Ordinary State-based Peridynamics Research Articles

A peridynamic compensated critical energy density criterion for mixed-mode fracturing in quasi-brittle materials

This study integrates a compensated critical energy density criterion (CCED) into the ordinary state-based peridynamics framework to enable detailed analysis of mode I, mode II, and mixed-mode fracturing. The proposed bond failure criterion builds upon the advantages of the traditional critical energy density (CED) to enable precise calculation of strain energy density. Additionally, the critical bond rotation criterion (CR) is employed to account for shear bonds that the CED might overlook. Using the peridynamic differential operator (PDDO) format in formulating the entire computational framework enhances both accuracy and numerical stability. The model’s validation is conducted through benchmark examples, including the mode I double cantilever beam test, the mode II compact shear test, and a mixed-mode mechanical fracturing test in pre-notched specimens with central borehole loading. Convergence studies, comparative analyses with finite element method simulations and theoretical solutions thoroughly examine the proposed model’s performance and accuracy.

Scratch-induced micro-fracture behaviour and translational failure of glass: An experimental and numerical study

The structural safety of aged glass members shows significant sensitivity to surface scratches. The distribution and geometrical features of scratch impacts the glass fracturing. In this study, the fracture behaviour of scratched thin glass pane was investigated by the ordinary state-based peridynamic (OSB-PD) theory. Coaxial double ring test was first conducted to identify the mechanical and fracture behaviour of scratched glass plates that serve as the benchmark. The applicability of simplifying and reconstructing the actual scratches in numerical models was then examined. The refined three-dimensional fracture process from micro-crack coalescence to macro-propagation of featured cracks can be well described by the developed PD model. The propagation of in-plane radial cracks was found prior to median cracks along the thickness for scratched glass. Crack branch was observed due to shallow scratch depth and the primary crack branching point lies near the edge of the scratch. The transition from scratch-induced failure to strength-induced failure due to particular scratch spatial location was also tackled, which benefits the accurate performance assessment of scratched glass.

An Improved Ordinary State-Based Peridynamic Model for Granular Fractures in Cubic Crystals and the Effects of Crystal Orientations on Crack Propagation.

Material anisotropy caused by crystal orientation is an essential factor affecting the mechanical and fracture properties of crystal materials. This paper proposes an improved ordinary state-based peridynamic (OSB-PD) model to study the effect of arbitrary crystal orientation on the granular fracture in cubic crystals. Based on the periodicity of the equivalent elastic modulus of a cubic crystal, a periodic function regarding the crystal orientation is introduced into peridynamic material parameters, and a complete derivation process and expressions of correction factors are given. In addition, the derived parameters do not require additional coordinate transformation, simplifying the simulation process. Through convergence analysis, a regulating strategy to obtain the converged and accurate results of crack propagation paths is proposed. The effects of crystal orientations on crack initiation and propagation paths of single-crystal materials with different notch shapes (square, equilateral triangle, semi-circle) and sizes were studied. The results show that variations in crystal orientation can change the bifurcation, the number, and the propagation path direction of cracks. Under biaxial tensile loading, single crystals with semi-circular notches have the slowest crack initiation time and average propagation speed in most cases and are more resistant to fracture. Finally, the effects of grain anisotropy on dynamic fractures in polycrystalline materials under different grain boundary coefficients were studied. The decrease in grain anisotropy degree can reduce the microcracks in intergranular fracture and the crack propagation speed in transgranular fracture, respectively.

Simulating earth deformation evolution caused by groundwater pumping through ordinary state-based Peridynamics method

Earth deformation including land subsidence and the related earth fissures caused by the over-exploitation of groundwater resources is threatening countries all over the World. At present, the simulation of earth deformation of aquifer system is mainly based on continuum and contact mechanics. These approaches lead to two main challenges, i.e. the need of knowing the location of possible earth fissures in advance and using meshes that must be adjusted during the rupture propagation. Here, ordinary state-based Peridynamics method is firstly introduced to simulate the evolution of continuous and discontinuous earth deformation due to groundwater withdrawal. A failure criteria accounting for shear and tensile strain simultaneously is proposed. The approach is used to simulate the evolution of land subsidence and earth fissures in compressible sedimentary soils with buried rigid ridges. Two case studies are reproduced: one is a test carried out in laboratory and the other one concerns Guangming village, Yangtze River delta. In both cases, the presence of a shallow buried bedrock crossing the alluvial deposits is responsible for the initiation and growth of earth fissure that a satisfactory captured by the model. For the lab experiment, a 0.1 m deep fissure appearing above the rocky tip is computed. In Guangming village area, results show that some small fissure groups were initiated. Then, three main earth fissures developed rapidly with the maximum depth of 32.5 m. This study proposes a consistency modelling framework to simulate continuum earth deformation and discontinuous fissures due to groundwater withdrawal in specific geological settings, and supply a reliable scientific tool for early warning of this geo-hazard.

