Discrete Lattice Model Research Articles

Size effect on compressive strength of foamed concrete: Experimental and numerical studies

This study investigates the size effect on the compressive strength of foamed concrete at the mesoscale level combining X-ray computed tomography (X-CT) and a discrete lattice model. Image segmentation techniques and X-CT were employed to obtain virtual specimens comprising hydrated cement paste and air voids. The lineal-path function and pore size distribution was used to characterise the air void structure. A two-dimensional lattice fracture model of foamed concrete considering different wet densities was established. The model was verified experimentally at a wet density of 700 kg/m3 and then used to predict the strengths of specimens with wet densities of 600 and 800 kg/m3. Square and rectangular specimens (slenderness ratio = 2) with widths of 10, 20, 40, 70.7, and 100 mm were investigated. Results show that the air void structure significantly influences the observed size effect on the compressive strength in the investigated size range. A random forest regressor was used to predict the compressive strength of the foamed concrete; the regressor yielded satisfactory results. Finally, existing analytical size effect models were used to fit the simulated strength. Although good fitting was achieved, special attention should be given to the applicable range and physical meaning of fitted empirical parameters.

Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response

This research presents a novel approach to modeling fracture propagation using a discrete lattice element model with embedded strong discontinuities. The focus is on enhancing the linear elastic response within the model followed by propagation of fractures until total failure. To achieve this, a generalized beam lattice element with an embedded strong discontinuity based on the kinematics of a rigid-body spring model is formulated. The linear elastic regime is refined by correcting the stress tensor at nodes within the domain based on the internal forces present in lattice elements, which is achieved by introducing fictitious forces into the standard internal force vectors to predict the right elastic response of the model related to Poisson’s effect. Upon initiation of the first fractures, the procedure for the computation of the fictitious stress tensor is terminated, and the embedded strong discontinuities are activated in the lattice elements for obtaining an objective fracture and failure response. This transition ensures a shift from the elastic phase to the fracture propagation phase, enhancing the predictive capabilities in capturing the full fracture processes.

A discrete lattice model with axial and angular springs for modeling fracture in fiber-reinforced composite laminates

In this research, we developed a new 2D triangular lattice model with axial and angular springs to simulate fracture problems in composites. The inclusion of angular springs in the model formulation resolves the constraint related to the choice of the shear modulus, which is present in other lattice models that only consider axial springs. Moreover, it accounts for progressive damage in composites by effectively integrating a linear softening constitutive law. This model is based on equating the strain energy stored in a representative hexagonal cell to its corresponding continuum solid. Four numerical examples demonstrate the ability of this method to predict elastic constants and dynamic and quasi-static fracture problems in unidirectional composites. The results will be compared with the numerical and empirical results documented by the other researchers.

Determination of flexoelectric coefficients of higher-order continuum theories from CsCl lattice model

Flexoelectricity is an electromechanical coupling phenomenon between inhomogeneous deformation and electric polarization. The classical theory, which relies on the strain, is insufficient for representing flexoelectricity. Instead, a higher-order continuum theory that employs a measure capable of expressing local inhomogeneous deformation is necessary. In the present work, we concatenate higher-order continuum theories with a CsCl lattice structure consisting of charged particles and determine the resulting constitutive coefficients for higher-order continuum theories. First, we derived continuum internal energy density functions for dielectrics from the conservation laws. We consider two different higher-order continuum theories: coupled stress and strain gradient theories. Second, we present a procedure for determining the coefficients of higher-order continuum theories by using a discrete lattice model under the assumption that the internal energies of the lattice and continuum models are the same for identical deformations. To validate the coefficients, three sample problems are examined by presenting their analytical solutions. The results of sample problems demonstrate that the two higher-order theories can provide quantitative information about the size and flexoelectric effects. In particular, the strain gradient theory predicts the deformation of the lattice structure more accurately.

