Peridynamic Model Research Articles

Nonlocal Nernst-Planck-Poisson System for Modeling Electrochemical Corrosion in Biodegradable Magnesium Implants

AbstractThis paper provides a comprehensive derivation and application of the nonlocal Nernst-Planck-Poisson (NNPP) system for accurate modeling of electrochemical corrosion with a focus on the biodegradation of magnesium-based implant materials under physiological conditions. The NNPP system extends and generalizes the peridynamic bi-material corrosion model by considering the transport of multiple ionic species due to electromigration. As in the peridynamic corrosion model, the NNPP system naturally accounts for moving boundaries due to the electrochemical dissolution of solid metallic materials in a liquid electrolyte as part of the dissolution process. In addition, we use the concept of a diffusive corrosion layer, which serves as an interface for constitutive corrosion modeling and provides an accurate representation of the kinetics with respect to the corrosion system under consideration. Through the NNPP model, we propose a corrosion modeling approach that incorporates diffusion, electromigration and reaction conditions in a single nonlocal framework. The validity of the NNPP-based corrosion model is illustrated by numerical simulations, including a one-dimensional example of pencil electrode corrosion and a three-dimensional simulation of a Mg-10Gd alloy bone implant screw decomposing in simulated body fluid. The numerical simulations correctly reproduce the corrosion patterns in agreement with macroscopic experimental corrosion data. Using numerical models of corrosion based on the NNPP system, a nonlocal approach to corrosion analysis is proposed, which reduces the gap between experimental observations and computational predictions, particularly in the development of biodegradable implant materials.

Traction‐associated peridynamic model and non‐uniform discretization simulation of plane axisymmetric problems

AbstractUsing the recently developed traction‐associated peridynamic motion equation (TPME), we investigate the plane axisymmetric problems of elastic deformation. The polar coordinate form of TPME is derived for a plane axisymmetric problem. The transfer function matching with prototype microelastic (PM) model is determined by the inverse method in a one‐dimensional case, and then it is directly extended to plane axisymmetric elasticity problems and plane problems. We promote a strategy to deal with the horizon in the non‐uniform discretization scheme. Based on this strategy, the non‐uniform discretization scheme is used to solve the elastic deformation of a hollow circular plate subjected to uniform inner and outer pressures. Total process from elastic deformation to failure of the hollow circular plate under tension is simulated completely. The results show that TPME does not only need volume and surface corrections, but also can deal with the complex traction boundary conditions conveniently.

Image-Based Peridynamic Modeling-Based Micro-CT for Failure Simulation of Composites

By utilizing computed tomography (CT) technology, we can gain a comprehensive understanding of the specific details within the material. When combined with computational mechanics, this approach allows us to predict the structural response through numerical simulation, thereby avoiding the high experimental costs. In this study, the tensile cracking behavior of carbon–silicon carbide (C/SiC) composites is numerically simulated using the bond-based peridynamics model (BB-PD), which is based on geometric models derived from segmented images of three-dimensional (3D) CT data. To obtain results efficiently and accurately, we adopted a deep learning-based image recognition model to identify the kinds of material and then the pixel type that corresponds to the material point, which can be modeled by BB-PD for failure simulation. The numerical simulations of the composites indicate that the proposed image-based peridynamics (IB-PD) model can accurately reconstruct the actual composite microstructure. It can effectively simulate various fracture phenomena such as interfacial debonding, crack propagation affected by defects, and damage to the matrix.

Bond-associated non-ordinary state-based peridynamics for simulating damage evolution of thermal barrier coatings in aero-engine turbine blades

The failure mode of thermal barrier coatings (TBC) systems in aeroengine turbine blades is very complex because of the harsh service conditions. A peridynamic (PD) model is established to simulate the damage evolution of TBC with uniform thermally grown oxide (TGO) growth under cycle load. The peridynamic differential operator is introduced to solve the zero-energy mode, and thermo-elastic deformation is considered. Moreover, the influence of high-temperature holding time, initial oxide layer thickness, and interface morphology on the evolution of the stress distribution and interface damage is discussed. The newly proposed PD model can effectively capture the interface cracking of TBC systems and it is conducive to the study of the failure of TBC systems.

