Peridynamics Research Articles

A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain non-local gradients of higher-order accuracy. We show, however, that the improved quadrature rule does not suffice to handle instability issues that have proven problematic for the correspondence model-based PD. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability without introducing artificial stabilization parameters. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, as the degree of non-locality (i.e., the horizon) approaches zero. Problems from linear elastostatics are utilized to verify the accuracy and stability of our approach. It is shown that the bond-associative approach improves the robustness of RK-PD and GMLS-PD formulations, which is essential for practical applications. The higher-order, bond-associated model can obtain second-order convergence for smooth problems and first-order convergence for problems involving field discontinuities, such as curvilinear free surfaces. In Part II of this paper we use our unified PD framework to (a) study wave propagation phenomena, which have proven problematic for the state-based correspondence PD framework; (b) propose a new methodology to enforce natural boundary conditions in correspondence PD formulations, which should be particularly appealing to coupled problems. Our results indicate that bond-associative methods accompanied by higher-order gradient corrections provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

Multi-scale peridynamic formulations for chloride diffusion in concrete

A multi-scale peridynamic (PD) model of concrete with three material phases is developed to study the influences of the properties of the interfacial transition zone (ITZ) on chloride diffusion. In the PD model, the quadtree algorithm is employed to generate the multi-scale grid and two correction schemes based on the background grid are developed to calibrate the volume and length of PD bonds. A comparison with the uniform grid PD model and the finite element model is carried out to validate the proposed multi-scale PD model. The results of the multi-scale PD model are consistent with that of the two models. In addition, the effects of ITZ properties, including ITZ diffusivity and ITZ thickness, on chloride diffusion are investigated. These results demonstrate that the penetration depth and the chloride concentration increase with the increasing ITZ diffusivity and ITZ thickness, and the increase rate of chloride concentration decreases with the exposed time.

Crack development assessment using modal analysis in peridynamic theory

AbstractIf structural damage remains undetected and is allowed to grow, structure's load-bearing capacity deteriorates, which can lead to costly repairs or in extreme cases its collapse. Modal analysis is widely used to detect structural damage because, when damage, such as cracks, is introduced, structure's geometrical and/or mechanical properties change, and these changes can be used for damage detection. Peridynamics is a non-local alternative to the continuum mechanics theory that represents forces and displacements using integral equations, which are defined even with discontinuous displacement fields, thus making this theory an attractive option for damage modeling. In this paper, authors verify peridynamic (PD) modal analysis against finite-element (FE) results, and validate it against experimental modal analysis results. The modal solver was implemented in the open-source program Peridigm and four different damage configurations were considered for verification and validation. The results show close agreement between the PD and the FE results, and the PD and the experimental results. Moreover, PD modal frequencies are shown to have similar accuracy to experimental data as the FE results. It is also shown that the frequency shifts are comparable between all three types of modal analysis. The PD mode shapes agreed well with both the FE and the experimental mode shapes at all considered damage configurations. Furthermore, the change in mode shapes from the introduced damage is similar in all three analyses.

Abaqus implementation of dual peridynamics for brittle fracture

A dual peridynamics (PD) for brittle fracture is implemented in the commercial software Abaqus by means of UEL/VUEL and UMAT/VUMAT subroutines, thereby, the peridynamic finite element method (PDFEM) will come into being. To this end, the concept of the peridynamic node and element is defined, subsequently, the peridynamic element internal force vector and mass matrix for the explicit PDFEM and the peridynamic element stiffness matrix for the implicit PDFEM are derived and constructed for the explicit and the implicit finite element implementation of the dual peridynamics in Abaqus, respectively. Additionally, a dual-based coupling approach is developed to couple the PDFEM and FEM to optimize the computational efficiency and accuracy of the pure PDFEM. Several numerical examples involving the static and dynamic crack propagation are investigated and the satisfactory results show the availability of the PDFEM and the hybrid model between PDFEM and FEM. Moreover, the finite element implementation of peridynamics in Abaqus may pave the way to an increased application of peridynamics to the practical engineering problems.

