Peridynamics Research Articles

A bond-based peridynamics model based on variable material properties for modeling elastoplastic behavior

Crack tip singularities and material discontinuities are taken to be the main challenges for analyzing cracks in fracture mechanics problems. Peridynamic (PD) theory has been recently introduced as a nonlocal extension of continuum mechanics capable of improved modeling of progressive damage and rupture in cracked structures. In this theory, the internal forces have been characterized by nonlocal interaction between the pairs of the material points whose displacements or displacement derivatives are discontinuous. Given that the bond-based PD (BB-PD) has not been satisfactorily employed for predicting ductile fracture, the main purpose of this work is to present a new PD plasticity model in conjunction with the Variable Material Property (VMP) method, aiming to provide the possibility of elastoplastic modeling in BB-PD. In the present work, the quasi-static loading has been selected for simulation and the von-Mises yield criterion has been utilized for describing the plastic deformation. The loadings are applied as the displacement is controlled and outputs from the PD model are compared with the results achieved from ABAQUS software based on the continuum mechanic assumptions. To validate the accuracy of the present formulation, two example problems of the plate with a central crack and a central hole subjected to tensile load are examined. The results of von-Mises stress, the plastic zone size, horizontal and vertical displacement, and equivalent plastic strain have been verified in comparison with the outputs calculated by the finite element method (FEM), evidencing good accuracy of the VMP-based PD elastoplastic model in predicting the desired parameters. To predict the formation of spontaneous cracks and crack growth trajectory, two more examples that include a plate with a central hole, as well as, a plate with two edge cracks and a double hole are discussed. The obtained results are in good agreement with those from the experiment and ABAQUS software.

An Efficient Jacobi Spectral Collocation Method with Nonlocal Quadrature Rules for Multi-Dimensional Volume-Constrained Nonlocal Models

In this paper, an efficient Jacobi spectral collocation method is developed for multi-dimensional weakly singular volume-constrained nonlocal models including both nonlocal diffusion (ND) models and peridynamic (PD) models. The model equation contains a weakly singular integral operator with the singularity located at the center of the integral domain, and the numerical approximation of it becomes an essential difficulty in solving nonlocal models. To approximate such singular nonlocal integrals in an accurate way, a novel nonlocal quadrature rule is constructed to accurately compute these integrals for the numerical solutions produced by spectral methods. Numerical experiments are given to show that spectral accuracy can be obtained by using the proposed Jacobi spectral collocation methods for several different nonlocal models. Besides, we numerically verify that the numerical solution of our Jacobi spectral method can converge to its correct local limit as the nonlocal interactions vanish.

A dissipation informed peridynamic model for dynamic brittle fracture

In this paper, we propose a dissipation informed peridynamic (PD) model for dynamic brittle fracture. The viscosity is introduced in for dissipation compensation to recover the physical mechanism of fracture. The general type of deformation in solids is considered and the bulk viscosity force state is nonlocally constructed from the bond level as a minimum variation of conventional PD. Rather than adjusting the artificial coefficient, the signal processing method, i.e. the wavelet method is employed for viscosity evolution. With the wavelet method, the spurious stress waves can be identified and the dissipation deficiency can be naturally quantified. The proposed model is then verified by a rigorous validation simulation against the experiment results, which is still missing in the PD literature. The real loading condition is modeled to reproduce the ‘quasi-steady-state crack propagation’ problem and twenty different cases are included to cover the whole fracture stage. The results demonstrate that the model is capable to recover the physical mechanism and capture the dynamic brittle fracture. New insights into the dynamic brittle fracture are also gained by investigating the viscosity effect and deriving the crack motion equation.

