Particle finite element analysis of the granular column collapse problem

Abstract

The problem of granular column collapse is investigated by means of an axisymmetric version of the particle finite element method (PFEM). The granular medium is represented by a simple rate-independent plasticity model and the frictional contact between the granular flow and its rigid basal surface is accounted for. In the version of the PFEM developed for this study, the governing equations of the boundary value problem are cast in terms of an optimization problem and solved using mathematical programming tools. The agreement between model and experiment is generally satisfactory, quantitatively as well as qualitatively. However, the friction angle of the granular material, as well as the exact interface conditions between the base and granular material, are shown to have a relatively significant influence on the results.

Euler-Lagrangian equations

The equivalence between the optimization problem (13) and the governing equations described in Sect. 3 are verified by showing that the Euler-Lagrange equations associated with (13) do indeed reproduce the governing equations. To this end, the inequality constraints in (13) are first converted into equality constraints by subtraction of positively restricted slack variables $\varvec{s}$:

$$\begin{aligned}&\underset{\Delta \varvec{u}}{\text {min}}\,\,\underset{(\varvec{\sigma },\varvec{r})_{n+1}}{\text {max}} \langle \varvec{\sigma }_{n+1}, {{\varvec{L}} (\Delta {\varvec{u}})}\rangle _\varOmega - \langle \tilde{\varvec{b}},\Delta \varvec{u} \rangle _\varOmega - \langle \tilde{\varvec{t}},\Delta \varvec{u}\rangle _\varGamma \nonumber \\&-\,\frac{1}{2}\Delta t^2\langle \varvec{r}_{n+1},{\tilde{\rho }}{}^{-1}\varvec{r}_{n+1}\rangle _\varOmega + \langle \varvec{r}_{n+1},\Delta \varvec{u}\rangle _\varOmega +\beta \,\Sigma _{I \in \mathcal {C}}\ln s^I\nonumber \\&\text {subject to} F(\varvec{\sigma }_{n+1}) +\varvec{s} = 0 \end{aligned}$$

(27)

where $\beta$ is an arbitrarily small constant and $\mathcal {C}$ is the set of stress integration points. The logarithmic barrier function eliminates the need to make explicit reference to the non-negativity requirement on s. The standard Lagrange multiplier technique then applies, i.e. the solution to (27) is found by requiring stationarity of the following functional:

$$\begin{aligned} J&= \langle \varvec{\sigma }_{n+1},\varvec{L}(\Delta \varvec{u})\rangle _\varOmega - \langle \tilde{\varvec{b}},\Delta \varvec{u} \rangle _\varOmega - \langle \tilde{\varvec{t}},\Delta \varvec{u}\rangle _\varGamma \nonumber \\&-\, \frac{1}{2}\Delta t^2\langle \varvec{r}_{n+1},{\tilde{\rho }}{}^{-1}\varvec{r}_{n+1}\rangle _\varOmega + \langle \varvec{r}_{n+1},\Delta \varvec{u}\rangle _\varOmega \nonumber \\&+\,\beta \,\Sigma _{I \in \mathcal {C}}\ln s^I -\Delta \varvec{\lambda }(F(\varvec{\sigma }_{n+1})+\varvec{s}) \end{aligned}$$

(28)

where $\varvec{\Delta }\varvec{\lambda }$ are Lagrange multipliers. The stationary conditions are given by

$$\begin{aligned}&{\displaystyle \frac{\delta J}{\delta \Delta \varvec{u}}} = \left\{ \begin{array}{lll} \varvec{S}^\mathsf{{\tiny T} }\varvec{\sigma }_{n+1} + \tilde{\varvec{b}} = \varvec{r}_{n+1},\,\,\text {in}\,\varOmega \\ \varvec{N}^\mathsf{{\tiny T} }\varvec{\sigma }_{n+1} = \tilde{\varvec{t}},\,\,\text {on}\,\varGamma \end{array} \right. \end{aligned}$$

(29)

$$\begin{aligned}&{\displaystyle \frac{\delta J}{\delta \varvec{r}_{n+1}}} = -\Delta t^2 \tilde{\rho }{}^{-1}\varvec{r}_{n+1} + \Delta \varvec{u} = \varvec{0}\end{aligned}$$

(30)

$$\begin{aligned}&{\displaystyle \frac{\delta J}{\delta \varvec{\sigma }_{n+1}}} = \varvec{\nabla }\varvec{u} - \Delta \varvec{\lambda }\nabla _\sigma F(\varvec{\sigma }_{n+1}) = \varvec{0}\end{aligned}$$

(31)

$$\begin{aligned}&{\displaystyle \frac{\delta J}{\delta \Delta \varvec{\lambda }}} = F(\varvec{\sigma }_{n+1}) +\varvec{s} = \varvec{0}\end{aligned}$$

(32)

$$\begin{aligned}&{\displaystyle \frac{\delta J}{\delta s^I}} = {\displaystyle \frac{\beta }{s^I}} -\Delta \lambda ^I= 0 \,\Rightarrow \, s^I\Delta \lambda ^I=\beta ,\,\,I\,\in \,\mathcal {C} \end{aligned}$$

(33)

For $\beta \rightarrow 0$, these equations are easily verified as being the governing equations for the problem at hand. Note that the flow rule in the above equations is associated, i.e. the flow potential is equal to the yield potential $F$. This is a consequence of the variational formulation and is not possible to circumvent directly. In this paper we use the approach suggested in [37] of replacing the original yield function by an ‘effective’ one which, when used as a flow potential, yields the correct plastic strains. This approach was shown to produce reliable results for a range of static small deformation problems and in this paper we demonstrate that it is equally reliable for dynamic problems involving very large deformations.