2.3.1. Flow Rule

The flow rule defines the change rate of various strain rate components during plastic deformation of the material, which is expressed in Equation (6):

$$
\kappa\_{ij}^p = \lambda \frac{\partial \mathcal{g}}{\partial \sigma\_{ij}} \tag{6}
$$

where g is a scalar function of the invariants of the deviating stress and is called plastic potential, and λ is a positive proportionality constant. When g is equal to the yield function *f* σ*ij* , Equation (6) is the associated flow rule and can be rewritten as Equation (7) [24]:

$$\frac{\dot{\varepsilon}\_x^p}{\sigma\_x^{\prime}} = \frac{\dot{\varepsilon}\_y^p}{\sigma\_y^{\prime}} = \frac{\dot{\varepsilon}\_z^p}{\sigma\_z^{\prime}} = \frac{\dot{\chi}\_{xy}^p}{2\pi\_{xy}} = \frac{\dot{\chi}\_{yz}^p}{2\pi\_{yz}} = \frac{\dot{\chi}\_{zx}^p}{2\pi\_{zx}}\tag{7}$$

where . ε*<sup>x</sup>* can be expressed in Equation (8):

$$
\dot{\varepsilon}\_x = \langle \sigma\_x - \frac{1}{3} (\sigma\_x + \sigma\_y + \sigma\_z) \rangle \dot{\lambda}. \tag{8}
$$

The . <sup>ε</sup><sup>y</sup> and . ε<sup>z</sup> have similar equations. Similarly, the shear strain rate can be expressed as:

$$
\dot{\varepsilon}\_{xy} = \frac{\dot{\mathcal{V}}\_{xy}}{2} = \tau\_{xy}\dot{\lambda}.\tag{9}
$$

The . <sup>ε</sup>yz and . εzx have similar equations. The scaling factor in Equations (8) and (9) can be derived from the work-hardening criterion as follows: .

$$
\dot{\lambda} = \frac{3\overline{\varepsilon}}{2\overline{\sigma}}\tag{10}
$$

where effective strain rate epsilon . ε is:

$$
\dot{\overline{\varepsilon}} = \sqrt{\frac{2}{3} \overline{\{\varepsilon\_{ij}\dot{\varepsilon}\_{ij}\}}}.\tag{11}
$$
