3. Integrating the behavioral relationship

To achieve numerical integration of the law of behavior, time discretization is performed and an implicit Euler scheme is adopted, considered appropriate for elastoplastic behavior relationships. From now on, the following notations will be used: \({A}^{-},A\text{et}\Delta A\) represent respectively the values of a quantity \(A\) at the beginning and at the end of the time step in question as well as its increment during the step. The problem is then as follows: knowing the state at time \({t}^{-}\) as well as the deformation increments \(\Delta \varepsilon\) and temperature \(\Deltat\), determine the state at time \(t\) as well as the constraints \(\sigma\).

Initially, the variations in characteristics with respect to temperature are taken into account, noting that:

\({\sigma }^{H}\text{=}\frac{K}{{K}^{-}}{\sigma }^{{H}^{-}}+K\text{tr}(\Delta \varepsilon -\Delta {\varepsilon }^{\text{th}})\) [éq3-1]

\(\tilde{\sigma }=\frac{\mu }{{\mu }^{-}}{\tilde{\sigma }}^{-}+2\mu (\Delta \tilde{\varepsilon }-\Delta {\varepsilon }^{p})\) [éq3-2]

\(X=\frac{C}{{C}^{-}}{X}^{-}+C\Delta {\varepsilon }^{p}\) [éq3-3]

In view of the equation [eq], it can be seen that the hydrostatic behavior is purely elastic. Only the treatment of the deviatoric component is delicate. To simplify future writing, we introduce \({\tilde{s}}^{e}\) the \(\tilde{\sigma }-X\) difference in the absence of an increment of plastic deformations, so that:

\(\tilde{\sigma }-X=\underset{{\tilde{s}}^{e}}{\underset{\underbrace{}}{\frac{\mu }{{\mu }^{-}}{\tilde{\sigma }}^{-}-\frac{C}{{C}^{-}}{X}^{-}\text{+}2\mu \Delta \tilde{e}}}-(2\mu \text{+}C)\Delta {\varepsilon }^{p}\) [éq3-4]

The flow equations [eq] and [eq] and the coherence condition [eq] are written once discretized and noting that \(p\text{=}\lambda\):

\(\Delta {\varepsilon }^{p}\text{=}\frac{3}{2}\Deltap \frac{\tilde{\sigma }\text{-}X}{{(\tilde{\sigma }\text{-}X)}_{\text{eq}}}\) [éq3-5]

\(F\text{<=}0\Deltap \text{>=}0F\Deltap \text{=}0\) [éq3-6]

The treatment of the coherence condition [eq] is classical. We start with an elastic test (\(\Deltap \text{=}0\)) which is indeed the solution if the plasticity criterion is not exceeded, that is to say if:

\(F\text{=}{s}_{\text{eq}}^{e}\text{-}R({p}^{-})\text{<=}0\) [éq3-7]

Otherwise, the solution is plastic (\(\Deltap \text{>}0\)) and the consistency condition [eq] is reduced to \(F\text{=}0\). To solve it, we start by showing that we can reduce ourselves to a scalar problem by eliminating \(\Delta {\varepsilon }^{p}\). In fact, taking into account [eq] and [eq], we see that \(\Delta {\varepsilon }^{p}\) is collinear to \({\tilde{s}}^{e}\) because:

\(\Delta {\varepsilon }^{p}\text{=}\frac{3}{2}\frac{\Deltap }{{(\tilde{\sigma }-X)}_{\text{eq}}}\left[{\tilde{s}}^{e}\text{-}(2\mu \text{+}C)\Delta {\varepsilon }^{p}\right]\) [éq3-8]

Moreover, according to [eq], the standard of \(\Delta {\varepsilon }^{p}\) is equal to:

\({(\Delta {\varepsilon }^{p})}_{\text{eq}}\text{=}\frac{3}{2}\Deltap\) [éq3-9]

We therefore immediately deduce the expression for \(\Delta {\varepsilon }^{p}\) as a function of \(\Deltap\):

\(\Delta {\varepsilon }^{p}=\frac{3}{2}\Deltap \frac{{\tilde{s}}^{e}}{{s}_{\text{eq}}^{e}}\) [éq3-10]

Now all that remains is to replace \(\Delta {\varepsilon }^{p}\) with its expression [eq] in the equation [eq] we get:

\(\tilde{\sigma }\text{-}X\text{=}{\tilde{s}}^{e}\left[1\text{-}\frac{\frac{3}{2}(2\mu \text{+}C)\Delta p}{{s}_{\text{eq}}^{e}}\right]\)

by bringing \(\tilde{\sigma }\text{-}X\) into equation \(F\text{=}0\), we come back to a scalar equation in \(\Delta p\) to be solved, namely:

\(\mid {s}_{\text{eq}}^{e}\text{-}\frac{3}{2}(2\mu \text{+}C)\Deltap \mid \text{-}R({p}^{\text{-}}\text{+}\Deltap )=0\) [éq3-11]

Insofar as the function \(R\) is positive, which we will now admit, there is a \(\Delta p\) solution to this equation, characterized by:

\(\frac{3}{2}(2\mu \text{+}C)\Delta p\text{+}R({p}^{-}\text{+}\Delta p)\text{=}{s}_{\text{eq}}^{e}\) [éq3-12]

Where \(0<\Delta p<\frac{2}{3}\frac{{s}_{\text{eq}}^{e}}{2\mu \text{+}C}\)

Note that within the interval specified in [eq], the solution is unique. For details on solving this equation, refer to [R5.03.02].

The particular case of plane stresses is studied in § 6.