Lattice Discrete Particle Model Research Articles

An efficient static solver for the lattice discrete particle model

AbstractThe lattice discrete particle model (LDPM) has been proven to be one of the most appealing computational tools to simulate fracture in quasi‐brittle materials. Despite tremendous advancements in the definition and implementation of the method, solution strategies are still limited to dynamic algorithms, resulting in prohibitive computational costs and challenges related to solution accuracy for quasi‐static conditions. This study presents a novel static solver for LDPM, introducing fundamental innovation: (1) LDPM constitutive laws are modified to provide continuous response through all possible strain/stress states; (2) an adaptive arc‐length method is proposed in combination with a criterion to select the sign of the iterative load factor; (3) an adaptive limit‐unloading–reloading path switch algorithm is proposed to restrict oscillations in the global stiffness matrix. Extensive validation of the proposed approach is presented. Numerical results demonstrate that the static solver exhibits satisfactory convergence rates, significantly outperforming available dynamic solutions in computational efficiency.

Clustering-enhanced Lattice discrete particle modeling for quasi-brittle fracture and fragmentation analysis

Clustering-enhanced Lattice discrete particle modeling for quasi-brittle fracture and fragmentation analysis

Numerical study on the behavior of polypropylene fiber reinforced concrete subjected to moderate temperature variations using LDPM theory

This paper presents the mechanical behavior of macro-synthetic fiber-reinforced concrete with polypropylene fibers (MSFRC) under varying temperature environments. The Lattice Discrete Particle Model (LDPM), a meso-scale model for concrete, is used for the plain concrete study. An extended version of LDPM taking into account the effect of fibers in a discrete way, LDPM-F, is applied for the MSFRC study. The model is calibrated based on the experimental data at room temperature (20 °C). Temperature effects on the MSFRC constituents, fiber and concrete, are individually adjusted inside the LDPM-F formulation based on literature data and available models. Subsequently, predictive three point bending simulations on MSFRC are performed at different temperatures (−30 °C, −15 °C, 0 °C, 40 °C and 60 °C) and the results are validated against available experiments. The results show a generally good agreement between simulations and tests.

Dissipation mechanisms of crack-parallel stress effects on fracture process zone in concrete

The effect of crack-parallel stresses on the fracture properties of quasi-brittle materials has recently received significant attention in the fracture mechanics community. A new experiment, the so-called gap test, was developed to reveal this effect. While the finite element crack band model (CBM) with the physically realistic Microplane damage model M7 was quite successful in capturing the damage and fracture during the gap test, some questions remain, particularly the near doubling of the fracture energy at moderate crack parallel compression, which was underestimated by about 30%. Presented here is an in-depth meso-mechanical investigation of energy dissipation mechanisms in the Fracture Process Zone (FPZ) during the gap test of concrete, an archetypal quasi-brittle material. The Lattice Discrete Particle Model (LDPM) is here used to simulate the quasi-brittle material at the mesoscale, which is the length scale of major heterogeneities. The LDPM can capture accurately the frictional sliding, mixed-mode fracture, and FPZ development. The model parameters characterizing the given mix design are first calibrated by standard laboratory tests, namely the hydrostatic, unconfined compression, and four-point bending (4PB) tests. The experimental data used characteristic of the given mix design are calibrated by the Brazilian split-cylinder tests and by gap tests of different sizes with and without crack-parallel stresses. The results show that crack-parallel stresses affect not only the length but also the width of the FPZ. It is found that the energy dissipation portion under crack-parallel compression is significantly larger than it is under tension, which is caused by micro-scale frictional shear slips, as intuitively suggested in previous work. For large compressive stresses, the failure mode changes to inclined compression-shear bands consisting of axial splitting microcracks. Several complications experienced in the numerical modeling of gap tests are also discussed, and the solutions provided.

