Circular Particle Models (PM) are a class of discrete elements which has been increasingly used for detailed analysis in rock and concrete structures. There have been few applications to masonry, but the potential of these techniques appears significant, due to their proven ability to simulate fracture processes through random particle assemblies representing quasi-brittle materials at the grain scale. The present chapter presents the fundamentals of this approach and reviews some previous applications of PM models to masonry. The model capabilities are first exemplified by simple models involving a few irregular blocks formed by particles. Irregular stone masonry wall specimens under compression and under in-plane shear loading are then presented. In these models both the units and the mortar are represented by circular particles, and failure processes through the joints or through joints and stones are analyzed. The main issues regarding the use of these models are finally discussed.
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Two fundamental numerical approaches towards masonry analysis are possible: macro-modelling as an equivalent homogeneous medium and micro-modelling of the individual components, joints and the units. Both approaches have specific strengths and engineering roles. While continuum models have been developed which simulate complex geometric patterns and material behaviour (e.g. Stefanou et al. 2015), the alternative discontinuum or micro-models allow a detailed representation of the internal structure, and have become a powerful tool for research on the fundamental behaviour of geo-materials.
The discrete element method (DEM) provides a numerical implementation framework for discontinuum modelling.The simplest element geometry that can be assumed is that of circular particles in 2D or spheres in 3D. Cundall (1971), the original paper that proposed DEM, already considered circular particle models in addition to rigid polygonal blocks. Cundall & Strack (1979) extended the circular particle approach and proposed it for the analysis of granular media, namely soils. In other fields of engineering and physics, the same concept was pursued by several researchers under different designations, namely Molecular Dynamics (e.g., Pöschel & Schwager, 2005).
Replacing the frictional contacts between particles by bonds with cohesive and tensile strength, circular particle assemblies were employed to simulate materials such as rock, concrete or asphalt pavement, in the early 1990s, Meguro and Hakuno (1989), Potyondy and Cundall (1996), Schlangen and Garboczi (1997), Chang and Meegoda (1997).
Studies of fracture propagation in these geo-materials have shown the ability of these bonded-particle models to reproduce the types of phenomena observed in laboratory experiments, such as uniaxial tension, uniaxial compression and triaxial tests. The random nature of particle assemblies allows the propagation of cracks to develop in a natural manner, replicating the irregularity of the structure of geo-materials. Circular or spherical particles may also be associated to build macro-particles with an arbitrary form. Rigid or breakable bonds can be assumed to link the component particles, thus allowing different levels of elaboration of the analysis. The interaction forces, however, are still obtained by the elementary contacts between the pairs of adjacent circular particles.
More recently, 3D rigid spherical particle models have been proposed for rock, Matsuda and Iwase (2002), Potyondy and Cundall (2004), Monteiro Azevedo and Lemos (2013), and for concrete, Lilliu and Van Mier (2003), Hentz et al. (2004). Models based on the rigid spring block method adopting 3D Voronoi shape polyhedra have also been developed for concrete, Nagai et al. (2005) and Berton and Bolander (2006). Numerical models that follow complex polyhedral based discrete element contact interaction and include the particle deformation by considering the inner block finite element mesh have been recently proposed for rock fracture, Gao and Stead (2014), Hamdi et al. (2014). A review of discrete modelling techniques for fracturing processes in rock can be found in Lisjak and Grasseli (2014).
Particle models, and more generally DEM models, embody an approach to describe the fundamental behaviour of materials by means of simple constitutive assumptions, realized by elementary contact models between the individual particles. Complexity of macroscopic behaviour arises from the combination of these mechanisms over a large random assembly. Procedures have been developed to calibrate the micro-properties to obtain the desired response, typically by comparison with the results of laboratory tests on samples. For masonry structures, particle models have the ability to represent the natural irregularity of their geometry and material structure, and therefore to simulate the complex patterns of cracking and failure development, through the units or the joints, which characterize their response.