An adaptive contact model involving friction based on peridynamics

In peridynamics (PD), various existing contact models are difficult to resolve the contradiction between identifying contact surfaces and calculating contact forces accurately. This contradiction poses a challenge to the quantitative analysis of impact, extrusion, friction, and closure cracks. A friction contact model in PD based on family material particles was proposed to realize the adaptive identification of contact surfaces. The boundary normal vector was derived from the directional characteristics of the boundary reaction force. According to the dynamic equations of the boundary material particles, the proposed contact model achieves the precise calculation of the contact forces. In addition, an improved bond-based peridynamic model with shear bonds (improved PDS model) was extended to a three-dimensional form to achieve accurate deformation analysis of spatially complex deformable bodies. By combining the improved PDS model with the proposed contact model, the non-objectivity of the relative velocity is eliminated and the computational accuracy of the contact model is greatly improved. Quantitative analysis of the spatially deformed cantilever beam proves that the 3D improved PDS model possesses high accuracy of displacement calculation. Several examples with non-negligible friction were conducted to verify the reliability and efficiency of the proposed contact model. The results show that the proposed contact model can significantly improve the accuracy of displacement analysis and damage prediction in contact problems with non-negligible friction effects.

A discrete surface correction method for bond-based peridynamics

The peridynamic (PD) material parameters at boundaries where the neighborhood is incomplete exhibit a softer material response compared to bulk material points. This effect is called the surface or skin effect. This study presents a new surface correction method for bond-based peridynamics that calculates the PD material parameters discretely during numerical simulations. The discretely calculated material parameters can easily account for the effect of newly formed free surfaces, i.e., crack surfaces, that can occur during the simulation. Hence, the proposed surface correction technique provides a single context for a surface correction method that can be applied to static problems and dynamic problems. In numerical simulations, the meshfree discretization technique enhanced with the HHB algorithm is utilized for the spatial discretization to improve accuracy in the numerical integration over the spatial domain. In addition, a half-step time integration scheme is used in temporal discretization. The results obtained using the proposed surface correction method are compared with the surface correction methods available in the literature. It is shown that the proposed surface correction method is more accurate in the static and dynamic PD analyses by considering the missing neighbors along the problem boundaries and the newly emerging boundaries during the crack propagation.

3-D coupled peridynamics and discrete element method for fracture and post-fracture behavior of soil-like materials

The present paper investigates a 3-D numerical analysis method, combining peridynamics (PD) and the discrete element method (DEM), which can simultaneously handle both the fracture and post-fracture behavior of materials assumed to be soils. With geomaterials, it is very important to treat the process from fracture to post-fracture in a continuous manner. The bond-based PD model is employed to reproduce the behavior from the initiation of cracks to fracturing. At the same time, the DEM model is also introduced so that the contact forces among the fragmented pieces can be calculated after the fracturing. The computational process of the physical contact between the wall element and the computational nodes of the PD is also solved based on the DEM so that plane geometry boundaries can be used. For the failure criteria of the connection between the computational nodes of the PD, a bilinear type of bond model is introduced to obtain a nonlinear stress–strain relationship and numerical stability when the calculation process switches from PD to the DEM. To verify the proposed approach, numerical simulations of the diametral compression test are performed on flawless cylindrical specimens, and the results are compared with theoretical solutions for the stress distribution inside the specimens. Based on the results, a simulation of the compressive failure of multiple soil-like crushable objects is performed as an example of the application of the coupled PD-DEM. The obtained results show that the proposed method will serve as a computational framework that can treat the continuous fracture behavior of soil-like materials in three dimensions.

An orthotropic peridynamic shell model for linear elastic deformation and crack propagation

Peridynamics is a non-local mechanics theory that can handle discontinuities such as cracks. This study presents an orthotropic peridynamic (PD) shell model based on the micro-beam bond. Under the homogenization assumption, eight PD material parameters expressed by four independent material constants are used to describe the axial, bending, and torsional deformations of the micro-beam bond. The eight PD material parameters can be derived by the rule of equating the strain energy density (SED) of the PD shell model and that of the classical continuum mechanics (CCM) shell model. To improve the computational efficiency and solve the surface effects of PD models, the morphing method is used to establish the PD-CCM shell coupling model. Several numerical examples of crack propagation in orthotropic materials are executed, and the results demonstrate the effectiveness of the proposed model.

