Ordinary State-based Peridynamics Research Articles

Elastoplastic Constitutive Modeling for Reinforced Concrete in Ordinary State-Based Peridynamics

ABSTRACTA non-local, ordinary state-based, peridynamic elastoplastic model is formulated to numerically simulate the fracture of reinforced concrete materials. Several basic definitions are first discussed to avoid confusion; and then, a detailed derivation of the force vector state is presented, leading to a unified expression of force state for one-, two- and three-dimensional elasticity problems. Furthermore, an ordinary state-based peridynamic (OSB PD) elastoplastic analysis approach is developed for both plastic compressible and incompressible materials, including the constitutive relationship, the yield function, the consistency condition and the plasticity flow rule. The peridynamic predictions of a quasi-static deformation of the steel rods are in good agreement with the analytical solution. Moreover, the OSB PD plasticity is verified by analyzing a square plate with or without a central hole suffering different loading-unloading paths. Finally, a two dimensional reinforced concrete clamped beam subjected to impact loading is simulated with the proposed OSB PD elastoplasticity, demonstrating its capability in capturing the damage characteristics and structural failure behavior. Simulation results show good accuracy of the peridynamics in simulating elastoplastic problems.

An in-depth investigation of critical stretch based failure criterion in ordinary state-based peridynamics

This study presents an in-depth investigation of the critical stretch based failure criterion in ordinary state-based peridynamics for both static and dynamic conditions. Seven different cases are investigated to determine the effect of the failure parameter on peridynamic forces between material points and dilatation. Based on crack opening displacement (COD) results from both peridynamics and finite element analysis, it was found that one of the seven cases provides the best agreement between the two approaches. This particular case is further investigated by considering the influence of the discretisation and the horizon sizes on COD and crack propagation speeds. Moreover, PD predictions of COD for PMMA material is analysed with the theory of dynamic fracture mechanics and compared with the fracture experiments. It is shown that the peridynamic model can correctly model, simulate and predict the behaviour of the crack under different loading conditions. Furthermore, the presented PD models capture accurate fracture phenomena, specifically the crack path, branching angles and crack propagation speeds, which are in good agreement with experimental results.

Fluid-elastic structure interaction simulation by using ordinary state-based peridynamics and peridynamic differential operator

The fluid-structure interaction phenomenon is often encountered in the ocean engineering field. In the present work, a non-local numerical model is developed for the simulations of weakly compressible viscous fluid and elastic structure interactions. The peridynamic theory is adopted for both the structure and fluid modelling. The elastic structure is described by using the ordinary state-based peridynamics, while the fluid is modelled by utilizing the peridynamic differential operator. Furthermore, the updated Lagrangian description is adopted for the fluid including the relative deformation gradient expressed by the peridynamic differential operator. The fluid-structure interface and its normal direction are calculated via the gradient of a colour function, which varies with the fluid motion and structure deformation. Besides, the interaction force exerted from fluid to structure is constrained to be always perpendicular to the moving interface. Hence the fluid motion and structural deformation are predicted simultaneously. The validation of the developed model is conducted through the simulation of a water dam break with a rubber gate. The good agreement between the peridynamic and the experiment results demonstrates the capability of the current model for solving fluid-elastic structure interaction problems.

Determination of horizon size in state-based peridynamics

Peridynamics is based on integro-differential equations and has a length scale parameter called horizon which gives peridynamics a non-local character. Currently, there are three main peridynamic formulations available in the literature including bond-based peridynamics, ordinary state-based peridynamics and non-ordinary state-based peridynamics. In this study, the optimum horizon size is determined for ordinary state-based peridynamics and non-ordinary state-based peridynamics formulations by using uniform and non-uniform discretisation under dynamic and static conditions. It is shown that the horizon sizes selected as optimum sizes for uniform discretisation can also be used for non-uniform discretisation without introducing significant error to the system. Moreover, a smaller horizon size can be selected for non-ordinary state-based formulation which can yield significant computational advantage. It is also shown that same horizon size can be used for both static and dynamic problems.

