Non-ordinary State-based Peridynamics 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.

Coupled field modelling of thermomechanical response in materials using peridynamic heat transport model and non-ordinary state-based peridynamics

Predicting the mechanical response of materials subjected to transient heat processes is crucial in various engineering applications. This paper introduces a novel peridynamic (PD) coupled field model formulated within the framework of Non-Ordinary State-Based Peridynamics (NOSBPD). This model offers the capability to simulate the coupled thermo-mechanical behavior of materials by incorporating both thermal transport and mechanical deformation analyses. To verify the model’s accuracy, a benchmark problem involving a steel plate undergoing transient thermal expansion is investigated. This model presents a valuable tool for various engineering applications involving coupled thermo-mechanical processes.

Peridynamic modelling of elastic and viscoelastic behaviour in polycrystalline ice: A study using NOSBPD and PDCHT

This study explores the elastic and viscoelastic behaviour of ice using the Non-Ordinary State-Based Peridynamic (NOSBPD) framework. A primary objective is to extend and validate the applicability of NOSBPD in modelling the complex responses of ice under various conditions. The study employs the Peridynamic Computational Homogenization Theory (PDCHT) to determine the critical threshold of grain count necessary to induce an effectively isotropic response in polycrystalline S2 ice. Results are consistent with previous findings from Finite Element Method (FEM) and Bond-Based Peridynamics (BBPD) studies.Furthermore, the viscoelastic response of ice is investigated by integrating a viscoelastic constitutive model into the NOSBPD framework. A benchmark problem of a viscoelastic ice sample subjected to tensile stress is simulated, with results compared against FEM simulations conducted in ANSYS Mechanical. The findings show good agreement, validating the NOSBPD framework's capability to capture time-dependent viscoelastic behaviour of ice accurately.The study contributes to the field of ice mechanics by demonstrating the robustness and versatility of NOSBPD in modelling both elastic and viscoelastic responses of ice. These advancements enhance the credibility and applicability of peridynamics (PD) as a powerful tool for simulating complex material behaviours, paving the way for further research and practical applications in ice engineering.

Peridynamics modelling of projectile penetration into concrete targets

A non-ordinary state-based peridynamics (NOSB-PD) model is proposed to simulate the projectile penetration into concrete targets. In this model, the Kong-Fang concrete material model recently proposed is firstly implemented into the NOSB-PD framework to describe the complex dynamic behavior and failures in concrete material subjected to penetration loading, and then an improved point-to-volume discrete frictional contact model is proposed to simulate the physical interaction between projectile and target. After the mesh-free discretization and explicit time integration, the proposed NOSB-PD model is used to numerically predict two sets of projectile penetration experiments into low-strength and high-strength concrete targets. And numerical predictions are found to be in good agreements with corresponding test data including penetration depth, projectile deceleration, deformation of projectile and failures in concrete targets.

Computational modeling of near-fault earthquake-induced landslides considering stochastic ground motions and spatially varying soil

Landslides represent a large-deformation process influenced by various factors of uncertainty, such as types of ground motion (GM), randomness of GMs, and spatial variability of soils. The behavior of landslides is profoundly affected, making their assessment and measurement challenging. This paper proposes a computational approach for simulating the large-deformation process of landslides based on three-dimensional (3D) non-ordinary state-based peridynamics (NOSBPD). The analysis results of NOSBPD indicate that the mean runout distance triggered by pulse-like ground motions (PLGMs) is 16% greater than that induced by non-pulse ground motions (NPGMs), suggesting that PLGMs exhibit higher destructiveness. Besides, this paper establishes a runout distance assessment framework by considering the stochastic nature of PLGMs. This framework allows for the evaluation of the specific risk probability of landslides caused by stochastic PLGMs that match the target spectrum specified by codes. By introducing the theory of random fields and implementing a coupled procedure in peridynamics, we conducted a detailed analysis of the impact of spatial heterogeneity on the evolution process and consequences of landslides. Additionally, compared to two-dimensional (2D) analysis, the mean runout distance obtained from 3D analysis increased by 27.5%. This suggests that 2D analysis may underestimate the consequences of landslides. The findings of this study can serve as a scientific foundation for predicting the extent and scope of landslides triggered by near-fault earthquakes.

