Ordinary State-based Peridynamics Research Articles

Fracture simulation of viscoelastic membranes by ordinary state-based peridynamics

Despite a load below the elastic limit is applied on a viscoelastic material, the material may fail after a long duration of constant loading because of the time-dependent viscous deformations. In this regard, a viscoelastic material model in the ordinary state-based peridynamic framework is proposed to capture crack propagation in polymeric water treatment membranes. The deformation state is decoupled into dilatational and distortional parts, and it is assumed that the dilatational part of deformation is elastic, while the distortional part is considered as viscoelastic, whose behaviour can be represented by the Prony series. First, we verify our implementation with FEM results for a benchmark case. Afterwards, the crack propagation is studied by the viscoelastic ordinary state-based peridynamic model.

A unified non-local fluid transport model for heterogeneous saturated porous media

A unified non-local fluid transport model is proposed for heterogeneous saturated porous media. First, a non-local governing equation of fluid transport is presented based on the Non-ordinary State-based Peridynamics (NOSB-PD), and a novel flux vector state and a non-local Darcy’s law is proposed to illustrate non-local fluid transport effects in heterogeneous saturated porous media. The proposed flux state vector ensures the non-local governing equation of fluid transport model in heterogeneous saturated porous media degenerates into the local one when the horizon of each material point approaches to zero. A non-local pressure gradient at each material point can be obtained by averaging all the flux state vectors in its horizon, which unifies the weak and strong discontinuities of pressure in heterogeneous saturated porous media in a consistent way. Second, a variational formulation of the unified non-local fluid transport model is developed. This variational formulation enables the proposed model to deal with complex boundary conditions, which is difficult or even impossible for current fluid transport models based on Bond-based Peridynamics (BB-PD) and Ordinary State-based Peridynamics (OSB-PD). Third, a fully implicit algorithm combined with the Newton–Raphson method is introduced to solve the general non-linear problems in the heterogeneous saturated porous media. The penalty function method is employed to eliminate the zero-energy mode oscillation in NOSB-PD. Further, numerical examples demonstrate that the proposed model is accurate and can well capture the weak and strong discontinuities of pressure at the material interfaces and crack surfaces in the heterogeneous saturated porous media. Numerical examples also indicate that the model can effectively eliminate the zero-energy mode oscillation inherently rooted in the NOSB-PD, and thus reaches a stable numerical solution.

Brittle Fracture Modeling Using Ordinary State-Based Peridynamics with Continuous Bond-Breakage Damage

Brittle Fracture Modeling Using Ordinary State-Based Peridynamics with Continuous Bond-Breakage Damage

A two-dimensional ordinary state-based peridynamic model for plastic deformation based on Drucker-Prager criteria with non-associated flow rule

The plastic behaviors of geomaterials often obey non-associated flow rule. In this paper, a two-dimensional (2D) ordinary state-based peridynamic (OSB-PD) increment model with shear deformation based on Drucker-Prager (D-P) criterion with non-associated flow rule is proposed to study the plastic behavior of geomaterials. An incremental form of peridynamic (PD) strain energy density for plane elastic problems is first established. The material parameters are determined when the incremental PD strain energy density is equal to the incremental strain energy density in classical mechanics. The PD force density for the OSB-PD theory with shear deformation in plane elastic cases is derived accordingly. The relationship between the incremental second invariant of the stress tensor dJ2, the first invariant function of stress tensor dI1 in the classical mechanics and PD are established in this paper, and the Drucker-Prager yield surface is expressed in the framework of peridynamics. To avoid the excessive plastic dilation, the non-associated flow rule is proposed, in which isotropic and deviatoric plastic elongation are employed to describe the plastic behaviors of a bond. Also, the equivalent stress and equivalent plastic strain for the proposed PD model are given. An integration algorithm for the proposed increment model is developed. Numerical examples including circular opening in rocks are performed, and the PD results are compared with that obtained by the finite element method or analytical solution. Those numerical simulations have verified the validity of the proposed PD model to predict the elasto-plastic behavior of geomaterials.

