# Extended limit analysis of porous media with non-associated DruckerPrager matrix: a bipotential approach and numerical validation

Long Cheng, Abdelbacet Oueslati, Jia Yun, Géry de Saxcé, Djimedo Kondo

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Long Cheng, Abdelbacet Oueslati, Jia Yun, Géry de Saxcé, Djimedo Kondo. Extended limit analysis of porous media with non-associated DruckerPrager matrix: a bipotential approach and numerical validation. 12e Colloque national en calcul des structures, CSMA, May 2015, Giens, France. hal-01516618

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# Extended limit analysis of porous media with non-associated Drucker-Prager matrix: a bipotential approach and numerical validation

L. Cheng<fnref n="1" />, A. Oueslati<fnref n="2" />, J. Yun<fnref n="2" />, G. de Saxcé<fnref n="2" />, D. Kondo<fnref n="1" />

**Abstract** — We first recall the basic elements of the non-associated (NA) Drucker-Prager plasticity model and then present the corresponding extended limit analysis theorems. Application of the latter to porous materials allows to establish a closed-form expression of the macroscopic criterion and of the non-associated flow rule. The established results, for the porous material with a NA matrix, recover available models as particular cases. Finally, the new model is assessed and compares well with predictions by means of Finite Element computations carried out on a cell.

**Mots clés** — Bipotential theory, Nonlinear homogenization, Extended limit analysis, Ductile porous materials, Non-associated plasticity, Drucker-Prager matrix.

## 1 Introduction

Following the pioneering work of Gurson, Guo and al. [Guo et al.(2008)] have proposed a macroscopic plastic model for ductile porous materials with pressure-sensitive and dilatant matrix obeying to the normality law (associated material). The present study aims at extending this limit analysis based model to porous materials for which the solid matrix obeys to a non-associated plastic flow rule (dilatancy angle differs from the friction one). In this case, as the normality property fails, the classical limit analysis cannot be applied. The usual approach of non associated plasticity is based on a yield function and a plastic potential (to represent the flow rule). Although it is intensively considered in the literature, this approach is in fact not very efficient in formulating variational methods required for limit analysis procedures. For this reason, the second last author had proposed several years ago a suitable modeling framework based on the bipotential, a function of both dual variables, the plastic strain rate and stress tensors (for numerical aspects see for instance [Hjiaj et al.(2003)]). On this ground, we take advantage of the bipotential-based variational framework to propose a macroscopic model for porous materials with a non-associated matrix. The procedure is implemented for a hollow sphere model by adopting simple suitable trial velocity fields and trial stress fields. The macroscopic yield function, obtained in a parametric form, is porosity $f$ and dilatancy angle dependent. Also, as in [Maghous et al.(2009)] (who had used a modified secant moduli approach), the non-associated character of the macroscopic flow rule is demonstrated. The established results are supported by results from Finite Elements computations performed in the context of non associated plasticity.

## 2 A bipotential formulation of the non-associated Drucker-Prager model

In this section, we will briefly recall the main ingredients of the non-associated Drucker-Prager model and present the corresponding bipotential-based formulation. For more details on the application of the bipotential theory to the non-associated Drucker-Prager model, the reader is referred for instance to [Hjiaj et al.(2003)]. The non-associated Drucker-Prager model is classically formulated by first considering a yield criterion in the form:

$$F(\sigma) = \sigma_e + 3\alpha\sigma_m - \sigma_0 \le 0, \quad (1)$$

where $\sigma_e$ is the equivalent stress, $\sigma_m$ the mean stress, $\sigma_0 > 0$ the shear cohesion stress of the material and $\alpha$ the pressure sensitivity factor related to the friction angle $\phi$ by: $\tan \phi = 3\alpha$. Owing to the non-

<footnote n="1">¹ IJLRDA, UMR7190 CNRS, UPMC, 4, Place Jussieu, Paris, {long.cheng,djimedokondo}@upmc.fr</footnote>

<footnote n="2">² LML, UMR8107, Université de Lille 1, Cité Scientifique, Villeneuve d'Ascq, {abdelbace.Oueslati, jia.yun, gery.desaxce}@univ-lille1.fr</footnote>
associated character of the model, a plastic potential is also required:

$$G(\sigma) = \sigma_e + 3\beta\sigma_m \quad (2)$$

where $\beta$ ($\beta \le \alpha$) depends on the dilatancy angle $\psi$ through $\tan \psi = 3\beta$. The plastic admissibility condition corresponding to this law classically reads:

$$H(d) = \beta d_{eq} - d_m \le 0$$

in which $d$ represents the plastic strain rate. For $\psi = \phi$, the normality rule is recovered

For the non-associated Drucker-Prager model, the bipotential, which represents the plastic dissipation power (by volume unit), takes the form [Hjiaj et al.(2003)]:

$$b(d, \sigma) = \begin{cases} \frac{\sigma_0}{\alpha} d_m + (\beta - \alpha) \left(3\sigma_m - \frac{\sigma_0}{\alpha}\right) d_{eq} & \text{if } F(\sigma) \le 0 \text{ and } H(d) \le 0 \\ +\infty & \text{otherwise} \end{cases}, \quad (3)$$

in which, $d$ is the strain rate tensor, $\sigma$ the stress tensor. The finite value of $b(d, \sigma)$ will be denoted for further consideration by $b_0(d, \sigma)$.