From atomic-scale to mesoscale: A characterization of geopolymer composites using molecular dynamics and peridynamics simulations

A multiscale approach that couples molecular dynamics (MD) and peridynamics (PD) simulations was implemented to study the mechanical behavior of geopolymer composites (GC) consisting of geopolymer binder (GB), calcium silicate hydrate (CSH) and quartz. Using MD, intrinsic mechanical properties such as bulk modulus, shear modulus and strain energy release rates were determined for the constituent phases as well as the respective two-phase systems. The MD generated properties were then used to determine the input parameters for PD simulations. Both bond-based and state-based PD models were considered for simulating the GC. The first part of the PD study consisted of characterizing the properties of mesoscale porous GB. Interestingly, the predicted modulus of porous GB at porosities corresponding to experimental values compared well with experimental observations based on nanoindentation (NI). NI tests were also conducted using PD simulation to predict the hardness of GB, which was also consistent with experimental data. Finally, PD simulations of mesoscale GC consisting of multiple GB, CSH and quartz (010) domains showed that the presence of quartz (010) increases its strength. It was also observed that the strength of GC increased with increasing domain-sizes of the constituent phases, which can be compared to inverse Hall-Petch effects in nano-crystalline materials.

A Unified, Stable, and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part II: Wave Propagation and Enforcement of Stress Boundary Conditions

The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain non-local gradients of higher-order accuracy. We show, however, that the improved quadrature rule does not suffice to handle instability issues that have proven problematic for the correspondence model-based PD. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability without introducing artificial stabilization parameters. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, as the degree of non-locality (i.e., the horizon) approaches zero. Problems from linear elastostatics are utilized to verify the accuracy and stability of our approach. It is shown that the bond-associative approach improves the robustness of RK-PD and GMLS-PD formulations, which is essential for practical applications. The higher-order, bond-associated model can obtain second-order convergence for smooth problems and first-order convergence for problems involving field discontinuities, such as curvilinear free surfaces. In Part II of this paper, we use our unified PD framework to (a) study wave propagation phenomena, which have proven problematic for the state-based correspondence PD framework, and (b) propose a new methodology to enforce natural boundary conditions in correspondence PD formulations, which should be particularly appealing to coupled problems. Our results indicate that bond-associative methods accompanied by higher-order gradient corrections provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

A smoothed particle hydrodynamics–peridynamics coupling strategy for modeling fluid–structure interaction problems

Fluid–structure interaction (FSI) is a multiphysics problem with diverse application scenarios ranging from the aeroelasticity of aircraft wings and the structural response of marine platforms, to blood circulation through aortic valves. Solving the FSI problems is challenging, predominantly because of the complex time-variant geometry in the fluid–structure interface. To alleviate the difficulties encountered by the grid-based methods in tracking and meshing the fluid–structure interface involving large deformation and structure failures, in this study, we propose a meshfree framework that couples smoothed particle hydrodynamics (SPH) with peridynamics (PD) for the numerical modeling of FSI problems. This SPH–PD framework involves a four-step partitioned coupling procedure, where the key point is utilizing the SPH moving ghost particles as a medium. An original data transfer scheme is developed based on the partnerships built between the SPH moving ghost particles and the PD particles. The structure failures are incorporated in the PD model by adopting a damage law of maximum bond stretch. The SPH–PD method is validated against existing experimental and numerical data. Through the use of this model, we successfully simulate the low-velocity fluid–structure impact and capture the large deformation with local failures.

Peridynamic modeling of composite laminates with material coupling and transverse shear deformation

This study presents a new state-based PeriDynamic (PD) model of a composite laminate with arbitrary laminate layup; it captures all types of material couplings in the presence of transverse shear deformation. It consists of normal and shear bonds instead of fiber and matrix bonds to allow arbitrary fiber orientations. Also, rotational degrees of freedom are included in the equilibrium equations in order to evaluate the transverse shear angle and curvature. Mindlin-Reissner plate theory, employed in this model, permits the use of existing composite constitutive law which leads to the coupling between different deformation modes. In addition, a quasi-local boundary condition applied only to the first row of material points avoids the approximation of nonlocal boundary values. The accuracy of this model is verified against benchmark solutions, and validated by comparison with experimental results.

