Peridynamic Equation Research Articles

On peridynamic acoustics

We consider a nonlocal (peridynamic) version of the classical forced wave equation. This scalar three-dimensional equation contains a weight function (the “micromodulus”) and a length parameter (the “horizon”) that have to be selected. We investigate various properties (the locality limit as the horizon shrinks, plane waves and group velocity), paying attention to how these properties depend on the choice of the micromodulus. We solve the forced peridynamic equation in the static case (avoiding divergent integrals) and in the time-harmonic case (with a radiation condition, when needed).

Re‐derivation and mathematical analysis for linear peridynamics model for arbitrary Poisson ratio's material

AbstractThis paper is concerned with the modeling and mathematical analysis of linear peridynamic model for arbitrary Poisson ratio's material. Based on the fundamental laws of dynamics, we re‐derive the bond‐based peridynamic model for anisotropic materials by relaxing certain assumptions. Through this process, we draw several significant conclusions, such as the relationship between the equivalent strain energy density hypothesis and the convergence of the peridynamic operator to the classical Navier operator. Additionally, the well‐posedness of time‐dependent peridynamic equations of motion is established. Finally, some necessary conditions for the material stability of anisotropic material are given.

IMPACT OF SPECTRAL CHARACTERISTICS OF HYPERSINGULAR OPERATORS ON SOLUTIONS OF PERIDYNAMICS PROBLEMS

Hypersingular operators play a key role in the mathematical modeling of non-local interactions, particularly in the field of peridynamics, where they provide a powerful tool for understanding material behavior under stress. This paper investigates the spectral characteristics of hypersingular operators and their impact on the solutions of peridynamic equations. Special attention is given to the spectral decomposition of these operators to gain insights into their stability, convergence, and computational efficiency in solving complex problems related to fracture mechanics and material deformation.

Iterated Crank–Nicolson Method for Peridynamic Models

In this paper, we explore the iterated Crank–Nicolson (ICN) algorithm for the one-dimensional peridynamic model. The peridynamic equation of motion is an integro-differential equation that governs structural deformations such as fractures. The ICN method was originally developed for hyperbolic advection equations. In peridynamics, we apply the ICN algorithm for temporal discretization and the midpoint quadrature method for spatial integration. Several numerical tests are carried out to evaluate the performance of the ICN method. In general, the ICN method demonstrates second-order accuracy, consistent with the Störmer–Verlet (SV) method. When the weight is 1/3, the ICN method behaves as a third-order Runge–Kutta method and maintains strong stability-preserving (SSP) properties for linear problems. Regarding energy conservation, the ICN algorithm maintains at least second-order accuracy, making it superior to the SV method, which converges linearly. Furthermore, selecting a weight of 0.25 results in fourth-order superconvergent energy variation for the ICN method. In this case, the ICN method exhibits energy variation similar to that of the fourth-order Runge–Kutta method but operates approximately 20% faster. Higher-order convergence for energy can also be achieved by increasing the number of iterations in the ICN method.

Peridynamic modelling of dynamic damage and fragmentation of cracked solids during impact contact

Dynamics damage of solids exhibit unique characteristics that are absent in static analysis. The presence of pre-existing cracks further increases the complications. To address this issue, the present study focuses on modelling dynamic damage of cracked solids during impact contact. The model is formulated based on the state-based peridynamics. It incorporates a modified damage model and employs the short-range force approach to capture the impact damage. The damping force is introduced to eliminate numerical oscillation around cracks. Then, the adaptive explicit time integration is introduced to efficiently solve the peridynamic equations. Model performance is evaluated through benchmark tests. Next, we simulate impact between two spheres with pre-existing cracks. Our focus lies in comprehending the mechanisms by which material properties and crack patterns affect damage. Further sensitivity analysis reveals that the kinetic energy of the impact system is significantly affected by the pre-existing cracks and geometric parameters of the cracks, including their number, length, angle, and distribution. The interplay of crack interaction and the gradual connection of the damaged region plays a vital role in the phenomena. The presented study offers insights into damage evolution during impact contact and provides guidelines for assessing impact damage in structures with pre-existing defects.

Peridynamic equation with boundary traction

Peridynamic equation with boundary traction

Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.

