Bond-based Peridynamics Research Articles

An extended fictitious node method for surface effect correction of bond-based peridynamics

The surface effect of peridynamics seriously affects the accuracy of numerical results near the boundary. The fictitious node method can modify the surface effect of peridynamics, but the previous method (Le and Bobaru, 2018) is only applicable to problems with only stress boundary conditions. This paper proposes an extended fictitious node method for deriving the fictitious node displacement constraint equations that apply to the displacement boundary conditions, based on the corresponding equivalence of stress components along with the boundary normal and tangential directions on the left and right sides of the boundary node. Consequently, the extended fictitious node method can handle problems with mixed boundary conditions of stress and displacement constraints. Then, this method is verified through numerical examples with mixed boundary conditions. The results after the surface correction show good agreements with those of the finite element method or with analytical solutions.

Refined Simulation of Cracked Reinforced Concrete Beams Based on Enhanced Bond-Based Peridynamics

Peridynamics (PD) has been widely used in simulating the crack behaviors of brittle material due to its extraordinary capacity in analyzing deformations with discontinuities. In this paper, an enhanced bond-based peridynamics (BPD) model is proposed to study the crack behaviors of reinforced concrete (RC) beams. A modified nonlinear concrete bond model is presented to simulate the tensile-compressive softening and improve the model’s convergence. A novel steel bond model is developed considering the nonlinear behaviors in both axial and shear directions. A coupled axial–shear interaction (ASI) model is adopted to simulate the bond-slip behaviors between concrete and steel. Furthermore, reformulated BPD equilibrium equations are modified for implicit static analyses. A gradually weakening fictitious element (GWFE) approach is presented to improve the stability in implicitly solving the BPD equations. The algorithms are implemented in an open-source finite element (FE) software, OpenSees, for BPD analysis. Experiments are conducted for three RC beams with different shear span to depth ratios under concentrated vertical loads to verify the enhanced BPD model. The comparative studies are performed between experimental and simulated results, and nonlinear responses of RC beam are investigated, e.g. the responses of deflection-shear force, strain distribution on stirrups, the shear resistance, and the propagation of cracks. The results show that as the shear span-to-effective depth ratio increases, the capacity provided by RC beam decreases while stirrups gradually provide more capacity instead. It is also found that the yield deflection increases significantly with the growing shear span-to-effective depth ratio.

Peridynamic investigation of dynamic damage behaviors of PBX confined in spherical steel shells

The bond-based peridynamics (PD) modeling is employed to investigate the dynamic damage response of polymer bonded explosive (PBX) confined in spherical steel shells under impact loading. The softening behavior of PBX is reasonably well captured by a bilinear constitutive model defined for PD bonds. The simulation results indicate that the damage behaviors of PBX depend on the impact speed. At the low impact speed, the damage of PBX is characterized by the absence of the longitudinal cracks. In contrast, the presence of longitudinal cracks leads to the serious damage of spherical shell of PBX at the high impact speed. The dynamics of stress wave such as reflection and transmittance at the solid boundary are analyzed based on the stress wave theory, providing a comprehensive understanding of how the external confinement affects dynamic damage response of PBX. The effects of Young's modulus of steel shells on the damage of PBX are also examined. The simulation results illustrate that the damage of PBX depends on the property of the external steel shell, but is almost independent on the property of the internal steel shell. These interesting phenomena can be well understood based on the dissipation of impact energy of the flyer. The presented computational results offer rigorous basis for the full understanding of the dynamic damage response of PBX in the more complicated confinement conditions.

Peridynamics simulation of impact failure in glass plates

This paper aims at presenting numerical simulation of dynamic failure process of Duran 50 glass plates subjected to high-velocity impact loads using a bond-based peridynamics (PD). In simulation, several affecting parameters including impact velocity, impact angle, impact contact area, glass plate thickness, and porosity percentage in glass plate are considered. The effects of these parameters on the damage, reaction force, and coefficient of restitution are examined. The functional relationship among the reaction force on bullet, the bullet velocity and the plate thickness is established. In addition, the functional relationship among the damage percentage of plate, the bullet velocity and the plate thickness is constructed. Thus, the reaction force on the bullet and the damage percentage of the plate can be directly evaluated with the functional relationships in the practice applications. PD simulated results are validated by existing experimental data with a good agreement. Through the simulated results, the following conclusions are obtained (1) a strong correlation exists between the coefficient of restitution observed experimentally and the model predictions; (2) the sensitivity to the velocity and sheet thickness in large-sized bullets is more than that in small-sized ones; (3) the damage, maximum reaction force, and coefficient of restitution almost change linearly with increasing the velocity, plate thickness and impact contact area; and (4) the effect of the porosity percentage on damage and reaction force is negligible.

