7. Behaviors VISC_IRRA_LOG and GRAN_IRRA_LOG#

The model presented in this chapter describes the viscoplastic behaviors 1D VISC_IRRA_LOG and GRAN_IRRA_LOG (creep and enlargement under irradiation of M5 and Zircaloy-4 alloys) for modeling fuel assemblies, and applicable to elements of multi-fiber bars and beams.

7.1. Formulation of the model#

The equations are as follows:

math:: `{begin {array} {c} {c} {dot {mathrm {epsilon}}}} ^ {mathit {VP}} =dot {mathrm {epsilon}}}frac {mathrm {epsilon}}}frac {mathrm {epsilon}}}frac {mathrm {sigma}}} {left|mathrm {sigma}right|}\ dot {epsilon}}\ dot {epsilon}}frac {mathrm {epsilon}}}frac {mathrm {epsilon}}}frac {mathrm {epsilon}}}epsilon}} =left|mathrm {sigma}right|mathrm {.} left ({e} ^ {frac {-Q} {-Q} {T}}right)mathrm {.} dot {mathrm {Phi}}}left (frac {Amathrm {omega}} {1+mathrm {omega}mathrm {Phi}}} +B-Cmathrm {omega}} {e} {e} {e} ^ {-mathrm {omega} t}right)\frac {dot {mathrm {sigma}}right)\frac {dot {mathrm {sigma}}}} {E} =dot {mathrm {epsilon}}} - {dot {mathrm {epsilon}}}} ^ {mathit {VP}} - {dot {mathrm {mathrm {epsilon}}}}}} ^ {mathrm {epsilon}}} - {mathit {v}}} - {mathit {th}}} - {mathit {th}}}}end {array} `

These relationships are deduced from creep tests FLETAN and REFLET [8] for various neutron flux values.

The coefficients are provided under the keyword VISC_IRRA_LOG or GRAN_IRRA_LOGde DEFI_MATERIAU and \(\Phi\) is the neutron fluence (integral of the flux with respect to time).

\({\varepsilon }^{g}\) represents the magnification deformation under flow. It is only taken into account in behavior GRAN_IRRA_LOG and is expressed in the form:

\({\varepsilon }^{g}(t)\mathrm{=}f(T,{\Phi }_{t}(x,y,z))\)

Notes:

Neutron fluence \({\Phi }_{t}(x,y,z)\) is necessarily expressed in \({10}^{20}n\mathrm{/}{\mathit{cm}}^{2}\). By convention in DEFI_MATERIAU [U4.43.01], if the value provided under the FLUX_PHI keyword is equal to 1, the fluence field is used for the behavior. Otherwise, the value provided in DEFI_MATERIAU is used as a constant neutron flux.
It is a node field defined as a command variable in the command AFFE_MATERIAU .
Attention : The irradiation field is incremental and corresponds to the irradiation history (stored as an internal variable — see below) to which is added the increment of the fluence field coming from the control variable.

7.2. Internal variables#

Three internal variables:

\(\mathit{V1}\): the cumulative viscoplastic deformation: \({\varepsilon }_{p}\);
\(\mathit{V2}\): memorization of the irradiation history (fluence).
\(\mathit{V3}\): the enlargement deformation: \({\varepsilon }^{g}\).

7.3. Implicit integration#

By direct implicit discretization of behavioral relationships, we obtain:

