清华大学断裂力学讲义第二章-Griffith断裂理论
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大家好
1
第二章：Griffith理论
热力学（回顾） Griffith理论 能量释放率G
G的实验测量—柔度标定 Griffith理论在非理想脆性材料中的修正
一些讨论பைடு நூலகம்
2
热力学
热力学第零定律：
若两个热力学系统均与第三个系统处于热平衡状态，此两个系统也必 互相处于热平衡。
热平衡的标志为系统的各个部分温度相等
Legendre变换
f x
g p
p df dx
200 year portrait debacle
Adrien-Marie Legendre Louis Legendre
L f x g p
max x
px
f
x
px* f x*
where d px* f x* 0 dx
在热力学里，使用Legendre变换主要的目的是：将一个函数与所含有的一个自变量，转换为一个新函数与所含有
封闭系统：系统与环境之间只有能量交换，没有物质交换。
内能
U S,V
焓
H S, P U PV
Helmholtz 自由能
F T,V U TS
Gibbs 自由能
GT, P U PV TS
min U
min H min F
达到平衡状态
min G
能量最小原理是热力学第二定律的另一种表述。
5
Legendre变换
热力学第一定律：
对于非孤立系统，系统的总能量始终是守恒的。
dU Q Wmech Wextra
U是状态量，Q、Wmech、Wextra是过程量（路径依赖） 系统又有往能量极小演化的趋势
似乎有矛盾，怎么回事？
3
热力学第二定律：
Clausius
不可能把热量从低温物体传递到高温物体 而不产生其他影响。
1920. In 1917, together with G.I. Taylor, he published a pioneering paper on the use of soap films in solving torsion problems, and in 1920 he published his famous paper on the theory of brittle fracture. He then worked on the design theory of gas turbines. Griffith was Head of the Engine Department of the Royal Aircraft Establishment in 1938 and joined Rolls Royce as research engineer in 1939. He worked first on conceptual design of turbojet engines and later on vertical takeoff aircraft design. He retired in 1960 but continued working as a consultant for Rolls Royce. He died on 13 October 1963.
的一个新自变量，此新自变量是旧函数对于旧自变量的偏导数；将旧函数减去新自变量与旧自变量的乘积，得到的
差就是新函数。 Legendre变换可以用来在各种热力势（thermodynamic potential）之间作转换。
6
Griffith理论
Alan Arnold Griffith (1893-1963). He was born in London on 13 June 1893. He earned his B.Eng. in mechanical engineering in 1914, M.Eng. in 1917, and D.Eng. in 1921, all from the University of Liverpool. In 1915, he entered the Royal Aircraft Factory (later known as the Royal Aircraft Establishment), and advanced through a workshop traineeship followed by other positions to become senior scientific officer in
US 4aBg Total energy
g Surface energy
Kelvin-Planck
不可能从单一热源吸收能量，使之完全变 为有用功而不产生其他影响。
引入S熵
Q TdS
Stotal 0
热力学第二定律揭示了系统在保持总能量不变情况下的发展方向
热能区别于其他能量形式
很多能量都最终耗散转化为热能
事实上系统演化是一个熵增的过程
4
能量最小原理：
对于具有定常体积、外参量和熵的封闭系统，系统总的内能将趋向减小， 当达到平衡状态时，总的内能达到极小值。
Biographical Memoirs of Fellows of the Royal Society, Vol. 10, (Nov., 1964), pp. 117-1367 http://www.jstor.org/stable/769315
Griffith理论
椭圆孔的应力分布（弹性力学解）
Energy balance concept
Difference in elastic energy between the cracked sheet and the uncracked sheet
UE
2a2B E
Plane strain Homework（作业题）
Cracked sheet has the free surface
Charles Inglis, 1913
A
1
2a b
A 1 2
a
b2 a
尖锐的裂纹
A 2
a
for a
C.E. Inglis, Stress in a plate due to the presence of cracks and sharp corners, 19138.
