Most viscous-plastic sea ice models use the elliptical yield curve. This yield curve has a fundamental flaw: it excludes acute angles between deformation features at high resolution. Conceptually, the teardrop and parabolic lens yield curves offer an attractive alternative. These yield curves feature a non-symmetrical shape, a Coulombic behavior for the low-medium compressive stress, and a continuous transition to the ridging-dominant mode. We show that the current formulation of the teardrop and parabolic lens viscous-plastic yield curves with normal flow rules results in negative or zero bulk and shear viscosities and, consequently, poor numerical convergence and representation of stress states on or within the yield curve. These issues are mainly linked to the assumption that the constitutive equation applicable to the elliptical yield curve also applies to non-symmetrical yield curves and yield curves with tensile strength. We present a new constitutive relation for the teardrop and parabolic lens yield curves that solves the numerical convergence issues naturally. Results from simple uni-axial loading experiments show that we can reduce the residual norm of the numerical solver with a smaller number of total solver iterations, resulting in significant improvements in numerical efficiency and representation of the stress and deformation field. These yield curves lead to smaller angles of failure, in agreement with theoretical predictions, and are good candidates to replace the elliptical yield curve in high-resolution pan-arctic sea ice simulation.