Tresca criterion

Therefore, the Tresca criterion has to be modified with the introduction of hydrostatic pressure. The simplest way is a linear dependence ... 

Therefore, both modified criteria (hexagon and ellipse) appear now shifted to the low values of oq, but are still symmetrical between oq and Mises criterion gives a better fit with experimental results than the Tresca criterion. 

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... 

Well-known yield criteria are the Tresca criterion and the Von Mises criterion. Discussion of this subject falls beyond the scope of this book, but a clear description is presented in, e.g. the monograph of Ward and Hadley (1993). If stresses increase above a certain value yield will occur. For metals this critical value is almost independent of pressure, whereas for polymers it is strongly dependent on pressure. An example is shown in Fig. 13.72 for PMMA in Sect. 13.5.4. 

In spite of the relative simplicity of the Tresca criterion, conditions for shear yielding in isotropic polymers are best summarized by the von Mises criterion (11),... 

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... 

Figure 14.9 Section of the yield surface in the plane 03 = 0 choosing the Tresca criterion (hexagonal envelope) and von Mises criterion (elliptical envelope) for polystyrene. The points correspond to experiments performed under pure shear (gi = -CJ2), biaxial tension (oj, 03 > 0), and uniaxial tension and compression. (From Ref. 13.)...

A standard yield criterion, such as the modified von Mises criterion or the modified Tresca criterion, can be used to predict the yield stress in other modes of testing (such as uniaxial compression, plane strain compression and simple shear), from the value of ay(T) in uniaxial... 

A criterion alternative to Tresca criterion is von Mises criterion. In terms of principal stresses, this criterion states that at the onset of yield,... 

Likewise, the Tresca criterion can be represented by a three-dimensional enclosure, this time in the form of a hexagonal section cylinder concentric with the von Mises cylinder. When viewing along the axis of the cylinder, the apices of the hexagonal section will coincide with the circular section (see Fig. 4) along the lines of projection of the axes. 

Keywords crazing, shear yielding, dilatational stress, Tresca criterion, von Mises criterion, necking, void nucleation, tie molecules, yielding. 

The Tresca criterion works well for polycrystalline materials. It has been observed that the following criterion proposed by van Mises is somewhat better (23) ... 

The differences between the two criteria is usually 15%. The Tresca criterion is often used because its form is simpler and because for engineering applications it gives a more conservative prediction for shear failure. 

Compare this to the prediction of ay/2 from the Tresca criterion. The yield criteria for both the Tresca and Von Mises theories are shown graphically in Figure 6. For simplicity, the plots are shown for conditions of plane stress (ie 03 = 0). We can see that the Von Mises criterion describes an ellipse in stress space, with the Tresca criterion consisting of a series of straight lines bounded by the Von Mises limits. 

ModifiGd Tr SCa CritGrion. A simple way to modify the Tresca criterion to allow for a pressure dependence is to make the critical shear yield stress a linear function of hydrostatic pressure ... 

The yield point in compression a was measured for various values of applied tensile stress 02. The results, shown in Figure 11.16, give Oi = —110.0 + 13.65ct2s where both o and 02 are expressed as true stresses in units of MPa. The results therefore elearly do not fit the Tresca criterion, where 0 - 02 = constant at yield neither do they fit a von Mises yield criterion. They are, however, consistent with a Coulomb yield criterion with r = 47.4 — 1.58(/n. 

According to the maximum shear stress theory, the maximum shear equals the shear stress at the elastic limit as determined from the uniaxial tension test. Here the maximum shear stress is one half the difference between the largest (say principal stresses. This is also known as the Tresca criterion, which states that pelding takes place when... 

Let us first consider the case of Tresca criterion. We further assume that (7i and (72 have the same sign. Then, following Eq. (3.1), we have... 

