北科大研究生斷裂力學(xué)Chapter5 Some Special Topics

1、1 Chapter 5 Special Topics on Fracture Mechanics Xuejun CHEN Department of Applied Mechanics University of Science and Technology Beijing Most updated version: April, 2014 2 Interfacial fracture mechanics l The background/examples of interfacial fracture coating/film decohesion interfacial engineeri

2、ng 3 Interfacial fracture mechanics l A body consists of two materials bonded at an interface. On the interface there is a crack. When the applied load reaches a critical level, the crack either extends along the interface, or kinks out of the interface. l For the crack extending on the interface, t

3、he energy release rate is still defined as the reduction in the total potential energy of the body associated with the crack advancing by unit area. All the familiar methods to determine it still apply. 1 1 2 11 2 2 2 22 2 11 2 11 E E E E Plane strain moduli lThe energy release rate characterizes th

4、e amplitude of the load. The critical condition for the extension of the crack also depends on the mode of the load. To characterize the mode, the field near the tip of the interfacial crack is needed. 4 l The Williams field (The stresses around a fault or crack in dissimilar media, Bull. Seismol. S

5、oc. America, 49 (1959):199-204.) By solving an eigenvalue problem, Williams discovered that the singular field around the tip of the interfacial crack is not square-root singular, but takes a new form. At a distance r ahead the tip of the crack, the stresses on the interface are given by: 2212 . 2 i

6、 Kr i r At a distance r behind the tip of the crack, the two faces of the crack move relative to each other by the displacements: 21 12 112 . 2 1 2cosh i Krr i EEi Interfacial fracture mechanics Note: For homogeneous materials, the stress field around crack tip is independent of materials constants,

7、 but it is not true for bi-materials. 5 A mathematical identity exp(ln )coslnsinln. i rirrir The oscillatory index is dimensionless and depends on the elastic constants of both materials: 1 1 2 1 2 2 34 1 ln. 2 341 When the two materials have identical elastic constants, the constant vanishes, and t

8、he singular field around the tip of the crack on the interface is similar to that around the tip of a crack in a homogeneous material. The stress intensity factor K is complex-valued (corresponding to two real numbers), which is of dimension: 1 2 stress length. i Interfacial fracture mechanics 2212

9、. 2 i Kr i r 6 l The amplitude of the stress intensity factor is defined by: .KKK l This real-valued quantity (magnitude) has the familiar dimension: 1 2 stress length. l Indeed, K relates to the energy release rate G as (Malyshev and Salganik, 1965): 2 2 12 111 . 2cosh K G EE Interfacial fracture m

10、echanics 2212 . 2 i Kr i r 7 l The range of The constant is monotonic in 1/2. When material 1 is much more compliant than material 2, in the limit 1/20, we obtain: 1 1 ln 34. 2 When material 1 is much stiffer than material 2, in the limit 1/2, we obtain: 2 1 ln 34. 2 If Poissons ratios of the two ma

11、terials are restricted in the interval 0, 0.5, the constant is bounded in the interval: 1 ln30.175. 2 If Poissons ratios of the two materials are restricted in the interval -1, 0.5, the constant is bounded in the interval: 1 ln70.31. 2 Interfacial fracture mechanics 1 1 2 1 2 2 34 1 ln. 2 341 8 l Ph

12、ase angle of the stress intensity factor One can write complex-valued stress intensity factor in terms of its amplitude and phase angle: exp.KKi The amplitude is related to the energy release rate G, and specifies the magnitude of the load. The phase angle specifies the mode of the load. l Oscillato

13、ry stresses 2212 expln. 22 i KKr iiir rr Separating the real and the imaginary parts, we obtain that: 22 12 cosln, 2 sinln. 2 K r r K r r Thus, the Williams field predicts that the stresses are oscillatory as r approaches the tip of the crack. Interfacial fracture mechanics 9 l The Williams field pr

14、edicts that, at a distance r behind the tip of the crack, the two faces of the crack move relative to each other by the displacement: 21 12 1 2 12 112 2 1 2cosh 112 explntan2. 2 1 4cosh i Krr i EEi Kr iiri EE When =0, this expression is similar to that for a crack in a homogenous material. When does

