Yield under different stress states

Most mechanics-of-solids textbooks analyse the necking instability that occurs in a tensile test. The analysis will be extended here to deal with products made by the stable propagation of a neck (textile fibres and the Tensar soil stabilising grids shown in Fig. 8.4). The analysis starts with two assumptions  

While the specimen is extending uniformly (Fig. 8.5), the true tensile 

Taking natural logarithms and using Eq. (8.1) gives Differentiation leads to 

At the peak force (position A in Fig. 8.5a), there are two possibilities for the next strain state Elastic unloading along path AU, and further plastic straining along the path AN. A non-uniform strain state develops, as parts of the specimen elastically unload, and the plastic strain in one region increases to form a neck. The plastic deformation of the neck is partially driven by elastic energy release from the rest of the specimen. The condition that A is at the maximum in the force-extension or force-strain curve can be written 

As the force is the product of the cross-sectional area A and the true stress T, defined as F/A, this condition becomes 

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Table 3. Yield stress under different stress states as determined by means of the residual strain method for PP-based polymers. 

The deformation mode affects the dominant failure mechanism by imposing different stress states on the specimen. For example, at a given temperature and deformation rate, the proclivity to fail by brittle fracture (not to be tough ) is much greater under plane strain tension than under simple shear. Another example is that many thermosets fail by brittle fracture under uniaxial tension while they undergo shear yielding under uniaxial compression. 

The simplified failure envelopes differ little from the concept of yield surfaces in the theory of plasticity. Both the failure envelopes (or surfaces) and the yield surfaces (or envelopes) represent the end of linear elastic behavior under a multiaxial stress state. The limits of linear elastic... 

Under dilatational stresses and in contact with solvents, polymers exhibit a cavitational mode of plasticity called environmental crazing. This phenomenon occurs at small strains in the order of a few percent well below the yield point of the polymer. Environmental crazes are normally observed at the surface of a specimen where the penetrating solvent produces a polymer-solvent mixture. Environmental crazing has been extensively discussed in the literature (see e.g. However, one basic problem in studying this phenomenon arises from the fact that the macroscopic state of the sample at craze initiation may differ considerably from the local one which is, in general, poorly defined. 

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... 

Because of the difference in form between Eqs. (2) and (3), the mechanisms of deformation and fracture change with the state of stress. For example, polystyrene yields by shear band formation under ccm ression, but crazes and frachues in a brittle matmer under tensile loading. Chants in failure nwchanian with state of stress are e cially important in particulate conqx tes, since the second phase can alter the local state of stress in the surrounding matrix. 

It was stated in section 6.2.3 that a dislocation will start to move if a sufficiently large shear stress acts on the slip system. This stress value is called critical resolved shear stress Tcrit- It is not equal to the yield strength rp of an isotropic material under shear loading because in the latter case different slip systems have to be activated that are usually not parallel to the shear stress. For a single crystal, the yield criterion (cf. section 3.3.1) is... 

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