CDPM2

CDPM2 Model for Concrete Material

References

1. 10.1016/j.ijsolstr.2013.07.008

2. 10.1016/j.ijsolstr.2005.05.038

Syntax

material CDPM2 (1) [2-16]
material CDPM2ISO (1) [2-16]
material CDPM2ANISO (1) [2-16]
# (1) int, unique material tag
# [2] double, elastic modulus, default: 3E4
# [3] double, poissons ratio, default: 0.3
# [4] double, tension strength (positive), default: 3.0
# [5] double, compression strength (positive), default: 30.0
# [6] double, q_h0, initial hardening factor, default: 0.3
# [7] double, h_p hardening ratio, default: .01
# [8] double, d_f, default: .85
# [9] double, a_h, hardening related parameter, default: 0.08
# [10] double, b_h, hardening related parameter, default: 0.003
# [11] double, c_h, hardening related parameter, default: 2.0
# [12] double, d_h, hardening related parameter, default: 1E-6
# [13] double, a_s, ductility related parameter, default: 5.0
# [14] double, e_ft, tension softening parameter, default: 5E-4
# [15] double, e_fc, compression softening parameter, default: 5E-4
# [16] double, density, default: 0.0

Remarks

1. The isotropic damage is implemented.

2. The anisotropic damage is implemented.

3. For detailed explanations of parameters, please refer to 10.1016/j.ijsolstr.2013.07.008.

4. The default CDPM2 uses isotropic damage, which is equivalent to CDPM2ISO.

5. The token CDPM2ANISO uses anisotropic damage.

6. If damage is activated, both tension and compression use the exponential damage model, the degradation is controlled by parameters [14] and [15]. The characteristic length can be accounted for by modifying them.

Details of implementation can be seen in the corresponding section in Constitutive Modelling Cookbook.

Recording

This model supports the following additional history variables to be recorded.

variable label	physical meaning
DT	tensile damage
DC	compressive damage

Examples

The isotropic damage uses the following expression for the final stress $\mathbf{\sigma}$,

$$\mathbf{\sigma}=(1-\omega_c)(1-\omega_t)\mathbf{\bar{\sigma}}.$$

The anisotropic damage uses the following expression,

$$\mathbf{\sigma}=(1-\omega_c)\mathbf{\bar{\sigma}}_c+(1-\omega_t)\mathbf{\bar{\sigma}}_t.$$

In the above expressions, $\mathbf{\bar{\sigma}}$ is the effective stress (undamaged), $\mathbf{\bar{\sigma}}_c$ and $\mathbf{\bar{\sigma}}_t$ are compressive and tensile part of the effective stress. They are computed via eigen decomposition of the effective stress tensor. $\omega_c$ and $\omega_t$ are the compressive and tensile damage variables, respectively.

Both damage types have physical implications. Depending on the damage type, the model parameters may be adjusted differently.

The reference strains $e_{ft}$ and $e_{fc}$ affect the degradation of strength.

The following are two examples, using different values of $e_{ft}$ and $e_{fc}$, while all other parameters are the default values.

compressive

tensile

Iso-error Map

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

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 = 3e4
yield_stress = 30

with ErrorMap(
   f"material CDPM2 1 {young_modulus} .3 6. {yield_stress} .3 .01 .85 .08 .003 2. 1E-6 5. 15E-4 5E-4",
   ref_strain=yield_stress / young_modulus,
   ref_stress=yield_stress,
   contour_samples=20,
) as error_map:
   error_map.contour("cdpm2.uniaxial", center=(-2, -2), size=1)
   error_map.contour("cdpm2.biaxial", center=(-2, 0), size=1)

absolute error uniaxial

absolute error biaxial