Bond-based Peridynamic Model Research Articles

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-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.

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.

Investigation of fracture patterns and effect on capacity in all-solid-state batteries based on peridynamic electro-chemo-mechanical model

The evolution of complex fracture patterns of composite electrodes in all-solid-state batteries (ASSBs) during electrochemical cycles is one of the main phenomena driving capacity degradation. Such fracture patterns in active materials usually involve cracks with various orientations. While conventional wisdom usually treats fracture as ultimate failure of materials in many scenarios, the effect of crack in batteries is worth further investigation. Here, a concurrently coupled electro-chemo-mechanical model based on peridynamics has been developed to study fracture patterns and electrochemical performances of composite anodes during processes of charge and discharge. The framework consists of a classic bond-based peridynamic mechanical model, an electrochemical model, and a coupling technique to reflect the interaction between them in a bidirectional manner. The bond-based peridynamic mechanical model considers the active material’s elastic–plastic deformation and the solid electrolyte’s elastic deformation. The fracture energies of the active material, the solid electrolyte, and the interface are also incorporated into this model. The electrochemical model considers the volume expansion of the active material caused by the intercalation of lithium ions and the Butler–Volmer relation of charge transfer on the interface. Various parametric analyses are performed using this modeling framework to investigate the effects of external pressure, fracture energies of active material and solid electrolyte, and Young’s modulus on fracture patterns and electrochemical performances. It is found that different material parameters result in distinct fracture patterns in the active material with radial cracks or circumferential cracks, thus profoundly affecting the rates of charge and discharge. This work demonstrates the importance of considering the complex fracture patterns in electrode materials of batteries and provides insights for selecting more suitable electrode materials to mitigate capacity degradation.

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 conjugated bond-based peridynamic model for laminated composite materials

The bond-based peridynamic (BB-PD) theory is inadequate in describing the four independent engineering material constants of composite lamina, which results in fixed values its Poisson's ratios and shear moduli. This study presents a novel lamina conjugated BB-PD model encompasses not only the fiber bond and matrix bond, but also the transverse bond and conjugated bond pair. By defining these four basic PD bonds, the improved conjugated BB-PD model can overcome the restriction of material properties. The determination of the four basic PD micro-modulus parameters is achieved through the utilization of the equivalent relationship of strain energy density. In addition, the lamina conjugated BB-PD model is extended to deal with composite laminates with different stacking sequences. The present study validates the accuracy and applicability of the conjugated BB-PD model for composite materials by conducting simulations of the elastic deformation and progressive damage behavior of unidirectional laminae and laminates. Moreover, the proposed model provides an elucidation of the damage modes and failure mechanisms of laminated composite materials subjected to uniaxial loading.

Peridynamic modeling of step-path failure in rock slopes

This work is devoted to investigating the step-path failure of rock slopes using peridynamic theory. To this end, an extended bond-based peridynamic model that can explicitly distinguish between tensile and shear cracks from mixed-mode fracture phenomena, previously proposed by the authors, is first introduced. Then, the fracture processes of gypsum specimens with various triple-flaw distributions under uniaxial compression are simulated at a laboratory scale. Tensile, shear, and mixed tensile-shear failure modes are reproduced, and the feasibility of the numerical tool is verified via comparison with experimental observations. Finally, the progressive failure of rock slopes at the field scale is analyzed, in which crack initiation, propagation, coalescence, and eventual structural damage are well captured. The types of crack propagation and coalescence predicted in this study contribute to understanding step-path failure mechanisms in experimental research and nature.

An optimized unibond dual-parameter peridynamic model for deformation and fracture simulation of quasi-brittle materials