A bond-augmented stabilized method for numerical oscillations in non-ordinary state-based peridynamics

Non-ordinary state-based peridynamics has been widely used due to its capability to combine non-local theory with conventional material models. However, it suffers from numerical oscillations in calculations. The fundamental reason is that the approximate deformation gradient causes the non-unique mapping between deformation states and force states. To address this issue, the present work utilizes the penalty function method to reformulate the deformation gradient for each individual bond. The force state corresponding to a single bond is then constructed based on this newly developed bond-augmented deformation gradient in the framework of least action principle. This bond-augmented force state has a strong dependence on the deformation of that particular bond, thus well resolving the non-unique mapping issue. The resulting bond-augmented stabilized method does not introduce an additional local bond-associated horizon or extra force state termed as that in the previous studies and does not require tedious parameter adjustments. Numerical results show that the proposed bond-augmented stabilized method can significantly remove numerical oscillations and have good accuracy and convergence. Moreover, the proposed bond-augmented stabilized method is also proved to readily capture crack paths.

Peridynamics contact model: Application to healing using phase field theory

This paper presents a contact algorithm in non-ordinary state-based peridynamics (NOSB-PD) framework and its application in healing phase field (HPF) theory. The idea here is to exploit the nonlocality in PD to model the contact between two deformable bodies. The proposed method does not require artificial short-range force or additional degrees of freedom, e.g., Lagrange multiplier, which are often used in the conventional classical and PD contact models. In the regions where contact is likely to take place, neighbor search is performed at every time step and extra particles entering the horizon are identified. These extra particles form artificial bonds which transfer forces only in compression. This feature is essential for rebound simulation. Envisioning surface-to-surface bonding as the opposite process of cracking, we propose an HPF theory to model high velocity impact induced bonding between two material bodies. Here an HPF based bonding criterion is postulated and integrated with the proposed contact algorithm. The solution of the governing equations is obtained using the fourth order Runge-Kutta method. The efficacy of the proposed contact algorithm is demonstrated by simulations of impact and rebound of two plates, two cylinders, and cylinder with rectangular prism and comparing the energy time histories with finite element solutions obtained using ANSYS®. The potential of the proposed HPF theory is demonstrated through simulation of impact of two plates where the plates rebound when the impact velocity is below a critical value, and above that velocity, the plates join with each other.

Fracture analysis of hyperelastic membrane using bond-associated non-ordinary state-based peridynamics

In this paper, a non-ordinary state-based peridynamics model (NOSB PD) has been proposed to simulate the large deformation and failure analysis of hyperelastic membranes. To account for the curving horizon of the membrane, a non-local membrane theory has been developed by approximating it as a flat surface and applying the plane stress assumption locally. Thus, the membrane structure is simulated using a single layer of material points, which simplifies implementation, improves efficiency, and avoids volume locking. The Gent model has been employed to simulate the hyperelastic material constitutively, as it has an elegant mathematical framework and can achieve the entire range of stretches. Furthermore, Bond-associate NOSB PD is utilized to overcome zero-energy modes without affecting the material properties. A modification has been addressed to overcome the boundary effect by adding a complementary force density to the boundary nodes. The accuracy of large deformation and incompressibility of the developed models are verified by the corresponding finite element analyses. Finally, the proposed method is demonstrated by presenting some benchmark examples, and the results illustrate the accuracy and efficiency of the proposed method for the large deformation, fracture, and off-plane tearing analyses of hyperelastic membranes.

