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A technique is proposed for reproducing particle size distributions in three‐dimensional simulations of the crushing and comminution of solid materials. The method is designed to produce realistic distributions over a wide range of loading conditions, especially for small fragments. In contrast to most existing methods, the new model does not explicitly treat the small‐scale process of fracture. Instead, it uses measured fragment distributions from laboratory tests as the basic material property that is incorporated into the algorithm, providing a data‐driven approach. The algorithm is implemented within a nonlocal peridynamic solver, which simulates the underlying continuum mechanics and contact interactions between fragments after they are formed. The technique is illustrated in reproducing fragmentation data from drop weight testing on sandstone samples. [ABSTRACT FROM AUTHOR]

This paper introduces a particle‐based framework for simulating the behavior of elastoplastic materials and the formation of fractures, grounded in Peridynamic theory. Traditional approaches, such as the Finite Element Method (FEM) and Smoothed Particle Hydrodynamics (SPH), to modeling elastic materials have primarily relied on discretization techniques and continuous constitutive model. However, accurately capturing fracture and crack development in elastoplastic materials poses significant challenges for these conventional models. Our approach integrates a Peridynamic‐based elastic model with a density constraint, enhancing stability and realism. We adopt the Von Mises yield criterion and a bond stretch criterion to simulate plastic deformation and fracture formation, respectively. The proposed method stabilizes the elastic model through a density‐based position constraint, while plasticity is modeled using the Von Mises yield criterion within the bond of particle paris. Fracturing and the generation of fine fragments are facilitated by the fracture criterion and the application of complementarity operations to the inter‐particle connections. Our experimental results demonstrate the efficacy of our framework in realistically depicting a wide range of material behaviors, including elasticity, plasticity, and fracturing, across various scenarios. [ABSTRACT FROM AUTHOR]

In engineering applications, the phenomenon of cracking is often accompanied by a coupled multiphysics effect. Peridynamics (PD) is an effective approach for solving cracking problems, but currently, no general PD model accounts for the coupling of multiple physical fields. In this work, we develop a PD model of coupled deformation, heat conduction, species diffusion, and chemical reactions. First, we establish the equations for mass, linear momentum, and energy. Then we establish fully coupled constitutive laws that interpret the interactions between the various fields and formulate evolution equations that govern the flux of species and heat. These laws and equations are developed based on the inequality of energy dissipation and the principles of chemical kinetics. Species diffusion and chemical reactions are treated as separate processes to study their effects on the Helmholtz free energy density of solids and the subsequent formation and propagation of cracks. In addition, certain coupling coefficients are calibrated by equating the corresponding physical quantities in the PD model with those in continuum mechanics. Four specific cases are simulated to validate the model, including redistribution of vacancies in ceramics, hydrogen traps, and embrittlement in metals. • A thermo-chemo-mechanically coupled peridynamics model is developed. • The expansion factor is defined to reveal its effect. • The chemical kinetics in a nonlocal form is derived. • The different effects of species diffusion and chemical reactions on cracks are investigated. [ABSTRACT FROM AUTHOR]

As a type of non-pneumatic wheel, mechanical elastic wheel (MEW) can is resistant to deformation and continue to be used when rubber is damaged by the bullet. In order to prove the deformation resistance ability of MEW, the model of rubber damaged by bullet based on the non-ordinary state-based peridynamic is proposed for the first time. By considering the two conditions with bullet penetrating and staying in rubber respectively, the model is simulated and predicted. The results show that the simulations about the model of rubber damaged by bullet is consistent with experiments, the validity of the model is verified. It explains the physical process of penetrating bullet hole being smaller than the bullet diameter, and explains the reasons for the rubber forming bulge after bullet staying in rubber. The MEW damaged by bullet still has the running ability, the damage wheel does not have obvious secondary damage after running, and MEW has the deformation resistance ability after bullet damage. It has reference significance for the research of rubber tearing problem.

