Multi-scale modelling of granular materials: numerical framework and study on micro-structural features
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Abstract
A multi-scale model for the analysis of granular systems is proposed, which combines the principles of a coupled FEM–DEM approach with a novel servo-control methodology for the implementation of appropriate micro-scale boundary conditions. A mesh convergence study is performed, whereby the results of a quasi-static biaxial compression test are compared with those obtained by direct numerical simulations. The comparison demonstrates the capability of the multi-scale method to realistically capture the macro-scale response, even for macroscopic domains characterized by a relatively coarse mesh; this makes it possible to accurately analyse large-scale granular systems in a computationally efficient manner. The multi-scale framework is applied to study in a systematic manner the role of individual micro-structural characteristics on the effective macro-scale response. The effect of particle contact friction, particle rotation, and initial fabric anisotropy on the overall response is considered, as measured in terms of the evolution of the effective stress, the volumetric deformation, the average coordination number and the induced anisotropy. The trends observed are in accordance with notions from physics, and observations from experiments and other DEM simulations presented in the literature. Hence, it is concluded that the present framework provides an adequate tool for exploring the effect of micro-structural characteristics on the macroscopic response of large-scale granular structures.
Keywords
Particle aggregates Multi-scale modelling FEM–DEM coupled simulations Micro-structural features1 Introduction
The intrinsic influence of the discrete micro-structure of granular materials on their effective material properties and structural response is nowadays well recognized. The morphology, material evolution and mechanical interactions at the particle scale all contribute to the observed macroscopic non-linear failure and deformation behaviour. Multi-scale approaches provide an ideal tool for the modelling of granular systems, as they allow to directly incorporate the complex behaviour of the discrete micro-structure into the response of large-scale structural problems. This is typically done by coupling the discrete element method (DEM), which accurately represents the complex particle behaviour at the micro scale [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], to the finite element method (FEM), which enables to efficiently solve boundary value problems at the macro scale. As a general principle, each integration point in the macro-scale FEM model is connected to a corresponding DEM micro-scale model via the application of adequate homogenization relations. In specific, a macroscopic deformation measure is imposed on the granular micro-structure through the definition of appropriate boundary conditions [11, 12]. The DEM model is solved in turn, providing the particle contact forces in the granular assembly. These forces are subsequently translated into a macroscopic stress measure, which is required to solve the boundary value problem at the structural level. Several examples of coupled FEM–DEM approaches for granular materials have been presented in the literature. In [13, 14] a quasi-static multi-scale method is formulated within the framework of small deformations, whereby the role of the particle microstructure on the effective frictional failure response of macroscopic samples is analysed, with a special focus on the initiation of strain localization. In [15] a small-strain multi-scale framework is proposed that elegantly computes the mechanical response for various monotonic and cyclic loading problems, whereby drained as well as undrained conditions are considered. In [16] this framework is applied for developing multi-scale insights into classical geomechanical problems, such as retaining wall and footing problems. Coupled FEM–DEM approaches are commonly validated by analysing the macroscopic structural response in experimental tests typical for granular media, such as a biaxial compression test [11, 15, 17, 18, 19], a slope stability test [20], or a (cyclic) shear test [15].
In the current communication a novel multi-scale framework is presented for granular materials, which employs the formulation and implementation of the micro-scale boundary conditions recently published in [12]. This formulation is based on the first-order homogenization approach originally proposed in [11], which includes important aspects that are usually ignored in other homogenization methods for particle systems, namely (i) the Hill–Mandel micro-heterogeneity condition that enforces consistency of energy at the micro- and macro scales, (ii) the influence of particle rotations in the formulation of micro-to-macro scale transitions, and (iii) a rigorous generalization of the multi-scale relations within the theory of finite deformations. The implementation of the micro-scale boundary conditions is performed with a servo-control algorithm that uses a feedback principle similar to that of algorithms applied in control theory of dynamic systems. The servo-control algorithm has several attractive features compared to other methods used for implementing micro-scale boundary conditions. Firstly, from the computational viewpoint the algorithm is relatively simple to implement. Secondly, it can be implemented at the level of the interface communicating information between the macro-scale FEM and micro-scale DEM models, whereby modifications of the FEM and DEM source codes are not needed. The algorithm can therefore be easily combined with commercial software, whose source codes generally are not available to the user. Thirdly, in contrast to the often-used penalty method, the servo-control methodology preserves the physical meaning of the homogenized stress measure derived from the granular assembly. Further, the limit case at which the micro-scale boundary conditions are met exactly is rigorously retrieved from the formulation, see [12] for more details.
