State-based Peridynamic Model Research Articles

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.

An Improved Ordinary State-Based Peridynamic Model for Granular Fractures in Cubic Crystals and the Effects of Crystal Orientations on Crack Propagation.

Material anisotropy caused by crystal orientation is an essential factor affecting the mechanical and fracture properties of crystal materials. This paper proposes an improved ordinary state-based peridynamic (OSB-PD) model to study the effect of arbitrary crystal orientation on the granular fracture in cubic crystals. Based on the periodicity of the equivalent elastic modulus of a cubic crystal, a periodic function regarding the crystal orientation is introduced into peridynamic material parameters, and a complete derivation process and expressions of correction factors are given. In addition, the derived parameters do not require additional coordinate transformation, simplifying the simulation process. Through convergence analysis, a regulating strategy to obtain the converged and accurate results of crack propagation paths is proposed. The effects of crystal orientations on crack initiation and propagation paths of single-crystal materials with different notch shapes (square, equilateral triangle, semi-circle) and sizes were studied. The results show that variations in crystal orientation can change the bifurcation, the number, and the propagation path direction of cracks. Under biaxial tensile loading, single crystals with semi-circular notches have the slowest crack initiation time and average propagation speed in most cases and are more resistant to fracture. Finally, the effects of grain anisotropy on dynamic fractures in polycrystalline materials under different grain boundary coefficients were studied. The decrease in grain anisotropy degree can reduce the microcracks in intergranular fracture and the crack propagation speed in transgranular fracture, respectively.

A stabilized state-based peridynamic heat conduction model for interface thermal resistance problems

As the demand for micro/nano-scale electronic devices and composite materials grows, numerical modeling of interfacial thermal resistance problems has become an integral part of the design for these devices and materials. In this work, a stabilized state-based peridynamic heat conduction model considering the Kapitza thermal resistance is proposed. A stabilized scheme is first developed to eliminate the numerical oscillations in the state-based peridynamics. The main idea is to formulate the temperature gradient for each individual bond by penalty function. The corresponding heat flow state is then constructed based on this newly developed bond-augmented temperature gradient in the framework of the Lagrangian formalism. The stabilization scheme is combined with the embedded discontinuous peridynamic concept to capture the temperature jump conditions of interface thermal resistance. The effectiveness of the proposed model is demonstrated by numerical examples. It paves the way for investigating the interfacial thermal resistance problems of materials with complex heat transfer constitutive relations.

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.

A non-ordinary state-based peridynamic model for creep–fatigue behavior and damage evolution

A peridynamic creep–fatigue model is proposed to study creep–fatigue mechanical responses and damage behaviors. A non-unified constitutive model is reformulated in a nonlocal form in the peridynamic-based framework. A probabilistic peridynamic damage criterion is proposed for characterizing creep–fatigue damage. The simulated mechanical responses are in good agreement with the results of creep–fatigue experiments conducted at high temperatures, which confirms the accuracy of the model. In addition, the damage evolution and morphology are well characterized by the model, and the damage mechanism is revealed. The model exhibits excellent potential for simulating nonlinear damage behaviors.

An ordinary state-based peridynamic model for granular fracture in polycrystalline materials with arbitrary orientations in cubic crystals

Granular fracture holds significant implications in material mechanics. However, the previous studies in ordinary state-based peridynamic (OSB-PD) framework often neglect the internal crystalline structure of particles or only consider limited crystal orientation. To address this gap, a novel OSB-PD model for granular fracture within polycrystalline materials is proposed, in which the periodic functions are incorporated in the PD strain energy density, taking into account the inherent random orientation in cubic crystals. By comparing energy density from PD and the classical continuum mechanics, four PD material parameters are defined. Moreover, the corresponding surface correction method in the global coordinate system is also proposed. Several numerical examples including fracture analysis of polycrystalline materials are conducted to validate the effectiveness of the proposed method. The proposed ordinary state-based peridynamic model offers a fresh perspective for investigating granular fracture behaviors within polycrystalline materials.

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.

Elastoplastic peridynamic formulation for materials with isotropic and kinematic hardening

We present an ordinary state-based peridynamic model in 2D and 3D consistent with rate-independent J2 plasticity with associated flow rule. The new contribution is the capability of the elastoplastic law to describe isotropic, kinematic and mixed hardening. The hardening formulations follow those available in the literature for classical elastoplasticity. The comparison between the results obtained with the peridynamic model and those obtained with a commercial FEM software shows that the two approaches are in good agreement. The extent of the plastic regions and von Mises stress computed with the new model for 2D and 3D examples match well those obtained with FEM-based solutions using ANSYS.

