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Modelling of fracture process in brittle materials using a lattice model
Published in Günther Meschke, René de Borst, Herbert Mang, Nenad Bićanić, Computational Modelling of Concrete Structures, 2020
The lattice model is a simple approach to the fracture behaviour in quasi-brittle materials but very useful in studying and understanding the phenomenon of the crack formation. Owing to it, novel (stronger and better) engineering materials can be developed. By using an elastic-purely brittle local fracture law at the particle level of the material, global softening behaviour is obtained. The lattice simulations yield a significant size effect in nominal strength, i.e. the strength increases with decreasing specimen size and increasing size of micro-structure (expressed by the beam length). The heterogeneous lattice model used in the paper requires for the brittle material composed of one component only 3 material parameters (p, E, εmin) and 4 geometric parameters related to the distribution, quantity and length of beams (g, s, α and rmax). The obtained results of crack patterns are qualitatively in agreement with experimental ones for concrete. However, the lattice stress-strain outcomes are too brittle. To decrease the material brittleness, a non-local approach can be used.
Simulation of Crystalline Nanoporous Materials and the Computation of Adsorption/Diffusion Properties
Published in T. Grant Glover, Bin Mu, Gas Adsorption in Metal-Organic Frameworks, 2018
Lattice models are extremely useful because of their computational efficiency, while still maintaining a high degree of physical realism. For example, from LGAs, it is possible to derive the macroscopic Navier–Stokes equations. The atomistic nature of the model makes it easy to include some interesting phenomena like diffusion and reaction. However, lattice gases are more difficult to extend to three dimensions. Also, LGA have high inherent numerical noise. The lattice Boltzmann methodology [105] solves much of these limitations by using ensemble averages and an approximation to the collision operator using a 3D cubic lattice, while still maintaining the particle nature of the methodology. Note that, by assuming a statistical model of pair collisions, the Liouville equation Equation 6.26 can be shown to reduce to the Boltzmann equation [106]. The lattice Boltzmann method models the streaming and collision of molecules only statistically, that is, it does not model the microscopic structure of the fluid and, therefore, is easily applicable to model macroscopic fluid flows. But because the method still retains a particle nature, incorporation of irregular boundary conditions, reactions, and other complexities which are hard in continuum approaches are relatively simple in the lattice Boltzmann formalism.
MuIti Degree-of-Freedom Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
Perfectly periodic discrete structures are sometimes called lattice models because, historically, such spring-mass systems looked like lattices to the physicists who used them to model the interactions of atoms in a solid. Here, a 10 degree-of-freedom structure undergoing longitudinal motion (along the axis of the structure) is formulated and some numerical results are presented and discussed.
Thermodynamics of quantum lattice system with local multi-well potentials: dipole ordering and strain effects in modified Blume–Emery–Griffiths model
Published in Phase Transitions, 2019
Lattice models play an important role as one of the extensions of statistical physics on new objects and phenomena in the field of condensed matter physics (especially, the physics of solid state). Relatively simple quantum lattice models often demonstrate a variety of phase states and sophisticated phase diagrams. As a widely used example, one can mention the description of thermodynamics and study of the order-disorder phase transitions in the crystals with locally anharmonic structure elements by the quantum lattice models. Local anharmonic potentials with multi-minima shapes commonly occur in this case. The various localizations of particles correspond to different quantum states (configurations of structure elements). For systems with two-well local potentials (e.g. crystals with hydrogen bonds), this approach leads to the transverse Ising model (known in the theory of ferroelectrics as the de Gennes model [1]). In the case of the three-well symmetrical potential and at the same conditions an appropriate lattice model corresponds to the Blume–Emery–Griffiths (BEG) model [2]. The model can be applied to description of crystals belonging to the SnPS family (with the possible partial substitutions SnPb and SSe, see [3]) which are an example of such objects.
Boundary equation from a lattice model and modification of the Peierls equation
Published in Philosophical Magazine, 2022
The discrete lattice model of solids is often appropriate to describe materials behaviour when length scale is about the angstrom range. However, in practice discrete modelling and continuum modelling may be seamlessly linked to solve problems in defects and deformation that would otherwise be difficult to solve [4]. An effort is presented here to obtain the basic features of discrete modelling and to explore how to extend the elastic continuum theory to make the link more seamlessly.
An overview on advances in computational fracture mechanics of rock
Published in Geosystem Engineering, 2021
Mojtaba Mohammadnejad, Hongyuan Liu, Andrew Chan, Sevda Dehkhoda, Daisuke Fukuda
Lattice models, which are also known as dynamic lattice network models (DLNM), are relatively simpler, modern techniques among other discontinuum methods. The basic concept is similar to BPM, where material can be represented as a collection of interacting discrete masses. As illustrated in Figure 11, the medium compromises of a set of either regular or irregular distributed point masses, which interact through simple zero-size spring/beam with ability to transfer forces. Although the technique is not new, its application in dynamic fracture modelling is a recent development. This method enjoys two main advantages of continuum and discontinuum methods in terms of being flexible and computationally efficient (Cundall, 2011). Different types of cells can be developed into the lattice model, allowing for model heterogeneity. In this technique, fracturing is simulated based on a linear elastic analysis with spring deletion when the force exceeds a threshold. A comprehensive explanation of this technique and its application in fracture mechanics can be found through the studies conducted by Schlangen (1995), Schlangen and Garboczi (1997), Bolander and Sukumar (2005), Slepyan (2005), Grassl, Bazant, and Cusatis (2006) and Quintana-Alonso and Fleck (2010). Different formulations of this technique were employed by researchers to investigate rock fracturing process. Song and Kim (1994) developed a DLNM and simulated fracturing process due to blasting. In the proposed model, the rock heterogeneity was assigned as a random stiffness of springs and the system was considered to follow linear elastic model. Zhao (2010) developed a distinct lattice spring model in which material was modelled through an un-uniform distribution of masses interacting via distributed bonds. A new algorithm based on the lattice model was also introduced into PFC by Cundall (2011) to improve the flexibility and efficiency of the method by removing the contact detection process. In the proposed method, the material is modelled by a series of springs which link masses. This method was successfully employed to simulate rock failure and rock fracturing from blasting (Cundall, 2011; Onederra, Chitombo, Cundall, & Furtney, 2009; Poulsen, Adhikary, Elmouttie, & Wilkins, 2015). Despite all merits of the lattice models, they suffer from difficulties in model calibration and practical-scale modelling.