Biaxial yielding

Fig. 8.16 Biaxial yield locus for PVC ( , x), PC ( ), PS (T), and PMMA (+), with all yield stresses normalized by the tensile yield stress at the same temperature. The calculated elliptical locus uses a friction coefficient of — 0.23 (from Raghara et al. (1973) courtesy of Springer).

Figure 8.16 shows a biaxial yield contour for principal stresses a and 02 (with (73 = 0) of PVC, PC, PS, and PMMA at room temperature, normalized with the uniaxial yield stresses of these four polymers (Raghava et al. 1973). Thus, the center of the typical elliptical von Mises yield contour is displaced toward the biaxial compression quadrant characteristic of the pressure-dependent yield formalisms of Coulomb (1773) and Mohr (1900). For PC, discussed above, a model friction coefficient /if = 0.297 is obtained from eq. (8.40). In contrast, the displaced elliptical yield contour of experiments shown in Fig. 8.16 was obtained with a best fit of /if = 0.23, which gives a somewhat larger discrepancy than that for PC alone, given through eqs. (8.36) above. This could be attributed to the fact that the Oy used for normalization is itself dependent on (Tm. 

The uniaxial yield strength and the theoretical biaxial yield strength in a 2 1 stress field. 

Using von Mises Theory from equation 4.58, the probabilistic requirement, P, to avoid yield in a ductile material, but under a biaxial stress system, is used to determine the reliability, R, as ... 

The Bertrand lens, an auxiliary lens that is focused on the objective back focal plane, is inserted with the sample between fully crossed polarizers, and the sample is oriented to show the lowest retardation colors. This will yield interference figures, which immediately reveal whether the sample is uniaxial (hexagonal or tetragonal) or biaxial (orthorhombic, monoclinic, or triclinic). Addition of the compensator and proper orientation of the rotating stage will further reveal whether the sample is optically positive or negative. 

The maximum permitted value for the combined biaxial stress is kST, where S is the specified minimum yield strength per para. PL-3.7.1(a), T is the temperature derating factor per Table PL-3.7.1(g), and k is defined in (b) and (c) below. 

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. 

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.)...

The distinction is easy if the microstructure is sufficiently coarse to yield interference figures for the individual domains. However, sufficiently large domains are uncommon in the case of LCPs. Magnetic or electric fields are sometimes used to coarsen the micro structure (15.161. though this is only reliable if one can be certain that the field does not induce artificial biaxiality. 

The failure function can be measured directly in a number of ways. Some are rather complex and still under development, like the new plane strain biaxial tester with flexible boundaries30, but the simplest method so far is the uniaxial compression test. Only the version developed by Williams et al,24 gives results close to those obtained indirectly with the Jenike shear cell, the other versions yield relative measurements only. 

The mono-substituted PEIs melt from a saniditic layered structure into a normal nematic phase whereas the disubstituted presumably yield a biaxially oriented nematic melt. The disubstituted PEIs isotropise from the melt around 200 °C lower than the monosubstituted of equal chain length. As the second substituent should increase chain stiffness rather than reduce it, Kricheldorf suggests that interactions between temporarily coplanar aromatic n systems, particularly donor-acceptor interactions, can make an efficient contribution to the stabilisation of nematic phases. If this hypothesis is correct then a greater variety of non-linear monomers may be regarded as building blocks for liquid crystal polymers. 

In the case of both EVAL EP-E105 and SOARNOL D resins, attempts to uniaxially orient to 2.5X were unsuccessful, as the films split and cracked upon drawing. Reducing the stretch rate and increasing the orientation temperature yielded no significant effect. As such, biaxial orientation of these two EVOH films was not attempted. 

Substitution of Eqs. (7.15 a to c) and (7.16 a to c) into Eq. (7.17 a to c) yields the full expression for the relaxed stress. Assuming linear-elastic behaviour of the material these equations can then be substituted into the biaxial Hooke s law and solved for the relaxed normal strains er and ee at point P(l ,a). The measuring procedure involves the following steps (Vishay Precision Group, 2010) ... 

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