3. Digital integration

With regard to the integration of the law of behavior, that is to say the calculation of internal variables and stresses with a given (mechanical) deformation history, there is no longer any need to distinguish whether it is a local formulation or a gradient formulation. In fact, the impact of this last one is limited to modifying the work hardening function according to (). Here we will continue to rate it \(R(\kappa )\) without distinction.

On the other hand, we will distinguish the plastic and viscoplastic cases, even if it can be shown that the latter is also limited to a correction of the threshold function in the discretized equations.

Finally, in the presence of a Lüders plateau, we chose to simplify the resolution by reducing ourselves to the situation without a Lüders plateau. Indeed, due to the uniqueness of the solution of the discretized problem, it is located either on the plateau or beyond. We start by looking for it on the set: we are then brought back to the case of perfect plasticity (a particular case of function \(\widehat{R}\)). If the solution obtained in this way proves to be beyond the plate, it is not taken into account (this is not the solution of the problem) and the resolution is repeated but this time looking for the solution on part \(\widehat{R}(\kappa )\) of the work hardening function, that is to say without worrying about the Lüders plate (by uniqueness, we are then certain that this new solution is effectively beyond the plate). This is why the method of solving in the absence of a Lüders plateau is simply described below.

3.1. Discretized equations

The time discretization of behavioral equations is based on an implicit Euler schema (even in viscoplasticity), that is to say that the various variables of the problem are expressed at the final moment of the time step in question. We note \({Q}_{n}\) the value of a quantity \(Q\) at the start of the time step, \(\mathrm{\Delta }Q\) its increment during the time step and (simply) \(Q\) its value at the end of the time step. The mechanical state at the start of time step \(({\epsilon }_{n},{{\epsilon }^{p}}_{n},{\kappa }_{n})\) is assumed to be known as well as the deformation increment \(\mathrm{\Delta }\epsilon\) (and therefore also the deformation \(\epsilon\)). It is then a question of calculating the increments of internal variables \({\mathrm{\Delta }{\epsilon }^{p}}_{n}\) and \(\mathrm{\Delta }\kappa\), as well as the constraint at the end of the time step \(\sigma\).

The discretization of the stress-strain relationship () is written as follows:

(3.1)\[\mathrm{\sigma }=K\mathit{tr}(\mathrm{\epsilon }-{{\mathrm{\epsilon }}^{p}}_{n}-\mathrm{\Delta }{\mathrm{\epsilon }}^{p})\mathit{Id}+2\mathrm{\mu }\mathit{dev}(\mathrm{\epsilon }-{{\mathrm{\epsilon }}^{p}}_{n}-\mathrm{\Delta }{\mathrm{\epsilon }}^{p})\]

Regardless of the flow regime, regular () or singular (), we note that the trace of the plastic deformation increment is zero, so that the same is true for plastic deformation at any time (isochoric plastic deformation). The relationship () is simplified and an elastic stress (or elastic test stress) can be shown there, noted \({\sigma }^{e}\) and which is known:

(3.2)\[\mathrm{\sigma }={\mathrm{\sigma }}^{e}-2\mathrm{\mu }\mathrm{\Delta }{\mathrm{\epsilon }}^{p}\text{;}{\mathrm{\sigma }}^{e}=K\mathit{tr}(\mathrm{\epsilon })\mathit{Id}+2\mathrm{\mu }\mathit{dev}(\mathrm{\epsilon }-{{\mathrm{\epsilon }}^{p}}_{n})\]

In the case of a regular flow, the temporal discretization of () is written as:

(3.3)\[{\mathrm{\Delta }\mathrm{\epsilon }}^{p}=\frac{3}{2}\frac{\mathrm{\Delta }\mathrm{\kappa }}{{\mathrm{\sigma }}_{\mathit{eq}}}\mathit{dev}({\mathrm{\sigma }}^{e}-2\mathrm{\mu }\mathrm{\Delta }{\mathrm{\epsilon }}^{p})\]

After a few algebraic manipulations, we deduce:

(3.4)\[{\mathrm{\sigma }}_{\mathit{eq}}={\mathrm{\sigma }}_{\mathit{eq}}^{e}-3\mathrm{\mu }\mathrm{\Delta }\mathrm{\kappa }\]

