Bond-based Peridynamics Research Articles

Dynamic numerical simulation of ice-propeller milling based on GPU parallel computing

To integrate high-performance GPU parallel computing with the peridynamics method and enhance the computational efficiency of numerical simulations for ice-propeller milling, thus providing better data support for the design of propellers in ice-covered areas, a GPU-based parallel peridynamics computational approach was developed on CUDA in this study. The approach was built upon the bond-based peridynamics theory and CUDA programming framework, and its validity was confirmed using test cases involving an airfoil cutting ice and an ice ball impacting a rigid wall. A corresponding three-dimensional GPU parallel computational program was created for the ice-propeller milling process, and the computational code was optimized, resulting in a 24-fold increase in computational efficiency. Utilizing the high-performance computational code, the influence of sea ice elastic modulus and propeller pitch on the mechanical performance of the blades was investigated. The computational results revealed that the ice loads on the blades increased with rising elastic modulus and decreased with increasing pitch, and a larger pitch led to more sea ice being milled away.

Three-dimensional peridynamic simulation on triaxial compression failure mechanical behavior of cylindrical marble specimen containing pre-existing fissures

A series of triaxial compression experiments and analyses were conducted out by using the three-dimensional bond-based peridynamics (BB-PD) model to investigate the mechanical behavior, deformability, and failure behavior of marble samples under different confining pressures. This study assessed the impact of confining pressure on mechanical performance of the intact marble, calibrated PD model micro-parameters using experimental data, and predicted the fissure behavior of marble specimens with single or parallel fissures. Through detailed analysis of crack evolution, crack pattern, displacement fields, and principal stress fields, the research revealed that the ductility of marble depended on the confining pressure, especially in the presence of fissures. Increasing confining pressure facilitated a transition from splitting failure to conjugate shear failure modes and induced stress concentration at fissure tips, leading to wing cracks and secondary cracks initiation. The presence and number of fissures were found to directly influence the mechanical behavior and failure process complexity of marble. A higher number of fissures reduced brittleness and strength while enhancing ductility and variety of failure modes. Seven distinct failure patterns of marble were identified under triaxial compression, illustrating the combined effect of confining pressure and fissures on marble specimen failure modes. This work can provide a theoretical and numerical basis for fissured rocks fracture in deep engineering.

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.

Modelling the frost cracking behavior in clayey soils: A peridynamic approach

Frost cracking is one of the primary causes of deterioration in frozen soil structures, yet few relevant numerical studies have been reported, and the simulation of frost cracking in soils remains challenging due to inadequate consideration of reasonable simulation algorithms. Numerous experimental studies have identified frost heave and desiccation shrinkage as the principal cause of frozen soil cracking. On this basis, this study presents a peridynamic (PD) model that considers the coupled effects of frost heaving and desiccation shrinkage for simulating frost cracking in soils during freezing process. The heat conduction equation is reformulated using the peridynamic differential operator (PDDO). The variation of thermal parameters for soils is addressed using the thermal enthalpy method, equivalent homogeneous method, and linear release assumption of latent heat. The frost-heaving load induced by pore water is represented using an equivalent displacement load. The multiphysics solution using PDDO and bond-based peridynamics (BBPD) considering freezing heave and desiccation shrinkage is developed for the first time. By simulating the frost cracking of a two-dimensional soil strip after model validations, the resulting crack pattern closely resembles the experimental observation. It indicates that the present model can capture the phase transition interface (PTI) and cracking behaviors of frozen soils.

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.

