Plastic yield stress

At temperatures sufficiently below the glass transition and under stresses well below the plastic yield stress to be defined later, all polymers exhibit reversible elastic behavior, which is quite often anisotropic, particularly when it relates to a polymer product that has undergone substantial prior deformation processing. 

Note that this is substantially smaller than a typical plastic yield stress for... 

In reality most solids in contact under macroscopic loads undergo irreversible plastic defonnation. This is caused by the fact that at high nonnal forces the stresses in the bulk of the solid below the contact points exceed the yield stress. Under these conditions the contact area expands until the integrated pressure across the contact area is equal to the nonnal force. Since the pressure is equal to the yield strength of the material cr, the plastic contact area is given by... 

Bingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, Ty, but flow like a. simple Newtonian fluid once the applied shear exceeds this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley (1926), Oldroyd (1947) and Casson (1959). 

The functions v,aij,Sij v) represent the velocity, components of the stress tensor and components of the rate strain tensor. The dot denotes the derivative with respect to t. The convex and continuous function describes the plasticity yield condition. It is assumed that the set... 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

The distance from the crack tip, along the X-axis, at which the von Mises equivalent stress falls below the yield stress, defines the size of the plastic zone, r. For the plane stress case of unconstrained yielding, which corresponds to the free surface of the specimen in Figure 4, this gives... 

In tougher materials the minimum thickness required by equation 15 can become excessive. In such cases the /-integral test is an attractive alternative. Because this test considers the stress distribution around the crack inside the plastic 2one, it can be used to obtain a vahd toughness measurement in a thinner specimen, because more extensive plastic yielding does not invaUdate the analysis. The equivalent of equation 15 for the /-integral test is... 

If the sum of the mechanical allowances, c, is neglected, then it may be shown from equation 15 that the pressure given by equation 33 is half the coUapse pressure of a cylinder made of an elastic ideal plastic material which yields in accordance with the shear stress energy criterion at a constant value of shear yield stress = y -... 

For most hydrardic pressure-driven processes (eg, reverse osmosis), dense membranes in hoUow-fiber configuration can be employed only if the internal diameters of the fibers are kept within the order of magnitude of the fiber-wall thickness. The asymmetric hoUow fiber has to have a high elastic modulus to prevent catastrophic coUapse of the filament. The yield-stress CJy of the fiber material, operating under hydrardic pressure, can be related to the fiber coUapse pressure to yield a more reaUstic estimate of plastic coUapse ... 

Of the models Hsted in Table 1, the Newtonian is the simplest. It fits water, solvents, and many polymer solutions over a wide strain rate range. The plastic or Bingham body model predicts constant plastic viscosity above a yield stress. This model works for a number of dispersions, including some pigment pastes. Yield stress, Tq, and plastic (Bingham) viscosity, = (t — Tq )/7, may be determined from the intercept and the slope beyond the intercept, respectively, of a shear stress vs shear rate plot. 

Plastic Forming. A plastic ceramic body deforms iaelastically without mpture under a compressive load that produces a shear stress ia excess of the shear strength of the body. Plastic forming processes (38,40—42,54—57) iavolve elastic—plastic behavior, whereby measurable elastic respoase occurs before and after plastic yielding. At pressures above the shear strength, the body deforms plastically by shear flow. 

Non-Newtonian fluids include those for which a finite stress 1,. is reqjiired before continuous deformation occurs these are c ailed yield-stress materials. The Bingbam plastic fluid is the simplest yield-stress material its rheogram has a constant slope [L, called the infinite shear viscosity. 

Viscosity has been replaced by a generahzed form of plastic deformation controlled by a yield stress which may be determined by compression e)meriments. Compare with Eq. (20-48). The critical shear rate describing complete granule rupture defines St , whereas the onset of deformation and the beginning of granule breakdown defines an additional critical value SVh... 

The process zone is a measure of the yield stress or plasticity of the material in comparison to its brittleness. Yielding within the process zone may take place either plastically or by dimise microcracking, depending on the brittleness of the material. For plastic yielding, / is also referred to as the plastic zone size. 

The closer one approaches to the tip of the crack, the higher the local stress becomes, until at some distance r from the tip of the crack the stress reaches the yield stress, o,, of the material, and plastic flow occurs (Fig. 14.2). The distance r is easily calculated by setting crio ai = o y in eqn. (14.1). Assuming r to be small compared to the crack length, a, the result is... 

This competition between mechanisms is conveniently summarised on Deformation Mechanism Diagrams (Figs. 19.5 and 19.6). They show the range of stress and temperature (Fig. 19.5) or of strain-rate and stress (Fig. 19.6) in which we expect to find each sort of creep (they also show where plastic yielding occurs, and where deformation is simply elastic). Diagrams like these are available for many metals and ceramics, and are a useful summary of creep behaviour, helpful in selecting a material for high-temperature applications. 

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