A novel kinematic-constraint-inspired non-ordinary state-based peridynamics

The non-ordinary state-based peridynamics (NOSBPD) is attractive because of its ability to incorporate available constitutive models. However, conventional NOSBPD suffers from numerical oscillation induced by zero-energy mode. Although the recently proposed bond-associated deformation gradient performs well with controlling the issues, the size of the bond-associated horizon is undetermined. The work is aimed to introduce kinematic constraints to redefine the classical bond-associated deformation gradient. Based on the kinematic constraints imposed by the symmetry of the bond-associated deformation gradient and non-local kinematic measures, a well-defined bond-associated horizon is derived for 1D, 2D, and 3D structures. In this way, a novel kinematic-constraint-inspired non-ordinary state-based peridynamics (KC-NOSBPD) is proposed, which completely differs from the previous continuum-kinematics-inspired peridynamics (CPD) extended from classical bond-based peridynamics (BBPD). The linear and angular momentum conservation is firstly and rigorously proved. The novel proposed model is proved capable of avoiding unphysical deformation modes that typically beset conventional formulation through several analytical examples. In addition, bond-associated stress-state-based failure criteria are proposed for fracture analysis. Several benchmarking tests verify the effectiveness of the proposed formulation in eliminating spurious oscillation of numerical solution and highlight the significance of kinematic constraints. The final crack pattern of sandstone specimens coincides with experimental observation, illustrating its suitability for solving crack propagation.

Peridynamic-based investigation of the cracking behavior of multilayer thermal barrier coatings

Numerical simulations of the cracking behavior of the top layers of multilayer thermal barrier coatings (TBCs) can effectively reveal the failure mechanisms of the TBCs. Current finite element method (FEM)-based simulation means have been applied to solve certain simple cracking problems in TBCs; however, they cannot effectively describe complex cracking problems in TBCs such as coalescence, intersection, and interference among multiple cracks. Peridynamic (PD), a newly developed mechanical theory, has been widely studied to provide analysis for cracking problems in TBCs. In this paper, a numerical model of TBCs is built by the bond-based PD (BB-PD) theory. Complex cracking behaviors, such as spontaneous crack propagation at both interfacial and internal regions, coalescence, and interference among multiple cracks, are simulated under isothermal cooling and gradient cooling conditions. In addition, the effects of interfacial roughness and calcium–magnesium–alumina–silicate (CMAS) inclusions on the cracking behavior are discussed. The results show that the PD model accurately captures complex cracking behaviors observed via scanning electron microscopy (SEM). Given the ability of the model for analyzing discontinuities in TBCs, it can help to further clarify the fracture mechanisms of TBCs.

Improved peridynamics approach for the progressive fracture of marine concrete

Crack initiation and propagation in marine concrete play a significant role in the security and durability of concrete in ocean construction. Peridynamics (PD), which reformulates classical continuum mechanics in terms of spatial integral equations instead of partial differential equations, has been used successfully to match the fracture patterns observed in experiments. However, the classical bond-based peridynamics (BPD) describe brittle fracture instead of quasi-brittle fracture; this includes the failure of quasi-brittle solid marine concrete. Further, the models have significant surface effects, and the effect of the internal length on nonlocal long-range forces is not reflected. The current work proposes an improved BPD approach to overcome the three drawbacks of fracture analyses of marine concrete. A novel kernel function is incorporated, an effective PD surface effect correction method is used, and the BPD bond stiffness at the boundary is modified. Then, the constitutive force function of the BPD is established based on the linear and nonlinear mechanical behaviours of the progressive fracture of ocean concrete materials, and a corresponding failure criterion is given. The validity and accuracy of the proposed method are demonstrated via numerical analyses and experiments. The proposed method can fully capture the nonlinear deformation and progressive fracture of marine concrete.