Investigation on mixed mode I/II crack propagation in nitrate ester plasticized polyether propellant: Experimental and numerical study

The fracture of solid propellant is predominantly attributed to the existence of mixed mode cracks, so it is essential to investigate the the fracture behavior of solid propellant with mixed mode I/II crack. This paper presents fracture characteristics of nitrate ester plasticized polyether (NEPE) propellant under different crack inclination angles (β = 30°–90°). Based on the combination of a drawing machine and a high-speed camera, the mechanical response, crack propagation velocity and crack-path morphology were investigated. The critical equivalent stress intensity factor Keqc was calculated to assess the fracture toughness of the NEPE propellant, and a potential simplified criterion related to the stress intensity factor was proposed. The experimental results demonstrated that the NEPE propellant with mixed mode I/II crack exhibited blunting fracture phenomena during crack propagation, resulting in fluctuating crack propagation velocity. As the crack inclination angle decreases, the fracture toughness of the NEPE propellant increases and then decreases, and the value of Keqc reaches its maximum at β = 45°. Furthermore, numerical studies based on bond-based peridynamic (BBPD) were performed by modeling the crack propagation process of the NEPE propellant, including the crack phase field diagram and the load–displacement curve of the NEPE propellant. The simulation results were then compared with the experiments.

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.

Peridynamic modelling of cryogenic deuterium pellet fragmentation for shattered pellet injection in tokamaks

Abstract Shattered pellet injection (SPI) is a promising method for controlling plasma disruptions in tokamaks. In this study, we present numerical modelling of the fragmentation of cryogenic deuterium pellets within the context of SPI, using the peridynamic (PD) theory. A dedicated in-house code has been developed, leveraging the meshfree method and GPU parallelization. The mechanical properties of cryogenic solid deuterium are obtained from available literature, and calibrated based on the shatter threshold along with the remaining solid mass fraction after shatter. The results from the bond-based peridynamics (BB-PD) successfully reproduce the main experimental results reported in the literature, both qualitatively and quantitatively.

Peridynamics-based model of composite lamina with progressive variations in mechanical properties

An analytical model of the fiber-reinforced composite lamina is constructed based on bond-based peridynamics (PD), which utilizes variable matrix bonds and fiber bonds to evaluate the static properties of matrix and fiber materials, respectively. The bond stiffness of the matrix is determined by trigonometric functions, and fibers are represented by constants. The proposed PD model describes the continuous variations in the static properties of composites as a function of fiber angle. Both the conventional analytical method and the PD method are used to solve the failure-free elastic deformation issue. The two methods estimate maximum errors in the displacement of material points in the x and y directions of 0.01 % and 4.86 %, respectively, indicating good consistency. The failure propagation of the preexisting crack layer under the axial tensile load is simulated and the obtained failure modes are in good agreement with experimental results. The proposed composite lamina model simulates the influence of fiber direction on overall behavior through various matrix bond stiffness. Breaking through the limitation of constant matrix bond stiffness in conventional models, is conducive to promoting the further development of bond-based composite PD models.

A new insight to investigate the strain distribution, evolution, and crack development of cement-stabilized coral aggregate

Cement-stabilized coral aggregate (CSCA) can be used as a base material for roads and airports in island engineering. In this study, four-point bending (FPB) tests were conducted on CSCA with different prefabricated crack depths, and distributed fiber optic sensor (DFOS) was used. The testing process of the CSCA beams was simulated using an improved bond-based peridynamic (BBPD) model that combines the bilinear damage principle and fracture parameters random field. The results indicate that DFOS has the potential to monitor strain evolution and detect crack in CSCA. The strain distribution in the spanwise direction of the beam during the plastic phase presents a single peak, which can be described by a Pearson-VII function. The fracture location can be predicted from the positions of the tensile strain peaks. The neutral axis begins to move gradually toward the compressed part of the specimen after the external load greater than 0.6 Pmax. The compression modulus of the CSCA material peaks is 4500 MPa, that is three times larger than its tensile modulus and can satisfy the requirement of semi-rigid base layer construction of island roads. The improved BBPD model is able to effectively simulate the strain-softening behavior of CSCA materials with consideration of the effect of material inhomogeneity on the actual fracture path. The peak load prediction error was less than 7 %. The R2 between the numerically predicted load-displacement curves and the test results were bigger than 0.95. The improved BBPD method can predict the actual fracture location of the specimen with a horizontal position deviation of less than 2 cm. The bending fracture process of CSCA are further explored from both experimental and numerical simulations by this study.

