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物理代写|断裂力学代写Fracture mechanics代考|ME836 The Mohr’s circle

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物理代写|断裂力学代写Fracture mechanics代考|The Mohr’s circle

As seen before, the state of stresses depends on the loads, body geometry, and the coordinate system orientation. Since the coordinate system orientation is arbitrarily chosen, the state of stresses may have an endless number of equivalent components, one for any possible orientation. This means that for the same body, with the same loads, the state of stresses changes if the orientation of the coordinate system is changed, but the stress tensor must be equivalent. Fig. $1.9$ intuitively shows this idea; notice that the body is the same, so is the load, but the stress tensor changes when the orientation of the coordinate system is changed. In the left figure, the $x$ axis is parallel to the applied force, so the stress tensor has only the component $\sigma_{x x}$, which corresponds to uniaxial tension. But, if the coordinate system is rotated around the $z$ axis, as in the right figure, the reaction forces resulting of $P$ on the cube’s faces produce four stress components, two of tension and two of shear on the $x, \gamma$ plane, thus giving a plane stress state.

The procedure to calculate the state of stresses when the orientation of the coordinate system is rotated is called stress transformation. The stress transformation equations in two dimensions are

$$\begin{gathered} \sigma_{x^{\prime} x^{\prime}}=\frac{\sigma_{x x}+\sigma_{\gamma y}}{2}+\left(\frac{\sigma_{x x}-\sigma_{\gamma \gamma}}{2}\right) \operatorname{Cos} 2 \theta+\tau_{x y} \operatorname{Sin} 2 \theta \ \sigma_{\gamma^{\prime} \gamma^{\prime}}=\frac{\sigma_{x x}+\sigma_{\gamma y}}{2}-\left(\frac{\sigma_{x x}-\sigma_{y \gamma}}{2}\right) \operatorname{Cos} 2 \theta-\tau_{x y} \operatorname{Sin} 2 \theta \ \tau_{x y^{\prime} y^{\prime}}=\frac{\sigma_{y \gamma}-\sigma_{x x}}{2} \operatorname{Sin} 2 \theta+\tau_{x y} \operatorname{Cos} 2 \theta \end{gathered}$$
Where the symbol (‘) indicates the stress component after rotation. Notice that:
$$\sigma_{x x}+\sigma_{\gamma \gamma}=\sigma_{x^{\prime} x^{\prime}}+\sigma_{\gamma^{\prime} \gamma^{\prime}}$$
This is known as first stress invariant, which can be extended to three dimensions and states that the sum of the normal stress components is constant.

物理代写|断裂力学代写Fracture mechanics代考|Yield criteria

In uniaxial tension, it is known that when a material reaches its yield strength, it will start to plastically strain. However, in practical situations, it is common to find a combined state of stresses that make plastic strain initiate at a stress different to the yield strength. The way to calculate whether there is yielding under a combined state of stresses is called Yield Criterion. The two most well-known yield criteria are Tresca’s and Von Mises’s, which are described next:

Tresca’s criterion: Tresca’s criterion, known also as the maximum shear stress criterion, establishes that the plastic strain will initiate when the maximum shear stress surpasses a critical value. According to Mohr’s circle, the maximum shear stress is the difference between the maximum and minimum principal stresses, thus, Tresca’s criterion is expressed by the following equation:
$$\sigma_1-\sigma_3=\sigma_0$$
Von Mises’s criterion: This criterion states that yielding starts when the effective stress reaches a critical value, and is expressed as
$$\sigma_0=\frac{1}{\sqrt{2}} \sqrt{\left(\sigma_1-\sigma_2\right)^2+\left(\sigma_2-\sigma_3\right)^2+\left(\sigma_1-\sigma_3\right)^2}$$

物理代写|断裂力学代写Fracture mechanics代考|The Mohr’s circle

$$\begin{gathered} \sigma_{x^{\prime} x^{\prime}}=\frac{\sigma_{x x}+\sigma_{\gamma y}}{2}+\left(\frac{\sigma_{x x}-\sigma_{\gamma \gamma}}{2}\right) \operatorname{Cos} 2 \theta+\tau_{x y} \operatorname{Sin} 2 \theta \ \sigma_{\gamma^{\prime} \gamma^{\prime}}=\frac{\sigma_{x x}+\sigma_{\gamma y}}{2}-\left(\frac{\sigma_{x x}-\sigma_{y \gamma}}{2}\right) \operatorname{Cos} 2 \theta-\tau_{x y} \operatorname{Sin} 2 \theta \ \tau_{x y^{\prime} y^{\prime}}=\frac{\sigma_{y \gamma}-\sigma_{x x}}{2} \operatorname{Sin} 2 \theta+\tau_{x y} \operatorname{Cos} 2 \theta \end{gathered}$$

$$\sigma_{x x}+\sigma_{\gamma \gamma}=\sigma_{x^{\prime} x^{\prime}}+\sigma_{\gamma^{\prime} \gamma^{\prime}}$$

物理代写|断裂力学代写断裂力学代考|屈服准则

.

Tresca准则:Tresca准则又称最大剪应力准则，它规定当最大剪应力超过一个临界值时，塑性应变将启动。根据莫尔圆，最大剪应力为最大主应力与最小主应力之差，因此Tresca准则用以下公式表示:
$$\sigma_1-\sigma_3=\sigma_0$$
Von Mises准则:该准则表示有效应力达到临界值时开始屈服，表示为
$$\sigma_0=\frac{1}{\sqrt{2}} \sqrt{\left(\sigma_1-\sigma_2\right)^2+\left(\sigma_2-\sigma_3\right)^2+\left(\sigma_1-\sigma_3\right)^2}$$

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。