Hybrid FEM and peridynamic simulation of hydraulic fracture propagation in saturated porous media

This paper presents a hybrid modeling approach for simulating hydraulic fracture propagation in saturated porous media: ordinary state-based peridynamics is used to describe the behavior of the solid phase, including the deformation and crack propagation, while FEM is used to describe the fluid flow and to evaluate the pore pressure. Classical Biot poroelasticity theory is adopted. The proposed approach is first verified by comparing its results with the exact solutions of two examples. Subsequently, a series of pressure- and fluid-driven crack propagation examples are solved and presented. The phenomenon of fluid pressure oscillation is observed in the fluid-driven crack propagation examples, which is consistent with previous experimental and numerical evidences. All the presented examples demonstrate the capability of the proposed approach in solving problems of hydraulic fracture propagation in saturated porous media.

Experiment and peridynamic simulation on cracking behavior of red sandstone containing a single non-straight fissure under uniaxial compression

It is widely accepted that the strength, deformability, and failure behaviors of rock material are strongly influenced by widespread natural fissures. Research focusing on the crack initiation, propagation, and coalescence behavior in fissured rock material is important to ensure the reliability and predictability of rock engineering. However, the types and forms of pre-existing fissures are complex and variable; thus, mechanical experiments have rarely been conducted on rock samples containing non-straight prefabricated open fissures. In this research, the strength, deformability, and failure behaviors of sandstone samples containing a single non-straight fissure were systematically investigated by a series of uniaxial compression experiments and numerical simulations. First, the influence of the fissure angle (α) on the strength and deformation characteristics of red sandstone containing a single non-straight fissure was evaluated. Then, the ordinary state-based peridynamics model was used to investigate the fracture behavior of red sandstone specimens with respect to various α values. The results indicate that for α = 0°–15° and 75°–90°, tensile cracks first initiated from the convex points of the prefabricated fissure, and the crack initiation position transferred from the convex point to the tip of the prefabricated fissure for α = 30°–60°. In addition, for the α = 0°–90°, the cracks initiated in the form of tensile cracks, and the ultimate failure modes were mixed tensile and shear failure. Secondary cracks always appeared in the form of shear crack. Moreover, for α ≥ 75°, fewer cracks were initiated before the peak strength was reached; afterward, the crack initiations were more prolific, and more acoustic emission counts were observed.

Influence of Different Types of Small-Size Defects on Propagation of Macro-cracks in Brittle Materials

The presence of defects in the structure requires noticeable attention and understanding of fracture mechanisms in brittle materials has to be established. Defects in the form of holes, macro- and micro-cracks are the main interest of this paper. This work investigates the dual role of holes and micro-crack arrays on toughening and degradation mechanisms in concrete structures. An ordinary state-based peridynamics (PD) model is utilized to analyse the fracture problem at the micro-level. The application of PD shows its advantage in crack-hole, macro- and micro-crack interaction problems since PD can accurately predict the contribution of defects on structural behaviour. The study of the three-point bending problem with five types of holes existing in the structure showed the crack arrest phenomena at the hole boundary and the “attraction” of the crack to propagate towards the hole. For the study of the macro- and micro-cracks interaction problem, various cases of the micro-crack distribution and inclination angles are considered and validated with analytical studies. The PD quasi-static simulations show good agreement with analytical solutions. Moreover, PD dynamic solutions show the capability of PD to capture complex crack propagation paths. It is observed that the presence of micro-cracks and holes ahead of the main crack can suppress its further propagation as well as have an influence on the crack propagation direction. The numerical results demonstrate the efficiency of the PD modelling of multiple crack interaction problems.