Evaluation of dynamic behaviour of porous media including micro-cracks by ordinary state-based peridynamics

Reliable evaluation of mechanical response in a porous solid might be challenging without any simplified assumptions. Peridynamics (PD) perform very well on a medium including pores owing to its definition, which is valid for entire domain regardless of any existed discontinuities. Accordingly, porosity is defined by randomly removing the PD interactions between the material points. As wave propagation in a solid body can be regarded as an indication of the material properties, wave propagation in porous media under an impact loading is studied first and average wave speeds are compared with the available reference results. A good agreement between the present and the reference results is achieved. Then, micro-cracks are introduced into porous media to investigate their influence on the elastic wave propagation. The micro-cracks are considered in both random and regular patterns by varying the number of cracks and their orientation. As the porosity ratio increases, it is observed that wave propagation speed drops considerably as expected. As for the cases with micro-cracks, the average wave speeds are not influenced significantly in random micro-crack configurations, while regular micro-cracks play a noticeable role in absorbing wave propagation depending on their orientation as well as the number of crack arrays in y-direction.

An In-depth Investigation of Bimaterial Interface Modeling Using Ordinary State-based Peridynamics

This study presents an in-depth investigation of interface modeling techniques in ordinary state-based peridynamics (OSB-PD). By performing numerical examples, the accuracy of each technique is investigated by comparison with the finite element solutions for simple bimaterial interface problems. Subsequently, the capabilities of OSB-PD in treating material interfaces with complicated geometries are demonstrated by solving various problems with a single inclusion or multiple inclusions. Finally, a cracked plate containing an inclusion is considered to study effects of the inclusion on dynamic stress intensity factor (DSIF). The OSB-PD shows its huge potential in modeling a structure with multiple defects due to the effectiveness and ease of implementation.

Thermo-Mechanically Coupled Peridynamic Modelling of Fracture Formation in Solid Oxide Fuel Cell Anode

Fracture formation can be observed in brittle yttria-stabilized zirconia (YSZ) in Ni-YSZ composite anode during high-temperature operation of solid oxide fuel cells (SOFCs). It is quite difficult to predict the fracture formation in solid materials accurately based on the classical continuum mechanics theory, because of the singularity problems in partial differential equations. In this work, an ordinary state-based-peridynamic (OSB-PD) model is proposed to predict the fracture formation in SOFC anode based on real microstructures reconstructed using focused ion beam-scanning electron microscopy (FIB-SEM) dual-beam technique. The key factors determining anode microstructure strength is analyzed based on the simulation results upon the comparison among different anode microstructures. The thermo-mechanical coupling effect is investigated by considering the influence of both external mechanical loads and temperature distribution.

Morphological characteristics of crack branching in asphalt mixtures under compression

Crack branching is a multi-crack propagation problem. Singular crack tip elements are frequently encountered, and defining the multi-crack tips’ enrichment functions is difficult. To overcome these drawbacks, this study used an already developed ordinary state-based peridynamics method to simulate crack branching in AC-13, AC-20, and PAC-13 asphalt mixtures. These asphalt mixtures’ digital images were acquired by an X-ray computed tomography method. Next, the Prony series coefficients of the fine asphalt mixtures were obtained by the dynamic modulus test. After that, the fracture energy of the fine asphalt mixtures was measured by the semi-circular bending test. Finally, the measured asphalt mixture properties were input into the ordinary state-based peridynamics method in order to simulate crack branching in the AC-13, AC-20, and PAC-13 asphalt mixtures. The simulation results indicate that the diagonal crack first emanates in asphalt mixtures. Then, the diagonal crack continues to branch into a wing crack, splitting crack, and branched crack. These four types of cracks can generate eight crack branching categories in asphalt mixtures. The crack branching trajectory was quantified by the fractal dimension, whose value ranged from 1.14 to 1.64. The physical significance of the fractal dimension explains the changes of crack branching paths in the asphalt mixtures.

