Lattice Spring Model Research Articles

Lattice Spring Model for Irregular Interface Based on an Adaptive Location Strategy

Lattice Spring Model for Irregular Interface Based on an Adaptive Location Strategy

Finite-time stability of any polyhedral sweeping process with uni-directional loading and application to elastoplastic models

A significant progress in the study of the dynamics of elastoplastic lattice spring models has been recently due to the development of a sweeping process framework where the dynamics of the stresses of springs is linked to a solution of the associated sweeping process. By using the sweeping process framework, a recent result by Gudoshnikov et al. (2022) ensures finite-time convergence of a lattice spring model under assumption that the vector g′(t) of the applied displacement controlled-loading lies strictly inside a suitable normal cone of the sweeping process. In this paper we drop the latter assumption by proving that g′(t) always belongs to a suitable normal cone when g′(t)=const (i.e. the loading is uni-directional) and when certain genericity condition holds. By using this result we establish that every elastoplastic lattice spring model with a generic associated polyhedron reaches a unique distribution of plastic deformations in finite time.

Photocontrol of pattern formation and hysteresis loops in polymer gels with host-guest interactions.

Patterns formed under external stresses are often critical for ensuring the functionality of soft materials. We focus on dynamic control of pattern formation and restructuring in hydrogels with host-guest interactions. We extend the three-dimensional gel Lattice Spring Model to capture the dynamics of photo-responsive hydrogels with pendant azobenzene moieties immersed in the α-cyclodextrin (α-CD) solution. While trans-azobenzene moieties are accommodated by the α-CD cavities forming inclusion complexes resulting in hydrogel swelling, exposure to UV irradiation induces trans-to-cis photoisomerization leading to decomplexation and matrix deswelling. We demonstrate swelling-induced patterns in confined samples upon adding the α-CD solution in the dark. Further, we show that spatiotemporal variations in UV irradiation effectively control patterns formed and hysteresis loops. We introduce soft confinements via illuminating specific regions of unconstrained samples with UV light. Our results indicate that features of patterns and hysteresis in systems' response to external conditions can be regulated via well-defined illumination patterns.

A micro-macro constitutive model for compressive failure of rock by using 4D lattice spring model

In this work, a micro-macro mechanics-based constitutive model is developed to describe the compressive failure process of rock. A micromechanics model is first developed based on the conceptual model of a micro cracked spring bond, where two micro parameters with clearly geometric meanings are introduced. By using the micromechanics model, it is possible to reproduce the whole nonlinear failure process of the rock under compressive loading, however, it fails to predict the reasonable quantitative relation between the crack initiation stress (σCI) and the direct tensile strength (DTS) that is commonly observed in laboratory tests. Therefore, a macroscopic failure criterion is introduced to determine the failure state of the spring bond instead of the spring deformation used in the micromechanics model. The numerical results indicate that the 4D-LSM with the micro-macro constitutive model can not only fully reproduce the whole nonlinear behavior of rock under different confining pressures, but also reproduce the quantitative relation between σCI and the DTS and the overall macroscopic nonlinear failure strength criterion defined by Hoek and Brown. The micro-macro constitutive model not only provides a feasible solution for better describing the complex nonlinear compressive failure process of rock but helps to better understand the internal failure mechanism of rock.

Verification of Lattice Spring Model for Modelling Composite Material

The use of lattice spring model (LSM) in modelling composite material is verified using static and dynamic loading test cases. The LSM results are compared against those from finite element method (FEM) implemented in ANSYS. Simple cubic (SC) lattice structure is employed in the LSM to model 3D solid body deformation. The material symmetry used to model composite material is orthotropic. In the current preliminary study, good agreement is found between the LSM and FEM results.

Realistic modeling of cemented granular materials with a lattice spring model by developing a digital 3D reconstruction approach

