Bond-based Peridynamics Research Articles

Damage and failure mechanism of sapphire under ballistic loading based on a modified bond‐based peridynamic model

AbstractTo investigate the dynamic mechanical behaviors of c‐plane and a‐plane sapphire, a novel anisotropic constitutive model and fracture criterion are constructed on the basis of bond‐based peridynamics (BB‐PD) theory. And the concept of strain is introduced in the framework of BB‐PD. After that, the simulations of a‐ and c‐planes of sapphire under spherical and cylindrical impact are conducted. In addition to capturing the distribution of strain and damage, the histories of various fracture types such as the primary fracture front and cracks are monitored for quantitative analysis. The numerical predictions are shown to agree well with previous experimental results, and further reveal the damage and failure mechanisms of sapphire. The different forms of contact between the projectile and the target strongly influence the stress waves generated and energy transferred ultimately affecting the damage development process. The crystal orientation dominates the appearance of anisotropic crack modes. Crack bending and deflection, as well as spalling‐like fracture, are associated with wave reflection and intersection. Moreover, an in‐depth examination of the observed wave splitting phenomenon is performed.

A peridynamic approach to simulating fatigue crack propagation in composite materials.

In this article, a numerical tool is proposed in the framework of bond-based peridynamics to simulate fatigue crack propagation in composite materials and structures. The cycle-dependent damage-cumulative model derived from Peerlings' law and applied to a bilinear constitutive law is used to evaluate the fatigue degradation of the bond stiffness. Several benchmark cases are studied to validate the proposed approach. Finally, static and fatigue crack propagations in composite systems with single or multi-inclusions are simulated to illustrate the capabilities and characteristics of the developed approach. This article is part of the theme issue 'Ageing and durability of composite materials'.

Modeling via peridynamics for large deformation and progressive fracture of hyperelastic materials

Hyperelastic materials are frequently observed in the natural world and engineering applications such as rubbers and animal soft tissues. Modeling large deformation and progressive fracture of these materials is challenging for grid-based methods due to the severe mesh distortions at crack tips. To address this problem, we proposed a novel meshfree framework using bond-based peridynamics (PD) based on finite deformation theory. The proposed modeling framework has the following novelties: (1) An original bond strain, derived from the spectrum representation of hyperelastic models is used to describe the nonlinear bond force-stretch relationship, which gives more concise and robust solutions to large deformation problems; (2) A numerical damping parameter inspired by the mass–dashpot–spring system is developed to endow the proposed framework with numerical advantages in terms of enhancing the stability of explicit time integration in modeling quasi-static problems; (3) It outperforms grid-based methods in capturing notable features like non-smooth crack surfaces, secondary damages, and materials fragmentations. The capability of our framework was successfully validated via several examples, including large deformation, crack initiation and propagation tests of specimens with various notches and eccentric holes. Besides, the numerical performance of our model to capture the crack deflection in laminate rubber specimens was also tested, with results showing good consistency with experimental observations. This framework is compatible with diverse hyperelastic models and sheds new light on modeling of elastomer-hydrogel composites and soft tissues.

A peridynamic differential operator-based scheme for the extended bond-based peridynamics and its application to fracture problems of brittle solids

In order to describe the mechanical behavior of solids with arbitrary Poisson’s ratio values, an extended bond-based peridynamic (XBB-PD) model was developed in the authors’ previous work by taking both the bond longitudinal and tangential deformations into account. In the current work, a peridynamic differential operator based implementation scheme is presented for the XBB-PD model. An energy-based failure criterion is introduced to describe the failure and crack propagation in brittle solids. First, two elastic examples are solved to validate the proposed approach. The m- and δ- convergence studies are carried out. Subsequentially, several numerical examples of crack propagation problems are simulated. All simulations show good agreement with the results reported in the references, demonstrating the flexibility and capabilities of the developed approach in the simulation of crack propagation in brittle solids.

