Posted on Categories:Fracture mechanics, 断裂力学, 物理代写

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

物理代写|断裂力学代写Fracture mechanics代考|The R-curve

The graphical representation of the energy criterion is known as R-curve, whose construction procedure is as follows. The $G$ and $R$ values are plotted in the vertical axis, and the crack length is plotted in the horizontal axis, where the initial crack size $\left(a_1\right)$ is plotted to the left and the crack extension ( $\Delta$ a) to the right. The $G, R$ axis intersects the crack size axis at $\Delta a=0$. As mentioned in the previous section, in plane stress condition,
$$G=\frac{K^2}{E}$$
For an infinite plate with a central crack under plane stress, the stress intensity factor is $K_I=\sigma \sqrt{ }(\pi a)$; therefore, $G$ becomes
$$G=\frac{\pi \sigma^2 a}{E}$$
Plotting the above equation on the R-curve shows that the variation of $G$ is a straight line with a slope equal to $\pi \sigma^2 / E$. If $R$ is constant and independent from the crack size, its plot is a horizontal straight line. The resulting R-curve is schematically shown in Fig. 3.4.

物理代写|断裂力学代写Fracture mechanics代考|Plane strain fracture toughness

It has been experimentally observed that fracture toughness $\left(K_C\right)$ varies with the thickness, as schematically shown in Fig. 3.7. The most important feature of this behavior is that $K_C$ is constant for thick specimens in plain strain condition, thus $K_C$ becomes material property identified as plane strain fracture toughness, represented by the symbol $K_{I C}$, since it is determined in crack opening Mode I. It is relevant to remark that the plane strain $K_{I C}$ is also the minimum value of $K_C$ with respect to the thickness, while the maximum $K_C$ occurs under plane stress and plane strain combinations. Although the exact cause for such behavior is not clear yet, it is likely to be due to the plasticity effects combined with the plastic zone size and shape, since these characteristics induce more energy consumption at the onset of instability and crack propagation.

Irwin introduced an expression to calculate the plane stress fracture toughness based on the plane strain fracture toughness that has the following form:
$$K c=K_{I C}\left[1+\frac{1.4}{B^2}\left(\frac{K_{I C}}{\sigma_0}\right)^4\right]^{0.5}$$

Furthermore, it has been experimentally found that the plane strain condition occurs at thicknesses $(B)$ that meet the condition:
$$B>2.5 K_{I C}^2 / \sigma_0$$

物理代写|断裂力学代写骨折力学代考| r曲线

. r曲线 .物理代写|

$$G=\frac{K^2}{E}$$

$$G=\frac{\pi \sigma^2 a}{E}$$

物理代写|断裂力学代写断裂力学代考|平面应变断裂韧性

Irwin引入了基于平面应变断裂韧性计算平面应力断裂韧性的表达式，其形式如下:
$$K c=K_{I C}\left[1+\frac{1.4}{B^2}\left(\frac{K_{I C}}{\sigma_0}\right)^4\right]^{0.5}$$

$$B>2.5 K_{I C}^2 / \sigma_0$$

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。

Posted on Categories:Fracture mechanics, 断裂力学, 物理代写

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

物理代写|断裂力学代写Fracture mechanics代考|The crack tip opening displacement

The crack opening displacement (COD) is the separation of the fracture surfaces along the load line, while its corresponding value at the crack tip is referred as the crack tip opening displacement (CTOD). In perfect elastic conditions, the CTOD should be zero, but the plastic zone causes a stretching of the material at the crack tip, so the CTOD is greater than zero as schematically depicted in Fig. $2.22 .$

To calculate the COD, the displacement in the vertical direction $(v)$ given by the following equation:
$$v=2(1+v) \frac{K_I}{E} \sqrt{\frac{r}{2 \pi}} \sin \frac{\theta}{2}\left[2-2 v+\cos ^2 \frac{\theta}{2}\right]$$
is calculated by substituting $\theta=\pi$ and $r=-a$ (to avoid misunderstandings, the polar coordinate $r$ is replaced by $x$, and using $K_I=\sigma \sqrt{ }(\pi a)$; by doing so, the following expression for $\mathrm{COD}$ is obtained:
$$\mathrm{COD}=2 v=\frac{4 \sigma}{E} \sqrt{a^2-x^2}$$
where $E$ is the Young’s modulus and $\sigma$ is the applied stress. According to the previous formula, at the crack tip, $x=a$, the COD shall be zero, which is not correct, but using the Irwin’s correction for the crack size due to the presence of the plastic zone, the crack size is replaced by the effective crack size, thus:
$$a_{e f f}=a+r_p$$
Substituting $a_{e f f}$ into the COD equation:
$$\mathrm{CTOD}=\frac{4 \sigma}{E} \sqrt{\left(a+r_p^\right)^2-a^2}=\frac{4 \sigma}{E} \sqrt{2 a r_p^}$$
Substituting $r_p=(1 / 2 \pi)\left(K^2 / \sigma_0^2\right)$ and $a=(1 / \pi)\left(K^2 / \sigma^2\right)$ the CTOD becomes
$$\mathrm{CTOD}=\frac{4}{\pi} \frac{K_I^2}{\mathrm{E} \sigma_0}$$
where $E^{\prime}=E$ for plane stress and $E^{\prime}=\left(1-v^2\right) / E$ for plane strain. The following example shows the typical magnitude of the CTOD in common metallic materials.

