returnmapping

Return mapping is a numerical integration technique used in computational plasticity to update the constitutive equations of elastoplastic materials during a time step. The method is designed to enforce the plasticity yield condition by projecting a trial elastic stress back onto the yield surface, hence the term "return mapping."

During a step, an elastic predictor is formed by assuming the material deforms elastically with the current state; the corresponding trial stress is compared with the yield function using the current internal variables. If the trial stress lies inside the yield surface, the step is purely elastic and no plastic strain increment is necessary. If the yield condition is violated, a plastic correction is computed by solving for the plastic multiplier (or increment) that drives the stress to the yield surface while obeying the flow rule and hardening law. The plastic strains and internal variables (such as isotropic or kinematic hardening) are updated accordingly.

For common models, the correction is often performed along the direction of the deviatoric stress (radial return) for J2 plasticity with isotropic hardening, so the updated stress lies on the von Mises surface. More general yield functions may require non-radial return mappings. The update is usually obtained by a Newton-Raphson solve for the plastic multiplier, and the algorithm also yields a consistent tangent operator used in the global Newton iterations of a finite element solver.

Applications and considerations: Return mapping is widely used in metal forming, stamping, and structural analyses where implicit time integration is preferred for robustness. The method provides unconditional yield constraint satisfaction and stable integration, but complex yield surfaces may require multiple return iterations or specialized algorithms.