Non-ordinary State-based Peridynamics Research Articles

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.

Stabilized non-ordinary state-based peridynamics with irregular nodal distribution

Peridynamics is a nonlocal continuum mechanics model that is employed in fracture simulation, and its numerical applicability can be extended using irregular arrangements in the nodal distribution. This study investigates the effect of irregular grids on the numerical accuracy of stabilized non-ordinary state-based peridynamics to establish an appropriate range of peridynamic geometric parameters for irregular grids with geometrically nonlinear formulations. To this end, the peridynamic geometric parameters of irregular grids are estimated by interpolating a piecewise graph of the parameter versus the number of neighbor nodes in the regular grid. In addition, a direct tensile test is performed on a two-dimensional square plate, and the displacement results are compared with the analytical solution to determine the appropriate range of the geometric parameters. Using the parameter range, a large deflection of a beam is simulated in the explicit dynamic analysis. Overall, the results of the horizontal and vertical displacements are consistent with those obtained using the finite element method. The numerical models are verified to be generalizable for large displacements and rotations.

Numerical investigation on fracture characteristic and failure mechanism of rock-like materials with intermittent flaws under compressive-shear loading

An extended non-ordinary state-based peridynamic (NOSB-PD) theory is utilized to investigate the fracture characteristics of rock-like materials containing intermittent flaws under compressive-shear loading. The stress-based failure criteria, i.e., the maximum tensile stress criterion and the Mohr-Coulomb criterion, are integrated into the NOSB-PD theory to respectively determine the tensile failure and shear failure of flawed rock-like materials. The distribution characteristics of maximum principal stress and shear stress acquired by the NOSB-PD theory are employed to analyze the fracture failure mechanism of flawed rock-like materials under compressive-shear loading. The effects of inclination angle of the central flaw on the initiation and coalescence modes of flawed rock-like specimens are investigated. The numerical results of flawed rock-like materials are in good agreement with the experimental results. The extended NOSB-PD has the ability to simulate the fracture processes under compressive-shear loading.

PD-FEM chemo-thermo-mechanical coupled model for simulation of early-age cracks in cement-based materials

The hydration process during the hardening of early-age cement-based materials can lead to a considerable volume change, consequently causing considerable deformation, and even fracturing. In the present study, an efficient PD-FEM chemo-thermo-mechanical coupling approach is proposed with the aim of modeling fracturing in early-age concrete structures, taking into account coupled processes, such as thermal transfer, hydration heat, creep, evolution of strength, fracturing as well as hindering effect of cracks on the temperature and hydration fields. The coupled chemo-thermo-mechanical problem is solved in a staggered manner within the classical finite element (FE) and the non-ordinary state-based peridynamics (NOSBPD) framework. By doing so, fracturing in early-age concrete structures can be easily accommodated by NOSBPD featuring an elegant treatment of discontinuities. The accuracy of the proposed approach is evaluated by a benchmark problem with experimental and existing numerical solutions. The capability of the proposed approach for simulating the complex crack initiation and propagation in early-age ring and T-shaped concrete structures is demonstrated. The interaction and competition between different stress inducing mechanisms of hydration heat, autogenous shrinkage, and creep are analyzed.

Large deformation and crack propagation analyses of hydrogel by peridynamics

In this paper, the large tensile deformation and Mode-I crack propagation of hydrogel are effectively simulated by the bond-associated (BA) non-ordinary state-based (NOSB) peridynamics (PD). Based on the nonlocal theory and meshless characteristic, the fracture involving large deformation of hydrogel is solved with simple implementations. In the analysis, the Gent model is first introduced into NOSB PD as the constitutive model to describe the stress-stretch response of hydrogel under a large tensile stretch. The bond-associated scheme is applied to overcome zero-energy modes and numerical oscillation in conventional NOSB PD. Effective stretch criterion is applied to capture the crack propagation. With the explicit dynamic solver, the tensile deformation of a hydrogel sheet with a hole and the crack propagation process of a hydrogel sheet with a pre-notch under pure shear are analyzed. The predicted stress-stretch responses are in good agreement with the experimental observations, which demonstrates the effectiveness and efficiency of the developed PD approach for predicting large deformation and crack propagation of hydrogel.

