Reference problem ===================== Geometry --------- It is a material point, representative of a state of homogeneous stresses and deformations. Material properties ------------------------ Properties for modeling A, crystal law MONO_DD_CFC ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Isotropic elasticity coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Shear modulus: :math:`\mu =80000.\mathrm{MPa}`, Poisson's ratio :math:`\nu =0.3` Young's module: :math:`E=\mu \ast 2\ast (1.+\nu )` Coefficient of law MONO_DD_CFC ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`A=0.13` :math:`B=0.005` :math:`\alpha =0.35` :math:`\beta ={2.5410}^{\text{-7}}(2.54\mathrm{Angström})` :math:`Y=2.5{10}^{\text{-7}}\mathrm{mm}(2.5\mathrm{Angstrom})` :math:`{\tau }_{f}=20.` :math:`n=5.` :math:`\dot{{\gamma }_{0}}={10}^{\text{-3}}` :math:`{\rho }_{\mathit{ref}}={\rho }_{\mathit{tot}}=1.2{10}^{6}{\mathit{mm}}^{\text{-2}}` is the initial total density, used to calculate the corrected interaction matrix, whose coefficients change with the total dislocation density. The interaction matrix is composed only of 1: :math:`\mathit{H1}=\mathit{H2}=\mathit{H3}=\mathit{H4}=\mathit{H5}=1.0`, The family of sliding systems is octahedral (:math:`\text{CFC}`). Each internal variable representing the dislocation density of a sliding system is initialized to :math:`{\rho }_{0}\ast {b}^{2}` with :math:`{\rho }_{0}={10}^{\text{5}}{\mathit{mm}}^{\text{-2}}=\frac{{\rho }_{\mathit{tot}}}{12}` Properties for B modeling, crystal law MONO_DD_CC ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Isotropic elasticity coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Poisson's Ratio :math:`\nu =0.35` Young's module: :math:`E=(236-\mathrm{0,0459}T)\text{GPa}` Coefficient of law MONO_DD_CC ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Two sets of coefficients are used depending on the case: .. csv-table:: "Case 1 (formulation 1)", "Case 2 (formulation 2)" ":math:`\text{DELTA1}=0` (formulation 1), :math:`\text{TEMP}=300K` :math:`\text{D\_LAT}=1000\text{mm}` :math:`\text{K\_BOLTZ}=8.62{10}^{\text{-5}}` :math:`\text{GAMMA0}={10}^{\text{-3}}{s}^{\text{-1}}` :math:`\text{TAU\_0}=363\text{MPa}` :math:`\text{TAU\_F}=20\text{MPa}` :math:`\text{RHO\_MOB}={10}^{\text{5}}{\text{mm}}^{\text{-2}}` :math:`\text{K\_F}=30\text{K\_SELF}=100` :math:`\text{B}=2.48{10}^{\text{-7}}\text{mm}` :math:`\text{N}=20` :math:`\text{DELTAG0}=0.84` :math:`\text{BETA}=0.2` :math:`\text{D}={10}^{\text{-5}}\text{mm}` :math:`\text{GH}={10}^{\text{11}}`, :math:`\text{Y\_AT}={10}^{\text{-6}}\text{mm}` The internal variables representing the dislocation density are initialized to :math:`{\rho }_{0}={10}^{\text{5}}{\mathrm{mm}}^{\text{-2}}` "," :math:`\text{DELTA1}=1` (formulation 2), :math:`\text{TEMP}=50K` :math:`\text{D\_LAT}=1000\text{mm}` :math:`\text{K\_BOLTZ}=8.62{10}^{\text{-5}}` :math:`\text{GAMMA0}={10}^{\text{-6}}{s}^{\text{-1}}` :math:`\text{TAU\_0}=363\text{MPa}` :math:`\text{TAU\_F}=0` :math:`\text{RHO\_MOB}={10}^{\text{5}}{\text{mm}}^{\text{-2}}` :math:`\text{K\_F}=75\text{K\_SELF}=100` :math:`\text{B}=2.48{10}^{\text{-7}}\text{mm}` :math:`\text{N}=50` :math:`\text{DELTAG0}=0.84` :math:`\text{BETA}=0.2` :math:`\text{D}={10}^{\text{-5}}\text{mm}` :math:`\text{GH}={10}^{\text{11}}`, :math:`\text{Y\_AT}=2{10}^{\text{-6}}\text{mm}` The internal variables representing the dislocation density are initialized to :math:`{\rho }_{0}={10}^{\text{5}}{\mathrm{mm}}^{\text{-2}}`, except for the main system (number 5): :math:`{\rho }_{0}={10}^{\text{6}}{\mathrm{mm}}^{\text{-2}}`" In both cases, the interaction matrix is constructed from the following values: :math:`\text{H1}=0.1024,\text{H2}=0.7,\text{H3=H4=H5=H6}=0.1` The family of sliding systems is cubic (:math:`\text{CC}`). Properties for C modeling ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Orthotropic elasticity coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Here, the elasticity is orthotropic cubic, and therefore defined by 3 coefficients: :math:`{y}_{1111}\mathrm{=}244000.\mathit{MPa}` :math:`{y}_{1122}\mathrm{=}96000.\mathit{MPa}` :math:`{y}_{1212}\mathrm{=}74000.\mathit{MPa}` We then have: :math:`{\nu }_{\text{LT}}\mathrm{=}{\nu }_{\mathit{TN}}\mathrm{=}{\nu }_{\text{LN}}\mathrm{=}\nu \mathrm{=}\frac{1}{(1+\frac{{y}_{1111}}{{y}_{1122}})}` :math:`` :math:`{E}_{L}\mathrm{=}{E}_{T}\mathrm{=}{E}_{N}\mathrm{=}{y}_{1111}\frac{(1\mathrm{-}3{\nu }^{2}\mathrm{-}2{\nu }^{3})}{(1\mathrm{-}{\nu }^{2})}` :math:`{G}_{\text{LT}}\mathrm{=}{G}_{\mathit{TN}}\mathrm{=}{G}_{\text{LN}}\mathrm{=}{y}_{1212}` Note: the :math:`{\mu }^{\mathit{loca}}` coefficient used for localization is equal to :math:`74000` Mpa. Coefficient of law MONO_DD_FAT ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`{\tau }_{f}\mathrm{=}44.9\mathit{MPa}` :math:`\dot{{\gamma }_{0}}=4.{10}^{\text{-11}}{s}^{\text{-1}}` :math:`\beta ={2.5410}^{\text{-7}}\mathrm{mm}(2.54\mathrm{Angström})` :math:`n=73.5` :math:`\text{UN\_SUR\_D}\mathrm{=}0.` :math:`{g}_{\mathit{c0}}\mathrm{=}1.33{10}^{\text{-6}}\mathit{mm}` :math:`K\mathrm{=}37.14` The interaction matrix is characterized by the following five coefficients (cf. [:ref:`R5.03.11 `]): :math:`\mathit{H1}=0.1236` :math:`\mathit{H2}=0.633` :math:`\mathit{H3}=0.1388` :math:`\mathit{H4}=0.1236` :math:`\mathit{H5}=0.0709` The family of sliding systems is octahedral. The internal variables representing dislocation density are initialized to :math:`{\rho }_{0}\mathrm{\times }{b}^{2}` with :math:`{\rho }_{0}=1.77{10}^{\text{6}}{\mathrm{mm}}^{\text{-2}}` Properties for D-modeling, crystal law MONO_DD_CC_IRRA ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Isotropic elasticity coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Poisson's Ratio :math:`\nu =0.35` Young's module: :math:`E=(236-\mathrm{0,0459}T)\text{GPa}` Coefficient of law MONO_DD_CC_IRRA ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. csv-table:: ":math:`\text{TEMP}=250K` :math:`\text{D\_LAT}=1000\text{mm}` :math:`\text{K\_BOLTZ}=8.62{10}^{\text{-5}}` :math:`\text{GAMMA0}={10}^{\text{-3}}{s}^{\text{-1}}` :math:`\text{TAU\_0}=363\text{MPa}` :math:`\text{TAU\_F}=20\text{MPa}` :math:`\text{K\_F}=30\text{K\_SELF}=100` :math:`\text{B}=2.48{10}^{\text{-7}}\text{mm}` :math:`\text{N}=20` :math:`\text{DELTAG0}=0.84` :math:`\text{D}={10}^{\text{-5}}\text{mm}` :math:`\text{GH}={10}^{\text{11}}` :math:`\text{Y\_AT}={10}^{\text{-6}}\text{mm}`, :math:`\text{A\_IRRA}=0.3,\text{XI\_IRRA}=4.0,` The internal variables representing dislocation density are initialized to :math:`{\rho }_{0}={10}^{\text{5}}{\mathit{mm}}^{\text{-2}}=\frac{{\rho }_{\mathit{tot}}}{12}` with :math:`{\rho }_{\mathit{tot}}=\mathrm{1,2}{10}^{\text{6}}{\mathit{mm}}^{\text{-2}}` :math:`\text{RHO\_MOB}={\rho }_{0}` is the mobile density by sliding system." The interaction matrix is constructed from the following values :math:`\mathit{H1}\mathrm{=}0.1024,\mathit{H2}\mathrm{=}0.7,\mathit{H3}\mathrm{=}0.1,\mathit{H4}\mathrm{=}0.1,\mathit{H5}\mathrm{=}0.1\mathit{H6}\mathrm{=}0.1,` The family of sliding systems is cubic (:math:`\text{CC}`). Properties for modeling E, crystal law MONO_DD_CFC_IRRA ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Isotropic elasticity coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Shear modulus: :math:`\mu =80000.