Peridynamic Theory Research Articles

A new bond model in peridynamics theory for progressive failure in cohesive brittle materials

In this study, a new bond damage model is proposed in the framework of the bond-based peridynamics theory to better describe the progressive failure process in cohesive brittle materials. The new bond model is able to account for the progressive failure and residual strength of bonds. The peridynamics theory is further coupled with the classical finite element method in order to efficiently describe failure processes at a large scale of structures. Several benchmark tests concerning mode-I and mixed-mode fractures in concrete structures are studied to verify the efficiency of the coupled peridynamics-finite element method. The obtained numerical results are in good agreement with the experimental data for both overall force–displacement responses and crack propagation patterns. It is found that the new bond damage model significantly improves the prediction of progressive failure process in cohesion brittle materials with respect to the classical elastic-brittle bond model.

Crack propagation and dynamic properties of coal under SHPB impact loading: Experimental investigation and numerical simulation

To investigate the crack propagation and dynamic mechanical properties of coal under high strain rate loading, 24 sets of Brazilian disk (BD) coal specimens with vertical and horizontal beddings were made, and tests were conducted by a split Hopkinson pressure bar (SHPB) system. Under different impact velocities, a high-frame and high-resolution camera was employed to capture the fracture process, and two high-dynamic strain gauges were used to record the stress pulse signals simultaneously. Using one-dimensional stress wave theory, the dynamic mechanical properties of coal with different bedding directions under different impact velocities were analyzed and discussed. Experimental results indicate that bedding directions not only have a major influence on dynamic mechanical properties such as dynamic tensile strength, strain rate and strain energy but also have a great influence on the crack propagation path. Then, based on the image processing technique and fractal method, cracks conforming to the fractal have been proven, and the results further illustrate that the fractal dimension of cracks on the coal surface increased in the fracture process under SHPB loading. Finally, a numerical model based on bond-based Peridynamic theory was proposed to simulate crack propagation and dynamic mechanical properties of coal under the SHPB test, and the displacement-strain-stress fields were also calculated, which further reveal the fracture mechanism and dynamic behavior of coal under different impact loading conditions.

Studies on Quasi-Static and Fatigue Crack Propagation Behaviours in Friction Stir Welded Joints Using Peridynamic Theory

The friction stir welding (FSW) technology has been widely applied in aircraft structures. The heterogeneity of mechanical properties in weld and the hole in structure will lead the crack to turn. Peridynamics (PD) has inherent advantages in calculating crack turning. The peridynamic theory is applied to study the crack turning behaviour of FSW joints in this work. The compact tension (CT) samples with and without a hole are designed. The crack propagation testing under quasistatic and fatigue loads are performed. The peridynamic microplastic model is used and a three-stage fatigue calculation model is developed to simulate the quasistatic fracture and the fatigue crack growth. The results predicted by the peridynamic models are compared with the experimental ones. The effects of welding direction on quasistatic and fatigue crack propagation behaviours are investigated and the effect of hole position on crack path geometry is also studied. It is shown that the crack turning in FSWed CT samples can be captured by the peridynamic microplastic and the three-stage fatigue calculation models. The peridynamic crack growth rates agree with the experimental results. For CT specimen without a hole, the crack turns into the weld zone where the material is softer. The effect of welding direction on crack growth rates is not obvious. For CT sample with a hole, the crack propagation direction has been mainly controlled by the hole location and the welding direction has a slight effect on crack path.

Study on Transient Thermal Response for Functionally Graded Materials Based on Peridynamic Theory

The transient heat conduction formula of functionally graded materials (FGMs) is presented based on peridynamics (PD). The simplified micro-heat conductivity model for FGMs is proposed and the numerical discretization and the peridynamic numerical formation are also illustrated. A FORTRAN program is coded to implement calculations. The accuracy of the program is verified by comparing the FEM and analytical results with PD solution. The FGM rectangle plate composed by titanium alloy coating zirconium oxide is performed to calculate temperature fields. The effects of material gradient, porosity and temperature load on thermal response are studied. It is shown that the ceramic proportion of FGMs is increased with an increasing material shape parameter and the thermal shielding performance of FGMs is also improved. The effect of the porosity on thermal response is more and more significant with the increasing time step. The increasing temperature load only affects the temperature response of FGM ceramic area. The thickness of temperature distribution area is increased with the increasing of heat conduction time.

