Failure theories

In many cases, a product fails when the material begins to yield plastically . In a few cases, one may tolerate a small dimensional change and permit a static load that exceeds the yield strength. Actual fracture at the ultimate strength of the material would then constitute failure. The criterion for failure may be based on normal or shear stress in either case. Modes of failure include excessive elastic deflection or buckling. The actual failure mechanism may be quite complicated each failure theory is only an attempt to explain the failure mechanism for a given class of materials. In each case a safety factor is employed. However, with proper part design, these failures are eliminated or can be permitted since part performance is met. 

An example of a theory is the Griffith theory. It expresses the strength of a material in terms of crack length and fracture surface energy. Brittle fracture is based on the idea that the presence of cracks determines the brittle 

These failures could be due to the variability of the plastic material and/or fabrication control of equipment. Applying an approach such as the IVoubleshooting Guide reviewed, can direct you to the solution. 

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The calculated loading stress, L, on a component is not only a function of applied load, but also the stress analysis technique used to find the stress, the geometry, and the failure theory used (Ullman, 1992). Using the variance equation, the parameters for the dimensional variation estimates and the applied load distribution, a statistical failure theory can then be formulated to determine the stress distribution, f L). This is then used in the SSI analysis to determine the probability of failure together with material strength distribution f S). 

The failure determining stresses are also often loeated in loeal regions of the eomponent and are not easily represented by standard stress analysis methods (Sehatz et al., 1974). Loads in two or more axes generally provide the greatest stresses, and should be resolved into prineipal stresses (Ireson et al., 1996). In statie failure theory, the error ean be represented by a eoeffieient of variation, and has been proposed as C =0.02. This margin of error inereases with dynamie models and for statie finite element analysis, the eoeffieient of variation is eited as Q = 0.05 (Smith, 1995 Ullman, 1992). 

The formulations for the failure governing stress for most stress systems can be found in Young (1989). Using the variance equation and the parameters for the dimensional variation estimates and applied load, a statistical failure theory can be formulated for a probabilistic analysis of stress rupture. 

Of all the theories dealing with the prediction of yielding in complex stress systems, the Distortion Energy Theory (also called the von Mises Failure Theory) agrees best with experimental results for ductile materials, for example mild steel and aluminium (Collins, 1993 Edwards and McKee, 1991 Norton, 1996 Shigley and Mischke, 1996). Its formulation is given in equation 4.57. The right-hand side of the equation is the effective stress, L, for the stress system. 

Materials sueh as east-brittle metals and eomposites do not exhibit these uniform properties and require more eomplex failure theories. Where the properties Su. and Su of a brittle material vary greatly (approximately 4 1 ratio), the Modified Mohr Theory is preferred and is good predietor of failure under statie loading eonditions (Norton, 1996 Shigley and Misehke, 1989). 

The simplified failure envelopes are not derived from physical theories of failure in which the actual physical processes that cause failure on a microscopic level are integrated to obtain a failure theory. We, instead, deal with phenomenological theories in which we ignore the actual failure mechanisms and concentrate on the gross macroscopic events of failure. Phenomenological theories are based on curve-fitting, so they are failure criteria and not theories of any kind (the term theory implies a formal derivation process). 

For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as uniaxial yielding. Tlie ratio of the measure yield stress to the effective stress is known as the factor of safety. 

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

The designer must familiarize himself with the various types of stress and loadings in order to accurately apply the results of analysis. The designer must also consider some adequate stress or failure theory in order to combine stresses and. set allowable stress limits. It is against this failure mode that he must compare and interpret stress values, and define how the stresses in a component react and contribute to the strength of that part. 

As stated previously, stresses are meaningless until compared to some stress/failure theory. The significance of a given stress must be related to its location in the vessel and its bearing on the ultimate failure of that vessel. Historically, various tlieories have been derived to combine and measure stres.ses against the potential failure mode. A number of stress theories, also called yield criteria, are available for describing the effects of combined stresses. For purposes of this book, as these failure theories apply to pressure vessels, only two theories will be discussed. 

Design Philosopliy, 1 Stress Analysis, I Stress/Failure Theories, 2 Failures in Pressure Vessels, 5 lx)adings, 6 Stress, 7... 

British thermal unit (Btu) A British thermal unit is the energy needed to raise the temperature of 1 lb of water ]°F (0.6°C) at sea level. As an example, one lb of solid waste usually contains 4,500 to 5,000 Btu. Plastic waste contains greater Btu than other materials of waste. See calorie energy consumption heat, brittle Easily broken, damaged, disrupted, cracked, snapped. See design-failure theory, Griffith metal fracture. 

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