AbstractThree‐dimensional (3D) geometric reconstruction is a fundamental requirement to realistically predict the mechanical behavior of cemented granular materials (CGMs). In this work, a four‐phase geometric reconstruction method for CGMs, involving aggregate, cement, in‐between interfacial transition zone (ITZ), and void phases, is developed by using digital image processing techniques and an optimization algorithm. The reconstruction method includes four steps. (1) The planar outlines of aggregates are extracted from a two‐dimensional (2D) aggregate image (e.g., gravels, pebbles, and crushed stones) based on the image segmentation algorithm. (2) Through spatial geometric transformations on the acquired planar outlines, the 3D aggregates are generated, whose reliability is verified by comparison with those obtained by a 3D scanner. (3) According to the volume ratio of each phase, the initial four‐phase model is reconstructed by randomly placing aggregates into the user‐defined domain with considering the size distribution of aggregates, followed by the generation of the ITZ represented by their surrounding minimum envelope surface, and the insertion of voids in the rest cement phase. (4) The final realistic CGM is reconstructed by optimizing the spatial distribution of the void and cement phases based on a simulated annealing algorithm. Afterwards, the rationality of the reconstructed CGM is verified by comparing two‐point probability functions between the physical model and the reconstructed model. With the aid of a four‐dimensional lattice spring model (4D‐LSM), the feasibility of the reconstructed CGM to reproduce its mechanical behavior is demonstrated by numerical examples and a comparison with existing experimental or other numerical solutions.

High‐performance computing of 3D blasting wave propagation in underground rock cavern by using 4D‐LSM on TianHe‐3 prototype E class supercomputer

AbstractParallel computing assigns the computing model to different processors on different devices and implements it simultaneously. Accordingly, it has broad applications in the numerical simulation of geotechnical engineering and underground engineering, of which models are always large‐scale. With parallel computing, the computing time or the memory requirements will be reduced by splitting the original domain of the numerical model into many subdomains, which is thus named as the domain decomposition method. In this study, a cubic and equal volume domain decomposition strategy was utilized to realize the parallel computing on the distributed memory system of four‐dimensional lattice spring model (4D‐LSM) based on the message passing interface. With a more efficient communication strategy introduced, this study aimed at operating an one‐billion‐particle model on a supercomputer platform. The preprocessing procedure of the parallelized 4D‐LSM was restructured and the particle generation strategy suitable for the supercomputer platform was employed to minimize the time consumption in preprocessing and calculation. On this basis, numerical calculations were performed on TianHe‐3 prototype E class supercomputer at the National Supercomputer Center in Tianjin. Two field‐scale three‐dimensional blasting wave propagation models were carried out, of which the numerical results verify the computing power and the advantage of the parallelized 4D‐LSM in the simulation of large‐scale three‐dimension models. Subsequently, the time complexity and spatial complexity of 4D‐LSM and other particle discrete element methods were analyzed.

Flow-Induced Locomotion of a Flexible Filament in the Wake of a Cylinder in Non-Newtonian Flows

The problem of a flapping filament mounted to a free cylinder provides insight into computational fluid dynamics. We study the extent to which the length of the flexible filament and the non-Newtonian power-law fluids should be implemented to stabilize the flapping of an immersed structure in the flow with the Reynolds number of 100. We propose a hybrid model that includes an explicit Lattice Spring Model and Immersed Boundary non-Newtonian Lattice Boltzmann Method to simulate the behavior of the filament in the vicinity of a fluid flow. We validate this hybrid model using bench-marked problems, where a non-Newtonian fluid flow passes over a fixed cylinder in an unconfined channel and in a Newtonian flow over a cylinder containing a flexible fin. In the result section, we study the effects of fluid and structural characteristics on the motion of the filament and cylinder. When shear-thinning, Newtonian, and shear-thickening fluids are used, the results show that the cylinder keeps its minimum distance to the channel centerline at a specific length. This is due to the flapping of the filament generating the low-pressure region downstream from the cylinder and filament structure. We also analyze the effects of the bending stiffness. We find that from the lowest bending stiffness of 0.003 to the highest of 0.007, flexibility has no impact on the cylinder's center of mass, when the filament length is set as large as the critical length. The vibration amplitude and frequency of the filaments are also measured in different fluids and in varied filament stiffnesses. When n = 0.7, the bending stiffness of 0.007 causes a high-frequency vibration with the highest amplitude, whereas a bending stiffness of 0.005 achieves the minimum amplitude.

Different types of spatial correlation functions for non-ergodic stochastic processes of macroscopic systems.

Focusing on non-ergodic macroscopic systems, we reconsider the variances [Formula: see text] of time averages [Formula: see text] of time-series [Formula: see text]. The total variance [Formula: see text] (direct average over all time series) is known to be the sum of an internal variance [Formula: see text] (fluctuations within the meta-basins) and an external variance [Formula: see text] (fluctuations between meta-basins). It is shown that whenever [Formula: see text] can be expressed as a volume average of a local field [Formula: see text] the three variances can be written as volume averages of correlation functions [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text]. The dependences of the [Formula: see text] on the sampling time [Formula: see text] and the system volume V can thus be traced back to [Formula: see text] and [Formula: see text]. Various relations are illustrated using lattice spring models with spatially correlated spring constants. .

