Chapter 3

Energy Release Rate

Griffith Postulation

Inglis found a solution for stress concentration effect in an elliptical hole.
英格利斯（Inglis）在椭圆孔中找到了应力集中效应的解。

According to Inglis solution, for a sharp crack where , will be much greater than the strength of the material, which is contrary to actual observation.
根据英格利斯解，对于尖锐的裂纹，其中，将远远大于材料的强度，这与实际观察相反。

Griffth Postulation

- A crack would not grow unless certain amount of energy is released while cracking to overcome the energy required to form two new surfaces.
- 除非在开裂时释放一定量的能量以克服形成两个新表面所需的能量，否则裂纹不会增长。

- From this postulation, a crack growth condition is stated:
- 根据这个假设，陈述了裂纹扩展条件：

(Total energy released when cracking (Released Energy) The energy required to form two new crack surfaces (Surface Energy))
（裂纹释放的总能量（释放能量）形成两个新裂纹表面所需的能量（表面能））

3.1.1 A simple model

Released Energy:

- It has been shown that for thin plates.
- 已经表明，对于薄板。

Thus,

Surface Energy:

- To grow a crack, new surfaces must be created by breaking the bonds that hold atoms together. The bond strength is the result of the attractive forces between atoms.
- 要长出裂缝，必须通过破坏将原子固定在一起的键来创建新的表面。键强度是原子之间吸引力的结果。

- The energy required to break the bonds along the growing crack surface is called “Surface Energy”.
- 沿不断增长的裂纹表面断裂键所需的能量称为“表面能”。

= Surface Energy per unit area (J/m²)
= 单位面积表面能 （J/m ² ）

is a material property
是一种材料属性

- The surface energy required in the plate is
- 板中所需的表面能为

two new surfaces are created.
将创建两个新曲面。

Table 3.1 Surface energy for some materials
表 3.1 某些材料的表面能

- From (2.1) and (2.2), it can be observed that
- 从（2.1）和（2.2）可以看出，

- At critical crack size, the rate of and against the incremental crack length () become the same:
- 在临界裂纹尺寸下，增量裂纹长度（）的速率和相对速率变得相同：

- Beyond this point, the crack will grow by itself.
- 超过这一点，裂缝会自行扩大。

- Thus the condition for self-crack growth is
- 因此，自裂纹生长的条件是

which means
这意味着

- The critical crack length is
- 临界裂纹长度为

- The critical stress :
- 临界应力：

※ For plane strain case
※ 平面应变情况

Energy release rate (Irwin, 1956)

Griffith provided the energy approach for cracking process. However, no useful parameter that can be used to determine crack expansion is suggested.
Griffith为裂解工艺提供了能量方法。但是，没有建议使用可用于确定裂纹扩展的有用参数。

Irwin suggested Energy Release Rate as a useful parameter for that purpose.
Irwin建议将能量释放率作为此目的的有用参数。

Energy Release Rate is a characterization of the driving force for crack growth.
能量释放速率是裂纹扩展驱动力的表征。

In general, when a crack grows in size:
一般来说，当裂缝变大时：

Elastic Strain Energy is changed ()

Stiffness of the component decreases
部件的刚度降低

Energy is consumed to create two new surfaces.
消耗能量以创建两个新表面。

External loads do work to the components ()
外部负载对组件 （） 起作用

For an incremental increase in the crack area , the total potential energy change is :
当裂纹面积逐渐增加时，总势能变化为：

The Energy Release Rate is
能量释放速率为

Energy is released from the system as the potential energy is decreased. With (B = thickness)
随着势能的降低，能量从系统中释放出来。带 （B = 厚度）

※ If the crack moves rapidly, some energy is being consumed to impart kinetic energy to the crack portions. In this case
※ 如果裂纹快速移动，则会消耗一些能量来将动能传递给裂纹部分。在这种情况下

where = kinetic energy

Energy Release Rate in the Forms of Compliance change
合规变化形式的能量释放率

Now consider a double cantilever beam (DCB)
现在考虑双悬臂梁 （DCB）

DCB with a constant load P (load controlled)

u: tip displacement of a cantilever
U：悬臂的尖端位移

${''W''}_{\mathrm{<ms\; style="font-family:\; Cambria\; Math;\; min-height:\; 14pt;\; font-size:\; 14pt;">ext</ms>}}''=Pu''$ Thus,

Now, (3.9)

G is determined by finding the rate of compliance change w.r.t a.
G 是通过求 w.r.t a 的合规性变化率来确定的。

The external work done by the constant load P → i.) half goes to deform the beam to a higher curvature increasing the strain energy, and ii.) the other half is released for the crack growth.
恒定载荷 P 所做的外部功→ i.）一半用于将梁变形为更高的曲率，增加应变能，以及 ii.）另一半被释放用于裂纹生长。

