Since An < Au, if m is low, the strain rate outside the neck will become negligibly low. For example, let the neck region have a cross-sectional area of 90% of that outside the neck. Ifm=0.02, ˙εu/ε˙n=(0.9)50=5×10−3. If, however,m=0.5, then ε˙u/ε˙n=(0.9)2 =0.81, so that although the unnecked region deforms slower than the neck, its rate of straining is still rather large.
5.3. EFFECT OF INHOMOGENEITIES 59
5.9. Relative strains in unreduced,εb, and reduced,εa, sections of a stepped tensile specimen for various levels ofm, assuming no strain hardening.
to realistic maximum values ofεawill be lower than those used in Figure5.10. Also, the experimental values are affected by difficulties in maintaining constant temperature over the length of the bar as well as a constant strain rate in the deforming section.
Nevertheless, the agreement between theory and experiments is striking.
Figure5.11shows the dependence of flow stress of the Al-Cu eutectic alloy at 520◦C upon strain rate. The effect of strain is not important here because strain hardening
5.10.Limiting strains,ε∗b, in unreduced sections of stepped tensile specimens as a function ofmand f. Values of percent elongation corresponding toε∗bare indicated on the right together with data from Figure5.8.
5.11.Dependence of flow stress of hot-worked Al-Cu eutectic alloy on strain rate at 520◦C. The curves are for different grain sizes.LAB is mean free path within the grains ofκand CuAl2phases andLIBthe mean free path between interphase boundaries. From D. A. Holt and W. A. Backofen,Trans. Q. ASM, 59 (1966), p. 755.
is negligible at this high temperature. Figure5.12shows the corresponding value ofm as a function of strain rate. At the higher strain rates,mis typical of thermally activated slip. At lower strain rates, deformation mechanisms other than slip prevail. Here there are two schools of thought. One school∗†maintains that deformation occurs primarily by diffusional creep withvacancies migrating from grain boundaries normal to the tensile axis to those parallel to it (i.e., diffusion ofatoms from boundaries parallel to the tensile axis to boundaries perpendicular to it). Such diffusion causes the grains to elongate in the tensile direction and contract laterally. Whether diffusion is through the lattice or along grain-boundary paths, the strain rate should be proportional to the applied stress and inversely related to the grain size. If diffusion were the only mechanism, m would equal one (Newtonian viscosity) but it is lowered because of the slip contribution to the overall strain. The other school‡ attributes the high rate sensitivity to the role of grain-boundary sliding (shearing on grain boundaries).
Although grain-boundary sliding alone would be viscous (m=1), it must be accom- panied by another mechanism to accommodate compatibility at triple points where the grains meet. Either slip or diffusion could serve as the accommodating mechanism.
Both models explain the need for a very fine grain size, high temperature, and low strain rate, but the diffusional-creep model does not explain why the grains remain equiaxed after large deformations.
∗ W. A. Backofen,Deformation Processing(Reading, MA: Addison-Wesley, 1972), pp. 217–20.
† A. H. Cottrell,The Mechanical Properties of Matter(New York: John Wiley, 1964), p. 202.
‡ M. F. Ashby and R. A. Verrall,Acta Met., 21 (1973), p. 149.
5.3. EFFECT OF INHOMOGENEITIES 61
5.12.Variation of strain-rate sensitivity,m=(∂lnσ/∂ln ˙ε)ε,T, for the Al-Cu eutectic at 520◦C. Compare with Fig.5.11. Note that the strain rate for peakmincreases with decreasing grain size. From D. A.
Holt and W. A. Backofen,op. cit.
One practical problem is the tendency for grain growth during superplastic defor- mation because of the high temperature and long times dictated by the slow strain rates and because of the deformation itself.∗ Such grain growth lowers them-value,† and increases the flow stress, causing the overall superplastic formability to deteriorate.
The effects of grain growth are the most important at very slow strain rates (because of the longer times), and it has been suggested that the decrease in m at low strain rates is caused by such grain growth. Because finely distributed second-phase particles markedly retard grain growth, most superplastic alloys have two-phase microstruc- tures. Among the alloys that exhibit superplasticity are Zn-22% Al and a range of steels (eutectoid) and Sn-40% Pb, Sn-5% Bi, Al-33% Cu (eutectics). Other alloys include Cu-10% Al, Zircaloy, severalα-βtitanium alloys, and some superalloys (γ−γ with carbides.)
Aluminum-base superplastic alloys are of considerable interest in the aerospace industry. It has been found that with controlled size and distribution of inclusions it is possible to generate very fine grain sizes by recrystallization and to prevent excessive grain growth during superplastic forming. Recently it has been found that very fine grain ceramics can be superplastically formed.
Superplasticity of some two-phase ceramics has been studied. These include zirconia-alumina, zirconia-mullite, alumina doped with magnesia, and so forth. These observations suggest that commercial forming of ceramics is a possibility.
∗ S. P. Agrawal and E. D. Weisert in 7th North American Metalworking Research Conf. (Dearborn, MI: Soc. Mfg. Engrs., 1979), pp. 197–204.
† The lowm-values often observed at very low strain rates have been attributed in part to grain coarsening during these experiments.