Dynamic Crack Propagation and Its Interaction With Micro-Cracks in an Impact Problem

AbstractThe dynamic fracture behavior of brittle materials that contain micro-level cracks should be examined when material subjected to impact loading. We investigated the effect of micro-cracks on the propagation of macro-cracks that initiate from notch tips in the Kalthoff–Winkler experiment, a classical impact problem. To define predefined micro-cracks in three-dimensional space, we proposed a two-dimensional micro-crack plane definition in the bond-based peridynamics (PD) that is a non-local form of classical continuum theory. Randomly distributed micro-cracks with different number densities in a constant area and number in expending area models were examined to monitor the toughening of the material. The velocities of macro-crack propagation and the time required for completing fractures were considered in several predefined micro-cracks cases. It has been observed that toughening mechanism is only initiated by exceeding a certain number of micro-cracks; therefore, there is a positive correlation between the density of predefined micro-cracks and macro-crack propagation rate and, also, toughening mechanism.

The computational framework for continuum-kinematics-inspired peridynamics

Peridynamics (PD) is a non-local continuum formulation. The original version of PD was restricted to bond-based interactions. Bond-based PD is geometrically exact and its kinematics are similar to classical continuum mechanics (CCM). However, it cannot capture the Poisson effect correctly. This shortcoming was addressed via state-based PD, but the kinematics are not accurately preserved. Continuum-kinematics-inspired peridynamics (CPD) provides a geometrically exact framework whose underlying kinematics coincide with that of CCM and captures the Poisson effect correctly. In CPD, one distinguishes between one-, two- and three-neighbour interactions. One-neighbour interactions are equivalent to the bond-based interactions of the original PD formalism. However, two- and three-neighbour interactions are fundamentally different from state-based interactions as the basic elements of continuum kinematics are preserved precisely. The objective of this contribution is to elaborate on computational aspects of CPD and present detailed derivations that are essential for its implementation. Key features of the resulting computational CPD are elucidated via a series of numerical examples. These include three-dimensional problems at large deformations. The proposed strategy is robust and the quadratic rate of convergence associated with the Newton–Raphson scheme is observed.

Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models

Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.

Peridynamic simulation of finite elastic deformation and rupture in polymers

This study concerns the finite elastic deformation and rupture in styrene-butadiene rubber (SBR) and Polydimethylsiloxane (PDMS) sheets under quasi-static loading conditions. The weak form of peridynamic (PD) equilibrium equation is derived based on Anand’s model and Talamini-Mao-Anand (TMA) model for polymers with slight compressibility. The nonlocal deformation gradient tensor is computed in a bond-associated domain of interaction using the PD differential operator. This approach is free of oscillations and spurious zero energy modes that are commonly observed in the PD correspondence models. Also, it permits the direct imposition of natural and essential boundary conditions. The fidelity of this PD model for predicting large deformation, progressive damage and rupture is established by comparison with experimental measurements and previous simulations of SBR and PDMS sheets with defects under displacement controlled tensile loading.

A new fatigue reliability analysis method for steel bridges based on peridynamic theory

A fatigue reliability model based on the peridynamic (PD) theory was proposed to analyze the fatigue performance of steel bridges in this study. Different from previous methods based on classical continuum mechanics, the PD model does not require additional criteria to guide the growth of crack or damage. In particular, the influence of fatigue threshold value was considered in a systematic formula, and the response surface method was utilized to analyze fatigue failure probability. The proposed model was applied to complex fatigue phenomena such as spontaneous crack nucleation, crack branching, and fatigue failure under biaxial cyclic loadings where fatigue crack paths interact in complicated ways. The fatigue crack growth pattern, fatigue life, and fatigue reliability of specimens considering the biaxial or uniaxial fatigue loadings were analyzed. The accuracy of the fatigue model was verified through a series of comparisons with the experimental, analytical, and FEM results. The effectiveness of the proposed method was also demonstrated in the application to the fatigue reliability analysis of the Ting Kau Bridge.