Hydro-thermo-mechanical coupled peridynamic modeling of freeze–thaw fracture of concrete

Hydro-thermo-mechanical coupled peridynamic modeling of freeze–thaw fracture of concrete

Peridynamics model of torsion-warping: Application to lattice beam structures

This paper presents a finite deformation beam model based on Simo-Reissner theory in peridynamics (PD) framework to deal with torsion induced warping deformation. Seven degrees of freedom, viz. three translational, three rotational, and one warping amplitude are considered at each material point. The governing equations of the beam are obtained by employing global balance of linear and angular momenta in conjunction with Simo's assumption on the deformation field. The relation between PD resultant force, moment, bi-moment, and bi-shear states with their classical counterparts is established using the constitutive correspondence method. Numerical implementation strategy is furnished for both quasi-static and dynamic cases. The solution for quasi-static load is obtained through the Newton-Raphson method. The proposed model is validated against finite element solutions considering cantilever beam and lattice structures. Quasi-static deformation responses of 3×3×3 octet and single unit compression-torsion lattice structures are presented further to demonstrate the effectiveness of proposed beam model. A new bond breaking criterion is proposed based on critical stretch, critical relative rotation, and critical relative warping amplitude and failure of the compression-torsion lattice structures under compressive load is simulated. The Newmark-beta method is utilized to solve the governing equations for dynamic loading. Numerical simulations include dynamic analysis of octet and compression-torsion lattice structures.

A modified mixed-mode Timoshenko-based peridynamics model considering shear deformation

The original two-dimensional bond-based peridynamic (BBPD) framework, which only considers the pairwise forces (compression and tension) between two material points, is extended by incorporating the effect of shear deformation in the calculations and its influence on the failure of the bonds. To this end, each bond is considered as a short Timoshenko beam, and by doing so, the traditional BBPD is enhanced into a more comprehensive model known as multi-polar peridynamic (MPPD). The proposed novel approach explicitly considers the shear influence factor used in Timoshenko beams and introduces a strain-based shear deformation failure criterion. The model is then validated against two benchmark experimental tests (i.e., a standard pure mode I edge crack, and a Kalthoff-Winkler configuration) reported in the literature under in-plane dynamic loading and plane stress conditions. In most cases, the developed model is shown to be more accurate in predicting the crack paths obtained from the experimental results when compared to other theoretical methods delineated in the literature. Furthermore, a noticeable change in crack branching and crack path is observed in a study on the effects of Poisson's ratio and the loading rate. This investigation also demonstrated that the proposed MPPD model can accommodate materials with Poisson's ratios up to 1/3, expanding the range beyond the traditional BBPD limitations.

Coupled field modelling of thermomechanical response in materials using peridynamic heat transport model and non-ordinary state-based peridynamics

Predicting the mechanical response of materials subjected to transient heat processes is crucial in various engineering applications. This paper introduces a novel peridynamic (PD) coupled field model formulated within the framework of Non-Ordinary State-Based Peridynamics (NOSBPD). This model offers the capability to simulate the coupled thermo-mechanical behavior of materials by incorporating both thermal transport and mechanical deformation analyses. To verify the model’s accuracy, a benchmark problem involving a steel plate undergoing transient thermal expansion is investigated. This model presents a valuable tool for various engineering applications involving coupled thermo-mechanical processes.

A fast computational framework for the linear peridynamic model

A fast computational framework for the linear peridynamic model

Numerical study of the effects of loading parameters on high-energy gas fracture propagation in layered rocks with peridynamics

The objective of this study is to investigate the influence of loading parameters on the propagation pattern of high-energy gas fractures in layered rock formations. To this end, a peridynamic model for brittle rock accounting for material heterogeneity was proposed. The ability of the model to simulate dynamic fractures was validated through laboratory experiments, and the homogeneity coefficient for the critical elongation rate was calibrated. On this basis, a numerical model of high-energy gas fracturing in layered rocks containing interfaces was constructed. Simulations were conducted to analyse high-energy gas fracturing from cylindrical intact boreholes and perforated boreholes under varying loading parameters. The results indicate that as the loading rate increases, the number of radial fractures surrounding the borehole gradually increases, whereas the influence of in-situ stress on fracture propagation diminishes. When the loading rate is fixed, both an increase in the peak pressure and a decrease in the decay rate are conducive to enhancing the propagation length of fractures. The propagation speed of fractures significantly decreases when they reach an interface but recovers after they penetrate it. Fractures tend to penetrate an interface when the angle of approach is closer to a right angle, and the direction of fracture propagation can be controlled through a perforation design. These findings provide valuable insights into the selection and optimization of loading parameters for reservoir stimulation via high-energy gas fracturing.

