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# 物理代写|断裂力学代写Fracture mechanics代考|MAE561 The crack tip opening displacement

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## 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

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