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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|Hooke’s law

The deformation behavior of a specific material is determined experimentally. These experiments are designed such that only one of the stress components and the corresponding strain dominates the problem. This state is known as a simpleloading state.

For linear, isotropic materials tensile loading of a slender test specimen, i.e., the simple-tension test, reveals two fundamental material properties. The relationship between the normal stress and the normal strain is found by conducting a simple-tension test, as follows:
$$\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z$$
where $E$ is the elastic modulus of the material, also referred to as the Young’s modulus. The relationship between the longitudinal strain $\varepsilon_{l}$ and the transverse strain $\varepsilon_{t}$ represents the Poisson’s ratio, the second material property,
$$v=-\frac{\varepsilon_{t}}{\varepsilon_{l}}$$
The simple-shear test reveals the relationship between the shear strain and the shear stress,
$\tau_{i j}=G \gamma_{i i} \quad$ for $\quad i, j=x, y, z \quad$ and $\quad i \neq j$
where $G$ is the shear modulus, or modulus of rigidity. For a linear, elastic, isotropic material the following relationship holds:
$$G=\frac{E}{2(1+v)}$$

## 数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

In previous sections it was indicated that, in general, the stress and strain tensors at a point have nine independent components each, if we do not take into account the symmetries. Therefore, the possibility exists for all of these 18 components to be interrelated. In it most general form, the linear elastic constitutive law, also known as generalized Hooke’s law, can be expressed as follows:
$$\sigma_{i j}=c_{i j r s} \varepsilon_{r s}$$
where the subscripts $i, j, r, s=x, y, z$ and the coefficients $c_{i j r s}$ are empirically determined. Note that the tensor notation is used in expressing Eq. (2.57) where $\sigma$ and $\varepsilon$ are second order tensors and $c_{i j r s}$ is a fourth order tensor [7]. Repeated indices imply summation, such that for $\sigma_{x x}$ the most general form of the Hooke’s law would be,
\begin{aligned} \sigma_{x x}=& c_{x x x x} \varepsilon_{x x}+c_{x x x y} \gamma_{x y}+c_{x x z z} \gamma_{x z}+c_{x x y x} \gamma_{y x}+c_{x x y y} \varepsilon_{y y}+c_{x x y z} \gamma_{y z}+c_{x x z x} \gamma_{z x} \ &+c_{x x z y} \gamma_{z y}+c_{x x z} \varepsilon_{z z} \end{aligned}

It can easily be deduced that 81 material properties would be required in case of an anisotropic material with no-symmetries in the strain and stress tensors. In matrix notation, Eq. (2.57) can be expressed as follows:
$${\sigma}=[E]{\varepsilon}$$
where $[E]$ is an $81 \times 81$ elasticity matrix and ${\sigma}$ and ${\varepsilon}$ are $9 \times 1$ vectors.

# 有限元方法代考

## 数学代写|有限元方法代写Finite Element Method代考|Hooke’s law

$$\sigma_{i i}=E_{\varepsilon_{i i}} \quad \text { for } \quad i=x, y, z$$

$$v=-\frac{\varepsilon_{t}}{\varepsilon_{l}}$$

$\tau_{i j}=G \gamma_{i i}$ 为了 $i, j=x, y, z$ 和 $i \neq j$

$$G=\frac{E}{2(1+v)}$$

## 数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

$$\sigma_{i j}=c_{i j r s} \varepsilon_{r s}$$

$$\sigma_{x x}=c_{x x x x} \varepsilon_{x x}+c_{x x x y} \gamma_{x y}+c_{x x z z} \gamma_{x z}+c_{x x y x} \gamma_{y x}+c_{x x y y} \varepsilon_{y y}+c_{x x y z} \gamma_{y z}+c_{x x z x} \gamma_{z x} \quad+c_{x x z y} \gamma_{z y}+c_{x x z} \varepsilon_{z z}$$

$$\sigma=[E] \varepsilon$$

## 有限元方法代写

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## 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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