Peridynamic analysis of rolling contact fatigue crack propagation in rail welding joints with pore defects

Pore defects are prevalent in rail welding joints and significantly contribute to the propagation of fatigue cracks. This study develops a peridynamic (PD) model that incorporates the characteristics of pore defects to analyze their impact on rolling contact fatigue behavior. Initially, compact tension (CT) fatigue tests were performed to derive and validate the bond fatigue failure model specific to rail weld materials. Subsequently, pore defects were modeled as holes in the CT specimens, with experimental results being compared to PD simulation outcomes for validation. Finally, a wheel-rail contact PD model was constructed to investigate the mechanisms of fatigue crack propagation in rail welding joints affected by pore defects.

Peridynamic modelling of elastic and viscoelastic behaviour in polycrystalline ice: A study using NOSBPD and PDCHT

This study explores the elastic and viscoelastic behaviour of ice using the Non-Ordinary State-Based Peridynamic (NOSBPD) framework. A primary objective is to extend and validate the applicability of NOSBPD in modelling the complex responses of ice under various conditions. The study employs the Peridynamic Computational Homogenization Theory (PDCHT) to determine the critical threshold of grain count necessary to induce an effectively isotropic response in polycrystalline S2 ice. Results are consistent with previous findings from Finite Element Method (FEM) and Bond-Based Peridynamics (BBPD) studies.Furthermore, the viscoelastic response of ice is investigated by integrating a viscoelastic constitutive model into the NOSBPD framework. A benchmark problem of a viscoelastic ice sample subjected to tensile stress is simulated, with results compared against FEM simulations conducted in ANSYS Mechanical. The findings show good agreement, validating the NOSBPD framework's capability to capture time-dependent viscoelastic behaviour of ice accurately.The study contributes to the field of ice mechanics by demonstrating the robustness and versatility of NOSBPD in modelling both elastic and viscoelastic responses of ice. These advancements enhance the credibility and applicability of peridynamics (PD) as a powerful tool for simulating complex material behaviours, paving the way for further research and practical applications in ice engineering.

Thermally-induced fracture in the oxide scale of T91 ferritic/martensitic steel after exposure to oxygen-saturated liquid lead–bismuth eutectic

Ensuring the continuity and stability of the oxide scale to separate T91 steel from lead-bismuth eutectic (LBE) is an effective method for limiting corrosion in the lead-based fast reactor system. In this study, we specifically investigated the impact of cooling thermal stress on the stability and damage of the oxide scales of T91 steel when exposed to oxygen-saturated liquid lead–bismuth eutectic at high temperatures. A corrosion test was conducted on T91 steels exposed to stagnant oxygen-saturated LBE at 500 °C. Experimental results indicated that oxide damage exists without any external force, including interface cracks and through-scale cracks perpendicular to the interface. We developed a three-layer peridynamic model of T91-(Fe,Cr)3O4-Fe3O4 to simulate the deformation and failure of oxides under a cooling process from high temperature to room temperature. The simulation results show that a temperature drop of about 200 °C from 500 °C can cause through-oxide cracks in the oxide scale, and subsequent cooling can lead to the propagation of interfacial cracks. Additional modifications were introduced in the model to account for the porosity of the oxide scale. Parametric investigations illustrate the effects of oxide scale thickness and porosity on the resulting crack patterns.

