/MAT/LAW124 (CDPM2)

Where the Haigh-Westergaard coordinates in stresses space are considered:

${\overline{\sigma }}_v=\frac{\text{tr}(\sigma )}{3}$ , $\overline{\rho }=\sqrt{2J_2}=\sqrt{s:s}$ , $\overline{\theta }=\frac{1}{3}\arccos \left(\frac{3\sqrt{3}}{2}\frac{J_3}{J_2^{3/2}}\right)$

Where,

${\kappa }_p$

Hardening variable

$f_c$

Limit strength in compression

$f_t$

Limit strength in tension

$m_0$

A parameter considering the effect of eccentricity $e_{cc}$

(2)

$$m_0=\frac{3\left(f_c^2-f_t^2\right)}{f_c{f}_t}\frac{e_{cc}}{e_{cc}+1}$$

Figure 1. CDPM2 model yield function shape (from Grassl)

Then $q_{h1}$ and $q_{h2}$ are the two hardening functions defined with (Figure 2):(3)

$$q_{h1}\left({\kappa }_p\right)=\left\{\begin{array}{ccc}{q}_{h0}+\left(1-q_{h0}\right)\left({\kappa }_p^3-3{\kappa }_p^2+2{\kappa }_p\right)& if& {\kappa }_p<1\\ 1& if& {\kappa }_p\ge 1\end{array}\right.$$

(4)

$$q_{h2}\left({\kappa }_p\right)=\left\{\begin{array}{ccc}1& if& {\kappa }_p<1\\ 1+H_p\left({\kappa }_p-1\right)& if& {\kappa }_p\ge 1\end{array}\right.$$

Figure 2. Hardening functions shape (from Grassl)

Here, $q_{h0}$ is the initial hardening defined so that $0<q_{h0}<1$ . $H_p$ is the hardening modulus whose recommended values are 0.01 without strain rate dependency (IRATE = 1), and 0.5 with rate effects (IRATE = 2). The evolution of the internal variable ${\kappa }_p$ is detailed below.

With

$m_g=A_g{B}_g{f}_c\exp \frac{{\overline{\sigma }}_v-q_{h2}\raisebox{1ex}{$f_t$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{B_g{f}_c}$

$A_g=\frac{3f_t{q}_{h2}}{f_c}+\frac{m_0}{2}$

$B_g=\frac{\left(\raisebox{1ex}{$q_{h2}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\left(1+\raisebox{1ex}{$f_t$}\!\left/ \!\raisebox{-1ex}{$f_c$}\right.\right)}{\ln A_g-\ln \left(2D_f-1\right)-\ln \left(3q_{h2}+\raisebox{1ex}{$m_0$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)+\ln \left(D_f+1\right)}$

Where, $D_f$ is the dilation parameter.

with $x_h=\left\{\begin{array}{ccc}{A}_h-\left(A_h-B_h\right)\exp \left(\raisebox{1ex}{$-R_h\left({\overline{\sigma }}_v\right)$}\!\left/ \!\raisebox{-1ex}{$C_h$}\right.\right)& \text{if}& R_h\left({\overline{\sigma }}_v\right)\ge 0\\ E_h\exp \left(\raisebox{1ex}{$R_h\left({\overline{\sigma }}_v\right)$}\!\left/ \!\raisebox{-1ex}{$F_h$}\right.\right)+D_h& \text{if}& R_h\left({\overline{\sigma }}_v\right)<0\end{array}\right.$

Where, $‖{\dot{\epsilon }}_p‖$ is the Euclidian norm of the increment of the plastic strain tensor.

The inelastic strains can be obtained from damage history variable using the following equations:

${\epsilon }_{inel}^t={\kappa }_{dt1}+{\omega }_t{\kappa }_{dt2}$ and ${\epsilon }_{inel}^c={\kappa }_{dc1}+{\omega }_c{\kappa }_{dc2}$

The damage history variable finally enables to update the corresponding damage variables.

