Rigid Body Spring Network Research Articles

Developments to the Bonded Block Modeling technique for Discrete Element simulation of transversely isotropic rocks

Developments to the Bonded Block Modeling technique for Discrete Element simulation of transversely isotropic rocks

Rigid‐body‐spring network model for failure simulation of reinforced‐concrete members under various loading rates

AbstractA method was developed to simulate the failure of reinforced concrete (RC) subjected to high‐loading rates using a three‐dimensional (3D) rigid‐body‐spring network (RBSN) model. The viscoplastic damage (VPD) and viscoelastic damage (VED) models were used to consider the rate dependence of the concrete. The rate dependence of the reinforcing bars was expressed using an empirical formula. The reinforcing bars were represented by a computationally efficient, semi‐discrete method within the 3D RBSN that allowed for bar positioning, irrespective of the background mesh used to represent the concrete. The residual deformation of the RC structure was expressed based on considerations of the plastic strains of the reinforcing bars. Various simulations were conducted to verify the developed model, including four‐point bending test simulations of RC beams subjected to high loading rates. A comparison of the result to those of experiments and previous simulations demonstrated the importance of the rate dependence of the reinforcing bar behavior within the simulation framework. Furthermore, RC slabs subjected to high loading rates were simulated to verify that the crack patterns and residual deformation of the RC slabs, as well as the rate dependence of the reinforcing bars, were suitably expressed in comparison with experimental results.

Mesh‐size independent rigid‐body spring network for simulation of the dynamic fracture behavior of concrete

AbstractIn this study, a numerical approach is developed for simulating the rate‐dependent behaviors of concrete without mesh sensitivity. The rheological units (e.g., dashpot) are integrated with the six directional springs in the rigid‐body spring network (RBSN) elements to reflect the rate‐dependent behavior of concrete. Previously, the viscoplastic damage model was associated with the elemental degrees of freedom. However, in the present approach, a viscoelastic constitutive law is newly defined for the normal direction as a function of the strain rate, from which the internal forces can be updated from the regularized elemental stresses. Such improvements are validated through the numerical simulations of the direct tensile test and spalling test using the modified split‐Hopkinson pressure bar (SHPB). The simulated results show consistent structural responses without mesh dependence on the strain rate. In addition, the influences of the softening curve shapes on the crack localizations are examined. It is shown that the rheological parameters can be optimized and determined for consistent applications through virtual experiments. The proposed numerical approach enables the mesh construction procedure without concerning the mesh‐size sensitivity due to the strain rate, and various applications are expected for the simulations of detailed numerical models of concrete materials and structures under high loading rates.

A breakable grain-based model for bi-modular rocks

This article proposes a breakable grain scheme and a method to achieve bi-modular elastic material behavior in grain-based models of rocks using the Hybrid Lattice/Discrete Element Method; a version of the Discrete Element Method that employs Rigid Body Spring Network interactions. First, the paper presents both contributions and two validation exercises regarding bi-modular elastic behavior. Subsequently, a set of comprehensive parametric analyses defines the impact that numerical input parameters have on macroscale results, including: crack initiation and damage stresses; unconfined compression and direct tensile strengths ; and material’s characteristic length. Of note, the analyses investigate the influence of heterogeneity, based on the application of the Weibull probability distribution to compressive and tensile Young moduli and inter and intragranular strength parameters. Afterwards, simulations based on three theoretical rock types show the capabilities of the Hybrid LDEM with the proposed contributions in representing typical macroscale properties of rocks observed in laboratory tests. Lastly, visual comparisons between grain breakage processes from numerical simulations and laboratory thin-section analyses indicate consistent correspondence, which evidences the potential application of the developed methods to future research focused on rocks’ microcracking phenomena. This work concludes that the proposed breakable grains and bi-modular material behavior schemes widen the range of macroscale properties that grain-based models can produce while preserving simplicity. • A novel breakable grain-based model is proposed for bi-modular rocks. • The model is based on the Hybrid Lattice/Discrete Element Method. • Breakable grain approach operates by effectively slicing the rigid particles. • Bi-modular elastic behavior is realized by a fictitious stress scheme. • The methods widen the range of macroscale properties that grain-based models produce.

