Non-ordinary State-based Peridynamics Research Articles

A novel kinematic-constraint-inspired non-ordinary state-based peridynamics

The non-ordinary state-based peridynamics (NOSBPD) is attractive because of its ability to incorporate available constitutive models. However, conventional NOSBPD suffers from numerical oscillation induced by zero-energy mode. Although the recently proposed bond-associated deformation gradient performs well with controlling the issues, the size of the bond-associated horizon is undetermined. The work is aimed to introduce kinematic constraints to redefine the classical bond-associated deformation gradient. Based on the kinematic constraints imposed by the symmetry of the bond-associated deformation gradient and non-local kinematic measures, a well-defined bond-associated horizon is derived for 1D, 2D, and 3D structures. In this way, a novel kinematic-constraint-inspired non-ordinary state-based peridynamics (KC-NOSBPD) is proposed, which completely differs from the previous continuum-kinematics-inspired peridynamics (CPD) extended from classical bond-based peridynamics (BBPD). The linear and angular momentum conservation is firstly and rigorously proved. The novel proposed model is proved capable of avoiding unphysical deformation modes that typically beset conventional formulation through several analytical examples. In addition, bond-associated stress-state-based failure criteria are proposed for fracture analysis. Several benchmarking tests verify the effectiveness of the proposed formulation in eliminating spurious oscillation of numerical solution and highlight the significance of kinematic constraints. The final crack pattern of sandstone specimens coincides with experimental observation, illustrating its suitability for solving crack propagation.

Non-ordinary state-based peridynamic fatigue modelling of composite laminates with arbitrary fibre orientation

For the safe application of carbon fibre-reinforced polymer (CFRP) materials in engineering structures, fatigue life prediction and failure modelling are essential. In this paper, an application of a nonordinary state-based peridynamics (NOSB-PD) approach for the fatigue failure prediction of multilayer CFRP was presented. To characterize the structure of composite laminates, matrix and fibre bonds were introduced within the plies, and normal and shear bonds were introduced between the plies. The fatigue simulation model of composite materials was established by combining the fatigue theory of peridynamics and Paris-Erdogan law, and a method to obtain model parameters was proposed. To demonstrate the accuracy of the model, tension and fatigue experiments were conducted. By comparing the results of tensile and fatigue experiments, it was shown that the model developed in this paper can accurately predict the failure process of multidirectional CFRP laminate under the action of tensile or fatigue loading. The crack initiation and propagation for different laminae during the tensile and fatigue processes can be obtained from the proposed model.

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.

Imposition of local boundary conditions in peridynamics without a fictitious layer and unphysical stress concentrations

This study introduces a general approach for the imposition of local boundary conditions in non-ordinary state-based peridynamics (NOSB PD) to eliminate the displacement kinks near the boundary without a fictitious layer under quasi-static loading conditions. It identifies the underlying reason for the unphysical displacement kinks. Under an imposed linear displacement field, the NOSB PD equilibrium equation is not satisfied near the boundary due to the unsymmetric horizon of material points. However, the equilibrium equation derived by using the PD differential operator is satisfied at such material points. Therefore, the material body is divided into three regions to satisfy the equilibrium equations and to impose displacement and tractions boundary conditions. This approach does not deviate from the original NOSB PD; however, it provides a simple solution to eliminate the displacement kink near the boundary, which leads to unphysical stress concentrations. Its efficacy is demonstrated by considering elastic rectangular and square plates subjected to various types of boundary conditions leading to homogeneous as well as nonhomogeneous deformations. The creep response of a rectangular plate further proves the robustness of the present approach. Also, a quasi-static crack propagation from a pre-existing crack in a square plate under mode-I, mode-II and mixed-mode loading conditions demonstrates its capability for failure prediction based on the critical stretch criteria. Finally, its applicability for 3D analysis is demonstrated by considering a rectangular prism under applied stretch and normal stress.