A computational approach to integrate three-dimensional peridynamics and two-dimensional higher-order classical elasticity theory for fracture analysis

AbstractAs a nonlocal alternative of classical continuum theory, peridynamics (PD) is mathematically compatible to discontinuities, making it particularly attractive for failure prediction. The PD theory on the other side can be computationally demanding due to its nonlocal interactions. A coupling between PD and refined higher-order finite element method (FEM) integrates their salient features. The present study proposes a computational approach to couple three-dimensional peridynamics with two-dimensional higher-order finite elements based on classical elasticity. The bond-based PD modeling is considered in a region where damage might appear while refined finite element modeling is used for the remaining region. The refined finite elements employed in this study are based on the 2D Carrera Unified Formulation (CUF), which provides 3D-like accuracy with optimized computational efficiency. The coupling between PD and FEM is achieved through the Lagrange multiplier method which permits physical consistency and compatibility at the interface domain. An adaptive convergence check algorithm is also proposed to achieve predetermined accuracy in the solution with minimum computational effort. Simulations of quasi-static tension tests, wedge splitting tests and L-plate cracking tests are carried out for verification. In-depth analysis shows that the present approach can reproduce the linear deformation, material degradation and crack propagation in an effective way.

Research on Crack Propagation of Nitrate Ester Plasticized Polyether Propellant: Experiments and Simulation.

To understand the fracture properties of the nitrate ester plasticized polyether (NEPE) propellant, single-edge notched tension (SENT) tests were carried out at room temperature (20 °C) under different tensile rates (10-500 mm/min). The mechanical response, crack morphology, evolution path, and crack propagation velocity during the fracture process were studied using a combination of a drawing machine and a high-speed camera. The mode I critical stress intensity factor KIc was calculated to analyze the tensile fracture toughness of the NEPE propellant, and a criterion related to KIc was proposed as a means of determining whether the solid rocket motors can normally work. The experimental results demonstrated that the NEPE propellant exhibited blunting fracture phenomena during crack propagation, resulting in fluctuating crack propagation velocity. The fracture toughness of the NEPE propellant exhibited clear rate dependence. When the tensile rate increased from 10 mm/min to 500 mm/min, the magnitude of the critical stress intensity factor increased by 62.3%. Moreover, numerical studies based on bond-based peridynamic (BBPD) were performed by modeling the fracture process of the NEPE propellant, including the crack propagation speed and the load-displacement curve of the NEPE propellant. The simulation results were then compared with the experiments.

A peridynamic-informed deep learning model for brittle damage prediction

Physics-informed Neural Network (PINN) has been introduced recently to predict and understand complex physical phenomena by directly incorporating feedback from governing equations. While PINNs offer remarkable capabilities, challenges arise when addressing discontinuities and heterogeneity, potentially compromising accuracy and reliability in prediction outcomes. In this study, a novel approach that combines the principles of peridynamic (PD) theory with PINN is presented to predict quasi-static damage and crack propagation in brittle materials. To achieve high prediction accuracy and convergence rate, the linearized PD governing equation is enforced in the PINN’s residual-based loss function. The proposed PD-INN is able to learn and capture complicated displacement patterns associated with different geometrical parameters, such as pre-crack position and length. The computational method presented in this study can be implemented to any PD constitutive model like Bond-Based PD or State-Based PD as well as any horizon factor. Several enhancements like cyclical annealing schedule and deformation gradient aware optimization technique are proposed to ensure the model would not get stuck in its trivial solution. The model’s performance assessment is conducted by monitoring the behavior of loss function throughout the training process. The results obtained through PD-INN demonstrate a satisfactory level of agreement in 2D and 3D when compared with the ground truth deformation from high-fidelity techniques like PD direct numerical method and Extended finite element method (X-FEM). Our results show the ability of the nonlocal PD-INN to predict damage and crack propagation accurately and efficiently.