Hybrid lattice/discrete element method for bonded block modeling of rocks

The Hybrid Lattice/Discrete Element Method is proposed for Bonded Block Modeling of rocks. The method is a combination of the Rigid Body Spring Network Lattice Model with the Discrete Element Method. The main advantages of the new method are: (i) it operates based on either a homogeneous or heterogeneous material assumption, with heterogeneity being represented by means of the Weibull probability distribution applied to elasticity and strength parameters; and (ii) it provides for a direct material calibration scheme for rock models based on both assumptions, which does not require trial-and-error iterations. Thirteen rock types, including igneous, metamorphic, and sedimentary ones were simulated in the two-dimensional version of the method considering both assumptions, and their numerical macroscale properties were compared to the laboratory ones. Results show that while deformability and peak strength properties were accurately represented by the homogeneous models, linear elastic-brittle behavior manifested. On the other hand, the heterogeneous models presented non-linear stress–strain behavior and provided reasonable matches for the following laboratory properties: Young modulus, Poisson’s ratio, crack initiation stress, crack damage stress, unconfined compression strength, and direct tensile strength.

Lattice modelling of gravity and stress-driven failures of rock tunnels

Abstract In this paper, an investigation is performed regarding the applicability of the Rigid Body Spring Network method (RBSN) to the analysis of hard rock tunnels in massive and fractured rock masses. For this objective, the method is enhanced with a new meshing scheme and a Cohesive Zone Model (CZM) formulation based on an exponential softening law. While the new meshing scheme provides for models containing complex Discrete Fracture Networks, the CZM affords realistic representations of fracturing phenomena. In order to demonstrate the robustness of the contributions, a verification model and three study cases are presented. The verification concerns the analysis of an elastic jointed medium with a circular hole, from which stresses and displacements are calculated and checked against Finite Element solution. The study cases are based on the real Monte-Seco and Mine-By tunnels and on a hypothetical tunnel. Simulations are performed, and the results are compared to either observed in situ behavior or Finite Element analyses. In the end, the cases show that the enhanced RBSN can realistically represent both gravity and stress-driven types of failures that are encountered in hard rock tunneling. Additional results highlight the method’s simple material calibration procedure and the practicality of the new meshing scheme.

Simulating hydraulic fracturing processes in laboratory-scale geological media using three-dimensional TOUGH-RBSN

Abstract In this context, recent developments in the coupled three-dimensional (3D) hydro-mechanical (HM) simulation tool TOUGH-RBSN are presented. This tool is used to model hydraulic fracture in geological media, as observed in laboratory-scale tests. The TOUGH-RBSN simulator is based on the effective linking of two numerical methods: TOUGH2, a finite volume method for simulating mass transport within a permeable medium; and a lattice model based on the rigid-body-spring network (RBSN) concept. The method relies on a Voronoi-based discretization technique that can represent fracture development within a permeable rock matrix. The simulator provides two-way coupling of HM processes, including fluid pressure-induced fracture and fracture-assisted flow. We first present the basic capabilities of the modeling approach using two example applications, i.e. permeability evolution under compression deformation, and analyses of a static fracturing simulation. Thereafter, the model is used to simulate laboratory tests of hydraulic fracturing in granite. In most respects, the simulation results meet expectations with respect to permeability evolution and fracturing patterns. It can be seen that the evolution of injection pressure associated with the simulated fracture developments is strongly affected by fluid viscosity.