A Penalty Method for Coupling of Finite‐Element and Peridynamic Models

AbstractA classical continuum mechanics model no longer fulfills its basic assumptions, when the deformations are not smooth or discontinuous. Peridynamics (PD) as an integral formulation could better overcome these issues. Therefore, it is better suited to solve fracture problems such as crack initiation, its growth and formation of crack patterns. In PD the magnitudes of internal forces at a material point depend on the collective interaction of all material points within a subdomain called horizon. To take advances from the formulation, typically it is implemented in a mesh‐free form. In that case, it is easy to separate interactions between different material points. As consequence, a relatively high resolution is required to describe continuous deformation functions and, consequently computationally expensive. Therefore, the application of mesh‐free PD in undamaged regions requires an unnecessary high effort for receiving sufficient accurate results. In contrast, the finite element method (FEM) as a classical continuum mechanics‐based approach, is very efficient, if continuous stress distributions can be assumed. Consequently, a computational cost reduction can be achieved, if the PD is applied in the local damaged zone only and the remaining area is modeled and analyzed with the classical FEM. Realization of such an approach requires a sufficient accurate coupling of the PD and the FEM subdomains.A brief overview about different coupling approaches is given. Then the general approach of an Arlequin based method is presented. Here the coupling is achieved via an overlapping of PD and FEM areas. The coupling equations are developed and realized with help of the Penalty method. The Penalty method is considered as an alternative procedure to the Lagrange multiplier strategy without creating new unknowns, which essentially simplifies the coupled PD‐FEM approach. The validity and limits of the proposed technique are demonstrated through analysis, where the choice of the Penalty number is discussed as well as artificial wave oscillations, caused by the coupling area.

Peridynamic modeling of damage and non-shock ignition behavior of confined polymer bonded explosives under impact loading

A mechanical-thermal-chemical coupled peridynamic (PD) modeling for analyzing the damage and non-shock ignition behavior of polymer bonded explosives (PBXs) subject to impact loading has been presented. The friction and heat conduction and Arrenhenius law are embedded into the PD theory, ensuring that the simulation model is capable of capturing major features of dynamic damage response of PBX. A PD material model considering plasticity and the asymmetric behaviors of PBX under tensile and compressive loadings is proposed in this study. For a ring-shaped PBX 9501 sample confined by the steel structures, the simulation results indicate that the PBX 9501 in the regions towards the impact face and support face suffers from the serious damage. Meanwhile, the corresponding average temperatures in these two regions are also higher than their counterparts in other regions due to the more intense interaction between PBX 9501 and the steel confinement structures. The ignition time decreases with the increasing impact velocity within the range of 40–120 m s−1. The internal steel shell response is analyzed according to its velocity history within a representative box. It is found that the deformation process of internal steel shell is closely related to the complicated flow behaviors of PBX 9501.

Peridynamic elastic waves in two-dimensional unbounded domains: Construction of nonlocal Dirichlet-type absorbing boundary conditions

The focus of this paper is on application of peridynamics (PD) to propagation of elastic waves in unbounded domains. We construct absorbing boundary conditions (ABCs) derived from a semi-analytical solution of the PD governing equation at the exterior region. This solution is made up of a finite series of plane waves, as fundamental solutions (modes), which satisfy the PD dispersion relations. The modes are adjusted to transmit the energy from the interior region (near field) to the exterior region (far field). The corresponding unknown coefficients of the series are found in terms of the displacement field at a layer of points adjacent to the absorbing boundary. This is accomplished through a collocation procedure at subregions (clouds) around each absorbing point. The proposed ABCs offer appealing advantages, which facilitate their application to PD. They are of Dirichlet-type, hence their implementation is relatively simple as no derivatives of the field variables are required. They are constructed in the time and space domains and thus application of Fourier and Laplace transforms, cumbersome for nonlocal models, is not required. At the discrete level, the modes satisfy the same numerical dispersion relations of the near field, which makes the far-field solution compatible with that of the near field. We scrutinize the performance of the proposed ABCs through several examples. Our investigation shows that the proposed ABCs perform stably in time with an appropriate level of accuracy even in problems characterized by highly-dispersive propagating waves, including crack propagation in semi-unbounded brittle solids.