In this work, we present a peridynamic-based simulation method for modeling quasi-static fracture propagation in isotropic and anisotropic rock within the framework of peridynamic least square minimization (PDLSM). The original isotropic elastic PDLSM is further extended to investigate fracture propagation in anisotropic materials in this study. The proposed AN-PDLSM model integrates an anisotropic model in fracture mechanics to analyze the failure process of anisotropic rocks. An important advancement in this research lies in the incorporation of the maximum energy release rate criterion (MERR) into the PDLSM framework for the first time. This enhancement enables accurately determining crack propagation and the associated crack angles. The proposed model utilizes the energy release rate calculated through the J-integral method to assess bond breakage, and it employs a mesh-independent, piecewise linear fracture model to describe crack propagation. The proposed method fully combines the merits of traditional fracture mechanics with the unique capabilities of peridynamics. To demonstrate the effectiveness of the proposed model, a simulation of fracture evolution in isotropic plates subjected to semi-circular bending tests is presented and compared with experimental results. It is shown that the proposed model accurately replicates fracture trajectories in isotropic specimens. In the context of anisotropic rock, the effect of a weak coefficient on crack morphology is discussed in order to obtain a suitable value. Additionally, the impact of bedding angles on fracture paths through our proposed model is also explored, revealing excellent agreement with experimental results. The findings in this research demonstrate that the proposed AN-PDLSM model is exceptionally proficient at capturing the intricate, oscillating crack paths observed in anisotropic rock materials. [ABSTRACT FROM AUTHOR]

The numerical simulation of crevice corrosion of a titanium alloy upon exposure to an aggressive marine environment is presented and discussed. The model parameters were calculated using the IR voltage theory and a one-dimensional calibration peridynamic model. The values of pH inside the crevice, having openings of 0.01, 0.05 and 0.1 mm, were calculated. A crevice corrosion simulation model, for a crevice of 0.01 in width and 4.95 mm in depth, was built, and a full-scale simulation was conducted using the model. Experiments were conducted to verify the simulation results. The chosen test pieces having crevice openings of 0.01, 0.05 and 0.1 mm were soaked in EXCO solution of 65 °C for 5, 10 and 15 days. The test results showed the pH inside the crevice initially decreased with depth and reached a minimum value when the titanium ions were saturated. Thereafter, the value of pH increased with depth. Comparing results of the simulation with experiment, results of corrosion depth within the crevice were in good consistence. Depth of pits as a direct consequence of crevice corrosion had an exponential relationship with distribution of pits. The IR voltage decrease was proven to be an important factor that exerts an influence on depth of corrosion due to crevice corrosion. [ABSTRACT FROM AUTHOR]

The stress of rock-like materials first increases and then decreases with an increase in the strain, and finally become damaged under tensile or compressive loads. It is not suitable to use the traditional bond-based peridynamics model to simulate the crack propagation of rock-like materials. Based on the bond-based peridynamics theory, a constitutive model has here been proposed that can reflect the characteristic of the stress with an increase in strain (i.e., that the stress of the rock-like materials first increases and then decreases, and finally fails). This makes up for the weakness with the traditional bond-based peridynamics theory, which fails to reflect the stress-strain change in rock-like materials. The strain energy density of the proposed constitutive model of rock-like materials has been derived and compared against the classical elasticity theory to obtain the model constants with respect to the elasticity moduli for plane stress conditions. Based on the here proposed constitutive model for rock-like materials, a numerical solution program for rock-like materials has been written using the Fortran language. For different loading conditions, the crack propagation process for an intact specimen has been simulated and compared with experimental results. This was also the situation for a specimen with a pre-existing flaw, a single non-straight flaw, and with three pre-existing flaws for plane stress conditions. The numerical simulation results were in good agreement with the corresponding experimental results. By using this proposed constitutive model, it was possible to simulate and predict the mechanical properties of rock-like materials, and the process of crack initiation, propagation, and coalescence under different loading conditions. The results from the present study can thus provide a reference for practical engineering. [ABSTRACT FROM AUTHOR]