Explosion damage analysis of concrete structure with bond-associated non-ordinary state-based peridynamics

Explosion damage analysis of concrete structure with bond-associated non-ordinary state-based peridynamics

A viscoelastic model of geometry-constraint-based non-ordinary state-based peridynamics with progressive damage

A viscoelastic model of geometry-constraint-based non-ordinary state-based peridynamics with progressive damage

A computational homogenization framework for non-ordinary state-based peridynamics

Peridynamic theory has been shown to possess the capabilities of describing phenomena that theories based on partial differential equations are not capable of describing. These phenomena include nonlocal interactions and presence of singularities in system responses. To exploit the capabilities offered by peridynamics in the homogenization of heterogenous media, a nonlocal computational homogenization theory based on peridynamic correspondence model (non-ordinary state-based peridynamics) is proposed. To set the development of the theory on a rigorous mathematical framework and to ensure consistency with the nonlocal nature of the peridynamic theory, a nonlocal vector calculus was used in the analysis of the nonlocal homogenization theory. The proposed theory is a two-scale micro–macro-homogenization strategy in which the constitutive relation at the macroscale is derived from explicit solution of a nonlocal volume constraint problem at the microscale. To justify the coupling between the two scales, nonlocal analogues of the stress and strain average theorems as well as the Hill–Mandel macrohomogeneity condition were derived. Validation of the proposed theory is achieved via numerical solution of Representative Volume Elements (RVE) from composite materials and comparing the results with those obtained by means of established methodologies.

A unified non-local fluid transport model for heterogeneous saturated porous media

A unified non-local fluid transport model is proposed for heterogeneous saturated porous media. First, a non-local governing equation of fluid transport is presented based on the Non-ordinary State-based Peridynamics (NOSB-PD), and a novel flux vector state and a non-local Darcy’s law is proposed to illustrate non-local fluid transport effects in heterogeneous saturated porous media. The proposed flux state vector ensures the non-local governing equation of fluid transport model in heterogeneous saturated porous media degenerates into the local one when the horizon of each material point approaches to zero. A non-local pressure gradient at each material point can be obtained by averaging all the flux state vectors in its horizon, which unifies the weak and strong discontinuities of pressure in heterogeneous saturated porous media in a consistent way. Second, a variational formulation of the unified non-local fluid transport model is developed. This variational formulation enables the proposed model to deal with complex boundary conditions, which is difficult or even impossible for current fluid transport models based on Bond-based Peridynamics (BB-PD) and Ordinary State-based Peridynamics (OSB-PD). Third, a fully implicit algorithm combined with the Newton–Raphson method is introduced to solve the general non-linear problems in the heterogeneous saturated porous media. The penalty function method is employed to eliminate the zero-energy mode oscillation in NOSB-PD. Further, numerical examples demonstrate that the proposed model is accurate and can well capture the weak and strong discontinuities of pressure at the material interfaces and crack surfaces in the heterogeneous saturated porous media. Numerical examples also indicate that the model can effectively eliminate the zero-energy mode oscillation inherently rooted in the NOSB-PD, and thus reaches a stable numerical solution.

A low cycle fatigue cracking simulation method of non-ordinary state-based peridynamics

We established a simulation method for low cycle fatigue elastoplastic fracture under the framework of the Non-ordinary State-based peridynamics. It used mixed hardening models and failure criteria based on accumulated plastic strain. Furthermore, the numerical solution method of the elastoplastic constitutive equation was given. In order to verify the simulation method, the simulations of 316L steel strain control symmetric cyclic test and Q235 low cycle fatigue failure test were conducted, which showed good agreement with the experimental results. Our findings demonstrated the suitability of the proposed simulation method for damage fracture analysis with elastoplastic low cycle fatigue.

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.

Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method

The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate.

Bridging the gap between local and nonlocal numerical methods—A unified variational framework for non-ordinary state-based peridynamics

The paper aims to develop a unified variational framework to bridge the gap between the non-ordinary state-based peridynamics (NOSB-PD) and the classical continuum mechanics (CCM). First, a new force state vector is proposed by introducing the first Piola–Kirchhoff stress. This new force state vector enables the stress divergence of each material point to be expressed by averaging all the force state vectors in its support domain. The new force state vector also ensures the mathematical consistency between the strong form of PD and CCM when the horizon of a material point approaches to zero. Second, the displacement and traction boundaries in CCM are transformed into the non-local fictious boundary layers in PD, and a non-local Gauss’s formulation is presented by transforming the displacement and traction boundaries in CCM into the non-local fictious boundary layers in PD, and this formulation unifies the variational framework and boundary conditions of PD and CCM. Third, a fully implicit algorithm is developed to obtain the general nonlinear problems such as fracture and large deformation of solid materials. Further, a penalty method is employed to eliminate the zero-energy mode oscillation inherently observed in NOSB-PD, and the penalty force and penalty stiffness matrix are derived for the proposed implicit algorithm and numerical implementation. Numerical results demonstrate that the proposed method is accurate and can well capture the fracture and large deformation of solid materials. Results also indicate that the method can effectively prevent the zero-mode oscillations inherently observed in the original NOSB-PD, and thus ensures the computational stability.

