State-based Peridynamic Model Research Articles

A Stress Tensor-based Failure Criterion for Ordinary State-based ‎Peridynamic Models‎

A Stress Tensor-based Failure Criterion for Ordinary State-based ‎Peridynamic Models‎

Simulation of highly nonlinear materials based on a stabilized non-ordinary state-based peridynamic model

Peridynamics (PD) is an effective method to solve discontinuity problems that involve cracking or crushing behaviors. In the non-ordinary state-based PD (NOSBPD), the so called zero-energy mode may cause inaccuracy particularly when the material is highly nonlinear. This paper proposed a novel stabilized NOSBPD modeling method to mitigate the inaccuracy and instability of NOSBPD solutions. A correction force for a PD point is defined as the difference between an internal force obtained by stress equilibrium equation and that obtained by the force states of the points within its horizon. The correction force is applied on the PD points to be corrected, e.g., that on the boundary of a PD model. Three examples are presented demonstrating the proposed method's effectiveness in static and dynamic analyses of both linear or highly nonlinear models, i.e., 3D cap plasticity concrete model and 2D multi-yield surface soil model. The predicted responses (e.g., displacements, stresses, strains at representative points) are analyzed and compared with those without force correction for both linear and highly nonlinear cases. The proposed method is demonstrated to be an effective method for mitigating the inaccuracy and instability of the NOSBPD solutions.

A two-dimensional ordinary state-based peridynamic model for surface fatigue crack propagation in railheads

Based on ordinary state-based peridynamic theory, a 2D peridynamic model has been established to investigate fatigue crack propagation in railheads. The proposed model is verified in terms of rail deformation under a quasi-static load and the ductile material-related fatigue failure model. Good agreements have been achieved between a finite element model and the experimental results. With the proposed model, the effects of the initial crack angle, initial crack length and wheel-rail friction coefficient on crack propagation in railheads are studied. This research provides a new method for studying crack propagation in railheads.

Peridynamic Model for Single-Layer Graphene Obtained from Coarse-Grained Bond Forces

An ordinary state-based peridynamic material model is proposed for single-sheet graphene. The model is calibrated using coarse-grained molecular dynamics simulations. The coarse-graining method allows the dependence of bond force on bond length to be determined, including the horizon. The peridynamic model allows the horizon to be rescaled, providing a multiscale capability and allowing for substantial reductions in computational cost compared with molecular dynamics. The calibrated peridynamic model is compared to experimental data on the deflection and perforation of a graphene sheet by an atomic force microscope probe.

Static and dynamic mechanical behaviors of cracked Mindlin plates in ordinary state-based peridynamic framework

An ordinary state-based peridynamic (PD) model based on the Mindlin plate theory is presented to deal with fracture mechanics problems. Static and dynamic stress resultant intensity factors regarded as the primary fracture parameters for plate structures are evaluated by the displacement extrapolation method. Owing to the PD surface effect, however, the accuracy of the displacement field near crack surfaces is significantly affected. Therefore, the arbitrary horizon domain method is adopted to correct the surface effect. It derives the variable PD parameters to properly describe mechanical behaviors for each material point. Several numerical examples are investigated to examine the performance of the presented method. It indicates that the PD Mindlin plate model incorporated with the arbitrary horizon domain method validly minimizes the influence of the PD surface effect and provides an effective approach to evaluate static and dynamic moment intensity factors.

Fracture simulation of viscoelastic membranes by ordinary state-based peridynamics

Despite a load below the elastic limit is applied on a viscoelastic material, the material may fail after a long duration of constant loading because of the time-dependent viscous deformations. In this regard, a viscoelastic material model in the ordinary state-based peridynamic framework is proposed to capture crack propagation in polymeric water treatment membranes. The deformation state is decoupled into dilatational and distortional parts, and it is assumed that the dilatational part of deformation is elastic, while the distortional part is considered as viscoelastic, whose behaviour can be represented by the Prony series. First, we verify our implementation with FEM results for a benchmark case. Afterwards, the crack propagation is studied by the viscoelastic ordinary state-based peridynamic model.

Implicit non-ordinary state-based peridynamics model for linear piezoelectricity

In the present work, the development and analysis of linear piezoelectricity in the peridynamics framework is presented. The model is developed using non-ordinary state-based Peridynamics. The equations of motion are derived using Hamilton’s principle and the correspondence material model is developed by establishing the connection between the classical theory of linear piezoelectricity and the newly proposed peridynamic linear piezoelectricity framework. An implicit implementation is proposed for two-dimensional linear piezoelectricity. Moreover, for the stabilization of the proposed scheme, the bond-associated correspondence model is used. We demonstrate the capabilities of the proposed model by presenting some 2 D numerical examples.

