Yield Tresca

The simplest yield criterion is that of Tresca. This criterion states that yielding will occur when the maximum shear stress on any plane in the tested sohd reaches its critical value [20] ... 

A yielding criterion gives critical conditions (at a given temperature and strain rate) where yielding will occur whatever the stress state. Two main criteria, originally derived by Tresca and von Mises for metals, can be applied to polymers (with some modifications due to the influence of hydrostatic pressure) ... 

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... 

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. 

Double-grooved specimens were used to study the failure of PC, PC/PE, PET, ABS, and HIPS during transitions from plane stress to plane strain. The yield behavior of PC is consistent with a von Mises-type yield criterion plane strain reduces its elongation. The yield behavior of PC/PE is consistent with a Tresca-type yield criterion plane strain appears to be relieved by voiding around the PE particles. PET undergoes a ductile-to-brittle transition its behavior is consistent with a von Mises-type yield locus intersected by a craze locus. The yield behavior of ABS and HIPS is not significantly affected by the plane-stress-to-plane-strain transition. Plane strain alone does not necessarily cause brittleness. 

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... 

PC/PE. In the case of PC/PE, plane strain alone does not produce significant changes in the yield stress and the deformation behavior. Its yield locus in the tension-tension quadrant is therefore either very nearly a quarter circle or similar to a Tresca locus. The exact shape of the locus can be determined only by much more elaborate biaxial tests. This material is not very notch sensitive compared with PC. The energy to break in a notched Izod impact test is 15 ft-lb/inch for Vs-inch thick bars and 11 ft-lb/inch for 4-inch bars whereas for PC the latter figure is about 2 ft-lb/inch. This reduction in notch sensitivity over pure PC appears to be related to the material s ability to void internally, probably relieving the plane strain. 

ABS and HIPS. The yield stress vs. W/t curves of ABS and HIPS are very similar. They are somewhat surprising because the yield stresses reach their respective maximum values near the W/t (or W/b) where plane strain predominates. This behavior is not predicted by either the von Mises-type or the Tresca-type yield criteria. This also appears to be typical of grafted-rubber reinforced polymer systems. A plausible explanation is that the rubber particles have created stress concentrations and constraints in such a way that even in very narrow specimens plane strain (or some stress state approaching it) already exists around these particles. Consequently, when plane strain is imposed on the specimen as a whole, these local stress state are not significantly affected. This may account for the similarity in the appearance of fracture surface electron micrographs (Figures 13a, 13b, 14a, and 14b), but the yield stress variation is still unexplained. 

In the model derived by McClintock and Irwin the shape and size of the plastic zone were calculated by a combination of the stress field at the crack tip (e.g. Eq. (2)) with a yield criterion (e.g. von Mises, Tresca). This leads to the well known dog-bone type of plastic zone showing the influence of stress state. Its form is often approximated to by a circle of radius Tp, where... 

Two comments should be made first, the yield criteria involved e.g. von Mises, Tresca are based on yielding in response to shear stresses secondly the above solution — although established on physical grounds — is mathematically not exact. 

The yield criterion first suggested for metals was Tresca s criterion, which proposes that in isotropic materials yield occurs when the maximum shear stress X reaches a critical value (10). ... 

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. 

Experimental data show that neither the Tresca nor the von Mises criterion adequately describes the shear yielding behavior in polymers. 

Figure 14.8 shows stress-strain curves for polycarbonate at 77 K obtained in tension and in uniaxial compression (12), where it can be seen that the yield stress differs in these two tests. In general, for polymers the compressive yield stress is higher than the tensile yield stress, as Figure 14.8 shows for polycarbonate. Also, yield stress increases significantly with hydrostatic pressure on polymers, though the Tresca and von Mises criteria predict that the yield stress measured in uniaxial tension is the same as that measured in compression. The differences observed between the behavior of polymers in uniaxial compression and in uniaxial tension are due to the fact that these materials are mostly van der Waals solids. Therefore it is not surprising that their mechanical properties are subject to hydrostatic pressure effects. It is possible to modify the yield criteria described in the previous section to take into account the pressure dependence. Thus, Xy in Eq. (14.10) can be expressed as a function of hydrostatic pressure P as... 

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.)...

Our geometric model of the crystal is most appropriate for polycrystals since we have hypothesized that any and all planes and slip directions are available for slip (i.e. the discrete crystalline slip systems are smeared out) and hence that slip will commence once the maximum shear stresses have reached a critical value on any such plane. This provides a scheme for explicitly describing the yield surface that is known as the Tresca yield condition. In particular, we conclude that yield occurs when... 

The value of a can be determined from a yield criterion. The criterion of Tresca yields a = 2 (This has been used in fig.l). The criterion of v. Mises yields a = V3. A more refined analysis of the Taylor type yields... 

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... 

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. 

Consider first the geometrical character of the stress. A multiaxial stress can be characterized by the three principal stresses Sj, S2, and S3, listed in descending value. Shear yielding depends primarily upon the difference between Sj and S3. The Tresca yield condition, Sj - S3 = Y (applicable to metals), has been modified for polymers (for which the shear yield stress Y increases with hydrostatic pressure) (24). 

The maximum shear stress criterion, also called Tresca s criterion, would predict failure when the shear stress in the shaft equals the shear yield stress (determined by a tensile test). By stress resolution, the shear stress in a tensile test is equal to the normal stress divided by two. Hence the shear stress to produce yield for the material of interest here would be Oq/2, so that failure of the shaft is predicted when... 

These stresses are related by a yield criterion. According to Tresca s yield criterion, the most tensile principal stress ci is related to the most compressive principal stress (Ti by... 

The Mohr circle representation (Fig. 9.6c) is a graphical method of relating stress components in different sets of axes. When the axes in the material rotate by an angle B, the diameter of the circle rotates by an angle 2 B. If the material yields, the circle has radius k, the constant in the Tresca yield criterion. The axes of the Mohr diagram are the tensile and shear stress components. Thus, in the left-hand circle, representing the stresses at A in Fig. 9.6b, the ends of the horizontal diameter are the principal stresses. The principal axes are parallel and perpendicular to the notch-free surface. There is a tensile principal stress Ik parallel to the surface, and a zero stress perpendicular to the surface. The points at the ends of the vertical diameter represent the stress components in the a)3 axes, rotated by 45° from the principal axes. In the a/3 axes, the shear stresses have a maximum value k, and there are equal biaxial tensile stresses of magnitude = k (the coordinate of the centre of the circle). 

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

Tresca and R. von Mises criteria are for isotropic materials. In 1948, Rodney Hill provided a quadratic yield criterion for anisotropic materials. A special case of this criterion is von Mises criterion. In 1979, Hill proposed a non-quadratic yield criterion. Later on several other criteria were proposed including Hill s 1993 criterion. Rodney HiU (1921-2011) was bom in Yorkshire, England and has tremendous contribution in the theory of plasticity. 

In the second form the von Mises criterion expresses directly the fact that the yield depends equally on the three shear stresses (cTj — oy)/2. A somewhat simpler criterion, the Tresca yield criterion, makes the slightly different assumption that yield takes place when the largest of these three shear stresses reaches a critical value. The surface in cr-space that represents the criterion is therefore defined by the six equations... 

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