A chirality-dependent peridynamic model for the fracture analysis of graphene sheets

Based on the bond-based peridynamic (BPD) method, a chirality-dependent peridynamic (CDPD) model is proposed for the fracture analysis of single layer graphene sheets (SLGS). The peridynamic (PD) parameters in the CDPD model are derived by analyzing the stress-strain relations of SLGS at atomic scale, and thus the CDPD model can provide a multiscale insight into the fracture analysis of SLGS with considering the effect of chirality. In order to increase the efficiency of CDPD simulations of large atomic-scale systems, a special coarse-grain (CG) technique is employed by considering the chiral structure of SLGS in the CDPD simulations. The effect of grid type, grid orientation and grid spacing on the numerical convergence is investigated. The proposed CDPD model is validated by comparing with the available molecular dynamics (MD) results, and the comparisons demonstrate the importance of considering the atomistic structure of SLGS when PD theory is applied in the fracture analysis because the chirality dominates the propagating directions of cracks. The crack propagations in SLGS with various chiralities are studied using the CDPD model. The numerical results indicate that the fracture of SLGS is significantly dependent on the chirality. Especially, examples of large-sized SLGS (750 nm × 750 nm) with about 20 million atoms in the corresponding fully atomistic systems demonstrate the applicability and effectiveness of the proposed CDPD model in the simulation of microscale structures of SLGS without loss of chiral features. To study such large systems is almost impractical for the fully atomistic simulation. This study also provides a significant insight for extending the PD model to the studies on chirality-related materials at microscale.

A Peridynamic Model for Analyzing Fracture Behavior of Functionally Graded Materials Used as an Interlayer

Functionally graded materials (FGMs) can be used as an interlayer. The interlayer connects two homogeneous materials and slows mutations in material properties, owing to the variability of parameters and microstructure. In this numerical simulation, a bond-based peridynamic (PD) model was used to perform dynamic fracture analysis of a continuous beam and a sandwiched structure with an FGM interlayer. Some simulation results of two convergence studies, which include grid refinement (m-convergence) and the reducing radius of PD horizon ( $$\delta $$ -convergence), were discussed. Under a four-point bending load, the propagation behaviors of single pre-crack in the continuous beam with the FGM interlayer at different locations are simulated to compare with the experimental results. For the sandwiched structure, the problems of a single crack and double cracks were considered. The numerical results show that the crack can deflect toward areas where the energy release rate is low. Additionally, the crack on the weak side for symmetric cracks only has fracture behavior, and if the double cracks are asymmetrical, all these cracks have fracture behaviors and propagation of the initial crack in the stiff region has a higher sensitivity than in the compliant region.

Peridynamic micromechanical modeling of plastic deformation and progressive damage prediction in dual-phase materials

In this study, a state-based Peridynamic (PD) microstructure model is developed to simulate the damage initiation and propagation and plastic deformation in multi-phase materials. The dual-phase (DP) steel is utilized as a material example. A two-dimensional PD model is developed to predict the response of three representative volume elements (RVEs) under tension static loading. RVEs are extracted from Scanning electron microscopy (SEM) images by image processing. Von Mises yield criterion with hardening is used to model plastic deformation. Failure criteria are achieved by relating PD total work needed for eliminating the interactions to the critical energy release rate. A new approach is also proposed for boundary conditions and assigning interface region properties. The failure mechanisms such as martensite cracking, ferrite, and martensite interface decohesion, are captured in the analyses. It is also found that damage initiation occurs in narrow cross-section of martensite grains, ferrite/martensite interfaces, and trapped ferrite phases. The predicted deformations, stress-strain curves, and damage patterns are also compared with the available experimental results. It is shown that the achieved results are in good agreement with the experimental results.