Dynamic fracture analysis of functionally graded materials using ordinary state-based peridynamics

Functionally graded materials are regarded as a special kind of composites capable of eliminating material interfaces and the delamination problems. Stress discontinuity can be avoided owing to smooth composition of the functionally graded ingredients. In this study, a recently emerged effective non-local continuum theory for solving fracture problems in solids and structures, peridynamics, is employed to simulate dynamic wave propagation as well as crack propagation in functionally graded materials. Specifically, the ordinary state-based formulation is adopted. The ordinary state-based formulation is slightly modified for the modelling of functionally graded materials. The averaging technique is employed to determine peridynamic parameters associated with the material properties. Firstly, a benchmark problem is considered to validate the present implementation of ordinary state-based peridynamics for brittle fracture of homogeneous materials. Then, the wave propagation in the functionally graded materials under impact loading is simulated. Finally, dynamic crack propagation in the functionally graded materials is studied. The evaluated crack paths and the displacement waves are compared with reference works including numerical and experimental results. Good agreement between the reference and present results is achieved. It is shown that a simple modification of ordinary state-based formulation has led to simulate dynamic fracture of functionally graded materials.

Seismic Response Analysis for Ordinary State-Based Peridynamics in a Linear Isotropic Elastic Material

Earthquakes may induce the failure of soil-structures built with a geo-material. It is therefore a crucially important aspect to study how a crack initiates and propagates in these soil-structures to analyze and improve their earthquake resistance. Peridynamics, although a powerful method for such an analysis, has not been applied to seismic response analysis of soil-structures on a viscous boundary surface that can consider an input seismic wave. To apply seismic response analysis to ordinary state-based peridynamics (OSB-PD), this paper proposed a fictitious viscous boundary layer and Rayleigh damping components constructed with time derivative of force density vectors. OSB-PD was semi-verified through the comparison of calculated response acceleration via the proposed method and via the dynamic finite element method (FEM). The discussions were stressed on the effect of Poisson’s ratio, the frequency, and the Rayleigh damping components relative to the calculated response acceleration in both methods. Moreover, the seismic response acceleration at the top of the analysis model was calculated through the proposed method using an actual seismic acceleration time history, and then the result was compared with that calculated by the dynamic FEM. The results of both methods in the seismic response analysis agreed well with each other. From these results, we verified that the fictitious viscous layer can represent incoming/outgoing acceleration wave from the bottom of the analysis domain, and that Rayleigh damping can be expressed using the time derivative of force density vectors.

Coupling of peridynamics with finite elements for brittle crack propagation problems

In this work, a new coupling model of finite element method (FEM) and ordinary state-based peridynamics (OSBPD) is proposed to accurately predict brittle fracture under quasi-static and dynamic loading. PD grids are directly coupled with FEM meshes without an overlapped zone. The transferred forces between PD grids and FEM meshes are reciprocal and balanced inherently, which avoids the complexity of the transitional zone-based coupling models and the non-equilibrium phenomenon of the structural force reported in the sharing nodes-based coupling model. More importantly, the proposed coupling model is free of significant influence of ghost forces and significant wave reflection at the coupling interface. Also, the surface effect of PD can be approximately removed. The accuracy of the proposed coupling model is verified through comparing with the FEM results and pure OSBPD results.

Dynamic crack arrest analysis by ordinary state-based peridynamics

Dynamic fracture analysis for the crack arrest phenomenon is performed by ordinary state-based peridynamics formulation and discretization employing transition bond concept. Double cantilever beam specimen is chosen for our numerical evidence purpose. The analysis consists of two main phases namely, generation and application (prediction) phases. In the generation phase, the dynamic stress intensity factors of propagating and arrested cracks are estimated by the present formulation for given crack path histories, and a good agreement is achieved. As for the application phase, dynamic stress intensity factors as well as total crack lengths after crack arrests are in good agreement with the experiments. Moreover, the influence of transition bond concept on the crack arrest behavior is investigated and it is found that the transition bond is very efficient in the simulation of the crack arrest problem such that premature arrests of cracks are observed without transition bond cases.

