Peridynamic Equation Of Motion Research Articles

Peridynamic correspondence model with strain gradient elasticity for microstructure dependent size effects

This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

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.

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.

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.

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.

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.

On the Computational Derivation of Bond-Based Peridynamic Stress Tensor

The concept of ‘contact stress’, as introduced by Cauchy, is a special case of a nonlocal stress tensor. In this work, the nonlocal stress tensor is derived through implementation of the bond-based formulation of peridynamics that uses an idealised model of interaction between points as bonds. The method is sufficiently general and can be implemented to study stress states in problems containing stress concentration, singularity, or discontinuities. Two case studies are presented, to study stress concentration around a circular hole in a square plate and conventionally singular stress fields in the vicinity of a sharp crack tip. The peridynamic stress tensor is compared with finite element approximations and available analytical solutions. It is shown that peridynamics is capable of capturing both shear and direct stresses and the results obtained correlate well with those obtained using analytical solutions and finite element approximations. A built-in MATLAB code is developed and used to construct a 2D peridynamic grid and subsequently approximate the solution of the peridynamic equation of motion. The stress tensor is then obtained using the tensorial product of bond force projections for bonds that geometrically pass through the point. To evaluate the accuracy of the predicted stresses near a crack tip, the J-integral value is computed using both a direct contour approximation and the equivalent domain integral method. In the formulation of the contour approximation, bond forces are used directly while the proposed peridynamic stress tensor is used for the domain method. The J-integral values computed are compared with those obtained by the commercial finite element package Abaqus 2018. The comparison provides an indication on the accurate prediction of the state of stress near the crack tip.

An extended ghost interlayer model in peridynamic theory for high-velocity impact fracture of laminated glass structures

The peridynamic theory is advantageous for problems involving damage since the peridynamic equation of motion is valid everywhere, regardless of existing discontinuities, and an external criterion is not necessary for predicting damage initiation and propagation. However, the current peridynamic modeling methods utilize regular or quasi-regular lattice structures, which poses difficulties in modeling laminated structures with alternating thin interlayers and thick ply layers. This study proposes an extended version of a nonlocal ghost interlayer model for simulating laminated structures with multi-body interactions. Using numerical studies, the proposed numerical schemes are verified. It is further confirmed that the proposed schemes are valid for simulating the high-velocity impact fracturing behavior of laminated glass.

Peridynamics simulation of surrounding rock damage characteristics during tunnel excavation

The improved peridynamics is introduced to analyze the stability of surrounding rock in tunnel excavation process. By introducing short-range repulsive force into peridynamic equation of motion, the tensile and compression failure characteristics of rock materials is simulated. In order to simulate the process of tunnel excavation, the material point dormancy method is proposed in this paper. By simulating the distribution characteristics of excavation damage zone (EDZ) during the excavation of a circular tunnel with high stress difference, the relationship between the distribution position of EDZ and the direction of maximum principal stress is revealed. The v-shaped notches developed on the tunnel periphery at the direction of perpendicular to the maximum principal stress. The predicted EDZ distribution characteristics is consistent with the results of previous studies. And the displacement field of surrounding rock after tunnel excavation is also in good agreement with FEM simulation result. The simulation results show that this method not only has good stability, but also has high computational efficiency. By analyzing the deformation, damage and failure characteristics of surrounding rock under different buried depths, different lateral pressure coefficients, different excavation methods and different section shapes, the evolution process of rock instability caused by tunnel excavation and unloading is revealed. Which provides reference for the design and optimization of surrounding rock support scheme in deep buried tunnels.

Family Member Search Algorithms for Peridynamic Analysis

Peridynamic equation of motion is usually solved numerically by using meshless approaches. Family search process is one of the most time-consuming parts of a peridynamic analysis. Especially for problems which require continuous update of family members inside the horizon of a material point, the time spent to search for family members becomes crucial. Hence, efficient algorithms are required to reduce the computational time. In this study, various family member search algorithms suitable for peridynamic simulations are presented including brute-force search, region partitioning, and tree data structures. By considering problem cases for different number of material points, computational time between different algorithms is compared and the most efficient algorithm is determined.

Numerical methods for the nonlocal wave equation of the peridynamics

In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering because seems to provide an effective approach to modeling mechanical systems avoiding spatial discontinuous derivatives and body singularities. In particular, we will consider the linear model of peridynamics in a one-dimensional spatial domain. Here we will review some numerical techniques to solve this equation and propose some new computational methods of higher order in space; moreover we will see how to apply the methods studied for the linear model to the nonlinear one. Also a spectral method for the spatial discretization of the linear problem will be discussed. Several numerical tests will be given in order to validate our results.

Peridynamic Modeling of Hyperelastic Membrane Deformation

This study concerns the development of peridynamic (PD) strain energy density functions for a Neo-Hookean type membrane under equibiaxial, planar, and uniaxial loading conditions. The material parameters for each loading case are determined by equating the PD strain energy density to that of the classical continuum mechanics. The PD equations of motion are derived based on the Neo-Hookean model under the assumption of incompressibility. Numerical results concern the deformation of a membrane with a defect in the form of a hole, a crack, and a rigid inclusion under equibiaxial, planar, and uniaxial loading conditions. The PD predictions are verified by comparison with those of finite element analysis.