An ordinary state-based peridynamic model for granular fracture in polycrystalline materials with arbitrary orientations in cubic crystals

Granular fracture holds significant implications in material mechanics. However, the previous studies in ordinary state-based peridynamic (OSB-PD) framework often neglect the internal crystalline structure of particles or only consider limited crystal orientation. To address this gap, a novel OSB-PD model for granular fracture within polycrystalline materials is proposed, in which the periodic functions are incorporated in the PD strain energy density, taking into account the inherent random orientation in cubic crystals. By comparing energy density from PD and the classical continuum mechanics, four PD material parameters are defined. Moreover, the corresponding surface correction method in the global coordinate system is also proposed. Several numerical examples including fracture analysis of polycrystalline materials are conducted to validate the effectiveness of the proposed method. The proposed ordinary state-based peridynamic model offers a fresh perspective for investigating granular fracture behaviors within polycrystalline materials.

The Boundary Element Method for Ordinary State-Based Peridynamics

The Boundary Element Method for Ordinary State-Based Peridynamics

Fracture mechanics investigation for 2D orthotropic materials by using ordinary state-based peridynamics

Fracture behaviors of orthotropic plates are studied by using ordinary state-based peridynamic (OSPD) theory. Based on OSPD, a novel nonlocal formulation of interaction integral is proposed by considering the material orthogonality for the fracture parameter evaluation. By employing peridynamic differential operator, the partial differential terms in the formulation can be transformed into corresponding spatial integral form, which contributes to the calculations of stress intensity factors within the framework of OSPD. It has built up a relationship between classical theory and peridynamic theory. Static and dynamic fracture parameters are carefully evaluated. The crack propagation directions are predicted by prototype microelastic brittle (PMB) criterion in OSPD and maximum circumferential stress (MCS) criterion in classical theory. Several pre-cracked plates with orthotropic material are examined, and results are validated by comparing against the reference solutions. The relationship between fiber orientation and crack inclined angle are also examined. Meanwhile, the crack inclination determined by PMB criterion and MCS criterion are compared and discussed. Accuracy of the peridynamic orthotropic model and proposed nonlocal interaction integral are discussed in detail.

A coupled thermo-mechanical peridynamic model for fracture behavior of granite subjected to heating and water-cooling processes

Thermal damage and thermal fracture of rocks are two important indicators in geothermal mining projects. This paper investigates the effects of heating and water-cooling on granite specimens at various temperatures. The laboratory uniaxial compression experiments were also conducted. Then, a coupled thermo-mechanical ordinary state-based peridynamic (OSB-PD) model and corresponding numerical scheme were developed to simulate the damage of rocks after the heating and cooling processes, and the change of crack evolution process was predicted. The results demonstrate that elevated heating temperatures exacerbate the thermal damage to the specimens, resulting in a decrease in peak strength and an increase in ductility of granite. The escalating occurrence of thermal-induced cracks significantly affects the crack evolution process during the loading phase. The numerical results accurately reproduce the damage and fracture characteristics of the granite under different final heating temperatures (FHTs), which are consistent with the test results in terms of strength, crack evolution process, and failure mode.

Numerical simulation of impact process between spherical ice and a rigid plate based on the ordinary state-based peridynamics

Ice sphere impacts are an important issue in the study of composite materials, polar engineering, and aerospace engineering. The traditional mechanics numerical methods, rooted in classical continuum mechanics, like the Finite Element Method (FEM), grapples with the challenge of spatial derivatives not existing at the crack tip or crack surfaces, rendering it unsuitable for addressing discontinuity-related problems such as fracture and failure. Developed particle method peridynamics shows great ability in handling such discontinuous problems. This paper establishes an ice sphere impact model using the peridynamics method. To verify the accuracy of our model, we compared the simulated results with the tested results of the high-speed impacting ice sphere and obtained that the predicted impact load and crack propagation of ice spheres are consistent well with the experiment results. An isolated damage zone was found to appear at the top region when the surface crack just extends across the hemispherical plane. This phenomenon happens due to that the impact wave propagating along the sphere center axis reaches the top end faster than the surface wave, making spalling damage occurred at the top region, and the encounter between the reflected wave and the surface wave would make the spalling damage more severe.