Peridynamic is a nonlocal theory, which is an effective tool for predicting fracture in solids. The unibond dual-parameter peridynamic (UDPD) model can break through the Poisson's ratio limitation compared with the bond-based peridynamic model. However, the traditional algorithm for the integrated volume is designed according to the analytical value of partial length in the one-dimensional model, which will produce non-negligible errors for the two- and three-dimensional models. Besides, the classical UDPD model is restricted to ideal brittle materials rather than quasi-brittles, and the long-range force effect is generally omitted for simplified analysis. In this work, an optimized UDPD model is developed to overcome the above drawbacks in simulating the deformation and fracture of quasi-brittle materials. First, the novel algorithm to obtain a more accurate integrated volume is introduced into the UDPD model. Second, considering the long-range force effect and the deformation characteristics, the nonlinear constitutive force function coupled with the cosine type influence function is developed for the UDPD model. Then, the corresponding bond stiffnesses are derived. The proposed model is verified through several representative numerical applications, which suggests that the predicted results match well with the experimental observations and former simulations. It is indicated that the proposed model reveals a significant improvement in the convergence and precision of results, which can not only retain the stability of the classical UDPD model, but also precisely analyze the deformation behavior and damage evolution process of quasi-brittle materials.

Nlfem: a Flexible 2d FEM Python Code for Nonlocal Convection-Diffusion and Mechanics

In this work, we present the mathematical foundation of an assembly code for finite element approximations of nonlocal models with compactly supported, weakly singular kernels. We demonstrate the code on a nonlocal diffusion model in various configurations and on a two-dimensional bond-based peridynamics model. Further examples can be found in D’Elia et al. (Math Models Methods Appl Sci 31(08):1505–1567, 2021). The code nlfem is published under the GNU General Public License (for details, see, e.g., https://www.gnu.org/licenses/gpl-3.0.de.html) and can be freely downloaded at https://gitlab.uni-trier.de/pde-opt/nonlocal-models/nlfem.

Multiphysics modeling of subcritical crack growth in glass

Subcritical crack growth starting from flaws on glass surfaces/edges causes slow degradation in glass strength. This phenomenon involves multiple physics and is very challenging to predict. In this work, we introduce a novel multiphysics model for subcritical crack growth in glass. This model considers water-assisted stress relaxation, stress-dependent chemical reaction between water and glass, diffusion of water to the crack tip, and reaction-driven slow crack growth. Stress relaxation is achieved by using a Prony series in the bond-based peridynamic viscoelastic model, and a new bond breakage criterion based on the concentration of reaction product is introduced. A displacement-loaded double beam with a pre-crack is used to obtain the crack velocity vs. stress intensity factor (v-K) curve. With proper simplifications and assumptions, v-K curves obtained from the model successfully reproduced their typical patterns in experiments. This model will help us better understand fundamental mechanisms and make predictions for subcritical crack growth in glass.

A New Bond-Based Peridynamic Model with the Ability to Model Elastoplastic Behavior

A New Bond-Based Peridynamic Model with the Ability to Model Elastoplastic Behavior

Peridynamic simulation for the damage patterns of glass considering the influence of rate-dependence and pre-defects

This study presents a novel rate-dependent bond-based peridynamic model for predicting the mechanical behavior of brittle aluminosilicate glass under quasi-static and dynamic flexure loading. To achieve this, the damage criterion is modified with reference to the concept of strain rate in the classical continuum mechanics (CCM). Additionally, a peridynamic (PD) model with inhomogeneous critical stretch based on the Weibull distribution is proposed to account for the randomly distributed defects in glass and investigate their effect on crack initiation and propagation. The accuracy and rationality of the improved PD model are verified by comparing the calculated strength and fracture modes with the experimental results under quasi-static Ball-On-Ring (BOR) and dynamic low-velocity impact tests. The results show that the rate effect increases the damage threshold of glass, enhancing strength and damage ratio. Notably, flexural failures induced by tensile stress are observed under both static and dynamic loading. The rate-dependent inhomogeneous PD model indicates that glass inhomogeneity reduces strength while increasing crack propagation velocity (CPV). The introduction of pre-existing defects alters the symmetrical crack propagation pattern typically observed in homogeneous materials, resulting in a more randomized crack propagation path. Overall, the results can provide a promising approach for predicting the mechanical behavior of brittle materials.