物理代写|断裂力学代写Fracture mechanics代考|The energy criterion

As seen in Chapter 2, Griffith postulated that a crack grows at the expense of the stored elastic energy, demonstrating that fracture is a process that initiated when the energy conversion rate is equal to or greater than the energy demand to make the crack grow. Making use of such ideas, $\operatorname{Irwin}^1$ analyzed the conditions that led to crack propagation, but instead of using the fracture surface energy, he developed an approach based on the loaddisplacement behavior of an ellastically strained cracked body. Irwin called this reasoning the energy criterion.

To derive the energy criterion, consider an elastically strained plate, with a crack of length $a$, an applied load $P$, and a crack opening displacement along the load line $2 v$. Under such conditions, the curve $P$ versus $v$ is as shown in Fig. 3.1.

The line $\mathrm{OA}$ represents the $P$ versus $v$ behavior for the initial crack size (a), and the line $\mathrm{OB}$ represents the behavior after a crack extension $\Delta a$. The inverse slope of the line $P$ versus $v$ is the compliance, represented by the symbol $C$. As the crack extends, two cases may be observed:

Constant load: If at reaching the point $\mathrm{A}$, the load $P$ is fixed, and the crack has an extension $\Delta a$, the point of coordinates $(P, v)$ moves to B along a horizontal line, increasing $v$ while $P$ remains constant. If the plate is unloaded and reloaded, the $P$ versus $v$ record will be the line $\mathrm{OB}$, which has a lower slope than line OA, thus increasing the compliance.

物理代写|断裂力学代写断裂力学代考|裂纹尖端开口位移

$$v=2(1+v) \frac{K_I}{E} \sqrt{\frac{r}{2 \pi}} \sin \frac{\theta}{2}\left[2-2 v+\cos ^2 \frac{\theta}{2}\right]$$

$$\mathrm{COD}=2 v=\frac{4 \sigma}{E} \sqrt{a^2-x^2}$$

$$a_{e f f}=a+r_p$$

$$\mathrm{CTOD}=\frac{4 \sigma}{E} \sqrt{\left(a+r_p^\right)^2-a^2}=\frac{4 \sigma}{E} \sqrt{2 a r_p^}$$

$$\mathrm{CTOD}=\frac{4}{\pi} \frac{K_I^2}{\mathrm{E} \sigma_0}$$

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。

Posted on Categories:Fracture mechanics, 断裂力学, 物理代写

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

物理代写|断裂力学代写Fracture mechanics代考|Stress concentration

According to the definition of stress, a reduction in the cross-section area means an increment of the stress, in an amount proportional to the magnitude of the area reduction. However, the sharp changes of crosssection, as well as discontinuities, such as holes, grooves, gouges, etcetera, increase the stress beyond the magnitude given by the sole reduction of the cross-section area. This phenomenon is called stress concentration and is of great technological importance, because it increases the local stress above nominal applied stresses. The magnitude of the stress increment due to the stress concentration is called stress concentration factor, defined by the symbol $K_T$, therefore
$$K_T=\frac{\sigma_{\max }}{\sigma_{a p p}}$$
Where $\sigma_{\max }$ is the maximum stress value at the root of the stress concentrator, and $\sigma_{a p p}$ is the applied stress. One of the first stress concentration factors was obtained by Inglis in 1913 , for an infinite thin panel with an elliptic hole in the center. The expression for $K_T$ is
$$K_T=1+2\left(\frac{a}{b}\right)$$

物理代写|断裂力学代写Fracture mechanics代考|Definitions and basic concepts of fracture

Fracture is defined as the separation or fragmentation of a solid under the action of stresses through a process of creation of new surfaces that are the fracture surfaces. Normally, to fracture a material after yielding, it is necessary to increase the stress until a crack nucleation mechanism takes place, followed by crack growth and final separation. Depending on the load conditions, body geometry and the mechanical properties of the material, it may be necessary to increase the load after the crack initiation has started to further cause fracture, whereas in other cases, it will be sufficient to reach the point of crack initiation to have the crack to growing spontaneously. A very important circumstance is that the fracture can initiate from a preexisting crack or a severe stress concentrator, so the nucleation stage is suppressed, and the fracture process consists only of the crack propagation and final separation stages.

It is important to point out that in order to break a solid component, it is not necessary that the initiation and crack propagation conditions be present throughout the entire body’s volume, but since the fracture process occurs on a plane, it is enough to meet the fracture conditions in that single plane. This is similar to the chain principle that says, “To break a chain, it is enough to break a single link.” The fact that a fracture may initiate on a localized plane, or more precisely in a narrow region, where there may be stress concentrators or preexisting flaws, suggests that it may take place at stresses lower than the design stresses, all of this provide the fracture characteristics of being sudden, unexpected and, very often, catastrophic.

Depending on the amount of plastic strain before failure, two types of fractures are recognized, as shown in Fig. 1.16.

Brittle fracture: The strain in most of the body’s volume is elastic and the extent of plastic strain is negligible; so, if after the fracture the fragments are reassembled, the component will have the same initial geometry.

Ductile fracture: It is the fracture after an appreciable amount of plastic strain in most of the body’s volume and consequently, there is a permanent change of geometry. In components under tensile stresses, the plastic deformation close to the fracture surface will show a lateral contraction called neck.

Quasibrittle fracture: The plastic deformation zone is concentrated close to the fracture surface, so the fracture component displays a brittle appearance, but the fracture mechanism is essentially ductile.

物理代写|断裂力学代写骨折力学代考|应力集中

$$K_T=\frac{\sigma_{\max }}{\sigma_{a p p}}$$

$$K_T=1+2\left(\frac{a}{b}\right)$$

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。