## 数学代写|有限元方法代写Finite Element Method代考|GENG5514

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law for isotropic materials with symmetric stress and strain tensors

In case the material is elastically isotropic and the stress and strain tensors are symmetric the material behavior can be characterized with two material constants,
E: Elastic modulus or Young’s modulus
v: Poisson’s ratio
For a three-dimensional problem, it can be shown that the following relationships exist between the stresses and strains,
\begin{aligned} \varepsilon_{x x} & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y} & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z} & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \tau_{x y} & =G \gamma_{x y} \ \tau_{y z} & =G \gamma_{y z} \ \tau_{z x} & =G \gamma_{z x} \end{aligned}
where shear modulus $G=E / 2(1+v)$.

Note that Eq. (2.61a) can be inverted and expressed as follows:
\begin{aligned} \sigma_{x x} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{x x} \ \sigma_{y y} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{y y} \ \sigma_{z z} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{z z} \ \tau_{x y} & =\mu \gamma_{x y} \ \tau_{y z} & =\mu \gamma_{y z} \ \tau_{z x} & =\mu \gamma_{z x} \end{aligned}
where, the Lamé constants are defined as follows:
\begin{aligned} & \lambda=\frac{v E}{(1+v)(1-2 v)} \ & \mu=G \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Effects of initial stress/strain and thermal strain

Thermal stress in a one-dimensional problem: Consider a long and slender bar of length $L$ and initial temperature $T^{(0)}$. If the temperature of the bar is changed by $\Delta T$, material points in the bar would experience thermal strain proportional to the temperature change,
$$\varepsilon^{(t h)}=\alpha \Delta T$$
the proportionality constant $\alpha$ is a material property known as the coefficient of thermal expansion with units of $\mathrm{K}^{-1}$ or $\left({ }^{\circ} \mathrm{C}\right)^{-1}$. If the bar is not constrained on its ends, its length will change by an amount,
$$\Delta L=\int_0^L \alpha \Delta T d x$$
but no internal stress will develop.

On the other hand if both ends of the bar are constrained, internal forces and hence stress will develop in the bar. If such constraint conditions exist, the thermal stress in the bar can be found from Hooke’s law as follows:
$$\sigma^{(t h)}=E \alpha \Delta T$$
Next, consider a constrained bar subjected to external forces and change of temperature. The total strain in this bar can be found by using the superposition of the mechanical component of the strain and the thermal strain,
$$\varepsilon=\frac{\sigma}{E}+\varepsilon^{(t h)}=\frac{\sigma}{E}+\alpha \Delta T$$
The inverse of this relation gives the corresponding total stress,
$$\sigma=E(\varepsilon-\alpha \Delta T)$$
Generalized stress-strain relations with thermal effects: For materials with isotropic material properties temperature change only causes normal strain in the material. The stress-strain relations for a three-dimensional isotropic material subjected to a temperature change $\Delta T$ are expressed as follows [8]:
\begin{aligned} \varepsilon_{x x}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \gamma_{x y} & =\frac{\tau_{x y}}{G} \ \gamma_{y z} & =\frac{\tau_{y z}}{G} \ \gamma_{z x} & =\frac{\tau_{z x}}{G} \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law for isotropic materials with symmetric stress and strain tensors

E:弹性模量或杨氏模量
v:泊松比

\begin{aligned} \varepsilon_{x x} & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y} & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z} & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \tau_{x y} & =G \gamma_{x y} \ \tau_{y z} & =G \gamma_{y z} \ \tau_{z x} & =G \gamma_{z x} \end{aligned}

\begin{aligned} \sigma_{x x} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{x x} \ \sigma_{y y} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{y y} \ \sigma_{z z} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{z z} \ \tau_{x y} & =\mu \gamma_{x y} \ \tau_{y z} & =\mu \gamma_{y z} \ \tau_{z x} & =\mu \gamma_{z x} \end{aligned}

\begin{aligned} & \lambda=\frac{v E}{(1+v)(1-2 v)} \ & \mu=G \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Effects of initial stress/strain and thermal strain

$$\varepsilon^{(t h)}=\alpha \Delta T$$

$$\Delta L=\int_0^L \alpha \Delta T d x$$

$$\sigma^{(t h)}=E \alpha \Delta T$$

$$\varepsilon=\frac{\sigma}{E}+\varepsilon^{(t h)}=\frac{\sigma}{E}+\alpha \Delta T$$

$$\sigma=E(\varepsilon-\alpha \Delta T)$$

\begin{aligned} \varepsilon_{x x}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \gamma_{x y} & =\frac{\tau_{x y}}{G} \ \gamma_{y z} & =\frac{\tau_{y z}}{G} \ \gamma_{z x} & =\frac{\tau_{z x}}{G} \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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 be found as follows:
\begin{aligned} & \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 \ & \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{aligned}

## 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 x 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 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.

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

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

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

\begin{aligned} & \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 \ & \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{aligned}

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

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

\begin{aligned} \sigma_{x x}= & c_{x x x x} \varepsilon_{x x}+c_{x x x y} \gamma_{x y}+c_{x x x 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 z} \varepsilon_{z z} \end{aligned}

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

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|MECH3300

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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 be found as follows:
\begin{aligned} & \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 \ & \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{aligned}

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

As stated in the introduction to Section 2.2, when a deformable body is subjected to external effects such as external forces and/or imposed displacements on its boundary, its shape will change and internal forces will develop throughout its volume. The level of deformation for given external effects depends on the material of the deformable body. Constitutive relations are empirically obtained, material specific relationships between the stress and the strain in the body. Here we are primarily interested in linear elastic relationships.

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,
$$\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j$$
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|Strain compatibility conditions

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

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

\begin{aligned} & \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 \ & \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{aligned}

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

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

$$v=-\frac{\varepsilon_t}{\varepsilon_l}$$

$$\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j$$

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

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。