Yield stress criteria

Since spallation is controlled by the response to tensile stress pulses, the measurements of yield behavior were performed in uniaxial tension rather than in shear, and a tensile yield stress criterion was required. Bouwens-Crowet et ah (6) rearranged Equation 1 to give an expression for the uniaxial-tension yield stress [Pg.201]

If it is assumed that yield and subsequent plastic flow of the material occurs in accordance with the maximum shear stress criterion, then /2 may be substituted for in the above and subsequent equations. For the shear strain energy criterion it may be assumed, as a first approximation, that the corresponding value is G j fz. Errors in this assumption have been discussed (11). 

If the sum of the mechanical allowances, c, is neglected, then it may be shown from equation 15 that the pressure given by equation 33 is half the coUapse pressure of a cylinder made of an elastic ideal plastic material which yields in accordance with the shear stress energy criterion at a constant value of shear yield stress = y -... 

In some cases where the ASME Code woidd not require pressure relief protection, the 1.5 Times Design Pressure Rule is apphcable. This rule is stated as follows Equipment may be considered to be adequately protected against overpressure from certain low-probability situations if the pressure does not exceed 1.5 times design pressure. This criterion has been selected since it generally does not exceed yield stress, and most Ukety would not occur more frequently than a hydrostatic test. Thus, it will protect against the possibility of a catastrophic failure. This rule is applied in special situations which have a low probability of occurrence but which cannot be completely ruled out. 

If the extruder is to be used to process polymer melts with a maximum melt viscosity of 500 Ns/m, calculate a suitable wall thickness for the extruder barrel based on the von Mises yield criterion. The tensile yield stress for the barrel metal is 925 MN/m and a factor of safety of 2.5 should be used. 

The von Mises criterion relates the tensile yield stress of a material to a state of multi-axial stress in a component made from the material. In a cylinder (the... 

Further pressure will cause it to fail and release the process fluid. The yield stress is taken as a design criterion beyond which failure will occur. Table 3.1-1 provides some representative values. 

Lower values of the yield stress measured in tension compared to those measured in compression suggest that the effect of pressure, which is important for polymers, is not accounted for in this criterion. Hence, appropriate correction has to be made in order to account for the effect from external pressure. The most frequent version of pressure-dependent yield criterion is the modified von Mises criterion [20] ... 

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. 

AGS may be expressed as proportional to compressive yield stress), yt (fracture strain), the plastic zone size, and the square of the concentration factor, K. The influence of hydrostatic pressure was taken into account with a modified von Mises criterion (Chapter 12). 

In order to estimate the onset yield stress of the material, three common criteria are introduced. The Tresca criterion is based on maximum shear stress and is given as... 

No significant difference is found for the predictions of the yield stresses from these three criteria. However, Tresca s criterion is more widely used than the other two because of its simplicity. When two solid spherical particles are in contact, the principal stresses along the normal axis through the contact point can be obtained from the Hertzian elastic... 

It can be proved from Eq. (2.156) that, for materials with Poisson s ratio of 0.3 (which is true for most solids), the maximum shear stress oz — or occurs at z/rc = 0.48. Consequently, according to Tresca s criterion, the yield stress Y in a simple compression is 0.62 p0. Therefore, when the hardness or the yield stress Y of the particle material is less than 0.62 times the maximum contact pressure, the sphere will, most likely, undergo plastic deformation. From the elastic collision of two solid spheres, the maximum contact pressure is given by Eq. (2.134). Thus, the relation between the critical normal collision velocity, Ui2Y. and the yield stress is given by... 

Another criterion is used with the Vickers hardness test after penetration of a pyramid-shaped diamond under stress, the diameter of the indent is measured after removal of the diamond. The hardness is defined as the applied force divided by the area of the indent. This is again a measure of the permanent deformation, or, possibly, of the yield stress. 

With respect to Hertzian contact stress, it is recommended that in the absence of actual data, a should be no greater than twice the yield stress of the ear material. Since the yield stress is given as 250 N/mm2, that criterion is satisfied in this example. 

The tensile yield stress variation as a function of W for a material which has a von Mises-type yield locus is illustrated schematically in Figure 5. This variation is caused by the fact that as the width of the specimen increases, the biaxiality also increases toward the asymptotic value at plane strain. If the material obeys the von Mises yield criterion exactly, the plane strain yield stress should be 15% higher than it would be for simple tension. On the other hand, if the material obeys the Tresca yield criterion, the plane strain yield stress should be identical... 

Equation (14.11) can be compared with Eq. (14.9), which corresponds to the Tresca criterion. According to Eq. (14.9) the shear yield stress is one-half the tensile yield stress, whereas Eq. (14.11) predicts that the shear yield stress is 1 /VI times the tensile yield stress. 

For macroscopically isotropic polymers, the Tresca and von Mises yield criteria take very simple analytical forms when expressed in terms of the principal stresses cji, form surfaces in the principal stress space. The shear yield surface for the pressure-dependent von Mises criterion [Eqs (14.10) and (14.12)] is a tapering cylinder centered on the applied pressure increases. The shear yield surface of the pressure-dependent Tresca criterion [Eqs (14.8) and (14.12)] is a hexagonal pyramid. To determine which of the two criteria is the most appropriate for a particular polymer it is necessary to determine the yield behavior of the polymer under different states of stress. This is done by working in plane stress (ct3 = 0) and obtaining yield stresses for simple uniaxial tension and compression, pure shear (di = —CT2), and biaxial tension (cti, 0-2 > 0). Figure 14.9 shows the experimental results for glassy polystyrene (13), where the... 

A sample of polypropylene tested at 30°C and 10 s shows a yield stress of 35 MPa in uniaxial tension and 38 MPa under uniaxial compression. Calculate the hydrostatic pressure that must be superimposed in order to reach yield stress of 80 MPa. Assume that the material obeys the pressure-dependent von Mises criterion. 

In order to check experimentally the size criterion given in Eq. (1), we analysed the evolution of the yield stress Oy with strain rate for both materials. Compression tests were carried out with cylinders of 10 mm height and 8 mm diameter. An Instron tensile/compression test machine was used with prescribed clamp speeds of 6.10 - 60 mm/min., resulting in initial strain rates of lO Vs lO Vs. The resulting yield stress varied from 60 to 130 MPa for PMMA and 50 to 65 MPa for PC. 

In equation (1), Tocto corresponds to the shear yield stress under zero pressure and a is a pressure coefficient, which quantifies the yield stress sensitivity to pressure. Such a yield criterion has previously been shown to hold for epoxy resins under a wide range of pressure, temperature and strain rate conditions [10, 11]. The two parameters, Tocto and a were found to be 44 MPa and 0.173 respectively from the uniaxial and plane strain compression results reported in table I. 

Since the stresses are singular at the crack tip, then clearly the yield oiterion is exceeded in some zone in the crack tip region. If this zone is assumed to be small, then it will not greatly disturb the elastic stress field so that the extent of the plastic zone will be defined by the elastic stresses. If it is assumed that the Von Mises yield criterion is applicable (a reasonable first approximation for polymers), then the shape and e of the plastic zone may be derived from the stresses given in Eq. (15). As-sumii a state of plane strain so that the transverse stress is given by 1/(0 + oee)> then for a yield stress of Oy, the dastic zone radius becomes ... 

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