Peridynamics modelling of projectile penetration into concrete targets

A non-ordinary state-based peridynamics (NOSB-PD) model is proposed to simulate the projectile penetration into concrete targets. In this model, the Kong-Fang concrete material model recently proposed is firstly implemented into the NOSB-PD framework to describe the complex dynamic behavior and failures in concrete material subjected to penetration loading, and then an improved point-to-volume discrete frictional contact model is proposed to simulate the physical interaction between projectile and target. After the mesh-free discretization and explicit time integration, the proposed NOSB-PD model is used to numerically predict two sets of projectile penetration experiments into low-strength and high-strength concrete targets. And numerical predictions are found to be in good agreements with corresponding test data including penetration depth, projectile deceleration, deformation of projectile and failures in concrete targets.

Re‐derivation and mathematical analysis for linear peridynamics model for arbitrary Poisson ratio's material

AbstractThis paper is concerned with the modeling and mathematical analysis of linear peridynamic model for arbitrary Poisson ratio's material. Based on the fundamental laws of dynamics, we re‐derive the bond‐based peridynamic model for anisotropic materials by relaxing certain assumptions. Through this process, we draw several significant conclusions, such as the relationship between the equivalent strain energy density hypothesis and the convergence of the peridynamic operator to the classical Navier operator. Additionally, the well‐posedness of time‐dependent peridynamic equations of motion is established. Finally, some necessary conditions for the material stability of anisotropic material are given.

A modified bond-based peridynamic approach for rigid projectile perforation on concrete slabs

Peridynamic (PD) has a unique advantage in describing the crack growth and fragmentation of brittle materials. Concerning the dynamic behaviors and failure patterns of concrete slabs under projectile perforations, a modified bond-based PD approach maintaining both the easy implementation and computational stability characteristics was firstly developed from the following three aspects, (i) a rate-dependent PD constitutive model was proposed for describing the dynamic behaviors of concrete; (ii) a progressive damage criterion considering the tension-compression anisotropy, softening behavior, and strain rate effect of concrete was incorporated to more accurately reproduce the damage and failure of concrete; (iii) an improved micro-modulus function related to bond length was introduced to reveal the internal length effect of bond force. Then, numerical simulations of projectile perforation on concrete slabs by utilizing the developed modified bond-based PD approach, as well as the corresponding sensitivity analyses of discretization parameters including horizon size and particle spacing were performed. Based on the recommended horizon size and particle spacing, the predicted residual velocity of projectile and failure patterns of concrete slabs exhibited an excellent agreement with the test data. Furthermore, by comparisons of the traditional bond-based PD and classical finite element methods, the superiority of developed approach in describing the perforation damage of concrete targets against projectile impact was demonstrated. Finally, the modified bond-based PD approach was employed to blind simulate the projectile normal and oblique perforating multi-layered spaced concrete target plates. It was found that the modified PD model reasonably predicted the terminal ballistic trajectory, deflection angle, and residual velocity of projectile, as well as the failure patterns of target plates. The present work provides a new way to predict the terminal ballistic effect of projectile and dynamic behaviors of concrete slabs.

Peridynamics-fueled convolutional neural network for predicting mechanical constitutive behaviors of fiber reinforced composites

Despite advancements in predicting the constitutive relationships of composite materials, characterizing the effects of microstructural randomness on their mechanical behaviors remains challenging. In this study, we propose a data-driven convolutional neural network (CNN) to efficiently predict the stress-strain curves containing three key material features (Tensile strength, modulus, and toughness) of fiber reinforced composites. Firstly, stress-strain curves for composites with arbitrary fiber distributions were generated using experimentally validated peridynamics (PD) model. Principal component analysis (PCA) was then employed to learn these curves in a lower-dimensional space, reducing computational costs. Subsequently, these reduced data, along with randomly distributed microstructural features, were used to train, validate, and evaluate the CNN models. The combined CNN and PCA model accurately predicted stress-strain curves with maximum errors of 2.5 % for tensile strength, 10% for modulus, and 20 % for toughness. Furthermore, data augmentation and Mean Squared Error (MSE) as a loss function significantly enhanced the model's prediction accuracy. Our findings indicated that DenseNet121 outperformed other CNN models in predicting the properties of fiber-reinforced materials, further demonstrating the effectiveness of the proposed model. This work successfully demonstrates the applicability of a data-driven CNN approach to predict stress-strain relations for engineering materials with intricate heterogeneous microstructures, paving the way for data-driven computational mechanics applied in composites.