An efficient peridynamics-based statistical multiscale method for fracture in composite structures

This paper proposes a peridynamics-based statistical multiscale (PSM) framework to simulate the macroscopic structure fracture with high efficiency. The heterogeneities of composites, including the shape, spatial distribution and volume fraction of particles, are characterized within the representative volume elements (RVEs), and their impact on structure failure are extracted as two types of peridynamic parameters, namely, statistical critical stretch and equivalent micromodulus. At the microscale level, a bond-based peridynamic (BPD) model with energy-based micromodulus correction technique is introduced to simulate the fracture in RVEs, and then the computational model of statistical critical stretch is established through micromechanical analysis. Moreover, based on the statistical homogenization approach, the computational model of effective elastic tensor is also established. Then, the equivalent micromodulus can be derived from the effective elastic tensor, according to the energy density equivalence between classical continuum mechanics (CCM) and BPD models. At the macroscale level, a macroscale BPD model with the statistical critical stretch and the equivalent micromodulus is constructed to simulate the fracture in the macroscopic homogenized structures. The algorithm framework of the PSM method is also described. Two- and three-dimensional numerical examples illustrate the validity, accuracy and efficiency of the proposed method.

A bond-based peridynamics modeling of polymeric material fracture under finite deformation

We have developed a bond-based peridynamics model to study the fracture of polymer networks under finite deformation. Our approach is based on the Kuhn–Grün model of randomly jointed polymers with extensible chains, which allows the rupture of polymers through the chain scission mechanism. We introduced a novel mesoscale potential function that intrinsically incorporates an Arruda–Boyce type model into a nonlocal continuum framework with the purpose of modeling the mechanical response of general polymeric materials and the fracture of polymers. We have demonstrated through a series of numerical examples, both two-dimensional and three-dimensional, that the proposed model is robust and efficient in modeling crack growth in polymeric materials. The model is carefully validated with experimental data and numerical results obtained on the basis of existing continuum mechanics polymer models. We have shown that the proposed peridynamic model is accurate and stable in various configurations and loading modes, such as mode I and shear crack opening. In comparison to existing continuum polymer models and fracture modeling methods, our present technique has the advantages of being theoretically simple, rigorous, and computationally fast. The numerical results show that the nonlocal polymer model can easily capture the fracture process in polymeric materials without complex ad hoc modeling parameters under finite deformation, and it has the potential to become a convenient simulation tool for modeling the polymer failure process in the design and optimization of polymeric materials.

On Peierls–Rice–Beltz nonlocal continuum model and simulations of mesoscale dislocations (slips) and shear cracks

In this work, we developed a cohesive Peierls–Rice–Beltz nonlocal continuum theory, or a bond-based Peierls–Rice–Beltz peridynamics model, and apply it to model mesoscale dislocation motions or crystal slips and shear cracks in crystal solids, which is based on a meso-scale Peierls–Rice–Beltz cohesive potential function (Xu and Needleman, 1993). By doing so, we have successfully developed a bond-based cohesive peridynamics model with intrinsic stress/strain measures as well as consistent and built-in correspondence macro-scale constitutive relations. The main novelties of this work are:(1) We have made the once-conceptual Peierls–Rice–Beltz model as a multiscale computational Peierls–Rice–Beltz model that can quantify both surface energy and unstable stacking fault energy in any given numerical simulations at mesoscale. By adjusting the ratio of the surface energy and unstable stacking fault energy, we can model opening cracks, shear cracks, and mixed mode fracture at ease. (2) We have developed a bond-based peridynamics model that can capture mesoscale dislocations or lattice slips, shear crack, and mixed-mode fractures, and (3) We have shown that the proposed method is able to model inelastic fracture and simulate ductile fracture of small scale yielding in the nonlocal cohesive continua. Several numerical examples have been presented, in which we show that the proposed approach is a simple, efficient and effective method to model inelastic fracture in the nonlocal cohesive media.

A Bond-Based Peridynamic Model with Matrix Plasticity for Impact Damage Analysis of Composite Materials

The prediction of damage and failure to fiber-reinforced polymer composites in extreme environments, particularly when subjected to impact loading, is a crucial issue for the application and design of protective structures. In this paper, based on the prototype microelastic brittle (PMB) model and the LaRC05 composite materials failure model, we proposed a bond-based peridynamic (BB-PD) model with the introduction of plastic hardening of the resin matrix for fiber-reinforced polymer composites. The PD constitutive relationships of the matrix bond and interlayer bond under compressive loading are considered to include two stages of linear elasticity and plastic hardening, according to the stress-strain relationship of the resin matrix in the LaRC05 failure model. The proposed PD model is employed to simulate the damage behaviors of laminated composites subjected to impact loading. The corresponding ballistic impact tests of composite laminates were carried out to observe their damage behaviors. The PD prediction results are in good agreement with the ballistic experimental results, which can verify the correctness and accuracy of the PD model developed in this study in describing the impact damage behaviors of composite materials. In addition, the characteristics and degree of damage in composite laminates are analyzed and discussed based on this PD model. The difference in the impact resistance of composite laminates with different stacking sequences is also studied using the numerical simulation results.

