2. Continuous model

2.1. Behavioral equations: local plastic model

It is a VonMises threshold plasticity model and classical isotropic work hardening. The (mechanical) deformation \(\epsilon\) breaks down additively into a plastic part \({\epsilon }^{p}\) and an elastic part \({\epsilon }^{e}\), the latter being linked to the stress by the following elastic relationship:

(2.1)\[\mathrm{\sigma }=K\mathit{tr}({\mathrm{\epsilon }}^{e})\mathit{Id}+2\mathrm{\mu }\mathit{dev}({\mathrm{\epsilon }}^{e})\text{};\text{}3K=\frac{E}{1-2\mathrm{\nu }}\text{}2\mathrm{\mu }=\frac{E}{1+\mathrm{\nu }}\]

where :math:`` and :math:`` respectively designate the trace and the deviator of a second-order tensor.

We now introduce the threshold function, where the work hardening function :math:`` has already been introduced in ():

(2.2)\[F(\mathrm{\sigma },\mathrm{\kappa })={\mathrm{\sigma }}_{\mathit{eq}}-R(\mathrm{\kappa })\text{};\text{}{\mathrm{\sigma }}_{\mathit{eq}}=\sqrt{\frac{3}{2}\mathit{dev}(\mathrm{\sigma })\mathrm{:}\mathit{dev}(\mathrm{\sigma })}\]

We remain within the framework of plasticity with positive work hardening so that we require that the \(R(\kappa )\) work hardening function be increasing.

The evolution of plastic deformation is governed by the flow equation. When the threshold function is differentiable with respect to \(\sigma\), the flow direction is normal to the threshold surface:

(2.3)\[\mathit{si}\text{}{\mathrm{\sigma }}_{\mathit{eq}}\ne 0\text{}{\dot{\mathrm{\epsilon }}}^{p}=\dot{\mathrm{\kappa }}N\text{};\text{}N=\frac{3}{2}\mathit{dev}\frac{(\mathrm{\sigma })}{{\mathrm{\sigma }}_{\mathit{eq}}}\]

In the singular case where \({\sigma }_{\mathit{eq}}=0\), the notion of derivative is generalized via notions of convex analysis (sub-gradient). The flow direction is only subject to the following condition:

(2.4)\[\mathit{si}\text{}{\mathrm{\sigma }}_{\mathit{eq}}=0\text{}\sqrt{\frac{2}{3}}\Vert {\dot{\mathrm{\epsilon }}}^{p}\Vert \le \dot{\mathrm{\kappa }}\text{;}\mathit{tr}({\dot{\mathrm{\epsilon }}}^{p})=0\]

Finally, the evolution of the work-hardening variable is fixed by the consistency condition:

(2.5)\[\dot{\mathrm{\kappa }}\ge 0\text{;}F(\mathrm{\sigma },\mathrm{\kappa })\le 0\text{;}\dot{\mathrm{\kappa }}F(\mathrm{\sigma },\mathrm{\kappa })=0\]

2.2. Taking viscosity into account

In the presence of viscosity, plastic flow is delayed. More exactly, the speed of evolution of work hardening is no longer fixed by the consistency condition () but it is now a function of the intensity of the threshold function according to Norton’s law:

(2.6)\[\dot{\mathrm{\kappa }}={\left(\frac{⟨F(\mathrm{\sigma },\mathrm{\kappa })⟩}{K}\right)}^{n}\]

where McAuley’s square brackets \(⟨\mathrm{.}⟩\) refer to the positive part. The equations () to () remain unchanged. When \(K\to 0\), we find the elastoplastic model.

2.3. Taking into account the work-hardening gradient

A non-local formulation with an internal variable gradient is suitable when the \(\nabla \kappa\) work hardening gradient becomes important. It consists in introducing an additional term \({\mathrm{\Phi }}^{\mathit{grad}}\) into free energy that reflects the interactions between neighboring material points, see [R5.04.01]:

(2.7)\[{\mathrm{\Phi }}^{\mathit{grad}}(\nabla \mathrm{\kappa })=\frac{1}{2}c{\left(\nabla \mathrm{\kappa }\right)}^{2}\]

Using the relaxed augmented formulation presented in [R5.04.01], the impact on the behavioral relationship is reflected in a modification of the thermodynamic force associated with the work hardening variable and therefore, in practice, in a modification of the work hardening function:

(2.8)\[\stackrel{~}{R}(\mathrm{\kappa })=R(\mathrm{\kappa })+r\mathrm{\kappa }-\mathrm{\varphi }\text{;}\mathrm{\varphi }=\mathrm{\lambda }+\mathit{ra}\]

where \(\lambda\) is the Lagrange multiplier associated with relaxation, \(r>0\) the increase coefficient and \(a\) the interpretation of the work-hardening variable at the structural scale. These three quantities are data as far as the behavioral relationship is concerned. In the end, the work hardening function is corrected by an affine term, which does not therefore add complexity compared to the numerical processing of the model. On the other hand, the sign and the amplitude of \(\varphi\) not being fixed, this explains why it is necessary to take into account the case of a singular flow for which \({\sigma }_{\mathit{eq}}=0\).