An improved bond-based peridynamic model with shear bonds for eliminating rigid body rotation

Peridynamics (PD) is a kind of nonlocal theory based on displacement calculation of material particles around a horizon. The calculation of bond force, which determines the distribution of displacement field, is the key to damage and fracture simulation. Due to introduction of tangential stiffness, the bond-based PD models with shear bonds (PDS models), put forward higher requirements for the calculation of bond force. Because PDS models cannot remove the false shear deformation of bonds caused by rigid body rotation, the calculation of bond force may be inaccurate. To eliminate the influence of rigid body rotation on the calculation of bond force, an improved PDS model is proposed. Based on the principle of minimum energy and the nonlocality of peridynamics, the formula of rigid body rotation is derived. By deducting the influence of rigid body rotation, the improved PDS model ensures the accuracy of bond force calculation. In addition, a new failure criterion, strain-based failure criterion, is developed to improve the accuracy of PD model in calculating fracture energy. The results of deformation and fracture examples with finite rigid body rotation illustrate the necessity of removing the effects of rigid body rotation, and prove that the improved PDS model has high precision in the calculation of elastic displacement and prediction of cracking process. Finally, verification of the developed strain-based failure criterion was tackled with dynamic crack examples.

Coupling Approaches for Classical Linear Elasticity and Bond-Based Peridynamic Models

Local-nonlocal coupling approaches provide a means to combine the computational efficiency of local models and the accuracy of nonlocal models. This paper studies the continuous and discrete formulations of three existing approaches for the coupling of classical linear elasticity and bond-based peridynamic models, namely (1) a method that enforces matching displacements in an overlap region, (2) a variant that enforces a constraint on the stresses instead, and (3) a method that considers a variable horizon in the vicinity of the interfaces. The performance of the three coupling approaches is compared on a series of one-dimensional numerical examples that involve cubic and quartic manufactured solutions. Accuracy of the presented methods is measured in terms of the difference between the solution to the coupling approach and the solution to the classical linear elasticity model, which can be viewed as a modeling error. The objective of the paper is to assess the quality and performance of the discrete formulation for this class of force-based coupling methods.

Peridynamic stress is the static first Piola–Kirchhoff Virial stress

The peridynamic stress formula proposed by Lehoucq and Silling (2008) is cumbersome to be implemented in numerical computations. In this paper, we show that the peridynamic stress tensor has the exact mathematical expression as that of the first Piola–Kirchhoff static Virial stress originated from the Irving–Kirkwood–Noll formalism (Irving and Kirkwood, 1905; Noll, 1955) through the Hardy–Murdoch procedure (Hardy, 1982; Murdoch, 1983), which offers a simple and clear expression for numerical calculations of peridynamic stress.Several numerical verifications have been carried out to validate the accuracy of the proposed peridynamic stress formula in predicting the stress states in the vicinity of the crack tip and other sources of stress concentration. By evaluating the peridynamic stress in the prototype microelastic brittle (PMB) material model of bond-based peridynamics, it is found that the PMB material model may exhibit nonlinear constitutive behaviors at large deformations. The stress fields calculated through the proposed peridynamic stress formula show good agreements with these calculated by using finite element methods, analytical solutions, as well as experimental data, demonstrating the merit of the peridynamic stress formula in predicting stress states for material failure problems.

A General Numerical Method to Model Anisotropy in Discretized Bond-Based Peridynamics

This work presents a novel and general method of determining bond micromoduli for anisotropic linear elastic bond-based peridynamics. The problem of finding a discrete distribution of bond micromoduli that reproduces an anisotropic peridynamic stiffness tensor is cast as a least-squares problem. The proposed numerical method is able to find a distribution of bond micromoduli that is able to exactly reproduce a desired anisotropic stiffness tensor provided conditions of Cauchy’s relations are met. Examples of all eight possible elastic material symmetries, from triclinic to isotropic are given and discussed in depth. Parametric studies are conducted to demonstrate that the numerical method is robust enough to handle a variety of horizon sizes, neighborhood shapes, influence functions and lattice rotation effects. Examples of a 2D single edge cracked plate are used to demonstrate the use of the proposed method for practical problems.

Three-dimensional dynamic and quasi-static crack growth by a hybrid XFEM-peridynamics approach

This paper aims at presenting a hybrid computational strategy and its detailed implementation for simulation of crack propagation problems in three-dimensional (3D) solids. The key idea of the hybrid approach lies in the combination of extended finite element method (XFEM) and bond-based peridynamics (PD), which takes excellent features of the high computation efficiency by the XFEM and the flexibility by the PD in dealing with crack growth problems. The PD theory is restricted to the region where crack tips are located and cracks grow, while the XFEM theory is applied in the rest of the specimen. One of the merits of this integrated approach allows cracks to grow naturally without any fracture criteria, and it thus enhances the computation efficiency as compared to the pure PD. It additionally removes most of the problems due to the surface effects typical of nonlocal methods. Explicit integration scheme and implicit solving scheme are used for crack propagation problems under dynamic and quasi-static loading conditions. Numerical examples with 3D dynamic and quasi-static crack propagation cases are considered to show the accuracy and performance of the developed method. Reproducing the shape of the experimentally observed crack is demonstrated. For dynamic crack propagation, we additionally explore the change of the crack propagation speed during the cracking process, and the crack propagation speeds from the present method match well with the reference solutions.