In these different equations, $h$ is a parameter that can be used to avoid mesh size dependency. When IREG = 1, no regularization method is used, and $h$ is set to 1. In that case, the critical value $w_f$ becomes a critical strain with no dimension. Otherwise, if IREG = 2, the Hillerborg’s regularization method ² (also called Crack Brand method ³) is used, and $h$ equals the initial element size. Then, the critical value $w_f$ becomes a critical displacement homogeneous to a displacement. Hillerborg’s regularization method is to ensure that the tensile fracture energy denoted $G_f^t$ remains constant no matter what element size is used (Figure 6).

Figure 6. Uniaxial tension test with bilinear damage on two different mesh sizes with IREQ = 0 (left); IREQ = 1 (right)

Regarding compression damage, only the exponential evolution shape is available without any regularization method (Figure 7). Mesh size dependency is assumed to be less sensitive.(12)

$${\omega }_c=1-\exp \left(-\frac{{\epsilon }_{inel}^c}{e_{fc}}\right)$$

Figure 7. Uniaxial compression test on a single unit element . with $e_{fc}=5\text{×}{10}^{(-4)}$

$f_t^{rate}={\alpha }_{rate}{f}_t$ , $f_c^{rate}={\alpha }_{rate}{f}_c$ and $f_{t1}^{rate}={\alpha }_{rate}{f}_{t1}$

Where, ${\alpha }_c$ is the compression factor defined above in damage history variables equations in Comment 3.

There is then a different strain rate dependency between tension and compression as concrete is more sensitive to strain rate effect in tension than in compression. The two dynamic increased factor for both tension and compression are computed with:

${\alpha }_{rate}^t=\left\{\begin{array}{ccc}1& \text{for}& {\dot{\epsilon }}_{\max }\le 30\times {10}^{-6}{\text{s}}^{-1}\\ {\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{t0}}\right)}^{{\delta }_s}& \text{for}& 30\times {10}^{-6}{\text{s}}^{-1}<{\dot{\epsilon }}_{\max }\le 1{\text{s}}^{-1}\\ {\beta }_s{\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{t0}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}& \text{for}& 1{\text{s}}^{-1}\le {\dot{\epsilon }}_{\max }\end{array}\right.$ with ${\delta }_s=\frac{1}{1+8\raisebox{1ex}{$f_c$}\!\left/ \!\raisebox{-1ex}{$f_{c0}$}\right.}$ , $\log {\beta }_s=6{\delta }_s-2$

${\alpha }_{rate}^c=\left\{\begin{array}{ccc}1& \text{for}& {\dot{\epsilon }}_{\max }\le 30\times {10}^{-6}{\text{s}}^{-1}\\ {\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{c0}}\right)}^{1.026{\alpha }_s}& \text{for}& 30\times {10}^{-6}{\text{s}}^{-1}<{\dot{\epsilon }}_{\max }\le 30{\text{s}}^{-1}\\ {\gamma }_s{\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{c0}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}& \text{for}& {\text{30s}}^{-1}\le {\dot{\epsilon }}_{\max }\end{array}\right.$ with ${\alpha }_s=\frac{1}{5+9\raisebox{1ex}{$f_c$}\!\left/ \!\raisebox{-1ex}{$f_{c0}$}\right.}$ , $\log {\gamma }_s=6.56{\alpha }_s-2$

The equivalent deviatoric strain rate $\dot{\epsilon }$ is used to compute the DIF in the equations above.

No parameters need to be identified for strain rate effect. You only need to set the flag IRATE to 2. Figure 11 shows the expected tendency of strain rate effect on the CDPM2 behavior. By increasing the strength limits in tension/compression, the dissipated energy during failure is also affected which is often observed experimentally.

Figure 11. Uniaxial tests in tension/compression with strain rate dependency (IRATE = 1)

${\epsilon }_i=\frac{f_t\left({\left(1.16f_c\right)}^2-f_c^2\right)}{1.16f_c\left(f_c^2-f_t^2\right)}$ and $e_{cc}=\frac{1+{\epsilon }_i}{2-{\epsilon }_i}$