A multiscale model for mechanical and fracture behavior of calcium-silicate-hydrate: From molecular dynamics to Peridynamics

Fully understanding the fracture behavior of concrete is a challenging work since concrete is a complex multiscale composite material with a heterogeneous structure at different length scales ranging from nanoscale to macroscale. A new multiscale model framework was proposed in the present study to explore the mechanical properties, including Young's modulus E and ultimate tensile strength UTS, and fracture behavior of calcium-silicate-hydrate (C-S-H) from the nanoscale to mesoscale. The nanoscale C-S-H globule models were built and simulated via molecular dynamics (MD) simulations and then the obtained mechanical properties were employed as the input parameters for the mesoscale Peridynamics (PD) simulations to acquire the E, UTS, and fracture behavior of C-S-H gel. The results reveal that at nanoscale, the mechanical performance of C-S-H is anisotropic: As the size of C-S-H model increases, the E and UTS of C-S-H decrease because the extension of crack in a larger C-S-H model will release more stored energy. At mesoscale as the packing fraction φ increases, the E and UTS of C-S-H gel increase. The resulted E and UTS are in line with former simulations and experimental data implying the effectiveness and accuracy of the proposed multiscale model.

Non-Local Formulation of Heat Transfer with Phase Change in Domains with Spherical and Axial Symmetries

Describing heat transfer in domains with strong non-linearities and discontinuities, e.g. propagating fronts between different phases, or growing cracks, is a challenge for classical approaches, where conservation laws are formulated as partial differential equations subsequently solved by discretisation methods such as the finite element method (FEM). An alternative approach for such problems is based on the non-local formulation; a prominent example is peridynamics (PD). Its numerical implementation however demands substantial computational resources for problems of practical interest. In many engineering situations, the problems of interest may be considered with either axial or spherical symmetry. Specialising the non-local description to such situations would decrease the number of PD particles by several orders of magnitude with proportional decrease of the computational time, allowing for analyses of larger domains or with higher resolution as required. This work addresses the need for specialisation by developing bond-based peridynamic formulations for physical problems with axial and spherical symmetries. The development is focused on the problem of heat transfer with phase change. The accuracy of the new non-local description is verified by comparing the computational results for several test problems with analytical solutions where available, or with numerical solutions by the finite element method.

Simulation Prediction and Experiment of Brittle Damage of Cemented Carbide Microgroove Turning Tools Based on Peridynamics

Brittle damage is a key factor restricting tool life extension. The peridynamic (PD) theory was applied to explain and predict the brittle damage of the near-field of the cutting edge of a cemented carbide microgroove turning tool (CCMTT) for the first time in this study, and the PD modeling of the complex surface was realized. The results showed that the PD modeling accuracy of the CCMTT can reach ±3.4%. The displacement of material points in the near-field of the cutting edge of the CCMTT is caused by the combined effect of the external load and the internal interaction force, and the former is dominant. There is no linear relationship between the displacement and the calculation time; instead, there are fluctuations and a maximum increase in the material point displacement in the main cutting direction. Only microdisplacements of material points in the near-field of the cutting edge occur under the given cutting conditions. The accumulation of microcracks caused by microdisplacement does not reach the transition threshold to form macrocracks. This agrees well with the experimental results, and the relative error can be controlled within 3.2%.