Peridynamics for unguided crack growth prediction under mixed-mode loading

Peridynamics enables complex crack path prediction in structures without resorting to a special treatment of mathematical singularities as in the case of classical continuum mechanics. The peridynamic equations of motion is valid everywhere even in the presence cracks. Thus, failure process is included as part of the constitutive model for crack initiation and propagation in an unguided manner. Also, peridynamics correctly redistributes the load in the presence of complex and multiple cracks for accurate prediction of residual strength of structures. This study concerns the prediction of crack propagation paths in isotropic materials under complex loading conditions by using peridynamics. Crack growth process is controlled by a single failure parameter referred to as critical stretch which can be expressed in terms of the fracture toughness of the material. The demonstration problems concern crack growth in a four-point shear specimen and a compact tension test specimens with a sink and a miss hole. All of these applications include a pre-existing crack under mixed-mode loading conditions. The simulations capture experimentally observed non-self-similar crack propagation paths.

Peridynamic theory of solids from the perspective of classical statistical mechanics

In this paper the classical statistical mechanics has been explored in order to develop statistical mechanical framework for peridynamics. Peridynamic equation of motion is known as upscaled Newton’s equation. The peridynamic system consists of finite number of nonlocally interacting particles at nano and meso scales. This particle representation of peridynamics can be treated in terms of classical statistical mechanics. Hence, in this work the phase space is constructed based on the PD particle from their evolving momentum pi and positions xi. The statistical ensembles are derived by defining appropriate partition functions. The algorithms for NVE and NPH implemented in the classical molecular dynamics are revisited for equilibrium peridynamic models. The current work introduces Langevin dynamics to the peridynamic theory through fluctuation–dissipation principle. This introduces a heat bath to the peridynamic system which eliminates the ambiguity with the role of temperature in a peridynamic system. Finally, it was seen that the homogenization of a peridynamic model with finite number of particles approaches to a conventional continuum model. The upscaled non-equilibrium peridynamics has potential applications in modeling wide variety of multiscale–multiphysics problems from nano to macro scale or vice versa.

An extended peridynamic approach for deformation and fracture analysis

An extended bond-based peridynamic approach is proposed for quantitative analysis of elastic deformation and simulation of both quasi-static and dynamic crack propagation. A continuous function is uniquely introduced in the commonly used peridynamic constitutive model by considering the internal length effect of long-range forces. Moreover, a local damping is incorporated into the peridynamic equations of motion, and the step-by-step loading algorithms and non-equilibrium criterion for particle systems are developed specifically for quasi-static problems. The effective calculation field of material points in the numerical peridynamic discretization is discussed as well. Analysis of quantitative elastic deformation and quasi-static fracture behavior of the model is verified through simulating the uniaxial tension and compression of a slab, bending and failure of a cantilever beam, and quasi-static propagation of mode I and mode II cracks in a plate. The numerical examples show that the present approach is capable of quantitatively calculating the elastic deformation and accurately predicting the critical failure load and crack propagation paths. To further demonstrate the capabilities of the proposed model, dynamic fracture of a brittle plate under high-rate loading is simulated, in which the experimentally-observed phenomena, such as successive branching, are captured naturally as a consequence of the solution.

An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis

An extended bond-based peridynamic (PD) approach is presented for quasi-static mechanical behavior and brittle failure analysis of materials and structures. A local artificial damping is introduced into the peridynamic equations of motion, and a step-by-step loading method and a non-equilibrium criterion are employed to calculate the elastic response quantitatively, simulate the crack initiation and propagation and predict the extreme failure load of structures. The effect of the magnitude of the local damping on the accuracy and efficiency of elastic calculation is investigated. In the proposed model, another important feature lies in the micromodulus of the pair-wise bond, which is described as a continuous function of the distance between two particles within the material horizon and is demonstrated to be more accurate when compared to the earlier PD models using a fixed stiffness constant to describe the interaction of particles. The qualitative and quantitative validity of the approach is established through simulating the mechanical response and fracture of a cantilever concrete beam, including the whole process from the earlier elastic deformation stage to crack initiation and propagation, and making comparisons to the analytical and other numerical data. To further demonstrate the capabilities of the proposed model, the crack propagation and fracture mechanism of a cantilever concrete beam with pre-existing notch is studied, and the effect of the notch location on the extreme failure load and fracture mode of the structure is examined.

An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory

The peridynamic theory is advantageous for problems involving damage since the peridynamic equation of motion is valid everywhere, regardless of existing discontinuities, and an external criterion is not necessary for predicting damage initiation and propagation. However, the current solution methods for the equations of peridynamics utilize explicit time integration, which poses difficulties in simulations of most experiments under quasi-static conditions. Thus, there is a need to obtain steady-state solutions in order to validate peridynamic predictions against experimental measurements. This study presents an extension of dynamic relaxation methods for obtaining steady-state solutions of nonlinear peridynamic equations.

Force flux and the peridynamic stress tensor

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.