Continuous model ============== In accordance with the rheological presentation, the stress :math:`\sigma` is the sum of two contributions: that coming from the elastoplastic branch :math:`{\sigma }^{p}` and that coming from the viscoelastic branch :math:`{\sigma }^{v}`: .. math:: :label: eq-1 \ mathrm {\ sigma} = {\ mathrm {\ sigma}}} ^ {p} + {\ mathrm {\ sigma}}} ^ {v} Elastoplastic branch ----------------------- It is a VonMises threshold plasticity model and classical linear kinematic work hardening. The (mechanical) deformation :math:`\epsilon` breaks down additively into a plastic part :math:`{\epsilon }^{p}` and an elastic part :math:`\epsilon -{\epsilon }^{p}`, the latter being linked to the stress of the elastoplastic branch by the following elastic relationship: .. math:: :label: eq-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 :math:`\mathit{sph}()` and :math:`` respectively denote the normalized spherical part and the deviator of a second-order tensor, while :math:`S` is the **standard** identity tensor of order two. For any tensor :math:`a` of order two, this is expressed by: .. math:: :label: eq-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 :math:`S` and :math:`\mathit{sph}()` rather than :math:`\mathit{Id}` and :math:`\mathit{tr}()` makes it possible to avoid the presence of :math:`3` and :math:`1/3` in the various expressions thanks to their standardized nature. We now introduce the threshold function, which involves a reminder constraint :math:`X=C{\epsilon }^{p}`: .. math:: : label: eq-4 F (\ mathrm {\ sigma}, {\ mathrm {\ epsilon}}} ^ {p}) = {\ vert\ mathrm {\ sigma} -C {\ mathrm {\ epsilon}}} ^ {p}\ epsilon}}} ^ {\ epsilon}}} ^ {\ epsilon}}} ^ {c}} -C {\ mathrm {\ epsilon}}} ^ {\ epsilon}}} ^ {\ epsilon}}} ^ {\ epsilon}}} ^ {\ epsilon}} ^ {\ epsilon}}} ^ {\ epsilon}} ^ {\ epsilon}}} ^ {\ epsilon}} ^ {\ epsilon}} ^ {\ epsilon}}} {\ Vert a\ Vert} _ {\ mathit {eq}}\ equiv\ sqrt {\ frac {3} {2}\ mathit {dev} (a)\ mathrm {:}\ mathrm {:}\ mathit {eq}}\ equiv\ sqrt {\ 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): .. math:: : label: eq-5 {\ dot {\ mathrm {\ epsilon}}}} ^ {p}} =\ frac {3} {2}\ dot {\ mathrm {\ kappa}}\ frac {N} {{\ green N\ green}}} _ {\ green N\ green\ green}} _ {\ 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 :math:`\dot{\kappa }` is determined by the consistency condition: .. math:: :label: eq-6 \ 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 .. _RefHeading___Toc 2097_2561604576: Viscoelastic branch ---------------------- It is a Maxwell model whose shock absorber is non-linear Norton type. The (mechanical) deformation :math:`\epsilon` breaks down additively into a viscous part :math:`{\epsilon }^{v}` and an elastic part :math:`\epsilon -{\epsilon }^{v}`, the latter being linked to the stress of the viscoelastic branch by the following elastic relationship: .. math:: :label: eq-7 {\ 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: .. math:: :label: eq-8 {\ 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 :math:`\gamma` the inverse exponent of :math:`n` to simplify the notations and :math:`V` introduced a fourth-order isotropic tensor, without a unit, which allows to "misalign" :math:`{\dot{\epsilon }}^{v}` with respect to :math:`{\sigma }^{v}`. 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: .. math:: :label: eq-9 {\ 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.