An enhanced lattice beam element model for the numerical simulation of rate-dependent self-healing in cementitious materials

This paper describes the development of a discrete lattice model for simulating structures formed from self-healing cementitious materials. In particular, a new approach is presented for simulating time dependent mechanical healing in lattice elements. The proposed formulation is designed to simulate the transient damage and healing behaviour of structures under a range of loading conditions. In addition, multiple and overlapping damage and healing events are considered. An illustrative example demonstrates the effects of varying the healing agent curing parameters on the computed mechanical response. The model is successfully validated using published experimental data from two series of tests on structural elements with an embedded autonomic self-healing system. The meso-scale model gives detailed information on the size and disposition of cracking and healing zones throughout an analysis time history. The model also provides an accurate means of determining the volume of healing agent required to achieve healing for all locations within a structural element. The importance of the information provided by the model for the design of self-healing cementitious material elements is highlighted.

Bayesian inference of mesoscale mechanical properties of mortar using experimental data from a double shear test

In this work, we propose Bayesian parameter estimation of a nonlinear mechanics based model describing the behaviour of mortar subjected to double shear test with externally bonded carbon fibre reinforced polymer (CFRP) plates. With the Bayesian approach, it is possible to identify mechanical material parameters of different phases of the mortar mesostructure, i.e. hardened cement paste, aggregates and interface transition zone (ITZ). Due to nonlinearity of the concerned problem, we use a novel sequential approach for the parameter inference, which does not require coupling between the finite element solver and software for the stochastic analysis. The model geometry and material mesostructure are learned based on micro-computed tomography (μCT) scans of the real specimen, whereas the unknown boundary conditions are assumed to be uncertain and are also identified from experimental data. Mortar is modelled through a discrete lattice model consisting of spatial Timoshenko beams with embedded discontinuities. The latter allows the description of the distinct stages in material degradation, from the appearance of microscopic material damage to its evolution into macroscopic cracks leading to localised failure.

Effect of microstructure heterogeneity shapes on constitutive behaviour of encapsulated self-healing cementitious materials

Self-healing cementitious materials with microcapsules are complex multiscale and multiphase materials. The random microstructure of these materials governs their mechanical and transport behaviour. The actual microstructure can be represented accurately with a discrete lattice model, but computational restrictions mean that the size of domain that can be considered with this approach is limited. By contrast, a smeared approach, based on a micromechanical formulation, provides an approximate representation of the material microstructure with low computational costs. The aim of this paper is to compare simulations of a microcapsule-based self-healing cementitious system with discrete-lattice and smeared-micromechanical models, and to assess the relative strengths and weaknesses of these models for simulating distributed fracture and healing in this type of self-healing material. A novel random field generation technique is used to represent the microstructure of a cementitious mortar specimen. The meshes and elements are created by the triangulation method and used to determine the input required for the lattice model. The paper also describes the enhancement of the TUDelft lattice model to include self-healing behaviour. The extended micromechanical model considers both microcracking and healing. The findings from the study provide insight into the relative merits of these two modelling approaches.

Experimentally validated meso-scale fracture modelling of foamed concrete

This work presents a study of mechanical properties of foamed concrete at the meso-scale based on a combination of X-ray computed tomography (XCT) technique and a discrete lattice type fracture model. The microstructure of the foamed concrete with different densities was obtained by XCT technique and binarized as two-phase (pore/solid) materials. The parameters (e.g., porosity, pore diameter and spacing distribution) of foamed concrete air‐void structure were characterized. The virtual specimens were subjected to computational uniaxial compression, Brazilian splitting and three-point bending test to calculate strengths and elastic modulus. The mechanical properties of solid phase were derived from the recent outcome of micromechanical models. Two types of element input parameters were used to investigate the influence of the input parameters on the simulated results. The modelling results (strength value and fracture pattern) were compared with the experiments. It shows that, without further calibration, the lattice model can predict the mechanical strength and crack pattern with good accuracy. The fracture toughness KIC was derived using three-point bending strength and the average pore diameter. The results indicate that the presence of air-void structure increases the brittleness and reduces the fracture toughness of the foamed concrete.

