Ordinary State-based Peridynamic Model Research Articles

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.

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.

Elastoplastic peridynamic formulation for materials with isotropic and kinematic hardening

We present an ordinary state-based peridynamic model in 2D and 3D consistent with rate-independent J2 plasticity with associated flow rule. The new contribution is the capability of the elastoplastic law to describe isotropic, kinematic and mixed hardening. The hardening formulations follow those available in the literature for classical elastoplasticity. The comparison between the results obtained with the peridynamic model and those obtained with a commercial FEM software shows that the two approaches are in good agreement. The extent of the plastic regions and von Mises stress computed with the new model for 2D and 3D examples match well those obtained with FEM-based solutions using ANSYS.

Nonlocal anisotropic model for deformation and fracture using peridynamic operator method

This paper proposes a nonlocal anisotropic model for deformation and fracture based on peridynamic operator method. The classic anisotropic theory is reformulated from its local form into nonlocal form by using peridynamic operator method. The final solution formulation is developed by employing variational principles and Lagrange's equations. In the case of the circular interaction domain, this model can be regarded as a novel anisotropic ordinary state-based peridynamic model. Moreover, we present for the first time the energetic failure criterion for the weak anisotropic fracture. Unlike the existing anisotropic peridynamic models that consider only two fracture toughness constants for orthogonal orientation, the fracture toughness in this criterion is an orientation-dependent function. Additionally, we develop an implicit static solution procedure to investigate corresponding crack propagation problems. Several numerical examples demonstrate the effectiveness and ability of the proposed model in handling the deformation and fracture behaviours of anisotropic materials.

Ordinary state-based peridynamic model for out-of-plane deformation and damage analysis of composite laminates

An ordinary state-based peridynamic (PD) model for predicting out-of-plane mechanical behaviour of composite laminates has been proposed, in which three types of PD bonds are used to describe the reinforcement properties. The non-local strain energy density and peridynamic formulations are obtained based on the principle of virtual work by using Total Lagrange formulation, and the force density is reformulated by interpolation technique. Since the Mindlin plate is regarded as the research object of this PD model, the transverse shear deformation of laminates is considered. Also, the dilatation term is included in the force density vector, and flaws in Poisson's ratio of materials can be overcome. In the proposed PD, the critical strain energy density rather than the critical curvature is adopted as the failure criterion. The capability of the developed PD model was demonstrated by the bending examples of composite laminates with different fiber orientations, and damage analysis was further conducted to demonstrate the strong capability of proposed PD model in replicating the damage process of laminates. In addition, a single-layer material point model (SLMPM) can be implemented in the proposed PD algorithm, and the computational efficiency of numerical models will be greatly improved.

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.

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.

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.

An extended ordinary state-based peridynamic model for nonlinear deformation and fracture

Peridynamic is a promising nonlocal continuum theory that reconstructs the equations of motion of solid mechanics using spatial integral equations, rendering it suitable for describing objects with discontinuities such as cracks. In this study, a novel extended ordinary state-based peridynamic model was developed for nonlinear deformation and fracture analysis, which established a general relationship with continuum-based parameters and permitted the selection of different influence functions. Based on the principle of virtual displacement, the complete derivations of the peridynamic parameters were presented for two and three-dimensional conditions. After that, the specific numerical scheme and algorithm implementation were summarized. The capability and accuracy of the proposed nonlinear model were verified by comparing with the experimental and finite element simulation results. Finally, several other numerical examples were provided to further demonstrate the applicability and robustness of the proposed model and its implementation.

An extended peridynamic model for analyzing interfacial failure of composite materials with non-uniform discretization

This work presents an extended ordinary state-based peridynamic model and approaches to analyze the fracture behavior of composite materials considering interfacial failure. An interface force term is introduced to characterize the interaction among different compositions in composite materials, which enables naturally the non-uniform discretization to be conveniently adopted, especially where sharp interfaces exist. The calculation of force states in different compositions is independent of each other, and the use of non-uniform discretization can improve computational efficiency significantly. Further, inspired by the cohesive zone model (CZM), the deformation coefficient and degradation behavior of interface bonds are derived by using the traction-separation law, and a mixed-mode fracture criterion for interface bonds based on the bond energy density is established as well. Validation of the proposed model and approach is established through a classical rectangular plate example, and then the proposed model is employed to analyze the failure of typical composites including laminated glass and concrete. The results show good agreement with corresponding experimental observations and it indicates that the proposed extended peridynamic model can accurately and efficiently capture and describe the fracture of composite materials with sharp interfaces.