Implementation of Johnson-Holmquist-Beissel model in four-dimensional lattice spring model and its application in projectile penetration

• The 4D-LSM is further developed by implementing JHB constitutive model. • The JHB-4D-LSM was verified by modelling Brazilian tests and dynamic spalling tests. • JHB-4D-LSM is applied to study penetration in both thin plate targets and thick B 4 C under various impact velocities. • Good agreement among numerical simulation and numerical/ analytic results demonstrates the capability of JHB-4D-LSM. The imbedded linear constitutive model in four-dimensional lattice spring model (4D-LSM) is not sufficient or accurate enough to solve the dynamic problems with high-speed impacts. In order to produce a more realistic prediction of dynamic fracture process in brittle materials, the capability of four-dimensional lattice spring model (4D-LSM) has been further extended by implementing Johnson-Holmquist-Beissel (JHB) model. This is achieved by defining the micromechanical constitutive model parameters of 4D-LSM through the macro-scale strength parameters in JHB model, with an incremental form of 4D-LSM being developed. The incremental JHB-4D-LSM model is verified by modelling dynamic Brazilian tests and dynamic spalling tests, with good agreement between modelling results by JHB-4D-LSM and theoretical/numerical solutions in literature being obtained. For the demonstration of model application, the proposed JHB-4D-LSM model is applied to simulate and analyse the problems of projectile penetration with various initial impact speeds. Reasonable numerical modelling results indicate that, the proposed JHB-4D-LSM model can be served a promising tool to predict projectile penetration behaviour of brittle materials.

An irregular lattice spring model: uniform elasticity, grid refinement and isotropic crack propagation

The Born lattice spring model (LSM) is implemented on an irregular network. This irregular LSM is directly mapped to linear elasticity theory such that the model can capture an elastically uniform system where the direction of resultant fracture propagation is no longer influenced by a regular lattice structure. Furthermore, this allows the density of nodes to be varied throughout the simulation while maintaining the correct elastic mapping. For the first time, the LSM can be refined around areas of interest such as heterogeneities or crack tips, while capturing the spatial cooperative effects of isotropic fracture propagation in heterogeneous materials.

Preliminary study on the effect of microstructure shape on impact compression dynamic fracture of two-dimensional brittle materials protective structures

Under the shock wave compression, the common cracks will rapidly expand and penetrate, resulting in a catastrophic fracture phenomenon, which can seriously threaten the structural design function of brittle material samples and limit their practical application range. By introducing and designing protective structures containing microstructures, the plasticity of the sample against impact can be significantly increased, effectively avoiding or slowing down the onset of failure. A lattice-spring model suitable for demonstrating the elastic properties and fracture simulation of brittle materials is used in this paper to investigate the effect of the shape of the microstructure on its impact properties. The calculation results show that the shape of the microstructure will affect the evolution characteristics and laws of the shock wave profile. Compared with the stress curves, it can be found that the microstructure with negative Poisson's ratio has greater impact plasticity. The analysis of the mesoscopic deformation characteristics of different microstructure shapes shows that the penetration and volume shrinkage deformation of the concave hexagon, the concave triangle and the fourth-order star-shaped microstructure are dominant, and the strain localization is significant. The research results of the microstructure, with negative Poisson's ratio properties, that can significantly enhance the macroscopic impact plasticity of brittle materials will help to improve their prevention in terms of impact failure and provide guidance for the optimal design of protective structures. The findings that microstructures with negative Poisson's ratio properties can significantly enhance the macroscopic impact plasticity of brittle materials, as revealed in this paper, will help to improve their prevention in terms of impact failure and provide guidance for the optimal design of protective structures.

A three-phase coupled numerical model for the hydraulic fracturing of rock

In this work, a three-phase coupled numerical model is developed for rock fracturing by integrating the lattice Boltzmann method (LBM) and the distinct lattice spring model (DLSM). The multiphase flow is handled by using the LBM based on the Shan–Chen potential, whereas the rock fracturing is solved by using the DLSM with a cohesive-like model. During the calculation, the solid and fluid models are executed in their own timeline independently, and the coupling is realized from boundary exchange within each step. The influence of fracturing on the multiphase distribution is considered by using a phase update operation. The three-phase model is verified against a few multiphase benchmark problems. Then, its ability to model the fracturing of rock is demonstrated through a comparison with the hydraulic fracturing test. Following this, the influence phase distribution along cavities on the hydraulic fracturing path was investigated by using the coupled model. Our results show that the three-phase coupled model can be a useful numerical tool for handling rock fracturing problems involving multiphase fluids.

