For J2 plasticity, the return is radial: $$ \sigma_n+1 = \sigma_n+1^trial - 3G \Delta\lambda \mathbfn $$ Where $\mathbfn$ is the unit normal to the yield surface and $G$ is the shear modulus.
In computational methods, these continuous laws must be discretized over finite time steps. This is where the "return mapping algorithm" becomes king. For J2 plasticity, the return is radial: $$
The resource will define the surface in stress space that separates elastic from plastic behavior. The most common standard is the , suitable for ductile metals. More advanced PDFs will cover: The resource will define the surface in stress
This guide explores the foundational theories, numerical algorithms, and practical applications commonly found in the authoritative text Computational Methods for Plasticity: Theory and Applications by de Souza Neto, Peric, and Owen. Core Theoretical Foundations For J2 plasticity
When you download a "computational methods for plasticity theory and applications pdf," you are essentially downloading a guide to the (also known as the closest point projection).