Forecasting water content dynamics in heterogeneous porous media has significant interest in hydrological applications; in particular, the treatment of infiltration when in presence of cracks and fractures can be accomplished resorting to peridynamic theory, which allows a proper modeling of non localities in space. In particular, in this framework, we introduce a suitable influence function able to provide a nonlocal interaction between material particles; moreover, we make use of Chebyshev transform on the diffusive component of the equation and then we integrate forward in time using an explicit method. We prove that the proposed spectral numerical scheme provides a solution converging to the unique solution in some appropriate Sobolev space. We finally exemplify on several different soils, also considering a sink term representing the root water uptake. • We introduce a nonlocal form of Richards' equation using a non-standard influence function. • We compute the solution to the semi-discretized Richards' equation with Dirichlet boundary conditions with spectral methods. • We prove that the solution to the semi-discretize scheme converges, in a suitable Sobolev space, to the exact solution. [ABSTRACT FROM AUTHOR]

Bi2Sr2Ca1Cu2O8+x (Bi-2212) wire is the potential material for the design of the high field superconducting magnet. However, compared with excellent current-carrying capacity, low mechanical strength can restrict the wide application of Bi-2212 wire. The superconducting filaments of Bi-2212 wire is sensitive to strain and is prone to fracture, which can lead to the degradation of critical current. Therefore, it is necessary to study the damage behavior of Bi-2212 round wire under external loads. In this paper, the three-dimensional mechanical damage and degradation of critical current in Bi-2212 composite wire under axial tensile load, radial compression load, bending and torsional loads are systematically investigated using ordinary state-based elastoplastic peridynamic (PD). The axial tensile simulations with the process of damage and thermal residual stress are in agreement with the experiment, and the critical current degradation with tensile strain is qualitatively predicted. The effect of plastic deformation of alloy on damage of filament bundles under radial compression load is explored. Finally, the simulations also reveal the damage characteristics of filament bundles in Bi-2212 round wire under torsional and bending loads. The above results are expected to provide a reference for application of Bi-2212 HTS round wire. [ABSTRACT FROM AUTHOR]

• A compensated critical energy density (CCED) criterion is proposed within peridynamics to optimize mixed-mode fracturing simulation in quasi-brittle materials. • CCED's robustness in modeling mode I, II, and mixed-mode fracturing is validated through benchmark tests. • A 90° approach angle between mechanically induced fractures and pre-existing natural fractures is commonly observed. 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. [ABSTRACT FROM AUTHOR]

• A newly developed geometry-constraint-based non-ordinary state-based peridynamic theory is adopted. • The development of new cracks of flawed rock-like materials under compression-dominating stresses is numerically predicted. • The crack propagation mechanism of flawed rock-like materials is revealed based on the dynamic evolution of stress fields. • The tensile and shear failure modes of brittle materials can be reasonably modeled. Comprehending the cracking evolution and failure mechanism of flawed rocks under compression-dominating stresses is essential to evaluating the stability of rock engineering. In this work, a newly developed geometry-constraint-based non-ordinary state-based peridynamic (GC-NOSBPD) theory that can eliminate the numerical oscillation is applied to predict the development of new cracks of flawed rocks under compression-dominating stresses. The failure of bonds between particles is judged by the bond-associated stress-based fracture criteria. The cracking evolution trajectories of semi-disc and disc specimens containing flaws are traced. The effects of flaw inclination angle on the cracking trajectories are analyzed. The cracking evolution paths of rock-like specimens with two flaws under uniaxial and biaxial compression are molded. The effects of confining stress on the growth of new cracks are investigated. The confining stress restrains the propagation of tensile cracks, but it is easy to promote the growth of shear cracks. The concentration and transfer effects of stress can plainly revel the failure mechanism of flawed rocks. The present numerical method is reasonable to predict the tensile and shear failure modes of flawed specimens under compression-dominating stresses. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a peridynamic (PD) model for the prediction of the viscoelastic creep deformation and failure model. The viscoelastic characteristic consists of several stages, namely primary creep, secondary creep, tertiary creep and fracture. A non-linear viscoelastic creep equation based on the internal state variable (ISV) theory covering four creep stages and PD equations are used. The viscoelastic equation is inserted into the PD equation to derive a PD model with two time parameters, i.e., numerical time and viscoelastic real time. The parameters of the viscoelastic equation are analyzed and optimized. A comparison between numerical and experimental data is performed to validate this PD model. The new PD model for nonlinear viscoelastic creep behavior is confirmed by an acceptable similarity between the numerical and experimental creep strain curves with an error of 15.85%. The nonlinearity of the experimental and numerical data is sufficiently similar as the error between the experimental and numerical curves of the secondary stage strain rate against the load is 21.83%. The factors for the errors are discussed and the variation of the constants in the nonlinear viscoelastic model is also investigated. [ABSTRACT FROM AUTHOR]