Numerical study on dynamic icebreaking process of an icebreaker by ordinary state-based peridynamics and continuous contact detection algorithm

The existence of ice and its interaction with structure greatly threaten the navigation of the icebreaker. Due to the damage characteristics of ice, the traditional numerical ways based on mesh method are challenging to solve ice-structure interaction. The nonlocal particle methods have increasingly gained popularity as an essential tool for calculating large deformation problems. In the present work, a meshfree particle-based computational model is developed to investigate the icebreaking process and dynamic icebreaking resistance. The model employs ordinary state-based Peridynamics theory to establish the constitutive relation for ice which is modelled as an isotropic homogeneous elastic material. Furthermore, a continuous contact detection algorithm is proposed, which can be applied to detect the contact process between irregular structures and particles, to study the process of ice-ship interaction. Two dimensional (2D) three-points bending of ice beam and 3D ice sphere impact are studied numerically to validate the method specifically in modelling ice failure. Then, the investigation on the continuous-mode icebreaking process is carried out. The comparison of the present numerical results with existing experimental data demonstrated the validity and accuracy of the present numerical model for analysing the icebreaking process and predicting the icebreaking resistance.

Numerical simulation of crack paths in asphalt mixture using ordinary state-based peridynamics

Cracking creates major distress in asphalt pavements. Because of mathematical complications, existing simulation efforts are incapable of simulating the crack branching and crack coalescence in asphalt mixtures. To overcome this deficiency, this paper developed a two-dimensional (2D) ordinary state-based peridynamics (OSB-PD) method to simulate the entire crack propagation phase, which includes crack propagation, crack branching and crack coalescence. The 2D asphalt mixture digital image was acquired via the X-ray computed tomography method. The Prony series coefficients of the fine aggregate mixture (FAM) were obtained from the dynamic modulus test. The fracture energy of the FAM was determined from the semicircular bending (SCB) test. The measured mixture properties were then input into the 2D OSB-PD numerical method to simulate the crack paths under the SCB test and the three-point bending (TPB) test conditions. The simulated crack paths matched well with those observed from the experiments (at 5% level of significance), thus validating the accuracy of the proposed 2D numerical simulation method. The validated method was finally implemented into the simulation of the crack paths in the laboratory-used asphalt mixture and subjected to compressive loading. The simulation results indicated that the diagonal crack exists when the crack propagates in an asphalt mixture. The diagonal crack then continues to branch and coalesce through the following possible paths until the failure of the asphalt mixture: (a) penetrating through the aggregate; (b) branching along the surface of aggregate in different directions; and (c) coalescing with each other.

A parameter to represent a local deformation mode and a fracture criterion based on the parameter in ordinary-state based peridynamics

It is known that a local deformation mode affects the fracture behavior. First, a parameter that represents the deformation mode is defined for a fracture criterion that can consider the deformation mode in ordinary state-based peridynamics. This parameter is analogous to stress triaxiality in conventional continuum mechanics. The calculated values of the parameter agree well with the theoretical values under uniaxial compression, pure shear, uniaxial tension, and equi-biaxial tension. Then, modified fracture strain by a function using the parameter is proposed as a fracture criterion. The intended fracture strains of uniaxial tensioned cubes and those of equi-biaxial tensioned cubes were well reproduced by calculations with the criterion. Further, the intended ratio of fracture strain of a uniaxial tensioned round bar to that of an equi-biaxial tensioned thin plate was reproduced by calculations with the criterion. In a pipe bending simulation, the crack propagation behavior was reproduced by the proposed criterion.

Fracture parameter analysis of flat shells under out-of-plane loading using ordinary state-based peridynamics

The present paper is devoted to numerical investigation on fracture parameters of cracked shells subjected to out-of-plane loading using ordinary state-based peridynamics (PD). The nonlocal deformation gradient and equivalent domain integral are introduced to evaluate fracture parameters. To reduce the surface effect and obtain more accurate results, the energy method and volume correction algorithm are considered. Meanwhile, the adaptive dynamic relaxation technique is employed to obtain steady-state solutions. From comparisons between PD results and reference solutions, the proposed PD shell model successfully evaluates fracture parameters in both single- and mixed-mode loading conditions.

