> For the complete documentation index, see [llms.txt](https://tlcfem.gitbook.io/suanpan-manual/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tlcfem.gitbook.io/suanpan-manual/material/material3d/druckerprager/nonlineardruckerprager.md).

# NonlinearDruckerPrager

Drucker-Prager Material Model

The Drucker-Prager model uses the following function as the yield surface.

$$
F(\sigma,c)=\sqrt{J\_2}+\eta\_yp-\xi{}c
$$

in which $$J\_2=\dfrac{1}{2}s:s$$ is the second invariant of stress $$\sigma$$, $$p=\dfrac{1}{3}( \sigma\_1+\sigma\_2+\sigma\_3)$$ is the hydrostatic stress, $$c(\bar{\varepsilon\_p})$$ is cohesion, $$\eta\_y$$ and $$\xi$$ are material constants.

Either associated or non-associated flow rule can be applied. The flow potential is defined as

$$
\Phi(\sigma,c)=\sqrt{J\_2}+\eta\_fp
$$

where $$\eta\_f$$ is another material constant. If $$\eta\_f=\eta\_y$$, the associative plasticity is defined so that the symmetry of stiffness matrix is recovered.

The accumulated plastic strain $$\bar{\varepsilon\_p}$$ is defined as

$$
\dot{\bar{\varepsilon\_p}}=\xi\gamma,
$$

where $$\gamma$$ is the plasticity multiplier.

## History Variable Layout

| location             | parameter                  |
| -------------------- | -------------------------- |
| `initial_history(0)` | accumulated plastic strain |

## Determination of Material Constants

There are quite a lot of schemes to determine the material constants used in Drucker-Prager model. Here a few are presented.

### Geomaterials

The friction angle $$\phi$$ and initial cohesion $$c\_0$$ shall be determined by experiments.

#### Outer Mohr-Coulomb

$$
\eta\_y=\dfrac{6\sin\phi}{\sqrt{3}(3-\sin\phi)},\qquad\xi=\dfrac{6\cos\phi}{\sqrt{3}(3-\sin\phi)}
$$

#### Inner Mohr-Coulomb

$$
\eta\_y=\dfrac{6\sin\phi}{\sqrt{3}(3+\sin\phi)},\qquad\xi=\dfrac{6\cos\phi}{\sqrt{3}(3+\sin\phi)}
$$

#### Plane Strain Fitting

$$
\eta\_y=\dfrac{3\tan\phi}{\sqrt{9+12\tan^2\phi}},\qquad\xi=\dfrac{3}{\sqrt{9+12\tan^2\phi}}
$$

### Concrete, Rock, etc.

To fit uniaxial tension and compression strength, the friction angle and cohesion shall be computed as

$$
\phi=\sin^{-1}\dfrac{f\_c-f\_t}{f\_c+f\_t},\qquad{}c=\dfrac{f\_cf\_t}{f\_c-f\_t}\tan\phi
$$

in which $$f\_t\ge0$$ and $$f\_c\ge0$$ are tension and compression strength respectively.

#### Uniaxial Tension/Compression

$$
\eta\_y=\dfrac{3\sin\phi}{\sqrt{3}},\qquad\xi=\dfrac{2\sin\phi}{\sqrt{3}}
$$

#### Biaxial Tension/Compression

$$
\eta\_y=\dfrac{3\sin\phi}{2\sqrt{3}},\qquad\xi=\dfrac{2\sin\phi}{\sqrt{3}}
$$


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation by asking a question.

Perform an HTTP GET request on the following URL with the `ask` and `goal` query parameters:

```
GET https://tlcfem.gitbook.io/suanpan-manual/material/material3d/druckerprager/nonlineardruckerprager.md?ask=<question>&goal=<user_goal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is what the user is ultimately trying to achieve, the reason they need the answer. Sharing it helps GitBook give you a better, more relevant answer. A goal is most helpful when it describes the outcome the user wants rather than restating the question. For example, with `ask=how do I create an API token`, a goal like `automate deployments from our CI pipeline` lets GitBook tailor the answer to that use case.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.