DCB with a fixed grip (displacement controlled)
带固定夹具的 DCB（位移控制）

Initial work stored:

$''U=''\frac{''1''}{''2''}''Pu''$

With the increase in the crack length, the cantilever acquires smaller curvature releasing strain energy. ()
随着裂纹长度的增加，悬臂获得更小的曲率，释放应变能。()

If the release is large enough to meet the demands of producing new surfaces, the crack will grow.
如果释放量足够大，可以满足产生新表面的需求，则裂纹会扩大。

Now, since $''\Delta ''{''W''}_{\mathrm{<ms\; style="font-family:\; Cambria\; Math;\; color:\; rgb(0,\; 0,\; 0);\; min-height:\; 14pt;\; font-size:\; 14pt;">ext</ms>}}''=P\Delta u''$=0,
现在，由于 $''\Delta ''{''W''}_{\mathrm{<ms\; style="font-family:\; Cambria\; Math;\; color:\; rgb(0,\; 0,\; 0);\; min-height:\; 14pt;\; font-size:\; 14pt;">ext</ms>}}''=P\Delta u''$ =0，

which is exactly the same with (3.9).
这与（3.9）完全相同。

Therefore, G is the same both for load-controlled and displacement-controlled.
因此，对于负载控制和位移控制，G 是相同的。

G in terms of compliance change provides convenient way for tests.
G在合规性变化方面为测试提供了方便的方法。

: will be shown in the next section
：将在下一节中显示

If both P and u change simultaneously, expression (3.9) is still valid.
如果 P 和 U 同时更改，则表达式 （3.9） 仍然有效。

The continuous curve can be treated as the integration of small steps which are load-controlled (displacement controlled)
连续曲线可以看作是负载控制（位移控制）的小步骤的积分

Finding G:

G in terms of Strain Energy Change
G 表示应变能变化

DCB with load-controlled (const. P)
带负载控制的 DCB（常量 P）

where

Thus,

→ half of the external work is used to increase the strain energy.
→一半的外部功用于增加应变能。

DCB with displacement controlled (const. u)
位移控制的 DCB（const. u）

In this case
在这种情况下

Thus,

→ already existing strain energy is decreased as the crack advances.
→已经存在的应变能随着裂纹的推进而降低。

Example 3.1 Determine G for an edge crack loaded as shown below:
例 3.1 确定加载的边缘裂纹的 G，如下所示：

where ,

Thus

Now

Energy Release Rate of DCB specimen
DCB试样的能量释放率

DCB specimen is frequently used to determine G for a specific material.
DCB试样经常用于测定特定材料的G。

The tip deflection of a cantilever with a free-end load P:
具有自由端载荷 P 的悬臂的尖端挠度：

For a DCB, same produces a displacement twice the deflection of one cantilever:
对于 DCB，同样产生的位移是悬臂挠度的两倍：

Using (a) in (3.9)

※ Energy Release Rate depends on the capacity of the body to store strain energy
※ 能量释放率取决于身体储存应变能量的能力

※ In this context “h” plays dominant role. The next dominant is B.
※ 在这种情况下，“h”起着主导作用。下一个占主导地位的是 B。

Example 3.2 Determine G of a DCB specimen whose thickness is 30mm, depth of each cantilever 12mm, and crack length 50mm. The pulling load is 15,405 N. The specimen is made of hardened steel with E = 207 GPa. From (3.10)
例 3.2 确定厚度为 30mm、每个悬臂深度为 12mm、裂纹长度为 50mm 的 DCB 试样的 G。拉力为 15,405 N。试样由淬火钢制成，E = 207 GPa。从 （3.10）

Inelastic Deformation Effect at Crack Tip
裂纹尖端的非弹性变形效应

The eqn (3.6) is valid for brittle materials.
方程 （3.6） 适用于脆性材料。

At the crack tip high stresses cause plastic deformation in most metals (ductile). A lot of energy is dissipated in the process of plastic deformation.
在裂纹尖端，高应力会导致大多数金属（延展性）发生塑性变形。在塑性变形过程中耗散了大量能量。

Irwin and Orowan modified Griffith expression to account for the dissipation.
Irwin和Orowan修改了Griffith的表达式，以解释耗散。

The eqn (3.6) is modified to
方程 （3.6） 修改为

where is the plastic work per unit area of surface created.
其中，每单位面积的表面所产生的塑性功。

※ For most structural metal . For example, for a mild steel, while
※ 适用于大多数结构金属。例如，对于低碳钢，而

