R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 80-89; DOI: 10.3221/IGF-ESIS.34.08 80 Focussed on Crack Paths A unified approach for static and dynamic fracture failure in solids and granular materials by a particle method Roberto Brighenti, Andrea Carpinteri, Nicholas Corbari University of Parma, Italy brigh@unipr.it , andrea.carpinteri@unipr.it , nicholas.corbari@gmail.com A BSTRACT . The material structure at the microscale reveals the particulate nature of solids. By exploiting the discrete aspect of materials, the so-called particle methods developed and applied to the simulation of solids and liquids have attracted the attention of several researchers in the field of computational mechanics. In the present paper, a particle method based on a suitable force potential is proposed to describe the nature and intensity of the forces existing between particles of either the same solid or different colliding solids. The formulation applies to problems involving both granular materials and solids interacting with granular materials. The above approach is applied to simulate different problems dealing with the 3D dynamic fracture and failure of solids. K EYWORDS . Particle method; Discrete finite element; Fracture, Contact , Dynamic failure. I NTRODUCTION he particle methods have attracted the attention of several researchers in the field of computational mechanics [1, 2]. It is straightforward when the material has to be described at the nanoscale where the molecular dynamic method can easily be applied [3]; however, this discrete description can successfully be applied even at the macroscale [4, 5]. Accordingly, a discrete approach can replace the traditional continuous modelling of a solid, and allows us to examine complex phenomena such as failure, fracture, and interaction with other bodies [4]. The formulation applies to problems involving both granular materials [6-8] and solids interacting with granular materials. The need to solve mechanical problems has solicited the development of several numerical techniques aimed at assessing the response of materials both during service and at their ultimate conditions. To this purpose, various computational tools have been proposed: the finite element method (FEM), the finite difference method, the finite volume method, the more recent boundary element method, meshless method [9] and natural element method [10], just to cite the main tools. Problems involving mechanical systems characterized by large displacements, high velocity impact, fracture, failure, fragmentation, clustering, fluid-solid interaction, and so on, cannot be solved through classical numerical techniques in the context of the so-called Lagrangian approach, because the severe distortion of the discretized domain leads to a lack of consistency between the numerical modelling and the physics problem. This issue has partially been overcome by using special remeshing techniques and the FE method, but that presents some drawbacks. The discretization through only nodal points – i.e. without mesh connectivity – represents a possible approach to solve this problem. In this context, the so-called Smoothed Particle Hydrodynamics (SPH) has been one of the first particle meshless methods in computational mechanics [11, 12]. The original idea of the developers of the above method lies in T

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