/MAT/LAW81 (DPRAG_CAP)

Plasticity is perfect unless an isotropic hardening is defined.

This law is LAG, ALE and EULER compatible.

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Definition

Comments

1. The yield surface is defined as:
   $$F=q-{\mathrm{r}}_c\left(p\right)\cdot \left(p\tan \varphi +c\right)=0$$

   Where,

   p

   Hydrostatic pressure $p=-\frac{1}{3}Tr\left(\mathit{\sigma }\right)$

   q

   von Mises stress $q=\sqrt{\frac{3}{2}\mathit{s}:\mathit{s}}$

   c

   Material cohesion (yield stress under pure shear loading) $p=0$

   $\varphi$

   Friction angle $\left(0\le \varphi \le \frac{\pi }{2}\right)$

   ${\mathrm{r}}_c\left(p\right)$ is used to create a smooth transition between the Drucker-Prager criterion and the cap hardening, and is defined such that ${\mathrm{r}}_c\left(p\right)=\left\{\begin{array}{l} 1 if p\le p_a\\ \sqrt{1-{\left(\frac{p-p_a}{p_b-p_a}\right)}^2} else if p_a\le p\le p_b\\ 0 else\end{array}\right.$

   Where,

   p_a

   Pressure triggering the yield surface to cap transition and is obtained with $p_a=\alpha p_b$

   p_b

   Cap limit pressure.

   The yield surface thus defined is plotted in the figure below:

   
   
   

   Note: In the figure above, a characteristic pressure denoted $p_0$ is emphasized. This pressure is obtained where the derivative of the yield function equals zero $\frac{\partial F}{\partial p}=0$ :

   $p_0=p_a+\frac{-\left(p_a\tan \varphi +c\right)+\sqrt{{\left(p_a\tan \varphi +c\right)}^2+8{\left(p_b-p_a\right)}^2{\tan }^2\varphi }}{4\tan \varphi }$

2. Plastic flow is governed by the non-associated flow potential G, as:

   $G=\left\{\begin{array}{l}q-p\tan \psi if p\le p_a\\ q-\tan \psi \left(p-\frac{{\left(p-p_a\right)}^2}{2\left(p_0-p_a\right)}\right) else if p_a\le p\le p_0\\ F else plastic flow associated on the cap\end{array}\right.$

3. If the two plasticity parameters $p_0$ and $c$ are defined as constant $p_b=p_{b0}$ and $c=c_0$ , the plastic response is perfect. Nevertheless, you can introduce an isotropic hardening using respectively the tabulated functions $fct\_I{D}_{P_b}$ and $fct\_I{D}_C$ . The cap limit pressure will then evolve following the plastic volumetric strain (chosen as positive in compression), whereas the material cohesion will evolve following the equivalent plastic strain. In that case, the two initial parameters $p_{b0}$ and $c_0$ becomes scales factors:

   $p_b\left({\epsilon }_v^p\right)=p_{b0}·f_{Pb}\left({\epsilon }_v^p\right)$ with ${\epsilon }_v^p=-Tr\left({\mathit{\epsilon }}^p\right)$

   $c\left({\epsilon }_{eq}^p\right)=c_0·f_c\left({\epsilon }_{eq}^p\right)$ with ${\epsilon }_{eq}^p=\sum _{t=0}^t\text{Δ}{\epsilon }_p$

4. It is worth noting that a maximum dilatancy can be defined. In this case, the effect of the pressure on the plastic flow becomes limited to compression:
5. By default, unlike the equivalent deviatoric plastic strain ${\epsilon }_{eq}^p$ , the plastic volumetric strain ${\epsilon }_v^p$ can increase (in compression) or decrease (in tension) so that it may become negative. This must be considered when defining the tabulated functions using this variable. Moreover, in that case, the yield surface can harden or soften. To avoid that, you can restrict the evolution of the volumetric plastic strain to a monotonically increase using the flag $I_{soft}=1$ . As a result:
   $${\epsilon }_v^p=\left\{\begin{array}{l}\sum _{t=0}^t\text{Δ}{\epsilon }_v^p if I_{soft}=0\\ \sum _{t=0}^{t}max \left(\text{Δ}{\epsilon }_v^p, 0\right) if I_{soft}=1\end{array}\right.$$
6. Like the cap limit pressure $p_b$ , the bulk and shear modulus might not be constant and may vary according to the volumetric plastic strain. As described in Comment 3, their evolution can be tabulated with:

   $K\left({\epsilon }_v^p\right)=K_0·f_K\left({\epsilon }_v^p\right)$

   $P\left({\epsilon }_v^p\right)=P_0·f_P\left({\epsilon }_v^p\right)$

7. You may want to consider that the pores of the material (concrete, soil) can be filled partially or completely with water, which can influence the mechanical behavior (especially under compressive loading). The porosity denoted $n$ represents the volume fraction of voids with respect to the total material volume:
   $$n= \frac{V_{void}}{V_{total}}$$

   In the elastic case, the void volume does not change. However, in plasticity, the porosity change according to the volumetric plastic strain is defined by:

   $$n=1-\left(1-n_0\right)e^{{\epsilon }_v^p-{\epsilon }_{v0}^p}$$
   Where,

   $n_0$

   Initial porosity

   ${\epsilon }_{v0}^p$

   Initial volumetric plastic strain

   When volumetric plastic strain increases, the pores of the material shrink and then the porosity decreases. The saturation in water is then higher. The saturation of pores in water corresponds to the volume fraction of water in the pores (void):

   $$S=\frac{V_{water}}{V_{void}}$$

   When the saturation is complete $S\ge 1$ the water pressure starts to influence the compressive behavior. The pore pressure corresponds to the volumetric water pressure that is denoted $u$ and is computed with:

   $$u=K_w\left(S-1\right)$$
   Where,

   $K_w$

   Bulk water modulus

   $\left(S-1\right)$

   Water volumetric strain, also denoted ${\mu }_w$

   A smooth transition is introduced to improve stability at the beginning of water pressure computation:

   • If ${\mu }_w<-tol$ , the water pressure equals zero,
   • If $\left|{\mu }_w\right|\le tol$ , $u=\frac{K_w}{4tol}{\left({\mu }_w+tol\right)}^2$ ,
   • If ${\mu }_w>-tol$ , $K_w{\mu }_w$ .

   In addition, a viscous pressure is added when ${\mu }_w>-tol$ , defined as:

   $$u_{vis}=-{\alpha }_v{\sqrt{K_w\rho V}}^{\frac{1}{3}}$$
   The obtained pore water pressure evolution is shown below:

   
   
   

   To consider the pore water pressure on the compressive mechanical behavior, the cap is shifted. The pressure then considered in the constitutive equation is substituted by a new one denoted $p^{\text{'}}$ such as:

   $$p^{\text{'}}\left\{\begin{array}{l}p if p<p_0\\ p_0 else if p-u\le p_0 \\ p-u else\end{array}\right.$$
   The cap shift then defined is plotted in the figure below:

   
   
   

8. The initial state of the pores is first defined by the initial porosity $n_0$ . Then, you can either specify an initial pore water pressure $u_0$ , or set an initial saturation $S_0$ :
   • If $u_0\ge 0$ , the initial saturation is recomputed with:
     $$S_0=1+\frac{u_0}{K_w}$$
   • If $S_0\ge 1$ , the pores are already saturated, and the initial pores water pressure is recomputed with:
     $$u_0=K_w\left(S_0-1\right)$$
   • If $0\le S_0<1$ , the initial pore water pressure is null $u_0=0$ .
9. The following user variables are available for post-treatment:
   • USR1 is the equivalent plastic strain ${\epsilon }_{eq}^p$
   • USR2 is the plastic volumetric strain ${\epsilon }_v^p$
   • USR3 is the cohesion c
   • USR4 is the cap limit pressure p_b
   • USR5 is the surface to cap pressure transition p _a
   • USR6 is the null derivative criterion pressure p₀
   • USR7 pore water pressure u
   • USR8 porosity n
   • USR9 saturation $S$
   • USR10 cap shift p^'