Peridynamic Differential Operator Research Articles

Bond-associated non-ordinary state-based peridynamics for simulating damage evolution of thermal barrier coatings in aero-engine turbine blades

The failure mode of thermal barrier coatings (TBC) systems in aeroengine turbine blades is very complex because of the harsh service conditions. A peridynamic (PD) model is established to simulate the damage evolution of TBC with uniform thermally grown oxide (TGO) growth under cycle load. The peridynamic differential operator is introduced to solve the zero-energy mode, and thermo-elastic deformation is considered. Moreover, the influence of high-temperature holding time, initial oxide layer thickness, and interface morphology on the evolution of the stress distribution and interface damage is discussed. The newly proposed PD model can effectively capture the interface cracking of TBC systems and it is conducive to the study of the failure of TBC systems.

A peridynamic compensated critical energy density criterion for mixed-mode fracturing in quasi-brittle materials

This study integrates a compensated critical energy density criterion (CCED) into the ordinary state-based peridynamics framework to enable detailed analysis of mode I, mode II, and mixed-mode fracturing. The proposed bond failure criterion builds upon the advantages of the traditional critical energy density (CED) to enable precise calculation of strain energy density. Additionally, the critical bond rotation criterion (CR) is employed to account for shear bonds that the CED might overlook. Using the peridynamic differential operator (PDDO) format in formulating the entire computational framework enhances both accuracy and numerical stability. The model’s validation is conducted through benchmark examples, including the mode I double cantilever beam test, the mode II compact shear test, and a mixed-mode mechanical fracturing test in pre-notched specimens with central borehole loading. Convergence studies, comparative analyses with finite element method simulations and theoretical solutions thoroughly examine the proposed model’s performance and accuracy.

A generalization of the Exner law for sediment nonlocal transport at bedform scale

Recent researches have highlighted the significance of nonlocal processes in understanding the morphodynamics and sediment transport across landscapes. Nonlocal processes in sediment transport refer to the influence of landscape properties beyond the immediate vicinity of a given point on the sediment flux. Existing nonlocal models, employing fractional operators, aim to capture global correlated nonlocality using a global convolution operator. Nevertheless, such models tend to disregard nonlocal phenomena occurring at regional scales, potentially resulting in substantial inaccuracies due to an inadequate representation of sediment transport processes at these scales. This study presents a novel and more comprehensive mathematical formulation of the nonlocal Exner law, leveraging the peridynamic differential operator (PDDO). The proposed regional nonlocal model incorporates nonlocal sediment transport processes by utilizing a pre-defined weight function and interaction domain within the framework of the PDDO. The novel regional nonlocal model effectively bridges the gap between local and global models by integrating both short-range and long-range interactions in sediment transport. Application reveals that that the regional nonlocal model provides a significantly enhanced accuracy in depicting profiles at the bedform scale compared to the local and the global models.

Nonlinear hydroelastic vibration of foamed concrete beams via peridynamic differential operator

Evaluating the hydroelastic responses of underwater cementitious structural elements is critical for ensuring the sustainability and durability of energy-saving marine infrastructures. Existing work on the hydroelastic analysis of porous structures has been mostly developed using the general elastic constitutive relation; however, it fails to capture the influence of saturation. To fill this knowledge gap, we for the first time propose a novel fluid-porous structure interactive model that incorporates the combined effects of hydrodynamic pressure and saturation-induced pore pressure. One more pioneering effort is to solve this nonlinear hydroelastic problem by introducing peridynamic differential operator (PDDO). It is worth noting that the introduction of PDDO removes the inherent drawback employing the local-theory based techniques, namely being prone to singularities arising from the presence of discontinuity. The accuracy and reliability of the proposed numerical framework are validated by comparing the results with the degraded model in the reported literature. Moreover, our results highlight that the angular frequencies are underestimated when ignoring the effect of saturation in foamed concrete beams. The presented method provides a profound understanding of the underwater structural dynamic monitoring that benefits the design of marine infrastructures.

A micropolar damage model for size-dependent concrete fracture problems and crack propagation simulated by PDDO method

Lots of experiments observe the size-dependent phenomena of concrete mechanical behaviors. In this paper, the micropolar theory is adopted to describe the size effects by introducing two size-related parameters into their constitutive equations, named coupling number N and characteristic length l. A micropolar damage model is built for size-dependent concrete fracture problems utilizing the bond-based peridynamic (PD) idea. That is, the material damage and fracture behaviors are determined by the local damage factors of the PD bonds. Then, the tensile strength of a concrete specimen is numerically estimated by the PD differential operator (PDDO) method. The influences of the size parameters N and l on the tensile strength are studied. By comparing with the transformed Bažant size-effect law and the fitting error analysis, the reasonable values of N and l are determined for a certain reinforced concrete composite. Finally, by using the determined micropolar damage model, the crack propagation paths in concrete members are numerically simulated.