Flow Lattice Model for the simulation of chemistry dependent transport phenomena in cementitious materials

This study presents the formulation and validation of a three-dimensional Flow Lattice Model (FLM) with application to the Hygro-Thermo-Chemical (HTC) model for analysis of moisture transport and heat transfer in cementitious materials. The FLM is a discrete transport model formulated in association with meso-mechanical models, such as the Lattice Discrete Particle Model. This enables the simulation of transport phenomena at the length scale at which the material exhibits intrinsic heterogeneity. The HTC theoretical formulation is based on mass and energy conservation laws, written using humidity and temperature as primary variables, and considering explicitly various chemical reactions, for example, cement hydration and silica fume reaction, as internal variables. In this work, the HTC formulation was extended to include the effect of temperature on the sorption isotherm. The FLM solutions were compared with those of a continuum finite element implementation of the HTC model and experimental data available from the literature; the overall agreement demonstrates the reliability of the proposed approach in reproducing phenomena such as cement hydration, self-desiccation and temperature-dependent moisture drying.

Lattice discrete particle modeling of the cycling behavior of strain-hardening cementitious composites with and without fiber reinforced polymer grid reinforcement

Strain-hardening cementitious composites (SHCC) has become increasingly prevalent in structural design. Compared to ordinary concrete, SHCC exhibits ultra-high tensile ductility, significant strain-hardening behavior, very fine multi-point cracking, high energy dissipation, and good durability. One robust model for simulating reinforced concrete behavior is the so-called Lattice Discrete Particle Model (LDPM), and specifically LDPM-F which includes the effect of fiber reinforcement. The present cycling constitutive law implemented in LDPM and the cycling fiber-bridging law in LDPM-F though cannot accurately capture the residual tensile plastic strain, loading–unloading path, and energy dissipation of SHCC during cyclic tension. To solve these concerns, the cycling tension–compression constitutive law and the nonlinear cycling fiber-bridging law were reformulated. Further, the new model was used to simulate the cycling tensile behavior of plain concrete, SHCC, and Fiber-Reinforced Polymer (FRP) grid reinforced SHCC (FRP-SHCC). Simulation results show that the multi-point cracking, crack widths, and ultra-high ductility properties are correctly captured by LDPM-F. In addition, LDPM-F with the modified cycling constitutive model can effectively simulate the cycling tension–compression behavior, accurately reproducing the stress–strain relationship, residual plastic strain, and overall dissipated energy. Finally, the simulated failure modes agree well with the actual fracture planes of these materials under cyclic tension.

Fracturing and collapse behavior of masonry vaulted structures: a lattice-discrete approach

This study presents a novel meso-scale approach to investigate the fracturing and collapse behaviors of unreinforced masonry vaulted structures induced by spreading supports. Traditionally, the behavior of masonry vaulted structures is investigated by resorting on limit analysis method, that is a limited approach as: (i) it tackles just the failure condition of the arched structural configuration as it assumes the simultaneous formation of hinges once the thrust line reaches the edge of the masonry structure, (ii) it completely ignores the damage propagation phenomenon, starting from the trigger of the first fracture up to the complete structural failure condition. The comprehension of this fracturing process is fundamental aiming to analyze intermediate damage states for the check of serviceability limit states and to individuate a more realistic structural crack distribution in ultimate conditions. This paper proposes a thorough understanding of the fracturing behavior of masonry vaults based on non-linear fracture mechanics concepts. For this purpose, the Lattice Discrete Particle Model (LDPM) is adopted to simulate a variety of stone masonry vaulted structures up to their collapse. LDPM simulates the behavior of masonry at the stone level. The interaction between stones that are bounded by weak layers of mortar is governed by specific constitutive equations describing tensile fracturing with strain-softening, cohesive and frictional shearing, and compressive response with strain-hardening. The formation of hinges, the activation of the mechanism and the kinematic mechanism are analyzed for three different types of vaults, namely groin, barrel and depressed vaults, and for six different slenderness. The first conclusion of this study is that LDPM can be used as an alternative tool to perform typical limit analysis for the assessment of safety of arches and vaults in ultimate conditions. Most importantly, LDPM is able to show that the fracturing process is a progressive phenomenon, the cracked surfaces are never strictly symmetric respect to the vertical axis and do not appear simultaneously. In particular, the features of non simultaneity and asymmetry of the cracks increases as the distance between the crown of the vault and the imposts increases, i.e. going from depressed vaults to groin vaults. Finally, the evolution of the fracturing process occurs more progressively and exhibits less pronounced asymmetry in the case of depressed vaults as compared to groin vaults for which, in turn, the damage appears to be more brittle and characterized by asymmetry in the cracks distribution.