Peridynamic analysis to investigate the influence of microstructure and porosity on fatigue crack propagation in additively manufactured Ti6Al4V

Additive Manufacturing (AM) has gained a lot of interest due to the freedom to produce complex metal geometries directly from the designed digital model. Despite the high potential of AM, the process induced imperfections, like pores and the microstructural changes due to the layer-by-layer manufacturing, made a significant impact on the fatigue resistance and crack growth behaviour. Consequently, this work aims to evaluate the effect of microstructure and existence of pores in additively manufactured Ti6Al4V on Fatigue Crack Growth (FCG) utilizing bond-based Peridynamics (PD) fatigue model. Employing the columnar granularity and different levels of porosities indicated a substantial impact on FCG rates.

A Quasi-Brittle damage model in the framework of Bond-based Peridynamics with Adaptive Dynamic Relaxation method

Peridynamics (PD) is a new tool, based on the non-local theory for modelling fracture mechanics, where particles connected through physical interaction used to represent a domain. By using the PD theory, damage or crack in a material domain can be shown in much practical representation. This study compares between Prototype Microelastic Brittle (PMB) damage model and a new Quasi-Brittle (QBR) damage model in the framework of the Bond-based Peridynamics (BBPD) in terms of the damage plot. An in-house code using Matlab was developed for BBPD with inclusion of both damage models, and tested for a quasi-static problem with the implementation of Adaptive Dynamic Relaxation (ADR) method in the theory in order to get a faster steady state solutions. This paper is the first attempt to include ADR method in the framework of BBPD for QBR damage model. This paper analysed a numerical problem with the absence of failure and compared the displacement with literature result that used Finite Element Method (FEM). The obtained numerical results are in good agreement with the result from FEM. The same problem was used with the allowance of the failure to happen for both of the damage models; PMB and QBR, to observe the damage pattern between these two damage models. PMB damage model produced damage value of roughly twice compared to the damage value from QBR damage model. It is found that the QBR damage model with ADR under quasi-static loading significantly improves the prediction of the progressive failure process, and managed to model a more realistic damage model with respect to the PMB damage model.

Quasi-static fracture analysis by coupled three-dimensional peridynamics and high order one-dimensional finite elements based on local elasticity.

This work investigates quasi‐static crack propagation in specimens made of brittle materials by combining local and non‐local elasticity models. The portion of the domain where the failure initiates and then propagates is modeled via three‐dimensional bond‐based peridynamics (PD). On the other hand, the remaining regions of the structure are analyzed with high order one‐dimensional finite elements based on the Carrera unified formulation (CUF). The coupling between the two zones is realized by using Lagrange multipliers. Static solutions of different fracture problems are provided by a sequential linear analysis. The proposed approach is demonstrated to combine the advantages of the CUF‐based classical continuum mechanics models and PD by providing, in an efficient manner, both the failure load and the shape of the crack pattern, even for three‐dimensional problems.

Analysis of heat transfer and water flow with phase change in saturated porous media by bond-based peridynamics

A wide range of natural and industrial processes involve heat and mass transport in porous media. In some important cases the transported substance may undergo phase change, e.g. from liquid to solid and vice versa in the case of freezing and thawing of soils. The predictive modelling of such phenomena faces physical (multiple physical processes taking place) and mathematical (evolving interface with step change of properties) challenges. In this work, we develop and test a non-local approach based on bond-based peridynamics which addresses the challenges successfully. Our formulation allows for predicting the location of the interface between phases, and for calculating the temperature and pressure distributions within the saturated porous medium under the conditions of pressure driven water flow. The formulation is verified against existing analytical solutions for 1D problems, as well as finite element transient solutions for 2D problems. The agreement found by the verification exercise demonstrates the accuracy of the proposed methodology. The detailed coupled description of heat and hydraulic processes can be considered as a critical step towards a thermo-hydro-mechanical model, which will allow, for example, description of the hydrological behaviour of permafrost soils and the frost heave phenomenon.