Peridynamics simulation of microparticle impact: Deformation, jetting, and the influence of surface oxide layer

Over the past decade, cold spraying (CS) has become a significant additive manufacturing technology for creating metallic structures and repairing damaged components. This study uses a novel peridynamic model to investigate the behavior of 12 μm copper particles impacting a copper substrate during CS deposition. The model examines deformation behavior, material jetting, and the effect of a native particle oxide layer. Two oxide thicknesses, 6 nm and 30 nm, were considered. The 6 nm oxide, typical for gas-atomized copper powders, had a negligible effect on deformation, while the 30 nm oxide showed a notable influence. For the 30 nm oxide, the experimentally observed coefficient of restitution (COR) deviated from theoretical predictions in the velocity range of 400–600 m/s due to cracking and fracturing of the oxide layer at higher impact velocities, reducing particle rebound velocity without sufficient metal-to-metal contact for permanent bonding. Permanent adhesion requires significant oxide fragmentation and separation. The study also identified threshold velocities for material jetting from the substrate and particle, showing that oxide presence delays particle jetting by impeding plastic deformation at the edges. Quantifying the oxide volume at the particle-substrate interface clarifies the effect of jetting on COR and bonding strength. The analysis includes effects of deflected jets at extremely high velocities (e.g., 870 m/s), compared with existing literature. Overall, the results demonstrate the efficacy of the peridynamics-based simulation in accurately representing material behavior during high-velocity impacts, effectively predicting oxide breakup, detachment, and removal in CS, and their effects on deposition characteristics.

Bond-Based Peridynamic Model for Tensile Deformation and Fracture of Polycarbonate and Polypropylene

Fracture mechanics has been a crucial aspect in the field of engineering science as technologies are rapidly growing nowadays. Various numerical methods have been developed to analyze fracture behaviour in different types of materials used in industries. Meanwhile, the application of polymers garners attention worldwide due to outstanding characteristics such as good strength, lightweight, and high temperature resistance, exemplified by polymers like polycarbonate (PC) and polypropylene (PP). Hence, failure aspects of such materials must be taken into consideration when conditions arise that may lead to failure, such as high-load impact, fatigue, and extreme temperatures. In this study, a bond-based Peridynamic model (PD) for the tensile behaviour, including fracture, of polymers has been developed. The PD model is constructed using the Centos software and encompasses both brittle and ductile fracture behaviours. Numerical results, including crack propagation, damage zone, and force-extension curves of notched specimens, are validated by comparison with experimental results of PC and PP. Through the validation process, PC specimens exhibit a difference percentage range for maximum load and rupture extension of 2.9% to 18.8% and 2.4% to 4.6%, respectively. PP specimens show a difference percentage range for maximum load and rupture extension of 31.2% to 43.5% and 0.9% to 30%, respectively. Consequently, the validation results indicate that the PD model for brittle specimens aligns more closely with experimental data compared to the PD model for ductile specimens.

Peridynamic study on thermomechanical damage of the rail during wheel idling

Wheel idling is an extreme sliding contact scenario that accelerates rail failure. Using temperature-dependent elastoplastic materials, a peridynamic (PD) model for wheel-rail contact was developed to investigate the thermomechanical damage behavior of rails during wheel idling. Numerical simulations of loading and unloading during wheel idling under different loads were performed, and the results of contact stress, temperature, plastic deformation, and material damage were obtained and analyzed. During loading, the temperature of the rail surface increases rapidly to over 1200 °C, which is well above the austenitization temperature. The intense thermal softening caused by high temperatures deteriorates the wheel-rail contact relationship and leads to a surge in plastic deformation. The softened surface material was continuously removed by the contact load, leading to severe wear damage, and the wear depth increased significantly with the load and idling time. The wear coefficients for the Archard wear model were derived using the wear results from the PD model. During unloading, the surface material cools rapidly at a rate of 1400°C/s through heat transfer into the rail, undergoing martensitic transformation and forming a brittle and hard white etching layer. The thickness of the WEL decreases as the wear depth increases.