17.3.1 Perzyna

The Perzyna viscoplastic model relates the viscoplastic strain rate vector to a specific function $\phi$ . This function usually depends on the current stresses and one or more state variables that include the stress-strain history. However, it is more convenient to write $\phi$ as a function of the yield function f and the internal state variable $\kappa$ . The direction of the viscoplastic strain vector is derived from a viscoplastic potential function g = g($\boldsymbol\sigma$,$\kappa$) as in rate-independent plasticity.

The Perzyna model defines the viscoplastic strain rate vector as

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{vp}}}_{} \gamma \langle \phi \kappa \rangle {\frac{{ \partial g}}{{ \partial \boldsymbol{\sigma}}}} \dot{{\lambda}} {\frac{{ \partial g}}{{ \partial \boldsymbol{\sigma}}}}$$ (17.241)

with the scalar $\gamma$ the fluidity parameter. A high value of $\gamma$ indicates a high strain rate effect, even at low values of $\phi$ . The notation $\langle$$\phi$(f,$\kappa$)$\rangle$ implies that

$$\langle \phi \kappa \rangle \begin{cases} 0 \qquad &\text{if} \quad f \leq 0 \\ [1ex] \phi ( f, \kappa ) \qquad &\text{if} \quad f > 0 \end{cases}$$ (17.242)

Simo at al. [96] have shown that Perzyna viscoplasticity cannot be combined with yield surfaces with singularities such as corners. In these cases the viscoplastic behavior at infinite times do not degenerate to the plastic behavior as would be expected. Therefore in DIANA only the Von Mises yield function is considered [§17.1.2].

DIANA comprises two types of Power Law viscoplastic functions $\phi$ , which look very similar.

$$\phi_{{1}}^{} \left(\vphantom{ \frac{f}{\sigma _{\mathrm{y}_{0}}} }\right. {\frac{{f}}{{\sigma _{\mathrm{y}_{0}}}}} \left.\vphantom{ \frac{f}{\sigma _{\mathrm{y}_{0}}} }\right)^{{\negthickspace n}}_{}$$ (17.243)

and

$$\phi_{{2}}^{} \left(\vphantom{ \frac{f}{\sigma _{\mathrm{y}}} }\right. {\frac{{f}}{{\sigma _{\mathrm{y}}}}} \left.\vphantom{ \frac{f}{\sigma _{\mathrm{y}}} }\right)^{{\negthickspace n}}_{}$$ (17.244)

In (17.243) the value of the yield function is divided by a fixed quantity: the initial yield stress $\sigma_{{\mathrm{y}_{0}}}^{}$ while in (17.244) it is divided by the current yield stress $\sigma_{{\mathrm{y}}}^{}$ that may depend on the equivalent plastic strain $\kappa$ . The evolution of the internal state variable is assumed to be given by a viscoplastic strain hardening hypothesis

$$\dot{{\kappa}} \sqrt{{ \tfrac{2}{3}\medspace \dot{\varepsilon }_{ij}^{\mathrm{vp}} \medspace \dot{\varepsilon }_{ij}^{\mathrm{vp}} }} \gamma \langle \phi_{{i}}^{} \rangle \dot{{\lambda}}$$ (17.245)