origin $O$ (a point whose location we understand in a deeper sense than any other point in space) and a set of three coordinate directions that we use to measure distance. Here we shall confine our attention to Cartesian coordinates, wherein the coordinate directions are mutually perpendicular. The location of a point is then given by its coordinates $\mathbf{x}=$ $\left(x_{1}, x_{2}, x_{3}\right)$. A point has a location independent of any particular coordinate system. The coordinate system is generally introduced for the convenience of description or numerical computation.

A curve is a one-dimensional geometric object whose size is characterized by its arc length. In a sense, a curve can be viewed as a sequence of points. $\mathbf{A}$ curve has some other interesting properties. At each point along a curve, the curve seems to be heading in a certain direction. Thus, a curve has an orientation in space that can be characterized at any point along the curve by the line tangent to the curve at that point. Another property of a curve is the rate at which this orientation changes as we move along the curve. A straight line is a curve whose orientation never changes. The curve C exemplifies the geometric notion of curves in space.

A surface is a two-dimensional geometric object whose size is characterized by its surface area. In a certain sense, a surface can be viewed as a family of curves. For example, the collection of lines parallel and perpendicular to the curve $C$ constitute a family of curves that characterize the surface $\oiiint$. A surface can also be viewed as a collection of points. Like a curve, a surface also has properties related to its orientation and the rate of change of this orientation as we move to adjacent points on the surface. The orientation of a surface is completely characterized by the single line that is perpendicular to the tangent lines of all curves that pass through a particular point. This line is called the normal direction to the surface at the point. A flat surface is usually called a plane, and is a surface whose orientation is constant.

A three-dimensional solid body is a collection of points. At each point, we ascribe some physical properties (e.g., mass density, elasticity, and heat capacity) to the body. The mathematical laws that describe how these physical properties affect the interaction of the body with the forces of nature summarize our understanding of the behavior of that body. The heart of the concept of continuum mechanics is that the body is continuous, that is, there are no finite gaps between points. Clearly, this idealization is at odds with particle physics, but, in the main, it leads to a workable and useful model of how solids behave. The primary purpose of hanging our whole theory on the concept of the continuum is that it allows us to do calculus without worrying about the details of material constitution as we pass to infinitesimal limits. We will sometimes find it useful to think of a solid body as a collection of lines, or a collection of surfaces, since each of these geometric concepts builds from the notion of a point in space.

## 物理代写|结构力学代写Structural Mechanics代考|Vectors

A vector is a directed line segment and provides one of the most useful geometric constructs in mechanics. A vector can be used for a variety of purposes. For example, in Fig. 2 the vector $\mathbf{v}$ records the position of point $b$ relative to point $a$. We often refer to such a vector as a position vector, particularly when $a$ is the origin of coordinates. Close relatives of the position vector are displacement (the difference between the position vectors of some point at different times), velocity (the rate of change of displacement), and acceleration (the rate of change of velocity). The other common use of the notion of a vector, to which we shall appeal in this book, is the concept of force. We generally think of force as an action that has a magnitude and a direction. Likewise, displacements are completely characterized by their magnitude and direction. Because a vector possesses only the properties of magnitude (length of the line) and direction (orientation of the line in space), it is perfectly suited to the mathematical modeling of things like forces and displacements. Vectors have many other uses, but these two are the most important in the present context.

Graphically, we represent a vector as an arrow. The shaft of the arrow gives the orientation and the head of the arrow distinguishes the direction of the vector from the two possibilities inherent in the line segment that describes the shaft (i.e., line segments $a b$ and $b a$ in Fig. 2 are both oriented the same way in space). The length, or magnitude, of a vector $\mathbf{v}$ is represented graphically by the length of the shaft of the arrow and will be denoted symbolically as $|\mathbf{v}|$ throughout the book.

## MATLAB代写

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。