Ordinary State-based Peridynamic Model Research Articles

A modified axisymmetric ordinary state-based peridynamics with shear deformation for elastic and fracture problems in brittle solids

Although axisymmetric problems are neither the plane stress nor plane strain problems, they can be solved in the two-dimensional (2-D) domain for convenience. In this paper, a modified axisymmetric ordinary state-based peridynamic model considering the shear deformation for linear elastic solids is proposed. By subtracting the rigid body rotation part from the total deformation, the shear deformation of a bond is considered. The bond force density vector is derived indirectly by equalizing the PD energy density and the classical strain energy density. The fictitious density in adaptive dynamic relaxation (ADR) method is derived. Two new damage criterions based on the maximum stretch/deviatoric strain are proposed. Several axisymmetrically three-dimensional (3-D) numerical examples are performed in the computational domains, where the present numerical results are compared with the existing analytical solutions, the classical finite element numerical results and axisymmetric ordinary state-based peridynamic results. The numerical results demonstrate this proposed method can simulate the initiation and propagation of cracks very well under the axisymmetrical conditions, and it has more accuracy than axisymmetric ordinary state-based peridynamic model.

Study of the Kalthoff–Winkler experiment using an ordinary state-based peridynamic model under low velocity impact

To overcome the limitation of constant Poisson’s ratio in bond-based peridynamics, an ordinary state-based peridynamic (OSB PD) model was applied to simulate dynamic crack propagation in Kalthoff–Winkler specimen subjected to a low velocity impact. A typical linear peridynamic solid material constitutive was implemented to characterize the materials’ behavior. This article focuses on capturing the underlying mechanisms of dynamic crack propagation; thus a wave propagation analysis was conducted, and the analysis was further verified by changing configurations of the specimen in horizontal and/or vertical directions. In addition, to evaluate the influence of parameters, a detailed parametric study of impact speed, plate thickness, and notch tip radius was performed. Crack initiation and propagation results agree well with experimental observations. Through parametric analysis, it was found that the impact speed has a remarkable influence on both crack propagation speed and crack path: increase of impact speed causes increased crack propagation speed and decreased crack angle; plate thickness has influence only on crack propagation speed: thicker plate leads to lower crack propagation speed; and notch radius has almost no influence on both of them.

Peridynamic simulation of thermal failure behaviors in rocks subjected to heating from boreholes

In this paper, a weakly coupled thermo-mechanical model within the framework of ordinary state-based peridynamics is proposed to investigate thermal cracking behaviors of rocks subjected to heating from boreholes. The weakly coupled thermo-mechanical ordinary state-based peridynamic model is decomposed into two parts, i.e., thermal conduction and mechanical deformation. In the first part, temperature distributions in solids are analyzed based on the heat conduction equations. While, in the second part, thermally-induced deformation and fracture can be simulated by the mechanical computations. Moreover, a multi-rate explicit time integration scheme is proposed to model thermal cracking phenomena in rocks to overcome different time-scale problems in multi-physical systems. A benchmark example with analytical solutions is firstly modeled to validate the correctness and accuracy of the proposed numerical method, and numerical convergence studies of the weakly coupled thermo-mechanical ordinary state-based peridynamics are also conducted. The present numerical results are in good agreement with the analytical solutions and the previous experimental data. Then, three numerical examples are performed to investigate thermally-induced cracking behaviors, i.e. cracking patterns and temperature and stress evolutions. The proposed numerical method not only provides a new tool for coupled thermal-mechanical fracture problems in geothermal engineering, but also reveals the mechanism of thermal cracking phenomena in rocks.

Peridynamic simulation of two-dimensional axisymmetric pull-out tests

In this study, a systematic pull-out simulation scheme in the formwork of two dimensional (2D) axisymmetric domain using peridynamics (PD) is presented. An axisymmetric ordinary state-based peridynamic model is used to model the pull-out deformation just like the axisymmetric stress element in the finite element method (FEM). A failure criterion based on the bond energy density is adopted to predict the interfacial debonding, while a new PD contact model in the 2D axisymmetric domain is proposed to simulate the friction sliding during the pull-out process. The PD pull-out simulation scheme based on the respective 2D axisymmetric model, interfacial failure criterion and contact model is quantitatively validated in comparisons with either the FEM or analytical solutions. The whole process of the pull-out test is simulated using the proposed PD scheme, and the influences of both the interfacial fracture toughness and friction coefficient on the bond-slip behavior are investigated. The developed PD pull-out modeling scheme is capable of effectively simulating the pull-out tests and predicting the bond-slip behavior.

An improved ordinary state-based peridynamic model for cohesive crack growth in quasi-brittle materials

Ordinary state-based peridynamics (OSBPD) is a well-established peridynamic (PD) model, but its applications are restricted in brittle fracture because the failure models proposed in OSBPD are mainly to describe brittle fracture rather than quasi-brittle fracture. This work attempts to propose a damage model to overcome this limitation for the investigation of damage and crack propagation in quasi-brittle materials. To this aim, a PD version of the cohesive zone model (CZM) is established in OSBPD for the fracture analysis of quasi-brittle materials. A degradation curve is proposed in the damage model to represent the cohesive effect in fracture process zone (FPZ), which is similar to the tension softening curve in CZM. The damage model can be uniquely determined once the tensile strength, the elastic modulus, the specific fracture energy, and tension softening constitutive law are given. Moreover, several representative examples of mode-I and mixed mode concrete fractures are simulated using OSBPD to validate the proposed damage model. The predicted load-displacement curves and crack paths in all the cases are in good agreement with the experimental results or other numerical results. In addition, a re-loading approach is proposed to obtain the descending part of load-displacement curve when the load-controlled loading procedure is adopted.