A peridynamic-informed neural network for continuum elastic displacement characterization

Physics-informed neural networks have attracted considerable attention for solving scientific and engineering problems because of their low data dependence and physical outputs. The most popular way to embed the laws of physics into neural networks is to apply partial differential equations. However, many complex mechanical problems, such as displacements near the phase interface in heterogeneous plates, are difficult to solve using differential equations expressed by higher-order derivatives. In this work, peridynamic (PD)-informed neural networks are established to characterize the displacement of homogeneous and especially heterogeneous elastic plates. The networks are directly constrained by the integro-differential equations of bond-based PD, thus avoiding higher-order derivatives. The networks are trained without labeled data, and the convergence of the training process is accelerated by a gradual-refinement sampling method. Four simulation experiments show that the displacements of two-dimensional plates with homogeneous or heterogeneous structures are globally approximated by the proposed network with small deviations compare to those obtained with peridynamic numerical solutions. Further investigations show that sampling refinement can significantly increase the prediction accuracy of the networks for complex heterogeneous plates.

Peridynamics for mechanism analysis of soil desiccation cracking: Coupled hygro-mechanical model, staggered and monolithic solution

Soil shrinkage cracking is a hygro-mechanical coupled phenomenon and it can significantly weaken the mechanical properties of soil and cause various potential natural disasters. This study proposes an improved bond-based peridynamics (PD) formulation, based on a comprehensive review of the hygro-mechanical peridynamic models. The proposed model with a certain influence function can be recast as an existing bond-based PD diffusion model. Also, an attenuation kernel function and discretized micro-modulus are adopted to revise the bond force density in the prototype micro-elastic brittle (PMB) model. For the numerical implementation, both monolithic and staggered schemes are adopted for the coupled hygro-mechanical problem. Specifically, the coupled stiffness matrix in the discretized equilibrium equation is constructed by using the Lagrange multiplier method for the monolithic scheme. While the direct integration method of explicit dynamics with an artificial damping term is adopted for the staggered scheme. For modeling of soil drying and cracking, two-dimensional soil rings, and two- and three-dimensional soil strips are simulated and demonstrated. The soil ring model verifies the accuracy of the coupled moisture-soil bond-based PD model through an explicit staggered scheme, and explores the effect of the relative thickness of rings on the number of cracks. The implicit monolithic solution is adopted to analyze the mechanisms of soil strip cracking. A detailed discussion on the scheme for controlling the maximum number of bond breakage is provided to guarantee convergency and accuracy with optimal efficiency. The soil strip cracking pattern and upward curling deformation agree well with experimental observations. The results show that the hygro-mechanical coupled peridynamic model has great advantages in solving soil drying and cracking. The present work shows a potential way in soil desiccation cracking investigation for real-scale simulation and fine mechanism exploration.

Metaheuristic-based crack detection in beam-type structures using peridynamics theory: A comparative study

This study is a first attempt to provide an effective tool for crack prediction in beam-type structures using the peridynamics (PD) theory and metaheuristic-based optimization. The norm of the difference between the response of the tested beam and that of the trial model (analyzed using PD), is minimized to find the position and depth of crack. Model calibration is discussed thoroughly. Five well-known metaheuristics as well as an upgraded charged system search algorithm are considered. Results reveal that the proposed procedure can effectively localize and detect the crack severity, even in noise contaminated cases.

Peridynamic Shell Model Based on Micro-Beam Bond

Peridynamics (PD) is a non-local mechanics theory that overcomes the limitations of classical continuum mechanics (CCM) in predicting the initiation and propagation of cracks. However, the calculation efficiency of PD models is generally lower than that of the traditional finite element method (FEM). Structural idealization can greatly improve the calculation efficiency of PD models for complex structures. This study presents a PD shell model based on the micro-beam bond via the homogenization assumption. First, the deformations of each endpoint of the micro-beam bond are calculated through the interpolation method. Second, the micro-potential energy of the axial, torsional, and bending deformations of the bond can be established from the deformations of endpoints. Finally, the micro moduli of the shell model can be obtained via the equivalence principle of strain energy density (SED). In addition, a new fracture criterion based on the SED of the micro-beam bond is adopted for crack simulation. Numerical examples of crack propagation are provided, and the results demonstrate the effectiveness of the proposed PD shell model.

