6. Viscoelastic regularization REGU_VISC

In the presence of hardware instabilities, resulting for example from damage, the solution no longer depends continuously on time (or loading parameters) but shows jumps. Piloting is a technique for overcoming these jumps, by allowing non-monotonic changes in the load to regain continuity. But these techniques are not applicable when physical time intervenes, whether through a history of control variables (thermal transient, for example) or in the behavior of the material (viscosity).

Viscous regularization is another approach to regain continuity. It consists in superimposing on the constraints resulting from the law of behavior \(\sigma\) a constraint of viscous origin \({\sigma }_{\mathit{vis}}\) in order to sufficiently penalize very rapid evolutions (whose limit is a jump in solution):

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: label: eq-148

{mathrm {sigma}} _ {mathit {tot}}} =mathrm {sigma} + {mathrm {sigma}} _ {mathit {vis}}text {;}text {;}frac {}}frac {mathit {tot}}} _ {mathit {tot}}} {dmathrm {epsilon}}} =frac {dmathrm {sigma}} {dmathrm {epsilon}}} +frac {d {mathrm {sigma}} _ {mathit {vis}}}} {dmathrm {epsilon}}} {dmathrm {epsilon}}

This is a digital technique, without a priory relationship with the viscous reality of the material. The law of viscoelastic behavior that governs viscoelastic stress \({\sigma }_{\mathit{vis}}\) is described in fascicle [R5.03.34].