The Boundary Element Method for Ordinary State-Based Peridynamics

The peridynamics(PD),as a promising nonlocal continuum mechanics theory,shines in solving discontinuous problems.Up to now,various numerical methods,such as the peridynamic mesh-free particlemethod(PD-MPM),peridynamic finite element method(PD-FEM),and peridynamic boundary element method(PD-BEM),have been proposed.PD-BEM,in particular,outperforms other methods by eliminating spurious boundary softening,efficiently handling infinite problems,and ensuring high computational accuracy.However,the existing PD-BEM is constructed exclusively for bond-based peridynamics(BBPD)with fixed Poisson’s ratio,limiting its applicability to crack propagation problems and scenarios involving infinite or semi-infinite problems.In this paper,we address these limitations by introducing the boundary element method(BEM)for ordinary state-based peridynamics(OSPD-BEM).Additionally,we present a crack propagationmodel embeddedwithin the framework ofOSPD-BEM to simulate crack propagations.To validate the effectiveness of OSPD-BEM,we conduct four numerical examples:deformation under uniaxial loading,crack initiation in a double-notched specimen,wedge-splitting test,and threepoint bending test.The results demonstrate the accuracy and efficiency of OSPD-BEM,highlighting its capability to successfully eliminate spurious boundary softening phenomena under varying Poisson’s ratios.Moreover,OSPDBEMsignificantly reduces computational time and exhibits greater consistencywith experimental results compared to PD-MPM.

A Non-Ordinary State-Based Peridynamic Formulation for Failure of Concrete Subjected to Impacting Loads

Strain hardening and strain rate play an important role in dynamic deformation and failure problems such as high-velocity impact cases.In this paper,a non-ordinary state-based peridynamic model for failure and damage of concrete materials subjected to impacting condition is proposed,taking the advantages of both damage model and nonlocal peridynamic method.The Holmquist-Johnson-Cook(HJC)model describing the mechanical character and damage of concrete materials under large strain,high strain rate and high hydrostatic pressure was reformulated in the framework of non-ordinary statebased peridynamic theory,and the corresponding numerical approach was developed.The proposed model and numerical approach were validated through simulating typical impacting examples and comparing the results with available experimental observations and results in literature.

The singularity in the state-based peridynamic solution of uniaxial tension

We solve the local uniaxial tension of an infinite rod in the framework of non-ordinary state-based peridynamics.The singular solutions of stress and displacement are acquired.When the influencing range of the window function approaches zero,these two solutions will return to the solutions of the classical elasticity.The analysis shows that the singularities of the solutions stem from such a feature of the window function that must be represented by a rapidly decreasing function in physics.Contrary to the classical elasticity,the stress solution of peridynamics is smoother than the displacement solution.In addition,a criterion used to select the window function is proposed in this paper.

A State-Based Peridynamic Formulation for Functionally Graded Euler-Bernoulli Beams

In this study,a new state-based peridynamic formulation is developed for functionally graded Euler-Bernoulli beams.The equation of motion is developed by using Lagrange’s equation and Taylor series.Both axial and transverse displacements are taken into account as degrees of freedom.Four different boundary conditions are considered including pinned support-roller support,pinned support-pinned support,clamped-clamped and clamped-free.Peridynamic results are compared against finite element analysis results for transverse and axial deformations and a very good agreement is observed for all different types of boundary conditions.

Finite Element Convergence for State-Based Peridynamic Fracture Models

We establish the a priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models.We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction.The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential.The hydrostatic potential can either be a quadratic function,delivering a linear force–strain relation,or a multi-well type that can be associated with the material degradation and cavitation.We first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space H2.We show that the finite element approximations converge to the H2 solutions uniformly as measured in the mean square norm.For linear continuous fi nite elements,the convergence rate is shown to be Ct Δt+Csh2/ε2,where𝜖is the size of the horizon,his the mesh size,and Δt is the size of the time step.The constants Ct and Cs are independent of Δt and h and may depend on ε through the norm of the exact solution.We demonstrate the stability of the semi-discrete approximation.The stability of the fully discrete approximation is shown for the linearized peridynamic force.We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.