A reformulated non-ordinary state-based peridynamic method for dynamic failure of ductile materials

A non-local correspondence peridynamic model is reformulated to describe the dynamic failure of ductile materials, under the framework of non-ordinary state-based peridynamics. To eliminate the zero-energy mode as well as to improve the numerical stability, the deformation gradient is updated on the bond level, which owns information including both the non-local deformation and the bond deviation. Peridynamic thermo-viscoplastic constitutive formulations are presented for the failure analyses of ductile materials. Moreover, a mixed strain-stress criterion describing the bond breaking and considering the temperature effect is presented on the bond level, in which the bonds can keep intact in calculations. Several comprehensive cases, including a notched tension test and Kalthoff-Winkler impact tests with different velocities, are investigated to verify the applicability of the proposed model for ductile fracture and impact failure. Numerical results, in terms of failure patterns, cracking switching and propagation, as well as adiabatic shear band propagation, show a reasonable agreement with corresponding experimental observations.

Modelling of viscoelastic materials using non-ordinary state-based peridynamics

This paper proposes a framework for implementing viscoelastic constitutive model from the classical continuum mechanics (CCM) theory within non-ordinary state-based peridynamics (NOSBPD). The motivation stems from the inadequacy of CCM to model very complex material behaviours such as initiation and propagation of cracks and nonlocal behaviour due to size effects. The proposed formulation leverages on the constitutive correspondence between NOSBPD and CCM to incorporate a CCM viscoelastic constitutive model based on hereditary integral into NOSBPD. The combination of hereditary constitutive model and NOSBPD effectively makes this formulation a nonlocal time–space viscoelastic framework where temporal nonlocality is incorporated by a hereditary viscoelastic model which stipulates that the behaviour of a material at any point in time depends on both the present action and the complete history of previous actions on the material, and spatial nonlocality on the other hand is incorporated via the nonlocal mechanism provided by the NOSBPD. For model validation, three benchmark problems were solved using the proposed framework. Results obtained were compared to results from analytical solution and solutions from referenced literature. In addition, parametric study was conducted to determine the influence of nonlocality on numerical prediction. Conclusions drawn from the validation studies presented are that the proposed framework is able to predict viscoelastic responses that agree well with local macro models as well as nonlocal micromodels/nanomodels as reported in the literature.

Investigating time-dependent behavior of rocks using kinematic-constraint-inspired non-ordinary state-based peridynamics

In this paper, the kinematic-constraint-inspired non-ordinary state-based peridynamic formulation (KC-NOSBPD) is proposed to investigate the time-dependent behaviors of rocks, which play a critical role in evaluating the long-term stability of the surrounding rocks around underground opening quantitatively. The peridynamic differential operator (PDDO) is introduced to KC-NOSBPD to improve the precision of bond-associated deformation gradient and to suppress the skin effect. On this basis, a unified creep-damage-rupture KC-NOSBPD model is proposed, in which the Burgers-creep viscoplastic constitutive law, a visco-elastoplastic constitutive model, is used to describe the rheological deformation of rocks. A unified damage-rupture framework based on a strain-based isotropic scalar damage variable is established to simulate the creep damage and subsequent creep rupture. The fidelity of this proposed model without damage is verified by comparing it with the benchmark solution for the creep response of a 2D rock specimen subjected to various load histories. Further, the proposed model is employed to simulate uniaxial and triaxial compression creep tests. These present numerical results agree well with the experimental observation, implying that the proposed model successfully captures the accelerating creep behavior of rocks. Moreover, the proposed model performs the time-dependent deformation and failure analysis for the surrounding rocks around the tunnel.