В настоящей работе доказывается единственность и существование решения задачи Коши для интегро-дифференциального уравнения, связанного с перидинамической моделью механики твёрдого тела с двумя пространственными переменными.

Copyright of Chinese Journal of High Pressure Physics is the property of Chinese Journal of High Pressure Physics Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Numerical simulations of the cracking behavior of the top layers of multilayer thermal barrier coatings (TBCs) can effectively reveal the failure mechanisms of the TBCs. Current finite element method (FEM)-based simulation means have been applied to solve certain simple cracking problems in TBCs; however, they cannot effectively describe complex cracking problems in TBCs such as coalescence, intersection, and interference among multiple cracks. Peridynamic (PD), a newly developed mechanical theory, has been widely studied to provide analysis for cracking problems in TBCs. In this paper, a numerical model of TBCs is built by the bond-based PD (BB-PD) theory. Complex cracking behaviors, such as spontaneous crack propagation at both interfacial and internal regions, coalescence, and interference among multiple cracks, are simulated under isothermal cooling and gradient cooling conditions. In addition, the effects of interfacial roughness and calcium–magnesium–alumina–silicate (CMAS) inclusions on the cracking behavior are discussed. The results show that the PD model accurately captures complex cracking behaviors observed via scanning electron microscopy (SEM). Given the ability of the model for analyzing discontinuities in TBCs, it can help to further clarify the fracture mechanisms of TBCs. [ABSTRACT FROM AUTHOR]

• Uncovering the inherent limitations of peridynamic model in predicting dynamic crack velocity. • The influence mechanism of size and stiffness of the nonlocal region on crack velocity is elucidated. • Exposing wave speed degradation rules in peridynamic and formulating a prediction formula for limiting crack velocity. • The crack velocity is regulated by the local stiffness and point density. Peridynamic, an innovative solid mechanics method in continuum theory, offers unique advantages for discontinuity modeling such as fracture simulation. This study addressed a puzzling phenomenon observed in peridynamic dynamic fracture modeling, where the predicted normalized limiting crack velocity remained nearly constant across different material parameters input. Delving into the mechanism influencing predicted crack velocity in peridynamic model, we systematically explored the impacts of stiffness distribution of the nonlocal region around the crack tip, through numerical simulations and theoretical analyses. Additionally, the implicit wave velocity degradation in the damage zone of the peridynamic model was uncovered. A prediction formula for the limiting crack velocity was then derived based on these analyses. Furthermore, the wave velocity degradation scheme and the proposed prediction model were validated through numerical instances, involving convergence studies and material density degradation investigation. This work contributes to a further understanding of the peridynamic model and provides new insights for crack velocity prediction in dynamic fracture scenarios. [ABSTRACT FROM AUTHOR]

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension n ∈ ℕ. [ABSTRACT FROM AUTHOR]

