Peridynamics Research Articles

Combining peridynamic and finite element simulations to capture the corrosion of degradable bone implants and to predict their residual strength

This paper proposes a computational framework to describe the biodegradation of magnesium (Mg)-based bone implants. It is based on a sequential combination of two models: an electrochemical corrosion model to compute the mass loss of the implant over several weeks combined with a mechanical model to assess its residual mechanical strength. The first model uses a peridynamic (PD) corrosion model to tackle the complex moving boundary of the corroding material in an efficient manner. The results of this corrosion simulation are mapped to a finite element (FE) model by way of a damage variable. Subsequently, the FE model is used for mechanical analysis. To use PD for such a complex problem, we proposed three innovative improvements compared to state-of-the-art PD models: (1) application of an adaptive multi-grid discretization in space and an implicit time-stepping algorithm enabling an efficient simulation of the complex implant geometry over prolonged periods, (2) novel non-local Dirichlet absorbing boundary conditions to truncate the simulation domain in the close neighborhood of the implant of interest without prohibitive losses of accuracy, and (3) selection of suitable non-local kernel functions and parameter calibration on the basis of experimental data by an evolutionary algorithm. We demonstrate that this framework can capture the loss of implant mass due to corrosion for typical alloys such as Mg-5Gd and Mg-10Gd. Moreover, we point out how this framework can be used in the future to predict the declining mechanical strength of bone screws subject to biocorrosion over several weeks.

A general and fast convolution-based method for peridynamics: Applications to elasticity and brittle fracture

A general and fast convolution-based method (FCBM) for peridynamics (PD) is introduced. Expressing the PD integrals in terms of convolutions and computing them by Fast Fourier Transform (FFT), the computational complexity of PD models drops from O(N2) to O(Nlog2N), with N being the number of discretization nodes. Initial neighbor identification and storing neighbor information is not required, and this means memory allocation scales with O(N) instead of O(N2), common for existing methods. FCBM is applicable to bounded domains with arbitrary shapes and boundary conditions via an “embedded constraint” (EC) approach. The formulation is shown for certain bond-based and state-based, linear and nonlinear, PD models of elasticity and dynamic brittle fracture, as applications. The method is verified on a 3D elastostatic problem and it is shown that the FCBM-PD reduces the computational time from days to hours and from years to days, compared with the original meshfree discretization for PD models. Large-scale computations of PD models are feasible with the new method, and its versatility is demonstrated by simulating, with ease, the difficult problem of cascading crack branching in a brittle plate.

A PD-FEM approach for fast solving static failure problems and its engineering application

A method of coupling of peridynamics (PD) and finite element method (FEM) for fast solving static failure problems is proposed. It is based on the force coupling method and adopts interface elements to couple the PD and FE subregions. By assembling the global matrix of the coupling model and using the adaptive dynamic relaxation (ADR) method to solve it, the static solution of elasticity and crack growth problems can be obtained quickly. By introducing short-range repulsive force and re-defining the coupling force, it is not only suitable for simulating tensile failure problems, but also suitable for compression failure problems. Through comparative analysis, the accuracy and effectiveness of this approach are verified. And then the method is successfully applied to simulate the excavation process of an underground cavern group, and the simulation of damage and failure process of engineering rock mass under complex tension–compression conditions has been realized. The stability characteristics of the surrounding rock are revealed by analyzing the damage and deformation evolution laws. It provides an effective numerical simulation method for predicting the excavation damage zone (EDZ) distribution characteristics and deformation law of the surrounding rock during the excavation of an underground cavern group. At the same time, compared with pure PD, the calculation efficiency is greatly improved, which is of great significance for the realization of engineering-scale simulation.

A hybrid meshfree discretization to improve the numerical performance of peridynamic models

