Cáceres11, 12


has developed a more general framework for


the concept of quality index, based on the notion that the maximum attainable ductility and strength of a tensile bar are determined by the onset of necking. The onset of neck- ing thus represents the maximum theoretical quality for any given alloy and temper. He began with the equation:


σ = K ε n Equation 2


which relates the true stress (σ) and true strain (ε) observed during a tensile test. K is a material constant, and n is the strain hardening coefficient:


Equation 3


Equation [2] represents the experimental curves reasonably well. A single value of K is used, and n is varied to represent different heat treatments. Examples of experimental flow curves are given in Figure 6 for an Al-7%Si-0.4%Mg alloy.


Curves 4 and 5 in Figure 6 are for T4 bars; curve 6 is for a casting aged 1 hr at 170C; and curves 1-3 are for samples aged 6 hr at 170C (a traditional T6 treatment). Dashed lines were calculated with K = 430 MPa and the values of n as indicated in the figure. Note that the solidification rate (or DAS) controls the elongation in these castings for each tem- per, but does not affect the yield strength or the strain hard- ening rate (value of n).


In the castings having the highest ductility, tensile failure involves necking. Necking will occur when the Considère criterion is met; or when:


Equation 4


Comparing equations [3] and [4], it is obvious that necking will occur when ε = n. In other words, the strain hardening exponent, n, determines the maximum uniform strain pos- sible in the tensile sample.


The condition where ε = n represents the maximum ductility. This is the best quality possible. Samples failing earlier have a lower ductility and quality. Cáceres consequently defined a relative quality factor by the relationship:


Equation 5


where the ≅ indicates that the difference between true and nominal strain has been ignored. Considering the level of ductility found in most casting alloys, this is an excellent ap- proximation. Equations [2] and [5] were used to generate the quality curves shown by solid lines in Figure 7, for a value of K equal to 430 Mpa. The dashed lines in Figure 7 show the lines of constant quality and yield strength originally pro- posed by Drouzy, Richard and Jacob. The theoretical curves (solid lines) are seen to correspond closely to the empirical (dashed) lines established by the French.


Cáceres12 also showed that the maximum possible quality


for a material (when q = 1) occurs when Q is numerically equal to 1.11 K. For Al-Si-Mg casting alloys 1.11K ≅ 1.11 x 430 ≅ 477 MPa. From the results shown above, this value is very close to what is found empirically in the best quality castings (Q=480-500 MPa).


Another indication of relative quality is the elongation to fracture. When this is equal to n, the elongation to fracture is 100% of that theoretically possible. The practical signifi- cance of this relationship may be seen by considering Figure 4 once more. The lower blue curve is for material aged two hours, which has a yield strength of 22,500 psi or 155 MPa. This yield strength corresponds to a value of n equal to 0.165 (16.5%). The elongation to fracture of these castings was nearly 16%, very close to the value expected from theory. There is a similar agreement for the two other curves shown (for aging times of 6 and 18 hours), between the value of the strain hardening coefficient and the elongation to fracture. In other words, these castings appear to have a quality close to the maximum theoretically possible (q ≅ 1).


True Plastic Strain


Figure 6. True stress-true strain curves for an Al-7%Si- 0.4%Mg alloy.12


12 Figure 7. The Quality Index map proposed by Cáceres.12 International Journal of Metalcasting/Winter 11


True Stress (MPa)


True Stress (ksi)


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