Analysis of the behavior of the masonry Medici tower resorting on a hybrid discrete-kinematic methodology

This study presents a novel integrated discrete-analytical approach for analyzing the collapse behavior of the masonry Medici tower (L'Aquila, Italy). Due to their slenderness, masonry towers are characterized by high susceptibility to seismic actions and several approaches can be adopted to analyze their seismic vulnerability. Generally, engineers-practitioners and researchers study the local and global collapse mechanisms based on simplified kinematic analysis, as prescribed by national and international construction codes or, alternatively, more sophisticated approaches such as nonlinear finite element methods have been adopted to simulate the response of masonry towers. Although successful in some applications, these methods are limited in accurately capturing crack distributions and fracture mechanisms. In fact, they completely ignore the damage propagation phenomenon, starting from the trigger of the fracture up to the complete structural failure condition, that is instead fundamental aiming to analyze intermediate damage states for the check of serviceability limit states or to individuate a more realistic structural crack distribution in ultimate conditions. This work proposes a hybrid discrete-kinematic approach: first, the Lattice Discrete Particle Model (LDPM), that simulates masonry at meso-scale, is used to individuate the actual collapse mechanism; next, the individuated cracked configuration is used in the kinematic analysis for the analysis in ultimate conditions. The results show that the collapse of the Medici tower due to the 2009 L'Aquila earthquake is well predicted by LDPM and the corresponding limit analyses demonstrate the efficiency of the proposed hybrid approach applied to this case study. Additional results point out that different load configurations, more specifically variations in the direction of the seismic action, provoke in certain cases a more diffused damage and a clear failure pattern can not be identified for kinematic analyses. In these cases, relying mainly on comprehensive numerical models, such as LDPM, is fundamental to study the fracturing process from the cracks trigger up to the ultimate complex collapse mechanism.

Discrete modeling of concrete failure and size-effect

Size-effect in concrete and other quasi-brittle materials defines the relation between the nominal strength and structural size when material fractures. The main cause of size-effect is the so-called energetic size-effect which results from the release of the stored energy in the structure into the fracture front. In quasi-brittle materials and in contrast to brittle materials, the size of the fracture process zone is non-negligible compared to the structural size. As a consequence, the resulting size-effect law is non-linear and deviates from the response predicted by linear elastic fracture mechanics. In order to simulate the size-effect, one needs to rely on numerical modeling to describe the formation, development and propagation of the fracture process zone. Although a number of models have been proposed over the years, it transpires that a correct description of the fracture and size-effect which accounts for boundary effects and varying structural geometry remains challenging. In this study, the Lattice Discrete Particle Model (LDPM) is proposed to investigate the effects of structural dimension and geometry on the nominal strength and fracturing process in concrete. LDPM simulates concrete at the aggregate level and has shown superior capabilities in simulating complex cracking mechanisms thanks to the inherent discrete nature of the model. In order to evaluate concrete size-effect and provide a solid validation of LDPM, one of the most complete experimental data set available in the literature was considered and includes three-point bending tests on notched and unnotched beams. The model parameters were first calibrated on a single size notched beam under three-point bending and on the mechanical response under unconfined compression. LDPM was then used to perform blind predictions on the load-crack mouth opening displacement curves of different beam sizes and notch lengths. Splitting test results on cylinders were also predicted. The results show a very good agreement with the experimental data. The quality of the predictions was quantitatively assessed. In addition, a discussion on the fracturing process and dissipated energy is provided. Last but not least, the Universal Size-Effect Law proposed by Bažant and coworkers was used to estimate concrete fracture parameters based on experimental and numerical data.