Efficient and accurate calculation of spatial integrals is of major interest in the numerical implementation of peridynamics (PD). The standard way to perform this calculation is a particle-based approach that discretizes the strong form of the PD governing equation. This approach has rapidly been adopted by the PD community since it offers some advantages. It is computationally cheaper than other available schemes, can conveniently handle material separation, and effectively deals with nonlinear PD models. Nevertheless, PD models are still computationally very expensive compared with those based on the classical continuum mechanics theory, particularly for large-scale problems in three dimensions. This results from the nonlocal nature of the PD theory which leads to interactions of each node of a discretized body with multiple surrounding nodes. Here, we propose a new approach to significantly boost the numerical efficiency of PD models. We propose a discretization scheme that employs a simple collocation procedure and is truly meshfree; i.e., it does not depend on any background integration cells. In contrast to the standard scheme, the proposed scheme requires a much smaller set of neighboring nodes (keeping the same physical length scale) to achieve a specific accuracy and is thus computationally more efficient. Our new scheme is applicable to the case of linear PD models and within neighborhoods where the solution can be approximated by smooth basis functions. Therefore, to fully exploit the advantages of both the standard and the proposed schemes, a hybrid discretization is presented that combines both approaches within an adaptive framework. The high performance of the developed framework is illustrated by several numerical examples, including brittle fracture and corrosion problems in two and three dimensions.

Modelling soil desiccation cracking by peridynamics

Understanding of desiccation-induced cracking in soil has improved over the last 20–30 years through experimental studies, but progress in predictive modelling of desiccation cracking has been limited. The heterogeneous structure of soils and the multi-physics nature of the phenomenon, involving emergence and propagation of discontinuities, make the mathematical description and analysis a challenging task. The authors present a non-local hydro-mechanical model for soil desiccation cracking capable of predicting crack initiation and growth. The model is based on peridynamics (PD) theory. Attempts to model soil desiccation cracking by PD are limited to a purely mechanical description of the process that involves calibration of the parameters. In contrast, the model presented in this paper describes soil desiccation cracking as a hydro-mechanical problem, where moisture flow and deformation are coupled. This allows for investigating and explaining the mechanisms controlling the initiation and propagation of discontinuities. The model is applied and tested against two sets of experimental data to explain the typical features of drying-induced cracking of clays. The validations use experimental parameters (Young's modulus, water retention characteristics) and avoid calibrations to test the accuracy of the model. The correlations between the shrinkage of soil clay, changes in displacement fields and crack growth are demonstrated. Crack initiation, propagation and ultimate crack patterns simulated by the model are found to be in very good agreement with experimental observations. The results show that the model can capture realistically key hydraulic, mechanical and geometry effects on clay desiccation cracking.

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.

Static and dynamic mechanical behaviors of cracked Mindlin plates in ordinary state-based peridynamic framework

An ordinary state-based peridynamic (PD) model based on the Mindlin plate theory is presented to deal with fracture mechanics problems. Static and dynamic stress resultant intensity factors regarded as the primary fracture parameters for plate structures are evaluated by the displacement extrapolation method. Owing to the PD surface effect, however, the accuracy of the displacement field near crack surfaces is significantly affected. Therefore, the arbitrary horizon domain method is adopted to correct the surface effect. It derives the variable PD parameters to properly describe mechanical behaviors for each material point. Several numerical examples are investigated to examine the performance of the presented method. It indicates that the PD Mindlin plate model incorporated with the arbitrary horizon domain method validly minimizes the influence of the PD surface effect and provides an effective approach to evaluate static and dynamic moment intensity factors.

ANSYS implementation of a coupled 3D peridynamic and finite element analysis for crack propagation under quasi-static loading

This study presents an approach to couple peridynamic (PD) and finite element method (FEM) for 3D deformation and crack propagation in ANSYS framework. The stiffness matrix of the PD element is derived by using the peridynamic least squares minimization (PD LSM) method based on the small deformation assumption. The PD governing equations are constructed by using MATRIX 27 element native to ANSYS. The coupling between MATRIX 27 elements and traditional finite elements is achieved through the coupled degrees of freedom (DOF) command available in ANSYS. The crack propagation is implemented by monitoring the number of broken bonds and updating the stiffness matrix of the PD element. The accuracy of the coupled PD-FE approach is verified by comparison against the 3D FE displacement predictions. The validity of the coupled PD-FE approach is established by simulating the wedge splitting test.