Nonlocal anisotropic model for deformation and fracture using peridynamic operator method

This paper proposes a nonlocal anisotropic model for deformation and fracture based on peridynamic operator method. The classic anisotropic theory is reformulated from its local form into nonlocal form by using peridynamic operator method. The final solution formulation is developed by employing variational principles and Lagrange's equations. In the case of the circular interaction domain, this model can be regarded as a novel anisotropic ordinary state-based peridynamic model. Moreover, we present for the first time the energetic failure criterion for the weak anisotropic fracture. Unlike the existing anisotropic peridynamic models that consider only two fracture toughness constants for orthogonal orientation, the fracture toughness in this criterion is an orientation-dependent function. Additionally, we develop an implicit static solution procedure to investigate corresponding crack propagation problems. Several numerical examples demonstrate the effectiveness and ability of the proposed model in handling the deformation and fracture behaviours of anisotropic materials.

Ordinary state-based peridynamic model for out-of-plane deformation and damage analysis of composite laminates

An ordinary state-based peridynamic (PD) model for predicting out-of-plane mechanical behaviour of composite laminates has been proposed, in which three types of PD bonds are used to describe the reinforcement properties. The non-local strain energy density and peridynamic formulations are obtained based on the principle of virtual work by using Total Lagrange formulation, and the force density is reformulated by interpolation technique. Since the Mindlin plate is regarded as the research object of this PD model, the transverse shear deformation of laminates is considered. Also, the dilatation term is included in the force density vector, and flaws in Poisson's ratio of materials can be overcome. In the proposed PD, the critical strain energy density rather than the critical curvature is adopted as the failure criterion. The capability of the developed PD model was demonstrated by the bending examples of composite laminates with different fiber orientations, and damage analysis was further conducted to demonstrate the strong capability of proposed PD model in replicating the damage process of laminates. In addition, a single-layer material point model (SLMPM) can be implemented in the proposed PD algorithm, and the computational efficiency of numerical models will be greatly improved.

A coupled thermo-mechanical peridynamic model for fracture behavior of granite subjected to heating and water-cooling processes

Thermal damage and thermal fracture of rocks are two important indicators in geothermal mining projects. This paper investigates the effects of heating and water-cooling on granite specimens at various temperatures. The laboratory uniaxial compression experiments were also conducted. Then, a coupled thermo-mechanical ordinary state-based peridynamic (OSB-PD) model and corresponding numerical scheme were developed to simulate the damage of rocks after the heating and cooling processes, and the change of crack evolution process was predicted. The results demonstrate that elevated heating temperatures exacerbate the thermal damage to the specimens, resulting in a decrease in peak strength and an increase in ductility of granite. The escalating occurrence of thermal-induced cracks significantly affects the crack evolution process during the loading phase. The numerical results accurately reproduce the damage and fracture characteristics of the granite under different final heating temperatures (FHTs), which are consistent with the test results in terms of strength, crack evolution process, and failure mode.

Peridynamic modeling of elastic-plastic ductile fracture

A peridynamics-based framework is established for the elastic-plastic ductile fracture analysis. In this frame, a new state-based peridynamic elastic-plastic model is presented using the novel bond extension state forms of yield function and plastic flow rule. The nonlinear energy release rate is computed by the extended peridynamic finite crack extension (PFCE) method, and an energy dissipation rate-based bond failure criterion is proposed for ductile crack growth modeling. Numerical methods for plastic states update and yield function solution are also given. Then, examples of plates with a center hole or a center crack, and compact tension (CT) tests are analyzed by the proposed models, and compared to those from theoretical and FEM solutions. The results demonstrate that the proposed peridynamic models can well capture the characteristics of elastic-plastic deformation and ductile fracture, including the plastic shape and size, energy distributions, nonlinear energy release rate, load-displacement and resistance curves, etc.

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.

A stabilized non-ordinary peridynamic model for linear piezoelectricity

Piezoelectric materials are used in a wide range of engineering applications as sensors, actuators, and transducers due to their intrinsic properties. However, analyzing these devices is not straightforward because of the material anisotropy and coupled electromechanical nature of the problems. In this study, a stabilized non-ordinary state-based peridynamic model for linear piezoelectricity is proposed. Fully coupled electromechanical simulation is conducted using an implicit method, and the tangent stiffness matrix is obtained via perturbation technique. Three two-dimensional electromechanical problems with combined static loading and various boundary conditions are used to validate the proposed technique. Then, the crack-opening displacements, and induced electric field on the crack-face are presented for a center-cracked PZT-5H plate. Hence, this study demonstrates that the proposed technique is capable of modeling linear piezoelectric materials and provides a promising method to study fracture and damage progression in piezoelectric materials.

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.

