IWPDF2023

Nomenclature of Yield Criteria for Isotropic Materials

H. Altenbach 1 , ∗ , V.A. Kolupaev 2 , P.L. Rosendahl 3

1 Lehrstuhl für Technische Mechanik, Institut für Mechanik (IFME), Fakultät für Maschinenbau, Ottovon-Guericke-Universität Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany 2 Mechanics & Simulation, Department of Plastics, Fraunhofer Institute for Structural Durability and System Reliability (LBF), Schloßgartenstr. 6, D-64289 Darmstadt, Germany 3 Structural Mechanics and Additive Manufacturing, Institut für Statik und Konstruktion ISM+D, Technische Universität Darmstadt, Karolinenplatz 5, D-64289 Darmstadt

∗ holm.altenbach@ovgu.de

Keywords: π plane, limit surface, systematization

The choice of the yield criterion is crucial for reliable material description and design results. Numerous yield criteria proposed over the last 150 years are hardly used because their utility is not obvious. In addition, the cost of material testing, parameter adjustment, and complexity of implementation often outweigh the benefits of accurate material description. There is no clear procedure for selecting the best criterion for a particular application. The mathematical expressions for the yield criteria can be very different, making it difficult to compare them directly for the best fit. However, possible shapes of yield criteria in the π -plane are limited by the convexity bounds. The upper and lower bounds are referred to as extreme yield figures. Extreme figures can take the shape of isogonal and isotoxal polygons of trigonal or hexagonal symmetry. Regular polygons are limit cases of the extreme yield figures [1, 3]. This work proposes a unique nomenclature of the criteria based on their geometric shapes and orientation in the π -plane, e.g., VON MISES ⃝ , IVLEV ˆ3 ,MARIOTTE 3 , TRESCA ˆ6 , SCHMIDTISHLINSKY 6 , SOKOLOVSKY 1ˆ2 , among others. Circumflex ˆ and macron refer to an upward pointing tip or upward facing flat base of the regular shape in the π -plane, respectively. The generalized yield criteria can be characterized by the regular polygons and the circle in the π -plane that they contain. There are known six criteria that are of interest: ˆ3 − ˆ6 − 3 , ˆ3 −⃝− 3 , ˆ3 − 6 − 3 , ˆ6 − 1ˆ2 − 6 , ˆ6 −⃝− 6 , ˆ6 − 12 − 6 . The criteria involving less than three of the basic geometries are edge cases and excluded from our discussion. Based on the introduced nomenclature, a verification standard for the yield criteria is developed and the number of the useful yield criteria is reduced to a few manageable cases. The C 0 and C 1 continuous criteria that contain five basic geometries ˆ3 − ˆ6 |⃝| 6 − 3 and ˆ6 − 1ˆ2 |⃝| 12 − 6 , and that satisfy the plausibility conditions [1] are significant. Their usage eliminates the need to develop and select specific criteria for classes of materials like alloys, polymers, etc. References [1] Altenbach, H., Kolupaev, V. A., General Forms of Limit Surface: Application for Isotropic Materials, in Altenbach, H., Beitelschmidt, M., Kästner, M. et al. (Eds.), Material Modeling and Structural Mechanics, Advanced Structured Materials, v. 161, Springer, Cham, 1-76, 2022.

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