(3.5)\[{\mathrm{\Delta }\mathrm{\epsilon }}^{p}=\frac{3}{2}\mathrm{\Delta }\mathrm{\kappa }\frac{\mathit{dev}({\mathrm{\sigma }}^{e})}{{\mathrm{\sigma }}_{\mathit{eq}}^{e}}\]

(3.6)\[\mathrm{\sigma }=\frac{1}{3}\mathit{tr}({\mathrm{\sigma }}^{e})\mathit{Id}+({\mathrm{\sigma }}_{\mathit{eq}}^{e}-3\mathrm{\mu }\mathrm{\Delta }\mathrm{\kappa })\frac{\mathit{dev}({\mathrm{\sigma }}^{e})}{{\mathrm{\sigma }}_{\mathit{eq}}^{e}}\]

The regular diet is only valid if \({\sigma }_{\mathit{eq}}>0\), according to ():

(3.7)\[3\mathrm{\mu }\mathrm{\Delta }\mathrm{\kappa }<{\mathrm{\sigma }}_{\mathit{eq}}^{e}\]

In the case of a singular flow, \({\sigma }_{\mathit{eq}}=0\). Given vonMises” definition of equivalent stress, this is equivalent to \(\mathit{dev}(\sigma )=0\). In particular, the constraint is simply written:

(3.8)\[\mathrm{\sigma }=\frac{1}{3}\mathit{tr}({\mathrm{\sigma }}^{e})\mathit{Id}\text{;}{\mathrm{\sigma }}_{\mathit{eq}}=0\]

Moreover, by injecting \(\mathit{dev}(\sigma )=0\) into (), we deduce the plastic deformation increment:

(3.9)\[2\mathrm{\mu }\mathrm{\Delta }{\mathrm{\epsilon }}^{p}=\mathit{dev}({\mathrm{\sigma }}^{e})\]

And the flow is subject to (), that is to say after discretization:

(3.10)\[\sqrt{\frac{2}{3}}\Vert {\mathrm{\Delta }\mathrm{\epsilon }}^{p}\Vert \le \mathrm{\Delta }\mathrm{\kappa }\]

By reporting () into (), it comes:

(3.11)\[{\mathrm{\sigma }}_{\mathit{eq}}^{e}\le 3\mathrm{\mu }\mathrm{\Delta }\mathrm{\kappa }\]

It may be noted that the conditions () and () make it possible to distinguish the regular case from the singular case. In particular, the separation value between the two regimes is introduced:

(3.12)\[{\mathrm{\kappa }}^{s}={\mathrm{\kappa }}_{n}+\frac{{\mathrm{\sigma }}_{\mathit{eq}}^{e}}{3\mathrm{\mu }}\]

3.2. Solving the discretized problem — Plastic case without viscosity

At this point, all that remains is to determine increment \(\mathrm{\Delta }\kappa\) via the consistency condition () which is written after discretization:

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

To do this, we start by considering the hypothesis of an elastic solution, that is to say \(\mathrm{\Delta }\kappa =0\) and \(\mathrm{\Delta }{\epsilon }^{p}\). In this case, \(\sigma ={\sigma }^{e}\) and this solution is valid based on () as long as \({F}^{e}=F({\sigma }^{e},{\kappa }_{n})\le 0\).

Now let’s say \({F}^{e}>0\). It’s about finding \(\mathrm{\Delta }\kappa >0\) such as \(F(\sigma ,\kappa )=0\). In the hypothesis of a regular flow (\({\kappa }_{n}<\kappa <{\kappa }^{s}\)), we can inject () into the expression of the threshold function which is then equal to:

(3.14)\[F(\mathrm{\sigma },\mathrm{\kappa })={\mathrm{\sigma }}_{\mathit{eq}}^{e}-3\mathrm{\mu }\mathrm{\Delta }\mathrm{\kappa }-R(\mathrm{\kappa })=\widehat{M}(\mathrm{\kappa })\]

As the work hardening function \(R\) is increasing, the function \(\widehat{M}\) is strictly decreasing. By the way, \(\widehat{M}({\kappa }_{n})={F}^{e}>0\). There is a (unique) solution in a regular flow regime if and only if \(\widehat{M}({\kappa }^{s})<0\). A Newton method with controlled terminals then makes it possible to determine it.