The first aim of this communication is to demonstrate how the servo-control algorithm for the micro-scale boundary conditions can be conveniently incorporated in a multi-scale FEM–DEM framework. Accordingly, the governing equations of the multi-scale framework are formulated, and their numerical implementation is validated by comparing the computational results obtained for a quasi-static biaxial compression test to those calculated by direct numerical simulations. The convergence behaviour of the numerical results under mesh refinement is analysed, and the heterogeneity of the mechanical response across the specimen height is explored. The second aim of this communication is to show how the FEM–DEM framework can be used for analysing the influence of micro-structural characteristics on the macroscopic response of a granular system. Using the biaxial compression test, the microscopic properties selected for the variation study are the particle contact friction, the particle rotation and the initial fabric anisotropy. The influence of these properties on the overall, macroscopic response is analysed by computing the evolution of the effective stress, the volumetric deformation, the average coordination number, and the induced fabric anisotropy. This study is essential for gaining confidence in the quality of the multi-scale formulation; nonetheless, most other works on coupled FEM–DEM modelling do not consider such a study, but refer to a specific example simulation for the validation of the proposed method.
The paper is organized as follows. Section 2 presents the numerical homogenization framework for particle aggregates by defining the macro-scale and micro-scale models and the scale transition relations. Section 3 discusses numerical implementation aspects. The explicit time integration scheme adopted for the macro-scale problem is outlined, and details are provided on the dynamic relaxation procedure applied for satisfying the equilibrium conditions, and on the servo-control algorithm used for defining the boundary conditions at the micro scale. The section ends with a presentation of the coupled FEM–DEM solution algorithm. In Sect. 4, the performance of the proposed multi-scale framework is analysed for a biaxial compression test by comparing the computational results to those obtained by direct numerical simulations. A mesh convergence study is performed, and the role of several micro-structural parameters on the macroscopic response is studied. Some concluding remarks are provided in Sect. 5.
In this communication the following notation will be used. The cross product and dyadic product of two vectors are denoted as \(\mathbf {a} \times {\mathbf {b}} = e_{ijk} a_i b_j {\mathbf {e}}_k\) and \( \mathbf {a}\otimes {\mathbf {b}} = a_i b_j {\mathbf {e}}_i \otimes {\mathbf {e}}_j \), respectively. Here \(e_{ijk}\) is the permutation symbol, \({\mathbf {e}}_i\), \({\mathbf {e}}_j\) and \({\mathbf {e}}_k\) are unit vectors in a Cartesian vector basis, and Einstein’s summation convention is used on repeated tensor indices. The inner products between two vectors and two second-order tensors are given by \(\mathbf {a} \cdot {\mathbf {b}} =a_i b_i\) and \( \mathbf {A}: \mathbf {B} = A_{ij} B_{ij}\), respectively. The action of a second-order tensor on a vector is indicated as \( \mathbf {A}\cdot {\mathbf {b}} = A_{ij} b_j {\mathbf {e}}_i \). The symbol \(\mathbf {\nabla }\) indicates the gradient operator with respect to the reference configuration, and |.| refers to the absolute value of a variable. Occasionally, field variables referring to the macroscopic scale are indicated by an overbar, for instance \( {\bar{\mathbf {F}}} \), in order to avoid misinterpretation.
The present study focuses on two-dimensional particle aggregates. Accordingly, the dimensions related to volume, area, stress and mass density are consistently presented in their reduced form as length\(^2\), length, force/length and mass/length\(^2\), respectively.
2 Multi-scale framework for particle aggregates
This section treats the main principles of a multi-scale homogenization strategy for granular structures. These principles ensue from transforming relevant theorems used in classical first-order homogenization theories [21, 22, 23, 24] from a continuous setting to a discrete setting. For more details on this aspect the reader is referred to [12, 25, 26].