A general-purpose, inelastic, rotation-free Kirchhoff–Love shell formulation for peridynamics

We present a comprehensive rotation-free Kirchhoff–Love (KL) shell formulation for peridynamics (PD) that is capable of modeling large elasto-plastic deformations and fracture in thin-walled structures. To remove the need for a predefined global parametric domain, Principal Component Analysis is employed in a meshfree setting to develop a local parameterization of the shell midsurface. The KL shell kinematics is utilized to develop a correspondence-based PD formulation. A bond-stabilization technique is employed to naturally achieve stability of the discrete solution. Only the mid-surface velocity degrees of freedom are used in the governing thin-shell equations. 3D rate-form material models are employed to enable simulating a wide range of material behavior. A bond-associative damage correspondence modeling approach is adopted to use classical failure criteria at the bond level, which readily enables the simulation of brittle and ductile fracture. Discretizing the model with asymptotically compatible meshfree approximation provides a scheme which converges to the classical KL shell model while providing an accurate and flexible framework for treating fracture. A wide range of numerical examples, ranging from elastostatics to problems involving plasticity, fracture, and fragmentation, are conducted to validate the accuracy, convergence, and robustness of the developed PD thin-shell formulation. It is also worth noting that the present method naturally enables the discretization of a shell theory requiring higher-order smoothness on a completely unstructured surface mesh.

Investigation of mixed-mode fracture of UHPC using a hybrid digital image correlation and peridynamics approach

Fracture of ultrahigh-performance concrete (UHPC) is a complicated process. It is difficult to simulate the whole process of crack initiation and crack propagation only using computational mechanics methods. The rapid advances of measurement technique in experimental mechanics make it feasible to set up a synergy between computational mechanics and experimental mechanics to explore the cracking mechanism of UHPC. Based on peridynamics (PD) and digital image correlation (DIC) technique, a hybrid experimental and computational method is proposed to investigate the damage and fracture characteristics of UHPC. Three-point bending tests of UHPC notched-beams were performed and the full-field displacements of the UHPC specimens were measured using DIC technique. The discrete PD model was reconstructed on the basis of the measured data. The damage process of UHPC specimens was simulated by combining the DIC displacement data and the PD model. The simulation results were compared with the experimental observations. The crack propagation paths of the simulation results match well with the experimental observations before the appearance of macro cracks. The results show that the proposed model possesses high accuracy in detecting damage and simulating crack propagation of mode I and mixed-mode fractures.

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.

A generalized hybrid smoothed particle hydrodynamics–peridynamics algorithm with a novel Lagrangian mapping for solution and failure analysis of fluid–structure interaction problems

Fluid–structure interaction problems (FSI) are omnipresent in a wide variety of engineering fields and includes highly non-linear physics such as violent free-surface flow, complex interfacial and inter-phase interactions, and deformable structures with the possibility of damage formations. Well-established mesh dependent approaches may suffer from dynamic mesh-refinement, special algorithms to track free surfaces and handle the interface and inter-phase physics. On the other hand, due to their inherent nature, particle based meshless methods, namely Smoothed Particle Hydrodynamics (SPH) and Peridynamics (PD), can lend themselves easily to tackling with the above-stated complex physics involved in the fluid and structure phases of the FSI problems, respectively. This study presents a novel hybrid algorithm to couple the SPH and PD in a robust and high-fidelity manner for modeling extreme FSI problems with and without multi-scale defects. The coupled methodology is integrated with a novel mapping strategy based on Lagrange polynomial interpolation, which allows for the solution procedure to utilize different particle resolutions for fluid and deformable structural media. The mapping strategy enables one to construct a higher-accuracy PD solver that can predict the dynamic behavior of solids and provide an ability to model and track the damage nucleation, propagation, and branching in the structure phase of FSI domains. The corrected weakly compressible SPH is used for pressure–velocity coupling in the fluid phase whereas the ordinary state based PD is utilized for modeling the deformable structure. The SPH and PD solvers of the algorithm are validated separately through solving several well-accepted benchmark test cases such as the collapse of a water column, large deformation of a cantilever beam, and damage propagation in a plate under impact loading (Kalthoff–Winkler experiment). After validating each solver, the hybrid coupled SPH–PD algorithm is applied to simulate various extreme FSI problems of literature such as the dam break with an elastic gate, collapse of the water column and its hydrodynamic impact on an elastic obstacle, and the entry of a V-shaped deformable beam on the free surface of water in a tank. A close agreement between the SPH–PD results and the literature data is achieved for each of the aforementioned cases. The algorithm is further applied to model the failure propagation in the elastic wall of a tank under hydrostatic pressure. A realistic failure behavior is recorded for the FSI setups with different initial states of the wall, i.e., the wall is assumed to be with and without initial defects of different geometrical orientations. In particular, this study proposes and implements a new perspective for coupling different particle methods to achieve high-fidelity and low-cost modeling of complex FSI problems, which may involve multi-scale defects (i.e., cracks and voids) and failure propagations.