In the case of a singular flow (\({\kappa }^{s}\le \kappa\)), \({\sigma }_{\mathit{eq}}=0\) and the consistency condition is simply written:

(3.15)\[F(\mathrm{\sigma },\mathrm{\kappa })=-R(\mathrm{\kappa })=0\]

Note that \(-R\) is a (strictly) decreasing function. Also, when \(\kappa \to +\mathrm{\infty }\), \(-R\) is negative. Indeed, in the local case, \(R(+\mathrm{\infty })\ge R(0)>0\). And in the non-local case, \(r>0\) so that the function \(R\) tends to infinity. There is therefore a (unique) solution in a singular flow regime if and only if \(-R({\kappa }^{s})=\widehat{M}({\kappa }^{s})\ge 0\). Again, a Newton method makes it possible to determine this solution.

Finally, there is a unique solution to the problem of integrating the law of behavior. The regime depends on the values \(\widehat{{M}^{e}}=\widehat{M}({\kappa }_{n})\) and \(\widehat{{M}^{s}}=\widehat{M}({\kappa }^{s})\), where \(\widehat{{M}^{e}}\ge \widehat{{M}^{s}}\):

• if \(0\ge \widehat{{M}^{e}}\ge \widehat{{M}^{s}}\): elastic solution;

• if \(\widehat{{M}^{e}}\ge 0\ge \widehat{{M}^{s}}\): plastic solution with regular flow;

• if \(\widehat{{M}^{e}}\ge \widehat{{M}^{s}}\ge 0\): plastic solution with a singular flow.

3.3. Solving the discretized problem: viscoplastic case

In the presence of viscosity, the consistency condition () is replaced by Norton’s law of evolution (). Once discretized, the latter is written:

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

And by reversing this relationship:

(3.17)\[{C}_{\mathrm{\Delta }t}{\mathrm{\Delta }\mathrm{\kappa }}^{q}=⟨F(\mathrm{\sigma },\mathrm{\kappa })⟩\text{;}{C}_{\mathrm{\Delta }t}=\frac{K}{{\mathrm{\Delta }t}^{1/n}}\text{;}q=\frac{1}{n}\]

It can still be expressed equivalently in the form of a Kuhn and Tucker relationship:

(3.18)\[\mathrm{\Delta }\mathrm{\kappa }\ge 0\text{;}{F}_{v}(\mathrm{\sigma },\mathrm{\kappa })\le 0\text{;}{F}_{v}(\mathrm{\sigma },\mathrm{\kappa })\mathrm{\Delta }\mathrm{\kappa }=0\]

(3.19)\[\text{où}{F}_{v}(\mathrm{\sigma },\mathrm{\kappa })=F(\mathrm{\sigma },\mathrm{\kappa })-{C}_{\mathrm{\Delta }t}{\mathrm{\Delta }\mathrm{\kappa }}^{q}={\mathrm{\sigma }}_{\mathit{eq}}-{R}_{v}(\mathrm{\kappa })\text{;}{R}_{v}(\mathrm{\kappa })=R(\mathrm{\kappa })+{C}_{\mathrm{\Delta }t}{(\mathrm{\kappa }-{\mathrm{\kappa }}_{n})}^{q}\]

After discretization, the consideration of viscosity is reduced to the addition of a term (positive and increasing) in the work hardening function.

We might be tempted to apply the same method of resolution as before but, unfortunately, this new term is not differentiable in \({\kappa }_{n}\) because \(q<1\) in general (\(n>1\)) .This is why we propose to start looking for the solution of the problem without viscosity (in a rough way) and we then switch to the solution with viscosity (in a crude way) and we then switch to the solution with viscosity by a Newton method by changing the variable \(\mathrm{\Delta }\kappa \to {C}_{\mathrm{\Delta }t}{\mathrm{\Delta }\kappa }^{q}\), thus eliminating the non-differentiability problem (and the steep slopes that are penalizing for a Newton method).