Modelling ductile damage in metals and alloys through Weyl condition exploiting local gauge symmetries

Local translational and scaling symmetries in space–time is exploited for modelling ductile damage in metals and alloys over wide ranges of strain rate and temperature. The invariant energy density corresponding to the ductile deformation is constructed through the gauge invariant curvature tensor by imposing the Weyl like condition. In contrast, the energetics of the plastic deformation is brought in through the gauge compensating field emerged due to local translation and attempted to explore the geometric interpretation of certain internal variables often used in classical viscoplasticity models. Invariance of the energy density under the local action of translation and scaling is preserved through minimally replaced space–time gauge covariant operators. Minimal replacement introduces two non-trivial gauge compensating fields pertaining to local translation and scaling. These are used to describe ductile damage, including plastic flow and micro-crack evolution in the material. A space–time pseudo-Riemannian metric is used to lay out the kinematics in a finite-deformation setting. Recognizing the available insights in classical theories of viscoplasticity, we also establish a correspondence of the gauge compensating field due to spatial translation with Kröner’s multiplicative decomposition of the deformation gradient. Thermodynamically consistent coupling between viscoplasticity and ductile damage is ensured through an appropriate degradation function. Non-ordinary state-based (NOSB) peridynamics (PD) discretization of the model is used for numerical implementation. We conduct simulations of uniaxial deformation to validate the model against available experimental evidence and to assess its predictive features. The model’s viability is tested in reproducing a few experimentally known facts, viz., strain rate locking in the stress–strain response, whose origin is traced to a nonlinear microscopic inertia term arising out of the space–time translation symmetry. Finally, we solved 2D and axisymmetric deformation problems for qualitatively validating the model’s viability. NOSB peridynamics axisymmetric formulation in finite deformation setup is also presented.

A neural network peridynamic method for modeling rubber-like materials

Peridynamic (PD) is a powerful tool for simulating the large deformation and failure process of many types of materials. However, its use in modeling rubber-like materials is limited due to the complex constitutive nature of the material, and low efficiency and numerical oscillation caused by PD. To address this issue, a neural network (NN) non-ordinary state-based peridynamics (NOSB PD) method is developed to model the large deformation and failure behavior of rubber-like materials. This method is free of the zero-energy modes, and can significantly improve the computational efficiency. Unlike the traditional NOSB PD method that formulates the force density vector based on the deformation gradient, this method uses a deep NN to map the bond related quantities to the force density vector. The accuracy and efficiency of the proposed method are demonstrated through a series of numerical examples. Additionally, this method can be applied to various hyperelastic materials for which analytical constitutive models exist.

Numerical study on dynamic cracking characteristic and mechanism of quasi-brittle materials under impulsive loading

This study focuses on investigating the dynamic cracking behaviors of quasi-brittle materials under impulsive loading using the extended non-ordinary state-based peridynamic (NOSB-PD) model. This extended model incorporates two stress-based failure criteria: Mohr-Coulomb and maximum tensile stress criteria, to represent shear and tensile fractures of quasi-brittle materials, respectively. By analyzing the evolution characteristics of the maximum principal stress obtained from the NOSB-PD model, the fracture mechanisms of quasi-brittle materials subjected to impulsive loading are explored. The research reveals that tensile failure predominantly governs the initiation and propagation processes of quasi-brittle materials under impulsive loading. The stress concentration effect, transfer effect, and dissipation effect observed in the maximum principal stress fields provide insights into the cracking mechanism during the failure processes of such materials. To validate the findings, the crack initiation modes and crack propagation paths obtained from the NOSB-PD model are compared with those obtained from the existing experimental and numerical approaches. The results demonstrate that the extended NOSB-PD model is effective in addressing fracture problems associated with quasi-brittle materials under impulsive loading conditions.