### 3 Extended limit analysis of porous media with a non-associated matrix

The bipotential formulation naturally opens the way to a variational formulation, allowing then an extension of limit analysis methods. We present here the main elements of this variational framework in the context of porous media. To this end, let us consider a reference cell $\Omega$ (whose boundary is denoted $\partial\Omega$) composed of a void $\omega$ and a non-associated matrix $\Omega_M = \Omega - \omega$. The external boundary of the cell is subjected to a uniform strain rate: $v = D \cdot x$, $x$ being the position vector at the boundary. The macroscopic stress is denoted $\Sigma$ while $D$ is the macrostrain rate. The solution to the homogenization problem consists in a variational formulation, more appropriate for simple approximations, thanks to relevant choice of trial fields and a minimization procedure. We have shown that this requires to solve a minimization problem whose Lagrangian function reads [Cheng et al.(2015), Cheng(2013)]:

$$\mathcal{L}(v, \sigma(\Sigma), \Lambda) = B_0(v, \sigma) + \Lambda \frac{1}{|\Omega|} \int_{\Omega_M} H(d) \, dV \quad (4)$$

$\Lambda$ being a positive multiplier. In (4), has been introduced the macroscopic finite valued bifunctional:

$$B_0(v', \sigma') = \frac{1}{|\Omega|} \int_{\Omega_M} b_0(d(v'), \sigma') \, dV - D : \Sigma. \quad (5)$$

defined for trial fields $v'$ and $\sigma'$. The local finite valued bifunctional, $b_0$, has been introduced in (3). The determination of the above Lagrangian function $\mathcal{L}$ requires a choice of a class of trial stress fields and of a trial velocity ones. The reader interested by this implementation must refer to [Cheng et al.(2015), Cheng(2013)]. Note that the macroscopic criterion and flow rule can be obtained after solving the minimization problem. Indeed:

* The macroscopic yield criterion is given by

$$\mathcal{F}(\Sigma(\phi, \psi, f)) = 0 \quad (6)$$

which is obtained by computing the stationarity of $\mathcal{L}$ with respect to the macrostrain rate $D$:

$$\frac{\partial \mathcal{L}}{\partial D}(\Lambda, \Sigma) = 0 \quad (7)$$

It has been shown that the above macroscopic criterion depends not only on the porosity $f$ and the friction angle $\phi$, but also on the dilatancy angle $\psi$ of the matrix.
* The non-associated macroscopic flow rule

$$D = D_m \mathbf{1} + \frac{3}{2} \frac{S}{\Sigma_e} D_e = D_e \left( \frac{D_m}{D_e} \mathbf{1} + \frac{3}{2} \frac{S}{\Sigma_e} \right) \quad (8)$$

$D_m$ is the hydrostatic part of $D$ while $D_e$ represents the equivalent strain rate and $S$ the deviatoric part of the macroscopic stress tensor. The ultimate step to obtain this flow rule lies therefore in the determination of $D_m/D_e$, which is given by the macroscopic admissibility condition

$$\frac{1}{|\Omega|} \int_{\Omega_M} H(d) \, dV = 0. \quad (9)$$

The above bipotential-based variational approach of porous media has been fully developed in [Cheng(2013), Cheng et al.(2015)] by considering a hollow sphere model made up of a spherical void embedded in a rigid-plastic matrix obeying to a non-associated Drucker-Prager model. Some predictions of the resulting analytical model will be now illustrated.

# 4 Illustration and numerical validation of the established criterion

In this section, the predictions of the established macroscopic criterion in non-associated cases (with different values of dilatancy angle $\psi$) will be first compared with the associated one in subsection 4.1. The expected influence of the non-associated features is clearly indicated. Next, Finite Element Method (FEM) based limit analysis computations are performed in subsection 4.2 and their results allow to assess the obtained theoretical criterion.

## 4.1 Preliminary illustration of the established criterion

We aim now at illustrating the macroscopic criterion both in associated and non-associated cases. The matrix pressure sensitivity is characterized by the friction angle $\phi$ and the dilatancy one $\psi$ for the Drucker-Prager model. We provide here on figure 1 illustration of the established criterion for a porosity $f = 0.2$ and friction angle $\phi = 30^{\circ}$. The corresponding associated case $\psi = \phi = 30^{\circ}$ is denoted AC, while two non-associated cases are considered; they are respectively defined by dilatancy angles $\psi = 15^{\circ}$ (NAC1) and $\psi = 5^{\circ}$ (NAC2).

