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[GeoMechanicsApplication] Replace two images with matrix equations by…
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… proper math blocks (#12801)

* Replace the two image of equations by a proper math block
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avdg81 authored Oct 29, 2024
1 parent 3ecd8ee commit 0643bd0
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12 changes: 10 additions & 2 deletions applications/GeoMechanicsApplication/custom_processes/README.md
Original file line number Diff line number Diff line change
Expand Up @@ -89,11 +89,19 @@ When the overconsolidation ratio ("OCR") and optionally the unloading-reloading

$$K_0 = OCR.K_0^{nc} + \frac{\nu_{ur}}{1 - \nu_{ur}} ( OCR - 1 )$$

![Initial effective stress tensor](initial_effective_stress_tensor.png)
```math
\sigma^{'}_{initial} = \begin{bmatrix} {K_0 \sigma^{'}_{zz}} & 0 & 0 \\
0 & {K_0 \sigma^{'}_{zz}} & 0 \\
0 & 0 & {\sigma^{'}_{zz}} \end{bmatrix}
```

Alternatively, when the pre-overburden pressure "POP" is specified, the initial stress tensor becomes:

![Initial effective stress tensor with POP](initial_effective_stress_tensor_with_POP.png)
```math
\sigma^{'}_{initial} = \begin{bmatrix} {K_0^{nc} (\sigma^{'}_{zz} + POP )} - \frac{\nu_{ur}}{1 - \nu_{ur}} POP & 0 & 0 \\
0 & {K_0^{nc} (\sigma^{'}_{zz} + POP)} - \frac{\nu_{ur}}{1 - \nu_{ur}} POP & 0 \\
0 & 0 & {\sigma^{'}_{zz}} \end{bmatrix}
```

When the optional unloading-reloading Poisson's ratio is omitted, a default value $\nu_{ur} = 0$ is used such that the correction term drops to 0.

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