In this work, we modeled the brittle fracture of shell structure in the framework of Peridynamics Mindlin-Reissener shell theory, in which the shell is described by material points in the mean-plane with its drilling rotation neglected in kinematic assumption. To improve the numerical accuracy, the stress-point method is utilized to eliminate the numerical instability induced by the zero-energy mode and rank-deficiency. The crack surface is represented explicitly by stress points, and a novel general crack criterion is proposed based on that. Instead of the critical stretch used in common peridynamic solid, it is convenient to describe the material failure by using the classic constitutive model in continuum mechanics. In this work, a concise crack simulation algorithm is also provided to describe the crack path and its development, in order to simulate the brittle fracture of the shell structure. Numerical examples are presented to validate and demonstrate our proposed model. Results reveal that our model has good accuracy and capability to represent crack propagation and branch spontaneously. [ABSTRACT FROM AUTHOR]

Copyright of Bulletin of the Chinese Ceramic Society is the property of Bulletin of the Chinese Ceramic Society Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The accurate understanding of the behavior and analysis of the structures is necessary to appropriate design of them. Over the past decades, various methods have been proposed for the analysis of structures that each of them include some deficiencies. The peridynamic is a new method for solving the problems. This method overcomes the deficiencies and limitations of conventional methods such as the finite element method due to its unique capabilities. Nowadays, it uses as a powerful method to solve problems. After the proposition of peridynamic theory in 2000, various problems have been solved. However, very limited work has been done in the solving longitudinal vibration of axial members and modeling of lattice structures. In this paper, the behavior of this type of members was examined by the peridynamics method and its capability to take into account the static and dynamic loading, as well as temperature variations, was shown using MATLAB software. The peridynamic results are in good agreement with analytical results. Also, for certain states of the problem, it is impossible or difficult to find an answer by analytical methods or other methods, while using the peridynamic method can easily be answered. Finally, the peridynamic model has been extended to truss modeling and the ability of this method to analyze any types of trusses were shown.

• A non-local absorbing boundary conditions is constructed in peridynamic theory. • Motion equation of peridynamic theory is re-derived considering material damping. • Algorithm to realize the non-local absorbing boundary conditions in peridynamics is proposed. • Elastic wave propagation in semi-unbounded and unbounded solids are simulated. In order to simulate the elastic wave propagation in unbounded solids, this paper introduces a non-local method to construct absorbing boundary conditions in the bond-based peridynamic theory. To construct non-local absorbing boundary conditions in the form of absorbing layers with increasing damping, we first re-derived the particle motion equilibrium equation of the bond-based peridynamic theory, based on the principle of virtual work and Lagrange's equation on the premise of considering material damping. We also give the expression for the inter-particle damping force density which satisfies the balance laws of linear momentum and angular momentum. This paper also provides a process and algorithm for realizing the non-local absorbing boundary conditions in the bond-based peridynamic theory, and adopts the Fortran language for computer program compilation. Finally, we simulate the elastic wave propagation in semi-unbounded and unbounded solids as numerical examples. The results show that such non-local absorbing boundary conditions can stably absorb and attenuate incident elastic waves. Since it is derived in the time domain, this method does not require frequency-domain operations, such as Fourier and Laplace transformations, nor does it require wave field splitting. [ABSTRACT FROM AUTHOR]

This article presents a new formulation of bond‐based peridynamic theory (PD) encompassing material texture heterogeneity through fractal geometry concepts and transforming all relevant variables from Euclidean space to fractal space. The heterogeneity incorporation is realized through the introduction of a linear transformation fractal coefficient (LTFC). By considering a mean value for all LTFCs of all spatial coordinates. For example, x, y, z, and using the relationship between fractal dimension Df and the Hurst exponent H. We present a fractal version of the bond‐based PD method that the equation of motion is conformed to fractal space. Validation of the formulation is accomplished in two steps: first, the effect of irregular mass distribution is characterized by the fractal dimension of the fracture surface roughness through experimental results of a series of single‐edge notched tensile tests of a ductile material. In the next step, the proposed formulation is calibrated with the fractal dimension obtained experimentally. The results are compared with the classical bond‐based PD formulation and the finite element analysis results. The obtained results in this article indicate that the irregular mass distribution of material can be introduced into particle‐based numerical methods using an adequate fractal exponent. The fractal version provides a good approximation of the fracture behavior of materials. [ABSTRACT FROM AUTHOR]

В настоящей работе рассматривается задача с начальными данными для нелинейного интегро-дифференциального уравнения, связанного с перидинамической моделью. Доказывается существование и единственность решения.