A two-dimensional ordinary state-based peridynamic model for plastic deformation based on Drucker-Prager criteria with non-associated flow rule

The plastic behaviors of geomaterials often obey non-associated flow rule. In this paper, a two-dimensional (2D) ordinary state-based peridynamic (OSB-PD) increment model with shear deformation based on Drucker-Prager (D-P) criterion with non-associated flow rule is proposed to study the plastic behavior of geomaterials. An incremental form of peridynamic (PD) strain energy density for plane elastic problems is first established. The material parameters are determined when the incremental PD strain energy density is equal to the incremental strain energy density in classical mechanics. The PD force density for the OSB-PD theory with shear deformation in plane elastic cases is derived accordingly. The relationship between the incremental second invariant of the stress tensor dJ2, the first invariant function of stress tensor dI1 in the classical mechanics and PD are established in this paper, and the Drucker-Prager yield surface is expressed in the framework of peridynamics. To avoid the excessive plastic dilation, the non-associated flow rule is proposed, in which isotropic and deviatoric plastic elongation are employed to describe the plastic behaviors of a bond. Also, the equivalent stress and equivalent plastic strain for the proposed PD model are given. An integration algorithm for the proposed increment model is developed. Numerical examples including circular opening in rocks are performed, and the PD results are compared with that obtained by the finite element method or analytical solution. Those numerical simulations have verified the validity of the proposed PD model to predict the elasto-plastic behavior of geomaterials.

Ordinary state-based peridynamic shell model with arbitrary horizon domains for surface effect correction

An ordinary state-based peridynamic (PD) shell model with arbitrary horizon domains is presented. In computations of the PD parameters, the model removes the complete horizon assumption when treating the PD surface effect. In this study, the membrane, bending, and transverse shear deformations in the PD framework are taken into account. To examine fracture mechanics behavior, crack opening displacements and strain energy densities, which correspond to physical values of the PD failure criteria, are investigated under single- and mixed-mode fracture conditions. Several numerical examples are considered to evaluate fracture behaviors for shell structures subjected to in-plane or out-of-plane loading. The PD surface effect near domain boundaries or crack surfaces is validly treated by using the presented method. In addition, a crack propagation analysis is performed to estimate the effectiveness of the presented method.

An ordinary state-based peridynamic elastoplastic 2D model consistent with J2 plasticity

A new ordinary state-based peridynamic model (OSB-PD) in 2D consistent with J2 plasticity using a novel decomposition for force and extension states is presented. A novel strategy for testing the consistency of an OSB-PD formulation for elastoplasticity is introduced. In contrast with other similar models, the new elasto-plastic OSB-PD model is objective and works for large rotations. Two rate-independent yield functions equivalent to J2 plasticity with associated flow rules are formulated based on force states and on the strain energy density. An efficient return mapping algorithm for peridynamic formulations in plasticity is proposed. The model is verified against results obtained with Abaqus for the classical formulation using two examples, one featuring plastic deformations under large rotations. The evolution of the plastic zone computed with the new model is found to match that from the classical model computed with Abaqus. With a simple damage model, a demonstrative plane stress example for shear localization and full rupture captures the same failure mechanism as observed experimentally: an initial almost symmetrical shear banding, evolves into a single, predominant shear band, followed by full rupture.

Bond-associated non-ordinary state-based peridynamic model for multiple spalling simulation of concrete

The non-ordinary state-based peridynamic (NOSB PD) model has the capability of incorporating existing constitutive relationships in the classical continuum mechanics. In the present work, we first develop an NOSB PD model corresponding to the Johnson–Holmquist II (JH-2) constitutive damage model, which can describe the severe damage of concrete under intense impact compression. Besides, the numerical oscillation problem of the NOSB PD caused by zero-energy mode is analyzed and hence a bond-associated non-ordinary state-based peridynamic (BA-NOSB PD) model is adopted to remove the oscillation. Then, the elastic deformation of a three-dimensional bar is analyzed to verify the capability of BA-NOSB PD in eliminating the numerical oscillation. Furthermore, concrete spalling caused by the interaction of incident compression wave and reflected tension wave is simulated. The dynamic tensile fracture process of concrete multiple spalling is accurately reproduced for several examples according to the spalling number and spalling thickness analysis, illustrating the approach can well simulate and analyze the concrete spalling discontinuities.

