The effect of the third invariant in computational plasticity

In this paper, we examine the influence of the third invariant in computational plasticity. For this purpose we consider the extended Leon model, an elasto‐plastic model for concrete materials which accounts for the difference of shear strength in triaxial compression and triaxial extension. Consequently, the deviatoric trace of the loading surface is no longer circular like in von Mises and Drucker‐Prager plasticity. In the limit it approaches the triangular shape of the Rankine condition of maximum direct stress. Thereby, elliptic functions describe the out‐of‐roundness of the circular trace in terms of C¹‐continuous functions of the Lode angle. The algorithmic aspects of the third invariant considerably complicate the computational implementation since the radial return method of J₂‐plasticity does no longer maintain normality leading to loss of deviatoric associativity. The paper will focus on the computational issues near the three regions with high curvature at the compressive meridians with special attention on the lack of convergence of the plastic return algorithm and its slow rate of convergence in these regions. The algorithmic discussion at the constitutive level will be augmented by the axial plane‐strain compression test in order to illustrate the effect of the third invariant at the structural level of finite element analysis.