Modelling of Contact Damage in Brittle Materials Based on Peridynamics

As a typical brittle material,glass is widely used in construction,transportation,shipbuilding,aviation,aerospace and other industries.The unsafe factors of glass mainly come from its rupture.Thus,establishing a set of prediction models for the cracks growth of glass under dynamic load is necessary.This paper presents a contact damage model for glass based on the ordinary state-based peridynamic theory by introducing a contact force function.The Hertz contact(nonembedded contact)problem is simulated,and the elastic contact force is determined by adjusting the penalty factor.The proposed model verifies the feasibility of penalty-based method to simulate the contact problem of glass.The failure process of glass specimen under impact is simulated,where two loading methods,the drop ball test and the split Hopkinson pressure bar are considered.Numerical results agree well with the experimental observations,thereby verifying the effectiveness of the proposed model.

Bond-associated non-ordinary state-based peridynamic model for multiple spalling simulation of concrete

The non-ordinary state-based peridynamic(NOSB PD)model has the capability of incorporating existing constitutive relationships in the classical continuum mechanics.In the present work,we first develop an NOSB PD model corresponding to the Johnson–Holmquist II(JH-2)constitutive damage model,which can describe the severe damage of concrete under intense impact compression.Besides,the numerical oscillation problem of the NOSB PD caused by zero-energy mode is analyzed and hence a bond-associated non-ordinary state-based peridynamic(BA-NOSB PD)model is adopted to remove the oscillation.Then,the elastic deformation of a three-dimensional bar is analyzed to verify the capability of BA-NOSB PD in eliminating the numerical oscillation.Furthermore,concrete spalling caused by the interaction of incident compression wave and reflected tension wave is simulated.The dynamic tensile fracture process of concrete multiple spalling is accurately reproduced for several examples according to the spalling number and spalling thickness analysis,illustrating the approach can well simulate and analyze the concrete spalling discontinuities.

Meso-Scale Modeling for Effective Properties in Continuous Fiber-Reinforced Composites by State-Based Peridynamics

This study demonstrates a homogenization approach via a modified state-based peridynamic(PD)method to predict the effective elastic properties of composite materials with periodic microstructure.The procedure of modeling the PD unit cell(UC)of continuous fiber-reinforced composite is presented.Periodic boundary conditions are derived and implemented through the Lagrange multiplier method.A matrix-dominated approach for modeling the interphase properties between dissimilar materials is proposed.The periodicity and continuity assumptions are employed to determine the stress and strain fields,as well as the effective elastic properties.The PD-UCs of square and hexagonal packs as well as the 0/90 laminate microstructure are modeled and compared with the analytical,numerical and experimental results from the literature.Good agreement of predicted effective properties can be observed.Unlike other PD homogenization approaches,the effective material properties can be directly and individually obtained from simple loading conditions.

A Fast State-Based Peridynamic Numerical Model

The peridynamic(PD)theory is a reformulation of the classical theory of continuum solid mechanics and is particularly suitable for the representation of discontinuities in displacement fields and the description of cracks and their evolution in materials,which the classical partial differential equation(PDE)models tend to fail to apply.However,the PD models yield numerical methods with dense stiffness matrices which requires O(N^(2))memory and O(N^(3))computational complexity where N is the number of spatial unknowns.Consequently,the PD models are deemed to be computationally very expensive especially for problems in multiple space dimensions.State-based PD models,which were developed lately,can be treated as a great improvement of the previous bond-based PD models.The state-based PD models have more complicated structures than the bond-based PD models.In this paper we develop a fast collocation method for a state-based linear PD model by exploring the structure of the stiffness matrix of the numerical method.The method has an O(N)memory requirement and computational complexity of O(N log N)per Krylov sub-space iteration.Numerical methods are presented to show the utility of the method.

Peridynamic Meso-scale Modeling for Degradation in Transverse Mechanical Properties of Composites with Micro-void Defects

The effect of micro-void on transverse stiffness and strength for fiber-reinforced composites subjected to load perpendicular to fiber is studied using the state-based peridynamic(PD)theory.The heterogeneous microstructure with micro-voids are discretized with irregular and non-uniform grids on mesoscale.The PD representative volume element with randomly distributed and sized micro-voids has been established with periodic boundary conditions.A parametric study was performed to evaluate the effects of void fraction and void size on transverse properties.It was found that the transverse modulus decreases with the void fraction in the range of 0.5–2%.A smaller void has a greater impact on the transverse modulus than a larger void under the same void content.The crack growth path for the multi-fiber model subjected to transverse tension was analyzed.The PD predictions show that cracks initiate at the interface between fiber and matrix,then deflect to the micro-voids nearby.The PD predictions match well with the analytical and experimental observations available in the literature.