Adaptive coupling of non-ordinary state-based peridynamics and classical continuum mechanics for fracture analysis

In this study, we proposed a coupled model between non-ordinary state-based peridynamics (NOSPDs) and classical continuum mechanics (CCMs) using the Morphing strategy for both quasi-static and dynamic fractures. We leveraged a novel strategy to approximate the deformation energy of the NOSPD model, which can unify the existing BPD and OSPD cases. We established the unified governing equations for NOSPDs and CCMs and applied the generalized strength criteria in the coupling model to activate the NOSPDs only when necessary. The critical stretch is recalibrated for the NOSPD model, and generalized strength-based bond failure criteria are proposed to determine bond failure. The model was numerically implemented using the finite element method (FEM) supplemented with discrete elements in the NOSPD subdomain. Parallel in-house code is written on graphics processing units (GPU) using the Compute Unified Device Architecture (CUDA). The 1-, 2- and 3-dimensional (D) numerical results demonstrated the accuracy and efficiency of the proposed method. Notably, the relative errors due to coupling during wave propagation in a 1D bar and uniform deformation in a 2D plate were less than 0.0375% and 0.14%, respectively. Further, the observed crack propagation speed aligned with the reference value.

A coupling algorithm of ordinary and non-ordinary state-based peridynamic models for fracture analysis in brittle and ductile materials

In this work, a novel coupling algorithm of the ordinary state-based peridynamic (OSB-PD) model and non-ordinary state-based peridynamic (NOSB-PD) model is provided under the peridynamic framework. The salient advantages of both OSB-PD and NOSB-PD models can be simultaneously inherited in this coupling model and approach. The feasibility of coupling these two models is first analyzed by demonstrating the numerical equivalence of their constitutive formulations in isotropic elastic and elastoplastic problems, and the detailed numerical implementation strategy of coupling formulation is given as well. With the coupling model in this study, the main calculation procedure of numerical simulations is accomplished in OSB-PD, while the damage and failure analysis is conducted in NOSB-PD. Moreover, different failure criteria are introduced in this coupling model to describe the damage and failure of brittle and metallic materials. Several typical examples, including the mode I and mixed mode I/II fracture of brittle materials, and the ductile fracture and impact failure of metallic materials, are conducted to demonstrate the robustness and accuracy of the presented coupling model and approach.

Numerical analysis on failure of sheet metals with non-ordinary state-based peridynamics

In this paper, a general anisotropic plastic model for sheet metals is proposed in the non-ordinary state-based peridynamics. The anisotropy plasticity of metallic material is described by the Hill48 yield criterion, and the associated flow rule and non-linear mixed-isotropic-kinematic hardening are included as well. Accordingly, the Newton-Raphson method is used here, which is more efficient than the dichotomy method. Meanwhile, a widely used continuum damage model is coupled with a new bond-breaking method based on the classic Hash principles. A detailed algorithm for numerical implementation is provided for a better understanding. The proposed model is verified by several typical examples and shows its great potential in describing anisotropic plasticity-ductile deformation and failure of sheet metal. In the last, the efficiency of the proposed failure criterion is analyzed and discussed through the numerical analysis of a uniaxial tensile failure test.

Three-dimensional modeling and analysis of anisotropic materials with quasi-static deformation and dynamic fracture in non-ordinary state-based peridynamics

A general three-dimensional anisotropic model is proposed within the non-ordinary state-based peridynamic framework in this work. Three-dimensional peridynamic formulations for anisotropic solids are formulated, and two types of stress rate solutions, Green-Naghdi and Jaumann stress rate are implemented in this model, respectively. Moreover, the equivalent bond stress and corresponding equivalent bond strain are presented and calculated. The 3D-Hashin failure criterion is used among the numerical analysis of anisotropic materials through the equivalent bond strain. Various typical cases are performed by the proposed anisotropic peridynamic model and approach, including the numerical analyses of quasi-static deformation and dynamic fracture for anisotropic materials. The effectiveness and applicability of the proposed model are verified by precisely characterizing the quasi-static deformation and dynamic failure behavior.

