Convexity of a function or set is an often needed and important mathematical property. In the case of yield functions (or elastic ranges) in terms of stresses, almost all empirical and mechanism-based yield functions have this property. However, requiring positive plastic dissipation does not necessarily exclude non-convex yield functions, which is confirmed by the fact that non-convex yield functions are observed experimentally, although this rarely happens. We therefore ask whether this nice mathematical property reflects a physical material property. This is investigated in an elastic–plastic, small strain, 2D setting. It appears that, at least in this setting, no specific material property can be attributed to the convexity of the yield function.
NaghdiPM. Stress-strain relations in plasticity and thermoplasticity. In: Proceedings of the Second Symposium on Naval Structural Mechanics. Oxford: Pergamon Press, 1960, 121–169.
8.
WilkinsMStreitRReaughJ.Cumulative-strain-damage model of ductile fracture: Simulation and prediction of engineering fracture tests. Technical Report UCRL-53058, Lawrence Livermore Laboratory, University of California, CA, 1980.
9.
DannemeyerS.Zur Veränderung der Fliefläche von Baustahl bei mehrachsiger plastischer Wechselbeanspruchung. PhD Thesis, Technische Universität Braunschweig, Germany, 1999.
10.
OrtizMRepettoE.Nonconvex energy minimization and dislocation structures in ductile single crystals. J Mech Phys Solid1999; 47: 397–462.
11.
KraskaM.Textursimulation bei groen inelastischen Verformungen mit der Technik des repräsentativen Volumenelements (RVE). PhD Thesis, TU Berlin, Germany, 1998.
12.
MieheCSchotteJLambrechtM.Homogenization of inelastic solid materials at finite strains based on incremental minimization principles: Application to the texture analysis of polycrystals. J Mech Phys Solid2002; 50(10): 2123–2167.
13.
GlügeRWeberMBertramA.Comparison of spherical and cubical statistical volume elements with respect to convergence, anisotropy, and localization behavior. Comput Mater Sci2012; 63: 91–104.
14.
GlügeR.Generalized boundary conditions on representative volume elements and their use in determining the effective material properties. Comput Mater Sci2013; 79: 408–416.
15.
KraskaMBertramA.Simulation of polycrystals using an FEM-based representative volume element. Tech Mech1996; 16(1): 51–62.
16.
SimoJHughesT.Computational inelasticity. New York, NY: Springer-Verlag, 1998.
17.
DunneFPetrinicN.Introduction to computational plasticity. Oxford: Oxford University Press, 2009.
18.
AlfanoGDe AngelisFRosatiL. A numerical algorithm for plane stress elastoplasticity. In: OwenDOateEHintonE (eds.) Computational plasticity: Fundamentals and applications (Proceedings of the Fifth International Conference on Computational Plasticity). Barcelona, Spain: CIMNE International Center for Numerical Methods in Engineering, 343–348.
19.
PedrosoDShengDSloanS.Stress update algorithm for elastoplastic models with nonconvex yield surfaces. Int J Numer Meth Eng2008; 76(13): 2029–2062.