Nonlocal numerical simulation of thermoelectric coupling field by using peridynamic differential operator

Nonlocal numerical simulation of thermoelectric coupling field by using peridynamic differential operator

A peridynamic differential operator modeling approach for ceramic matrix composites microstructure with uniform or non-uniform discretization

Based on the theory of peridynamic differential operator (PDDO), a novel three-dimensional (3D) peridynamic (PD) model for isotropic and orthotropic material is proposed to investigate the elastic mechanical behaviors and stress distribution of Ceramic Matrix Composites (CMC) microstructure. The non-local expressions of strain and stress tensors are derived by employing PDDO and the classical constitutive equations. The bond forces in interface region crossing two different constituent materials are obtained by converting the partial differential terms into the non-local forms. The weak form of PD equations of motion is derived by using the principle of virtual work and PDDO. The validity of this approach is verified by predicting the elastic response and stress distributions of isotropic and orthotropic materials under uniaxial tension and pure shear deformation. Finally, simulations are conducted on a CMC microstructure with fiber and matrix under periodic boundary conditions. It is demonstrated that the current approach can effectively build 3D CMC microstructure with uniform or non-uniform discretization, and accurately predict the stress fields and effective elastic properties.

Peridynamics thermomechanical coupling simulation of damage in engineered cementitious composite-concrete bonding specimens under high temperature

The current application of Peridynamics (PD) models in addressing thermally induced failures within Engineered Cementitious Composites (ECC) and concrete bonding specimens is relatively limited. This study introduces a novel semi-discrete PD model for ECC and a random heterogeneous PD model for concrete. These models are grounded in the microscopic constitutive relationships between ECC and concrete materials and incorporate the strain softening behavior of cement-based materials. The effectiveness of the interface model is evaluated through splitting tensile tests on ECC-concrete bonding specimens, employing a series model alongside PD surface effects for material interface correction. Considering the performance degradation of materials at elevated temperatures, a PD thermo-mechanical coupling model, which integrates PD thermal diffusion and motion equations, is proposed to analyze the failure of ECC and concrete under high thermal loads. The validity of both the thermo-mechanical coupling and concrete heterogeneity models is confirmed via thermal cracking tests on heated concrete columns subjected to drilling. Lastly, considering the dynamic response of fire, the Navier-Stokes equation are reformulated into a non-local form utilizing the Peridynamic differential operator (PDDO) to simulate fire environment. A thermo-fluid-mechanical coupling model is developed, and applied to analyze ECC and concrete bonding specimens’ failure behaviors under fire scenarios.

Peridynamic differential operator for stress analysis of imperfect functionally graded porous sandwich beams based on refined zigzag theory

This study focuses on the stress analysis of imperfect functionally graded porous (FGP) sandwich beams using the Peridynamic Differential Operator (PDDO) and Refined Zigzag Theory (RZT). Functionally graded materials (FGMs) can be found in diverse engineering applications since they offer smooth transitions in the mechanical properties of distinct materials, unlike traditional composite materials. Micro-voids and porosities may appear inside the FGMs due to technical challenges during the manufacturing process of such materials. Therefore, understanding the stress variations of the FG sandwich beams with porosities/micro-voids, which are called imperfect FGMs, under loads is of vital importance. In order to achieve the porous configuration in the FG structures, the modified rule of mixtures is applied to calculate the effective material properties of FGMs. The RZT is very suitable for stress analysis of laminated composites, especially for thick and moderately thick beams. It contains only four kinematic variables and eliminates the use of the shear correction factors. In this study, the equilibrium equations of the RZT are solved by means of the PDDO for the stress analysis of imperfect FG sandwich beams having uniform and non-uniform porosity distributions. The PDDO transforms the differential equations into their nonlocal form, namely integral form, providing highly accurate approximations of the derivatives. A comprehensive PDDO analysis is performed for the investigation of the influence of the material variation, and even and uneven porosity distributions on the stress variations of the imperfect FG sandwich beams. It is noted that when the FG core exhibits soft material distributions, the effect of porosity is evident. The even type porosity distribution pattern is more influential on the axial displacement and stress variations in comparison with those of the uneven type porosity distribution pattern.

