.. _J2-flow:

The J2 plasticity theory
~~~~~~~~~~~~~~~~~~~~~~~~

The J2 plasticity model is based on the following considerations:

    * Additive decomposition of the strain tensor. In this case one assumes that the strain tensor :math:`\varepsilon`  can be decomposed into an elastic and a plastic part :math:`\varepsilon=\varepsilon^e + \varepsilon^p`.

    * Elastic stress response. The stress tensor :math:`\sigma`  is related to the elastic strain by means of a stored-energy function :math:`W` according to the relation :math:`\sigma=\partial W / \partial \varepsilon`. In the particular case of linear elasticity, :math:`W` is a quadratic form of the elastic strain, i.e. :math:`W=\frac{1}{2} \varepsilon^e : \mathbb{C} : \varepsilon^e` , where :math:`\mathbb{C}` is the elastic stiffness tensor which is assumed to be constant. Then, the stress tensor can be written in the form :math:`\sigma = \mathbb{C} : (\varepsilon - \varepsilon^p)` . 

    * The yield condition is defined by means of the so called yield criterion function :math:`f(\sigma,q)` , where :math:`q` is a vector of internal variables. The admissible states of the material :math:`{\sigma,q}` are constrained as to fulfil the inequality :math:`f(\sigma,q) \leq 0`. A typical choice of internal variables for metal's plasticity is :math:`q={\xi,\beta}` , where :math:`\xi` is the equivalent plastic strain that defines the isotropic hardening of the Von Mises yield surface, and :math:`\beta` defines the center of the Von Mises yield surface in the stress deviatoric space. The resulting J2-plasticity model has the following yield condition:
    .. math::
       \begin{aligned}
       \eta=dev(\sigma) - \beta, \quad tr(\beta)=0
       \end{aligned}
       :label: yield1

    .. math::
       \begin{aligned}
       f(\sigma,q) = \sqrt{\eta:\eta} - \sqrt{ K\left(\xi \right)}
       \end{aligned}
       :label: yield2

    * The flow rule and hardening law for a J2-plasticity model is given by:
    .. math::
       \begin{aligned}
       d\varepsilon^p = \frac{\gamma \, \eta}{\sqrt{\eta : \eta}}
       \end{aligned}
       :label: FlowRule1
    
    .. math::
       \begin{aligned}
       d \beta = \gamma \frac{2}{3} H(\xi) \frac{\eta}{\sqrt{\eta : \eta}}
       \end{aligned}
       :label: FlowRule2

    .. math::
       \begin{aligned}
       d \xi = \gamma \sqrt{\frac{2}{3}}
       \end{aligned}
       :label: FlowRUle3

Further explanations can be found in [Simo_1998]_