An ordinary state-based peridynamic model for the fracture of zigzag graphene sheets.

This study develops an ordinary state-based peridynamic coarse-graining (OSPD-CG) model for the investigation of fracture in single-layer graphene sheets (SLGS), in which the peridynamic (PD) parameters are derived through combining the PD model and molecular dynamics (MD) simulations from the fully atomistic system via energy conservation. The fracture failure of pre-cracked SLGS under uniaxial tension is studied using the proposed PD model. And the PD simulation results agree well with those from MD simulations, including the stress-strain relations, the crack propagation patterns and the average crack propagation velocities. The interaction effect between cracks located at the centre and the edge on the crack propagation of the pre-cracked SLGS is discussed in detail. This work shows that the proposed PD model is much more efficient than the MD simulations and, thus, indicates that the PD-based method is applicable to study larger nanoscale systems.

Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems

An ordinary state-based peridynamic model is developed for transient fully coupled thermoelastic problems. By adopting an integral form instead of spatial derivatives in the equation of motion, the developed model is still valid at discontinuities. In addition, the ordinary state-based peridynamic model eliminates the limitation on Poisson’s ratio which exists in bond-based peridynamics. Interactions between thermal and structural responses are also considered by including the coupling terms in the formulations. These formulations are also cast into their non-dimensional forms. Validation of the new model is conducted by solving some benchmark problems and comparing them with other numerical solutions. Thin plate and block under shock-loading conditions are investigated. Good agreements are obtained by comparing the thermal and mechanical responses with those obtained from boundary element method and finite element solutions. Subsequently, a three-point bending test simulation is conducted by allowing crack propagation. Then a crack propagation for a plate with a pre-existing crack is investigated under pressure shock-loading condition. Finally, a numerical simulation based on the Kalthoff experiment is conducted in a fully coupled manner. The crack propagation processes and the temperature evolutions are presented. In conclusion, the present model is suitable for modelling thermoelastic problems in which discontinuities exist and coupling effects cannot be neglected.

An axisymmetric ordinary state-based peridynamic model for linear elastic solids

In this study, a new axisymmetric ordinary state-based peridynamic (PD) model for axisymmetric problems of linear elastic solids is presented. A fracture criterion based on the PD bond energy density is proposed. Adaptive dynamic relaxation (ADR) method is adopted to obtain equilibrium solutions, and a viable fictitious density of the model is derived and proven to be valid for implementation in the ADR method. Performance and validity of the proposed axisymmetric PD model are demonstrated by three kinds of numerical problems, i.e., compression tests, pull-out deformation, and indentation fracture. In the compression tests with focus on constant-strain deformations, the displacements predicted by the present model are compared with classical analytical solutions, and the results show good agreements. Both m- convergence and δ-convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases. The recovery of the Poisson’s ratios in the model is tested in detail as well. The capability of capturing general non-uniform axisymmetric deformation by the proposed model is verified in the pull-out analysis. The equilibrium displacement fields predicted by the present model agree very well with those by the finite element method. The peridynamic evaluation of strains and stresses also show good match with the finite element ones. The proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data. The model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state-based peridynamics.

An extended state-based peridynamic model for damage growth prediction of bimaterial structures under thermomechanical loading

An extended ordinary state-based peridynamic model considering thermomechanical loading is presented to predict damage growth of bimaterial structures, such as cermet. In this new model, the three-dimensional (3D) and two-dimensional (2D) (both plane stress and strain) cases are all considered. As examples, 2D bimaterial beams and 3D thick plates are analyzed under thermal loading and three-point bending. m-convergence and δ-convergence are discussed in the cases of 2D verification, and comparison of displacement with finite element model shows great accuracy of the extended model. Damage growth (in term of crack propagation) of bimaterial beams due to incremental thermal loading and three-point bending is investigated. The new model successfully captures interface crack propagation in bimaterial beams under thermal loading as well as crack growth within substrate material and at bimaterial interface under quasi-static and impact loading. Distribution of elastic strain energy density is analyzed during dynamic crack propagation under impact loading.

Generalization of the ordinary state-based peridynamic model for isotropic linear viscoelasticity

This paper presents a generalization of the original ordinary state-based peridynamic model for isotropic linear viscoelasticity. The viscoelastic material response is represented using the thermodynamically acceptable Prony series approach. It can feature as many Prony terms as required and accounts for viscoelastic spherical and deviatoric components. The model was derived from an equivalence between peridynamic viscoelastic parameters and those appearing in classical continuum mechanics, by equating the free energy densities expressed in both frameworks. The model was simplified to a uni-dimensional expression and implemented to simulate a creep-recovery test. This implementation was finally validated by comparing peridynamic predictions to those predicted from classical continuum mechanics. An exact correspondence between peridynamics and the classical continuum approach was shown when the peridynamic horizon becomes small, meaning peridynamics tends toward classical continuum mechanics. This work provides a clear and direct means to researchers dealing with viscoelastic phenomena to tackle their problem within the peridynamic framework.