Nonlocal modelling of piezoelectric solid with cracks based on peridynamic differential operator

The paper establishes a nonlocal model for piezoelectric solids using the Peridynamic differential operator. The method transforms the displacement-electric potential equation, constitutive equation, and boundary conditions of piezoelectric solids from a local differential form to a nonlocal integral form. The solution for the displacement and electric potential distributions in piezoelectric solids is obtained using variational principles and the Lagrange multiplier method. The correctness and effectiveness of the model are validated through a comparison of the deformation analysis of piezoelectric solid square plates with analytical solutions. Analysis of piezoelectric solid square plates with edge cracks proves the applicability of this model for solving discontinuous problems. Furthermore, the paper explores the suitability of the model in analysing problems involving multiple cracks in piezoelectric solid plates, highlighting its capability to address defective piezoelectric solid problems. This model provides a new perspective for solving problems involving defects under electro-mechanical coupling conditions.

Meshless Generalized Finite Difference Method Based on Nonlocal Differential Operators for Numerical Simulation of Elastostatics

This study proposes an innovative meshless approach that merges the peridynamic differential operator (PDDO) with the generalized finite difference method (GFDM). Based on the PDDO theory, this method introduces a new nonlocal differential operator that aims to reduce the pre-assumption required for the PDDO method and simplify the calculation process. By discretizing through the particle approximation method, this technique proficiently preserves the PDDO’s nonlocal features, enhancing the numerical simulation’s flexibility and usability. Through the numerical simulation of classical elastic static problems, this article focuses on the evaluation of the calculation accuracy, calculation efficiency, robustness, and convergence of the method. This method is significantly stronger than the finite element method in many performance indicators. In fact, this study demonstrates the practicability and superiority of the proposed method in the field of elastic statics and provides a new approach to more complex problems.

An Eulerian–Lagrangian particle method for weakly compressible viscous flows using peridynamic differential operator

This study proposes a peridynamic differential operator (PDDO)-based Eulerian–Lagrangian hybrid particle method for weakly compressible viscous flows. The PDDO is utilized to transform the governing partial differential equations into their integral form; hence, the issue of local non-differentiability is eliminated. Both the Eulerian and Lagrangian formulations of particle methods for solving the governing integral equations are derived and then combined to propose the hybrid method. In the proposed method, the Eulerian and Lagrangian formulation is utilized to solve the governing equations in the inner computational domain and in areas with free surfaces, respectively. The results of these two domains are then merged to arrive at the solution. In such a way, the hybrid particle method effectively balances the computational demands and applicability. The numerical stability and interpolation consistency of the proposed method are shown by introducing the laminar viscosity model. A detailed numerical procedure is provided involving the boundary conditions and time-stepping strategy. The proposed method is validated by several benchmark problems. Furthermore, the method was applied to several benchmark problems including the hydrostatic test, Taylor–Green vortex, and numerical wave generation. The results of these numerical examples suggest that the proposed method is computationally less demanding while maintaining accuracy.

Analyzing non-isothermal phase transition problems with natural convection using peridynamic differential operator

In this study, a developed model for non-isothermal phase transition with natural convection is proposed by using peridynamic differential operator (PDDO). The dimensionless governing equations of heat source approach and vorticity-stream function approach are reconstructed into the non-local integral form. The Euler forward difference is used for time integration. The application of the developed PDDO model extends to simulations involving pure phase transition, pure natural convection, and non-isothermal phase transition coupled with natural convection. The validity and accuracy of the developed non-isothermal phase transition model are verified by comparing the PDDO simulation results with results in published literature. This research provides a new alternative approach for simulating phase transition and convection problems.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization

The population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic equations, it becomes imperative to establish a high-resolution scheme to reduce numerical diffusion and numerical dispersion, thereby ensuring accurate crystal size distribution. This paper uses Euler’s first-order explicit (EE) method–Peridynamic Differential Operator (PDDO) to solve 2D PBE, namely, the EE method for discretizing the time derivative and the PDDO for discretizing the internal crystal-size derivative. Five examples, including size-independent growth with smooth and non-smooth distributions, size-dependent growth, nucleation, and size-independent/dependent growth for batch crystallization are considered. The results show that the EE–PDDO method is more accurate than the HR method and that it is as good as the fifth-order Weighted Essential Non-Oscillatory (WENO) method in solving 2D PBE. This study extends the EE–PDDO method to the simulation of 2D PBE, and the advantages of the EE-PDDO method in dealing with discontinuous and sharp front problems are demonstrated.

