Maximum Normal Stress Theory

By definition, a brittle material does not fail in shear failure oeeurs when the largest prineipal stress reaehes the ultimate tensile strength, Su. Where the ultimate eompressive strength, Su, and Su of brittle material are approximately the same, the Maximum Normal Stress Theory applies (Edwards and MeKee, 1991 Norton, 1996). The probabilistie failure eriterion is essentially the same as equation 4.55. 

When the material behavior is brittle rather than ductile, the mechanics of the failure process are much different. Instead of the slow coalescence of voids associated with ductile rupture, brittle fracture proceeds by the high-velocity propagation of a crack across the loaded member. If the material behavior is clearly brittle, fracture may be predicted with reasonable accuracy through use of the maximum normal stress theory of failure. Thus failure is predicted to occur in the multi-axial state of stress when the maximum principal normal stress becomes equal to or exceeds the maximum normal stress at the time of failure in a simple uniaxial stress test using a specimen of the sane material. 

Maximum normal stress theory (Lame-Navier) Failure occurs when the... 

Comparison Between Theory and Experiment Comparisons between theory and experiment have been made for many materials. Shown in Fig. 2.20 are the graphs in stress space for the equations for the three theories given above. Also shown is experimental data on five different metals as well as four different polymers. It will be noted that cast iron, a very brittle material agrees well with the maximum normal stress theory while the ductile materials of steel and aluminum tend to agree best with the von Mises criteria. Polymers tend to be better represented by von Mises than the other theories. 

The maximum shear-stress theory has been found to be suitable for predicting the failure of ductile materials under complex loading and is the criterion normally used in the pressure-vessel design. 

The maximum intensity of stress allowed will depend on the particular theory of failure adopted in the design method (see Section 13.3.2). The maximum shear-stress theory is normally used for pressure vessel design. 

The maximum principle stress theory (Rankine s theory) states that the largest principle stress component, 03, in the material determines failure regardless of the value of normal or shearing stresses. The stability criterion is formulated as... 

When a beam is bent, a continuous gradation of stress occurs from a maximum tensile stress on one surface through a neutral axis to a maximum compressive stress on the other surface. It is the maximum tensile stress and strain that are calculated. Because of the geometry differences and the fact that in bending tests the surface stress rather than a homogeneous stress is considered, values for strength and modulus cannot be simply equated with those from normal tensile tests, although in theory they are equal. 

There are conditions of loading a product that is subjected to a combination of tensile, compressive, and/or shear stresses. For example, a shaft that is simultaneously bent and twisted is subjected to combined stresses, namely, longitudinal tension and compression, and torsional shear. For the purposes of analysis it is convenient to reduce such systems of combined stresses to a basic system of stress coordinates known as principal stresses. These stresses act on axes that differ in general from the axes along which the applied stresses are acting and represent the maximum and minimum values of the normal stresses for the particular point considered. There are different theories that relate to these stresses. They include Mohr s Circle, Rankine s, Saint Venant, Guest, Hencky-Von Mises, and Strain-Energy. 

Note the good agreement between predictions and experimental data for both the steady-state shear stress and first normal stress difference in Fig. 11.7. These predictions substantially improve on those of the Doi-Edwards and DEMG theories, especially in removing, or nearly removing, the maximum in shear stress as a function of shear rate. They also show that with... 

There are numerous theories based on structural models of suspensions [Mikami, 1980]. Wildemuth and Williams [1984] considered that the maximum packing volume fraction, ( ), is a function of normalized shear stress, CTij = cTjj / M, where M is a numerical parameter. The authors derived the relation ... 

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