\mathrm{MPa}`, Poisson's ratio :math:`\nu =0.3` Young's module: :math:`E=\mu \ast 2\ast (1.+\nu )` Coefficient of law MONO_DD_CFC ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`A=0.13` :math:`B=0.005` :math:`\alpha =0.35` :math:`\beta ={2.5410}^{\text{-7}}(2.54\mathrm{Angström})` :math:`Y=2.5{10}^{\text{-7}}\mathrm{mm}(2.5\mathrm{Angstrom})` :math:`{\tau }_{f}=20.` :math:`n=5.` :math:`\dot{{\gamma }_{0}}={10}^{\text{-3}}` :math:`{\rho }_{\mathit{ref}}={10}^{5}{\mathit{mm}}^{\text{-2}}` :math:`\begin{array}{c}{\alpha }^{\mathit{loops}}=\mathrm{0,1}{\varphi }^{\mathit{loops}}=5.9{10}^{\text{-6}}{\alpha }^{\mathit{voids}}=0{\rho }^{\mathit{voids}}=1.e3\\ {\rho }_{\mathit{sat}}=0{\varphi }_{\mathit{sat}}=0.04{\xi }_{\mathit{irra}}=10{\zeta }_{\mathit{irra}}={10}^{7}\end{array}` with :math:`{\rho }_{0}=\frac{{10}^{\text{5}}}{12}{\mathit{mm}}^{\text{-2}}` The interaction matrix is characterized by the following five coefficients (cf. [:ref:`R5.03.11 `]): :math:`\mathit{H1}=0.124` :math:`\mathit{H2}=0.625` :math:`\mathit{H3}=0.137` :math:`\mathit{H4}=0.122` :math:`\mathit{H5}=0.07` , The family of sliding systems is octahedral (:math:`\text{CFC}`). The internal variables representing the dislocation density are initialized to :math:`{\rho }_{0}\ast {b}^{2}` Those linked to irradiation have as initial values: :math:`{\rho }_{s}^{\mathit{loops}}=7.4{10}^{\text{-13}}{b}^{2}` :math:`{\phi }_{s}^{\mathrm{voids}}=0.001` Boundary conditions and loads ------------------------------------- Loading for models A, B (case 1), and D ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The load is subject to imposed constraints: :math:`\sigma ={\sigma }_{0}n\otimes n` with :math:`{\sigma }_{0}\mathrm{=}100\mathit{MPa}` and :math:`\mathrm{n}\mathrm{=}{(0.09667365,0.48336824,0.87006284)}^{T}` Hence the components of the imposed stress tensor: :math:`{\sigma }_{\mathrm{xx}}=0.93457943925233633` :math:`{\sigma }_{\mathrm{yy}}=23.364485981308412` :math:`{\sigma }_{\mathrm{zz}}=75.700934579439235` :math:`{\sigma }_{\mathrm{xy}}=4.6728971962616823` :math:`{\sigma }_{\mathrm{xz}}=8.411214953271027` :math:`{\sigma }_{\mathrm{yz}}=42.056074766355138` Charging for modeling B (case 2) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The load is in imposed deformations: :math:`\mathrm{dt}{\epsilon }_{\text{zz imposée}}=3{10}^{\text{-4}}{s}^{\text{-1}}` and :math:`{\varepsilon }_{\mathrm{zz}}(\mathrm{tmax})=\mathrm{0,27}` with :math:`\mathrm{tmax}=900s` The orientation of the single crystal is [:ref:`-1,4,9 <-1,4,9>`]. Loading for C modeling ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The load is in imposed deformations: :math:`{\epsilon }_{\text{zz imposée}}\mathrm{=}0.001t` from :math:`t\mathrm{=}0s` to :math:`t\mathrm{=}45s` Charging for modeling E ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The load is in imposed deformations: :math:`{ϵ}_{\text{zz imposée}}=0.05t` from :math:`t\mathrm{=}0s` to :math:`t=1s` Initial conditions -------------------- Zero stresses and deformations.