思路总结

本节内容为自己看完整章后总结出的思路性文字，介绍本章的大体思维路线。
第三章介绍了单一型裂纹的断裂判据，本章在此基础上，介绍复合型裂纹的断裂判据。
脆性材料也就是线弹性材料，在应力的计算可以采用叠加原理。
单一型裂纹的扩展方向为裂纹本身的方向，而复合型的，要解决两个问题。

• 裂纹扩展方向（往哪个方向扩展）

• 裂纹扩展判据（什么情况下会扩展） :::info 这里的方法论上，就是典型的科学假设的方法。给出基本假设，推导相关理论，实验求取参数或者验证理论的正确性。 :::

  最大周向拉应力理论

  基本假设

• 裂纹沿着最大周向拉应力的方向扩展

• 最大拉应力达到临界值后扩展

Ⅰ Ⅱ型复合裂纹的推导

受力状态

既有正应力，也有切应力。

分析

解耦合，假定Ⅰ 型裂纹K由正应力决定，与剪应力无关，Ⅱ型裂纹K由剪应力决定，与正应力无关。
得到：

应力公式转换成极坐标，在按照线性叠加原理，直接相加。这里不给出公式了。

根据假设（1），得到：

1. 如果
2. 那么就是后面括号等于零。

接下来的分析有意思了！
解出来(4.1.10)的值，然后计算：

根据假设（2），得到：

这里的为材料的临界值，可以通过Ⅰ 型裂纹的来确定。 :::info 关于断裂准则的宏观理解
断裂准则的公式左边为一物理量，是变量，右边是一常数，一般与材料有关。
在本节中，裂纹为复合裂纹，公式左边是一个复合物理量，右边是一个常数，这个常数与裂纹的类型无关，也就是说这个常数既适用于单一型的裂纹，也适用于复合型的裂纹，那么，我们就可以根据单一型的裂纹来推导计算或者实测该常数。 :::

1. 我们用纯Ⅰ 型裂纹来分析。

，根据(4.1.9)，可以得到。
那么，公式(4.1.10)变成：

达到临界值后

整理(4.1.10)和(4.1.13)得到：

公式(4.1.14)就是最大周向拉应力理论建立的Ⅰ 、Ⅱ型复合裂纹的断裂准则。

1. 我们用纯Ⅱ型来分析

解得：，代入(4.1.14)得到：

接下来分析裂纹初始角度的作用。把(4.1.5)代入(4.1.9)得到：

那么，给出就可以算出。
将代入到(4.1.14)可以求出临界应力。

能量释放率理论

与格里菲斯的能量理论思路类似，都是裂纹尖端释放的能量等于形成新表面的能量。不同的是裂纹尖端裂纹的扩展方向不一定是沿着裂纹面的方向。

基本假设

1. 方向：裂纹沿着最大能量释放率的方向扩展。

1. 该方向的能量释放率达到临界值。

Ⅰ-Ⅲ型复合

对于Ⅰ-Ⅲ复合裂纹，试验证明裂纹沿着原方向扩展。
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根据上小结相同的思路，用单一裂纹形式的来推导临界值，也就是从一般到特殊的原理。

整理可得

含有Ⅱ型

这个分析很有意思，运用到应力强度因子的概念以及多次极限的方法。
对于含有Ⅱ型的复合裂纹，一二，或者一二三。需要考虑角度加以分析。下图所示，有主裂纹和支裂纹，并且两套坐标系。

纯Ⅰ ，纯Ⅱ型能量计算，如果沿着裂纹本身方向扩展。

如果支裂纹沿着本身平面扩展，那么有，上面的横线代表支裂纹。

假设支裂纹的尖端应力场趋近于原裂纹的应力场，即：

由第三章可知，当，有

由应力定义应力强度因子

仿照上面两式，可以定义支裂纹的应力强度因子，就是上面两式，相关变量上面加横线，不再写出。
然后再令支裂纹长度趋于0。重要
得到支裂纹的应力强度因子的起始值。

上面的计算都是准备计算，目的是得到支裂纹尖端的能量表达式。
把公式(4.2.10)的符号代换，可以得到支裂纹的能量表达式。

然后可以根据假设1，列出方程了。

然后用到了一个重要技巧，在裂纹尖端收敛，那么可以令其近似等于极限值，这一技巧在建立应力强度因子的概念的时候就用到了。重要
(4.2.15)代入(4.2.17)得到

根据前面章节内容，有

(4.2.18)继续推导可得

解方程

1. 括号里面等于0，

如此计算得到的角度的能量释放率为
:::danger 此处计算存疑，书上给出的结果是上式，但是我自己手算和Mathematica计算得到的结果不是。 ::: 与（4.2.10）的原平面扩展的比较，，所以，达不到最大的能量释放率，该解废弃。

1. ，该解等价于，也就是说，与最大周向应力准则等价！！重要

然后就是推导临界值的过程。最后得到的断裂准则为，与最大周向应力的完全相同（4.1.14）。

应变能密度因子理论

该理论由Sih（薛昌明）在1973年基于应变能密度场的概念而提出的，该理论计算简单，适用性广，与一些脆性断裂的实验吻合的很好。
弹性体变形后，在体内储存了应变能，单位体积的应变能称为应变能密度，对线弹性体而言，应变能密度为

E为弹性模量，泊松比，剪切弹性模量。
对于复合裂纹，应力按照线性叠加原理叠加，得到复合裂纹的裂纹尖端应力计算公式，这里不再给出。
代入（4.3.1）计算应变能密度

上述都是应变能密度场的理论，接下来引入应变能密度因子S。
令

S这个量，与有关，可以表征场的特性。在此，分析下场的概念，场物理量是在基本物理量的基础上再分析得到的，比如在断裂力学中，应力应变场是基本物理量，而K因子等是在此基础上的新物理量，它表征场的强度。

基本假设

1. 裂纹扩展方向为应变能密度因子极小值的方向。

1. 极小值达到或者超过临界值，裂纹扩展。

:::info 此处为啥是极小值，之前的理论中都是极大值？
因为在这里，应变能密度是一个阻止扩展的力量，裂纹扩展需要给予超过应变能的能量，所以是极小值，而之前的理论都是计算裂纹的驱动力。 :::

公式推导

1. 纯Ⅰ 型

1. 纯Ⅱ型

当时，不满足二阶导数条件，所以取。
此时

达到临界值时

得到

1. 纯Ⅲ型

相同的分析方法，得到