Stresses intensity factor

1 Central Crack in an Infinite Plate under Biaxial Tension (Griffith Problem) 

The case of an infinitely large thin plate, containing a central through-thickness crack of length 2a, subjected to remote, uniform biaxial tension is considered (Fig. 3.3). The boundary conditions are as follows  

For this simple case, a solution may be obtained through examination of the boundary conditions. To satisfy the traction-free boundary condition along the crack surfaces and to account for stress intensification at the crack tip i.e., at x = a), the stress function Z(z) would need to be of the following form  

Specifically, because Z(z) for -a x a would be imaginary and Oyy = 91cZ(z) along y = 0, Eqn. (3.24), Oyy would be zero and satisfy the traction-free condition along the crack. At x = a, the function would tend to infinity and, thereby, satisfy the required stress intensification. To additionally satisfy the remote traction boundary conditions, the following form of Z(z) is chosen to be a possible solution  

It is seen that, under this assumption, the stress function Z(z) satisfies the boundary conditions and is a solution to the problem. The assumption of A = 0, however, needs to be verified further through a consideration of the displacements (see Eqn. (3.27)). For the moment, the assumption is deemed to be correct, and the process for obtaining the stress intensity factor is considered. 

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The computational process of analysis is hidden from the user, and visually the analysis is conducted in terms of M-02-91 or R6 [6] assessment procedure On the basis of data of stress state and defect configuration the necessary assessment parameters (limit load, stress intensity factor variation along the crack-like defect edge) are determined. Special attention is devoted to realization of sensitivity analysis. Effect of variations in calculated stress distribution and defect configuration are estimated by built-in way. 

Therefore, the magnitude of the stress at small distances from the crack tip is a function of the crack length, a, and the remotely appHed stress. O. Close to the crack tip (r ft) the stress can be scaled usiag a parameter called the stress intensity factor, K (9—11) ... 

D. P. Rooke and D. J. Cartwright, Compendium of Stress Intensity factors. Her Majesty s Stationery Office, London, 1974. 

Y. Murakami, ed.. Stress Intensity factors Handbook, Pergamon Press, Oxford, U.K., 1987. 

Lack of accepted stress intensity factors for internally pressurized components has, until recently, limited this appHcation. The factors are a function of the size and shape of both cracks and high pressure components as well as modes of loading (91). Stress intensity factors can be derived analytically for some simple geometries, but most require the appHcation of advanced numerical methods (105—107). Alternatively they may be deterrnined experimentally (108). 

Another important appHcation of LEFM is the rate of growth of a fatigue crack under cycHc loading. This is also controlled by the stress intensity factor through an equation of the following form (110) ... 

One aspect of pressure vessel design which has received considerable attention in recent years is the design of threaded closures where, due to the high stress concentration at the root of the first active thread, a fatigue crack may quickly initiate and propagate in the radial—circumferential plane. Stress intensity factors for this type of crack are difficult to compute (112,113), and more geometries need to be examined before the factors can be used with confidence. 

For a single-value toughness material, dT/dc = 0. Accordingly, if the applied stress intensity factor is always increasing with crack length, equation 4 is always satisfied. Thus, the condition for fracture is equation 5, where is given by the applied loading conditions. 

Figure 7 shows these results schematically for both twist and tilt crack deflections. Thus, for the stress intensity factor required to drive a crack at a tilt or twist angle, the appHed driving force must be increased over and above that required to propagate the crack under pure mode 1 loading conditions. Twist deflection out of plane is a more effective toughening mechanism than a simple tilt deflection out of plane. 

Toughening for whisker-reinforced composites has been shown to arise from two separate mechanisms frictional bridging of intact whiskers, and pullout of fractured whiskers, both of which are crack-wake phenomena. These bridging processes are shown schematically in Figure 13. The mechanics of whisker bridging have been addressed (52). The appHed stress intensity factor is given by ... 

The term a Tra crops up so frequently in discussing fast fracture that it is usually abbreviated to a single symbol, K, having units MN m " it is called, somewhat unclearly, the stress intensity factor. Fast fracture therefore occurs when... 

K,- = lG,. = fracture toughness (sometimes critical stress intensity factor). Usual units MN m ... 

K[ is the mode I stress intensity factor which serves as a scalar multiplier of the crack tip stress field. 

Correspondingly, the effective stress intensity factor range, may be expressed as... 

In metals, inelastic deformation occurs at the crack tip, yielding a plastic zone. Smith [34] has argued that the elastic stress intensity factor is adequate to describe the crack tip field condition if the inelastic zone is limited in size compared with the near crack tip field, which is then assumed to dominate the crack tip inelastic response. He suggested that the inelastic zone be 1/5 of the size of the near crack tip elastic field (a/10). This restriction is in accordance with the generally accepted limitation on the maximum size of the plastic zone allowed in a valid fracture toughness test [35,36]. For the case of crack propagation, the minimum crack size for which continuum considerations hold should be at least 50 x (r ,J. 

Fracture mechanics analysis requires the determination of the mode I stress intensity factor for a surface crack having a circular section profile. Here the circular section flaw will be approximated by a semi-elliptical flaw. 

Irwin [23] developed an expression for the mode I stress intensity factor around an elliptical crack embedded in an infinite elastic solid subjected to uniform tension. The most general formulation is given by ... 

Fig. 14. Mode I stress intensity factor versus crack depth for small flaw fraeture tests on H-451 graphite...

This is the probability that failure will occur due to the propagation of one tip of the initial defect c under stress o, where is the critical stress intensity factor of the filler particle and a is the filler particle size. 

J. C. Newman, Jr. A Review and Assessment of the Stress Intensity Factors for Surface Cracks," Part-Through Crack Fatigue Life Prediction, ASTM STP 687, J.B. Chang, Ed, American Society for Testing and Materials, 1979, pp. 16 42. 

Baney, J.M., Hui, C.Y. and Cohen, C., Experimental investigations of a stress intensity factor based description of the adhesion of viscoelastic materials. Langmuir, 17(3), 681-687 (2001). 

Step 2. After a contact time t, the material is fractured or fatigued and the mechanical properties determined. The measured properties will be a function of the test configuration, rate of testing, temperature, etc., and include the critical strain energy release rate Gic, the critical stress intensity factor K[c, the critical... 

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