Ordinary state-based peridynamics for nonlinear hardening plastic materials' deformation and its fracture process

In this paper, a modelling for ordinary state-based peridynamics to simulate plastic deformation is presented. This modelling can simulate nonlinear-hardening plastic mode, what's more, the perfect plastic model and linear-hardening plastic model can also be simulated in a simple way too. The expression of equivalent plastic stress in the formulation of nonlocal ordinary state-based peridynamics is established. The yield function is constructed based on the Von Mises yield criteria. Then the way to calculate the equivalent plastic stress in every step is established. And all the formulations have been discussed both in 2D case and 3D case. The validity of this modelling is established by comparing the results of a plate under tension or pushing simulated by peridynamics and FEM in 2D case, and the plate under tension is the plastic material with power-hardening law, the plate under pushing is the plastic material with linear-hardening law.

A modified axisymmetric ordinary state-based peridynamics with shear deformation for elastic and fracture problems in brittle solids

Although axisymmetric problems are neither the plane stress nor plane strain problems, they can be solved in the two-dimensional (2-D) domain for convenience. In this paper, a modified axisymmetric ordinary state-based peridynamic model considering the shear deformation for linear elastic solids is proposed. By subtracting the rigid body rotation part from the total deformation, the shear deformation of a bond is considered. The bond force density vector is derived indirectly by equalizing the PD energy density and the classical strain energy density. The fictitious density in adaptive dynamic relaxation (ADR) method is derived. Two new damage criterions based on the maximum stretch/deviatoric strain are proposed. Several axisymmetrically three-dimensional (3-D) numerical examples are performed in the computational domains, where the present numerical results are compared with the existing analytical solutions, the classical finite element numerical results and axisymmetric ordinary state-based peridynamic results. The numerical results demonstrate this proposed method can simulate the initiation and propagation of cracks very well under the axisymmetrical conditions, and it has more accuracy than axisymmetric ordinary state-based peridynamic model.

Coupling of FEM and ordinary state-based peridynamics for brittle failure analysis in 3D

This paper presents a method to couple FEM meshes with ordinary state-based peridynamic grids. It generates a computational tool that is as efficient as computer methods based on classical continuum mechanics and as flexible as peridynamics in dealing with crack propagation. Moreover, it removes most of the problems due to the surface effects typical of nonlocal methods such as OSB-PD. The adaptive dynamic relaxation approach is used to solve linear elastic and crack propagation problems. The method is successfully applied to a complex crack propagation case in 3 D and is able to accurately reproduce the shape of the experimentally observed crack.

Peridynamic simulation of thermal failure behaviors in rocks subjected to heating from boreholes

In this paper, a weakly coupled thermo-mechanical model within the framework of ordinary state-based peridynamics is proposed to investigate thermal cracking behaviors of rocks subjected to heating from boreholes. The weakly coupled thermo-mechanical ordinary state-based peridynamic model is decomposed into two parts, i.e., thermal conduction and mechanical deformation. In the first part, temperature distributions in solids are analyzed based on the heat conduction equations. While, in the second part, thermally-induced deformation and fracture can be simulated by the mechanical computations. Moreover, a multi-rate explicit time integration scheme is proposed to model thermal cracking phenomena in rocks to overcome different time-scale problems in multi-physical systems. A benchmark example with analytical solutions is firstly modeled to validate the correctness and accuracy of the proposed numerical method, and numerical convergence studies of the weakly coupled thermo-mechanical ordinary state-based peridynamics are also conducted. The present numerical results are in good agreement with the analytical solutions and the previous experimental data. Then, three numerical examples are performed to investigate thermally-induced cracking behaviors, i.e. cracking patterns and temperature and stress evolutions. The proposed numerical method not only provides a new tool for coupled thermal-mechanical fracture problems in geothermal engineering, but also reveals the mechanism of thermal cracking phenomena in rocks.