Ordinary state-based peridynamic plastic model with Drucker-Prager criterion considering geometric nonlinearity

In this paper, an ordinary state-based peridynamic (OSB-PD) plastic model is proposed to simulate the large deformation of geomaterials. The PD dilatation, which is equivalent to the volumetric strain in terms of the logarithmic strain in classical continuum mechanics (CCM), is proposed. The PD material parameters are obtained by equaling the assumed PD energy density to the energy density in terms of the Kirchhoff stress and logarithmic strain in CCM. The PD force, which is obtained by calculating the Frechet derivative of PD energy density, is naturally collinear with the deformed bond. The OSB-PD plastic model with the Drucker-Prager criterion considering geometric nonlinearity is subsequently established. Benchmark problems including the plate under large deformation and the l-shaped panel test are simulated, which has verified the proposed model.

Peridynamic investigation on crack propagation mechanism of rock mass during excavation of tunnel group in cold regions

An extended ordinary state-based peridynamic model was developed to investigate the crack propagation of surrounding rock due to tunnel excavation. The frost heave effect on the crack surface was described by an equivalent pressure density. Two numerical examples were conducted to verify the effectiveness and robustness of the proposed model. Moreover, the influence of frost heave pressure, buried depth and rock elastic modulus on damage characteristics of the surrounding rock was investigated. The results demonstrate that massive microcrack damage will be formed at the invert when the tunnel is excavated in a deeply buried rock mass or soft rock stratum. HIGHLIGHTS A peridynamic model of rock mass under excavation and frost heave was constructed. Damage characteristics of surrounding rock could be simulated efficiently. Frost heave load was exerted on the crack surface by an equivalent pressure density. The effects of frost heave load, buried depth and rock elastic modulus were studied.

Cosserat ordinary state-based peridynamic model and numerical simulation of rock fracture

The propagation and coalescence of rock cracks is the direct cause of instability in many geotechnical engineering structures. To predict the rock crack propagation trajectory and coalescence mode, an ordinary state-based peridynamic (OS-PD) based on the Cosserat continuum is proposed. Compared with classical OS-PD model considering only pairwise force, the effects of the pairwise moment between two material points is also considered in the proposed Cosserat ordinary state-based peridynamic (COS-PD) model. The rotational stiffness and rotation angle are introduced to characterize the microdeformation and microstructure of rock, and the expression of rotational stiffness coefficient is derived. Three numerical cases are presented to compare the performance of OS-PD model and COS-PD model. The effect of the rotational stiffness on crack growth is investigated. The results show that the rock failure process is accelerated by the introduction of pairwise moment, the crack propagation type and coalescence mode are more inclined to shear failure with the increase of rotation stiffness coefficient. Compared with experimental results, it is found that COS-PD model could extend classical OS-PD model and better capture the evolution process of the rock crack.

Extension of Ordinary State-Based Peridynamic Model for Nonlinear Analysis

Extension of Ordinary State-Based Peridynamic Model for Nonlinear Analysis

An Ordinary State-Based Peridynamic Model of Unidirectional Carbon Fiber Reinforced Polymer Material in the Cutting Process

Due to the complexity of the composite structure, analyzing the material failure process of carbon fiber reinforced polymers (CFRP) is fairly difficult, particularly for the machining process. Peridynamic theory, a new branch of solid mechanics, is a useful tool for dealing with discontinuities. This study presents an ordinary state-based peridynamic (OSB-PD) model for unidirectional CFRP material in the cutting process. In this model, angle tolerance is used to overcome the fiber angle limitation in a classical OSB-PD laminate method, and the short-range force approach is utilized to simulate the contact of the cutting tool and workpiece. The effectiveness of the supplied models is validated by tension and cutting tests. Finally, it can be indicated that the OSB-PD model is capable of predicting machined surface damage and cutting force, based on the comparison of simulation and experimental data.

