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

$\displaystyle \boldsymbol{\sigma} = \left(1-\omega_{\rm t}\right) \bar{\boldsym...
...gma}}_{\rm t} + \left(1-\omega_{\rm c}\right) \bar{\boldsymbol{\sigma}}_{\rm c}$ (163)

where $\bar{\boldsymbol{\sigma}}_{\rm t}$ and $\bar{\boldsymbol{\sigma}}_{\rm c}$ are the positive and negative parts of the effective stress tensor $\bar{\boldsymbol{\sigma}}$, respectively, and $\omega_{\rm t}$ and $\omega_{\rm c}$ 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

$\displaystyle \boldsymbol{\sigma} = \left(1-\omega_{\rm t}\right) \bar{\boldsymbol{\sigma}}$ (164)

The effective stress $\bar{\boldsymbol{\sigma}}$ is defined according to the damage mechanics convention as

$\displaystyle \bar{\boldsymbol{\sigma}} = \mathbf{D}_{\rm e} : \left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}_{\rm p}\right)$ (165)


The yield surface is described by the Haigh-Westergaard coordinates: the volumetric effective stress $\sigma_{\rm V}$, the norm of the deviatoric effective stress $\rho$ and the Lode angle $\theta$. The yield surface is

& f_{\rm p}(\bar{\sigma}_{\rm V},\bar\rho,\bar\...
...}}^2(\kappa_{\rm p}) q_{{\rm h2}}^2(\kappa_{\rm p})
\end{split}\end{displaymath} (166)

It depends also on the hardening variable $\kappa_{\rm p}$ (which enters through the dimensionless variables $q_{\rm h1}$ and $q_{\rm h2}$). Parameter $f_{\rm c}$ is the uniaxial compressive strength. For $q_{\rm h2} = 1$, the yield function is identical to the one of CDPM. The shape of the deviatoric section is controlled by the Willam-Warnke function

&r(\cos{\bar{\theta}}) = \frac{4(1-e^2)\cos^2{\...
...(2e-1)\sqrt{4(1-e^2)\cos^2{\bar{\theta}}+5e^2 -4e}}
\end{split}\end{displaymath} (167)

Here, $e$ is the eccentricity parameter. The friction parameter $m_0$ is given by

$\displaystyle m_0 = \dfrac{3 \left(f_{\rm c}^2 - f_{\rm t}^2\right)}{f_{\rm c}f_{\rm t}} \dfrac{e}{e+1}$ (168)

where $f_{\rm t}$ 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

$\displaystyle \mathbf{m} = \dfrac{\partial g}{\partial \bar{\boldsymbol{\sigma}...
...ial \bar{\rho}} \frac{\partial \bar{\rho} }{\partial \bar{\boldsymbol{\sigma}}}$ (169)

Taking into account that $\partial\bar{\sigma}_{\rm V}/\partial\bar{\boldsymbol{\sigma}}=\boldsymbol{\delta}/3$ and $\partial\bar{\rho}/\partial\bar{\boldsymbol{\sigma}}=\bar{\mbox{\boldmath $s$}}/\bar\rho$, restricting attention to the post-peak regime (in which $q_{\rm h1} = 1$) and differentiating the plastic potential (147), we rewrite equation (169) as

$\displaystyle \mbox{\boldmath$m$}$$\displaystyle = \dfrac{\partial g}{\partial \bar{\boldsymbol{\sigma}}} = \frac{...
...\frac{m_0}{\sqrt{6}\bar\rho} \right)\frac{\bar{\mbox{\boldmath$s$}}}{f_{\rm c}}$ (170)

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 $q_{\rm h1}$ and $q_{\rm h2}$ that appear in (143), (147) and (148) are functions of the hardening variable $\kappa_{\rm p}$. They control the evolution of the size and shape of the yield surface and plastic potential. The first hardening law $q_{\rm h1}$ is

$\displaystyle q_{\rm h1}(\kappa_{\rm p}) =
\left \{ \begin{array}{ll} q_{\rm h0...
...kappa_{\rm p} < 1$} \\
1 & \mbox{if $\kappa_{\rm p} \ge 1$}
\end{array}\right.$ (171)

The second hardening law $q_{\rm h2}$ is given by

$\displaystyle q_{\rm h2}(\kappa_{\rm p}) = \left \{ \begin{array}{ll} 1 & \mbox...
...m p} (\kappa_{\rm p} - 1) & \mbox{if $\kappa_{\rm p} \ge 1$}
\end{array}\right.$ (172)

The evolution law for the hardening variable,

$\displaystyle \dot{\kappa}_{\rm {p}} = \frac {\Vert \dot{\mbox{\boldmath$\varep...
...{\rm {h}}\left(\bar{\sigma}_{\rm V} \right)} \left(2\cos{\bar{\theta}}\right)^2$ (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 is initiated when the maximum equivalent strain in the history of the material reaches the threshold $\varepsilon_{0} = f_{\rm t}/E$. This expression is determined from the yield surface ( $f_{\rm p} = 0$) by setting $q_{\rm {h1}}=1$ and $q_{\rm {h2}} = \tilde{\varepsilon}/\varepsilon_{0}$. From this quadratic equation for $\tilde{\varepsilon}$, the equivalent strain is determined as