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.

Relationships between the material parameters of continuum-kinematics-inspired peridynamics and isotropic linear elasticity for two-dimensional problems

Continuum-kinematics-inspired Peridynamics (CPD) has been recently proposed as a geometrically exact formulation of peridynamics that is also thermodynamically and variationally consistent. CPD can capture the Poisson effect exactly, unlike the original formulation of peridynamics (PD). Due to its geometrically exact nature, CPD does not suffer from zero-energy modes and displacement oscillations that may be observed in state-based PD formulations. For a two-dimensional analysis, CPD builds upon one-neighbor and two-neighbor interactions. The one-neighbor interactions of CPD are equivalent to the bond-based interactions of the original PD formalism. Two-neighbor interactions, however, are key in CPD since the basic notions of classical continuum kinematics, namely length and area, are preserved exactly. The isotropic two-dimensional CPD formulation of non-local elasticity therefore involves two material constants, namely C1 and C2, associated with length and area, respectively. This manuscript aims to establish relationships between the material parameters of CPD and isotropic linear elasticity for an affine deformation in a two-dimensional setting. It is shown that each of the CPD material parameters can be expressed in terms of any pairs of isotropic linear elasticity constants, such as Lamé parameters. Finally, we establish the admissible ranges for CPD material parameters.

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.

Numerical Modeling on Crack Propagation Based on a Multi-Grid Bond-Based Dual-Horizon Peridynamics

Peridynamics (PD) is a novel nonlocal theory of continuum mechanics capable of describing crack formation and propagation without defining any fracture rules in advance. In this study, a multi-grid bond-based dual-horizon peridynamics (DH-PD) model is presented, which includes varying horizon sizes and can avoid spurious wave reflections. This model incorporates the volume correction, surface correction, and a technique of nonuniformity discretization to improve calculation accuracy and efficiency. Two benchmark problems are simulated to verify the reliability of the proposed model with the effect of the volume correction and surface correction on the computational accuracy confirmed. Two numerical examples, the fracture of an L-shaped concrete specimen and the mixed damage of a double-edged notched specimen, are simulated and analyzed. The simulation results are compared against experimental data, the numerical solution of a traditional PD model, and the output from a finite element model. The comparisons verify the calculation accuracy of the corrected DH-PD model and its advantages over some other models like the traditional PD model.

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.

The Performance of Five Extended Bond-Based Peridynamic Models on Crack Simulation

Peridynamics (PDs) is a nonlocal continuum mechanics theory used to solve fracture and failure problems. The classical bond-based peridynamics (BBPDs) involves one parameter to describe the microelastic bond stretch response, which makes the Poisson’s ratio restricted. In previous studies, kinds of extended bond-based peridynamic (EBBPD) models were proposed to deal with the limitation of Poisson’s ratio in BBPDs. In this paper, four benchmarks with different load conditions are designed to compare the performance of five EBBPD models and to investigate the effects of model parameters on crack propagation. Results show that compared with BBPDs, the crack patterns of the EBBPD models have the main form of BBPDs; on the other hand, the introduction of tangential stiffness makes the form of crack tend to shear failure, and the increase of tangential stiffness will enhance this effect. The crack patterns transition of wing cracks from tensile failure to mixed tensile–shear failure when the tangential stiffness increases. For the models with rotational stiffness, the rotational stiffness affects the width and propagation direction of the secondary crack and makes the trends to shear failure. Besides, the rotational stiffness affects the forms of crack coalescence and divides the results of these five models into three types.

A peridynamic model for crevice corrosion damage

A new peridynamic (PD) model for crevice corrosion damage is introduced and cross-validated with experimental results from the literature. Electrochemical mechanisms are reduced to a simple metal-ion concentration-dependent corrosion rate. Computations over domains with extreme aspect ratios like those seen in crevices are efficiently performed by modifying the PD formulation to accommodate arbitrary-shape horizons. The model is validated against experimental results on bolted washers. Simulations predict both the location and the depth of deep corrosion trenches that form at a distance from the crevice mouth. Far-field boundary conditions are used to autonomously reproduce the experimentally observed progression of crevice corrosion. The PD model results also show that the two factors considered here (diffusion-driven transport of dissolved metal ions, and dissolution being dependent on their concentration) are sufficient to predict the evolution of crevice corrosion.