Peridynamics modelling of dynamic tensile failure in concrete

Concrete as a brittle material suffers from tensile failures such as fracture and spalling under tension. These tensile failures are believed to be associated with the initiation, propagation, and coalescence of cracks. The purpose of this study is to present a numerical investigation on the dynamic tensile failures in concrete. For this purpose, the non-ordinary state-based peridynamics (NOSB-PD) theory is employed as an analysis tool to avoid the difficulties encountered in the finite element method in dealing with discontinuities such as cracks. Furthermore, to obtain the realistic stress field in the concrete and so as to update the nonlinear relationship between the force states and the displacement vectors required in the NOSB-PD theory, a well-developed rate-dependent plastic-damage model for concrete is incorporated. This model is modified from the Kong-Fang concrete model which has been verified to be suitable for the modeling of dynamic tensile failures in concrete within the classical continuum mechanics. Several numerical examples including dynamic fracture tests and spall tests are presented to demonstrate the performance of the current model. The results reveal that the dynamic fracture and spalling in concrete can be well captured and are observed to be greatly influenced by the loading rates. The predicted crack patterns of the concrete under different loading rates are found to be comparable to the corresponding experimental results.

Peridynamic modelling of blasting induced rock fractures

This paper presents a peridynamics-based computational approach for modelling blasting induced rock fractures. A new non-ordinary state-based peridynamics approach is proposed in conjunction with a Johnson-Holmquist (JH2) constitutive model to consider the pressure dependency, strain rate effect, and viscoplasticity of rocks under blasting loads. The fracturing process in a rock is assessed based on both the JH2 damage model and a tension failure model. The former evaluates the material response pertaining to excessive plastic strain and the latter is used to gauge failure based on tensile stress in consideration of strain rate effect. Detonation in the explosives is simulated using updated Lagrangian peridynamics in conjunction with Jones-Wilkins-Lee (JWL) equation of state. Simulations of single-hole blasting in granite rock are presented and compared with experimental records. The proposed approach is shown to capture reasonably well the plastic material failure surrounding the borehole as well as the tensile cracks on both radial and circumferential directions. Further sensitivity studies indicate that the intact strength parameters in the JH2 model and the tensile strength of material play a vital role in producing the obtained fracture patterns and should hereby be selected with care. The presented computational approach offers a rigorous basis for future development of versatile, multi-physics-integrated computational framework on rock blasting simulations.

Coupling of non‐ordinary state‐based peridynamics and finite element method with reduced boundary effect

AbstractIn this study, we propose a new numerical technique to couple non‐ordinary state‐based peridynamics (NOSB‐PD) and the finite element method (FEM), and improve the scheme by implementing an effective boundary imposition method and a stabilization method. This coupling scheme takes the mutual advantages of peridynamics and the FEM to solve fracture problems without the imposition of additional criteria, yielding enhanced computational efficiency. In addition, the coupling model brings the reduction of the boundary effect using the boundary imposition method. To combine the peridynamics and FEM, the model is partitioned into the peridynamic subregion and finite element subregion. Subsequently, the two subregions are bridged by interface elements, where peridynamic nodes are embedded. Two types of coupling schemes are developed and the boundary effect of the peridynamic subregion is analyzed in each coupling approach. Moreover, stabilization of the coupling method is implemented to control the zero‐energy mode inherent in NOSB‐PD to simulate fracture problems. The proposed methodology is verified by solving several quasi‐static problems in the one‐ and two‐dimensional domains, and fracture problems are solved. The crack paths predicted by the proposed coupling method are in good agreement with the results of the exact solution and the experiments.