Abstract The high‐pressure gas expansion‐induced deformation and dynamic fracture of coal are important parts of coal and gas outburst. To better understand the law of this process, laboratory experiments and numerical simulation are used to study the law of damage. A cavity with different pressures of CH4 or N2 was destroyed by a jack to achieve the rapid expansion of the gas and coal fracture inside. The particle size distribution of the coal particles before and after the experiment was measured, and the breakage ratio and the newly added surface area were calculated. The experimental results indicate that during the gas expansion process, the breakage ratio of coal and the newly added surface area clearly increase with the increase in gas pressure. Finally, a numerical model based on peridynamic theory was developed to simulate crack generation and the propagation of coal induced by the expansion of gases at different pressures. The numerical simulation results show that the higher the initial gas pressure is, the higher the number of failure units. Moreover, only when the gas pressure is large enough will the coal crack in various directions at the same time.

The peridynamic method had great advantages in solving the problem of crack propagation in rock material due to its nonlocal characteristics. However, the method also faced problems such as zero-energy mode and boundary effects. In order to solve the above problems, the paper first proved that the non-ordinary state-based peridynamic(NOSB-PD)method was equivalent to the Galerkin weak form with nodal integral scheme. The equation of solving the deformation gradient F in NOSB-PD method was extended to a more general form, namely the peridynamic differential operator(PDDO)approximation. Since the PDDO had the same displacement approximation with the reconstruction kernel particle method(RKPM), this paper compared the difference between the two methods in the approximations of displacement derivatives in detail. The RKPM-PD coupling method with higher accuracy was proposed in the paper. Several numerical examples proved the accuracy of the new method in predicting the dynamic crack propagation in rock. [ABSTRACT FROM AUTHOR]

Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Here, we apply Peridynamic (PD) enrichment to propagate cracks in the PUM global–local enrichment scheme. We apply linear elasticity globally and PD over local zones where fractures occur. The elastic fields provide appropriate boundary data for local PD simulations on a subdomain containing the crack tip to grow the crack. Once the updated crack path is found the elastic field in the body and surrounding the crack is updated using the PUM basis with an elastic field enrichment near the crack. The subdomain for the PD simulation is chosen to include the current crack tip as well as features that influence crack growth. This paper is part II of this series and validates the combined PD/PUM simulator against the experimental results. The results of numerical simulation show that we attain good agreement between experiment and simulation with a local PD subdomain that is moving with the crack tip and adaptively chosen size. • A novel Multiscale Fracture Model using Peridynamic Enrichment of Finite Elements with an Adaptive Partition of Unity • Validation against three experimental data sets. • In addition to the experiments, we validate our hierarchical coupling technique approach with a concurrent coupling approach proposed by Ni et al. using the same experimental data. [ABSTRACT FROM AUTHOR]