Investigation of the shear failure of rock joints using the four-dimensional lattice spring model

This research adopts the four-dimensional lattice spring model (4D-LSM) to investigate the shear strength and failure mechanism of irregular rock joints and the influence of rock heterogeneity. The 4D-LSM has been used for a wide range of rock engineering applications, however its ability for simulating direct shear tests has not been investigated in detail. The material strength parameters were calibrated using uniaxial compressive strength (UCS) and Brazilian tensile strength (BTS) tests on granite specimens while the joint parameters were calibrated from direct shear tests on saw-cut UCS specimens and rock joints containing a single triangular asperity. From UCS tests, a calibration method is proposed to adjust the excessive machine deformation due to the high brittleness of granite, which is further applied to the direct shear test results. It was found that the strength of rock joints is strongly influenced by the intact rock properties. To investigate the influence of joint geometry and rock heterogeneity, three joint asperity angles and five heterogeneous models were simulated using the calibrated 4D-LSM model. Simulation results were able to capture the influence of different joint asperity angles on the shear strength and failure mechanism as reported in previous experimental and numerical studies. With larger asperity angles, higher shear strengths were obtained. Meanwhile, rock heterogeneity represented by particles with lower strength will reduce the shear strength and alter the failure mechanism, especially at a high porosity. This strength reduction was observed to be greater for steep-angle asperities and becomes less significant for flatter asperities. Similar conclusions were also obtained based on the simulations of natural joint profiles. From these findings, it is suggested that the characterisation of rock heterogeneity is crucial to estimate the strength of rock joints. In addition to the roughness parameters , heterogeneity parameters such as porosity are fundamental for rock engineering applications. • The ability of the (4D-LSM) for simulating the direct shear of irregular rock joints is verified against experimental results. • A calibration is developed for 4D-LSM to model rock joints by integrating uniaxial compressive and direct shear tests. • The influence of rock joint surface on its shear strength is investigated numerically by using the calibrated 4D-LSM. • The influence of different rock heterogeneities on the shear strength of the joints is numerically studied.

Two-dimensional periodic structures modeling based on second strain gradient elasticity for a beam grid

Higher order gradient elasticity theories are widely applied to determine the wave propagation characteristics of micro-sized structures. The novelty of this paper, firstly, is using the Second Strain Gradient (SSG) theory to explore the mechanism of a micro-sized 2D beam grid. The strong formulas of continuum model including governing equations and boundary conditions are derived by using the Hamilton principle. Then, a valuable long-range Lattice Spring Model (LSM) is elaborated, providing a reasonable explanation for the model based on SSG theory. The dynamic continuum equations from LSM are calculated through the Fourier series transform approach. Finally, the dynamic properties of 2D beam grid are analyzed within the Wave Finite Element Method (WFEM) framework. The band structure and slowness surfaces, confined to the irreducible first Brillouin zone, are studied in frequency spectrum. The energy flow vector fields and wave beaming effects are discussed through SSG theory and Classical Theory (CT) of elasticity. The results show that the proposed approach is of significant potential for investigating the 2D wave propagation characteristics of complex micro-sized periodic structures.

Development of the four-dimensional lattice spring model for thermo-mechanical fracturing and large deformation of solids

Thermo-mechanical coupling is a key issue in the fields of engineering structure security, geothermal energy exploitation, metal hot working, crustal evolution, etc. This work aims to further develop the four-dimensional lattice spring model (4D-LSM) to address thermo-mechanical coupling problems involving fracturing and large deformations. To reproduce the thermal governing equations, a thermal pipe network partially or completely duplicates the corresponding mechanical lattice, where Fourier's law is solved at the pipe level. Furthermore, two different time steps are adopted in the calculations of the equations of force and heat to enable the coupled model to address both transient and steady-state thermo-mechanical coupling problems. The influence of the thermal field on the mechanical responses is addressed through the thermal expansion equation, and the influence of fracturing and recontacting on the thermal connections of the pipe is considered. By introducing thermal damage functions, the influence of temperature on material properties has been considered in the coupled model. A series of examples are evaluated to verify the correctness and accuracy of the coupled thermo-mechanical 4D-LSM. Next, several application examples are analysed to show the ability of the coupled model to simulate the thermal fracturing of heterogeneous materials and the large deformation behaviours of solids.