An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

Meshfree discretizations of state-based peridynamic models are attractive due to their ability to naturally describe fracture of general materials. However, two factors conspire to prevent meshfree discretizations of state-based peridynamics from converging to corresponding local solutions as resolution is increased: quadrature error prevents an accurate prediction of bulk mechanics, and the lack of an explicit boundary representation presents challenges when applying traction loads. In this paper, we develop a reformulation of the linear peridynamic solid (LPS) model to address these shortcomings, using improved meshfree quadrature, a reformulation of the nonlocal dilatation, and a consistent handling of the nonlocal traction condition to construct a model with rigorous accuracy guarantees. In particular, these improvements are designed to enforce discrete consistency in the presence of evolving fractures, whose a priori unknown location render consistent treatment difficult. In the absence of fracture, when a corresponding classical continuum mechanics model exists, our improvements provide asymptotically compatible convergence to corresponding local solutions, eliminating surface effects and issues with traction loading which have historically plagued peridynamic discretizations. When fracture occurs, our formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. We provide rigorous error analysis and demonstrate convergence for a number of benchmarks, including manufactured solutions, free-surface, nonhomogeneous traction loading, and composite material problems. Finally, we validate simulations of brittle fracture against a recent experiment of dynamic crack branching in soda-lime glass, providing evidence that the scheme yields accurate predictions for practical engineering problems.

Ordinary state-based peridynamics for geometrically nonlinear analysis of plates

This study presents a novel ordinary state-based peridynamic model for geometrically nonlinear analysis of plates. The nonlinear strain energy density and nonlinear equations of motion for a plate in peridynamics are obtained based on the principle of virtual displacements by using Total Lagrange formulation. The numerical procedure for geometrically nonlinear analysis of a plate is also provided. The accuracy of the proposed nonlinear peridynamic model is validated by considering large deformations of a plate subjected to bending and a plate subjected to transverse shear forces. To further demonstrate the capabilities of the proposed nonlinear model, damages on a plate with a single crack subjected to stretching and tearing, a plate with two parallel cracks subjected to tearing, and a plate subjected to torsional loading are predicted.

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

This paper proposes an axisymmetric ordinary state-based peridynamic model considering the thermal expansion effect for temperature-induced cracking of linear elastic solids. Both the force state and bond failure criterion of the model are built in axisymmetric domain. The direct calculation for peridynamic bond energy density of the axisymmetric model considering thermal terms is given. The adaptive dynamic relaxation method is adopted for the quasi-static peridynamic simulations. Validity and reliability of the proposed axisymmetric model are illustrated by numerical examples of cold-induced fracture in a pipe with an inner crack. It is shown that the displacement predicted by the proposed peridynamic model is in good agreement with the corresponding finite element method. The critical bond energy density criterion for thermal cracking is verified via comparisons with the critical fracture temperature loads computed from the virtual crack closure technique and released energy based on Griffith’s theory. The sensitivity of the present model on the peridynamic discretization is investigated and discussed in both the continuous deformation and cracking cases. The performance of the peridynamic model is further demonstrated for cold-induced fracture of double-layered pipes with inner cracks, and the cracking behaviors influenced by the interfacial toughness and number of inner cracks are captured, respectively. The present model is capable of solving axisymmetric thermal cracking problems of linear elastic solids effectively, and it can be applied in more advanced axisymmetric structures which involve more complicated cracking issues.

A physics-guided machine learning model for two-dimensional structures based on ordinary state-based peridynamics

This study presents a novel physics-guided machine learning model for two-dimensional structures based on ordinary state-based peridynamics. The linear relationships between the displacements of a material point and the displacements of its neighbours and the applied forces are obtained for the machine learning model by using linear regression. The numerical procedure for coupling the ordinary state-based peridynamic model and the machine learning model is also provided. The accuracy of the coupled model is verified by predicting deformations of a two-dimensional plate with circular cut-out subjected to tension and a two-dimensional representation of three points bending test. To further demonstrate the capabilities of the coupled model, damage predictions for a two-dimensional representation of a three-point bending test, a notched plate with a hole subjected to tension, a square plate with a pre-existing crack subjected to tension, and a plate with a pre-existing crack subjected to sudden loading are presented.

