laws:orthopla
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laws:orthopla [2023/12/01 16:24] hangbiao [Cohesion anisotropy with major principal stress orientation relative to bedding (IANISO = 0)] |
laws:orthopla [2024/01/23 12:15] (current) hangbiao |
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| ==== Hardening/softening ==== | ==== Hardening/softening ==== | ||
| See [[laws:epplasol|PLASOL]] law (above) | See [[laws:epplasol|PLASOL]] law (above) | ||
| - | ==== Cohesion anisotropy with major principal stress orientation relative to bedding (IANISO = 0) (Salehnia, 2015)((Salehnia, F. (2015) From some obscurity to clarity in Boom clay behavior: Analysis of its coupled hydro-mechanical response in the presence of strain localization. Thesis, Liège University.))==== | + | ==== Cohesion anisotropy with major principal stress orientation relative to bedding (IANISO = 0)==== |
| The material cohesion depends on the angle $\alpha_{\sigma_1}$ between the major compressive principal stress $\vec{\sigma_1}$ and the normal to the bedding plane $\vec{e_3}$ : | The material cohesion depends on the angle $\alpha_{\sigma_1}$ between the major compressive principal stress $\vec{\sigma_1}$ and the normal to the bedding plane $\vec{e_3}$ : | ||
| \[ | \[ | ||
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| :laws:schematic_view_of_the_angle_between_the_normal_to_bedding_plane_and_the_direction_of_major_principal_stress.png?150 |}}</imgcaption> | :laws:schematic_view_of_the_angle_between_the_normal_to_bedding_plane_and_the_direction_of_major_principal_stress.png?150 |}}</imgcaption> | ||
| - | Three cohesion values are defined ($c_{0^{\circ}}, c_{min}, c_{90^{\circ}}$), for major principal stress parallel $\alpha_{\sigma_1} = 0^{\circ}$ (perpendicular), perpendicular $\alpha_{\sigma_1} = 90^{\circ}$ (parallel) and with an angle of $\alpha_{\sigma_1, min}$ with respect to the normal to bedding plane (with respect to the bedding plane). Between those values, cohesion varies linearly with $\alpha_{\sigma_1}$. The mathematical expression of the cohesion is as follows: | + | Three cohesion values are defined ($c_{0^{\circ}}, c_{min}, c_{90^{\circ}}$), for major principal stress parallel $\alpha_{\sigma_1} = 0^{\circ}$ (perpendicular), perpendicular $\alpha_{\sigma_1} = 90^{\circ}$ (parallel) and with an angle of $\alpha_{\sigma_1, min}$ with respect to the normal to bedding plane (with respect to the bedding plane) (Salehnia, 2015)((Salehnia, F. (2015) From some obscurity to clarity in Boom clay behavior: Analysis of its coupled hydro-mechanical response in the presence of strain localization. Thesis, Liège University.)). Between those values, cohesion varies linearly with $\alpha_{\sigma_1}$. The mathematical expression of the cohesion is as follows: |
| \[ | \[ | ||
| c = \max \left[\left( \frac{c_{min} - c_{0^{\circ}}}{\alpha_{\sigma_1, min}} \right)\alpha_{\sigma_1} + c_{0^{\circ}} ; \left( \frac{c_{90^{\circ}} - c_{min}}{90^{\circ} - \alpha_{\sigma_1, min}} \right)\left( \alpha_{\sigma_1} - \alpha_{\sigma_1, min} \right)+ c_{0^{\circ}} \right] | c = \max \left[\left( \frac{c_{min} - c_{0^{\circ}}}{\alpha_{\sigma_1, min}} \right)\alpha_{\sigma_1} + c_{0^{\circ}} ; \left( \frac{c_{90^{\circ}} - c_{min}}{90^{\circ} - \alpha_{\sigma_1, min}} \right)\left( \alpha_{\sigma_1} - \alpha_{\sigma_1, min} \right)+ c_{0^{\circ}} \right] | ||
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| Considering cross-anisotropy, i.e. transverse isotropy, and refering the problem to the principal material axes implies $A_{ij} = 0$ for $i \neq j$, $A_{ii} = A_{11}+A_{22}+A_{33} = 0$, $A_{11} = A_{33}$ if the bedding plane is in ($e_1, e_3$) anisotropic plane, $A_{22} = -2A_{11}$, implying : | Considering cross-anisotropy, i.e. transverse isotropy, and refering the problem to the principal material axes implies $A_{ij} = 0$ for $i \neq j$, $A_{ii} = A_{11}+A_{22}+A_{33} = 0$, $A_{11} = A_{33}$ if the bedding plane is in ($e_1, e_3$) anisotropic plane, $A_{22} = -2A_{11}$, implying : | ||
| \[A_{ij}l_il_j = A_{l1}(1-3l_2^2)\] | \[A_{ij}l_il_j = A_{l1}(1-3l_2^2)\] | ||
| - | where $A_{11}$ is the component of the microstructure operator $A_{ij}$ in the isotropic (bedding) plane. The late expression for cohesion becomes : | + | where $A_{11}$ is the component of the microstructure operator $A_{ij}$ in the isotropic (bedding) plane. The late expression for cohesion becomes (Pardoen, 2015)((Pardoen, B. (2015) Hydro-mechanical analysis of the fracturing induced by the excavation of nuclear waste repository galleries using shear banding. Thesis, Liège University.)): |
| \[c= c_0 \left( 1+A_{l1}(1-3l_2^2) + b_1A_{l1}^2(1-3l_2^2)^2 + b_2A_{l1}^3(1-3l_2^2)^3 + … \right)\] | \[c= c_0 \left( 1+A_{l1}(1-3l_2^2) + b_1A_{l1}^2(1-3l_2^2)^2 + b_2A_{l1}^3(1-3l_2^2)^3 + … \right)\] | ||
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| |E3F|Final elastic Young modulus E($e_{3f}$)| | |E3F|Final elastic Young modulus E($e_{3f}$)| | ||
| |Gamma7|equivalent strain at which the Young's modulus has reduced to 0.7 times | | |Gamma7|equivalent strain at which the Young's modulus has reduced to 0.7 times | | ||
| + | |Aa|Fitting parameter | | ||
| ^ Line 8 (7G10.0) (Only if IECPS = 2 or 3) ^^ | ^ Line 8 (7G10.0) (Only if IECPS = 2 or 3) ^^ | ||
| |PSICPEAK| Peak of dilatancy angle for compressive paths (If IECPS=2 then PSICPEAK is the initial value of dilatancy angle| | |PSICPEAK| Peak of dilatancy angle for compressive paths (If IECPS=2 then PSICPEAK is the initial value of dilatancy angle| | ||
laws/orthopla.1701444240.txt.gz · Last modified: 2023/12/01 16:24 by hangbiao
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