Polymer yield criterion

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. 

It is always very useful to be able to predict at what level of external stress and in which directions the macroscopic yielding will occur under different loading geometry. Mathematically, the aim is to find functions of all stress components which reach their critical values equal to some material properties for all different test geometries. This is mathematically equivalent to derivation of some plastic instability conditions commonly termed as the yield criterion. Historically, the yield criteria derived for metals were appHed to polymers and, later, these criteria have been modified as the knowledge of the differences in deformation behavior of polymers compared to metals has been acquired [20,25,114,115]. 

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] ... 

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) ... 

The yield criteria of polymers have been reviewed by Ward (7) and more recently by Raghava et al. (8). Except for the craze yield criteria of Sternstein and Ongchin (9) and Bowden and Oxborough (10), most of the yield data can be described by a pressure-modified, von Mises-yield criterion. The corresponding yield surface is everywhere convex. A typical yield locus on the [Pg.103]

The shear component of the applied stress appears to be the major factor in causing yielding. The uniaxial tensile stress in a conventional stress-strain experiment can be resolved into a shear stress and a dilational (negative compressive) stress normal to the parallel sides of test specimens ofthe type shown in Fig. 11-20. Yielding occurs when the shear strain energy reaches a critical value that depends on the material, according to the von Mises yield criterion, which applies fairly well to polymers. 

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 ... 

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. 

One way of attempting to allow for the non-ideal properties of polymers in the yield criteria would be to assume that C or C in equation (8.6) or (8.8) was not a constant but some function C(c, J, p) or C (c, T,p), where e represents the strain-rate, T the temperature and p the pressure. A somewhat simpler approach, based on the so-called Coulomb yield criterion, has, however, been found useful and is described below. 

There have been two main contrasting approaches to the establishment of a yield criterion for oriented polymers. On the one hand we have those... 

Note that both (1) and (2) are symmetrical with respect to the subscripts X, y, z and 1, 2, 3 respectively, reflecting the isotropy of the material. Note further, that both of these equations are unchanged by increasing each normal stress by a constant amount, p, Le. if we write Gx- Gx+p,Gy- Gy+p,Gf- Gi+p, then eqn. (1) is unchanged. This feature, implying that hydrostatic pressure does not affect yield, is a necessary ingredient for a yield criterion for metals which deform mainly by slip processes at constant volume. It however serves only as a first approximation to the yield behaviour of isotropic polymers (see Ref. 6), for... 

In this section we will deal with attempts to describe the yield criterion for anisotropic polymers in terms of a critical resolved shear stress by analogy with the deformation of metal single crystals. Further discussion of the single crystal approach follows in section 11.3.2 where we discuss the structural reorganisation occurring within deformation bands. 

It is not yet possible to establish a clear link between the approach of Ward and co-workers to the problem of molecular reorientation, which is discussed above at length, with many theories in the literature which take as their starting point the crystalline nature of the material. Some of these ideas have already been discussed in relation to the yield criterion for anisotropic polymers, but it is worth looking at them again to assess how they relate to the structure of the material which has been deformed in the... 

The strength-differential effect is also reflected prominently in the multi-axial yield criteria which translate the multi-axial stress driving forces for yield into an equivalent uniaxial state of extension (tension) or simple shear <7se that is most relevant to the mechanisms governing plastic flow. In a more mechanistieally relevant statement for polymers, the multi-axial yield criterion of von Mises defines a uniaxial equivalent stress Oe (or a o-se) as... 

Most amorphous solids and many crystalline ones, particularly non-metals and polymers, exhibit a Coulomb-Mohr-type (Coulomb 1773 Mohr 1900) yield criterion or plastic-shear resistance such that this resistance on the best shear plane is dependent on the normal stress acting across the plane of shear, resulting in a dependence of the type... 

The Tresca yield criterion assumes that the critical shear stress is independent of the normal pressure on the plane on which yield is occurring. Although this assumption is valid for metals, it is more appropriate in polymers to consider the possible applicability of the Coulomb yield criterion [10], which states that the critical shear stress r for yielding to occur in any plane varies linerarly with the stress normal to this plane, i.e. 

A thin walled cylinder with closed ends of radius r and wall thickness is fabricated from a polymer with a yield stress in pure shear of k. Calculate the internal pressure required to produce yielding of the cylinder walls if the yield criterion under appropriate conditions of temperature and strain rate may be written as... 

In contrast to metals, the yield strength of polymers is different in compression and tension. Frequently, the yield strength in uniaxial compression is 20% to 30% larger than in uniaxial tension (see also section 8.4). To account for this, the von Mises yield criterion is augmented by terms that depend on the hydrostatic stress state. We will discuss two possible approaches. 

Fig. 3.25. Comparison of the yield criteria for polymers with the original von Mises yield criterion. The curves are given for identical Rp values...

The criterion (9.6) shows that amorphous glassy polymers yielding process is controlled by cluster structure stability loss (for more details see chapter four) and the condition of transition from shear to crazing can be written as follows [20] ... 

Mechanical Properties of Solid Polymers 12.23 The Coulomb Yield Criterion... 

A very simple yield criterion for anisotropic materials is the critical resolved shear stress of Schmid [14]. This is concerned with crystal slip. The law states that yield occurs when the resolved shear stress in the slip direction in the slip plane reaches a critical value. Although this law is extensively used in metal plasticity, it is of restricted application in polymers. 

Many studies of the yield behaviour of polymers have bypassed the question of strain rate and temperature and sought to establish a yield criterion as discussed in Section 12.2. In very general terms, such studies divide into two categories (1) those which attempt to define a yield criterion on the basis of determining yield for different stress states and (2) those which confine the experimental studies to an examination of the influence of hydrostatic pressure on the yield behaviour. 

We will see that this simple form of pressure-dependent yield criterion is more satisfactory than the Coulomb criterion when a representation is developed which includes the effects of temperature and strain rate on the yield behaviour. In physical terms, the hydrostatic pressure can be seen as changing the state of the polymer by compressing the polymer significantly, unlike the situation in metals where the bulk moduli are much larger ( 100 GPa compared... 

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