This model is an extension of the ConcreteDPM presented in 1.6.9. CDPM2 has been developed by Grassl, Xenos, Nyström, Rempling and Gylltoft for modelling the failure of concrete for both static and dynamic loading. It is is described in detail in [9]. The main differences between CDPM2 and ConcreteDPM are that in CDPM2 the plasticity part exhibits hardening once damage is active. Furthermore, two independent damage parameters describing tensile and compressive damage are introduced. The parameters of CDPM2 are summarised in Tab. 41

The stress for the anisotropic damage plasticity model (ISOFLAG=0) is defined as

(163) |

where and are the positive and negative parts of the effective stress tensor , respectively, and and are two scalar damage variables, ranging from 0 (undamaged) to 1 (fully damaged).

The stress for the isotropic damage plasticity model (ISOFLAG=1) is defined as

(164) |

The effective stress is defined according to the damage mechanics convention as

(165) |

Plasticity:

The yield surface is described by the Haigh-Westergaard coordinates: the volumetric effective stress , the norm of the deviatoric effective stress and the Lode angle . The yield surface is

It depends also on the hardening variable (which enters through the dimensionless variables and ). Parameter is the uniaxial compressive strength. For , the yield function is identical to the one of CDPM. The shape of the deviatoric section is controlled by the Willam-Warnke function

Here, is the eccentricity parameter. The friction parameter is given by

where is the tensile strength.

The flow rule (26) is split into a volumetric and a deviatoric part, i.e., the gradient of the plastic potential is decomposed as

Taking into account that and , restricting attention to the post-peak regime (in which ) and differentiating the plastic potential (147), we rewrite equation (169) as

The flow rule is non-associative which means that the direction of the plastic flow is not normal to the yield surface. This is important for concrete since an associative flow rule would give an overestimated maximum stress for passive confinement.

The dimensionless variables and that appear in (143), (147) and (148) are functions of the hardening variable . They control the evolution of the size and shape of the yield surface and plastic potential. The first hardening law is

The second hardening law is given by

The evolution law for the hardening variable,

(173) |

sets the rate of the hardening variable equal to the norm of the plastic strain rate scaled by a hardening ductility measure, which is identical to the one used for the CDPM.

Damage:

Damage is initiated when the maximum equivalent strain in the history of the material reaches the threshold . This expression is determined from the yield surface ( ) by setting and . From this quadratic equation for , the equivalent strain is determined as

(174) |

Tensile damage is described by a stress-inelastic displacement law. For linear and exponential damage type the stress value and the displacement value must be defined. For the bi-linear type two additional parameters and are required.

For the compressive damage variable, an evolution based on an exponential stress-inelastic strain law is used. The stress versus inelastic strain in the softening regime in compression is

(175) |

where is an inelastic strain threshold which controls the initial inclination of the softening curve. The use of different damage evolution for tension and compression is one important improvement over CDPM.

The history variables , , and depend on a ductility measure , which takes into account the influence of multiaxial stress states on the damage evolution. This ductility measure is given by

(176) |

where is

(177) |

and is a model parameter.

Strain rate:

Concrete is strongly rate dependent. If the loading rate is increased, the tensile and compressive strength increase and are more prominent in tension then in compression. The dependency is taken into account by an additional variable . The rate dependency is included by scaling both the equivalent strain rate and the inelastic strain. The rate parameter is defined by

(178) |

where is the continuous compression measure (= 1 means only compression, = 0 means only tension).

The functions and depend on the input parameter . A recommended value for is 10 MPa.

2019-03-19