Regularity and convergence results for nonlocal peridynamic equations with truncated tensor kernels

Regularity and convergence results for nonlocal peridynamic equations with truncated tensor kernels

Peridynamic analysis of curved elastic beams

Peridynamics is an extended continuum theory which operates with non-local deformation measures as well as long range internal force/moment interactions. The aim of this paper is to present and to analyze peridynamic governing equations for elastic curved beams. To this end the strain energy density is formulated as a function of two non-local deformation measures including the axial deformation and curvature. Applying the Lagrangian formalism, peridynamic equations of motion are derived. To analyze curved beams of finite length peridynamic boundary conditions are required. Examples of boundary conditions including simple support and clamped edge are presented by introducing a fictitious domain. Solutions to PD equations of motion are presented for various problems including static analysis of simply supported and clamped arc beams as well as modal analysis of a slender ring and a simply supported arc beam. To validate peridynamic formulations, solutions for displacements, natural frequencies and vibration modes are compared with those by the classical beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the derived equations of motion and proposed boundary conditions.

Analytical solutions of peridynamic equations. Part II: Elastic wave propagation

We use the separation of variables technique to construct analytical solutions for peridynamic models of dynamic elasticity. We show that, similar to the case of peridynamic models for transient diffusion, infinite series nonlocal solutions for peridynamic elasticity can be obtained directly from the solutions of the corresponding classical model by inserting “peridynamic/nonlocal factors” in the time-exponential part of the solution. The analytical solutions show that wave dispersion, caused by nonlocality, is contained in the horizon-dependent nonlocal factor. We obtain formulas for wave dispersion and group velocities for 1D and 2D peridynamic elastic wave propagation with three commonly-used peridynamic kernels. We observe interesting complexity in nonlocal solutions, generated by nonlocal wave dispersion, and “proportional” to the size of the nonlocal interaction region. Different from the diffusion case, as time goes to infinity, the nonlocal solution for elasticity does not converge to the classical one for a fixed horizon size, meaning that nonlocal effects persist in time. We solve several examples of wave propagation with Dirichlet boundary conditions and smooth or discontinuous initial conditions and compare these analytical solutions with those corresponding to the classical model, which is seen as a particular case of the PD model for horizon equal to zero. Interestingly, we find that in PD solutions, initial discontinuities in space persist at the same location, in time. While most of the analytical solutions we present here are formal, for some of the cases, we are able to prove uniform convergence of the series solutions. This work is the first presentation of a systematic analytical treatment of peridynamic problems in 2D finite domains.

ABAQUS and ANSYS Implementations of the Peridynamics-Based Finite Element Method (PeriFEM) for Brittle Fractures

In this study, we propose the first unified implementation strategy for peridynamics in commercial finite element method (FEM) software packages based on their application programming interface using the peridynamics-based finite element method (PeriFEM). Using ANSYS and ABAQUS as examples, we present the numerical results and implementation details of PeriFEM in commercial FEM software. PeriFEM is a reformulation of the traditional FEM for solving peridynamic equations numerically. It is considered that the non-local features of peridynamics yet possesses the same computational framework as the traditional FEM. Therefore, this implementation benefits from the consistent computational frameworks of both PeriFEM and the traditional FEM. An implicit algorithm is used for both ANSYS and ABAQUS; however, different convergence criteria are adopted owing to their unique features. In ANSYS, APDL enables users to conveniently obtain broken-bond information from UPFs; thus, the convergence criterion is chosen as no new broken bond. In ABAQUS, obtaining broken-bond information is not convenient for users; thus, the default convergence criterion is used in ABAQUS. The codes integrated into ANSYS and ABAQUS are both verified through benchmark examples, and the computational convergence and costs are compared. The results show that, for some specific examples, ABAQUS is more efficient, whereas the convergence criterion adopted in ANSYS is more robust. Finally, 3D examples are presented to demonstrate the ability of the proposed approach to deal with complex engineering problems.

Localized Chebyshev and MLS collocation methods for solving 2D steady state nonlocal diffusion and peridynamic equations

The localized Chebyshev collocation method and MLS collocation method are presented to obtain the solution of two-dimensional nonlocal diffusion and peridynamic equations. The Chebyshev polynomial and MLS interpolation techniques are used to construct shape functions in the frame of the collocation method. Low computational cost and high accuracy are the main advantages of these two methods for solving nonlocal diffusion and peridynamic equations. Several numerical examples are provided to show the validity and applicability of the proposed method with the regular and irregular domains. Numerical experiments indicate that the localized Chebyshev collocation method has high accuracy for nonlocal problems with continuous solutions. The MLS collocation method is more efficient and can maintain good behavior for problems with discontinuous solutions.

