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