On the peridynamic formulation for an orthotropic Mindlin plate under bending

Composite materials are becoming increasingly widespread across applications such as aerospace, automotive, wind and solar energy systems, on account of their superior specific strength and stiffness. There are still unresolved challenges in the structural analysis of composite materials in practical engineering applications, owing to their complex structures. Peridynamic theory is a robust technique for the analysis of composite structures; however, it is computationally costly. Structural idealizations are commonly applied in engineering applications to represent structures with a reasonable simplicity, thus reducing the computational cost. Idealization of a planar structure, such as a plate, reduces the number of peridynamic interactions to be solved. In this study, a peridynamic plate formulation of an orthotropic plate with transverse shear deformation is proposed. Peridynamic bond parameters are derived by localizing the peridynamic equations of motion and comparing them with equations of motion obtained using classical continuum mechanics. A failure theory is implemented in the formulation in order to perform crack propagation simulations. A MATLAB script is written using the proposed formulation for the solution of orthotropic plates under bending. The peridynamic solutions are obtained for both elastic deformations and failure prediction of pre-cracked plates, and are validated using finite-element analysis solutions.

Peridynamics simulation of crack propagation of ring-shaped specimen like rock under dynamic loading

The split Hopkinson pressure bar (SHPB) system with a spindle-shaped striker is widely used to test the characteristics of rock-like materials subjected to dynamic loading, and the crack propagation of ring-shaped specimens under dynamic loading is a research focus in rock mechanics. In this paper, a model of the SHPB system, including the striker, incident bar, specimen, and transmitted bar, is established based on peridynamics theory and is validated by comparing two kinds of stress waves obtained by a peridynamics simulation and experiment. Based on these results, the crack propagation of a ring-shaped specimen with an aperture is simulated, and the influences of the apertures of the specimen on the failure patterns are discussed. The results show that there are four cracks distributed on the specimen with an aperture of 24 mm: crack-1 first initiates at the end of the inner diameter near the incident bar, crack-2 then occurs at the end of the inner diameter near the transmitted bar, and then crack-3, also called a secondary crack, initiates on the upper and lower outer boundaries of the specimen; these secondary cracks are perpendicular to the other cracks. The sequence of starting velocities and average propagation velocities of cracks from high to low is crack-3, crack-2, and crack-1, and the failure of the ring-shaped specimen occurs due to tension. Controlled by the structure effect of the ring-shaped specimen, the loading rate and the peak stress decrease gradually, and the failure patterns of the specimens transform from splatted by one initial crack to broken by two cracks that are perpendicular to each other and then to asymmetrical failure with an increase in the apertures of the specimen. The comparison results of peak stresses and failure patterns between the peridynamics simulation and the experimental results verify the accuracy of peridynamics theory.

An application of non-ordinary state-based peridynamics theory in cutting process modelling of unidirectional carbon fiber reinforced polymer material

The usage of carbon-fiber-reinforced polymer (CFRP) composite keeps increasing in modern industries. However, because of the complexity of CFRP structure, the analysis for CFRP failure process, especially for machining process, is quite difficult. This paper presents a non-ordinary state-based peridynamic approach that can be applied to CFRP machining analysis. In order to describe the composite’s anisotropy, fiber bond and matrix bond is first introduced in this approach. For describing the failure characteristic of composite, the Maximum stress failure criteria and Hashin failure criteria is then incorporated into the state-based peridynamic approach. Finally, the contact state between tool and workpiece is detected by utilizing the penalty method in modeling the machining process. To demonstrate the effectiveness of the present method, tension and cutting experiments are conducted as well.

The third Sandia Fracture Challenge: peridynamic blind prediction of ductile fracture characterization in additively manufactured metal

In the context of the third Sandia Fracture Challenge (SFC3), the details of the blind predictions performed by the University of Texas team are provided in this article. Over the past two decades, the peridynamic theory has shown great promise in modeling autonomous crack nucleation and growth in materials. While peridynamics has been commonly applied to simulate failure of brittle materials, its ability in predicting ductile fracture has remained mostly untested. This fracture challenge was seen as an opportunity to assess the state of the art of the peridynamic theory in predicting the response of an additively manufactured 316L stainless steel bar with a complex geometry under the dynamic tensile experiments performed by Sandia National Laboratories. The performance of a recently proposed, generalized, ordinary finite deformation constitutive correspondence model, coupled with a recent state-based damage correspondence model was explored over this problem. For finite deformation material modeling, the classical elastoplastic framework of Simo was implemented within the ordinary correspondence theory. Damage modeling was achieved by incorporating the Johnson-Cook failure criterion using the damage correspondence framework. An iterative inverse technique was applied to calibrate the model parameters using the longitudinal and notched tensile tests data provided by Sandia National Laboratories. A blind prediction of the deformation and failure behavior of the SFC3 geometry was performed by embedding the calibrated model in a peridynamic simulation. Uncertainty was introduced into the model parameters to quantify material variability. The results are compared to the experiments conducted at Sandia National Laboratories. While our modeling approach led to qualitatively good results and a correctly predicted crack path, it underpredicted the load-carrying capacity of the structure and simulated an early fracture. Our post-experiment analysis identifies material instability issues associated with the model as the primary sources of error.