Dynamic simulation of aircraft electro-impulse de-icing using bond-based peridynamics

Electro-impulse de-icing is a lightweight type of mechanical de-icing system that consumes little energy, provides high reliability, and offers wide application prospects. In this study, the bond-based peridynamics theory was used to simulate the dynamic damage process and evolution law of aircraft electro-impulse de-icing. The ice layer was simplified as a brittle material that satisfies the linear elastic constitutive relation. A numerical analysis model of the ice layer, aircraft skin substrate, and their interface was established to simulate the dynamic responses under a high strain rate. Considering the anisotropy of ice adhesion on the substrate, the interfacial bonds were described as shear bonds and tensile bonds, and the critical elongations of the two were derived. The tensile and shear capacities of the ice on the substrate were then simulated using these critical elongations, and the peeling rates of single ice particles and interfacial ice layers were taken as indexes describing the efficacy of electro-impulse de-icing. The process of electro-impulse de-icing was then analyzed for an aluminum substrate with two adjacent clamped edges. Finally, the results of the peridynamic simulation were compared with those of existing experiments and finite element models to verify the effectiveness of the peridynamic approach.

Thermo-Mechanical Coupling Model of Bond-Based Peridynamics for Quasi-Brittle Materials.

The mechanical properties of quasi-brittle materials, which are widely used in engineering applications, are often affected by the thermal condition of their service environment. Moreover, the materials appear brittle when subjected to tensile loading and show plastic characteristics under high pressure. These two phenomena manifest under different circumstances as completely different mechanical behaviors in the material. To accurately describe the mechanical response, the material behavior, and the failure mechanism of quasi-brittle materials with the thermo-mechanical coupling effect, the influence of the thermal condition is considered in calculating bond forces in the stretching and compression stages, based on a new bond-based Peridynamic (BB-PD) model. In this study, a novel bond-based Peridynamic, fully coupled, thermo-mechanical model is proposed for quasi-brittle materials, with a heat conduction component to account for the effect of the thermo-mechanical coupling. Numerical simulations are carried out to demonstrate the validity and capability of the proposed model. The results reveal that agreement could be found between our model and the experimental data, which show good reliability and promise in the proposed approach.

An energetically consistent surface correction method for bond-based peridynamics

A novel surface correction method is proposed for bond-based peridynamics which ensures energy consistency with a classical reference body for general affine deformations. This method is validated for simple geometries and then applied to a typical surface-dominated problem, namely the indentation of a surface in the shallow to moderate-depth regime.

Application of Weibull fracture strength distributions to modelling crack initiation behaviour in nuclear fuel pellets using peridynamics

The thermomechanical behaviour of uranium dioxide nuclear fuel pellets irradiated in a pressurised water reactor has been simulated using a two-dimensional application of bond-based peridynamics implemented in the Abaqus commercial finite element software. Near-surface bond failure, and hence crack initiation, were modelled assuming a probabilistic (variable) failure strain described by a Weibull distribution – with bond failure, and hence crack propagation, in the bulk of the fuel pellets modelled assuming a deterministic (fixed) failure strain. The measured dependency of the number of radial pellet cracks on heat generation rate per unit length – which we show cannot be reproduced by the common assumption in pellet modelling of a deterministic failure strain throughout the pellet volume – was accurately predicted when a size-scaled Weibull distribution with a modulus of 5 was used. However, this low modulus value was associated with the prediction of some cracks initiating away from the pellet surface, which is unphysical. Use of a Weibull modulus of 10 avoided this simulation artefact while still reproducing the experimentally observed dependency with reasonable accuracy.

An improved four-parameter conjugated bond-based peridynamic method for fiber-reinforced composites

The classical bond-based peridynamic (BB-PD) theory has inherent limitations on material properties when dealing with composites. An improved lamina conjugated BB-PD model with four independent material parameters is established to study the fracture behavior of composite materials. The model defines four basic generalized PD bonds, including fiber bond, matrix bond, fiber-matrix conjugated bond pair and matrix-matrix conjugated bond pair. Using the equivalent relationship of elastic strain energy density, the analytical expressions of the four micro-modulus in the improved lamina conjugated BB-PD model are determined. These four micro-modulus parameters can describe the four independent engineering material constants of the lamina, which eliminates the restriction of Poisson’s ratio and shear elastic modulus for the composite materials. Moreover, an energy-based failure criterion is further proposed for the failure analysis of composite materials. In order to verify the capability and applicability of the proposed model, the displacement field and fracture behavior of the composite lamina with uniaxial tensile are simulated. In addition, the failure modes and crack propagation path of the laminae with different fiber orientations and initial crack angles are investigated by using the proposed model.