Locally resonant metamaterials utilizing dynamic directional amplification: An application for seismic mitigation

Locally resonant metamaterials (LRMs), with unit-cells exhibiting local resonance, present extraordinary wave propagation properties as a result of their spatial periodicity. However, these structures present certain limitations when wide bandgaps are sought, such as the need of heavy resonating masses. In this paper, a simple dynamic directional amplification (DDA) mechanism is proposed as a means to artificially increase inertia of the resonating mass. The DDA is realized by imposing kinematic constraints to the model's degrees of freedom (DoFs), hence inertia is increased by coupling the horizontal and vertical motion of a reference two-dimensional (2D) mass-in-mass lattice. A discrete element lattice model based on mass, stiffness, and damping elements, is used to establish dispersion behavior and frequency response, while results indicate broader bandgaps and significant improvements compared to the reference LRM. At last, a preliminary design of a DDA-metabarrier is presented as a proof of concept, showcasing the potential application of the mechanism as a seismic wave mitigation measure.

Lieb-Robinson bounds imply locality of interactions

Discrete lattice models are a cornerstone of quantum many-body physics. They arise as effective descriptions of condensed matter systems and lattice-regularized quantum field theories. Lieb-Robinson bounds imply that if the degrees of freedom at each lattice site only interact locally with each other, correlations can only propagate with a finite group velocity through the lattice, similarly to a light cone in relativistic systems. Here we show that Lieb-Robinson bounds are equivalent to the locality of the interactions: a system with k-body interactions fulfills Lieb-Robinson bounds in exponential form if and only if the underlying interactions decay exponentially in space. In particular, our result already follows from the behavior of two-point correlation functions for single-site observables and generalizes to different decay behaviours as well as fermionic lattice models. As a side-result, we thus find that Lieb-Robinson bounds for single-site observables imply Lieb-Robinson bounds for bounded observables with arbitrary support.

Multistability and transitions between spatiotemporal patterns through versatile Notch-Hes signaling

The Delta-Notch-Hes signaling pathway is involved in various developmental processes ranging from the formation of somites to the dynamic fine-grained patterns of cell types in developing or regenerating tissues. Such broad patterning capabilities rely in part on the versatile and tunable dynamics of the Notch-Hes feedback circuit eliciting both pulsatile and switching behaviors. This raises the theoretical issue of which specific spatiotemporal features emerge from lateral inhibition between cells that can display and transit between monostable, oscillatory and bistable regimes. To address this issue, I consider a discrete cell lattice model where intracellular dynamics is described by a phase-like variable and displays a typical cross-shaped phase diagram. Model analysis determines how the existence and stability of many spatially inhomogeneous and temporally synchronized states depends on key intracellular and intercellular parameters. It reveals a parameter-dependent multistability between those diverse spatiotemporal patterns, giving rise to tunable and robust developmental transition scenarios ensuring defect-free spatial patterns. Such broad repertoire and multistability of spatiotemporal patterns is corroborated with regulatory network modeling of the Delta-Notch-Hes pathway.

Random walk discrete element lattice boltzmann model for scalar transport in fluid and particle flows with strict scalar mass conservation

Scalar transport in fluid-particle systems can be found in many natural processes and engineering applications. Modeling these systems can be challenging due to complex moving boundaries involved, such as fluid-particle, solute-particle interfaces. One often overlooked issue is: the scalar mass within the fluid domain is not strictly conserved even the no-flux boundary condition is applied in many grid-based methods. Here we introduce a hybrid model for simulating scalar transport in particulate flows with strict scalar mass conservation. It is achieved by adapting the Random Walk Method (RWM) to simulate the scalar transport and coupled with the Discrete Element Lattice Boltzmann Method (DE-LBM) for fluid-particle interactions. A moving boundary condition is proposed to prevent the scalar from penetrating solid boundaries and special treatments are presented in detail. The proposed model is validated by a series of benchmarks and experiments, then the mass conservation for both RWM and grid-based methods are checked. Finally, the model is applied to simulate scalar (solute) dispersion with particles in an oscillator which demonstrate its capability on simulating fluid-particle-solute systems. Furthermore, it is straightforward to extend the proposed model for non-spherical particles and reactive boundary conditions.

