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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|Three-dimensional deformation

In general, material points in a deformable body are displaced in the threedimensional space when subjected to external effects. The deformation vector is expressed as follows:
$$\vec{u}=u_{x} \hat{i}+u_{y} \hat{j}+u_{z} \hat{k}$$
where $u_{x}, u_{y}$ and $u_{z}$ are the projections of $\vec{u}$ onto the $x, y$, and $z$ axes, respectively. Note that in case of three-dimensional deformation, these variables are functions of all three coordinate directions, i.e. $u_{x}=u_{x}(x, y, z), u_{y}=u_{y}(x, y, z)$ and $u_{z}=u_{z}(x, y, z)$. In vector notation, the deformation vector is given as follows:
$${u}=\left{\begin{array}{lll} u_{x} & u_{y} & u_{z} \end{array}\right}^{T}$$
It can be shown that, in three-dimensions, strain components are defined by the following relationships,
\begin{aligned} &\varepsilon_{x x}=\frac{\partial u_{x}}{\partial x}, \varepsilon_{y y}=\frac{\partial u_{y}}{\partial y}, \varepsilon_{z z}=\frac{\partial u_{z}}{\partial z} \ &\gamma_{x y}=\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{\partial x}, \quad \gamma_{y z}=\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}, \gamma_{z x}=\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{\partial z} \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

An elastic deformation should not cause holes in a deformable body that does not have any holes before deformation. Moreover, no material overlap should be predicted by the displacement field. The strain compatibility conditions ensure that these constraints are satisfied [7].

In a planar deformation, where $u_{x}=u_{x}(x, y), u_{y}=u_{y}(x, y)$ and $u_{z}=0$, consider the following combination of the strains,
$$\frac{\partial^{2} \varepsilon_{y y}}{\partial x^{2}}+\frac{\partial^{2} \varepsilon_{x x}}{\partial y^{2}}-\frac{\partial^{2} \gamma_{x y}}{\partial x \partial y}$$

Using the definitions given in Eq. (2.47) we will find,
$$\frac{\partial^{3} u_{y}}{\partial x^{2} \partial y}+\frac{\partial^{3} u_{x}}{\partial y^{2} \partial x}-\frac{\partial^{2}}{\partial x \partial y}\left(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y}\right)=0$$
Thus we see that the relationship (a) must be equal to zero. This is the strain compatibility equation for a two-dimensional deformation in the $x, y$ plane, which imposes a specific relationship between the strains and the strain-displacement relationships.

For three-dimensional deformations where $u_{x}=u_{x}(x, y, z), u_{y}=u_{y}(x, y, z)$ and $u_{z}=u_{z}(x, y, z)$ there are a total of six strain compatibility conditions. These can bé found as follows:
$$\begin{array}{ll} \frac{\partial^{2} \varepsilon_{y y}}{\partial x^{2}}+\frac{\partial^{2} \varepsilon_{x x}}{\partial y^{2}}-\frac{\partial^{2} \gamma_{x y}}{\partial x \partial y}=0 \ \frac{\partial^{2} \varepsilon_{y y}}{\partial z^{2}}+\frac{\partial^{2} \varepsilon_{z z}}{\partial y^{2}}-\frac{\partial^{2} \gamma_{y z}}{\partial z \partial y}=0 & \text { (2.52b) } \ \frac{\partial^{2} \varepsilon_{z z}}{\partial x^{2}}+\frac{\partial^{2} \varepsilon_{x x}}{\partial z^{2}}-\frac{\partial^{2} \gamma_{z y}}{\partial x \partial z}=0 \ 2 \frac{\partial^{2} \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \ 2 \frac{\partial^{2} \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \ 2 \frac{\partial^{2} \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right) \end{array}$$

# 有限元方法代考

## 数学代写|有限元方法代写Finite Element Method代考|Three-dimensional deformation

$$\vec{u}=u_{x} \hat{i}+u_{y} \hat{j}+u_{z} \hat{k}$$ $u_{z}=u_{z}(x, y, z)$. 在矢量符号中，变形矢量如下:
$\backslash 1 \mathrm{eft}$ 的分隔符缺失或无法识别

$$\varepsilon_{x x}=\frac{\partial u_{x}}{\partial x}, \varepsilon_{y y}=\frac{\partial u_{y}}{\partial y}, \varepsilon_{z z}=\frac{\partial u_{z}}{\partial z} \quad \gamma_{x y}=\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{\partial x}, \quad \gamma_{y z}=\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}, \gamma_{z x}=\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{\partial z}$$

## 数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

$$\frac{\partial^{2} \varepsilon_{y y}}{\partial x^{2}}+\frac{\partial^{2} \varepsilon_{x x}}{\partial y^{2}}-\frac{\partial^{2} \gamma_{x y}}{\partial x \partial y}$$

$$\frac{\partial^{3} u_{y}}{\partial x^{2} \partial y}+\frac{\partial^{3} u_{x}}{\partial y^{2} \partial x}-\frac{\partial^{2}}{\partial x \partial y}\left(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y}\right)=0$$

## 有限元方法代写

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## MATLAB代写

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