• A novel dynamic damage constitutive model of rock-like materials based on peridynamics is proposed. • The relationship between the micro-modulus constant and elastic modulus has been derived. • Dynamic mechanical properties and energy dissipation of argillaceous sandstone with different internal flaws are obtained. • The dynamic failure process of argillaceous sandstone specimens are captured by numerical simulation. The original bond-based peridynamic model addresses brittle fracture rather than progressive fracture. It also does not account for the impact of strain rate on bond strength and fails to incorporate the impact of internal length on non-local long-range forces. As a result, it is inadequate for simulating the dynamic failure of rock-like materials under impact loading. To address these limitations, a new approach involves a sextic polynomial attenuation function. This function captures the impact of internal length on non-local long-range forces and leads to the derivation of a corresponding micro-modulus expression. A damage function and a new failure criterion are integrated to describe the nonlinear mechanical behavior and progressive failure of rock-like materials. The influence of strain rate on bond dynamic strength is considered to reflect the strain rate effect in rock materials. The outcome is a newly proposed dynamic damage constitutive model that overcomes the three shortcomings of the original bond-based peridynamic model when simulating the dynamic failure of rock-like materials. Numerical examples are simulated based on this model, and their results are compared with the experimental data. This comparison allows further exploration of the volumetric fracturing behavior and mechanical properties of rock specimens with internal flaws in three dimensions when subjected to impact loading. The findings demonstrate the effectiveness of the proposed model in accurately simulating crack propagation and mechanical properties across varying strain rates in rock-like materials. Notably, the internal flaw's depth significantly influences the specimen's mechanical properties and energy utilization efficiency at a specific flaw angle α = 90 °. Complex three-dimensional cracks, consisting of shell-like, wing, anti-wing, and shear cracks, emerge due to the interconnection of various crack types. The shape of these cracks is intricately linked to the flaw angle and depth. Ultimately, this model solves the challenge of capturing internal crack growth during dynamic impact processes that are often difficult to observe in experimental tests. [ABSTRACT FROM AUTHOR]

This paper presents the peridynamic numerical method for nonlinear viscoelastic creep behaviour which consists of primary, secondary, tertiary creep stages and creep rupture. A nonlinear viscoelastic creep constitutive equation based on internal state variable (ISV) theory which covers four creep stages is examined. The viscoelastic equation is substituted into material parameter in the peridynamic equation to derive a new peridynamic method with two time parameters i.e. numerical time and real time. The parameters of the viscoelastic equation is analyzed and evaluated. In validating this peridynamic method, a comparison is made between numerical and experimental data. The peridynamic method for nonlinear viscoelastic creep behaviour (VE-PD) is approved by the good similarity between numerical and experimental creep strain curves with overall difference of 10.67%. The nonlinearity of experimental and numerical data is adequately similar as the error between experimental and numerical curves of secondary stage strain rate against load is 8.022%. The shapes of fractured numerical specimen show good resemblance with the experimental result as well.

Peridynamics (PD) is a nonlocal continuum theory based on integro-differential equations without spatial derivatives. The elongation fracture criterion is implicitly incorporated in the PD theory, and fracture is a natural outcome of the simulation. On the other hand, a new fracture criterion based on the crack opening displacement combined with peridynamic (PD-COD) is proposed. When the relative deformation of the PD bond between two particles reaches the critical crack tip opening displacement of the fracture mechanics, we assume that the bond force vanishes. The new damage rule of fracture criteria similar to the local damage rule in conventional PD is introduced to simulate the fracture. In this paper, first, a comparative study between XFEM and PD is presented. Then, four examples, i.e., a bilateral crack problem, double parallel crack, monoclinic crack, and the double inclined crack are given to demonstrate the effectiveness of the new criterion.

Perdynamic（PD） is used for analyzing crack propagation in engineering material fracture,the bonds constitute to the constitutive relation of the peridynamic,and the broken bonds represent to material fracture,but it can’t be used for the fatigue analysis.Based on the Miner’s linear cumulative damage theory,the bond fatigue break criterion was proposed instead of the old break criterion,the PD fatigue fracture model for high cycle fatigue was constructed.Finally,a bending fatigue fracture analysis was carried out on a standard involute gear,the results show that the fatigue fracture process for gear by PD fatigue fracture model is consistent with experiment,the PD fatigue fracture model provides a method which consists of life prediction and fatigue crack propagation.

