2.1.3.1. 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 \(\varepsilon\) can be decomposed into an elastic and a plastic part \(\varepsilon=\varepsilon^e + \varepsilon^p\).

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

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

(2.25)\[\begin{aligned} \eta=dev(\sigma) - \beta, \quad tr(\beta)=0 \end{aligned}\]
(2.26)\[\begin{aligned} f(\sigma,q) = \sqrt{\eta:\eta} - \sqrt{ K\left(\xi \right)} \end{aligned}\]

  • The flow rule and hardening law for a J2-plasticity model is given by:

(2.27)\[\begin{aligned} d\varepsilon^p = \frac{\gamma \, \eta}{\sqrt{\eta : \eta}} \end{aligned}\]
(2.28)\[\begin{aligned} d \beta = \gamma \frac{2}{3} H(\xi) \frac{\eta}{\sqrt{\eta : \eta}} \end{aligned}\]
(2.29)\[\begin{aligned} d \xi = \gamma \sqrt{\frac{2}{3}} \end{aligned}\]

Further explanations can be found in [Simo_1998]