Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 80-89; DOI: 10.3221/IGF-ESIS.34.08 81 the fact that the collective motion of particles in particle-wise systems (such as the movement of liquid, gas flow, particulate assemblies, etc.) may physically be modeled by the governing equations of classical Newtonian hydrodynamics. Such mesh-free Lagrangian methods are characterized by variable nodal connectivity; the material is discretized with interacting particles and the method transforms a discrete field into a continuum one by using a localized kernel function that operates as a smoothing interpolation. The method can be applied to discrete systems but also to continuum ones, where the nodal points are not directly associated with the single particle composing the matter [13-15]. The noticeable advantages of such a method became clear after its first introduction, suddenly leading to its extension to the solution of a wide range of physics and engineering problems, well beyond the initial classical hydrodynamics field [16, 16]. The discrete nature of solid matters is evident at the microscale or at the molecular level [17]. On the other hand, the matter is typically modeled as continuous medium at the macroscale, suggesting the use of continuous computational techniques such as the FEM. However, discrete models can conveniently be applied even in macroscopically continuum systems where each representative particle can be associated to a cluster of continuum elements. Discrete Element Methods (DEMs) sometimes also called Particle Methods (PMs), are numerical procedures to solve different engineering problems: the main assumption for this class of methods consists in the description of the material through an assemblage of separate discrete elements, such as atoms, molecules, grains, particle solid elements, etc. according to the scale of observation adopted [1, 17]. In the discrete approach, non-linear interactions between bodies and within bodies are numerically simulated in order to get their non-linear differential equations of motion. It is worth mentioning that the discrete nature of materials is well evident for some classes of them, such as the granular ones: geomechanical and powders materials can be studied by exploiting their noticeable granular nature [18, 19]. Discrete methods have usefully been applied to solve problems at the macroscale such as mineral processing, rock and concrete blasting and crushing, sand mechanics, failure and fracture of compact or granular bodies [20, 21], problems involving fluid materials like liquids and gases [16, 22]. A generic solid can also be assumed to have a particle structure with a proper nature of particles’ interaction forces. From this remark, the mechanical behaviour of a material can range from very incoherent behaviour (characteristic of fluids, granular, powders, etc) up to compact behaviour (typical of polycrystalline materials), respectively. Therefore, a discrete model can be used to simulate different classes of solids, by properly choosing the nature of the interaction forces between their discrete constituent elements. Such an approach permits to get the overall response of the material at the macroscale, which is the main interest of materials science and mechanics of materials. Moreover, the particle discretisation for a solid allows us to tackle the problem from a dynamic point of view, enabling to solve high strain rate, impact, large displacements and large strains problems involving history-dependent behaviour, plasticity, etc., that are more naturally expressed in a Lagrangian computational framework. It should also be considered that discrete methods allow us to deal with short range forces (such as contact forces) and long range forces (such as the electrostatic ones). That enables to represent the physical nature of materials at the small or nano scales, where interatomic forces obeying such a non-local interaction relationship exist. In the present paper, a discrete element approach for the mechanical simulation of either continuum or granular-like materials (formulated on the basis of the force potential interaction concept) is proposed and applied to analyse dynamic fracture and failure of both granular materials and compact solids. A brief introduction related to the forces existing in continuum solids and discrete incoherent aggregates – modeled through discrete particles – is presented by taking into account the dynamic nature and large strain characteristics of the problem. Finally, some examples demonstrating the versatility and the capability of the proposed approach are discussed. P OTENTIAL - FORCE FORMULATION FOR PARTICLE - PARTICLE INTERACTION s is recalled above, the particle approach is straightforward when the material is really composed by several discrete elements as at the nano/microscale (for both solids and granular-like matters), where the particles can immediately be identified as the atoms/molecules or grains in the matter. However, a continuous solid can also be idealized as a cluster of elements reciprocally joined by forces depending on the particles relative positions: if the particles tend to approach to each other, a repulsive force arises (typical of compact solids and granular materials), while an attractive force appears when they tend to move far away (typical of compact solids and cohesive granular materials). The above interacting forces can be thought to have a local nature when they are associated to the direct contact between elements (for instance when granular materials are examined), while they assume a longer range characteristic in the case A