Lattice discrete particle modeling of concrete under cyclic tension–compression with multi-axial confinement

Accurate modeling of concrete mechanical behavior under cyclic loading with different levels of confinement is crucial in design and analysis of structures subjected to seismic events. One robust model for simulating concrete behavior is the Lattice Discrete Particle Model (LDPM), a discrete model formulated at the scale of aggregate and composed of polyhedral cells connected through a lattice of nonlinear fracturing struts. To improve the performance of LDPM for the prediction of mechanical behavior under different cyclic loading schemes and multi-axial confinement, a comprehensive cycling constitutive model, a modified volumetric–deviatoric compression law, and a modified frictional behavior under compression are established in this study. An effective strain and a corresponding effective stress at the mesoscale are formulated and used to determine the stress–strain relationship under monotonic loading. In addition, the constitutive relationships related to the tension and shear stresses are established separately to ensure a smooth stress transition from tension to compression state. Two material parameters are used to control the stiffness attenuation: the residual plastic strain, and the energy dissipation of concrete during loading and unloading, under both tension and compression. Further, a new constitutive equation for the frictional behavior under compression is proposed to simulate the post-peak behavior under confinement. The proposed constitutive model is used to simulate the mechanical behavior and failure mode of concrete under tension–compression cycling and hydrostatic pressure. Triaxial tests under different levels of confinement and unconfined compression tests are also simulated. Model validation is performed using multiple data sets available in the literature on concretes of various strengths. Simulation results show that the established cyclic constitutive model can effectively characterize the mechanical response of concrete under different cycling loadings and confining pressures.

Mesoscopic discrete modeling of multiaxial load-induced thermal strain of concrete at high temperature

This work presents a mesoscopic discrete model of load-induced thermal strain (LITS) as part of the Lattice Discrete Particle Model at high temperature (LDPM-HT) that captures the experimentally observed deformations and mechanical responses of concrete heating up to 800 °C under multiaxial loads. In the proposed model, the LITS is decoupled into elastic strain increment due to thermal degradation, and thermo-mechanical strain at the mesoscale. As the most important component, the mesoscopic thermo-mechanical strain is decomposed into a normal and two shear components. The normal component in compression of the thermo-mechanical deformation at the mesoscale controls the macroscopic LITS in the load direction, while the mesoscopic thermo-mechanical strain components in normal tension and shear directions dominate the macroscopic LITS in the unloaded directions. The correctness and accuracy of the improved LDPM-HT are demonstrated by simulating two experimental investigations, namely a heating test up to 800 °C with uniaxial load and a heating test up to 250 °C with multiaxial loads.

Discrete modeling of deterministic size effect of normal-strength and ultra-high performance concrete under compression

Size effect of heterogeneous granular materials under tensile loading is a well known phenomenon and the subsequent size effect laws are widely available in the scientific literature. On the contrary, size effect in compression is scarcely studied, which, especially for concrete, is more significant for its brittle failure mode in practical applications. This paper investigates the deterministic size effect in compression of concretes including a normal strength concrete (NSC) and an ultra high performance concrete (UHPC). The Lattice Discrete Particle Model (LDPM) is selected to determine the compressive strength of a broad range of sizes utilizing experimental-based calibrated data sets. The simulated tests include both cylinders and prisms with the diameter/width range from 150 mm to 600 mm for normal strength concrete and from 50 mm to 200 mm for UHPC. The investigated aspect ratio ranges from 1 to 16. It is found that the simulated strengths show no size effect under low friction loading, and the magnitude of size effect decreases with rising aspect ratio under high friction loading. The size effect laws available in the literature are not able to fit the obtained compressive strength data and a new compressive size effect law formula named ComSEL3D is proposed in terms of diameter and aspect ratio. The correlation by ComSEL3D for the simulated strengths is above 0.99, which shows its excellent ability in describing and predicting the deterministic size effect behavior in compression for both concretes in cylindrical and prism shapes. The ComSEL3D also outperforms the available formulas in the literature in matching the experimental results of various concretes under compression.