Ordinary state-based peridynamics for geometrically nonlinear analysis of plates

This study presents a novel ordinary state-based peridynamic model for geometrically nonlinear analysis of plates. The nonlinear strain energy density and nonlinear equations of motion for a plate in peridynamics are obtained based on the principle of virtual displacements by using Total Lagrange formulation. The numerical procedure for geometrically nonlinear analysis of a plate is also provided. The accuracy of the proposed nonlinear peridynamic model is validated by considering large deformations of a plate subjected to bending and a plate subjected to transverse shear forces. To further demonstrate the capabilities of the proposed nonlinear model, damages on a plate with a single crack subjected to stretching and tearing, a plate with two parallel cracks subjected to tearing, and a plate subjected to torsional loading are predicted.

A physics-guided machine learning model for two-dimensional structures based on ordinary state-based peridynamics

This study presents a novel physics-guided machine learning model for two-dimensional structures based on ordinary state-based peridynamics. The linear relationships between the displacements of a material point and the displacements of its neighbours and the applied forces are obtained for the machine learning model by using linear regression. The numerical procedure for coupling the ordinary state-based peridynamic model and the machine learning model is also provided. The accuracy of the coupled model is verified by predicting deformations of a two-dimensional plate with circular cut-out subjected to tension and a two-dimensional representation of three points bending test. To further demonstrate the capabilities of the coupled model, damage predictions for a two-dimensional representation of a three-point bending test, a notched plate with a hole subjected to tension, a square plate with a pre-existing crack subjected to tension, and a plate with a pre-existing crack subjected to sudden loading are presented.

Peridynamic simulation of fracture mechanical behaviour of granite specimen under real-time temperature and post-temperature treatments

This study developed a fully coupled thermo-mechanical model within the framework of ordinary state-based peridynamics to investigate the thermal-mechanical properties and fracture characteristics of granite materials under real-time temperature (RT) and post-temperature (PT) treatments. Moreover, a modified multi-layer computational method was proposed to eliminate the effect of thermal gradient-induced cracks. In this method, the size of the thermal layer was set to be larger than that of the tested sample, with the additional area used to withstand the thermal shock. During the loading process, the thermal damage information was transplanted to the corresponding position of the mechanical layer with the data of the additional area cut off. It was verified that thermal cycling cracks and thermal gradient-induced cracks could be modelled and distinguished, and the stress-strain and cracking behaviours of both RT and PT samples could be properly simulated using the proposed method. First, two benchmark examples involving heat conduction in a square plate and uniaxial compression of a granite specimen were simulated to investigate the numerical convergence and calibrate the simulation parameters, respectively. Then, two numerical examples were used to investigate and compare the stress-strain behaviours, cracking patterns, and temperature evolutions of the RT and PT samples. A systematic comparison with the experimental results makes it possible to discuss and summarise the failure mechanism of granite specimens subjected to PT and RT tests.

Modeling and studies of fracture in functionally graded materials under thermal shock loading using peridynamics

In this study, an extended ordinary state-based peridynamic (OSBPD) model was proposed and applied to study the thermal shock damage and fracture of functionally graded materials (FGM) and structures. A modified term including thermal expansion coefficient is introduced into the conventional peridynamic model for investigating the thermal deformation of materials, and an influence function reflecting the gradient change of material properties is introduced into the force vector state in peridynamic model. The model and algorithm are validated by analyzing the cracking of a plate quenching in water and comparing with experimental results and available literature data. Furthermore, the effects of gradient distribution of elastic modulus, thermal expansion coefficient and fracture toughness of FGM on thermal damage and crack propagation are investigated. Numerical results show that the proposed model can effectively describe the failure process of functionally gradient materials under thermal shock loading. The distribution of elastic modulus, coefficient of thermal expansion and fracture toughness in FGM have great influence on the initiation and propagation of cracks, and reasonable gradient change form of materials can improve the thermal resistance ability of the structure.