A bond-based peridynamics model based on variable material properties for modeling elastoplastic behavior

Crack tip singularities and material discontinuities are taken to be the main challenges for analyzing cracks in fracture mechanics problems. Peridynamic (PD) theory has been recently introduced as a nonlocal extension of continuum mechanics capable of improved modeling of progressive damage and rupture in cracked structures. In this theory, the internal forces have been characterized by nonlocal interaction between the pairs of the material points whose displacements or displacement derivatives are discontinuous. Given that the bond-based PD (BB-PD) has not been satisfactorily employed for predicting ductile fracture, the main purpose of this work is to present a new PD plasticity model in conjunction with the Variable Material Property (VMP) method, aiming to provide the possibility of elastoplastic modeling in BB-PD. In the present work, the quasi-static loading has been selected for simulation and the von-Mises yield criterion has been utilized for describing the plastic deformation. The loadings are applied as the displacement is controlled and outputs from the PD model are compared with the results achieved from ABAQUS software based on the continuum mechanic assumptions. To validate the accuracy of the present formulation, two example problems of the plate with a central crack and a central hole subjected to tensile load are examined. The results of von-Mises stress, the plastic zone size, horizontal and vertical displacement, and equivalent plastic strain have been verified in comparison with the outputs calculated by the finite element method (FEM), evidencing good accuracy of the VMP-based PD elastoplastic model in predicting the desired parameters. To predict the formation of spontaneous cracks and crack growth trajectory, two more examples that include a plate with a central hole, as well as, a plate with two edge cracks and a double hole are discussed. The obtained results are in good agreement with those from the experiment and ABAQUS software.

Chemo-mechanical coupling bond-based peridynamic model for electrochemical corrosion and stress chemical corrosion

In this paper, a chemo-mechanical coupling bond-based peridynamic model considering stress-chemical corrosion (SCC) is first proposed to simulate chemical corrosion. The proposed model can also investigate the damage and cracking behaviors of materials under chemo-mechanical coupling condition, which is different from the previous electrochemical corrosion PD model only considering chemical corrosion condition. Furthermore, two examples of electrochemical corrosion and stress chemical corrosion are employed to illustrate the ability of the chemo-mechanical coupling bond-based peridynamic model. The present numerical results are in good agreement with the previous results obtained by the phase field method and the analytical solution, which further verifies the accuracy and ability of the proposed model. Finally, it is concluded that the more severe the chemical corrosion, the faster the crack growth rate and the deeper the chemical corrosion pits.

An adaptive contact model involving friction based on peridynamics

In peridynamics (PD), various existing contact models are difficult to resolve the contradiction between identifying contact surfaces and calculating contact forces accurately. This contradiction poses a challenge to the quantitative analysis of impact, extrusion, friction, and closure cracks. A friction contact model in PD based on family material particles was proposed to realize the adaptive identification of contact surfaces. The boundary normal vector was derived from the directional characteristics of the boundary reaction force. According to the dynamic equations of the boundary material particles, the proposed contact model achieves the precise calculation of the contact forces. In addition, an improved bond-based peridynamic model with shear bonds (improved PDS model) was extended to a three-dimensional form to achieve accurate deformation analysis of spatially complex deformable bodies. By combining the improved PDS model with the proposed contact model, the non-objectivity of the relative velocity is eliminated and the computational accuracy of the contact model is greatly improved. Quantitative analysis of the spatially deformed cantilever beam proves that the 3D improved PDS model possesses high accuracy of displacement calculation. Several examples with non-negligible friction were conducted to verify the reliability and efficiency of the proposed contact model. The results show that the proposed contact model can significantly improve the accuracy of displacement analysis and damage prediction in contact problems with non-negligible friction effects.