Process flow stress paths

These ideas have been developed extensively by Ward and co-workers to provide quantitative analyses of the mechanics of hydrostatic extrusion and die drawing for a range of polymers [100-103]. 

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Fig. 21. Schematic diagram of process flow stress paths for elements of material undergoing extrusion through a conical die of small semi-angle, and a uniaxial tensile test...

Figure 12.37 Process flow stress paths for hydrostatic extrusion and die drawing. (Reproduced with permission from Hope, RS. and Ward, I.M. (1981) An activated rate theory approach to the hydrostatic extrusion of polymers. J. Mater. Sci., 16, 1511. Copyright (2000) Hanser Publications.)...

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

It is important to stress that this result holds regardless of the path 3-month LIBOR takes in the future. To see this, we replicate the process described in Exhibit 3.5 once again with one important exception. Rather than remaining constant, we assume that 3-month LIBOR forward rates increase by 1 basis point per quarter until the floater s maturity. These calculations are displayed in Exhibit 3.6. As before, the present value of the floater s projected cash flows is 100. When the market s required margin equals the quoted margin, any increase/decrease in the floater s projected cash flows will result in an offsetting increase/... 

The performance of a material or a structure also depends on local variations due to processing conditions, such as cooling rate, shear stress, and melt-flow paths, which result in orientation and residual stress. Multiphase materials can exhibit even greater local variations in material properties than those observed in single phase polymers. A structure produced by a given process may possess significant morphological differences from test specimens. Ultimately, it may be necessary to test the impact resistance of components. 

The discussion up to this point has focused on the role of free surfaces and internal interfaces, such as grain boundaries, in mass diffusion. Surfaces produced internally in the material as a consequence of permanent deformation and damage induced by stress can also serve, in some cases, as paths along which enhanced atomic diffusion may occur. In amorphous solids undergoing active plastic flow, such increased atomic mobility along shear bands can result in the formation of nanocrystalline particles locally at the bands. An example of such crystallization process is illustrated in this section for the case of a bulk amorphous metallic alloy subjected to quasi-static nanoindentation at room temperature. 

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