Modelling of dynamic fracture in bulk superconductor during pulsed field magnetization using ordinary state-based peridynamics

Bulk superconductor is subjected to a combination of huge electromagnetic force and temperature rise owing to the rapid movement of magnetic flux during pulsed field magnetization (PFM). Meanwhile, the heat generated will decrease the shielding effect and limit the strength of trapped field in turn. In view of its intrinsic brittle property and existing defects in fabrication, the bulk is prone to fracture under the complicated electromagnetic and thermal loadings. In this paper, the fracture behavior and damage evolution of the bulk superconductor with two or more cracks during PFM are investigated. Based on the electromagnetic field and temperature obtained by solving governing equation of H-formulation and heat transfer equation, the mechanical behavior of bulk is simulated with ordinary state-based Peridynamic model. The dynamic stress intensity factors considering thermal stress at crack tips are studied for different external magnetic fields and crack angles. Then, the influences of thermal contact resistance and local heat source are discussed. Finally, crack propagation in the bulk samples without reinforcement is modelled during PFM.

A Multi-Yield-Surface Plasticity State-Based Peridynamics Model and its Applications to Simulations of Ice-Structure Interactions

Due to complex mesoscopic and the distinct macroscopic evolution characteristics of ice, especially for its brittle-to-ductile transition in dynamic response, it is still a challenging task to build an accurate ice constitutive model to predict ice loads during ship-ice collision. To address this, we incorporate the conventional multi-yield-surface plasticity model with the state-based peridynamics to simulate the stress and crack formation of ice under impact. Additionally, we take into account of the effects of inhomogeneous temperature distribution, strain rate, and pressure sensitivity. By doing so, we can successfully predict material failure of isotropic freshwater ice,iceberg ice, and columnar saline ice. Particularly, the proposed ice constitutive model is validated through several benchmark tests, and proved its applicability to model ice fragmentation under impacts, including drop tower tests and ballistic problems. Our results show that the proposed approach provides good computational performance to simulate ship-ice collision.

Strength damage behavior analysis of silicon carbide(SiC) ceramic cylindrical shells under hydrostatic pressure based on the state-based peridynamics

Ceramics are one of the most promising materials for pressure-resistant shells of submersibles due to their exceptional mechanical properties. However, ceramics are brittle materials, and the occurrence of strength damage is inevitably accompanied by the generation and propagation of cracks. A new state-based peridynamic damage model is established, the relation between critical elongation and critical energy release rate for tensile and compression fracture is deduced innovatively, to investigate the strength damage behavior of silicon carbide (SiC) cylindrical shells under hydrostatic pressure. The findings demonstrate that by increasing cylinder thickness, the structural damage pressure can be significantly improved. Moreover, the initial damage zones, are located on the internal surface of shell and induced by the compression fracture, and evenly scattered at 90° around the circular direction. As pressure increases, tensile damage zones emerge on the external surface of cylinder. The failure zones on the interior and exterior walls of the shell propagate along the circumferential direction firstly, then extend toward the middle surface until the damage zones completely penetrate the entire thickness, and the cylinder loses its load-bearing capacity.

Accelerating convergence of crack propagation simulation in peridynamic models via high-order temporal discretization

This work proposes a high-order temporal discretization for reformulations of both bond-based and state-based peridynamic models modeling, e.g. crack propagation. It is discovered from various experiments that for such discontinuous problems, commonly-used second-order methods are less stable and more computationally consuming than the high-order method, which indicates that the high-order temporal discretization not only improves the accuracy and the efficiency for conventional smooth problems, but also exhibits a promising behavior for some non-smooth models such as peridynamics.

A Modified Bond-Associated Non-Ordinary State-Based Peridynamic Model for Impact Problems of Quasi-Brittle Materials.

In this work, we have developed a novel bond-associated non-ordinary state-based peridynamic (BA-NOSB PD) model for the numerical modeling and prediction of the impact response and fracture damage of quasi-brittle materials. First, the improved Johnson-Holmquist (JH2) constitutive relationship is implemented in the framework of BA-NOSB PD theory to describe the nonlinear material response, which also helps to eliminate the zero-energy mode. Afterwards, the volumetric strain in the equation of state is redefined by the introduction of the bond-associated deformation gradient, which can effectively improve the stability and accuracy of the material model. Then, a new general bond-breaking criterion is proposed in the BA-NOSB PD model, which is capable of covering various failure modes of quasi-brittle materials, including the tensile-shear failure that is not commonly considered in the literature. Subsequently, a practical bond-breaking strategy and its computational implementation are presented and discussed by means of energy convergence. Finally, the proposed model is verified by two benchmark numerical examples and demonstrated by the numerical simulation of edge-on impact and normal impact experiments on ceramics. The comparison between our results and references shows good capability and stability for impact problems of quasi-brittle materials. Numerical oscillations and unphysical deformation modes are effectively eliminated, showing strong robustness and bright prospects for relevant applications.