Behaviour of BH Girder Composed of CIP and Precast concrete Subjected to Flexural Loading

This paper presents the behaviour of reinforced BH girder which is made of precast, and cast in place (CIP) subjected to flexural loading (bending). The girder is made of two types of concrete materials, precast and cast in place (CIP), which has different strengths. The girder is simply supported and loading force is given at the mid span of the beam during analysis. The main objective of the analysis and modelling is to studying the combined effect of CIP, precast and the reinforcing materials. Modelling process is carried out using Rigid-Body-Spring-Network (RBSN) and Abaqus commercial software. Cracking and failure behaviour of the girder is investigated. Performance of the BH girder (precast and CIP) is estimated based on the load carrying capacity. Deflection of the composite is measured along the span of the girder. From the analysis result; the ductility, load carrying capacity and crack initiation and crack propagation behaviour of BH girder is investigated.

Elastically-homogeneous lattice modelling of transversely isotropic rocks

Abstract In this paper, the Rigid Body Spring Network method is further developed to perform analyses of transversely isotropic rocks. Two contributions are proposed: (1) an extension of the fictitious stress approach and (2) the combined application of the Smooth Joint Model and a discontinuous failure criterion. While the first contribution enables the numerical method to perform elastically homogeneous analysis of transversely isotropic materials, the second provides for the proper manifestation of sliding failure on a discontinuity. In order to verify the competence of the contributions, validation analyses are performed and a case study is presented based on an argillite rock.

Extended Rigid Body Spring Network method for the simulation of brittle rocks

Abstract In this paper, the Rigid Body Spring Network method is extended in order to provide for a realistic representation of the fracturing processes of brittle rocks. To achieve this objective, three new contributions are given to the method: a Cohesive Zone Model; a representation of rock heterogeneity; and the consideration of preexisting natural microcracks. In order to verify the competence of the contributions proposed, a case study is presented based on a granite rock. Conventional laboratory tests are numerically performed and the results are compared to those obtained in laboratory in order to show the performance of the proposed model.

A discrete numerical method for brittle rocks using mathematical programming

A computational formulation of discrete simulations of damage and failure in brittle rocks using mathematical programming methods is proposed. The variational formulations are developed in two and three dimensions. These formulations naturally lead to second-order cone programs and can conveniently be solved using off-the-shelf mathematical programming solvers. Pure static formulations are derived so that no artificial damping parameters are required. The rock is represented by rigid blocks, with interfaces between blocks modelled by zero-thickness springs based on the rigid-body–spring network method. A modified Mohr–Coulomb failure criterion is proposed to model the failure of the interfaces. When the interface’ strength limits are reached, a microscopic crack forms and its strength is irreversibly lost. The microscopic elastic properties of the springs are related to the observed elastic behaviour of rocks with the developed empirical equations. The program is first validated with three simple tests. Then, numerical uniaxial and biaxial compression tests and the Brazilian tests are conducted. Furthermore, the proposed approach is employed to study the rock crack propagation and coalescence using cracked Brazilian disc test. The results are in good agreements with reported experimental data, which shows its potential in modelling mechanical behaviour of brittle rocks.

TOUGH–RBSN simulator for hydraulic fracture propagation within fractured media: Model validations against laboratory experiments

This paper presents coupled hydro-mechanical modeling of hydraulic fracturing processes in complex fractured media using a discrete fracture network (DFN) approach. The individual physical processes in the fracture propagation are represented by separate program modules: the TOUGH2 code for multiphase flow and mass transport based on the finite volume approach; and the rigid-body-spring network (RBSN) model for mechanical and fracture-damage behavior, which are coupled with each other. Fractures are modeled as discrete features, of which the hydrological properties are evaluated from the fracture deformation and aperture change. The verification of the TOUGH–RBSN code is performed against a 2D analytical model for single hydraulic fracture propagation. Subsequently, modeling capabilities for hydraulic fracturing are demonstrated through simulations of laboratory experiments conducted on rock-analogue (soda-lime glass) samples containing a designed network of pre-existing fractures. Sensitivity analyses are also conducted by changing the modeling parameters, such as viscosity of injected fluid, strength of pre-existing fractures, and confining stress conditions. The hydraulic fracturing characteristics attributed to the modeling parameters are investigated through comparisons of the simulation results.

