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A method capable of calculating the Yield Point Elongation (YPE) in an automated testing system was investigated and an algorithm for determining YPE was developed. One problem that had to be resolved before a YPE algorithm could be developed was that the definition of YPE as presented in ASTM E 6, Definitions of Terms Relating to Methods of Mechanical Testing, was based upon an informal definition of discontinuous yielding. Automation of the YPE calculation required a formal definition. Discontinuous yielding was formalized to be the region over which the stress-strain diagram exhibited points of inflection (i.e., points where the second derivative of the stress-strain diagram is equal to zero). The definition of YPE then became “the difference between the elongation at the yield point and the last inflection point.”
Given the formalized definition of discontinuous yielding, it became necessary to develop an algorithm for the calculation of first and second derivatives. This investigation centered on the use of difference equations and the use of third order polynomial approximations (cubic splines) to the sampled data. Since derivative calculations inherently increase any noise that is present in the system, the susceptibility of each of these algorithms to noise was examined by impressing a gaussian noise source onto a sinusoid. The cubic spline method was selected since it caused the least amount of noise amplification. The amount of noise rejected by the cubic spline increased with the size of the region that was fitted. However, this caused an increase in the error of the polynomial approximation. When applied to actual test data, the error in the polynomial approximation resulted in increased error in the location of the inflection points. This put a limit on the amount of noise that could be rejected.
Since the noise could not be completely eliminated, false inflection points were exhibited.Therefore it was necessary to identify and ignore these false inflections. This was accomplished by calculating the slopes at successive inflection points and comparing them to the slope of the line intersecting the origin and the point of ultimate load (i.e., the point at which the maximum load occurred after lower yield). If the slope at the inflection point was within certain bounds, specified as a percentage of the ultimate load line slope, then the inflection point was considered to be valid; otherwise, it was rejected. Once the false inflection points were eliminated, the location of the last inflection was identified and the yield point elongation was calculated.
yield point elongation (YPE), yield point, inflection point, cubic spline, discontinuous yielding, parametric equations, ultimate tensile strength
Senior electrical engineer, Advanced Electronic Development Group, Instron Corporation, Canton, MA