For hydrostatic loadings (traction and compression), the non-associated cases provide the same predictions as that of the associated one (see [Guo et al.(2008)]). This observation is in full agreement with the theoretical and numerical results already established in [Cheng et al.(2012)]. Furthermore, the non-associated cases show in general different yield loci with respect to the associated one: as expected, the yield surface for a non-associated case is lower than the associated one. Note that a decrease of the dilatancy angle leads to a weaker strength, the difference between the cases $\psi = 15^{\circ}$ and $\psi = 5^{\circ}$ being however slight. These results will be assessed in subsection 4.2 by means of numerical FE results.

<figure><img src="image_1.png" /><figcaption>Figure 1: Comparison of yield surfaces between the associated case (denoted AC) with dilatancy angle $\psi = 30^{\circ}$ and two non-associated cases (denoted NAC1 and NAC2) with $\psi = 15^{\circ}$ and $5^{\circ}$, respectively. Porosity: $f = 0.2$; friction angle $\phi = 30^{\circ}$.</figcaption></figure>
## 4.2 Numerical investigations of the macroscopic yield surface and plastic flow rule of the porous medium with a non associated matrix

For the FEM analysis, we consider an axisymmetric model of the spherical shell. Moreover, the numerical analysis is carried out in the context of non-associated elastoplasticity and small deformations. The computations are performed by means of ABAQUS/Standard software and a user subroutine MPC (Multi-Points Constraints).

Fig.2 displays the FEM results not only for the macroscopic limit stress, but also for the direction of plastic flow. As in subsection 4.1, the porosity $f = 0.2$ and friction angle $\phi = 30°$ are considered here. Also, the direction of plastic flow for the associated case and two non-associated cases are denoted DA($\psi = 30°$), DNA1 ($\psi = 15°$) and DNA2 ($\psi = 5°$), respectively. Moreover, the numerical yield surfaces, obtained by connecting each FEM point of plastic limit state, are indicated by SFA($\psi = 30°$), SFNA1($\psi = 15°$) and SFNA2($\psi = 5°$), respectively. Note that each FEM point has been obtained by performing computation at fixed stress triaxialities $\Sigma_m/\Sigma_e$ (equivalently at fixed $T_m/T_e$).

Coming now to the results, an excellent agreement between the DA and SFA is noted (see Fig.2), the plastic flow direction (DA) at each FEM point being normal to the yield surface (SFA). Concerning the plastic flow direction (DNA1 and DNA2) for the cases of non-associated matrix, a lack of normality to the corresponding yield surfaces (SFNA1 and SFNA2) is noted. These FEM results prove the non-associated character of the macroscopic flow rule in the case of a non-associated matrix. It must be noted that the lack of normality is more pronounced when the dilatancy angle $\psi$ is smaller.

<figure><img src="image_2.png" /><figcaption>Figure 2: FEM results: plastic flow directions (denoted DA for associated case, DNA1 and DNA2 for non-associated ones) and yield surface (denoted SFA for associated case, SFNA1 and SFNA2 for non-associated ones).</figcaption></figure>

## References

[Cheng et al. (2012)] Cheng, L., Jia, Y., Oueslati, A., de Saxcé, G., Kondo, D., 2012 Plastic limit state of the hollow sphere model with non-associated Drucker-Prager material under isotropic loading. Computational Materials Science, Vol. 62, 210-215.

[Cheng (2013)] Cheng, L. Homogenization of porous media with plastic matrix and non-associated flow rule by variational methods. Ph.D. Thesis, Univ. Lille 1, 2013.

[Cheng et al. (2015)] Cheng, L., Jia, Y., Oueslati, A., de Saxcé, G., Kondo, D., 2015 A bipotential-based limit analysis and homogenization of ductile porous materials with non-associated Drucker-Prager matrix. Journal of the Mechanics and Physics of Solids, In Press.

[Guo et al. (2008)] Guo, T.F., Faleskog, J., Shih, C.F. 2008. Continuum modeling of a porous solid with pressure-sensitive dilatant matrix. Journal of the Mechanics and Physics of Solids, 56, 2188-2212.

[Hjiaj et al. (2003)] Hjiaj, M., Fortin, J., de Saxcé, G., 2003. A complete stress update algorithm for the non-associated Drucker-Prager model including treatment of the apex. Int. Journ. of Eng. Sci., 41, 1109-1143.

[Maghous et al. (2009)] Maghous, S., Dormieux, L., Barthélémy, J.F., 2009. Micromechanical approach to the strength propoerties of frictional geomaterials. European Journal of Mechanics A/Solids, 28, 179-188.