Failure criteria discussion

In general, the calculation of solder joint fatigue has been based on a stress, strain, or energy value calculated for an intact solder joint. Most recently, these calculations have been performed through finite element analysis. The package is analyzed and the critical parameter, a stress, strain, or energy value, is incorporated into a failure theory. ITiese failure theories are, by and large, analytic expressions that are external to the finite element code. A few of these analytic failure theories that find extensive use with finite element analysis are briefly discussed. A detailed review of current failure theories is included in Ref 20. 

Originally developed for the low-cycle fatigue of traditional structural materials such as steel and nickel alloys, the Cofifln-Manson equation (Ref 21-23), has found application in the evaluation of solder joint fatigue fife  

A failure indicator was introduced (Ref 25, 26) that proposed that fatigue life was directly proportional to the inverse of the accumulated per-thermal-cycle matrix creep strain. The matrix creep criterion, which is consistent with the Monkman-Grant relation, takes the form  

Tensile tests have shown that the low-cycle fatigue fife of 96.5Sn-3Ag-0.5Cu dogbone specimens follow the Coffin-Manson relation shown in Eq 7 (Ref 37)  

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The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

For E-glass-epoxy, the Tsai-Hill failure criterion seems the most accurate of the criteria discussed. However, the applicability of a particular failure criterion depends on whether the material being studied is ductile or brittle. Other composite materials might be better treated with the maximum stress or the maximum strain criteria or even some other criterion. 

The analysis of stresses in the laminae of a laminate is a straight-fonvard, but sometimes tedious, task. The reader is presumed to be familiar with the basic lamination principles that were discussed earlier in this chapter. There, the stresses were seen to be a linear function of the applied loads if the laminae exhibit linear elastic behavior. Thus, a single stress analysis suffices to determine the stress field that causes failure of an individual lamina. That is, if all laminae stresses are known, then the stresses in each lamina can be compared with the lamina failure criterion and uniformly scaled upward to determine the load at which failure occurs. 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

The procedure of laminate strength analysis outlined in Section 4.5.2, with the Tsai-Hill lamina failure criterion will be illustrated for cross-ply laminates that have been cured at a temperature above their service or operating temperature in the manner of Tsai [4-10]. Thus, the thermal effects discussed in Section 4.5.3 must be considered as well. For cross-ply laminates, the transformations of lamina properties are trivial, so the laminate strength-analysis procedure is readily interpreted. 

Hence, the ratio Klc/Klcs may be directly and quantitatively related to the crack tip radius, g, at the onset of crack growth by assuming a failure criterion based upon the attainment of a critical stress acting at a certain distance ahead of the tip. A brief examination of Eq. (12) shows that it exhibits the same general trends with regard to rate and temperature dependence that were used successfully in the yield stress discussion to explain in a qualitative way the observed fracture behaviour. 

Saroglou (2007) studied the behaviour of anisotropic rocks, emphasizing on the classification of anisotropy and proposed a modified Hoek-Brown failure criterion (Saroglou Tsiambaos, 2008) to account for the effect of strength anisotropy. A discussion on the results of this... 

To suggest an alternative failure criterion, based on the assumption that all materials contain inherent flaws, linear elastic fracture mechanics applied to adhesive joints was introduced. Basic fracture mechanics approaches were discussed, as well as available test techniques and the influence of various test conditions. 

A single failure in the PMS does not prevent an actuation of the ESFs when the monitored condition reaches the preset value that requires the initiation of an actuation signal (see subsection 7.3.2.2.2 of Reference 6.1). The single failure criterion is met even when one division of the ESF coincidence logic is being tested, as discussed in subsection 7.1.2.9 of Reference 6.1, or when there is a bypass condition in connection with test or maintenance of the PMS. 

A Simple example as sketched In Fig. 5, Is carried out for comparing the efficiency and accuracy of some of the methods discussed In the sections above. Structural and load data are given In Table 3, The occurrence, I.e. development of a collapse mechanism is defined as the structural failure criterion. All possible failure modes are Illustrated In Fig. 5. 

Blister Tests. Blister tests are appealing for measuring the adhesion of thin layers or films, (for example, of paint), to a rigid substrate, because they resemble failure processes encountered in service. Also, they can be analyzed theoretically to yield values of the fracture energy Ga. But they take several different forms. When the pressurized debond (blister) is small in radius compared to the thickness of the over-lying layer, then the failure criterion is the same as that discussed previously under tensile tests, Eq. (20). When the blister radius is comparable to the thickness of the overlying layer, then the layer is deformed primarily in bending and the relation for the failure pressure II becomes... 

It is evident from the above discussions that the parameters Gic and K c do not provide a unique failure criterion but are usually functions of rate and temperature and, in some cases, also of joint geometry. However, a more general criterion may be formulated by considering the stress field at the tip of a crack. It has been shown [156] that, for a crack of tip radius, pt, and length, a, the stress o-n, normal to the axis of the crack at a small distance, r, ahead of the tip is given by ... 

As discussed in Appendix 2B, the RPS is designed to make an ATWS event very unlikely. The RPS has multiple (at least 3, usually 4) channels to meet the single failure criterion, permit sensor calibration during plant operation, and reduce the potential for spurious scrams. The RPS is specifically designed to be separate from plant control systems. 

In those systems to which the single failure criterion applies, if a channel is bypassed during plant operation for the purposes of maintenance, testing, repair or calibration, the remaining operable channels of the system should continue to meet the single failure criterion unless otherwise justified as discussed in paras 4.15-4.21 of this Safety Guide. 

As subsequently discussed in lead-based tin solders such as 96.5Pb-3.5Sn, work-hardening continues to increase the maximum stress for many cycles although extensive cracking is observed. Eventually, the reduction in ffmax due to cracking overcomes the contribution due to work hardening, causing the [Pg.226]

The suitability of a particular failure theory can only be determined by experiment for each material and for each type of failure being considered. It is important to remember that for each of the failure theories discussed above, a single material strength property is aU that is needed to define the failure criterion. Moreover, the value of the strength property is obtained from a single tensile test run on a sample of the material. 

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