※ For polymer, energy is dissipated for polymer chains near cracked surface to align themselves under the stresses. This energy is also several times higher than .
※ 对于聚合物，在裂纹表面附近的聚合物链在应力作用下会耗散能量。这个能量也比 高几倍。

To reflect these facts, (3.6) is rewritten as
为了反映这些事实，（3.6）改写为

where 𝛾 is the overall surface energy that may include plastic, viscoelastic or viscoplastic effects at the crack tip. The Griffith model is for linear elastic material. Therefore the global behavior of the structure must be elastic.
其中γ是总表面能，其中可能包括裂纹尖端的塑性、粘弹性或粘塑性效应。格里菲斯模型适用于线弹性材料。因此，结构的全局行为必须是弹性的。

Crack Resistance and Instability
抗裂性和不稳定性

A crack starts to grow when G = R=2γ ( Show this:$''G=-''\frac{''d\pi ''}{''dA''}'',''\frac{''d''{''E''}_{\mathrm{<ms\; style="font-family:\; Cambria\; Math;\; min-height:\; 14pt;\; font-size:\; 14pt;">s</ms>}}}{''dA''}''=2\gamma ,\; A=crack\; surface\; area''$). R is called CRACK RESISTANCE
当 G = R=2γ 时，裂缝开始增长（显示： $''G=-''\frac{''d\pi ''}{''dA''}'',''\frac{''d''{''E''}_{\mathrm{<ms\; style="font-family:\; Cambria\; Math;\; min-height:\; 14pt;\; font-size:\; 14pt;">s</ms>}}}{''dA''}''=2\gamma ,\; A=crack\; surface\; area''$ ）。R 称为 CRACK RESISTANCE

i. Brittle materials

R is almost constant w.r.t the change in crack size.
R几乎是恒定的，随着裂纹尺寸的变化。

For a center crack shown below $''G''''=''\frac{''\pi ''{''\sigma ''}^{\mathrm{<ms\; style="font-family:\; Cambria\; Math;\; min-height:\; 14pt;\; font-size:\; 14pt;">2</ms>}}''a''}{''E''}$
对于中心裂纹，如下 $''G''''=''\frac{''\pi ''{''\sigma ''}^{\mathrm{<ms\; style="font-family:\; Cambria\; Math;\; min-height:\; 14pt;\; font-size:\; 14pt;">2</ms>}}''a''}{''E''}$ 所示

Figure 3.1 Driving Force (G) curves for different values of
图 3.1 不同值的驱动力 （G） 曲线

(G is a function of the crack size)
（G是裂纹大小的函数）

When , it becomes unstable as G exceeds R when crack starts to grow.
当 时，当裂纹开始增长时，当 G 超过 R 时，它变得不稳定。

ii. Ductile materials

As the crack grows in size, the plastic zone at the crack tip increases and the resistance to crack opening increases.
随着裂纹尺寸的增大，裂纹尖端的塑性区增加，裂纹开裂的阻力增加。

When , the crack does not grow. (, )
当 时，裂缝不变大。(, )

For , the crack grows from to (). However, beyond point A, and it is stable.
对于 ，裂纹从 （） 增长。但是，在A点之外，它是稳定的。

For , the crack starts to grow by itself. (beyond point B, )
因为，裂缝开始自行扩大。（超出 B 点，）

The condition for stable crack growth:
稳定裂纹扩展的条件：

Unstable crack growth occurs when
当以下情况下，会发生不稳定的裂纹扩展

The Effect of Thickness of the Specimen
试样厚度的影响

In a thick plate, plane-stress state prevails near specimen surfaces. 在厚板中，试样表面附近存在平面应力状态。 In the interior, plane-strain state exists. 在内部，存在平面应变状态。 As the thickness increases the plane-strain state dominates. Beyond certain size of the thickness, the thickness effects are not felt anymore. 随着厚度的增加，平面应变状态占主导地位。超过一定尺寸的厚度，就不再感觉到厚度效应。

As B increases, .
随着 B 的增加，.

is called the Critical Energy Release Rate for modeⅠ and is considered a material property whose values can be found in handbooks.
称为模式I的临界能量释放速率，被认为是一种材料属性，其值可以在手册中找到。

For a thin plate, the stress state is plane-stress state. Therefore, if the handbook values of are used, the designs will be conservative.
对于薄板，应力状态是平面应力状态。因此，如果使用手册值，则设计将是保守的。

Plane-stress state in a thin plate
薄板中的平面应力状态

Large → Yielding (energy dissipation)
大→屈服（耗能）

Plane-strain state in a thick plate

Small → No yielding or Small yielding → easy fracture
小→无屈服或屈服小→易断裂

Table 3.2 Representative plane strain of some common materials
表3.2 一些常见材料的代表性平面应变