15、nt vanish, this expression indicates that, for some values of r, the two faces of the crack interpenetrate!(contradiction!traction-free on crack faces ) The component of the displacement normal to the plane of the interface is: 1 2 2 12 112 coslntan2. 2 1 4cosh Kr r EE Interfacial fracture mechanics

16、 10 l For the case of =0. 2212 . 2 K i r The stress field becomes square-root singular, namely: Interfacial fracture mechanics Separating the complex-valued K into the real and imaginary parts: . III KKiK One can obtain that: 22 12 , 2 . 2 I II K r K r The two parameters KI and KII measure the ampli

17、tudes of two fields. Consequently, we can treat a crack on an interface the same way as we treat a crack in a homogeneous material under mixed-mode loading. In reading the literature on interfacial fracture mechanics, one can simply set =0, because it is negligible by definition. 11 Example 1: isola

18、ted forces on a semi-infinite crack Interfacial fracture mechanics J.R. Rice, G.C. Sih. Plane problems of cracks in dissimilar media, J. Appl. Mech. (32)1965:418-423. l The complex-valued SIF determination For a given configuration of an interfacial crack, the complex-valued stress intensity factor

19、K is determined by solving the boundary-value problem within the theory of linear elasticity. 12 l Example 2: infinite plate with a crack subjected to stresses at infinity Interfacial fracture mechanics 13 l The propagation life is the portion of the total life spent growing a crack to failure. l Th

20、e initiation life encompasses the development and early growth of a small crack. Initiation life is usually assumed to be the portion of life spent developing an engineering crack, which is about 0.3mm for smaller components. Fatigue and life prediction l Fracture caused by cyclic loads is known as

21、fatigue. Since fatigue is an extremely complex phenomenon, we will only focus on the application of fracture mechanics to fatigue. l Fatigue may be divided into two phases: (i) the initiation of a crack and (ii) the extension of the crack. The extension of a crack in a body under a cyclic load can b

22、e characterized by using fracture mechanics, as described by Paris et al. (1961). 14 The stress amplitude Fatigue and life prediction l Characteristic of cyclic stress maxmin . 2 S The mean stress maxmin . 2 m The stress ratio min max .R The range of stress maxmin. The stress fluctuates as a functio

23、n of time. 15 Fatigue and life prediction l A typical S-N curve Ultimate tensile strength. If the amplitude of the stress equals the ultimate tensile strength, S=TS, the fracture occurs upon first loading. Endurance limit. there may exist an amplitude of the stress, called the endurance limit, EL. W

24、hen the amplitude is below the endurance limit, SEL, the bar can sustain infinite number of cycles without fracture. Cycle-to-fracture. When the load amplitude falls between the ultimate tensile strength and the endurance limit, ELSTS, the cycle-to-fracture Nf decreases as the load amplitude increas

25、es. 16 l The use of the S-N curve in design. The S-N curve is specific to each material. Once it is measured experimentally for a given material, one can use the curve to answer design questions. Fatigue and life prediction To predict the cycle-to-fracture if the amplitude of stress is known. To det

26、ermine the allowable amplitude of stress if a cycle-to-fracture is prescribed. l The Paris approach. The Paris approach assumes that a crack pre-exists in a body. Under a cyclic load, the crack grows a little bit each cycle. When the crack is long enough, the body fractures. The stress intensity fac

27、tor is the loading parameter on a crack. l Paris hypothesized that the crack extension per cycle, dN/da, is a function of K. ,. da fK R dN maxmin min max . KKK R ， in which: 17 l This function is a material property. For a given material, the function can be measured experimentally using a single sa

28、mple. Paris law is as follows: Fatigue and life prediction . mda AK dN 11 10,24.Am : double logarithmic axis 18 l Using the Paris law in design. In a large component, a small crack of initial size ai, has been detected. The component is subject to a cyclic stress between 0 and S. The component under

29、goes fast fracture when the crack size reaches af. How many cycles will the component last? Fatigue and life prediction Note that: .KSa The Paris law becomes: . m da ASa dN This is an ordinary differential equation that governs the function a(N). Assuming the exponent m = 4. Rearranging the above, we have 244 . da dN AS a An integration from ai to af gives the number of cycles to fast fracture: 24 111 . f if N ASaa 19 Fatigue and life prediction 24 111 . f if N ASaa Main observations: The