$\displaystyle \tilde{\varepsilon} = \dfrac{\varepsilon_0 m_0}{2} \left(\dfrac{\...
...}}{f_{\rm c}}\right)^2 + \dfrac{3 \varepsilon_0^2 \bar{\rho}^2}{2 f_{\rm c}^2}}$ (174)

Tensile damage is described by a stress-inelastic displacement law. For linear and exponential damage type the stress value $f_{\rm t}$ and the displacement value $w_{\rm f}$ must be defined. For the bi-linear type two additional parameters $f_{\rm t1}$ and $w_{\rm f1}$ 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

\sigma = f_{\rm t} \exp\left(-\dfrac{\varep...
...m fc}}\right) & \mbox{if $0 < \varepsilon_{\rm i}$}
\end{array}\end{displaymath} (175)

where $\varepsilon_{\rm fc}$ 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 $\kappa_{\rm dt1}$, $\kappa_{\rm dt2}$, $\kappa_{\rm dc1}$ and $\kappa_{\rm dc2}$ depend on a ductility measure $x_{\rm s}$, which takes into account the influence of multiaxial stress states on the damage evolution. This ductility measure is given by

$\displaystyle x_{\rm s} = 1+ \left(A_{\rm s}-1\right) R_{\rm s}$ (176)

where $R_{\rm s}$ is

$\displaystyle R_{\rm s} = \left \{ \begin{array}{ll} -\dfrac{\sqrt{6} \bar{\sig...
...rm V} \leq 0$} \\
0 & \mbox{if $\bar{\sigma}_{\rm V} > 0$} \end{array} \right.$ (177)

and $A_{\rm s}$ 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 $alpha_{\rm r}$. The rate dependency is included by scaling both the equivalent strain rate and the inelastic strain. The rate parameter is defined by

$\displaystyle \alpha_{\rm r}=(1-X) \alpha_{\rm rt} + X \alpha_{\rm rc}$ (178)

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

The functions $\alpha_{\rm rt}$ and $\alpha_{\rm rc}$ depend on the input parameter $f_{\rm c0}$. A recommended value for $f_{\rm c0}$ is 10 MPa.

Table 41: CDPM2 - summary.
Description CDPM2
Record Format con2dpm d(rn) # E(rn) # n(rn) # tAlpha(rn) # ft(rn) # fc(rn) # wf(rn) # [stype(in) #] [ft1(rn) #] [wf1(rn) #] [efc(rn) #] [ecc(rn) #] [kinit(rn) #] [Ahard(rn) #] [Bhard(rn) #] [Chard(rn) #] [Dhard(rn) #] [Asoft(rn) #] [helem(rn) #] [dilation(rn) #] [hp(rn) #] [isoflag(in) #] [rateflag(in) #] fcZero(in) # [yieldtol(rn) #] [newtoniter(in) #]
Parameters - d material density
  - E Young modulus
  - n Poisson ratio
  - tAlpha thermal dilatation coefficient
  - ft uniaxial tensile strength
  - fc uniaxial compressive strength
  - wf parameter that controls the slope of the softening branch to be used
  - stype allows to choose different types of softening laws, default value 1:
0 - linear softening
1 - bilinear softening
2 - exponential softening
  - ft1 parameter for the bilinear softening law (stype = 1) defining the ratio between intermediate stress and tensile strength, optional, default value 0.3
  - wf1 parameter for the bilinear softening law (stype = 1) defining the intermediate crack opening, optional, default value 0.15
  - efc parameter for exponential softening law in compression, optional, default value $100 \times 10^{-6}$
  - ecc eccentricity parameter, optional, default value 0.525
  - kinit initial value of hardening law, optional, default value 0.3
  - Ahard hardening parameter, optional, default value 0.08
  - Bhard hardening parameter, optional, default value 0.003
  - Chard hardening parameter, optional, default value 2
  - Dhard hardening parameter, optional, default value $1\times 10^{-6}$
  - Asoft softening parameter, optional, default value 15
  - helem element size $h$, optional (if not specified, the actual element size is used)
  - dilation dilation factor, optional, default value 0.85
  - hp hardening modulus, optional, default value 0.5
  - isoflag flag which allows for the use of only one damage parameter if set to 1, default value 0
  - rateflag flag which allows for the consideration of the effect of strain rates if set to 1, default value 0
  - fcZero Input parameter for modelling the effect of strain value. Recommended value $10$ MPa. Only if rateflag=1, fcZero has to be specified.
  - yieldtol tolerance for the implicit stress return algorithm, optional, default value $1\times 10^{-6}$
  - newtoniter maximum number of iterations in the implicit stress return algorithm, optional, default value 100
Supported modes 3dMat, PlaneStrain

Borek Patzak