Numerical and theoretical investigation of crack propagation in the windshield in collision with a dummy headform

Studies show that the impact of the pedestrian head and windshield is a significant contributing factor to casualties in traffic accidents. Accordingly, investigating the mechanical response of the windshield and its failure mechanism is of significant importance in enhancing the safety of pedestrians in car accidents. This study conducted the pedestrian protection head impact test to determine the fracture mode of the windshield, acquire the acceleration curve of the headform impactor, and accurately obtain the peak acceleration and Head Injury Criterion (HIC). Subsequently, to quickly predict HIC and facilitate structural selection in designing automobiles, a theoretical analysis model for predicting peak acceleration was established and validated against the test results. To analyze the damage mechanism in detail, finite element analysis software LS-DYNA was employed to simulate the crack propagation in the windshield based on the peridynamics theory. Through a comparison between the numerical simulation and experimental results, the accuracy of the bond-based peridynamics model was verified, accurately revealing its failure mode and mechanical response to impact. Meanwhile, the peak acceleration and HIC were effectively predicted, which provided an effective approach for vehicle safety evaluation and design.

A high-order multi-time-step scheme for bond-based peridynamics

A high-order multi-time-step scheme (MTS) for the bond-based peridynamic (PD) model, an extension of classical continuous mechanics widely used for analyzing discontinuous problems like cracks, is proposed. The MTS scheme discretizes the spatial domain with a meshfree method and advances in time with a high-order Runge–Kutta method. To effectively handle discontinuities (cracks) that appear in a local subdomain in the solution, the scheme employs the Taylor expansion and Lagrange interpolation polynomials with a finer time step size, that is, coarse and fine time step sizes for smooth and discontinuous subdomains, respectively, to achieve accurate and efficient simulations. By eliminating unnecessary fine-scale resolution imposed on the entire domain, the MTS scheme outperforms the standard STS scheme for PD by significantly reducing computational costs, particularly for problems with discontinuous solutions, as demonstrated by comprehensive theoretical analysis and numerical experiments.

A penetration correction contact model involving friction based on Peridynamics

Peridynamics employs an integral form to deal with the discontinuity within the object, showing great advantages in analyzing material failure. However, in Peridynamics, many existing contact methods have some shortcomings in accurately calculating contact forces, leading to some unreasonable results when the problem involves contact. A more accurate contact model based on particle penetration is proposed by modifying the traditional particle-to-particle contact model. The proposed contact model is verified through two typical benchmark tests, and the calculated results are compared with the traditional pin-ball contact model and short-range-force contact model. The results demonstrate the correctness and advantages of the proposed model. Bond-Based Peridynamics equipment with the proposed contact model is used to analyze the Brazilian tensile test of rocks. The elastic deformation and fracture evolution show a high agreement with the analytical solution and experimental results, demonstrating the effectiveness and robustness of the proposed contact model. Based on the Brazilian tensile test, a method to verify the correctness of contact models is proposed. The differences in numerical results obtained with the proposed contact model and the short-range force model are compared, and the mechanism underlying the differences are investigated. In addition, Graphics Processing Units (GPU) are used to implement parallel computation to improve efficiency. This research contributes to the understanding of the mechanism of Brazilian tensile test.