An improved ordinary state-based peridynamic model for cohesive crack growth in quasi-brittle materials

Ordinary state-based peridynamics (OSBPD) is a well-established peridynamic (PD) model, but its applications are restricted in brittle fracture because the failure models proposed in OSBPD are mainly to describe brittle fracture rather than quasi-brittle fracture. This work attempts to propose a damage model to overcome this limitation for the investigation of damage and crack propagation in quasi-brittle materials. To this aim, a PD version of the cohesive zone model (CZM) is established in OSBPD for the fracture analysis of quasi-brittle materials. A degradation curve is proposed in the damage model to represent the cohesive effect in fracture process zone (FPZ), which is similar to the tension softening curve in CZM. The damage model can be uniquely determined once the tensile strength, the elastic modulus, the specific fracture energy, and tension softening constitutive law are given. Moreover, several representative examples of mode-I and mixed mode concrete fractures are simulated using OSBPD to validate the proposed damage model. The predicted load-displacement curves and crack paths in all the cases are in good agreement with the experimental results or other numerical results. In addition, a re-loading approach is proposed to obtain the descending part of load-displacement curve when the load-controlled loading procedure is adopted.

Crack propagation and arrest modeling in peridynamics

Dynamic fracture and crack propagation phenomena are unstable, and especially brittle fracture of relatively thick plates in ship structures for large container ships has been remarked recently. Numerical simulation is convenient and powerful method for designing and evaluating the safety of structures. The recently developed ordinary state-based peridynamics (OSPD) is further enhanced to study elastodynamic propagating crack based on the dynamic stress intensity factors (DSIFs). Dynamic fracture analysis for simulating crack arrest phenomenon is performed by OSPD is discretization employing transition bond concept. Double-cantilever-beam specimen is chosen as the numerical model. In application phase, dynamic stress intensity factors as well as total crack lengths after crack arrests are in good agreement with the experiments and reference solution.

Evaluation of Dynamic Stress Intensity Factors Using Varying Horizon Size in Ordinary State-Based Peridynamics

Evaluation of Dynamic Stress Intensity Factors Using Varying Horizon Size in Ordinary State-Based Peridynamics

A computational approach based on ordinary state-based peridynamics with new transition bond for dynamic fracture analysis

The recently developed ordinary state-based peridynamics (OSPD) is further enhanced to study elastodynamic propagating crack based on the dynamic stress intensity factors (DSIFs). The displacement discontinuity such as a crack surface is represented by a bond-failure. Variations of the mixed-mode DSIFs with time are evaluated by the interaction integral method for the dynamic crack propagation. In terms of OSPD fracture modeling, numerical oscillation of DSIFs becomes a critical issue during the evolution of a crack. To overcome this numerical oscillation problem, we introduce a new model of bond-failure, the transition bond. The enhanced OSPD approach using the new transition bond model offers accurate and acceptable results, suppressing the numerical oscillation of responses and reflecting an effective approach. The effects of different types of transition bond are numerically analyzed. Accuracy of the DSIFs is examined employing the various damping parameters and effectiveness of the new PD fracture model is verified. The Kalthoff-Winkler impact test is considered for evaluating the mixed-mode DSIFs and the crack paths.

An axisymmetric ordinary state-based peridynamic model for linear elastic solids

In this study, a new axisymmetric ordinary state-based peridynamic (PD) model for axisymmetric problems of linear elastic solids is presented. A fracture criterion based on the PD bond energy density is proposed. Adaptive dynamic relaxation (ADR) method is adopted to obtain equilibrium solutions, and a viable fictitious density of the model is derived and proven to be valid for implementation in the ADR method. Performance and validity of the proposed axisymmetric PD model are demonstrated by three kinds of numerical problems, i.e., compression tests, pull-out deformation, and indentation fracture. In the compression tests with focus on constant-strain deformations, the displacements predicted by the present model are compared with classical analytical solutions, and the results show good agreements. Both m- convergence and δ-convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases. The recovery of the Poisson’s ratios in the model is tested in detail as well. The capability of capturing general non-uniform axisymmetric deformation by the proposed model is verified in the pull-out analysis. The equilibrium displacement fields predicted by the present model agree very well with those by the finite element method. The peridynamic evaluation of strains and stresses also show good match with the finite element ones. The proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data. The model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state-based peridynamics.