Micromechanics of Fiber–Concrete Interaction: Numerical Modeling and Experimental Validation

To simulate the behavior of fiber-reinforced concrete, an accurate modeling of the concrete and fiber phases as well as the fiber–concrete interaction is required. This study proposes a new model to account for fiber-matrix interactions based on micromechanics. The concrete matrix is simulated using the lattice discrete particle model (LDPM), a mesoscale model for heterogeneous materials. The fiber is explicitly represented using elastoplastic beam elements. The fiber–concrete interaction algorithm features a slideline model and a constitutive model to describe the bond-slip relation. The slideline model includes a tie algorithm for interactions between concrete and slideline, and a penalty constraint between the slideline and the fiber. The proposed model is examined by simulating different types of fiber pullout tests. These simulations examine the model validity in capturing the fiber–concrete interaction both in the direction orthogonal to the fiber and in the direction of pullout. Also, the proposed model is calibrated and validated by comparing numerical simulations with experimental data from the literature, including fiber pullout under confinement, pullout of hooked fiber, and pullout of inclined fiber that leads to matrix spalling. The good agreement between the simulation results and the experimental data in terms of force versus displacement curve demonstrates the effectiveness of the proposed approach in simulating fiber–concrete interaction. In addition, using the proposed model, the contributions from different toughening mechanisms can be quantitatively compared.

Critical Comparison of Phase-Field, Peridynamics, and Crack Band Model M7 in Light of Gap Test and Classical Fracture Tests

Abstract The recently conceived gap test and its simulation revealed that the fracture energy Gf (or Kc, Jcr) of concrete, plastic-hardening metals, composites, and probably most materials can change by ±100%, depending on the crack-parallel stresses σxx, σzz, and their history. Therefore, one must consider not only a finite length but also a finite width of the fracture process zone, along with its tensorial damage behavior. The data from this test, along with ten other classical tests important for fracture problems (nine on concrete, one on sandstone), are optimally fitted to evaluate the performance of the state-of-art phase-field, peridynamic, and crack band models. Thanks to its realistic boundary and crack-face conditions as well as its tensorial nature, the crack band model, combined with the microplane damage constitutive law in its latest version M7, is found to fit all data well. On the contrary, the phase-field models perform poorly. Peridynamic models (both bond based and state based) perform even worse. The recent correction in the bond-associated deformation gradient helps to improve the predictions in some experiments, but not all. This confirms the previous strictly theoretical critique (JAM 2016), which showed that peridynamics of all kinds suffers from several conceptual faults: (1) It implies a lattice microstructure; (2) its particle–skipping interactions are a fiction; (4) it ignores shear-resisted particle rotations (which are what lends the lattice discrete particle model (LDPM) its superior performance); (3) its representation of the boundaries, especially the crack and fracture process zone faces, is physically unrealistic; and (5) it cannot reproduce the transitional size effect—a quintessential characteristic of quasibrittleness. The misleading practice of “verifying” a model with only one or two simple tests matchable by many different models, or showcasing an ad hoc improvement for one type of test while ignoring misfits of others, is pointed out. In closing, the ubiquity of crack-parallel stresses in practical problems of concrete, shale, fiber composites, plastic-hardening metals, and materials on submicrometer scale is emphasized.

Generation of mesostructure for lattice discrete particle model

A preliminary study of two approaches for the internal structure generation utilized in the lattice discrete particle models (LDPM) [1] is presented. The presented methods used for particle generation and placement are meant to capture the internal structure of materials realistically. The first approach governs the positioning of the generated spherical particles by a gradient field to mimic the casting process. The second approach considers the non-spherical shape of individual particles, i.e. ellipsoidal particles. Therefore, the grain size, angularity, and flakiness can be controlled to match the real grain distribution closely.

A discrete numerical model for the effects of crack healing on the behaviour of ordinary plain concrete: Implementation, calibration, and validation

In the last decade the self-healing of cracks in cementitious materials has been gaining an increasing interest by both the concrete industry and the scientific community. Framed into the Horizon 2020 project ReSHEALience, the present research work aims to formulate a proposal for the numerical modelling of autogenous and stimulated autogenous healing in ordinary plain cement-based materials, whose composition is enriched, in the latter case, with crystalline admixtures. In this paper a meso-scale discrete model that also considers the healing process is presented, relying on the coupling and the enhancement of two models: the Hygro-Thermo-Chemical model, for the simulation of chemical, moisture and heat transport phenomena, and the Lattice Discrete Particle Model, for the mechanical part. The evolution of the healing phenomenon is implemented into the HTC discrete formulation, in order to simulate the degree of crack closure over time. The latter is then employed to capture how the self-repairing affects both moisture permeability and mechanical performances. Finally, the results of a laboratory campaign, carried out at the Politecnico di Milano, are used for calibrating and validating the model presented.