Investigation of heat transfer in cracked gun barrels

Heat transfer process significantly affects the life span of gun barrels as it directly associates to the thermoelastic dynamic responses, erosion, and wears of gun barrels. The previous studies regarding the heat transfer in gun barrels were developed in the realm of local theory that is expressed by partial differential equations with the condition of smoothly distributed physical fields, and have rarely studied the cracked gun barrels. However, on one hand, due to manufacturing errors, there inherently exists fractures in realistic gun barrels; on the other hand, the inner wall of the gun barrel wears due to the high-temperature and -speed burning gas generated during the interior ballistic process, as well as the high-speed friction from projectile’s movement, which severely lead more cracks and damages. Therefore, it is of great importance to investigate the heat transfer in cracked gun barrels. In this paper, a non-local heat transfer model for cracked gun barrels is formulated via the bond-based peridynamics. The governing equation is presented in the manner of integration, such that the heat process with the influence of cracks can be readily captured. Numerical results show that the heat energy severely accumulates on the crack surface and as the size of horizon increases, the non-local characteristics of the heat transfer appear to be more pronounced, resulting in a stiffer thermal response.

A spatial nonlocal Green-Naghdi generalized thermoelasticity model based on bond-based peridynamics: Transient responses of laminated structures subjected to thermal shock

The Green-Naghdi generalized thermoelastic model is firstly extended based on the peridynamics and the thermo-mechanical response of laminated structures under thermal shock is investigated in this paper. Firstly, the GN-II and GN-III models of peridynamics (PD-GN model), which can consider spatial nonlocality and discontinuity of problems, are derived. Then, the characteristics of the PD-GN-II model are studied. To validate the correctness of the method and PD-GN-II model in this work, the results are compared with the results of other investigators and FEM results. In addition, the separate effects of non-locality of different fields on the thermo-mechanical response are discussed. Aiming at the thermal coupling problems of laminated structures subjected to thermal shock, the influence of non-local effect, the types of thermal-bonds (weaken or broken thermal-bonds), arrangement of material, and damage location are investigated. It is concluded that the new spatial nonlocal generalized thermoelastic model can handle discontinuous problems well, and material design has significant influence on the thermo-mechanical response and damage evolution of structures. The PD-GN model established in this work is not only a novel spatial non-local generalized thermoelastic model, but also provides a new way to solve the fracture problems in non-Fourier heat conduction.

Coupling of bond-based peridynamics and continuous density-based topology optimization methods for effective design of three-dimensional structures with discontinuities

This study proposes continuous density-based three-dimensional topology optimization (TO) approaches developed by coupling the peridynamic theory (PD) with optimality criteria (OC) and proportional approach (PROP). These frameworks, abbreviated as PD-OC-TO and PD-PROP-TO, can be practically utilized to enhance the fracture toughness of the structures during the optimization process by taking critical regions into account as pre-defined cracks. Breaking the non-local interactions (bonds) between relevant PD particles enables us to readily model cracks. Utilizing this advantage, we solve several benchmark optimization problems including different numbers, positions, and alignments of the cracks. The major differences between the proposed methods are examined by comparing optimum topologies for various cracked scenarios. Moreover, the mechanical behaviour of the optimized structures is investigated under dynamic loads to prove the significant improvements achieved by the present approach in the final designs. The results of dynamic analyses reveal the viability of both PD-TO methods for increasing the fracture toughness of the structure in the optimization stage. Overall, the proposed approach is confirmed as a superior design and optimization tool for future engineering structures.Graphical abstract

Non-local modelling of freezing and thawing of unsaturated soils

A large part of the earth’s surface is covered by seasonally or permanently frozen soils. Considering the negative impact of climate change, future development of such regions can be underpinned by mathematical methods for accurate analysis of heat and moisture transport in freezing and thawing soils. Reported in this paper is a novel non-local formulation of water and heat transport in unsaturated soils. The formulation uses bond-based peridynamics (PD) and consists of a set of integral–difference formulations of energy and mass conservation. Specific features of freezing/thawing soils are incorporated by a combination of van Genuthcen and Clausius–Clapeyron relations. Computational results are compared with four sets of laboratory experiments to demonstrate the efficiency of the developed approach. The model can be used to analyse the effect of water flow on heat transfer in soils during thawing of permafrost soils. Further, it can be applied in modelling climate change effects, and can be used for construction of coupled physically justified models of frost heave.