2.1 Macro-scale problem
2.2 Micro-scale problem
2.3 Scale transition relations
2.3.1 Macro-to-micro: kinematics and boundary conditions
2.3.2 Micro-to-macro: macroscopic stress and Hill–Mandel condition
3 Numerical implementation
3.1 Macro-scale problem
3.1.1 Finite element formulation
3.1.2 Dynamic relaxation
3.2 Micro-scale problem
3.2.1 Dynamic relaxation
3.2.2 Servo-control algorithm for micro-scale boundary conditions
3.3 Multi-scale FEM–DEM coupling
Incremental-iterative nested multi-scale solution scheme for the coupled FEM–DEM framework
Macro | Micro |
---|---|
1. Initialization | |
\(\bullet \) Initialize the macroscopic model | |
\(\bullet \) Assign a discrete RVE to every IP | |
2. Next increment \( i_{inc} \ge 1 \) | |
\(\bullet \) Apply increment of the macroscopic external load | |
3. Next iteration \( i_{it} \ge 1 \) | |
\(\bullet \) Loop over all integration points | |
If increment \( i_{inc} = 1 \Rightarrow \) Set \(\varvec{\bar{F}}= \mathbf {I} \) and \( \varvec{\bar{P}}= \mathbf {0}\) | |
If increment \( i_{inc} > 1 \Rightarrow \) Compute deformation gradient \( \varvec{\bar{F}}\) \(\xrightarrow {~~ \varvec{\bar{F}}~~}\) | DEM simulation—see Table 2 |
\(\bullet \) Prescribe periodic boundary conditions | |
\(\bullet \) Dynamic relaxation | |
Store macroscopic stress \( \varvec{\bar{P}}\) \(\xleftarrow {~~\varvec{\bar{P}}~~}\) | \(\bullet \) Compute macroscopic stress \( \varvec{\bar{P}}\) |
4. Continue iteration | |
\(\bullet \) End IPs loop | |
\(\bullet \) Solve the macroscopic equation of motion (20) with iteratively updated damping coefficient (26) and time step (27) | |
\(\bullet \) Compute nodal velocities using (22) | |
\(\bullet \) Compute nodal displacements with (23) | |
5. Check for convergence (28) in terms of kinetic energy | |
\(\bullet \) If converged \( \Rightarrow \) go to next increment 2 | |
\(\bullet \) If not converged \( \Rightarrow \) go to next iteration 3 |
Algorithm for the solution of the DEM problem
DEM simulation. Increment \( i_{inc} \) | |
---|---|
1. Initialize boundary conditions by applying updated macro-scale deformation homogeneously | |
\({\mathbf {x}}_q=\varvec{\bar{F}}{\mathbf {X}}_q \quad \text {and } \quad \theta _q=0 \quad \text {for } \quad q=1\ldots ,Q\) | |
2. Dynamic relaxation until convergence criterion (31) is satisfied. | |
Obtain boundary forces and moments. | |
3. Update particle configuration | |
Partition the boundary into corner c and edge e particles | |
Calculate edge particles displacement \( \Delta \mathbf {u}_e \) and rotation \( \Delta \theta _e \) corrections via (33)-(34) | |
and corner particles rotation corrections \( \Delta \theta _c \) via (35)–(36) | |
4. Dynamic relaxation until convergence criterion (31) is satisfied. | |
Obtain boundary forces and moments. | |
5. Calculate residual | |
\( r_a =\sqrt{\sum _{e=1}^{E/2} (\Delta \mathbf {a}_e \cdot \Delta \mathbf {a}_e / \tilde{a}_e^2 + (\Delta {m}_e/\tilde{m}_e)^2 ) + (\Delta {m}_c/\tilde{m}_c)^2 }\) | |
where \( \tilde{a}_k=\frac{{M_k}{R_k}}{{{\Delta t}}^2}, \tilde{m}_k=\frac{{M_k}{R_k^2}}{{{\Delta t}}^2} \) with \( M_k \) and \( R_k \) the mass and radius of particle k; \( k \in \{c, e\} \). | |
6. Check for convergence: \( r_a \le \epsilon _a \) | |
6A. if converged \( \Longrightarrow \) Save current configuration, compute macroscopic stress \( \varvec{\bar{P}}\) with (14) and go to macro-scale simulation. | |
6B. if not converged \( \Longrightarrow \) Return to 3. |
4 Computational results
In this section the proposed FEM–DEM multi-scale framework is validated on a series of representative numerical simulations. A reference problem is defined first, for which a mesh convergence study is performed to establish the appropriate element size for the FEM model. Subsequently, the influence of various micro-structural properties on the macro-scale response is investigated.