Griffith理论
1
第二章：Griffith理论
热力学（回顾） Griffith理论 能量释放率G
G的实验测量—柔度标定 Griffith理论在非理想脆性材料中的修正
一些讨论பைடு நூலகம்
2
热力学
热力学第零定律：
若两个热力学系统均与第三个系统处于热平衡状态，此两个系统也必 互相处于热平衡。
热平衡的标志为系统的各个部分温度相等
Legendre变换
f x
g p
p df dx
200 year portrait debacle
Adrien-Marie Legendre Louis Legendre
L f x g p
max x
px
f
x
px* f x*
where d px* f x* 0 dx
在热力学里，使用Legendre变换主要的目的是：将一个函数与所含有的一个自变量，转换为一个新函数与所含有
封闭系统：系统与环境之间只有能量交换，没有物质交换。
内能
U S,V
焓
H S, P U PV
Helmholtz 自由能
F T,V U TS
Gibbs 自由能
GT, P U PV TS
min U
min H min F
达到平衡状态
min G
能量最小原理是热力学第二定律的另一种表述。
5
Legendre变换
热力学第一定律：
对于非孤立系统，系统的总能量始终是守恒的。
dU Q Wmech Wextra
U是状态量，Q、Wmech、Wextra是过程量（路径依赖） 系统又有往能量极小演化的趋势
似乎有矛盾，怎么回事？
3
热力学第二定律：
Clausius
不可能把热量从低温物体传递到高温物体 而不产生其他影响。
1920. In 1917, together with G.I. Taylor, he published a pioneering paper on the use of soap films in solving torsion problems, and in 1920 he published his famous paper on the theory of brittle fracture. He then worked on the design theory of gas turbines. Griffith was Head of the Engine Department of the Royal Aircraft Establishment in 1938 and joined Rolls Royce as research engineer in 1939. He worked first on conceptual design of turbojet engines and later on vertical takeoff aircraft design. He retired in 1960 but continued working as a consultant for Rolls Royce. He died on 13 October 1963.
的一个新自变量，此新自变量是旧函数对于旧自变量的偏导数；将旧函数减去新自变量与旧自变量的乘积，得到的
差就是新函数。 Legendre变换可以用来在各种热力势（thermodynamic potential）之间作转换。
6
Griffith理论
Alan Arnold Griffith (1893-1963). He was born in London on 13 June 1893. He earned his B.Eng. in mechanical engineering in 1914, M.Eng. in 1917, and D.Eng. in 1921, all from the University of Liverpool. In 1915, he entered the Royal Aircraft Factory (later known as the Royal Aircraft Establishment), and advanced through a workshop traineeship followed by other positions to become senior scientific officer in
US 4aBg Total energy
g Surface energy
Kelvin-Planck
不可能从单一热源吸收能量，使之完全变 为有用功而不产生其他影响。
引入S熵
Q TdS
Stotal 0
热力学第二定律揭示了系统在保持总能量不变情况下的发展方向
热能区别于其他能量形式
很多能量都最终耗散转化为热能
事实上系统演化是一个熵增的过程
4
能量最小原理：
对于具有定常体积、外参量和熵的封闭系统，系统总的内能将趋向减小， 当达到平衡状态时，总的内能达到极小值。
Biographical Memoirs of Fellows of the Royal Society, Vol. 10, (Nov., 1964), pp. 117-1367 http://www.jstor.org/stable/769315
Griffith理论
椭圆孔的应力分布（弹性力学解）
Energy balance concept
Difference in elastic energy between the cracked sheet and the uncracked sheet
UE
2a2B E
Plane strain Homework（作业题）
Cracked sheet has the free surface
Charles Inglis, 1913
A
1
2a b
A 1 2
a
b2 a
尖锐的裂纹
A 2
a
for a
C.E. Inglis, Stress in a plate due to the presence of cracks and sharp corners, 19138.
Griffith理论