Summary
An attempt is made to modify the plasticity theory for compressible materials and then to derive the upper bound theorem or the second extremum principle which incorporate normal velocity discontinuity; this discontinuity is considered to be a measure of fracture. The theorem is applied to predict the occurrence of fracture, central bursting in extrusion or drawing. The occurrence of fracture appears to depend on the value of the parameters in the theorem; these parameters may be related to ductility or workability of the material in question. The upper bound technique is then applied to multi-stage extrusion of a carbon steel for which the values of the parameters are given with some assumptions. It is thus shown that the stage at which central bursting occurs agrees with experimental results and that it apparently depends on the extrusion condition.
Übersicht
Es wird der Versuch unternommen, mit einer modifizierten Plastizitätstheorie für kompressible Werkstoffe den Obere-Schrankensatz bzw. das zweite Extremum-Prinzip herzuleiten, wenn die Normalgeschwindigkeit auf Sprungflächen unstetig ist. Diese Unstetigkeit wird als Indikator von Bruchvorgängen angesehen. Angewandt wird diese Hypothese auf die Vorhersage des Bruchs, d. h. der Materialtrennung im Kern beim Durchdrück- bzw. Durchziehverfahren. Das Auftreten von Brüchen erweist sich als abhängig von den Parametern des Ansatzes. Diese Parameter lassen sich mit der Umformbarkeit des Materials in Zusammenhang bringen. Angewandt wird das Obere-Schranken-Konzept auf das mehrstufige Durchpressen eines Kohlenstoff-Stahls, für den die Parameterwerte mit Hilfe einiger Parameterwerte festgelegt werden. Davon ausgehend wird gezeigt, daß die Preßstufe bei Materialtrennung mit experimentellen Ergebnissen übereinstimmt und offensichtlich von den Durchpreßbedingungen abhängt.
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Shima, S., Nose, Y. Development of upper bound technique for analysis of fracture in metal forming. Ing. arch 60, 311–322 (1990). https://doi.org/10.1007/BF00538832
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DOI: https://doi.org/10.1007/BF00538832