An approach of peridynamic modeling associated with molecular dynamics for fracture simulation of particle reinforced metal matrix composites

A peridynamic (PD) model that involves the interface constitutive relation constructed by molecular dynamics (MD) is proposed to simulate the mechanical properties of the particle reinforced metal matrix composites and reproduce the similar failure mode in the experiments. In our numerical simulation, the constitutive models of PD bond are proposed and employed to simulate the Al-Al and SiC-SiC systems. Then, a MD simulation is carried out to characterize the Al-SiC interface. The predicted results are validated by comparing with the existing researches. After that, a concluded constitutive model of interface bond is applied to PD simulations. The performance of the modified theoretical models is demonstrated by two quasi-static simulations with and without considering the strengthening effects that SiC inclusions have on Al-matrix. Besides, the proposed model is also applied to predict the failure behaviors of the overall composite system successfully. The obtained results indicate that the constructed approach is applicative in investigating both the quasi-static and dynamic problems.

A peridynamic approach to computation of elastic and entropic interactions of inclusions on a lipid membrane

The physics behind the interaction of inclusions on lipid membranes is relevant to the self-assembly of proteins that mediate exo- and endo-cytosis in cells. These interactions can be quantified in terms of free energy changes that can be calculated using Gaussian integrals since the potential energy of a lipid membrane with inclusions can be expressed as a quadratic form in terms of a stiffness matrix that includes the coupling of in-plane tension and out-of-plane deflections. The free energy calculation requires the evaluation of the determinant of a stiffness matrix appearing in the energy expression. However, computing the determinant of an extremely large stiffness matrix whose size is dictated by the characteristic length scale of the membrane and its molecular constituents poses computational challenges. This study presents a peridynamic (PD) approach to construct the stiffness matrix and evaluate its determinant accurately without any computational challenges. It specifically employs the PD differential operator for accurate evaluation of the determinant of the stiffness matrix which is a key step in the calculation of the partition function. The simultaneous use of fine and coarse grids along with PD interpolations eliminates the computational challenges for decreasing grid spacing. This is illustrated by reproducing the worm-like-chain force-extension relation for a fluctuating elastic rod. In the case of a membrane without any inclusions, the PD predictions converge and approach an analytical solution for the tension-area relation of a membrane as the grid spacing decreases. In the presence of inclusions, the PD predictions capture the expected deformation of the membrane as well as the behavior of relative free energy for varying spacing, shape and position of the inclusions. The magnitude of relative free energy increases with increasing number of inclusions, thus indicating higher degree of interaction in larger clusters. This approach is general and enables the exploration of the effect on membrane-inclusion interactions of different membrane geometries with curvature, boundary conditions and the contact angle between the membrane and inclusion.

A Peridynamic Computational Scheme for Thermoelectric Fields.

Thermoelectric materials are materials that involve the coexistence of heat flux and electric current in the absence of magnetic field. In such materials, there is a coupling among electric potential and temperature gradients, causing the thermoelectric effects of Seebeck and Peltier. Those coupling effects make the design and analysis of thermoelectric materials complicated and sophisticated. The main aim of this work is dealing with thermoelectric materials with discontinuities. Since heat and electric fluxes are undefined at the crack tip and the temperature and electric fields across the crack surface are discontinuous, it is better to apply peridynamic (PD) theory to capture such details at the crack tips. Hence, we propose in this paper a PD theory which is suitable in tackling such discontinuities in thermal and electric fields. In this study, the continuum-based electrical potentials and temperature fields are written in the form of nonlocal integrals of the electrical potentials and temperature that are effective whether we have discontinuities or not. To illustrate the consistency of the peridynamic technique, a number of examples were presented and witnessed that PD results were in good agreement with those results from the literature, finite element solutions and analytical solutions.

Axisymmetric peridynamic analysis of crack deflection in a single strand ceramic matrix composite

This study employs Peridynamic (PD) theory to predict crack deflection in axisymmetric ceramic matrix composites (CMC). It specifically employs the weak form of PD equations of motion which enables the direct imposition of natural and essential boundary conditions. The results indicate that the critical stress ratio between the coating and matrix have a strong influence on crack deflection in a CMC. Also, it was observed that the smaller this ratio is, the earlier the crack deflection in the fiber coating occurs.