A non-ordinary state-based Godunov-peridynamics formulation for strong shocks in solids

The theory and meshfree implementation of peridynamics has been proposed to model problems involving transient strong discontinuities such as dynamic fracture and fragment-impact problems. For effective application of numerical methods to these events, essential shock physics and Gibbs instability should be addressed. The Godunov scheme for shock treatment has been shown to be an effective approach for tackling these two issues but has not been considered yet for peridynamics. This work introduces a physics-based shock modeling formulation for non-ordinary state-based peridynamics, in which the Godunov scheme is introduced by embedding the Riemann solution into the force state, resulting in a shock formulation free of tuneable parameters. Several benchmark problems are solved to demonstrate the effectiveness of the proposed formulation for modeling problems involving shocks in solids.

A non-ordinary state-based peridynamic formulation for thermo-visco-plastic deformation and impact fracture

A three-dimensional (3-D) non-local thermo-visco-plastic solid model which considering the plastic hardening, loading rate effect, thermal softening, as well as the fracture behavior of materials was reformulated under the framework of non-ordinary state-based peridynamic theory, and a corresponding spatial integral-type numerical approach was developed together for simulating the impact-caused failure process. They were then employed to study the high-rate thermo-visco-plastic deformation and fracture of metallic materials when subjected to impact loading. The model and numerical approach were validated through simulation of the 3-D Kalthoff-Winkler impacting experiment. The numerical results, including the cracking initiation time and propagation orientation, crack propagation speed, as well as the failure mode and the equivalent stress distribution, agree well with experimental observations and the available numerical results in literature. The effect of the impacting velocity on the crack initiation time, the crack propagation and peak temperature around the crack tips in the target when subjected to impacting loads, were investigated further.

Improved method for zero-energy mode suppression in peridynamic correspondence model

The peridynamic correspondence model provides a general formulation to incorporate the classical local model and, therefore, helps to solve mechanical problems with discontinuities easily. But it suffers from zero-energy mode instability in numerical implementation due to the approximation of deformation gradient tensor. To suppress zero-energy modes, previous stabilized methods were generally more based on adding a supplemental force state derived from bond-based peridynamic theory, which requires a bond-based peridynamic micro-modulus. In this work, we present an improved stabilized method where the stabilization force state is derived directly from the peridynamic correspondence model. Hence, the bond-based peridynamic micro-modulus is abandoned. This improved method needs no extra constant to control the magnitude of stabilization force state and it is suitable for either isotropic or anisotropic materials. Several examples are presented to demonstrate its performance in simulating crack propagation, and numerical results show its efficiency and effectiveness.

Damage and Fracture Analysis of Bolted Joints of Composite Materials Based on Peridynamic Theory

Abstract It is clear that the advantages of fibre glass-reinforced plastics surpass those of steel, but the failure analysis of composite structures is much more complex than that of isotropic materials as composite materials may fail in a variety of ways. In order to simulate the damage and fracture of bolted joints of fibre reinforced composite, the bond-based peridynamic method suitable for elastic, brittle and anisotropic characteristics of composite material is used. The peridynamic model for composite laminate is validated by the finite element method. Then a peridynamic program of composite damage is applied to calculating the damage of bolted joint structure and the damage propagation process and failure mode of the structure is obtained.