Multiscale probabilistic modeling of a crack bridge in glass fiber reinforced concrete

The present paper introduces a probabilistic approach to simulating the crack bridging effects of chopped glass strands in cement-based matrices and compares it to a discrete rigid body spring network model with semi-discrete representation of the chopped strands. The glass strands exhibit random features at various scales, which are taken into account by both models. Fiber strength and interface stress are considered as random variables at the scale of a single fiber bundle while the orientation and position of individual bundles with respect to a crack plane are considered as random variables at the crack bridge scale. At the scale of the whole composite domain, the distribution of fibers and the resulting number of crack-bridging fibers is considered. All the above random effects contribute to the variability of the crack bridge performance and result in size-dependent behavior of a multiply cracked composite.

Plane stress problems using hysteretic rigid body spring network models

In this work, a discrete numerical scheme is presented capable of modeling the hysteretic behavior of 2D structures. Rigid Body Spring Network (RBSN) models that were first proposed by Kawai (Nucl Eng Des 48(1):29–207, 1978) are extended to account for hysteretic elastoplastic behavior. Discretization is based on Voronoi tessellation, as proposed specifically for RBSN models to ensure uniformity. As a result, the structure is discretized into convex polygons that form the discrete rigid bodies of the model. These are connected with three zero length, i.e., single-node springs in the middle of their common facets. The springs follow the smooth hysteretic Bouc–Wen model which efficiently incorporates classical plasticity with no direct reference to a yield surface. Numerical results for both static and dynamic loadings are presented, which validate the proposed simplified spring-mass formulation. In addition, they verify the model’s applicability on determining primarily the displacement field and plastic zones compared to the standard elastoplastic finite element method.

Impact of aggregate properties on the development of shrinkage-induced cracking in concrete under restraint conditions

Abstract Concrete specimens were prepared with the same mixture proportion except for their constituent coarse aggregates, namely, limestone and sandstone, that possess different inherent drying shrinkage values. The strain at a cross-section perpendicular to the drying direction under restricted and unrestrained conditions was observed using a digital image correlation method. It was confirmed that initiation and propagation of cracks were greatly affected by the types of aggregate. Analysis by the rigid-body spring network method, which explicitly incorporates aggregate shrinkage and properties of the interfacial transition zone (ITZ), revealed that aggregate shrinkage and fracture energy of the ITZ greatly influence fine-crack distribution and localization of cracks. It was elucidated that pure limestone with a small drying shrinkage value can reduce the number of visible cracks in concrete under restraint conditions since it allows fine cracks to form around coarse aggregate particles that absorb the localization of cracks, thus limiting wider cracks.

콘크리트 재료의 동적 물성 변화를 모사하기 위한 유변학적(Rheological)모델 개발 및 평가

In this study, the rheological models were introduced and developed to reflect rate dependent tensile behaviour of concrete. In general, mechanical properties(e.g. strength, elasticity, and fracture energy) of concrete are increased under high loading rates. The strength of concrete shows high rate dependency among its mechanical properties, and the tensile strength has higher rate dependency than the compressional strength. To simulate the rate dependency of concrete, original spring set of RBSN(Rigid-Body- Spring-Network) model was adjusted with viscous and friction units(e.g. dashpot and Coulomb friction component). Three types of models( 1) visco-elastic, 2) visco-plastic, and 3) visco-elasto- plastic damage models) are considered, and the constitutive relationships for the models are derived. For validation purpose, direct tensile test were simulated, and characteristics of the three different rheological models were compared with experimental stress-strain responses. Simulation result of the developed visco-elasto-plastic damage(VEPD) model demonstrated well describing and fitting with experimental results.

