# BilinearDP

Bilinear Drucker-Prager Model

## Syntax

```
material BilinearDP (1) (2) (3) (4) (5) (6) (7) (8) [9]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, poissons ratio
# (4) double, \eta_y
# (5) double, \eta_f
# (6) double, \xi
# (7) double, initial cohesion, c_0
# (8) double, hardening ratio/modulus, H
# [9] double, density, default: 0.0
```

## Theory

See more details on the formulation in the parent [page](/suanpan-manual/material/material3d/druckerprager/nonlineardruckerprager.md).

### Hardening Function

The cohesion develops linearly with the accumulated plastic strain,

$$
c=c\_0+H\bar{\varepsilon\_p},
$$

in which $$c\_0$$ is the initial cohesion (similar to the initial yield stress), $$H$$ is the hardening modulus, and $$\bar{\varepsilon\_p}$$ is the accumulated plastic strain.

### More

If one sets $$\eta\_y=\eta\_f=0$$, the model effectively becomes the von Mises model with the associative plasticity. In a uniaxial loading case, the yield function is then

$$
F(\sigma,c)=\sigma-\sqrt{3}\xi{}c.
$$

This leads to a yield stress $$\sigma\_y=\sqrt{3}\xi{}c\_0$$. The plastic hardening modulus is $$3\xi^2H$$. In terms of total strain and stress, the hardening ratio is $$3\xi^2H/(E+3\xi^2H)$$.

For the following model, one can compute the hardening ratio to be

$$
\dfrac{3\xi^2H}{E+3\xi^2H}=0.06204607451.
$$

One can validate this value by plotting the response.

```
node 1 5 -5 -5
node 2 5 5 -5
node 3 -5 5 -5
node 4 -5 -5 -5
node 5 5 -5 5
node 6 5 5 5
node 7 -5 5 5
node 8 -5 -5 5

material BilinearDP 1 1E4 .3 0 0 .58461851886189 5 645.1584849161

element C3D8 1 1 2 3 4 5 6 7 8 1

fix 1 1 1 2 5 6
fix 2 2 1 4 5 8
fix 3 3 1 2 3 4

displacement 1 0 -.2 3 5 6 7 8

hdf5recorder 1 Element E 1
hdf5recorder 2 Element S 1

step static 1
set fixed_step_size 1
set ini_step_size 1E-2
set symm_mat 0

analyze

save recorder 1 2

exit
```

## Iso-error Map

The following example iso-error maps are obtained via the following script.

```py
from plugins import ErrorMap
# note: the dependency `ErrorMap` can be found in the following link
# https://github.com/TLCFEM/suanPan-manual/blob/dev/plugins/scripts/ErrorMap.py

young_modulus = 1e5
ref_stress = 20

with ErrorMap(
    f"material BilinearDP 1 {young_modulus} 0.3 0.31 0.31 1.219 6.983 100",
    ref_strain=ref_stress / young_modulus,
    ref_stress=ref_stress,
    contour_samples=20,
) as error_map:
    error_map.contour("dp.uniaxial", center=(-2, 0.5, 0.5), size=1)
```

![absolute error uniaxial](/files/gHJz5xiOWZkIrQMKfEvl)


---

# Agent Instructions: Querying This Documentation

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

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

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

The question should be specific, self-contained, and written in natural language.
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.