laws:orthopla
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laws:orthopla [2023/11/28 17:58] hangbiao |
laws:orthopla [2024/01/23 12:15] (current) hangbiao |
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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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| ==== Vicoplasticity ==== | ==== Vicoplasticity ==== | ||
| See [[laws:epplasol|PLASOL]] \\ | See [[laws:epplasol|PLASOL]] \\ | ||
| - | Remark : For anisotropic Biot’s coeffcient, the deviatoric stress is calculated from the effective stresses (more details about this anisotropy are available in the definition of element CSOL2 and ISOL=9 in [[appendices:a8|Appendix 8]]). | + | Remark : For anisotropic Biot’s coeffcient, the deviatoric stress is calculated from the effective stresses (more details about this anisotropy are available in the definition of element CSOL2 and ISOL=9 in [[appendices:a7|Appendix 7]]). |
| ===== Availability ===== | ===== Availability ===== | ||
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| |:::| If NINTV = 0 : number of sub-steps is based on the norm of the deformation increment and on DIV| | |:::| If NINTV = 0 : number of sub-steps is based on the norm of the deformation increment and on DIV| | ||
| |ISOL| = 0 : use of total stresses in the constitutive law| | |ISOL| = 0 : use of total stresses in the constitutive law| | ||
| - | |:::| $\neq$ 0 : use of effective stresses in the constitutive law. See [[appendices:a8|Appendix 7]] | | + | |:::| $\neq$ 0 : use of effective stresses in the constitutive law. See [[appendices:a7|Appendix 7]] | |
| |ICBIF| = 0 : nothing| | |ICBIF| = 0 : nothing| | ||
| |:::|1 : Rice bifurcation criterion is computed (only for 2D plane strain analysis)| | |:::|1 : Rice bifurcation criterion is computed (only for 2D plane strain analysis)| | ||
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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| | + | |PSICPEAK| Peak of dilatancy angle for compressive paths (If IECPS=2 then PSICPEAK is the initial value of dilatancy angle| |
| - | |PSICLIM| | + | |PSICLIM| Limit value of dilatancy angle for compressive paths| |
| - | |RATPSI| | + | |RATPSI| Ratio between initial and peak of dilatancy angle| |
| - | |BPSI| | + | |BPSI| Value of EEQU for which PSIC=0.5 (PSICPEAK - PSICLIM) | |
| - | |PSIEPEAK| | + | |PSIEPEAK| Peak of dilatancy angle for extensive paths (If IECPS=2 then PSIEPEAK is the initial value of dilatancy angle) | |
| - | |PSIELIM| | + | |PSIELIM| Limit value of dilatancy angle for extensive paths| |
| - | |DECPSI| | + | |DECPSI| Value of EEQU when the dilatancy angle has been half decreased between its initial and final values| |
| - | ^ Line 9 (4G10.0) (Only if IDAM = 1) ^^ | + | ^ Line 9 (2G10.0) (Only if IDAM = 1) ^^ |
| - | |P| | + | |P|Parameter controlling the damage evolution rate| |
| - | |YD0| | + | |YD0|Initial threshold| |
| ===== Stresses ===== | ===== Stresses ===== | ||
| ==== Number of stresses ==== | ==== Number of stresses ==== | ||
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| ===== State variables ===== | ===== State variables ===== | ||
| ==== Number of state variables ==== | ==== Number of state variables ==== | ||
| - | = 43 : for 2D plane strain analysis with bifurcation criterion (ICBIF=1)\\ | + | = 48 : for 2D plane strain analysis with bifurcation criterion (ICBIF=1)\\ |
| - | = 31 : in all the other cases | + | = 36 : in all the other cases |
| ==== List of state variables ==== | ==== List of state variables ==== | ||
| |Q(1)| = 1 in plane strain state | | |Q(1)| = 1 in plane strain state | | ||
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| |Q(34)$\rightarrow$ Q(36)| = reserved for small strain stiffness (E1, E2, E3) | | |Q(34)$\rightarrow$ Q(36)| = reserved for small strain stiffness (E1, E2, E3) | | ||
| |Q(37)$\rightarrow$ Q(48)| = reserved for bifurcation | | |Q(37)$\rightarrow$ Q(48)| = reserved for bifurcation | | ||
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laws/orthopla.1701190687.txt.gz · Last modified: 2023/11/28 17:58 by hangbiao
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