A coupling algorithm of ordinary and non-ordinary state-based peridynamic models for fracture analysis in brittle and ductile materials

In this work, a novel coupling algorithm of the ordinary state-based peridynamic (OSB-PD) model and non-ordinary state-based peridynamic (NOSB-PD) model is provided under the peridynamic framework. The salient advantages of both OSB-PD and NOSB-PD models can be simultaneously inherited in this coupling model and approach. The feasibility of coupling these two models is first analyzed by demonstrating the numerical equivalence of their constitutive formulations in isotropic elastic and elastoplastic problems, and the detailed numerical implementation strategy of coupling formulation is given as well. With the coupling model in this study, the main calculation procedure of numerical simulations is accomplished in OSB-PD, while the damage and failure analysis is conducted in NOSB-PD. Moreover, different failure criteria are introduced in this coupling model to describe the damage and failure of brittle and metallic materials. Several typical examples, including the mode I and mixed mode I/II fracture of brittle materials, and the ductile fracture and impact failure of metallic materials, are conducted to demonstrate the robustness and accuracy of the presented coupling model and approach.

A fully coupled thermo-mechanical peridynamic model for cracking analysis of frozen rocks

In cold regions, the crack propagation and coalescence behaviors of frozen rocks with ice-filled flaws are very complex. Research on the failure mechanisms of such rock masses is essential to ensure the structural safety of geological engineering in low-temperature environments. Therefore, a fully coupled thermo-mechanical model considering the frost heave effect was constructed using the ordinary state-based peridynamic (OSBPD) theory, and the cracking characteristics of frozen rocks were analyzed. The constitutive equations of the OSBPD model were modified by introducing the coupling term for the thermal-structural interaction and an equivalent pressure density term for frost heave. An improved global convergence criterion was also proposed to achieve a fast convergence to the numerical solution. The correctness and validity of the proposed coupling model were confirmed using several numerical examples. Additionally, the effect of frost heave pressure on the cracking behavior and mechanical properties of frozen rocks was thoroughly discussed. The results show that (a) the frost heave pressure drives the rock to crack along the direction of prefabricated flaws, and (b) the ultimate compressive strength of frozen rocks exhibits a decreasing trend with an increase in frost heave pressure.

Modelling of dynamic fracture in bulk superconductor during pulsed field magnetization using ordinary state-based peridynamics

Bulk superconductor is subjected to a combination of huge electromagnetic force and temperature rise owing to the rapid movement of magnetic flux during pulsed field magnetization (PFM). Meanwhile, the heat generated will decrease the shielding effect and limit the strength of trapped field in turn. In view of its intrinsic brittle property and existing defects in fabrication, the bulk is prone to fracture under the complicated electromagnetic and thermal loadings. In this paper, the fracture behavior and damage evolution of the bulk superconductor with two or more cracks during PFM are investigated. Based on the electromagnetic field and temperature obtained by solving governing equation of H-formulation and heat transfer equation, the mechanical behavior of bulk is simulated with ordinary state-based Peridynamic model. The dynamic stress intensity factors considering thermal stress at crack tips are studied for different external magnetic fields and crack angles. Then, the influences of thermal contact resistance and local heat source are discussed. Finally, crack propagation in the bulk samples without reinforcement is modelled during PFM.

Closed-Form Wave Dispersion Relationships for Ordinary State-Based Peridynamics

Peridynamics (PD) is a new continuum mechanics formulation introduced to overcome limitations of classical continuum mechanics (CCM). This is mainly achieved by using integro-differential equations rather than partial differential equations. Another important difference of PD is its nonlocal nature with respect to local characteristic of CCM. Moreover, it has a length scale parameter, horizon, defining the range of nonlocal interactions between material points. This nonclassical behaviour also shows itself for dispersion relationships. As opposed to linear dispersion relationships for CCM, PD dispersion relationships are non-linear similar to the observed in experiments. In this study, closed-form dispersion relationships are provided for ordinary state-based peridynamics which is one of the most common PD formulations. Finally, derived closed-form solutions are used to demonstrate the dispersion relationships for various material systems including copper, gold, silver and platinum.