Lifetime of skyrmions in discrete systems with infinitesimal lattice constant

Topological protection of chiral magnetic structures is investigated by taking a two-dimensional magnetic skyrmion as an example. The skyrmion lifetime is calculated based on harmonic transition state theory for a discrete lattice model using various values of the ratio of the lattice constant and the skyrmion size. Parameters of the system corresponding to exchange, anisotropy and Dzyaloshinskii–Moriya interaction are chosen in such a way as to keep the energy and size of the skyrmion unchanged for small values of the lattice constant, using scaling relations derived from continuous micromagnetic description. The number of magnetic moments included in the calculations reaches more than a million. The results indicate that in the limit of infinitesimal lattice constant, the energy barrier for skyrmion collapse approaches the Belavin–Polyakov lower bound of the energy of a topological soliton in the σ-model, the entropy contribution to the pre-exponential factor in the Arrhenius rate expression for collapse approaches a constant and the skyrmion lifetime can, for large enough number of spins, correspond to thermally stable skyrmion at room temperature even without magnetic dipole–dipole interaction.

Research on the process of crack growth induced by explosion stress waves in rock masses with open joints

A joint in the medium has a significant impact on the propagation of explosion stress wave and the growth of explosion-induced crack. In this work, the crack growth experiment in the PMMA(polymethyl methacrylate) medium with vertical open joints under explosion loading was first carried out on the digital laser dynamic caustics experimental system (DLDC), in order to analyze the crack initiation, growth and arrest of wing cracks, and examine how the DSIF(dynamic stress intensity factor) at crack tip and the velocity of crack growth changed. The process of stress wave propagation and crack growth in this experiment was effectively inverted using the discrete lattice spring model (DLSM), the stress changes at the midpoint and endpoint of the joint were compared, and the acting characteristics of stress wave were analyzed here. Finally, the differences in velocity and acceleration of crack growth of wing cracks were compared and examined under different dip angles of joints.

A discrete lattice model for assessment of buildability performance of 3D‐printed concrete

AbstractIn this work, the lattice model is applied to study the printing process and quantify the buildability (i.e., the maximum height that can be printed) for 3D concrete printing (3DCP). The model simulates structural failure by incorporating an element birth technique, time‐dependent stiffness and strength, printing velocity, non‐uniform gravitational load, localized damage, and spatial variation of the printed object. The model can reproduce the plastic collapse failure modes reported in the literature. In this research, three main contributions for 3DCP modeling work can be found. A new failure criterion is proposed and adopted to improve the estimation of critical printing height; the element birth technique is utilized to mimic the continuous printing process and study the impact of non‐uniform gravitational load; variability of a printed structure is modeled through the inclusion of disorder during mesh generation and Gaussian distributions of material properties. Using this model, parametric analyses on non‐uniform gravitational load and material variation are conducted to assess their impact on the failure–deformation response and the critical printing height. Finally, the model is validated by comparison with two 3D printing experiments from the literature. The proposed lattice model can reproduce the correct failure‐deformation modes of two types of structures commonly used for buildability quantification: A 3D‐printed hollow cylinder and a square wall layout. Lattice modeling of the square structure yields a relative difference of around 10% with the experimental printing height. For the cylinder structure, the predicted radial deformation and corresponding height show good agreement with the experimental data; the model yields a 41.38% overprediction of the total number of printing layers, compared with the experimental data. Possible reasons for the quantitative discrepancy are discussed.

Vibrational motion of a vortex lattice induced near the surface of a type-II superconductor by an external low-frequency ac magnetic field

