Plasticity

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On this page we conduct numerical studies of plastic deformation by employing the von Mises plasticity model with isotropic hardening. The majority of the theory is explained in detail in Computational Methods for Plasticity - Theory and Applications by EA de Souza Neto . For a more detailed theory description, we strongly recommend reading the book.

Introduction

In physics and materials science, plasticity, also known as plastic deformation, is observed when the solid material undergoes permanent deformation in response to applied forces. There are different small-strain elastoplastic constitutive models of plasticity known. The most popular theories are the von Mises, Tresca, Mohr-Coulomb and Drucker-Prager models. The main difference between them is in the computation of yield condition - threshold value after which the deformation is irreversible.

This demonstration uses the von Mises model, with nonlinear isotropic hardening, due to the simplicity of its computational implementation and appropriateness for real-world applications.

Theory of plasticity

In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other. When additional external force $\bf f$ is applied to the body, the dynamics can be described by a second Newton's law \begin{equation} \ \nabla \cdot \boldsymbol{\sigma} + \bf{f} = 0, \end{equation} where $\boldsymbol \sigma $ is the stress tensor. Now contrary to the elastic response, where the relationship between the deformation $\bf u$ and stress $\boldsymbol \sigma$ is linear, the plastic response is a typical example of a non-linear relationship, because the stress tensor not only depends depends on the current deformation $\bf u$ but also on its history states. Generally speaking, for such highly non-linear problems, iterative numerical methods are used to solve the problem. Therefore, the residuum vector $\bf r$ is introduced and above equation is rewritten \begin{equation} \ \bf r (\bf u_{n + 1}) = \nabla \boldsymbol \sigma (\bf u_{n + 1}) - \bf f = 0. \end{equation}

TO BE FINISHED. As previously stated, all of the theory is thoroughly explained in Computational Methods for Plasticity - Theory and Applications by EA de Souza Neto, mainly chapters 4-7.