A mode-dependent energy-based damage model for peridynamics and its implementation

The mathematical modeling of failure mechanisms in solid materials and structures is a long standing problem. In recent years, peridynamics has been used as a theoretical basis for numerical studies of fracture initiation, evolution and propagation. In order to investigate damage phenomena numerically, suitable material and damage models have to be implemented in an efficient numerical framework. This framework should be highly parallelizable in order to cope with the computational effort due to the high spatial and, depending on the problem, temporal resolution required for high accuracy. The open-source peridynamic framework Peridigm offers a computational platform upon which new developments of the peridynamic theory can be implemented. Today, isotropic material models and a very simple damage model are implemented in Peridigm. This paper proposes three energy-based damage criteria. The implementation approach as well as the extension of Peridigm with these physically motivated models is described. The original criterion of Foster et al. is adapted for ordinary state based material. The other two criteria utilize the decomposition of peridynamic states in isotropic and deviatoric parts to account for the failure-mode dependency. The original criterion is verified by the numerical simulation of two mechanical problems. At first, a virtual double cantilever beam (DCB) experiment is performed to determine the energy release rate. This value is the fundamental material property required for the proposed criteria. Additionally, the DCB problem is then used to investigate the convergence of the numerical scheme implemented in Peridigm. In a second step, a model of a plate with a cylindrical hole under tensile loading is compared with an extended finite element method solution. Results of both numerical solutions are in good agreement. Finally, a fiber reinforced micro structure model is used to analyze the effect of the different criteria to the damage initiation and crack propagation under a more complex loading condition.

A multi-physics peridynamics-DEM-IB-CLBM framework for the prediction of erosive impact of solid particles in viscous fluids

In this paper, a new fully-resolved framework capable of capturing the fundamental physics of particle–fluid interactions, the collision of particles with solid surfaces, and the resulting damage is proposed. A coupled DEM-IB-CLBM, consisting of a discrete element method (DEM), an immersed boundary (IB) method, and a cascaded lattice Boltzmann method (CLBM), is used to fully resolve the interaction of the particles with the surrounding viscous fluid. The peridynamics theory is then implemented and used to predict the impact damage to the target material. This framework is validated by comparing the trajectory of a particle–wall collision event in a viscous fluid with the previous results in the literature. Furthermore, the variation of the restitution coefficient with the impact velocity is in a good agreement with the available experimental results. The influence of multiple impacts and the resulting surface damage on the fluid dynamics of the system is investigated. It is demonstrated that the method correctly predicts the expected effects of multiple collisions and impact angle variations on the surface damage.

Numerical simulation of dynamic fracture behavior in bulk superconductors with an electromagnetic-thermal model

Single-grain GdBCO bulk superconductors have significant applications potential due to their ability to trap stable and large magnetic fields. The internal fracture of these bulk superconductors caused by Lorentz force and thermal load is a key issue in their practical applications. The aim of this work presented here is to investigate the mechanical behavior of the bulk superconductor during pulsed field magnetization. The H-formulation and heat transfer equation are used to obtain the electromagnetic force and thermal load in the bulk with and without defects. Numerical simulations show strong local enhancement of the electromagnetic load at the crack tips. Moreover, dynamic stress intensity factors at the crack tips are presented based on the two-dimensional state-based peridynamic theory. In addition, the crack propagation path is predicted. Finally, different dynamic crack problems are considered to discuss the influence of crack, void, inclusion, and hole on the mechanical stability of the bulk. Results show that defects increase the risk of damage of superconducting bulks.

Decoupling Strength and Grid Resolution in Peridynamic Theory

The primary mechanism of damage evolution of brittle materials in peridynamic theory is based on individual pair-wise bond strain. A critical value for bond strain is derived based on linear elastic fracture mechanic theory. It is a function of the horizon, the radius of non-locality of the material. The horizon must be larger than the material point spacing but not too large which would significantly slow the simulation. Typical sizes used for the horizon range between three and six times the material point spacing, if a constant multiple is used it could be said that the critical strain is a function of the model’s material point grid resolution. This critical strain function ensures that the fracture toughness is independent of the horizon and grid resolution. This works well when modeled materials have pre-existing cracks. However, since the peridynamic fracture toughness manifests as a critical strain it imposes an artificial strength onto the modeled material that may not represent the real strength of the material. When the material strength is less than the model’s critical strain certain flaw insertion methods can be used to capture the strength behavior. However, if the material strength is larger than the critical strain then the flaw size that represents that strength is too small to model. To address this discrepancy, multi-resolution models have been previously used, having differently sized horizons, such that the region that is expected to nucleate failure is represented correctly while the bulk material is modeled with a coarser grid and larger horizon. Such a multiscale approach could be designed from the beginning of the simulation or exhibit adaptive refinement during crack propagation. Such multiscale approaches add significant complexity to the simulation framework and fundamental model descriptions. This paper introduces a method that is able to decouple the model strength from the horizon and grid resolution by using a refinement overlay technique. This overlay is virtual and does not change the explicit material resolution as adaptive techniques do.