Two basic theories of failure are used in the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section I, Section IV, Section 111 Division 1 (Subsections NC, ND, and NE), and Section VIII Division 1 use the maximum principal stress theory. Section ni Division 1 (Subsection NB and the optional part of NC) and Section VIII Division 2 use the maximum shear stress theory or the Tresca criterion. The maximum principal stress theory (sometimes called Rankine theory) is appropriate for materials such as cast iron at room temperature, and for mild steels at temperatures below the nil ductility transition (NDT) temperature (discussed in Section 3.7). Although this theory is used in some design codes (as mentioned previously) the reason is that of simplicity, in that it reduces the amount of analysis, although often necessitating large factors of safety. 

It is generally agreed that the von Mises criterion is better suited for common pressure vessels, the ASME Code chose to use the Tresca criterion as a framework for the design by analysis procedure for two reasons (a) it is more conservative, and (b) it is considered easier to apply. However, now that computers are used for the calculations, the von Mises expression is a continuous function and is easily adapted for calculations, whereas the Tresca expression is discontinuous (as can be seen from Figure 3.1). 

In terms of the stress intensity, S, Tresca criterion then reduces to... 

The in-plane normal stresses in a flat plate are 10 MPa and 60 MPa and the shear stress is 30 MPa. Find the stress intensity and the von Mises equivalent stress. What is the factor of safety corresponding to (a) Tresca criterion, and (b) von Mises criterion if the material yield strength is 150 MPa ... 

Within the context of pressure vessel design codes, the comparison of the allowable strength of the material is always done with respect to the stress intensities. This puts the comparison in terms of the appropriate failure theory either the maximum shear stress theory (Tresca criterion) or the maximum distortion energy theory (von Mises criterion). These failure theories have been discussed in some detail in Chapter 3. 

If we use the Tresca yield criterion, yielding occurs exactly at cri = Rp. With the von Mises yield criterion, the result is /i/2 [(tri — ctc) + cr f + a ] = Rp. Depending on the value of the circumferential stress, the axial stress at which yielding starts may be up to 15.5% larger than with the Tresca criterion (see... 

Finite element modelling on a macroscopic scale using the Tresca criterion matches the observed size and shape of the plastic zone and also the shape of the indentation stress-strain curves indicating that the physical characteristics of the microstructure determine its yield strength. The use of the Tresca criterion implies a zero coefficient of friction on the microstructural scale. The range of macroscopic yield stresses for the materials studied here is 750 MPa to 2000 MPa. 

Of the many theories developed to predict elastic failure, the three most commonly used are the maximum principal stress theory, the maximum shear stress theory, and the distortion energy theory. The maximum (principal) stress theory considers failure to occur when any one of the three principal stresses has reached a stress equal to the elastic limit as determined from a uniaxial tension or compression test. The maximum shear stress theory (also called the Tresca criterion) considers failure to occur when the maximum shear stress equals the shear stress at the elastic limit as determined from a pure shear test. The maximum shear stress is defined as one-half the algebraic difference between the largest and smallest of the three principal stresses. The distortion energy theory (also called the maximum strain energy theory, the octahedral shear theory, and the von Mises criterion) considers failure to have occurred when the distortion energy accumulated in the part under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension or compression test. 

An important drawback of the Tresca criterion is that it does not consider the effect of the normal stress on the damage progression (see Fig. 9.11). In fact, a component under uniaxial traction a and another under pure torsion t — all have the same equivalent stress Og but while the former loading condition results in the existence of a stress, cr , normal to the plane of Zmax opening the micro-crack as it forms, the latter condition has no normal stress and the micro-crack remains closed, as it can be easily seen from the respective Mohr circles shown in Fig. 9.5. This closure introduces friction and mechanical interlocking effects that retard the micro-crack growth. 

Although the relative simplicity of the Tresca criterion is rather attractive it is found that the criterion suggested by von Mises gives a somewhat better prediction of the yield behaviour of most materials. The criterion corresponds to the condition that yield occurs when the shear-strain energy in the material reaches a critical value and it can be expressed as a symmetrical relationship between the principal stresses of the form... 

---