State-Based peridynamics for thermomechanical modeling of fracture mechanisms in nuclear fuel pellets

This study presents a coupled thermomechanical ordinary state-based peridynamic (OSB-PD) model, including the effects of randomly varying strength parameter and crack gap heat transfer, to investigate the fracture mechanisms in nuclear fuel pellets. The three-dimensional (3D) models of fuel pellets are constructed through irregular non-uniform discretization. The model includes the random variability of the critical stretch of each bond based on normal distribution. The accuracy of the OSB-PD model is validated against the FEM results for the thermomechanical behavior of a fuel pellet without damage. Subsequently, the crack propagation and crack patterns of fuel pellet are studied at different power levels (10 ∼ 40 kW/m). The results show that radial and axial cracks mainly develop during the power rise stage while the circumferential cracks develop during power down stage. These three crack types are verified by comparing with the experimental observations. The number of cracks and fragments of the fuel pellet increases with the increase in power level. Moreover, the fuel pellet can be completely separated at the mid-height cross section when the power level is equal or greater than 25 kW/m.

A State-Based Peridynamic Flexural Fatigue Model for Contact and Bending Conditions.

To address flexural fractures and predict fatigue life, an ordinary state-based peridynamic (PD) fatigue model is proposed for the initiation and propagation of flexural fractures. The key to this model is to replace the traditional partial differential fracture model with a spatially integral peridynamic model. Based on the contact and slip theory, the nonlocal peridynamic contact algorithm is confirmed and the load transfer is through the contact area. With the 3D peridynamic J-integration and the energy-based bond failure criterion, the peridynamic fatigue model for flexural cracks' initiation and propagation is constructed. The peridynamic solid consists of a pair of gear contact surfaces and the formation and growth of flexural fatigue cracks evolved naturally over many loading cycles. The repeated load is transferred from the drive gear to the follower gear using the nonlocal peridynamic contact algorithm. The improved adaptive dynamic relaxation approach is used to determine the static solution for each load cycle. The fatigue bending crack angle errors are within 2.92% and the cycle number errors are within 10%. According to the experimental results, the proposed peridynamic fatigue model accurately predicts the location of the crack without the need for additional criteria and the fatigue life predicted by the simulation agrees quite well with the experimental results.

Generalized plastic ordinary state-based peridynamic model with shear deformation of geomaterials

Generalized plastic ordinary state-based peridynamic model with shear deformation of geomaterials

Numerical estimate of critical failure surface of slope by ordinary state-based peridynamic plastic model

In this paper, two-dimensional ordinary state-based peridynamic (OSB-PD) plastic model is coupled mechanically with the Drucker-Prager (D-P) criterion, aiming at numerically reproducing localized deformation and locating the critical failure surface of slopes. A non-physical (or negative) incremental plastic energy can be avoided under extreme non-uniform deformation in this model. The strength reduction method is adopted to estimate the critical failure surface and factor of safety. For PD strength reduction method, three evaluation criteria are introduced to estimate the critical damage state of slopes. Two numerical simulations including a classical slope model and centrifuge tests of sand slopes are performed by the proposed PD method. The critical failure surfaces of slopes are obtained by PD and compared with the results obtained by (i) the finite element method (FEM), (ii) the simplified Bishop method, and (iii) the centrifuge tests. The numerical, analytical and experimental results have collectively verified the proposed PD method.

Peridynamics simulation of shotcrete lining damage characteristics under freeze-thaw cycles in cold region tunnels

An improved ordinary state-based peridynamic (OSBPD) model is proposed to study the damage evolution of shotcrete lining under freeze-thaw cycles (FTCs) in cold region tunnels. The fracture criterion of bond in PD model is modified by introducing freeze-thaw damage function and by considering the difference of mechanical behaviors of shotcrete under tension and compression. In combination with material point dormancy method, a PD model is constructed to simulate the excavating and supporting process of tunnels in cold regions. In the numerical implementation, a global convergence criterion is employed to accelerate the solution convergence. For comparison, a finite difference method (FDM) model also is established to simulate the same excavating and supporting process. It shows that the results of the improved OSBPD model are consistent with that of the FDM, and the improved OSBPD model is more efficient. With the proposed model, some influence factors, such as tunnel buried depth, surrounding rock grade and section shape, on damage evolution of the lining structure under FTCs are originally investigated. The results demonstrate that accumulated freeze-thaw damage at the arch foot is more severe while buried depth increasing or rock elastic modulus decreasing.