A robust peridynamic computational framework for predicting mechanical properties of porous quasi-brittle materials

Prediction of the macroscopic mechanical properties and damage evolution of cement paste is significant; however, it poses challenges due to the complex multiphase heterogeneity and porosity of the hydration microstructure. This study presents a robust implicit-explicit hybrid computational framework based on the bond-based peridynamics for predicting crack evolution and macroscopic mechanical properties of porous quasi-brittle materials. Specifically, the prototype micro-elastic brittle (PMB) material model is improved by considering an attenuation kernel function and surface effect correction. Subsequently, an efficient implicit-explicit hybrid approach is developed for the solution of the strong form of peridynamic equation of motion. A critical damage index of the global system is specified to switch the solver from implicit to explicit. CEMHYD3D software is employed to generate the microstructure of hydration under uniaxial tension. Systematic simulations of the microstructure explain the influence of several computational strategies on the results, i.e., the approach of boundary enforcement, the geometry of porous microstructure, the loading rate in the explicit stage, and breaking the bonds across the voids or not. The simulations demonstrate that applying the improved bond-based peridynamic solver on hydration microstructures can capture their cracking behavior and macroscopic mechanical properties, e.g., stress–strain curve, peak stress, maximum tensile strain, Young’s modulus, and fracture energy release rate. This study also paves the way for peridynamic multiscale modeling of cement-based materials.

Improved peridynamic model and its application to crack propagation in rocks.

The conventional bond-based peridynamics (BB-PD) model is suitable for simulating the failure mode of homogeneous elastic-brittle materials. However, the strain hardening and subsequent strain softening characteristics of rocks under loading cannot be reflected. In addition, the fracture mechanisms of rock materials under tension and compression are completely different. To solve these problems, this paper proposes an improved BB-PD model using different fracture criteria in the tensile and compression stages of the bond based on previous improved models, and a critical failure condition obeying the Weibull distribution is introduced to reflect the heterogeneity of the rock. The crack propagation processes of an intact rock specimen, rock specimen with a single pre-existing flaw and rock specimens with two and three pre-existing flaws under compressive loading are simulated using the model, and its feasibility is verified by comparing with the results of previous laboratory tests. Next, the effects of the inclination angle and length on the wing crack propagation length are studied. Finally, the changes in the crack aggregation modes under different rock bridge inclination angles are simulated. Eight types of crack aggregation modes are found, and the conditions under which they may occur are analysed. The improved model proposed can effectively simulate the crack propagation and coalescing processes and has a wide application prospect for rock fracture simulations.

Effect of water-cement ratio and size on tensile damage in hardened cement paste: Insight from peridynamic simulations

This paper uses the improved bond-based peridynamics (BB PD) method to predict the crack evolution and macroscopic mechanical properties of the hardened cement paste at the micro-scale. Specifically, the prototype micro-elastic brittle (PMB) model is improved by considering an attenuation kernel function and the surface effect correction. Subsequently, the uniaxial tension simulations of hardened cement microstructures generated by μic software are systemically conducted to investigate the influences of the heterogeneity of microstructure, size effect, and water-cement ratio (w/c) on crack growth and mechanical properties, i.e., tensile strength and Young's modulus. The results show that the heterogeneity of the microstructure inevitably will cause the discreteness of the macroscopic properties of the numerical cement paste. Therefore, a statistical method is considered to analyze the mechanical properties of the hardened cement. Moreover, the tensile strength has a remarkable size effect, with the size increase of microstructure, the tensile strength decreases; the simulation results fit well with the existing analytical size effect model, i.e., Weibull size effect model, multifractal scaling law, and Bažant's size effect law. Besides, the higher ratio of w/c is likely to result in lower tensile strength, Young's modulus and premature damage to the cement paste. The predicted macroscopic mechanical properties are in good agreement with the previous simulation and experimental data, which demonstrates the reliability of the improved bond-based peridynamic method in studying the crack initiation and propagation of hardened cement paste.