A peridynamic model for coupled thermo-mechanical-oxygenic analysis of C/C composites with SiC coating

The oxidation induced by coating cracking is a general failure mode of high-temperature composites such as C/C and C/SiC. However, this failure mode has not been fully investigated due to the lack of a method suitable for both crack simulation and oxidation interface capture. In this paper, a coupled thermo-mechanical-oxygenic peridynamic model is proposed for SiC-coated C/C composites in an oxidizing environment. The ordinary state-based peridynamics is extended to plane strain thermoelastic problems for orthotropic media to calculate the crack distribution. Cracks in the coating are modeled as oxygen diffusion channels and the peridynamic oxidation equation is developed to simulate the oxidation of the C/C substrate. Furthermore, the peridynamic non-Fourier heat transfer equation is employed to consider the finite thermal propagation speed and the model is validated by experimental results. Numerical examples are presented to investigate the oxidation failure after thermal shock, non-Fourier thermal shock fracture, and stress oxidation failure of SiC-coated C/C composites.

Ordinary state-based peri‑ultrasound modeling for monitoring crack propagation in plate structures using sideband peak count-index technique

This work presents a nonlocal theory based on ordinary state-based (OSB) peridynamics for numerical modeling of crack propagation under loadings and peri‑ultrasound modeling for elastic wave propagation that interacts with generated cracks in 3-D plate structures. Propagating cracks in plate structures for two types of loading – tensile and shear are considered. For both loading cases, at six different times after the start of the loading the analysis is carried out to investigate how the results vary for these six different stages of crack propagation. An ultrasonic excitation is applied at a point on the surface of the plate to generate guided waves propagating through the plate structure and interacting with generated cracks at different stages of crack formation. The out-of-plane velocity fields are recorded at the receiving material point for further analysis. A relatively new and promising nonlinear ultrasonic (NLU) technique called sideband peak count-index (or SPC-I) is adopted to extract the nonlinear response due to the generated cracks. Results show that when the crack propagation is at its early stage the SPC-I value first increases, and then decreases with time at the advanced stages of crack propagation. This is because for both types of loading (tensile or shear) the newly generated cracks are small cracks (or micro-cracks) in the beginning and then they evolve into macro-cracks under increasing loadings. The observation for SPC-I trends is consistent with other numerical modelings and experimental observations reported in the literature, and such qualitative matching gives confidence in the proposed method. The proposed method combining the peri‑ultrasound modeling and SPC-I technique presented in this paper can provide a new insight for monitoring crack propagation and give some guidance for experimental structural health monitoring (SHM) applications.

Study of multiple impact behaviors of CFRP based on peridynamics

Impact damage of carbon fiber reinforced polymers (CFRP) composites is a common form of damage in composite structures. In this work, a model integrating the structure model of composites, failure model and contact model with friction based on ordinary state-based peridynamics (PD) theory is developed to simulate multiple impacts of CFRP. The simulation results of single impact are in good agreement with the existing experimental and finite element simulation results, validating the proposed PD model. Then the effects of impact energy, stacking sequences and impact angles on the CFRP damage behaviors under multiple impacts are thoroughly investigated. Multiple high energy impacts may cause a non-linear increase in the amount of damage. The peanut-shaped stress distributions induced by impact along different directions are typical of the stress responses, which usually correspond to different fiber directions. Moreover, the laminates with stacking sequences of different fiber directions have a better impact damage resistance, as they can hinder the propagation of stress waves across the layers. In addition, as the impact angle increases, the stress and damage become more pronounced. The insights gained shed light on the repeated impact damage mechanisms of CFRP.

Thermomechanical modeling of pellet-cladding interaction using state-based peridynamics

Pellet-Cladding Interaction (PCI) is a significant concern for the safe and reliable design of nuclear fuel rods. This study presents a modeling approach for investigating PCI employing the Ordinary State-Based PeriDynamics (OSB-PD) while considering irregular discretization to improve the modeling of curved boundaries of fuel rods with geometric precision. Unlike the existing PD models for fuel pellet, the present study considers contact and heat transfer between fuel pellet and cladding. Also, it presents a new frictional contact model with stick and slide friction. The heat conduction is modeled through gas, contact and radiation between pellet and cladding. In order to capture complex fragmentation and failure in pellet, material variability is considered by applying randomized critical stretch values with normal distribution. Specifically, it presents the effects of friction coefficient, gap size, and power level on crack patterns in fuel pellets while considering PCI. The PD predictions show that the number of major radial cracks remains constant as the friction coefficient increases; however, the number of circumferential cracks increases significantly. As the gap size increases, both the temperature in fuel pellet and the temperature difference across the gap increase. Also, the number of cracks in fuel pellet increases with the increasing value of gap size. The power level has a strong influence on the temperature gradient of the fuel rod, and the number of cracks in fuel pellet directly correlates with the power level.

