This report presents a study on consistent tangent operators for rate-independent elasto-plasticity, focusing on the development of tangent operators that ensure quadratic asymptotic convergence in Newton-type iterative solution schemes. The authors, Juan C. Simo and Robert L. Taylor, analyze the role of consistency between the tangent operator and the integration algorithm in preserving convergence properties. They consider both associative $ J_2 $ plasticity with nonlinear hardening rules and non-associative flow rules, and demonstrate that using the so-called elasto-plastic tangent with a radial return algorithm leads to suboptimal convergence rates. The paper introduces a return mapping algorithm, which is a numerical method for integrating the rate constitutive equations in elasto-plastic problems. This algorithm enforces the consistency condition at the end of each time step, ensuring that the stress point remains on the yield surface if no unloading occurs. The algorithm is shown to be effective for both isotropic and kinematic hardening rules, and is applied to a variety of numerical examples, including thick-walled cylinders, perforated strips, and thick hollow spheres. The authors derive an explicit expression for the tangent moduli consistent with the return mapping algorithm, which is crucial for maintaining the quadratic convergence rate of Newton's method. They compare this consistent tangent with the "continuum" elasto-plastic tangent, showing that the former provides better convergence properties, especially for large time steps. The study also considers non-associative plastic flow rules, demonstrating that the return mapping algorithm can be adapted to these cases as well. The numerical examples illustrate the practical importance of consistent tangent operators in Newton solution procedures, showing that using the consistent tangent leads to significantly better convergence behavior compared to the continuum tangent. The results highlight the importance of deriving tangent operators that are consistent with the integration algorithm used in the solution of the incremental problem. The study concludes that the consistent tangent operator is essential for achieving the quadratic rate of asymptotic convergence in Newton-type algorithms for rate-independent elasto-plasticity.This report presents a study on consistent tangent operators for rate-independent elasto-plasticity, focusing on the development of tangent operators that ensure quadratic asymptotic convergence in Newton-type iterative solution schemes. The authors, Juan C. Simo and Robert L. Taylor, analyze the role of consistency between the tangent operator and the integration algorithm in preserving convergence properties. They consider both associative $ J_2 $ plasticity with nonlinear hardening rules and non-associative flow rules, and demonstrate that using the so-called elasto-plastic tangent with a radial return algorithm leads to suboptimal convergence rates. The paper introduces a return mapping algorithm, which is a numerical method for integrating the rate constitutive equations in elasto-plastic problems. This algorithm enforces the consistency condition at the end of each time step, ensuring that the stress point remains on the yield surface if no unloading occurs. The algorithm is shown to be effective for both isotropic and kinematic hardening rules, and is applied to a variety of numerical examples, including thick-walled cylinders, perforated strips, and thick hollow spheres. The authors derive an explicit expression for the tangent moduli consistent with the return mapping algorithm, which is crucial for maintaining the quadratic convergence rate of Newton's method. They compare this consistent tangent with the "continuum" elasto-plastic tangent, showing that the former provides better convergence properties, especially for large time steps. The study also considers non-associative plastic flow rules, demonstrating that the return mapping algorithm can be adapted to these cases as well. The numerical examples illustrate the practical importance of consistent tangent operators in Newton solution procedures, showing that using the consistent tangent leads to significantly better convergence behavior compared to the continuum tangent. The results highlight the importance of deriving tangent operators that are consistent with the integration algorithm used in the solution of the incremental problem. The study concludes that the consistent tangent operator is essential for achieving the quadratic rate of asymptotic convergence in Newton-type algorithms for rate-independent elasto-plasticity.