Hydro-mechanical model for wetting/drying and fracture development in geomaterials

This paper presents a modeling approach for studying hydro-mechanical coupled processes, including fracture development, within geological formations. This is accomplished through the novel linking of two codes: TOUGH2, which is a widely used simulator of subsurface multiphase flow based on the finite volume method; and an implementation of the Rigid-Body-Spring Network (RBSN) method, which provides a discrete (lattice) representation of material elasticity and fracture development. The modeling approach is facilitated by a Voronoi-based discretization technique, capable of representing discrete fracture networks. The TOUGH–RBSN simulator is intended to predict fracture evolution, as well as mass transport through permeable media, under dynamically changing hydrologic and mechanical conditions. Numerical results are compared with those of two independent studies involving hydro-mechanical coupling: (1) numerical modeling of swelling stress development in bentonite; and (2) experimental study of desiccation cracking in a mining waste. The comparisons show good agreement with respect to moisture content, stress development with changes in pore pressure, and time to crack initiation. The observed relationship between material thickness and crack patterns (e.g., mean spacing of cracks) is captured by the proposed modeling approach.

Rigid-Body-Spring Network with Visco-Plastic Damage Model for Simulating Rate Dependent Fracture of RC Structures

The mechanical properties of concrete materials vary with the loading rate underdynamic conditions, which can influence the dynamic fracture behavior of structures. The ratedependency is reported as due to the microscopic mechanisms, such as a material inertia effectand the Stefan effect. In this study, the rigid-body-spring network (RBSN) is employed forthe fracture analysis, and the visco-plastic damage model is implemented to represent the rateeffect in this macroscopic simulation framework. The parameters in the Perzyna type visco-plastic formulation are adjusted through the direct tensile test with various loading rates asa preliminary calibration. As the loading rate increases, the strength increase is presented interms of the dynamic increase factor (DIF), and compared with the experimental and empiricalresults. Next, the flexural beam test is conducted for plain and reinforced concrete beams underslow and impact rates of loading. At the failure stage, different crack patterns are observeddepending on the loading rate. The impact loading induces the failure to be more localizedon the compressive zone of the beam, which is due to rather the rate dependent materialfeatures. In structural aspects, the reinforcement exerts stronger effects on reducing crack widthand improving ductility at the slow loading rate. The ductility is also evaluated through thecomparison of load-deformation curves until the final rupture of the beams. This study canprovide understandings of the structural rate dependent behavior and the reinforcing effectunder dynamic loadings.

Modeling of phase interfaces during pre-critical crack growth in concrete

Abstract Recent developments in the Rigid-Body-Spring Network (RBSN) modeling of concrete are presented, with candid assessments of research needs for the modeling of pre-critical crack growth under multi-axial stress conditions. Fracture of micro-concrete specimens is investigated through the combined use of such discrete (lattice) models, X-ray tomography, and imaging techniques. The specimens contain small amounts (10–50 wt.% fraction) of spherical glass aggregates. Acid-etching of the glass aggregate surfaces is done to vary the bond properties of the matrix–aggregate interface. Tomographic images of the unloaded specimens provide the initial configurations of the three-dimensional lattice models, which are based on a three-phase representation of the material meso-structure: fine-grained mortar matrix, glass aggregate inclusions, and matrix–aggregate interfaces. The numerical and physical test results agree well with respect to peak loads and the associated crack patterns. Material structure and interface properties affect pre-critical cracking, in accordance with expectations. Dependence of composite strength on aggregate content and arrangement is studied through simulations of large sets of nominally identical models, which differ only in random positioning of the aggregates.

Effects of Material and Structural System on Drying Shrinkage Behavior

Abstract This article aims to investigate the effects of material and structural system, specifically concrete strength and restrained level on drying shrinkage via the time dependent model. This model consists of Rigid-Body-Spring Network (RBSN) and modified solidification concept. In the analysis, the series of each effect were analyzed and investigated by varying concrete strengths and restrained levels represented for material and structural system, respectively. As the results, the time that the first crack occurs is not in direct proportion to the increase in concrete strength and restrained level but they have the same tendency. Therefore, the proper value of concrete strength and moisture humidity should be examined before designing the structure. Furthermore, it is not suitable to apply comparative results between tensile stress and tensile stress when basing a prediction on the cracking times, since tensile stress in the specimens is lower than the tensile strength at the cracking period. Also, stress histories and accumulated permanent damages in early concrete were founded to be the major effect of concrete cracking.