math:: `{begin {array} {c}mathrm {Delta} {mathrm {epsilon}}} ^ {mathit {VP}} =mathrm {Delta} pfrac {mathrm {delta} pfrac {mathrm {delta} pfrac {mathrm {delta} pfrac {mathrm {delta} pfrac {mathrm {delta} pfrac {mathrm {sigma}sigma}left ({t} ^ {text {-}}} +mathrm {Delta}} pfrac {mathrm {Delta} pfrac {mathrm {Delta} pfrac {left|mathrm {sigma}left ({t} ^ {text {-}} +mathrm {Delta} tright)right|}\ mathrm {Delta} p=left|left|left|mathrm {sigma}left ({t} ^ {text {-}}} +mathrm {Delta} {Delta} p=left|left|mathrm {delta} p=left|mathrm {delta} p=left|mathrm {delta} p=left|mathrm {delta} p=left|mathrm {sigma}right)right |left ({e} ^ {frac {-Q} {-Q} {T}}right)mathrm {.} left (frac {Amathrm {omega}}} {1+mathrm {omega}}mathrm {Phi} ({t} ^ {text {-}} +mathrm {Delta} t)}} +B-Cmathrm {Delta} t}}}} +B-Cmathrm {Delta} t)} +B-Cmathrm {Delta} t)} +B-Cmathrm {Delta} t)} +B-Cmathrm {Delta} t)} +B-Cmathrm {delta} t)} +B-Cmathrm {delta} t)} +B-Cmathrm {delta} t)} +B-Cmathrm {delta} t}Delta}mathrm {Phi}\frac {mathrm {sigma}} {E} -frac {{mathrm {sigma}} ^ {text {-}}}} {{E}}}} {{E}}}} {{E}}}} {{E} ^ {text {-}}} =mathrm {Delta}}} =mathrm {Delta}}} ^ {text {-}}} ^ {text {-}}}} {{E}}}} {{E} ^ {text {-}}}} {{E} ^ {text {-}}}} {{E} ^ {text {Delta}}} {mathrm {epsilon}} ^ {mathit {VP}}} -mathrm {Delta} {mathrm {epsilon}} ^ {g} -mathrm {Delta} {Delta} {delta} {delta} {delta}} {epsilon}} ^ {mathit {epsilon}}} ^ {mathit {th}}}\ text {with}\mathrm {Delta} {delta} {mathrm {Delta} {mathrm {Delta} {mathrm {epsilon}}} ^ {mathit {th}}}\ text {with}\mathrm {Delta} {Delta} {mathrm {Delta} thrm {epsilon}} ^ {mathit {th}}} =mathrm {alpha} (T)left (T- {T} _ {mathit {ref}}}right) -mathrm {alpha} ({T} ^ {text {-}})left (T- {T}})left (T- {T}} _ {mathit {ref}}right)\ mathrm {Delta} {mathrm {epsilon}}}}} ^ {epsilon}}}} ^ {epsilon}}} ^ {g} =f ({T}} =f ({T}} ^ {+}}}, {mathrm {Phi}}} _ {epsilon}}} _ {epsilon}}} ^ {text}} +}}) -f ({T} ^ {text {-}}}, {mathrm {Phi}} _ {t} ^ {text {-}})end {text {-}})end {array} `

We can solve these equations explicitly by asking: \({\sigma }^{e}=\frac{E}{{E}^{\text{-}}}{\sigma }^{\text{-}}+E(\Delta \varepsilon -\Delta {\varepsilon }^{g}-\Delta {\varepsilon }^{\mathrm{th}})\)

so the system is reduced to: \(\mathrm{\sigma }={\mathrm{\sigma }}^{e}-E\mathrm{\sigma }\left({e}^{\frac{-Q}{T}}\right)\mathrm{.}\left(\frac{A\mathrm{\omega }}{1+\mathrm{\omega }\mathrm{\Phi }}+B-C\mathrm{\omega }{e}^{-\mathrm{\omega }t}\right)\mathrm{\Delta }\mathrm{\Phi }\)

so the solution is obtained immediately: \(\mathrm{\sigma }=\frac{{\mathrm{\sigma }}^{e}}{1+E\left({e}^{\frac{-Q}{T}}\right)\mathrm{.}\left(\frac{A\mathrm{\omega }}{1+\mathrm{\omega }\mathrm{\Phi }}+B-C\mathrm{\omega }{e}^{-\mathrm{\omega }t}\right)\mathrm{\Delta }\mathrm{\Phi }}\)

and the tangent operator is written as: \(\frac{\partial \mathrm{\sigma }}{\partial \mathrm{\epsilon }}=\frac{E}{1+E\left({e}^{\frac{-Q}{T}}\right)\mathrm{.}\left(\frac{A\mathrm{\omega }}{1+\mathrm{\omega }\mathrm{\Phi }}+B-C\mathrm{\omega }{e}^{-\mathrm{\omega }t}\right)\mathrm{\Delta }\mathrm{\Phi }}\)