[GeomechanicsApplication] Add documentation for compression cap#14348
[GeomechanicsApplication] Add documentation for compression cap#14348mnabideltares wants to merge 2 commits intomasterfrom
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WPK4FEM
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WPK4FEM
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Hi Mohamed,
Please find my comments.
Wijtze Pieter
| ```math | ||
| \frac{\partial G_{cap}}{\partial \sigma_i} = \frac{1}{3} G_{,p} + \frac{3 G_{,q}}{2 q} \left(\sigma_i - p \right) | ||
| ``` |
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Here the derivative is done with , notation for the first time. That comes without explanation. I would prefer to still use \partial. That is what is on the left side too.
| ```math | ||
| A = 1 + b_1 a_2^2 | ||
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| ```math | ||
| B = -2 b_1 a_2 c_2 | ||
| ``` | ||
| ```math | ||
| C = b_1 c_2^2 - c_1 | ||
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| ```math | ||
| b_1 = 1 / X^2 | ||
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| ```math | ||
| c_1 = p_c^2 | ||
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| ```math | ||
| a_2 = \frac{6 \sin{\phi}}{3 - \sin{\phi}} | ||
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| ```math | ||
| c_2 = \frac{6 c \cos{\phi}}{3 - \sin{\phi}} | ||
| ``` |
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The many intermediate variable definitions are more a programming aid than making it more readable for me.
| c_2 = \frac{6 c \cos{\phi}}{3 - \sin{\phi}} | ||
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| A second order polynomial equation needs to be solved, and the minimum root needs to be selected, because $p \lt 0$ |
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The "because p < 0 " I don't understand. With large cohesion, I can create a situation where the cap-MC corner has positive p. Its more a takeaway from drawing the intersections that you want the one with lowest p.
| C = \frac{c_3}{c_1} \left( \frac{c_3 c_4}{c_1} + c_6 \right) - c9 | ||
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| Then, solving the recond order polynomial, it gives |
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| Then, solving the recond order polynomial, it gives | |
| Then, solving the second order polynomial, it gives |
| We define the following vectors, | ||
| ```math | ||
| \Delta \sigma = \left[\sigma_1 - \sigma_2 \; , \; \sigma_2 - \sigma_3 \; , \; \sigma_3 - \sigma_1 \right]^T | ||
| ``` | ||
| ```math | ||
| \Delta \sigma^{cor} = \left[\sigma_1^{cor} - \sigma_2^{cor} \; , \; \sigma_2^{cor} - \sigma_3^{cor} \; , \; \sigma_3^{cor} - \sigma_1^{cor} \right]^T | ||
| ``` |
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Naming these differences "Delta" is confusing for me. In return mapping the Delta sigma is usually a stress increment.
| + \lambda_{cap} C \frac{\partial G_{cap}}{\partial \sigma} | ||
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| Subsituting this traisl stresses in compression cap and Coulomb yield surfaces, it leads to two equations and two unknowns. |
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| Subsituting this traisl stresses in compression cap and Coulomb yield surfaces, it leads to two equations and two unknowns. | |
| Subsituting this trial stresses in compression cap and Coulomb yield surfaces, it leads to two equations and two unknowns. |
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| ### Return mapping from cap corner zone | ||
| The cap compression zone is the rigion where the trial principal stresses are, |
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| The cap compression zone is the rigion where the trial principal stresses are, | |
| The cap compression zone is the region where the trial (p, q) stress invariants are demarcated by the following two lines, |
| ### Return mapping from cap corner zone | ||
| The cap compression zone is the rigion where the trial principal stresses are, | ||
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| 1. above the line which passes from the cap corner point and in the direction normal to the flow function of the cap yield surface. |
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| 1. above the line which passes from the cap corner point and in the direction normal to the flow function of the cap yield surface. | |
| 1. above the line which passes from the cap corner point and in the direction normal to the cap flow function $G_cap$. |
| q - q_{corner} - \left( G_{cap,p}/G_{cap,q} \right) (p - p_{corner}) > 0 | ||
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| 2. Under the line which passes from the cap corner point and in the direction normal to the flow function of the Coulomb yield surface. |
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| 2. Under the line which passes from the cap corner point and in the direction normal to the flow function of the Coulomb yield surface. | |
| 2. Under the line which passes from the cap corner point and in the direction normal to the Coulomb flow function $G_MC$ |
| Subsituting the stresses with the mapped stresses, we get the following relations. | ||
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| ```math | ||
| A = \frac{\Delta \sigma^{cor} \cdot \Delta \sigma^{cor}}{2 X^2} + \frac{1}{9} \left( \sum_{i=1}^3{\sigma_i^{cor}} \right)^2 |
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Then \sum sigma_i / 3 looks like p^{cor}, but that would lead to a confusing mix of principal stresses and p,q stress invariants.
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📝 Description
The formulations for the compression cap and its intersection with Coulomb yield surface, and return mappping need to be documented.
🆕 Changelog