Stress intensity factor complex

The mechanics of fracture along bimaterial interfaces have been studied extensively. Excellent reviews have been published [18]. The stress and deformation field near the tip of a crack lying along a bimaterial interface can be uniquely characterized by means of the complex stress intensity factor K = Kl + iK2. K and K2 have the dimension (Pa m112 " ) and are functions of the sample geometry, applied loading and material properties, i = is the imaginary number and is a dimensionless material constant defined below. 

The corresponding complex stress intensity factor, which is more useful because... 

In this section, discussion focuses on the interface fracture mechanics and the details of crack trajectory predictions that are possible with numerical implementation of these concepts. According to the interface fracture mechanics theory discussed in Chapter 2, a crack at the interface between the adherend and adhesive can be represented by a sub-interface crack lying a small distance (St) below the interface and the complex stress intensity factors K and K2 for the interface crack are related to the conventional stress intensity factors and Ku for the sub-interface crack as... 

In order to obtain crack-tip quantities such as the strain energy release rate g, the complex stress intensity factor K, and the mode-mixity xp, the following procedure may be adopted first, the strain energy release rate Q is directly computed via a contour integral evaluation - the J-integral method, or the VCCT second, the modulus of K, can be computed from Equation (10) and third, the crack surface displacements may be substituted in Equation (8) and with the knowledge of e, the parameter a is computed. Finally, the stress intensity factors may be expressed as ... 

Composite materials have many distinctive characteristics reiative to isotropic materials that render application of linear elastic fracture mechanics difficult. The anisotropy and heterogeneity, both from the standpoint of the fibers versus the matrix, and from the standpoint of multiple laminae of different orientations, are the principal problems. The extension to homogeneous anisotropic materials should be straightfor-wrard because none of the basic principles used in fracture mechanics is then changed. Thus, the approximation of composite materials by homogeneous anisotropic materials is often made. Then, stress-intensity factors for anisotropic materials are calculated by use of complex variable mapping techniques. 

One of the simplest techniques to determine in a complex configuration is to use superposition to build up the solution from a set of simpler and known solutions. Clearly, the precision with which the superposed geometries replicate the final, more complex, structure will impact the accuracy of the final solution. Consider the situation shown in Fig. 8.24, in which cracks emanating from a circular hole is subjected to a biaxial stress. This solution can be broken down into two uniaxial stress solutions, and K. Thus, the total stress intensity factor is found by superposition, K=K, +K. A somewhat more complex configuration is shown in Fig. 8.25. The problem again involves a cracked circular hole but, in this case, it is being loaded along a semi-circular portion of the hole. The problem is asymmetric but, as shown, it can be found from the superposition of two symmetric solutions, i.e., K = K +K H. 

Figure 8.37 Complex void shapes have been modeled by a eircular eraek emanating from a spherical void. This allows the stress intensity factor to be calculated. For short cracks, the solution approaches that of a surface crack, while for long cracks that for a circular crack of radius (R+a).

Figure 8.43 Example of the use of compounding to determine the stress intensity factor for a complex configuration. (Adapted from Parker, 1981, reproduced courtesy of Chapman and Hall Publishers, London, UK.)...

The relationship between the cracking energy or the critical stress intensity factor and porosity is more complex, and this relationship was establish quahtatively by Beaudoin [94] (Fig. 5.41). These both parameters depertd also on the sample drying procedure, which is understandable in the light of water irtflnence on paste strength, related to the humidity of envirorrment where it was stored. 

The above examples outline the complexity of predicting the strength of ceramic materials, as the largest defect - or the one with the highest stress intensity factor - is rarely known a priori. However, it can be seen that the fracture toughness of a material remains a common factor in determining when fast fracture will occur. 

The second approach comes from the work of Irwin, who found that the stress field around a sharp crack in a linear-elastic material could be uniquely defined by a parameter named the stress-intensity factor, K, and stated that fracture occurs when the value of K exceeds some critical value, Kc. Thus, K is a stress-field parameter independent of the material, whereas Kc, often referred to as the fracture toughness, is a measure of a material property. However, this approach becomes far more difficult to employ when the crack is either in a thin adhesive layer or is located at the adhesive-substrate interface, and the reader is referred to the references for further details on this very complex issue. 

Rates of CF crack propagation are uniquely defined by the linear elastic fracture mechanics stress intensity factor range that combines the effects of applied load, crack size, and geometry 17,40. The similitude principle states that fatigue and CF cracks grow at equal rates when subjected to equal values of AK [6-S]. The dal N versus AK relationship may be complex however, an effective approach is based on a power (or Paris) relationship of the form [4/]... 

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