4.1 Definition of the reference problem
The macro-scale domain consists of a rectangular specimen of dimensions 10 mm \(\times \) 20 mm, supported vertically at the complete bottom edge and horizontally in the lower left corner node. The domain is discretized into \(n_e\) bilinear quadrilateral elements, with \( n_{ip}=4 \) integration points per element. The specimen is first subjected to isotropic compression with the stress magnitude \( \bar{\sigma }_0 = 0.15 \) MN/m applied in ten loading steps, see Fig. 3a. Next, a biaxial compression loading stage is initiated, whereby the vertical displacement \(\bar{d}\) is increased incrementally up to a vertical strain of \({\bar{\varepsilon }} = \bar{d}/h_{ic} = 10\%\) of the sample height \(h_{ic}\) obtained after isotropic compression, see Fig. 3b. The contribution by the gravitational loading to the sample response is relatively small, and therefore may be ignored. Four different FEM discretizations of the macro-scale domain are considered, as detailed in Sect. 4.2. The damping ratio and safety factor used in expression (27) are \(\xi =1.0\) and \(\gamma =0.5\), respectively^{2}.
Geometrical, physical and algorithmic model parameters at the macro and micro scales
Parameter | Value | Unit |
---|---|---|
Macro-scale | ||
Applied stress isotropic compression \( \bar{\sigma }_0 \) | 0.15 | MN/m |
Vertical strain biaxial compression \( {\bar{\varepsilon }} \) | 0.1 | – |
Tolerance kinetic energy \(\mathrm {tol}_{\bar{E}}\) | \(10^{-3}\) | – |
Damping ratio \(\xi \) | 1.0 | – |
Safety factor \(\gamma \) | 0.5 | – |
Micro-scale | ||
Polydispersity \(R_{max}/R_{min} \) | 1.5 | |
Anisotropy \( {\mathcal {A}}^0 \) | 0.02 | |
Coordination number \( {\bar{n}}^0 \) | 3.42 | |
Elastic normal stiffness \( k_{n} \) | \(1 \cdot 10^4 \) | N/m |
Elastic tangential stiffness \( k_{s} \) | \(2 \cdot 10^3 \) | N/m |
Friction coefficient \( \mu \) | 0.4 | – |
Density \( \rho \) | \( 2 \cdot 10^3 \) | kg/m\(^2\) |
Translational damping \( \alpha \) | 0.7 | – |
Rotational damping \( \beta \) | 0.7 | – |
Time increment \( \Delta t \) | \(10^{-6} \) | s |
Tolerance force \( \epsilon _{a} \) | \( 10^{-4}\) | – |
Gain force \( g_a M/\Delta t^2 \) | \(1 \cdot 10^2 \) | – |
Gain moment \(g_m M R^2/\Delta t^2 \) | \(2 \cdot 10^2\) | – |
Tolerance dynamic relaxation \(\mathrm {tol}_{E}\) | \(10^{-3}\) | – |
4.2 Mesh convergence study
Figure 4 illustrates that the average stress response of the coupled FEM–DEM models for all meshes considered is very close to the predictions of the direct numerical simulations, in particular in the pre-peak regime, \( 0 \le {\bar{\varepsilon }} \le 0.05 \). After this point, a moderate softening behaviour is observed, whereby the responses for the different meshes start to deviate from one other. The mesh size dependency of the FEM response in the post-peak regime is a well-known effect; to circumvent this problem in a multi-scale setting, kinematically enriched multi-scale frameworks have been proposed in the literature, see [38, 39]. Since in the present work the focus is mainly on the pre-peak regime of the macroscopic response, the application of these frameworks for granular systems is considered as a topic for future research.
