### 数学代写|有限元方法代写Finite Element Method代考|ENGR7961

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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|Stress transformations

The state of stress of the point $P^{\prime}$ is expressed by the stress tensor in the global, Cartesian, coordinates $(x, y, z)$ by Eq. (2.5). However, the choice of this coordinate system and the small volume $(d V=d x . d y . d z)$ is arbitrary. We would like to be able to transform the stress state between different orientations.
In order to develop these transformations, we consider an oblique crosssection of the hexahedron by a plane of arbitrary, but known orientation. This results in the tetrahedral volume on one side of the plane as shown in Fig. 2.3. The traction components $\vec{T}{x}, \vec{T}{y}, \vec{T}{z}$, and $\vec{T}{n}$, acting on the small triangular areas $\Delta A_{x}, \Delta A_{y}, \Delta A_{z}$ and $\Delta A_{n}$, respectively, must be in static equilibrium in order to keep the continuum whole. Let us look into the arbitrary plane in more detail before we state this equilibrium condition.

The orientation of the plane is identified by its outward unit normal $\vec{n}$, as shown in the figure. The unit normal is defined as follows:
$$\vec{n}=n_{x} \hat{i}+n_{y} \hat{j}+n_{z} \hat{k}$$
in Cartesian coordinate system. In vector notation, the unit normal is expressed as follows:
$${n}=\left{\begin{array}{lll} n_{x} & n_{y} & n_{z} \end{array}\right}^{T}$$
The components $n_{x}, n_{y}$, and $n_{z}$ are the direction cosines of $\vec{n}$ with respect to the $(x, y, z)$ axes, and have the property,
$$n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=1$$

## 数学代写|有限元方法代写Finite Element Method代考|Normal and shear components of tractions

Traction $\vec{T}{n}$ on an oblique plane $\vec{n}$ can be expressed by using the normal and shear components on the plane as follows: $$\vec{T}{n}=\sigma_{n n} \cdot \vec{n}+\sigma_{n t} \cdot \vec{t}$$
where the normal and tangential traction components are $\sigma_{n n}$ and $\sigma_{n t}$, respectively. These components can be found as follows:
$$\sigma_{n n}=\vec{T}{n} \cdot \vec{n} \quad \text { and } \quad \sigma{n t}=\vec{T}{n} \cdot \vec{t}$$ where $\vec{t}$ is a unit vector in the oblique the plane, which has the unit normal $\vec{n}$. The vector $\vec{t}$ has following components expressed in the global Cartesian system, $$\vec{t}=t{x} \hat{i}+t_{y} \hat{j}+t_{z} \hat{k} \quad \text { or } \quad{t}=\left{\begin{array}{lll} t_{x} & t_{y} & t_{z} \end{array}\right}^{T}$$
The normal component of the traction $\vec{T}{n}$ is found by using Eqs. (2.16) and (2.20) as follows: \begin{aligned} \sigma{n n}=& \vec{T}{n} \cdot \vec{n}=\left([\sigma]^{T} \cdot{n}\right) \cdot{n} \ =& {\left[\left(\sigma{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) \hat{i}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) \hat{j}\right.} \ &\left.+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) \hat{k}\right] \cdot\left(n_{x} \hat{i}+n_{y} \hat{j}+n_{z} \hat{k}\right) \end{aligned}
or, after rearranging,
\begin{aligned} \sigma_{n n}=&\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) n_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) n_{y} \ &+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) n_{z} \end{aligned}
Similarly, the tangential component of the traction $\vec{T}{n}$ is found as follows: \begin{aligned} \sigma{n t}=&\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) t_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) t_{y} \ &+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) t_{z} \end{aligned}
In fact, Eqs. (2.22) and (2.23) can be used to transform the stresses to any orientation $\vec{n}$ and $\vec{t}$.

## 数学代写|有限元方法代写Finite Element Method代考|Stress transformations

$$\vec{n}=n_{x} \hat{i}+n_{y} \hat{j}+n_{z} \hat{k}$$

${n}=|$ left $\left{\right.$ begin ${a r r a y}{| l}_{-}{x} \& n_{-}{y} \& n_{-}{z} \backslash$ lend ${a r r a y} \backslash r_{i g h t} \wedge \wedge{T}$

$$n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=1$$

## 数学代写|有限元方法代写Finite Element Method代考|Normal and shear components of tractions

$$\vec{T} n=\sigma_{n n} \cdot \vec{n}+\sigma_{n t} \cdot \vec{t}$$

$$\sigma_{n n}=\vec{T} n \cdot \vec{n} \quad \text { and } \quad \sigma n t=\vec{T} n \cdot \vec{t}$$

㸻引的法向分量 $\vec{T} n$ 是通过使用方程式找到的。(2.16) 和 (2.20) 如下:
$$\sigma n n=\vec{T} n \cdot \vec{n}=\left([\sigma]^{T} \cdot n\right) \cdot n=\quad\left[\left(\sigma x x n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) \hat{i}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) \hat{j}+\left(\tau_{x z}\right.\right.$$

$$\sigma_{n n}=\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) n_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) n_{y} \quad+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) n_{z}$$

$$\sigma n t=\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) t_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) t_{y} \quad+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) t_{z}$$

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