Comparative study of scale effect in concrete fracturing via Lattice Discrete Particle and Finite Discrete Element Models

We study the structural response of concrete weight coating of offshore pipelines subject to various loading conditions by utilizing the Lattice Discrete Particle Model and Finite Discrete Element Method in a comparative manner. We use laboratory experiments to validate our numerical simulations. We discuss the framework in detail, experiment setups, and outline the capabilities and limitations of both methods. A parametric study with a correlation to the size effect is also presented. We simulate Uniaxial Compressive Strength, Brazilian Disk and a three-point bending of a beam with a notch in both methods. We observe that strength and macroscopic fracture patterns fit well between numerical and experimental methods. We discuss the effectiveness of the methods concerning the transition of size effect, as well as comparing to the size effect law. Our statistical analysis quantifies the capability of capturing the size effect in both models. Capturing the size effect helps to increase the accuracy of predicting the structure response and performance which in turn a key factor in durability of concrete structures.

Stochastic lattice discrete particle modeling of fracture in pervious concrete

AbstractThis article presents a stochastic modeling approach for simulating the mechanical behavior of pervious concrete, based on novel extensions of the lattice discrete particle model. Selected digital images of the internal mesostructure, obtained from physical specimens, are used to survey material features and produce statistically representative descriptions of the pore networks. A procedure for estimating the statistical features of the mesostructure is proposed, and samples of a spatially correlated random field are utilized to numerically reproduce the distribution of the large, connected pores in the material. The numerical samples are linked to the topology of the lattice network by means of two novel techniques proposed herein: (1) a random placement procedure for the poly‐sized spheres representing coarse aggregate and (2) a ray tracing technique, which is used to evaluate the effective distributions of mass, stiffness, and strength of each lattice element according to the local distribution of porosity. Numerical results demonstrate how the proposed model is capable of simulating both the large scatter in strength and the variety of failure modes that are observed when testing physical specimens of pervious concrete. The proposed procedure is generally applicable to different types of materials with varying porosity, opening up new possibilities for the simulation of porous media by means of lattice discrete particle modeling techniques.

Introducing fracturing through aggregates in LDPM

We describe simple and practical innovative developments of the Lattice Discrete Particle Model (LDPM) that capture fracturing (failure) through aggregates in concrete. While the original LDPM treats aggregates as rigid bodies that do not fail, for realistic modeling of cracking observed in High Strength Concrete (HSC), the aggregate cracking must be included. We compared the results of our method with the scans of actual cracking and could obtain matching fracturing patterns. Moreover, good fits were found for HSC brittleness when comparing the simulation results with the experimental for models of a three-point bending beam with a notch.

Engineered Cementitious Composites using Chinese local ingredients: Material preparation and numerical investigation

Characterized with remarkable tensile ductility, toughness and multi-cracking behavior under tension, Engineered Cementitious Composites (ECC) with Chinese local ingredients was developed in this paper, aiming to reduce the cost and to match a tensile strain capacity of 4%. Compression, tension and bending tests were carried out to determine the mechanical performance of this composite material. In addition, a numerical study to characterize the tensile and flexural behaviors of ECC specimens was conducted based on Lattice Discrete Particle Model (LDPM), which was recently developed to simulate quasi-brittle materials through a three dimensional assemblage of spherical particles. The effect of fiber reinforcement was investigated using the so-called LDPM-F which accounts for dispersed fibers at the meso-level. Plain mortar specimens without fibers were tested to calibrate the relevant LDPM parameters, while the LDPM-F parameters related to the fiber-matrix interaction were identified by matching three-point bending test of ECC specimens. Results show that the validated ECC numerical model realistically predicts the tension response of dogbone specimens. In addition, typical multi-cracking phenomenon and strain-hardening behavior of ECC are successfully captured. Finally, the effect of fiber length on ECC tensile response was further studied, and the results suggest that the best tensile performance of ECC is obtained for 18-mm-long polyvinyl alcohol (PVA) fiber reinforcement.