A Coupled Thermomechanical Crack Propagation Behavior of Brittle Materials by Peridynamic Differential Operator

A Coupled Thermomechanical Crack Propagation Behavior of Brittle Materials by Peridynamic Differential Operator

Fracture mechanics investigation for 2D orthotropic materials by using ordinary state-based peridynamics

Fracture behaviors of orthotropic plates are studied by using ordinary state-based peridynamic (OSPD) theory. Based on OSPD, a novel nonlocal formulation of interaction integral is proposed by considering the material orthogonality for the fracture parameter evaluation. By employing peridynamic differential operator, the partial differential terms in the formulation can be transformed into corresponding spatial integral form, which contributes to the calculations of stress intensity factors within the framework of OSPD. It has built up a relationship between classical theory and peridynamic theory. Static and dynamic fracture parameters are carefully evaluated. The crack propagation directions are predicted by prototype microelastic brittle (PMB) criterion in OSPD and maximum circumferential stress (MCS) criterion in classical theory. Several pre-cracked plates with orthotropic material are examined, and results are validated by comparing against the reference solutions. The relationship between fiber orientation and crack inclined angle are also examined. Meanwhile, the crack inclination determined by PMB criterion and MCS criterion are compared and discussed. Accuracy of the peridynamic orthotropic model and proposed nonlocal interaction integral are discussed in detail.

Experimental and PDDO-based meshfree numerical study on ductile fracture behavior of steel plate

This paper presents a meshfree approach based on the coupling of peridynamic differential operator (PDDO) and continuum mechanics to predict the ductile fracture of the steel plates. The interpolation function of particles for a shell model is established by applying the PDDO in the 2D case. The weak form of the kinetic equations is derived by discretizing the motion variables. The Gurson-Tvergaard-Needleman (GTN) constitutive model is adopted to characterize the plasticity and ductile fracture behavior of material, in which the material parameters are determined by conducting uniaxial tensile tests and scanning electron microscope analysis. The ductile fracture test of the EH36 steel plate with a prefabricated crack is conducted to validate the proposed numerical approach. The results show that the established meshfree shell model is able to accurately simulate the bearing capacity, crack propagation rate, and ductile fracture characteristics of the structure.

Non-local modelling of multiphase flow wetting and thermo-capillary flow using peridynamic differential operator

AbstractInterfaces in multiphase flows are affected by surface tension, and when temperature gradients occur in the flow domain, tangential surface tensions along the interface also arise. As the behaviour of fluids contacting on a solid surface is also governed by surface tension, the description of the wetting phenomenon is challenging. Peridynamic differential operator (PDDO) can express partial differentials of any order by integral equations. Therefore, the governing equations for multiphase fluid motion, such as the Navier–Stokes equations and energy equations, can be reformulated in terms of integral equations. In this study, a novel non-local method is developed for modelling the multiphase fluid flow motion using the PDDO, and the thermal effect on surface tension force is considered. To describe the surface tension forces in the normal and tangential directions, the non-local form of the continuum surface force (CFS) model is presented. Besides, to overcome the inaccuracy of the unit normal vectors at the three-phase flow intersection region, an additional treatment for this region is presented. Finally, several benchmark multiphase fluid flow cases, such as square droplet deformation, surface wetting, and droplet migration in thermo-capillary flow are presented and validated. The results demonstrate that the developed non-local model can accurately capture the surface tension effect in multiphase fluid flow motion.

Static and dynamic analysis of couple-stress elastic structures by using peridynamic differential operator

In this paper, a peridynamic differential operator (PDDO) method is developed to analyze the static and dynamic responses of size-dependent elastic structures in the frame of a consistent couple stress theory (CCST). The displacement governing equations together with the boundary conditions of the couple-stress static problems are transformed from the local differential form to the nonlocal integral form through a fourth-order PDDO and then solved by a meshfree scheme. For dynamic cases, the time-domain problems are first transformed into frequency domain by Fourier transform, then solved by the PDDO meshless method in the frequency domain. The time-domain results are finally obtained by using an exponential window method (EWM). The size effects on the static and dynamic responses of some typical elastic problems are investigated by the developed PDDO method. A very high level of computational accuracy is achieved by comparing with the analytical solutions and the existing numerical results.

Numerical simulation of 3D fracture propagation problem with reproducing kernel peridynamic method

The non-ordinary state-based peridynamic (NOSB-PD) method has significant advantages in solving crack propagation problems due to its non-local characteristics, but it also faces problems such as numerical oscillations and boundary effects. In order to solve the above defects, the paper first discusses the NOSB-PD method within the Galerkin framework. After that, the peridynamic differential operator (PDDO) approximation was introduced by extending the calculation of the deformation gradient tensor F in NOSB-PD. Since the zero-order approximations in the PDDO approximation and the reproducing kernel (RK) approximation are identical, the distinctions between the two approximations have been thoroughly analysed. Although the two approximations show the same consistency, only the RK approximation satisfies the integrable condition. Therefore, a RK-PD coupling method is proposed in the paper, and an implicit iterative algorithm suited to the RK-PD method is presented as well. At last, several numerical examples demonstrate that the proposed method does not suffer from numerical oscillations and can be used to solve the three-dimensional crack propagation problems.