A finite element-peridynamic combined multiscale analysis strategy based on implicit integration scheme

Although peridynamics (PD) method can address discontinuity issues existing in civil engineering, it is much more computationally expensive than the conventional finite element (FE) method because of inherent shortcomings such as small grid spacing and explicit integration scheme. In this regard, a multiscale analysis method reformulated in an implicit integration scheme, named FE-PD method, is proposed, which integrates the PD method with the FE method to take the advantages of discontinuity solving capacity of the PD method and high computing efficiency of the FE method. This study commences with the introduction of the bond-based PD theory. Subsequently, the framework of the FE-PD method together with two newly developed interface elements is presented. Finally, how to implement the FE-PD method into an open-source FE software, OpenSees, is illustrated, and the proposed FE-PD method is verified by a benchmark problem involving an l-shaped concrete plate sustaining a vertical loading, as well as a 2D and a 3D cantilever beam sustaining uniaxial tensile and shear loadings, respectively. Besides, the FE-PD method is employed to simulate the bond-slip behavior of the rebar embedded in a solid concrete specimen. The investigation results confirm that the FE-PD method not only consumes relatively much less (e.g., about 70%) computing resources compared with the PD method but also well captures the crack propagation of concrete and the bond-slip behavior of the rebar embedded in a quasi-brittle concrete specimen.

A coupled SPH-PD model for fluid–structure interaction in an irregular channel flow considering the structural failure

Understanding fluid–structure interaction (FSI) is important because it dominates diverse natural phenomena and engineering problems. This paper presents an integrated particle model for FSI problems involving irregular channel flows and crack propagation in structures. The proposed model is implemented as follows: (1) we couple weakly compressible smoothed particle hydrodynamics (WCSPH) with bond-based peridynamics (BBPD) in a partitioned approach (this framework has a much simpler algorithm than the previously reported SPH-PD method); (2) we propose a novel periodic boundary conditions (PBCs) algorithm for modeling flows in non-regular channels; and (3) we incorporate crack propagation in structural responses under fluid dynamics, which was rarely considered in previous works. The new framework has been validated and illustrated to be effective and versatile in diverse FSI problems, including hydrostatic pressure-induced solid deformation, violent free-surface flows and channel flows interacting with flexible structures. Compared with conventional grid-based methods, this particle framework is more user-friendly, since no extra effort is required to update meshes, even when a discontinuity appears during the modeling process. The extendibility and potential of this framework is further demonstrated by the simulation of fluid-driven deformation and crack propagation in elastomers.

Progressive failure analysis of cylindrical shell under hydrostatic pressure based on an extended bond-based peridynamic method

A three-dimensional extended bond-based peridynamic (BPD) model, which considers the influence of internal length effect of long-range force on calculation accuracy, is proposed for progressive failure analysis of cylindrical shell subjected to hydrostatic pressure. The maximum strain criterion is introduced into the model to be a benchmark to judge whether bond breakage occurs as the hydrostatic pressure increase. Numerical results reveal that the initiation strength failure zones originated from the inner wall of the middle part of the cylinder, and are uniformly distributed at 90° in the circumferential direction. As the damage evolution continue, two new failure zones occur in the outer wall of the cylinder. Furthermore, the failure zones initiation in the inner and outer walls propagates toward to the middle surface of the cylindrical shell, and eventually they connect together. Throughout the failure evolution history, the failure degree of cylindrical shell behaves exponential growth with the loading linear increase. Once strength failure occurs, a small pressure increment would prompt the structure failure rapidly.

Seamless coupling of bond- and state-based peridynamic and finite element analyses

This study presents a variational approach to couple PeriDynamic (PD) and Finite Element (FE) analyses to take advantage of their salient features. The PD and FE domains share the same nodes along their interface without an overlap region and any constraint conditions. Therefore, this approach does not require a morphing or a blending function that facilitates coupling over a transition zone. Although this approach is general and not limited to any commercial FE software, the PD interactions for bond-based (BB), ordinary state-based (OSB) and non-ordinary state-based (NOSB) are modeled with MATRIX27 elements native to ANSYS, a commercial software, for the purpose of demonstration. It is a general framework to perform BB, OSB and NOSB PD coupled with traditional elements. It provides the ability to practicing engineers in industry for integrating PD into their FE based design tools. The PD domains can share a boundary with FE domains or be completely embedded in the FE domain. The resulting global system of equations includes the contributions arising from the PD nodes and FE nodes. These equations are solved simultaneously without requiring an iterative procedure through implicit methods. Therefore, it is a direct (seamless) coupling approach without an overlap zone. The results are free of displacement kinks; thus, unphysical stress concentrations do not appear near the interface between the PD and FE regions. Its accuracy is demonstrated by considering isotropic elastic plates subjected to various types of boundary conditions under quasi-static and dynamic loading conditions. Also, its accuracy is demonstrated for quasi-static crack propagation by considering double cantilever beam and three-point bend test specimens. The results from the coupled PD-FE approach agree well with those of FEM for all boundary and loading conditions.

