Posted on Categories:Elasticity, 弹性力学, 物理代写

物理代写|弹性力学代写Elasticity代考|EM505 Cartesian tensors

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物理代写|弹性力学代写Elasticity代考|Cartesian tensors

Scalars, vectors, matrices, and higher-order quantities can be represented by a general index notational scheme. Using this approach, all quantities may then be referred to as tensors of different orders. The previously presented transformation properties of a vector can be used to establish the general transformation properties of these tensors. Restricting the transformations to those only between Cartesian coordinate systems, the general set of transformation relations for various orders can be written as
\begin{aligned} a^{\prime} &=a, \text { zero order (scalar) } \ a_i^{\prime} &=Q_{i p} a_p, \text { first order (vector) } \ a_{i j}^{\prime} &=Q_{i p} Q_{j q} a_{p q}, \text { second order (matrix) } \ a_{i j k}^{\prime} &=Q_{i p} Q_{j p} Q_{k r} a_{p q r}, \text { third order } \ a_{i j k l}^{\prime} &=Q_{i p} Q_{j q} Q_{k r} Q_{l s} a_{p q r s}, \text { fourth order } \ \vdots \ a_{i j k k m}^{\prime} &=Q_{i p} Q_{j q} Q_{k r} \cdots Q_{m t} a_{p q r \ldots t}, \text { general order } \end{aligned}
Note that, according to these definitions, a scalar is a zero-order tensor, a vector is a tensor of order one, and a matrix is a tensor of order two. Relations (1.5.1) then specify the transformation rules for the components of Cartesian tensors of any order under the rotation $Q_{i j}$. This transformation theory proves to be very valuable in determining the displacement, stress, and strain in different coordinate directions. Some tensors are of a special form in which their components remain the same under all transformations, and these are referred to as isotropic tensors. It can be easily verified (see Exercise 1.8) that the Kronecker delta $\delta_{i j}$ has such a property and is therefore a second-order isotropic tensor. The alternating symbol $\varepsilon_{i j k}$ is found to be the third-order isotropic form. The fourth-order case (Exercise 1.9) can be expressed in terms of products of Kronecker deltas, and this has important applications in formulating isotropic elastic constitutive relations in Section $4.2$.

物理代写|弹性力学代写Elasticity代考|Principal values and directions for symmetric second-order tensors

Considering the tensor transformation concept previously discussed, it should be apparent that there might exist particular coordinate systems in which the components of a tensor take on maximum or minimum values. This concept is easily visualized when we consider the components of a vector as shown in Fig. 1.1. If we choose a particular coordinate system that has been rotated so that the $x_3$-axis lies along the direction of the vector, then the vector will have components $v={0,0,|v|}$. For this case, two of the components have been reduced to zero, while the remaining component becomes the largest possible (the total magnitude).

This situation is most useful for symmetric second-order tensors that eventually represent the stress and/or strain at a point in an elastic solid. The direction determined by the unit vector $\boldsymbol{n}$ is said to be a principal direction or eigenvector of the symmetric second-order tensor $a_{i j}$ if there exists a parameter $\lambda$ such that
$$\begin{gathered} a_{i j} n_j=\lambda n_i \ a_{11} n_1+a_{12} n_2+a_{13} n_3=\lambda n_1 \ a_{21} n_1+a_{22} n_2+a_{23} n_3=\lambda n_2 \ a_{31} n_1+a_{32} n_2+a_{33} n_3=\lambda n_3 \end{gathered}$$

物理代写|弹性力学代写弹性代考|笛卡尔张量

\begin{aligned} a^{\prime} &=a, \text { zero order (scalar) } \ a_i^{\prime} &=Q_{i p} a_p, \text { first order (vector) } \ a_{i j}^{\prime} &=Q_{i p} Q_{j q} a_{p q}, \text { second order (matrix) } \ a_{i j k}^{\prime} &=Q_{i p} Q_{j p} Q_{k r} a_{p q r}, \text { third order } \ a_{i j k l}^{\prime} &=Q_{i p} Q_{j q} Q_{k r} Q_{l s} a_{p q r s}, \text { fourth order } \ \vdots \ a_{i j k k m}^{\prime} &=Q_{i p} Q_{j q} Q_{k r} \cdots Q_{m t} a_{p q r \ldots t}, \text { general order } \end{aligned}

物理代写|弹性力学代写弹性代考|对称二阶张量的主值和方向

$$\begin{gathered} a_{i j} n_j=\lambda n_i \ a_{11} n_1+a_{12} n_2+a_{13} n_3=\lambda n_1 \ a_{21} n_1+a_{22} n_2+a_{23} n_3=\lambda n_2 \ a_{31} n_1+a_{32} n_2+a_{33} n_3=\lambda n_3 \end{gathered}$$

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。