Numerical simulation for ice breaking and water entry of sphere

A three-dimensional ice-water-structure interaction numerical model is developed to simulate the ice breaking and water entry of rigid spheres. The algorithm employs the boundary data immersion method (BDIM) to simulate the solid-fluid interactions, the volume of fluid (VOF) method to track the liquid-gas interface, and the bond-based peridynamics method to model the ice cover. For the water entry of spheres, a good agreement between numerical and experimental results has been obtained for the water-entry cavity evolution and the motion of spheres. For the interaction between a sphere and the ice cover, a good agreement between the present results and the previous numerical results has been obtained for the shape and number of cracks. The crack propagation path of the ice cover, the cavity shape, pressure and velocity field, as well as the motion of spheres during the ice breaking and water entry are analyzed. Results show that the ice load on spheres is much greater than water load during the ice breaking and water entry process. Effects of the ice thickness and elastic modulus of the ice cover, the impact velocity and the density of the spheres are also discussed.

A unified non-local fluid transport model for heterogeneous saturated porous media

A unified non-local fluid transport model is proposed for heterogeneous saturated porous media. First, a non-local governing equation of fluid transport is presented based on the Non-ordinary State-based Peridynamics (NOSB-PD), and a novel flux vector state and a non-local Darcy’s law is proposed to illustrate non-local fluid transport effects in heterogeneous saturated porous media. The proposed flux state vector ensures the non-local governing equation of fluid transport model in heterogeneous saturated porous media degenerates into the local one when the horizon of each material point approaches to zero. A non-local pressure gradient at each material point can be obtained by averaging all the flux state vectors in its horizon, which unifies the weak and strong discontinuities of pressure in heterogeneous saturated porous media in a consistent way. Second, a variational formulation of the unified non-local fluid transport model is developed. This variational formulation enables the proposed model to deal with complex boundary conditions, which is difficult or even impossible for current fluid transport models based on Bond-based Peridynamics (BB-PD) and Ordinary State-based Peridynamics (OSB-PD). Third, a fully implicit algorithm combined with the Newton–Raphson method is introduced to solve the general non-linear problems in the heterogeneous saturated porous media. The penalty function method is employed to eliminate the zero-energy mode oscillation in NOSB-PD. Further, numerical examples demonstrate that the proposed model is accurate and can well capture the weak and strong discontinuities of pressure at the material interfaces and crack surfaces in the heterogeneous saturated porous media. Numerical examples also indicate that the model can effectively eliminate the zero-energy mode oscillation inherently rooted in the NOSB-PD, and thus reaches a stable numerical solution.

Peridynamic simulation of dynamic fracture in functionally graded materials subjected to impact load

Dynamic crack propagation assessment in functionally graded materials (FGMs) with micro-cracks is accomplished using bond-based Peridynamics (PD). The dynamic fracture behaviour of various FGMs’ material models is studied in Kalthoff–Winkler experiment. Dynamic crack growth predictions and associated material damage of the specimen under dynamic loading conditions are considered. The effect of micro-cracks near macro-crack tips on the toughening mechanism is evaluated in terms of crack propagation velocities. Stochastically pre-located micro-cracks are modelled to obtain the toughening effect in the material. In addition, the velocities and time required for fracture are compared in different FGM cases. It is frankly found that if a crack propagates in the harder region of the specimen, velocities decrease and toughness increase in contrast to the softer region. Furthermore, micro-cracks around a macro-crack decelerate the crack propagation and enhance toughening mechanism in FGM body depending on gradation of material properties.

Peridynamic simulation on hydraulic fracture propagation in shale formation

The inhomogeneity of shale reservoirs, bedding properties and in-situ stress make crack propagation paths complex and diverse in hydraulic fracturing. However, the simulations on hydraulic fracture propagation in shale gas reservoirs still require an effective alternative numerical method. In this paper, a peridynamic (PD) fluid–solid coupling model was developed for the numerical simulation on the hydraulic fracturing process in bedding shale formation. Firstly, a modified bond-based peridynamic (BB-PD) theory was formulated by introducing normal bond, tangential bond and pressure bond. This removes the limitation of classical BB-PD theory on Poisson's ratio. Secondly, a fluid–solid coupling model was established by combining the modified BB-PD theory with pressure driven fluid transport in porous medium. Thirdly, a fracturing criterion for the modified BB-PD theory was derived from energy conservation. Fourthly, the PD fluid–solid coupling model was numerically discretized and verified by three-point bending tests on shale beam. Their accuracy was further checked through the Terzaghi's consolidation problem. Finally, numerical simulations were conducted to explore the generation of hydraulic fracturing induced fracture network in bedding shale formation under different dip angles and horizontal stress ratios. The simulation results showed that the horizontal stress ratio has a greater effect on fracturing pressure, and the dip angle plays a crucial role in the shape of fracture network. The increase of horizontal stress ratio leads to the increase of fracturing pressure.