An improved implicit non-ordinary state-based peridynamics model for modeling crack propagation and coalescence problems

A quasi-static solution for crack propagation and coalescence can provide an efficient way to predict both structure failure load and fracture pattern morphology. This paper proposes an improved implicit static non-ordinary state-based peridynamics (IINOSB-PD) solution procedure to study crack propagation problems through incorporating a stress-based criterion into the model. The comprehensive discretized processes and the approach to mitigate numerical oscillation are presented in this manuscript. A parameter sensitivity analysis is first performed to evaluate the effects of uncertain parameters in this procedure on numerical output. Subsequently, four numerical examples are investigated using the proposed IINOSB-PD to validate this procedure. The research findings indicate that the proposed IINOSB-PD can not only accurately predict material crack propagation and coalescence processes but also reduce the programming difficulties and avoid the traditional singular problem. These results provide valuable insights for the application of the IINOSB-PD method in related fields.

Stabilized state-based peridynamics for elasticity emanating from constrained Lagrangian

State-based peridynamics for elasticity paves the way to bridge the local theory with the inherent capabilities of general elasticity with non-local effects. However, the original non-ordinary state-based peridynamics with correspondence materiel models suffers from high-frequency oscillations due to zero-energy mode instability. To address this issue, we start from the fundamental Lagrangian with the introduction of a constraint potential energy that minimizes the numerical error and suppresses the zero-energy mode oscillations. A stabilized non-ordinary state-based peridynamics is then derived from this constrained Lagrangian that forms an integral–differential–algebraic system. In specifics, the equation of motion for particles is described by integral-differential equations while the constraint suppressing the zero-energy mode instability is expressed as algebraic equations (zero-error constraint). An energy–momentum conserving implicit time-stepping scheme for this integral–differential–algebraic system is then derived. End-to-end 2D and 3D simulations, including scalar wave propagation, crack path, and large deformation problems, are presented to show that the proposed method is promising for these applications in terms of accurately capturing the crack path and large deformation, efficiently eliminating the numeral oscillations, conserving the total energy for long duration simulations, and preserving the zero-error constraint.

Propeller-Ice Contact modelling with Peridynamics

Ice loads pose a significant threat to ice-class propellers in frozen waters, making it crucial to analyse the magnitude of ice loads. However, the complex mechanical properties of ice and its failure process make it challenging to investigate the correlation between ice-propeller contact phenomena and ice loads. This paper uses a PC3 class propeller blade to study the propeller-ice contact process. This study involves developing ice failure and propeller blade models based on ordinary and non-ordinary state-based peridynamics theory and validating them using experimental. Convergence analysis was performed before blade ice loads evaluation. Additionally, we investigate the effect of the pitch angle on the ice-milling process.

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.

Convergence study of stabilized non-ordinary state-based peridynamics for elastic and fracture problems

Non-ordinary state-based peridynamics simulates crack propagation while avoiding the imposition of additional criteria to describe the responses at discontinuities. However, an incomplete coverage by the sphere of a horizon near the boundaries and crack surfaces could cause the boundary effect. In this study, the boundary effects occurring in the stress concentration region are investigated in δ- and m-convergence tests for elastic deformation and crack propagation. Three types of boundary imposition methods are investigated in m-convergence and the method showing the fastest convergence is applied to fracture problems. The weighted average stress estimation is proposed for the damage criterion to reduce the variation in the crack paths in the convergence test, and the weight function is defined as a parameter that can change the shape of the function into three different forms: linear, bilinear, and constant. The fracture simulation is performed in an L-shaped panel with varying geometrical parameters. As a result, the bilinear weighted average stress estimation reduces the variation in the crack path compared to the conventional arithmetic average estimation method. Therefore, the proposed weighted average stress estimation method increases the accuracy of the crack path prediction and will enlarge the application area for fracture simulation using peridynamics.