Stress Invariant Criteria

Stress invariant criteria are based on the value of the hydrostatic stress ajj and the second invariant of the stress deviator 

Marin [20] suggested a criterion based on the values of the amplitude 2,0 and mean stress 72,m of the second invariant of the stress deviator According to Marin, fatigue failure would not occur when 

This criterion respects the hnding that a constant torsion stress does not affect fatigue strength while a constant normal stress does. The application to a fully reversed bending leads to 

The criterion was successively modified by Sines and Ohgi [22] to include an additional term to take care of nonlinear effects due to possible higher mean stresses 

Similar to the Sines criterion 9.38 is that formulated by Crossland [23] who considered the maximum hydrostatic stress aa nox rather than the medium one 

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This model is based on the mean features of the Mohr-Coulomb model and is expressed with stress invariants [Maleki (1999)] instead of principal stresses. Until plasticity is reached, a linear elastic behaviour is assumed. It is fully described by the drained elastic bulk and shear moduli. The yield surface of the perfectly plastic model is given by equation 7. Function 7i(0) is chosen so that the shape of the criterion in the principal stress space is close to the Lade criterion. 

Von Mises stress is originally formulated to describe plastic response of ductile materials. It is also applicable for the analysis of plastic failure for coal undergoing high strain rate. The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J2 reaches a critical value. In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, a scalar stress value that can be computed from the stress tensor ... 

The boundary of the crazed region coincided to a good approximation with contour plots showing lines of constant major principal stress 0, as shown in Figure 12.14(b) where the contour numbers are per unit of applied stress. At low applied stresses it is not possible to discriminate between the contours of constant cTi and contours showing constant values of the first stress invariant I = Oi+ a. However, the consensus of the results is in accord with a craze-stress criterion based on the former rather than on the latter and, as we have seen, the direction of the crazes is consistent with the former. 

Equations (14.10) and (14.12) give the pressure-dependent von Mises criterion. Also, for any state of stresses, P is an invariant given by the expression P = (l/3)(ai-I-Q2-1-cy3). On the basis of this expression, in a uniaxial tension test 02 = a3 = 0)... 

The von Mises yield criterion uses the principal invariants of the deviatoric stress tensor, J(, J2, and J3. Using equation (3.26), we find... 

The simplest theories of plasticity exclude time as a variable and ignore any feature of the behaviour, which takes place below the yield point. In other words, we assume a rigid-plastic material whose stress-strain relationship in tension is shown in Figure 12.9. For stresses below the yield stress there is no deformation. Yield can be produced by a wide range of stress states, not just simple tension. In general, it must therefore be assumed that the yield condition depends on a function of the three-dimensional stress field. In a Cartesian axis set, this is defined by the six components of stress, an, a22, <733, ayi, < 22 and a 31. However, the numerical values of these components depend on the orientation of the axis set, and it is crucial that the yield criterion be independent of the observer s chosen viewpoint the yield criterion must be invariant with respect to changes in the axis set. It is often convenient to make use of the principal stresses. If the material itself is such that its tendency to yield is independent of the direction of the stresses - that is if it is isotropic -then the yield criterion is a function of the principal stresses only... 

Since the SSSC value is symmetric with respect to vectors n and s, it is invariant of the choice of the fault plane from the two nodal planes. The condition of the maximum shear stress Eq. 30 is not fully correct and physically means that faults should obey the so-called Tresca failure criterion where faults are assumed to have zero friction. Only faults with zero friction can achieve maximum shear stress and satisfy Eq. 30 for faults with friction, the SSSC value is always reduced (see Fig. 11). 

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