Simulation of Brittle Materials Based on Ordinary State-based Peridynamics

We present a coupled model for brittle materials based on the ordinary state-based peridynamics. The ordinary state-based peridynamics theory is suitable for describing the plastic incompressibility problem and has good application prospects in the fracture damage simulation of brittle materials. The brittle material is a material with a compressive load-bearing capacity greater than the tensile load-bearing capacity and has the characteristics of incompressibility, which is consistent with the application scope of ordinary state-based peridynamics. The mechanical behavior of brittle materials is simulated and analyzed using the ordinary state-based peridynamics theory coupled with brittle materials, and the validity of the model is verified using the finite element method. The results show that the model can effectively capture the displacement field distribution of the brittle material, and the comparison of the finite element results displacement field is consistent. Finally, crack extension analysis is performed on the brittle material plate with prefabricated defects, where the cracks extend along the defect location, bifurcate, and finally penetrate the upper and lower surfaces of the material. The results show that the model can effectively predict the crack expansion path of the material, which provides a new research idea for the simulation of crack expansion in brittle materials.

Ordinary state-based peri-ultrasound modeling to study the effects of multiple cracks on the nonlinear response of plate structures

Since it is almost impossible to carry out a comprehensive parametric investigation experimentally for internal cracks with different geometry and orientation, a good numerical modeling and simulation technique is necessary to have a clear understanding of the physics of wave propagation and its interaction with cracks. Such investigation is helpful for structural health monitoring (SHM) with ultrasonic techniques. This work presents a nonlocal peri-ultrasound theory based on ordinary state-based (OSB) peridynamics for modeling elastic wave propagation in three-dimensional (3-D) plate structures containing multiple cracks. A relatively new and promising nonlinear ultrasonic technique called Sideband Peak Count – Index (or SPC-I) is adopted to extract the nonlinearity generated from the interactions between elastic waves and multiple cracks. Effects of three main parameters – the distance between the acoustic source and the crack, the crack spacing and the number of cracks are investigated using the proposed OSB peri-ultrasound theory together with the SPC-I technique. For each of these three parameters investigation, different crack thicknesses were considered − 0 mm (crack-free), 1 mm (thin crack), 2 mm (intermediate thickness) and 4 mm (thick crack); thin and thick cracks are defined after comparing the crack thickness value with the horizon size mentioned in the peri-ultrasound theory. It is found that for obtaining consistent results the acoustic source should be placed at least one wavelength away from the crack and crack spacings also play an important role on the nonlinear response. It is concluded that the nonlinear response diminishes when the cracks become too thick, and thin cracks show higher nonlinearity than that of thick cracks and no cracks. Finally, the proposed method which is combining the peri-ultrasound theory and SPC-I technique is used for monitoring cracks’ evolution process. The numerical modeling results are compared with the experimental findings reported in the literature. Consistent qualitative trends in SPC-I variations predicted numerically and obtained experimentally are observed, thus it gives confidence in the proposed method.

Characterization and mechanism of petaling failure under high-velocity impact using plastic state-based peridynamics

Petaling failure is one of the most prevalent modes for tubular members exposed to high-velocity impacts. Capturing progressive damage in real-time is crucial for elucidating the trigger mechanism and damage evolution. In this study, a three-dimensional plastic ordinary state-based peridynamics (POSB-PD) was developed to predict the material discontinuity of high-grade X80 steel pipelines under high-velocity impacts. Characteristics of the petaling mode were analyzed quantitatively using cluster analysis and statistical description. The impact processes of petaling formation, including crack propagation, stress triaxiality, bond stretch, and damage number, were systematically discussed. Penetration mechanisms of petaling formation were revealed based on bond breakage and cumulative damage. Intrinsic relationships between mesoscopic defects and macroscopic fractures were established, considering various fracture parameters. A simplified model of energy release rate was established to evaluate the critical fracture condition. These results provided insights into revealing the penetration mechanisms and establishing macro-micro failure linkages for ductile failures in metal structures.