Coupling of non‐ordinary state‐based peridynamics and finite element method for fracture propagation in saturated porous media

AbstractIn the present manuscript fracture propagation in a saturated porous medium is modeled based on the classical Biot theory, where solid skeleton and fluid flow are represented by separate two layers. The non‐ordinary state‐based peridynamics (NOSBPD) layer is employed to capture deformation including fracturing of the solid skeleton, while the fluid flow is controlled by the finite element method (FEM) layer. The interaction between the layers is realized by considering the effects of pore pressure from the FEM layer on the NOSBPD layer and, vice versa, the effect of the volumetric strain, porosity, and permeability variations from the NOSBPD layer on the FEM layer. The coupling terms retain their parent characteristics, that is, the interaction term in the momentum balance equation is approximated by the local FEM formulation whereas the interaction term in the mass balance equation is approximated by the nonlocal NOSBPD formulation. By doing so, the model retains the flexibility of coupling two independent discretizations. The coupled system is solved by a fully implicit solution scheme. The accuracy of the proposed method has been verified against available closed‐form solutions and published numerical approaches for the pressure‐ and fluid‐driven facture propagation problems.

A bond-based correspondence model and its application in dynamic plastic fracture analysis for quasi-brittle materials

This work presents a generalized bond-based peridynamic model named bond-based correspondence model to simulate the fracture. This model combines the advantages of bond-based peridynamics and the correspondence model (non-ordinary state-based peridynamics). The peridynamic pair-wise force is calculated by classical constitutive equations through the relation between peridynamic pair-wise force and stress. With specific function types of relation between peridynamic pair-wise force and stress, the bond-based correspondence model can degenerate to the existing bond-based peridynamic models, such as the extended peridynamic model with bond rotation and the micropolar peridynamics. Besides, this model is easy to be extended to plasticity, in this study, the small deformation von Mises elastoplastic model is developed. And the effects of plastic parameters on dynamic fracture are analyzed for quasi-brittle materials. Numerical examples illustrate the model’s capacity in solving the fixed Poisson’s ratio and simulating dynamic crack propagation. And the numerical results show that plastic parameters have a great impact on fracture propagation, including crack initiation and the speed of propagation.

Thermomechanics of damage in brittle solids: A peridynamics model

We propose a fully coupled model for damage in brittle solids undergoing thermomechanical deformation. The model is fully coupled in the sense that, just as the thermal field appears in the evolutions of mechanical deformation and damage, the deformation and damage fields also occur in the evolutions of the other two complementing state variables. Damage is presently described by a continuous field variable, an order parameter, that undergoes a smooth transition from the undamaged to a fully fractured state. The governing equations describing the evolutions of deformation, damage and temperature are in the form of coupled partial differential equations (PDEs). The constitutive relations and temperature evolution equation are derived consistent with the two laws of thermodynamics. Other than accounting for the thermal loads, the temperature evolution also incorporates the effects of local heat generation (entropy production due to evolving damage) at a crack tip and thermomechanical coupling (in the form of a dilatation term). In addition to heat transfer through conduction, the heat equation also incorporates heat flow through the radiation mode. The model is computationally implemented through non-ordinary state-based (NOSB) peridynamics (PD) considering the evolving discontinuities (e.g. the cracked surfaces) and the higher order derivatives appearing in the governing PDEs. In the PD implementation, a scalar entropy flux and hence a scalar heat flux, is introduced and a notion of entropy equivalence is exploited. The entropy equivalence furnishes a constitutive correspondence for the entropy and heat fluxes. In the PD formulation, the governing PDEs are converted to the integro-differential type. Validation exercises on the model’s performance against transient heat transfer in a silica tile experiment and Kalthoff’s experiment are carried out to demonstrate the predictive ability. By way of a more comprehensive assessment of the model, numerical simulation of a silica tile under fully coupled thermomechanical loading is also performed.

Dynamic delamination of laminated structure under impact load using Non-Ordinary State-Based Peridynamics

Dynamic delamination of laminated structure under impact load using Non-Ordinary State-Based Peridynamics

Revised non-ordinary state-based peridynamics and a new framework for coupling with finite element method

In this paper, we present a revised non-ordinary state-based peridynamics (RNOSBPD), which works for the arbitrary interaction domain and does not require surface correction and volume correction. For the spherical interaction domain, RNOSBPD recovers non-ordinary state-based peridynamics (NOSBPD) of the published work. This current work also presents a new approach to couple the RNOSBPD with FEM (RNOSBPD-FEM). This new coupling approach is straightforward and is easily introduced into commercial finite element codes. With this coupling method, special techniques, such as overlapping regions, interface elements, or fictitious nodes, that have published in the literature for coupling PD with FEM are not needed. Four static applications are performed and the results show the accuracy and efficiency of the RNOSBPD-FEM.