Functionally graded materials (FGMs) can be used as an interlayer. The interlayer connects two homogeneous materials and slows mutations in material properties, owing to the variability of parameters and microstructure. In this numerical simulation, a bond-based peridynamic (PD) model was used to perform dynamic fracture analysis of a continuous beam and a sandwiched structure with an FGM interlayer. Some simulation results of two convergence studies, which include grid refinement (m-convergence) and the reducing radius of PD horizon (δ -convergence), were discussed. Under a four-point bending load, the propagation behaviors of single pre-crack in the continuous beam with the FGM interlayer at different locations are simulated to compare with the experimental results. For the sandwiched structure, the problems of a single crack and double cracks were considered. The numerical results show that the crack can deflect toward areas where the energy release rate is low. Additionally, the crack on the weak side for symmetric cracks only has fracture behavior, and if the double cracks are asymmetrical, all these cracks have fracture behaviors and propagation of the initial crack in the stiff region has a higher sensitivity than in the compliant region. [ABSTRACT FROM AUTHOR]

A fast algorithm for the nonlocal unsteady problems was proposed, which can be employed in the numerical simulation of nonlocal diffusion and peridynamic. The surrogate model constructed by the proper orthogonal decomposition (POD) speeds up the process of solving equations by reducing the order of linear equations. Then, the accuracy and efficiency of the proposed algorithm was verified by numerical experiments. The results showed that this approach ensures accuracy while reduces the computational burden of the nonlocal model. [ABSTRACT FROM AUTHOR]

The high‐pressure gas expansion‐induced deformation and dynamic fracture of coal are important parts of coal and gas outburst. To better understand the law of this process, laboratory experiments and numerical simulation are used to study the law of damage. A cavity with different pressures of CH4 or N2 was destroyed by a jack to achieve the rapid expansion of the gas and coal fracture inside. The particle size distribution of the coal particles before and after the experiment was measured, and the breakage ratio and the newly added surface area were calculated. The experimental results indicate that during the gas expansion process, the breakage ratio of coal and the newly added surface area clearly increase with the increase in gas pressure. Finally, a numerical model based on peridynamic theory was developed to simulate crack generation and the propagation of coal induced by the expansion of gases at different pressures. The numerical simulation results show that the higher the initial gas pressure is, the higher the number of failure units. Moreover, only when the gas pressure is large enough will the coal crack in various directions at the same time. [ABSTRACT FROM AUTHOR]

Recently developed peridynamic (PD) simulations, allowing spontaneous crack nucleation and propagation, have been conducted to investigate the dynamic breakage behavior of impact spherical particles as a function of impact speeds. With the increase of impact speed, the Hertzian cracks, primary meridian cracks, secondary meridian cracks, circumferential cracks, and crack bifurcations are observed. The crack propagation speeds can reach 62% Rayleigh wave speed at impact speed of 20 m/s. Then with the further increase of impact speed, crack bifurcation is observed due to dynamic path instability, leading to the catastrophic failure of impact spheres. The evolution of damage ratio, defined as the ratio of broken bonds to total initial bonds in PD model, agrees with previous results using discrete element method (DEM), validating the current model. Furthermore, the comparison between computed and theoretic contact force is performed along with experimental measurements. The research provides additional insights into breakage mechanisms of particles. Unlabelled Image • Peridynamics is used to investigate dynamic breakage of impact spherical particles. • Breakage patterns of impact spherical particles are achieved. • Influence on crack propagation speed of impact speed is explored. • Role of crack nucleation and propagation during particle breakage is revealed. [ABSTRACT FROM AUTHOR]

The peridinamics theory, proposed by Silling in 2000, is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives and may cope with discontinuous displacement fields commonly occurring in fracture mechanics. Instead of spatial differential operators, integration over differences of the displacement field is used to describe the existing, possibly nonlinear, forces between particles of the solid body. Beside an overview of the peridynamics modelling and its application, results concerning the mathematical solution of the governing equation, which is a partial integro-differential equation with second-order time derivative, are presented. In this paper we consider well-posed Cauchy problem for the singular periodic integro-differential equation of peridinamics. In case of two-dimensional space the existence and uniqueness of solution are proved in Sobolev spaces. In multidimensional space the unique solvability of the problem in logarithmic scale of Hilbert spaces is proved. [ABSTRACT FROM AUTHOR]