Influences of temperature and impacting velocity on dynamic failure of laminated glass: Insights from peridynamic simulations

In this work, a non-local ordinary state-based peridynamic (PD) model for rate-dependent thermo-visco-elasticity was formulated to simulate the dynamic failure process of laminated glasses subjected to impact loads. The formulations of force states between two interacting material points within one ‘horizon’ were derived by taking the viscous and temperature-dependent mechanical responses of polyvinyl butyral (PVB) interlayers into consideration. A dynamic failure criterion used to determine the breakage of bonds was developed to model the failure process of glass sheets under impacting loads as well. The adhesion between glass and PVB layer was described by using a penalty-based method in numerical simulations. Two benchmark numerical examples, including (i) creep-recovery test of a PVB bar and (ii) dynamic fracture of a glass specimen subjected to different high strain rates, were performed to illustrate the accuracy and effectiveness of the proposed numerical model. After verification, the proposed model was employed to study the dynamic fracture of typical laminated glass plates under drop-weight loading conditions, in which numerical results show good agreement with experimental observations, and then the effects of temperature and impacting velocities on the impact resistance and failure mechanisms of laminated glass were investigated further.

Modeling and studies of fracture in functionally graded materials under thermal shock loading using peridynamics

In this study, an extended ordinary state-based peridynamic (OSBPD) model was proposed and applied to study the thermal shock damage and fracture of functionally graded materials (FGM) and structures. A modified term including thermal expansion coefficient is introduced into the conventional peridynamic model for investigating the thermal deformation of materials, and an influence function reflecting the gradient change of material properties is introduced into the force vector state in peridynamic model. The model and algorithm are validated by analyzing the cracking of a plate quenching in water and comparing with experimental results and available literature data. Furthermore, the effects of gradient distribution of elastic modulus, thermal expansion coefficient and fracture toughness of FGM on thermal damage and crack propagation are investigated. Numerical results show that the proposed model can effectively describe the failure process of functionally gradient materials under thermal shock loading. The distribution of elastic modulus, coefficient of thermal expansion and fracture toughness in FGM have great influence on the initiation and propagation of cracks, and reasonable gradient change form of materials can improve the thermal resistance ability of the structure.

A coupled hydro-mechanical non-ordinary state-based peridynamics for the fissured porous rocks

A novel coupled hydro-mechanical non-ordinary state-based peridynamics is proposed to investigate hydro-mechanical problems in fissured porous rocks. The hydraulic bond is introduced to the coupled numerical model to transfer the water flow stored in material points. The flow equations for fissured porous rocks are obtained in the framework of peridynamics. Then, the relationship between peridynamic hydraulic conductivity and classic hydraulic conductivity for one-dimensional and two-dimensional flow cases is established. Moreover, Biot's porous media theory is employed to determine the deformation caused by water pressure, which is introduced to the non-ordinary state-based peridynamic government equation. In addition, the effects of mechanical deformation and fracture on hydraulic conductivity are considered in the coupled hydro-mechanical non-ordinary state-based peridynamic model. Finally, three different numerical experiments are carried out to verify the correctness of the coupled hydro-mechanical peridynamic model proposed in this paper, and it is found that the present numerical results are in good agreement with the analytical solutions.

A state-based peridynamic formulation for functionally graded Kirchhoff plates

Functionally graded materials are a potential alternative to traditional fibre-reinforced composite materials as they have continuously varying material properties which do not cause stress concentrations. In this study, a state-based peridynamic model is presented for functionally graded Kirchhoff plates. Equations of motion of the new formulation are obtained using the Euler–Lagrange equation and Taylor’s expansion. The formulation is verified by considering several benchmark problems including a clamped plate subjected to transverse loading and a simply supported plate subjected to transverse loading and inclined loading. The material properties are chosen such that Young’s modulus is assumed to be varied linearly through the thickness direction and Poisson’s ratio is constant. Peridynamic results are compared against finite element analysis results, and a very good agreement is obtained between the two approaches.

Peridynamic modeling of composite laminates with material coupling and transverse shear deformation

This study presents a new state-based PeriDynamic (PD) model of a composite laminate with arbitrary laminate layup; it captures all types of material couplings in the presence of transverse shear deformation. It consists of normal and shear bonds instead of fiber and matrix bonds to allow arbitrary fiber orientations. Also, rotational degrees of freedom are included in the equilibrium equations in order to evaluate the transverse shear angle and curvature. Mindlin-Reissner plate theory, employed in this model, permits the use of existing composite constitutive law which leads to the coupling between different deformation modes. In addition, a quasi-local boundary condition applied only to the first row of material points avoids the approximation of nonlocal boundary values. The accuracy of this model is verified against benchmark solutions, and validated by comparison with experimental results.