A peridynamic plastic model based on von Mises criteria with isotropic, kinematic and mixed hardenings under cyclic loading

In this study, an elastoplastic model based on ordinary state-based peridynamic theory is presented. The von Mises yield criteria is used to describe plastic yielding and the equivalent plastic stretch is utilized as internal variable for the general form of isotropic hardening, the modified form of kinematic hardening and the mixed of isotropic and kinematic hardening. Like classical approach to plasticity, a plastic flow rule is proposed based on the yield function. The proposed plastic model is described in the thermodynamic framework and it is shown that the proposed plastic flow and hardening of the peridynamic model are satisfied the requirements of the second law of thermodynamics. The proposed plastic model is rate-independent and the quasi-static analysis is considered. A numerical approach is proposed for the elastoplastic model and Newton Raphson algorithm is used to solve the nonlinear peridynamic equations. The numerical validation of proposed peridynamic models are done by comparing the results of 2-D peridynamic models under cyclic loading with the results of finite element methods based on the classical local continuum mechanic assumptions. The numerical results show that the proposed PD elastoplastic model can predict plastic yielding and linear/nonlinear hardening of materials beyond initial yield stress accurately. Moreover, it is observed that the presented method it is capable of modeling the kinematic hardening and Bauschinger effect in cyclic loading too.

Nonlocal numerical simulation of low Reynolds number laminar fluid motion by using peridynamic differential operator

A considerable fluid load can cause local damages on the offshore structures, which may be a risk in the field of ocean engineering. Therefore, an accurate fluid motion prediction is a crucial issue in predicting the offshore structure motion. In this study, a non-local Lagrangian model is developed for Newtonian fluid low Reynold's number laminar flow. Based on the peridynamic theory, a peridynamic differential operator is recently proposed for directly converting the partial differential into its integral form. Therefore, the peridynamic differential operator is applied to convert the classical Navier-Stokes equations into their integral forms. The numerical algorithms are developed both in total and updated Lagrangian description. Finally, several benchmark fluid flow problems such as Couette flow, Poiseuille flow, Taylor Green vortex, shear-driven cavity problem and dam collapse problems are numerically solved. The simulation results are compared with the ones available in the published literature. The good agreements validate of the capability of the proposed non-local model for Newtonian fluid low Reynold's number laminar flow simulation.

A Non-Ordinary State-Based Peridynamic Formulation for Failure of Concrete Subjected to Impacting Loads

Strain hardening and strain rate play an important role in dynamic deformation and failure problems such as high-velocity impact cases. In this paper, a non-ordinary state-based peridynamic model for failure and damage of concrete materials subjected to impacting condition is proposed, taking the advantages of both damage model and non-local peridynamic method. The Holmquist-Johnson-Cook (HJC) model describing the mechanical character and damage of concrete materials under large strain, high strain rate and high hydrostatic pressure was reformulated in the framework of non-ordinary state-based peridynamic theory, and the corresponding numerical approach was developed. The proposed model and numerical approach were validated through simulating typical impacting examples and comparing the results with available experimental observations and results in literature.

A nonlocal peridynamics modeling approach for corrosion damage and crack propagation

Corrosion and its synergistic interactions with mechanical loading are a major cause of damage and failure of structural components. In this work, we present a recently developed peridynamics model for modeling corrosion damage and resulting crack propagation phenomena under synergistic effects of corrosion and mechanical loading. The approach is based on nonlocal peridynamics theory that replaces governing equations of classical continuum mechanics with integro-differential equations that are easy to solve across discontinuities like cracks. Here we introduce a micro-chemically sensitive peridynamic modeling approach developed for corrosion damage phenomena. First we introduce the theoretical approach and then simplify it to obtain a peridynamic model for mechanistic corrosion damage and crack propagation. Subsequently, numerical simulations are used demonstrate the capabilities of the peridynamic model for capturing mechanics of corroding solid. The framework is able to capture corrosion pitting, nucleation, crack path propagation under synergistic influence of corrosion and mechanical loading, without the need to re-mesh the domain or special numerical treatments. The developed peridynamic approach provides a physics-based alternative to conventional theories and enables crack propagation studies in corrosive environments.