The influence of the choice of the macroscopic mesh size on the effective response is further investigated by considering the evolution of the coordination number \( {\bar{n}} \) and the induced fabric anisotropy \( {\mathcal {A}} \) (both averaged over all macroscopic integration points), see Fig. 5a, b. As a general trend, it can be observed that the results are only slightly sensitive to the mesh adopted. The coordination number somewhat decreases with increasing deformation, due to the horizontal expansion of the macroscopic domain. The anisotropy initially increases, since the packing deforms stronger in the vertical direction than in the horizontal direction, in correspondence with the macroscopic biaxial loading conditions applied. The induced anisotropy becomes maximal at about the same deformation stage as at which the peak strength is reached. In the softening regime, a small decrease in anisotropy is observed, which is caused by a dilating particle structure that develops under progressive shear failure.
4.3 Influence of micro-structural parameters on the macroscopic response
The macroscopic geometry in Fig. 3 is discretized with the selected mesh of \(2 \times 4\) finite elements, whereby the particle contact friction, the particle rotation, and the initial fabric anisotropy are varied. In the analysis of the results the attention will be focused on the pre-peak regime of the macroscopic response, during which the FEM results are independent of the mesh size, see Sect. 4.2
4.3.1 Particle contact friction
4.3.2 Particle rotation
4.3.3 Initial anisotropy
The influence of the initial anisotropy of the granular micro-structure on the macro-scale response is assessed by considering, together with the reference particle packing defined in Table 3, two additional particle packings characterized by higher anisotropy values. Accordingly, the set of initial anisotropy parameters is \( {\mathcal {A}}^0 = \left[ 0.02, 0.05, 0.08 \right] \), and the corresponding microstructures and rose diagrams are sketched in Fig. 12.
5 Conclusions
In the present contribution a multi-scale model for the analysis of granular systems has been proposed, which combines the principles of a coupled FEM–DEM approach with a novel servo-control methodology for the implementation of appropriate micro-scale boundary conditions. A mesh convergence study has been performed, whereby the results of a quasi-static biaxial compression test were compared to those obtained by direct numerical simulations. The comparison demonstrated the capability of the multi-scale method to realistically capture the macro-scale response, even for macroscopic domains characterized by a relatively coarse mesh; this makes it possible to accurately analyse large-scale granular systems in a computationally efficient manner. The multi-scale framework has been applied to study in a systematic manner the role of individual micro-structural characteristics on the effective macro-scale response. The effect of particle contact friction, particle rotation, and initial fabric anisotropy on the overall response has been considered, as measured in terms of the evolution of the effective stress, the volumetric deformation, the average coordination number and the induced anisotropy. The trends observed are in accordance with notions from physics, and observations from experiments and other DEM simulations presented in the literature. Accordingly, it is concluded that the present framework provides an adequate tool for exploring the effect of micro-structural characteristics on the macroscopic response of large-scale granular structures.
Since the proposed multi-scale framework is based on first-order homogenization principles, it can only be adequately applied for problems whereby microscopic length scale effects do not influence the macroscopic response. Examples whereby this separation of length scales holds are static (and dynamic) problems in which significant strain localization remains absent, and dynamic problems in which the time-dependent response is composed of non-dispersive, slowly varying low-frequency components. The extension of the proposed multi-scale FEM–DEM scheme for applications related to strain localization and high-frequency wave propagation is a topic for future research.
Footnotes
Notes
Acknowledgements
The authors thank Dr. Ning Guo from Carleton University, Ottawa, Canada, and Dr. Lutz Gross from the University of Queensland, Brisbane, Australia, for helpful discussions and support on the use of Escript. Feedback provided by Joran Jessurun from the Eindhoven University of Technology, Netherlands, on numerical implementation issues within the ESyS-Particle code, is appreciated. The Netherlands Organization for Scientific Research (NWO) is acknowledged for providing access to the supercomputer facilities; the simulations were performed under NWO-Project 15421 Multi-scale modelling of granular materials, within the funding scheme “Computing Time National Computing Facilities Assessment Pilot Applications 2016”. The financial support of J.L. by the China Scholarship Council is gratefully acknowledged.
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