Analysis of dynamic crack propagation in two-dimensional elastic bodies by coupling the boundary element method and the bond-based peridynamics

A novel method for predicting dynamic crack propagation based on coupling the boundary element method (BEM) and bond-based peridynamics (BBPD) is developed in this work. The special feature of this method is that it can take full advantages of both the BEM and PD to achieve a higher level of accuracy and efficiency. Based on the scale of the structure and the location of cracks, the considered domain can be divided into a non-cracked region and a cracked region. For the non-cracked region, a meshfree boundary-domain integral equation method (meshfree-BDIEM) is employed in the analysis to reduce the dimension by one and to increase the computational efficiency. The bond-based PD is applied to simulate the cracked region, which can model the initiation and propagation of the cracks automatically. The boundary nodes from the BEM on the interfaces can interact with the material points from the PD directly. By using the displacement continuity and force equilibrium conditions on the interfaces, a combined model is obtained by merging the mass, stiffness, and force matrices from each region of the domain. Both the implicit and explicit BEM-PD coupled solution methods can deliver accurate results without inducing the ghost forces. Several benchmark problems of dynamic crack propagation have been modeled by using the method, which demonstrate that the developed BEM-PD coupled approach can be an efficient numerical tool to model the dynamic crack propagation problems.

Data-driven bond-based peridynamics with nonlocal influence function for crack propagation

In this paper, a data-driven method is proposed to define nonlocal influence function of bond-based peridynamics. In the proposed method, the data-driven algorithm coupling with Taylor series expansion and Adam optimization method shows strong stability and high accuracy in complex nonlocal influence function regression process. Moreover, two dataset acquisition methods including experimental dataset acquisition and the manufactured dataset are given out. Meanwhile, this algorithm is tested by the manufactured dataset, and the numerical results are in good agreement with the manufactured nonlocal influence function, which verifies the feasibility of the algorithm. Finally, two numerical examples are used to qualitatively and quantitatively illustrate its ability to regress the nonlocal influence function of bond-based peridynamics.

Investigation into the quasi-plasticity damage model for concrete with micropores based on peridynamics

The non-linear performance of concrete is determined by the heterogeneous micro-structure of concrete. The existing methods mainly focus on numerical simulation of concrete at meso-scale. In view of this, the micro-calculation model based on bond-based peridynamics (BBPD) was established by the MATLAB-ABAQUS co-simulation method. In this study, based on the concrete plastic damage model in ABAQUS, combined with the quasi-brittle damage model and the basic characteristics of BBPD, a quasi-plasticity damage model was proposed which is suitable for the simulation of concrete. The calculation results show that the micro-model based on BBPD can well reflect the mechanical properties of concrete, and the crack propagation path is affected by the heterogeneity of concrete. At the same time, the quasi-plasticity damage model was established in which the mechanical behaviour of concrete in compression/tension can be considered simultaneously. For the distribution of pores, with the reduction of large capillary pores, the crack path inside the concrete is significantly simplified, and the distribution of cracks is significantly reduced. It can improve the mechanical behaviour of the material by optimising the pore structure and reducing the distribution of large pores.

Homogenization of periodic microstructure based on representative volume element using improved bond-based peridynamics

In this paper, a homogenization method based on peridynamics is proposed, in which the rotation effect of bond is considered and the limitation of Poisson's ratio is overcome. Integral form is used as governing equation instead of differential form in peridynamics, so that it is suitable to conduct homogenization analysis involving defects. Two algorithms, general algorithm and simplified algorithm, are used to homogenize periodic materials in this study. Equivalent material properties are obtained from the displacement gradient tensor. The results are in good agreement with those of other homogenization methods. This method can also be applied to homogenize periodic materials with defects. The defects such as cracks and pores of materials are realized by breaking the bonds between material points. Through this study, a new method for calculating the equivalent properties of materials with defects can be provided.