Meshless numerical solution for nonlocal integral differentiation equation with application in peridynamics

The mid-point quadrature rule is often used to solve the nonlocal integral differentiation equations, in which a central material point is used to calculate the integral of the subdomain in the meshless geometry. However, it can only achieve the linear approximation and the first order convergence rate, and it is only accurate enough when the entire cell overlaps with the neighborhood of a material point. In this paper, to achieve the higher precision and the higher rate of convergence speed, a novel coupling method of reproducing kernel particle method and Gaussian-Legendre quadrature scheme is proposed to solve the nonlocal integral differentiation equations in the meshless geometry, in which the mid-point quadrature scheme is replaced with Gauss quadrature scheme to calculate the integral of the subdomain in the meshless geometry, the reproducing kernel particle method is employed to construct the shape functions for material points in the nonlocal model, and to approximate the value of the Gaussian quadrature node. Moreover, the meshless solution for the peridynamic equation is derived detailedly based on the proposed method. Compared with the traditional meshless solution for the nonlocal equations, the proposed solution has higher precision and higher rate of convergence speed in different problems. The numerical results demonstrate improved results in both mathematics and mechanical problems using the proposed method.

Some closed form series solutions to peridynamic plate equations

Peridynamics is a generalized continuum theory which is developed to account for long range internal force/moment interactions. Peridynamic equations of motion are of integro-differential type and only several closed-form analytical solutions are available for elementary structures. The aim of this paper is to derive analytical solutions to peridynamic Kirchhoff type (shear rigid) plates equations for both static and dynamic cases. Applying trigonometric series with respect to in-plane coordinates, solutions for the deflection function of a rectangular simply supported plate are derived. The coefficients in the series are presented in a closed analytical form such that the equations of motion are exactly satisfied. For the dynamic case the solution is derived applying the variable separation with respect to the time and the in-plane coordinates. Several numerical cases are presented to illustrate the derived solutions. Furthermore, results of peridynamic plate equations are compared against results obtained from the classical plate theory. A very good agreement between these two theories is obtained for the case of the small horizon sizes which shows the capability of the presented approach.

Analytical Solution of the Peridynamic Equation of Motion for a 2-Dimensional Rectangular Membrane

Analytical Solution of the Peridynamic Equation of Motion for a 2-Dimensional Rectangular Membrane

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.

Analytical Solution of 1-Dimensional Peridynamic Equation of Motion

Analytical Solution of 1-Dimensional Peridynamic Equation of Motion

Beam buckling analysis in peridynamic framework

Peridynamics is a non-local continuum theory which accounts for long-range internal force/moment interactions. Peridynamic equations of motion are integro-differential equations, and only few analytical solutions to these equations are available. The aim of this paper is to formulate governing equations for buckling of beams and to derive analytical solutions for critical buckling loads based on the nonlinear peridynamic beam theory. For three types of boundary conditions, explicit expressions for the buckling loads are presented. The results are compared with the classical results for buckling loads. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the current approach. The results show that with an increase of the horizon size the critical buckling load slightly decreases for the fixed overall stiffness of the beam.

A data‐driven bond‐based peridynamic model derived from group method of data handling neural network with genetic algorithm

AbstractIn the governing equation of motion of bond‐based peridynamics, the acceleration of a material point can be considered as the response function of all the displacements of material points in the horizon and a micro stiffness containing Young's modulus and length scale. A group method of data handling (GMDH) neural network is first developed to explicitly derive the discrete bond‐based peridynamic equation of motion based on measured data in this study rather than traditional complicated mathematical derivation. In order to discover the optimal structure more efficiently and to avoid exhaustive search, genetic algorithm is incorporated into GMDH structure. It is found that the prediction results obtained by GMDH model agree well with measured values both for training and testing data. Moreover, the derived equation of motion is expressed as the product of parameter composed of Young's modulus and length scale and linear combination of displacements of material points in the horizon, which is in accordance with the original bond‐based peridynamic formulation. Furthermore, numerical benchmarks associated with elastic deformation and crack problems are performed and compared with analytical solution or finite element analysis result to verify the validity and feasibility of the proposed model.

The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations

The radial basis function partition of unity (RBF-PU) method  was applied to obtain the numerical solution of 2D nonlocal diffusion and peridynamic problems. The main idea is to partition the original domain into several patches, use the RBF approximation on each local domain, and then give weighting to obtain the global approximation of the unknown function. The radial basis function method based on the strong form of the equation has many advantages, such as avoiding an additional layer of integral calculation, no need to deal with intersections of neighborhoods with the mesh, and easiness of implementation. The numerical results show that, this method can solve nonlocal diffusion equations and peridynamic equations accurately and efficiently.