Mixed-mode stress intensity factors evaluation of flat shells under in-plane loading employing ordinary state-based peridynamics

A recently developed peridynamic (PD) shell model based on the ordinary state-based peridynamic theory is adopted to evaluate stress intensity factors (SIFs) under in-plane loading. Strain and stress components are obtained by introducing the peridynamic differential operator. In order to evaluate mixed-mode SIFs, the domain form of the interaction integral is employed. The adaptive dynamic relaxation technique is utilized to obtain steady-state solutions, and the energy method is applied to reduce the PD surface effect. Several numerical examples are considered, including single- and mixed-mode fracture problems. All the PD results are compared with reference results to demonstrate the accuracy and effectiveness of the proposed approach. The present paper aims to examine the performance of the PD shell model in linear elastic fracture mechanics and provides an effective approach for SIFs evaluation.

A smoothed variable horizon peridynamics and its application to the fracture parameters evaluation

Peridynamics has attractive features for solving several fracture mechanics problems. On the other hand, to ensure the accuracy, a large amount of particles (material points) is required. Introduction of variable horizon is an alternative approach by changing the horizon size over the problem domain. In the present study, we propose a novel variable horizon concept. It is known that an undesired “ghost” force arises along the transition region of different horizons. In order to suppress the ghost force, the gradual variation in the horizon size over a certain region, called smoothing length, is introduced between different scale particle distributions. Efficiency of the smoothed variable horizon peridynamics is demonstrated by several numerical studies employing the ordinary state-based peridynamics. As a basic case, a linear displacement field is considered. It is observed that the proposed approach significantly reduces the ghost forces along the interface of different spatial discretizations. Additionally, the dynamic stress intensity factors of stationary cracks are carefully examined. The path independence of the fracture parameters in the variable horizon peridynamics is ensured. The efficiency of the presented method is then discussed. It is monitored that introduction of the smoothing length concept significantly reduces the computational costs in the peridynamic modeling.

An improved mesoscale damage model for quasi-brittle fracture analysis of concrete with ordinary state-based peridynamics

Ordinary state-based peridynamics (OSPD) is a promising computational method to analyze damage and fracture process because its equation of motion is still satisfied at the presence of discontinuities. Though OSPD has been introduced to evaluate mesoscopic behavior of concrete, there are many unsolved issues to tackle quasi-brittle behavior of mesoscopic constituents. In this paper, an improved mesoscale damage model for two-dimensional fracture analysis of concrete is proposed. Instead of modeling the interfacial transition zone (ITZ) with material points, the mechanical behavior of ITZ is embedded in the corresponding bonds of mortar and aggregates to improve modeling and computational efficiency. During the failure process, the damage initiation criterion is established based on the stress computed from the scalar force density; while the softening is characterized by the fracture energy and an exponential softening law. The proposed model is verified by simulating cracking of ITZ in uniaxial tension as well as crack propagation of concrete in three-point bending test. The prediction results show good agreements with either analytical or experimental results. Then, the model is employed to study mesoscale crack initiation and propagation of concrete under uniaxial tension, considering the distributions of mortar, aggregates, ITZ, and voids. Influences of the mesoscopic fracture properties on the macro constitutive relationship are also investigated through parametric studies. The improved mesoscale damage model presented is a simple yet effective tool for fracture analysis of concrete.

Elastoplastic Constitutive Modeling for Reinforced Concrete in Ordinary State-Based Peridynamics

ABSTRACTA non-local, ordinary state-based, peridynamic elastoplastic model is formulated to numerically simulate the fracture of reinforced concrete materials. Several basic definitions are first discussed to avoid confusion; and then, a detailed derivation of the force vector state is presented, leading to a unified expression of force state for one-, two- and three-dimensional elasticity problems. Furthermore, an ordinary state-based peridynamic (OSB PD) elastoplastic analysis approach is developed for both plastic compressible and incompressible materials, including the constitutive relationship, the yield function, the consistency condition and the plasticity flow rule. The peridynamic predictions of a quasi-static deformation of the steel rods are in good agreement with the analytical solution. Moreover, the OSB PD plasticity is verified by analyzing a square plate with or without a central hole suffering different loading-unloading paths. Finally, a two dimensional reinforced concrete clamped beam subjected to impact loading is simulated with the proposed OSB PD elastoplasticity, demonstrating its capability in capturing the damage characteristics and structural failure behavior. Simulation results show good accuracy of the peridynamics in simulating elastoplastic problems.