The theoretical description and experimental study of a reversible vibrational vortex motion induced in a type-II superconductor by a weak low-frequency ac magnetic field are discussed. The case where the external magnetic field and vortex lines are parallel to the surface of the superconductor is considered. The equilibrium positions of vortices near a surface, the amplitude of their vibrations, and the contribution of this process to the dynamic permeability ${\ensuremath{\mu}}_{v}$ are calculated. In these calculations, the interactions of vortices with each other, with the surface, and with pinning centers are considered within the discrete vortex lattice model. The calculations in the London approximation are compared with the results obtained within a more accurate model, including the spatial variation of the order parameter in the vortex core by means of a variational function. It is shown that the structural parameters of the vortex lattice near the surface depend on the accuracy of the description of the intervortex interactions, but the contribution ${\ensuremath{\mu}}_{v}$ appears to be model independent. It is important that this conclusion is valid in the case of the exponential approximation for the interaction between vortex rows. In this approximation, the problem under consideration is analytically examined, and formulas describing the dependences of the parameters of the vortex lattice and the contribution ${\ensuremath{\mu}}_{v}$ on the external dc magnetic field and the characteristics of a superconductor are derived. The theoretical results are compared with experimental data for a $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$ high-${T}_{c}$ superconducting single-crystal plate. The magnetic field is parallel to the plate, i.e., to the crystallographic $ab$ plane. A strong anisotropy of the properties and shape of the crystal is considered when analyzing the experimental data. A procedure for the determination of the experimental dependence ${\ensuremath{\mu}}_{v}(H)$ is described. It is shown that the reversible vibrations of vortices occur near the planes of the single-crystal plate in the direction of the crystallographic $c$ axis, whereas vortices enter its lateral ends and leave them (along the structural layers) even at a small magnetic field amplitude of about 1 Oe. It is found that the developed theoretical description reproduces the experimental data ${\ensuremath{\mu}}_{v}(H)$ in the entire studied range of temperatures and magnetic fields under the assumptions that pinning is isotropic and the length of interaction of vortices with pinning centers is close to the coherence length.

A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

Discrete lattice fracture modelling of hydrated cement paste under uniaxial compression at micro-scale

A combination of laboratory experiments and numerical simulations at multiple length scales can provide in-depth understanding of fracture behaviour of hydrated cement paste (HCP). To that end, the current work presents a numerical study on compressive failure of hydrated cement paste (HCP) at the micro-scale. Virtual specimens consisting of various phases were obtained using a combination of X-ray computed tomography and image segmentation techniques. The discrete lattice fracture model was used for the deformation and fracture analysis of the specimens subjected to uniaxial compression. The input local mechanical properties of each individual phase were taken from the literature in which a micro-scale compression test was conducted for the calibration of the same type model. The influence of slenderness ratios (1 and 2), water-to-cement ratios (0.3, 0.4 and 0.5 respectively) and lateral confinement of the specimen ends (free and restricted) on the failure behaviour were investigated. It has been shown by the current study that the stress–strain response cannot be completely separated from the used boundary conditions. The proposed model provides an effective tool to understand the compressive fracture behaviour of cement paste at the micro-scale.

3D thermo-hydro-mechanical coupled discrete beam lattice model of saturated poro-plastic medium

In this paper, we present a 3D thermo-hydro-mechanical coupled discrete beam lattice model of structure built of the nonisothermal saturated poro-plastic medium subjected to mechanical loads and nonstationary heat transfer conditions. The proposed model is based on Voronoi cell representation of the domain with cohesive links represented as inelastic Timoshenko beam finite elements enhanced with additional kinematics in terms of embedded strong discontinuities in axial and both transverse directions. The enhanced Timoshenko beam finite element is capable of modeling crack formation in mode I, mode II and mode III. Mode I relates to crack opening, mode II relates to in-plane crack sliding, and mode III relates to the out-of-plane shear sliding. The pore fluid flow and heat flow in the proposed model are governed by Darcy\'s law and Fourier\'s law for heat conduction, respectively. The pore pressure field and temperature field are approximated with linear tetrahedral finite elements. By exploiting nodal point quadrature rule for numerical integration on tetrahedral finite elements and duality property between Voronoi diagram and Delaunay tetrahedralization, the numerical implementation of the coupling results with additional pore pressure and temperature degrees of freedom placed at each node of a Timoshenko beam finite element. The results of several numerical simulations are presented and discussed.

Formulation of an efficient continuum mechanics-based model to study wave propagation in one-dimensional diatomic lattices

In this paper, we formulate an efficient continuum mechanics-based model on the basis of a discrete lattice model. First, the dispersion relation of a lattice wave in a one-dimensional diatomic crystal lattice is derived. Then, the second- and fourth-order continuum models are obtained from the differential-difference equations of motion by using the Padé approximations. The results show that the proposed fourth-order continuum model can predict the dispersion behaviour of the one-dimensional diatomic crystal lattice very well in the first Brillouin zone. Furthermore, the applicability of the present model to the prediction of the dispersion behaviour of the one-dimensional diatomic lattice with internal resonator and inerter is examined. Finally, the vibration frequencies of finite diatomic lattices are calculated by both the discrete and the proposed continuum models.