第1讲 断裂力学导论

B
Y
X
The potential or internal energy of the body is U p =U i +U a -U w
2a

Due to creation of new surface increase in surface energy is (2.17) U = 4a s The total elastic energy of the cracked plate is 2 2a 2 U t dA Fdy 4a s A 2E E

Griffith proposed that ‘There is a simple energy balance consisting of the decrease in potential energy with in the stressed body due to crack extension and this decrease is balanced by increase in surface energy due to increased crack surface’

The initial strain energy for the uncracked plate per thickness is 2 (2.14) U i dA A 2E On creating a crack of size 2a, the tensile force on an element ds on elliptic hole is relaxed from dx to zero. The elastic strain energy released per unit width due to introduction of a crack of length 2a is given by a where displacement U a 4 1 2 dx v 0 v a sin usin g x a cos E 2a 2 (2.15) Ua E
Rates, G,
s
2E s f 2 a(1 )
Elastic Strain energy released
(2.20) for plane strain
Ua
2a 2 E
Potential energy U release rate G = a Syrface energy/unit extension = (b) Crack length, a ac (a) Variation of Energy with Crack length (b) Variation of energy rates with crack length
4a
s
(a) Crack length, a
(2.19) for plane stress
1/ 2
Total energy
Stable
Unstable
The Griffith theory is obeyed by materials which fail in a completely brittle elastic manner, e.g. glass, mica, diamond and refractory metals.
Griffith theory establishes theoretical strength of brittle material and relationship between fracture strength f and flaw size ‘a’
B
Y
X
2a

Griffith’s Energy balance approach (Contd.)
The Aloha Boeing 737 Accident
On April 28, 1988, part of the fuselage of a Boeing 737 failed after 19 years of service. The failure was caused by fatigue (multi-site damage).
形形色色的断裂现象
断裂的扳手
脆性断裂
疲劳断裂
韧性试件的断裂
Liberty Ships in WWII
Facts on Liberty Ships
• There were 2,751 Liberty Ships manufactured between 1941-1945. Cracks propagated in 400 of these ships including 145 catastrophic failures; only 2 exist today which are sea-worthy. • The hulls of Liberty Ships fractured without warning, mainly in the North Atlantic. • These are the first ships mass produced with welds. • The low temperatures of the North Atlantic caused the steel to be brittle. • Fractures occurred mainly in the vicinity of stress raisers. • The problem may be prevented by employing higher quality steels and improvement of the design of the ship.
结构完整性和寿命评估系统
从强度理论到裂纹力学
应力状态和主应力参数
强度函数
• Some Criteria predict materials failure by yielding, others by fracture • A given material may fail by yielding or fracture strength depending on its properties and the state of stress • Failure criterion for isotropic material is expressed as the principal normal stresses have reached a critical failure strength (c ) f (1, 2, 3) = c
2a m 1 b
2b 2a

A
A
The ratio m is defined as the stress concentration factor, k t When a = b, it is a circular hole, then k t 3. When b is very very small, Inglis define radius of curvature as 2
b a
And the tip stress as
a m 1 2
Inglis理论推广到裂纹
玻璃纤维的断裂强度（示意图）
Griffith关于裂纹扩展的概念
Griffith’s Energy balance approach
•First documented paper on fracture (1920) •Considered as father of Fracture Mechanics
B
Y
X
2a

Griffith’s Energy balance approach (Contd.) Fdy, External work = U w (2.16)
where F= resultant force = area =total relative displacement
断裂力学 第一讲
导论：
什么是断裂力学？ 断裂力学研究什么？ 断裂力学和材料力学有什么联系和区别？ 从强度理论到断裂理论 断裂力学和破坏力学（failure )之间的关系
断裂力学
• 从材料力学、弹性力学发展起来 • 固体力学中最活跃的研究领域
– 实用性：大量的工程应用要求研究材料破坏； – 跨学科：力学、材料、物理、环境、分析计算方 法、测试技术； – 科学、工程结合
Inglis (1913) analyzed for the flat plate with an elliptical hole with major axis 2a and minor axis 2b, subjected to far end stress The stress at the tip of the major axis (point A) is given by
Griffith’s Energy balance approach (Contd.) A A Griffith laid the foundations of modern fracture mechanics by designing a criterion for fast fracture. He assumed that preexisting flaws propagate under the influence of an applied stress only if the total energy of the system is thereby reduced. Thus, Griffith's theory is not concerned with crack tip processes or the micromechanisms by which a crack advances.
常用的强度准则
• • • • • • • 最大主应力准则； 最大剪切应力准则； 最大主应变准则； 最大总应变能准则； 最大畸变应变能准则； 最大Von Mises应力准则； Etc.
孔洞导致应力集中 - Inglis理论
Fracture stress for realistic material
(2.18)
Load, P
a Ua E
P1
2 2
Crack begins to grow from length (a)
P2
) (a ) da + (a
Crack is longer by an increment (da)
v Displacement, v
Denoting as f during fracture
2E s f a
1/ 2
Energy, U
Griffith’s Energy balance approach (Contd.) The variation of U t with crack extension should be minimum = 2 U dU t 2 a y erg n 0 4 s 0 eE ac f r da E Su
The Aloha Boeing 737 Accident
一个不断发展的学科： 强度问题、裂纹问题、断裂问 题、疲劳、损伤、分析和规范、 结构完整性问题 ，等等
损伤分布
与裂纹共生的结构
工程问题: • 结构承载能力 (Structural safety and integrity). • 结构承载时间 (Durability). • 结构可信度 (Reliability). • 安全性评估 (Confidence level). 经济问题: • 建造成本和获取手段 (capital or acquisition cost); • 运行代价 (operational cost)，废弃代价 (disposal/recycling cost) • 成本优化和使用寿命优化 (Optimize capital costs and life cycle cost)