第讲 断裂力学导论

1、第讲 断裂力学导论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、工程结合形形色色的断裂现象韧性试件的断裂Facts on Liberty ShipsThere 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 ar

2、e 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

3、prevented by employing higher quality steels and improvement of the design of the ship.The Aloha Boeing 737 AccidentOn 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).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

4、完整性问题 ，等等损伤分布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 fractureA given material may fail by yielding or fracture strength depending on its properties and the state of stressFailure criterion for isotropic material is expressed as the principal

5、 normal stresses have reached a critical failure strength (c ) f (1, 2, 3) = c孔洞导致应力集中 - Inglis理论Fracture stress for realistic materialInglis (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 majo

6、r axis (point A) is given by The ratio is defined as the stress concentration factor, When a = b, it is a circular hole, thenWhen b is very very small, Inglis define radius of curvature as And the tip stress as 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A A Griffith laid the foundations of modern f

7、racture mechanics by designing a criterion for fast fracture. He assumed that pre-existing flaws propagate under the influence of an applied stress only if the total energy of the system is thereby reduced. Thus, Griffiths theory is not concerned with crack tip processes or the micromechanisms by wh

8、ich a crack advances. Griffiths Energy balance approach (Contd.)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 s

9、urfaceGriffith theory establishes theoretical strength of brittle material and relationship between fracture strength and flaw size a Griffiths Energy balance approach (Contd.)The initial strain energy for the uncracked plate per thickness is (2.14)On creating a crack of size 2a, the tensile force o

10、n an element ds on elliptic hole is relaxed from to zero. The elastic strain energy released per unit width due to introduction of a crack of length 2a is given by (2.15) Griffiths Energy balance approach (Contd.)External work = (2.16)The potential or internal energy of the body is Due to creation o

11、f new surface increase in surface energy is (2.17)The total elastic energy of the cracked plate is (2.18) Griffiths Energy balance approach (Contd.)The variation of with crack extension should be minimum Denoting as during fracture (2.19) for plane stress (2.20) for plane strainThe Griffith theory i

12、s obeyed by materials which fail in a completely brittle elastic manner, e.g. glass, mica, diamond and refractory metals. Griffiths Energy balance approach (Contd.)Griffith extrapolated surface tension values of soda lime glass from high temperature to obtain the value at room temperature as Using v

13、alue of E = 62GPa,The value of as 0.15 From the experimental study on spherical vessels he calculated as 0.25 0.28 However, it is important to note that according to the Griffith theory, it is impossible to initiate brittle fracture unless pre-existing defects are present, so that fracture is always

14、 considered to be propagation- (rather than nucleation-) controlled; this is a serious short-coming of the theory. Modification for Ductile MaterialsFor more ductile materials (e.g. metals and plastics) it is found that the functional form of the Griffith relationship is still obeyed, i.e. . However

15、, the proportionality constant can be used to evaluate gs (provided E is known) and if this is done, one finds the value is many orders of magnitude higher than what is known to be the true value of the surface energy (which can be determined by other means). For these materials plastic deformation

16、accompanies crack propagation even though fracture is macroscopically brittle; The released strain energy is then largely dissipated by producing localized plastic flow at the crack tip. Irwin and Orowan modified the Griffith theory and came out with an expressionWhere gprepresents energy expended i

17、n plastic work. Typically for cleavage in metallic materials gp=104 J/m2 and gs=1 J/m2. Since gp gs we have Strain Energy Release RateThe strain energy release rate usually referred to Note that the strain energy release rate is respect to crack length and most definitely not time. Fracture occurs w

18、hen reaches a critical value which is denoted . At fracture we have so that One disadvantage of using is that in order to determine it is necessary to know E as well as . This can be a problem with some materials, eg polymers and composites, where varies with composition and processing. In practice, it is usually more convenient to combine E and in a single fracture toughness parameter where . Then can be simply determined experiment