2. Continuous model

In accordance with the rheological presentation, the stress \(\sigma\) is the sum of two contributions: that coming from the elastoplastic branch \({\sigma }^{p}\) and that coming from the viscoelastic branch \({\sigma }^{v}\):

(2.1)\[ \ mathrm {\ sigma} = {\ mathrm {\ sigma}}} ^ {p} + {\ mathrm {\ sigma}}} ^ {v}\]

2.1. Elastoplastic branch

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

(2.2)\[ {\ mathrm {\ sigma}} ^ {p} = {C} ^ {p} = {C} ^ {p}\ mathrm {\ epsilon}} ^ {p}) = {C} ^ {p}) =3 {K} ^ {p}) =3 {K} ^ {p}\ mathrm {\ epsilon} - {\ mathrm {\ epsilon} - {\ mathrm {\ epsilon}} - {\ mathrm {\ epsilon}}} ^ {p}) =3 {K} ^ {p}) =3 {K} ^ {p}) =3 {K} ^ {p}) =3 {K} ^ {p}) =3 {K} ^ {p}}\ mathrm {\ epsilon} - {\ on}} ^ {p}) S+2 {\ mathrm {\ mu}}} ^ {mu}}} ^ {p}\ mathit {dev} (\ mathrm {\ epsilon}}} ^ {p}} ^ {p})\ text {})\ text {}})\ text {}};\ text {}};\ text {} 3 {K} {p} =\ frac {{E} ^ {p}}} {1-2 {\ mathron}} {1-2 {\ mathron}} {1-2 {\ mathrm}} {m {\nu}} ^ {p}}\ text {}}\ text {}} 2 {\ mathrm {\ mu}}} ^ {p} =\ frac {{E} ^ {p}}} {1+ {\ mathrm {\nu}}} {\ mathrm {\nu}}} ^ {p}}\]

where \(\mathit{sph}()\) and :math:`` respectively denote the normalized spherical part and the deviator of a second-order tensor, while \(S\) is the standard identity tensor of order two. For any tensor \(a\) of order two, this is expressed by:

(2.3)\[ S\ equiv\ frac {\ mathit {Id}} {\ sqrt {3}} {\ sqrt {3}}}\ text {;}\ mathit {sph} (a)\ equiv S\ mathrm {:} a=\ frac {\ mathit {tr} (a)\ frac {\ mathit {tr} (a)\ frac {\ mathit {tr} (a)\ frac {\ mathit {tr} (a)} {\ mathit {tr} (a)} {\ mathit {tr} (a)} {\ mathit {tr} (a)} {\ mathit {tr} (a)} {\ mathit {tr} (a)} {\ mathit {tr} (a)} {\ mathit {tr} (a)\ mathrm {:} a) S\]

Note that the use of \(S\) and \(\mathit{sph}()\) rather than \(\mathit{Id}\) and \(\mathit{tr}()\) makes it possible to avoid the presence of \(3\) and \(1/3\) in the various expressions thanks to their standardized nature.

We now introduce the threshold function, which involves a reminder constraint \(X=C{\epsilon }^{p}\):

\[\]

: label: eq-4

F (mathrm {sigma}, {mathrm {epsilon}}} ^ {p}) = {vertmathrm {sigma} -C {mathrm {epsilon}}} ^ {p}epsilon}}} ^ {epsilon}}} ^ {epsilon}}} ^ {c}} -C {mathrm {epsilon}}} ^ {epsilon}}} ^ {epsilon}}} ^ {epsilon}}} ^ {epsilon}} ^ {epsilon}}} ^ {epsilon}} ^ {epsilon}}} ^ {epsilon}} ^ {epsilon}} ^ {epsilon}}} {Vert aVert} _ {mathit {eq}}equivsqrt {frac {3} {2}mathit {dev} (a)mathrm {:}mathrm {:}mathit {eq}}equivsqrt {frac {3} {2}mathit {dev} (a)} (a)mathrm {:}:}mathit {eq}}

The evolution of plastic deformation is governed by the flow equation. The flow direction is normal to the threshold surface (associated plasticity):

\[\]

: label: eq-5

{dot {mathrm {epsilon}}}} ^ {p}} =frac {3} {2}dot {mathrm {kappa}}frac {N} {{green Ngreen}}} _ {green Ngreengreen}} _ {mathit {eq}}} =frac {epsilon}}}}frac {epsilon}}}}frac {epsilon}}}} ^ {p} =frac {epsilon}}}} ^ {p} =frac {epsilon}}}} ^ {p} =frac {epsilon}}}}frac {epsilon}}} {epsilon}}}} C {mathrm {epsilon}}} ^ {p})

Finally, the evolution of the plastic multiplier \(\dot{\kappa }\) is determined by the consistency condition:

(2.4)\[ \ dot {\ mathrm {\ kappa}}\ ge 0\ text {;}} F (\ mathrm {\ sigma}, {\ mathrm {\ epsilon}} ^ {p})\ le 0\ text {;}}\ dot {\ text {;}}\ dot {\ text {;}}\ dot {\ mathrm {\ kappa}} F (\ mathrm {\ epsilon}})\ le 0\ text {;;};}\ dot {\ mathrm {;}}\ dot {\ mathrm {\ kappa}} F (\ mathrm {\ epsilon}}) ^ {on}} p}) =0\]

2.2. Viscoelastic branch

It is a Maxwell model whose shock absorber is non-linear Norton type. The (mechanical) deformation \(\epsilon\) breaks down additively into a viscous part \({\epsilon }^{v}\) and an elastic part \(\epsilon -{\epsilon }^{v}\), the latter being linked to the stress of the viscoelastic branch by the following elastic relationship:

(2.5)\[ {\ mathrm {\ sigma}} ^ {v} = {C} ^ {v} = {C} ^ {v}\ mathrm {\ epsilon}} ^ {v}) = {C} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}\ mathrm {\ epsilon} - {\ mathrm {\ epsilon} - {\ mathrm {\ epsilon}} - {\ mathrm {\ epsilon}}} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}}\ mathrm {\ epsilon} - {\ mathrm {\ epsilon}} on}} ^ {v}) S+2 {\ mathrm {\ mu}}} ^ {v}} ^ {v}\ mathit {dev} (\ mathrm {\ epsilon}}} ^ {v}} ^ {v})\ text {v})\ text {})\ text {}};\ text {}};\ text {}};\ text {} 3 {K}} ^ {v} =\ frac {{E} ^ {v}}} {1-2 {\ mathron}} {1-2 {\ mathrm}} {v}} {1-2 {\ mathrm}} {m {\nu}} ^ {v}}\ text {}}\ text {}} 2 {\ mathrm {\ mu}}} ^ {v} =\ frac {{E} ^ {v}}} {1+ {\ mathrm {\nu}}} {\ mathrm {\nu}} {\\]

The evolution of viscous deformation is governed by Norton’s law of evolution:

(2.6)\[ {\ dot {\ mathrm {\ epsilon}}}} ^ {v}} =\ frac {1} {{\ mathrm {\ eta}}} ^ {\ mathrm {\ gamma}}}} {\ green {\ mathrm {\ sigma}}}} {\ mathrm {\ gamma}}}}} {\ green {\ mathrm {\ gamma}}}} {\ green {\ mathrm {\ gamma}}}} {\ green {\ mathrm {\ sigma}}}} {\ green {\ mathrm {\ sigma}}}} {\ green {\ mathrm {\ sigma}}}} {\ green {\ mathrm {\ sigma}}}} {\ green {\ mathrm {\ sigma}}}} {\ green {\ mathrm {\ sigma}} ^ {v}\ text {;}\ text {;}\ mathrm {\ gamma}\ equiv\ frac {1} {n}\ text {;} {\ Vert a\ Vert} _ {V}\ equiv\ sqrt {;}\ equiv\ sqrt {;}}\ equiv\ sqrt {;}}\ equiv\ sqrt {;}}\ equiv\ sqrt {;}}\ equiv\ sqrt {;}\ sqrt {;}\ sqrt {a\ sqrt {a\ mathrm {:} {:} a\ equiv v (1+ {\ mathrm {\nu}}} ^ {d})\ mathit {dev} (a) + (1-2 {\ mathrm {\nu}} ^ {d}})\ mathit {sph} (a) S\]

where we introduced \(\gamma\) the inverse exponent of \(n\) to simplify the notations and \(V\) introduced a fourth-order isotropic tensor, without a unit, which allows to « misalign » \({\dot{\epsilon }}^{v}\) with respect to \({\sigma }^{v}\).

2.3. Post-processing variables

Two variables can be combined in some post-treatment formulations to predict material ruin. These are cumulative plastic deformation and cumulative viscous deformation, respectively defined as follows:

(2.7)\[ {\ dot {\ mathrm {\ epsilon}}}} _ {\ mathit {cum}}} ^ {p} =\ sqrt {\ frac {2} {3} {\ dot {\ mathrm {\ epsilon}}}}}}} ^ {epsilon}}}}} ^ {epsilon}}}} {\ mathrm {\ epsilon}}}} ^ {epsilon}}}}\ text {;} {\ dot {\ mathrm {\ epsilon}}}} _ {\ mathit {cum}}} ^ {v} =\ sqrt {\ frac {2} {3} {\ dot {\ mathrm {\ epsilon}}}}} ^ {epsilon}}}}} ^ {epsilon}}}}} ^ {epsilon}}}} ^ {epsilon}}}} ^ {epsilon}}}} ^ {epsilon}}}} ^ {epsilon}}}} ^ {epsilon}}}} ^ {epsilon}}}} ^ {epsilon}}}} ^ {v}}}\]

For practical reasons, they are calculated at the end of the behavior integration phase and stored among the internal variables.