Stress octahedral shear

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. 

Fig. 11 Calculated surface profiles of the octahedral shear stress at yield assuming a modified Von Mises criterion (a), and of the octahedral shear stress for a glass/epoxy contact under gross sliding condition (b). The grey area delimits the region at the leading edge of the contact where the octahedral shear stress is exceeding the limit octahedral shear stress at yield (a is the radius of the contact area) (from [97])...

CTjj Stress tensor component Octahedral shear stress... 

Strain rate of 5x10" s , while the strain rates within the epoxy surface layer are in the order of 10 s under fretting conditions. Accordingly, the values of the octahedral shear stress at the onset of yield are probably underestimated. In addition to the limited viscoelastic response of the epoxy material at the considered frequency and temperature (tan 8 = 0.005 at 25°C and 1 Hz, table I), this analysis supports the validity of a global elastic description of the contact stress environment. 

The average stress, aoct, is the trace of the stress tensor corresponding to the volumetric part of the stress field, while the octahedral shear stress, t, involves only the shear components of the same field. 

Maximum Octahedral Shearing Stress Criterion (von Mises [4] Criterion)... 

This failure criterion is given in terms of the octahedral shearing stress, ft is identical to the maximum distortion energy criterion, except that it is expressed in stress versus energy units. The criterion, expressed in terms of the principal stresses, is given in Eqn. (2.9). 

Criteria 2, 5, and 6 are generally used for yielding, or the onset of plastic deformation, whereas criteria 1,3, and 4 are used for fracture. The maximum shearing stress (or Tresca [3]) criterion is generally not true for multiaxial loading, but is widely used because of its simplicity. The distortion energy and octahedral shearing stress criteria (or von Mises criterion [4]) have been found to be more accurate. None of the failure criteria works very well. Their inadequacy is attributed, in part, to the presence of cracks, and of their dominance, in the failure process. 

Before recognition of the importance of inherent flaws in a material, the analyst relied upon one of several stress or strain criteria to predict conditions for failure. These criteria are still useful as failure predictors when flaws are less than the critical size. Typical criteria of this type include maximum principal stress, maximum shear stress, maximum octahedral shear stress, and others depending generally on experimental evidence and experience (, These criteria hypothesize, respectively, that... 

The octahedral shear stress criterion, also called the maximum energy of distortion criterion or the Von Mises theory, would predict the following torque at yield. For the shaft the octahedral shear stress would be (from Equation 3)... 

Now that three separate values for the failure torque have been found for this shaft, the logical question Is which (If any) of the answers Is correct. The answer to this question depends very strongly on the nature of the material Investigated. For very brittle materials (e.g., cast unplastlclzed polystyrene), experiments have shown that the maximum principal stress criterion gives quite reasonable results. For ductile materials such as molded nylon, experimental evidence Indicates that either the maximum shear stress or octahedral shear stress criterion Is more appropriate. 

The octahedral shear stress criterion has some appeal for materials that deform by dislocation motion In which the slip planes are randomly oriented. Dislocation motion Is dependent on the resolved shear stress In the plane of the dislocation and In Its direction of motion ( ). The stress required to initiate this motion is called the critical resolved shear stress. The octahedral shear stress might be viewed as the "root mean square" shear stress and hence an "average" of the shear stresses on these randomly oriented planes. It seems reasonable, therefore, to assume that slip would initiate when this stress reaches a critical value at least for polycrystal1ine metals. The role of dislocations on plastic deformation in polymers (even semicrystalline ones) has not been established. Nevertheless, slip is known to occur during polymer yielding and suggests the use of either the maximum shear stress or the octahedral shear stress criterion. The predictions of these two criteria are very close and never differ by more than 15%. The maximum shear stress criterion is always the more conservative of the two. 

A calculation of the evolution of effective mean stress and the octahedral shear stress in borehole SG2 at a radial distance r= 3m has been made. Pore pressures were approximated by data collected in neighbouring boreholes. It was also assumed that normal stresses on the three normal planes are principal stresses. The results are given in Table 5... 

Table 5. Mean effective and octahedral shear stresses recorded at a radial distance of 3m in borehole SG2 Incremental Incremental Incremental...

Figure 9 shows the octahedral shear stresses generated by ice sheet loading and... 

Figure 9. Octahedral shear stress field at 19 ka before present.

This is the so-called von Mises yield criterion. It can also be written (see problem 8.1) in the form = VC/3, where Tgct is the so-called octahedral shear stress given by... 

Derive the von Mises yield criterion in the form tod = VC/3 from equation (8.6) and the definition of the octahedral shear stress Tod-... 

The von Mises yield criterion is often written in terms of the so-called octahedral shear stress Toct, where... 

Figure 11.20 The strain rate dependence of the octahedral shear stress Toct at atmospheric pressure using data from torsion (o), tension (A) and compression ( ). (Reproduced with permission from Duckett et al., Br. Polym. J., 10, 11 (1978))...

Voorhees analysis assumes that the creep-rupture life of a vessel under complex stressing is controlled by an equivalent stress, J, termed the shear-stress invariant. This average stress is also known as the octahedral shear stress, the effective stress, the intensity of stress, and the quadratic invariant. The theory for the biaxial-stress condition was developed by Von Mises (205), and this theory was further developed to apply to the triaxial-stress condition independently by Hencky (206, 207, 208) and by Huber (209). A derivation of the relationship between the equivalent stress, /, and the three principal stresses, /i, /2, and/s where /i > /2 > /s was given by Eichinger (210). The relationship between these stresses is ... 

To = octahedral shear stress under static loads... 

Maximum distortion energy (or maximum octahedral shear stress) theory (von Mises) Failure occurs when the maximum distortion energy (or maximum octahedral shear stress) at an arbitrary point in a stressed medium reaches the value equivalent to the maximum distortion energy (or maximum octahedral shear stress) at failure (yield) in simple tension... 

Development of the octahedral shear stress can be found in many texts and will not be given here. However, it is appropriate to note the geometry of the octahedral plane. That is, if a diagonal plane is identified for stressed element as shown in Fig. 2.19(a) such that the normal to the diagonal plane makes an angle of 54.7°, the stress state will be as shown in Fig. 2.19(b). The resultant shear stress on this octahedral plane, so named because there are eight such planes at a point, is the octahedral shear stress. 

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