Formation of shear bands in crushable and irregularly shaped granular materials and the associated microstructural evolution
Graphical abstract
Introduction
Granular materials have a wide spectrum of characteristics and phenomena that distinguish them from liquids and solids. One ubiquitous feature of granular materials is the formation of shear bands. Many geotechnical failures are characterized by bifurcation and spontaneous localization of deformation into rupture zones.
Experimental evidence has shown that the failure of plane strain (PS) specimens always occur along a well-defined shear plane [1], [2], [3]. When a specimen is subjected to conventional triaxial compression (CTC), it fails with either a localized shear plane or a bulging shape with no clearly defined shear bands [4]. Shear band formation and evolution also occur in true triaxial tests [5], [6], ring shear tests [7], plane strain extension [8], and static and cyclic torsional shear tests [9], [10]. These experimental studies have revealed that the shear band formation is influenced by several factors, including the porosity, the inherent and stress-induced anisotropy, the particle size and shape of the material, and the level of confining stress.
Despite extensive experimental studies, the fundamental mechanism of shear band formation and how the influencing factors affect the characteristics of shear bands is still a valuable research topic. Granular material has been recognized as a discrete system, and the overall material response is the result of various micromechanical processes, including particle sliding, particle rolling, and particle breakage. Therefore, the discrete element method (DEM) has been employed by numerous researchers to investigate shear bands or strain localization in granular materials. The DEM provides much richer information and deeper insight into the microscopic behavior of granular materials that are not otherwise obtained by experiments [11], [12], [13], [14], [15], [16], [17]. Special attention has been given to the influences of rolling resistance [18], [19], [20], particle shape [21], [22], inherent anisotropy [23], [24], and packing density [25] on shear banding. Previously published DEM studies on shear bands share several common features. The most prominent feature is that particle rotation is restricted by either a rolling resistance model or a particle clump or cluster. Incorporating irregular particle shapes into discrete modeling will give more realistic particle rotation behavior than using a rolling resistance model [22]. The second common feature is that particle breakage is usually not considered. Stress fluctuations in granular materials result from the combined effects of frictional instabilities and particle breakage, especially in granular materials composed of highly crushable particles or under relatively high stress levels. Thus, realistic modeling of shear banding must include grain fragmentation.
Despite its underlying simplified assumptions, the DEM has been instrumental in identifying key mechanisms in shear banding failure mode. However, several challenges affect the physical behavior that the DEM attempts to model, such as deformability, crushability, and shape representation. The combination of the finite element method (FEM) and the DEM has emerged in response to the need for a more realistic and robust numerical simulation technique; this combination is referred to as the combined FDEM [26]. The combined FDEM has been successfully applied to a wide range of engineering and scientific fields. Recent developments in particle shape representation [27], [28], [29] and cohesive crack model [30], [31], [32] have made combined FDEM an ideal tool for modeling irregularly shaped and crushable granular materials.
To the best of our knowledge, few studies have been focused on the shear banding in assemblies of irregularly shaped and crushable particles [33]. The aim of this study is to investigate shear band formation and the associated microstructural evolution, which could supplement previous shear banding research. In this study, a particle-scale representation is developed in which granular materials are viewed as an assembly of densely packed polygonal particles. Cohesive interface elements (CIEs) with zero-thickness, which behave like cohesive bonds in the DEM, are inserted into the finite element discretization associated with each particle to consider particle breakage. Although the boundary effect has been investigated by many researchers, we include it as a verification of the newly developed modeling technique for irregularly shaped and crushable granular materials. This paper takes a further step to explore the role of particle crushability in the shear banding of granular materials. In depth micromechanical analyses of the particle-scale information inside and outside the shear bands are presented, including the accumulated particle rotation, void ratio distribution, particle breakage behavior, granular temperature, and friction mobilization. Attempts are made to link shear banding to the microstructural evolution.
Section snippets
Combined finite discrete element method
A typical combined FDEM simulation comprises a large number of interacting bodies, each of which is discretized into a finite element mesh. An explicit central difference integration scheme is employed to solve the equations of motion for the discretized system and update the nodal coordinates at each time step. As the simulation progresses, these bodies can deform, translate, rotate, interact, and fracture or fragment when they satisfy certain failure criteria and thus produce new discrete
Numerical specimen preparation
Simulated biaxial tests are performed on a polydisperse assembly of polygonal particles. The equivalent particle diameter, d⁎, ranges from 1 mm to 4 mm, and the mean grain diameter d50⁎ is 2.5 mm, where d⁎ is defined as the diameter of a circle with an equivalent area to the irregularly shaped particle. Fig. 3a shows the particle size distribution. The particle assembly is prepared by initially generating a cloud of non-contacting polygonal particles using random deposition. The assembly consists
Differences in the macroscopic response
A review of experiments on strain localization demonstrates that several patterns of localization have been observed depending on the boundary conditions, including parallel and crossing shear bands as well as temporary or “non-persistent” modes of localization, which are localized regions that form during the test and eventually “disappear” [38]. Rigid and flexible boundary conditions are compared using the same particle assembly, input parameters, and loading procedures. The simulation
Difference in the macroscopic response
Five levels of particle crushability are considered using CIEs with UCS values of 40, 60, 80, 100, and 120 MPa, which correspond to assemblies with a range of crushability from high to low. Fig. 7 shows the macroscopic responses of the numerical biaxial tests in terms of the deviator stress and the volumetric strain versus axial strain. The figure qualitatively shows that the simulated stress-strain-dilation responses are typical of those observed in laboratory tests. The macroscopic responses
Conclusions
The Voronoi-based particle generation of realistic particle assemblies for discrete element modeling is a significant advance in particle shape representation. Subsequently, the cohesive crack model is introduced into the combined FDEM to simulate particle breakage based on damage mechanics. These two developments make the combined FDEM an ideal tool for modeling irregularly shaped and crushable granular materials. Although it is more computationally expensive than DEM, the development of
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (grant no. 51379161 and no. 51509190) and China Postdoctoral Science Foundation (2015M572195), and the Fundamental Research Funds for the Central Universities.
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