The Coupled Thermo-Chemo-Mechanical Peridynamics for ZrB2 Ceramics Ablation Behavior

The ablation of ultra-high-temperature ceramics (UTHCs) is a complex physicochemical process including mechanical behavior, temperature effect, and chemical reactions. In order to realize the structural optimization and functional design of ultra-high temperature ceramics, a coupled thermo-chemo-mechanical bond-based peridynamics (PD) model is proposed based on the ZrB2 ceramics oxidation kinetics model and coupled thermo-mechanical bond-based peridynamics. Compared with the traditional coupled thermo-mechanical model, the proposed model considers the influence of chemical reaction process on the ablation resistance of ceramic materials. In order to verify the reliability of the proposed model, the thermo-mechanical coupling model, damage model and oxidation kinetic model are established respectively to investigate the applicability of the proposed model proposed in dealing with thermo-mechanical coupling, crack propagation, and chemical reaction, and the results show that the model is reliable. Finally, the coupled thermo-mechanical model and coupled thermo-chemo-mechanical model are used to simulate the crack propagation process of the plate under the thermal shock load, and the results show that the oxide layer plays a good role in preventing heat transfer and protecting the internal materials. Based on the PD fully coupled thermo-mechanical model, this paper innovatively introduces the oxidation kinetic model to analyze the influence of parameter changes caused by oxide layer growth and chemical growth strain on the thermal protection ability of ceramics. The proposed model provides an effective simulation technology for the structural design of UTHCs.

Study on Crack Propagation Parameters of Tunnel Lining Structure Based on Peridynamics

The numerical simulation results utilizing the Peridynamics (PD) method reveal that the initial crack and crack propagation of the tunnel concrete lining structure agree with the experimental data compared to the Japanese prototype lining test. The load structure model takes into account the cracking process and distribution of the lining segment under the influence of local bias pressure and lining thickness. In addition, the influence of preset cracks and lining section form on the crack propagation of the concrete lining model is studied. This study evaluates the stability and sustainability of tunnel structure by the Peridynamics method, which provides a reference for the analysis of the causes of lining cracks, and also lays a foundation for the prevention, reinforcement and repair of tunnel lining cracks.

Refined three-dimensional simulation of ribbed bar pull-out tests based on an enhanced peridynamic model

Peridynamic (PD) theory can effectively model discontinuous problems, but it remains a challenging task for refined three-dimensional (3D) simulation of bond-slip behaviors in pull-out tests, particularly when the rib of rebar is considered. The mesh size of the model needs to be small enough (e.g., decimillimetre or 0.1 mm) to simulate various shapes of ribs. Therefore, the scale of the PD model is usually significantly large and the computational efficiency is low. This paper proposes an enhanced bond-based PD (BPD) model to perform a refined 3D simulation of bond-slip behaviors in the ribbed bar pull-out test. In order to reduce the computation scale and improve efficiency, a practical 3D coupled element is developed to couple the fine PD model and coarse finite element (FE) model, where the PD is used to model the regions in concrete and rebar that are close to the interface, while FE model is used to simulate the regions far away from the interface. A computational framework is developed for implicit and static analyses of the coupled PD-FE model. A parallel computing strategy is adopted to improve the computational efficiency, and enhanced concrete and steel BPD models are presented to accurately simulate the nonlinear behaviors and damage of concrete and ribbed rebars. The enhanced PD modeling approach is implemented in an open-source finite element software framework, OpenSees, and verified by a cantilever beam under uniaxial loading conditions and a pushover analysis of an RC column. Based on the enhanced PD, a refined 3D simulation of ribbed rebar pull-out tests is performed. The bond-slip behaviors are studied in detail, e.g., the crack propagation with different types of ribs, concrete strengths, and rebar properties. The enhanced PD modeling approach provides an effective tool for refined simulation of the complicated bond-slip behaviors in RC structures.