Further development of distinct lattice spring model for stability and collapse analysis of deep foundation pit excavation

In this work, the distinct lattice spring model (DLSM) is further developed for handling practical geotechnical problems. The Zienkiewicz-Pande (ZP) constitutive model is implemented into the DLSM for a better prediction of soil plasticity. The beam finite element is introduced to simulate supporting structures. The birth–death particle method is developed for the DLSM to simulate the multistage construction. The accuracy and stability of the enriched DLSM are verified against both analytical solutions and numerical results from well-established commercial software. Following this, the enriched DLSM is applied to analyze the stability of one actual deep foundation pit construction project, and its numerical prediction is compared with the actual monitoring data. Finally, a preliminary trial on the design of the foundation pit supporting structure is carried out by using the enriched DLSM. These applications show that the enriched DLSM can be an alternative numerical method for the stability and collapse analysis of deep foundation pit excavation.

ImageMech: From Image to Particle Spring Network for Mechanical Characterization

The emerging demand for advanced structural and biological materials calls for novel modeling tools that can rapidly yield high-fidelity estimation on materials properties in design cycles. Lattice spring model , a coarse-grained particle spring network, has gained attention in recent years for predicting the mechanical properties and giving insights into the fracture mechanism with high reproducibility and generalizability. However, to simulate the materials in sufficient detail for guaranteed numerical stability and convergence, most of the time a large number of particles are needed, greatly diminishing the potential for high-throughput computation and therewith data generation for machine learning frameworks. Here, we implement CuLSM, a GPU-accelerated compute unified device architecture C++ code realizing parallelism over the spring list instead of the commonly used spatial decomposition, which requires intermittent updates on the particle neighbor list. Along with the image-to-particle conversion tool Img2Particle, our toolkit offers a fast and flexible platform to characterize the elastic and fracture behaviors of materials, expediting the design process between additive manufacturing and computer-aided design. With the growing demand for new lightweight, adaptable, and multi-functional materials and structures, such tailored and optimized modeling platform has profound impacts, enabling faster exploration in design spaces, better quality control for 3D printing by digital twin techniques, and larger data generation pipelines for image-based generative machine learning models.

DLSM Simulation Analysis of the Influence of Blasting Construction on Adjacent Tunnels in Rock Mass with Discontinuities

In order to study the influence of blasting construction of rock mass with discontinuities on adjacent tunnels, a feasible blasting scheme is formulated to ensure the safety of adjacent tunnels during construction. Based on the water resources allocation project in the Pearl River Delta, a tunnel construction model is established by using continuous‐discontinuous medium mechanics calculation model, Distinct Lattice Spring Model (DLSM), to simulate the impact of blasting construction of the left tunnel on the right adjacent tunnel. According to the numerical simulation results, the stress wave transmission process of the left tunnel with blasting time t = 18 ms is analyzed. Furthermore, the effects of the positions of the discontinuity, the width of the discontinuity, and the angle between the discontinuity and the horizontal direction on the adjacent tunnel are studied. The modelling results show that the existence of discontinuities plays a certain role in improving the safety of adjacent tunnels in the blasting stage. The farther the distance between the discontinuity and the adjacent tunnel is, the smaller the impact of blasting on the adjacent tunnel is. In the process of blasting, the larger the width of discontinuity is, the more effective it is to block the influence of blasting stress wave on the adjacent tunnel. The smaller the angle between discontinuity and horizontal direction, the smaller the impact of blasting on adjacent tunnels.

A multifrequency ultrasonic approach to extracting static modulus and damage characteristics of rock

Understanding the mechanical behavior of rock is a fundamental requirement for many geological explorations . Based on the frequency dependence of primary and shear wave velocities (P- and S-wave velocities), this work proposed a new approach to obtaining the static modulus of rocks without using empirical correlations. Six group granite specimens are treated by different temperatures, then the P- and S-wave velocities of these specimens at various frequencies were tested. The static moduli of all the specimens were calculated by deriving the newly proposed critical P- and S-wave velocities with a classical formula. The results showed that the static moduli of all the specimens obtained from the ultrasonic approach complied with the initial modulus obtained from uniaxial compression tests , demonstrating the reasonability of the proposed ultrasonic approach. The reasonability of the proposed approach was further proven by using a distinct lattice spring model. Numerical results showed that the calculated modulus is consistent with the true value. In addition, the standard deviation of wave velocities at different frequencies was adopted to describe the damage characteristics in rock, which well explained the change in damage variable during the thermal process. The proposed approach provides a potential method to fast obtain the static modulus and damage characteristics of rock.