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Generalized Tresca yield condition as a family of elliptic curves with application to mode I crack problems

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Abstract

By appending a constant to the Weierstrass form of the conventional Tresca yield condition, a new yield condition is generated that allows a continuous transition of yield criteria spanning the Tresca to the von Mises and beyond. The Weierstrass form of the Tresca yield condition is defined by a cubic algebraic relationship between the second and third invariants of the deviatoric stress tensor. In general, the Weierstrass form is commonly associated with a family of plane curves called elliptic, which have special group properties. The additional parameter of the generalized Tresca yield condition is determined from an analytical expression that relates the yield strength of a material in tension to the yield strength of a material in pure shear. Solutions of plane stress, perfectly plastic, mode I crack problems are then obtained for the generalized Tresca yield condition for variations in this parameter. Through insight gained by varying this parameter, an analytical mode I perfectly plastic solution is proposed for the traditional Tresca yield condition under plane stress loading conditions. Unlike the corresponding mode I perfectly plastic solution for the von Mises yield condition, which has three distinct sectors in the upper half plane of symmetry, the solution for the Tresca yield condition requires a fourth distinct sector.

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  1. Mechanical and Civil Engineering, University of Evansville, Evansville, IN, 47722, USA

    David J. Unger

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  1. David J. Unger
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Unger, D.J. Generalized Tresca yield condition as a family of elliptic curves with application to mode I crack problems. Z. Angew. Math. Phys. 73, 184 (2022). https://doi.org/10.1007/s00033-022-01825-6

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  • DOI: https://doi.org/10.1007/s00033-022-01825-6

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