3.4. Tangent matrix

In the local case, it is a question of determining the tensor of order four \({H}_{\sigma ϵ}=\delta \sigma /\delta \epsilon\). In the non-local case, it is also necessary to build the second order tensors \({H}_{\sigma \varphi }=\delta \sigma /\delta \varphi\) and \({H}_{\kappa ϵ}=\delta \kappa /\delta \epsilon\) as well as the scalar \({H}_{\kappa \varphi }=\delta \kappa /\delta \varphi\).

The tangent operator depends on the flow regime. In the elastic case, it is reduced to:

(3.20)\[\mathrm{\delta }\mathrm{\sigma }=K\mathit{tr}(\mathrm{\delta }\mathrm{\epsilon })\mathit{Id}+2\mathrm{\mu }\mathit{dev}(\mathrm{\delta }\mathrm{\epsilon })\text{;}\mathrm{\delta }\mathrm{\kappa }=0\]

In the case of the singular regime, the expression is also very simple:

(3.21)\[\mathrm{\delta }\mathrm{\sigma }=K\mathit{tr}(\mathrm{\delta }\mathrm{\epsilon })\mathit{Id}\text{;}\mathrm{\delta }\mathrm{\kappa }={\left(\frac{\partial \stackrel{~}{{R}_{v}}}{\partial \mathrm{\kappa }}\right)}^{-1}\mathrm{\delta }\mathrm{\varphi }\]

where the work hardening function contains, where appropriate, terms related to non-local and to viscosity (hence the tilde and the index to recall this explicitly).

The regular flow regime leads to a more complex tangent operator. The use of a vonMises criterion leads to the following derivatives:

(3.22)\[\mathrm{\delta }{\mathrm{\sigma }}_{\mathit{eq}}=3\mathrm{\mu }N\mathrm{:}\mathrm{\delta }\mathrm{\epsilon }\]

(3.23)\[\mathrm{\delta }N=\frac{1}{{\mathrm{\sigma }}_{\mathit{eq}}}\left[2\mathrm{\mu }\mathit{dev}(\mathrm{\delta }\mathrm{\epsilon })-\mathrm{\delta }{\mathrm{\sigma }}_{\mathit{eq}}N\right]\]

By then applying derivatives of compound functions, we obtain:

(3.24)\[\mathrm{\delta }\mathrm{\sigma }=K\mathit{tr}(\mathrm{\delta }\mathrm{\epsilon })\mathit{Id}+2\mathrm{\mu }\left[1-\frac{3\mathrm{\mu }\mathrm{\Delta }\mathrm{\kappa }}{{\mathrm{\sigma }}_{\mathit{eq}}}\right]\mathit{dev}(\mathrm{\delta }\mathrm{\epsilon })+\frac{9{\mathrm{\mu }}^{2}\mathrm{\Delta }\mathrm{\kappa }}{{\mathrm{\sigma }}_{\mathit{eq}}}N\otimes N\mathrm{:}\mathrm{\delta }\mathrm{\epsilon }-3\mathrm{\mu }N\mathrm{\delta }\mathrm{\kappa }\]

(3.25)\[\mathrm{\delta }\mathrm{\kappa }={\left[3\mathrm{\mu }+\frac{\partial \stackrel{~}{{R}_{v}}}{\partial \mathrm{\kappa }}\right]}^{-1}\left[\mathrm{\delta }{\mathrm{\sigma }}_{\mathit{eq}}+\mathrm{\delta }\mathrm{\varphi }\right]\]

In the prediction phase (\(\epsilon ={\epsilon }_{n}\)), two specificities should be noted:

• In the presence of viscosity, \(\delta \kappa =0\) because the derivative :math:`` is infinite.

• The tangent operators are not continuous at the transitions between regimes (\(\widehat{{M}^{e}}\approx 0\) and \(\widehat{{M}^{s}}\approx 0\)) and in this case we opt for the domain corresponding to the regime of the previous step. On the other hand, if the mechanical state at the start of the time step is not in the vicinity of a transition for, then the regime is fixed as for the integration of the law.