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.

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.

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.

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.

Peridynamic Modeling of Brittle Fracture in Mindlin-Reissner Shell Theory

In this work,wemodeled the brittle fracture of shell structure in the framework of Peridynamics Mindlin-Reissener shell theory,in which the shell is described by material points in themean-plane with its drilling rotation neglected in kinematic assumption.To improve the numerical accuracy,the stress-point method is utilized to eliminate the numerical instability induced by the zero-energy mode and rank-deficiency.The crack surface is represented explicitly by stress points,and a novel general crack criterion is proposed based on that.Instead of the critical stretch used in common peridynamic solid,it is convenient to describe thematerial failure by using the classic constitutive model in continuum mechanics.In this work,a concise crack simulation algorithm is also provided to describe the crack path and its development,in order to simulate the brittle fracture of the shell structure.Numerical examples are presented to validate and demonstrate our proposed model.Results reveal that our model has good accuracy and capability to represent crack propagation and branch spontaneously.

Peridynamic含初始缺陷中心平行双裂纹扩展分析

为了更好地模拟裂纹扩展行为,一种基于非局部理论——近场动力学被提出,PD采用积分的方法避免了传统方法存在的问题,并采用相邻质点间距离和临界延伸率sc的变化来描述物体受力变化规律,更好地模拟了裂纹扩展及多裂纹融合。本文基于PD模拟了没有缺陷不同间距的中心平行双裂纹扩展路径,然后采用降低质点间键的强度,模拟了含随机分布初始缺陷(气孔,夹杂,未熔合)的中心平行双裂纹扩展路径。得到了初始缺陷对不同纵向间距中心平行双裂纹扩展融合及背离路径的影响规律。

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.

基于态基近场动力学的碳化硅陶瓷圆柱壳体断裂损伤分析

碳化硅(SiC)陶瓷材料具有优异的力学性能,广泛应用于各种工程部件中.然而,陶瓷是脆性材料,当发生脆断时会造成巨大的危害.因此对其断裂行为的准确预测显得极为重要.为了研究SiC陶瓷壳体在外载荷作用下的断裂损伤机理,首先通过准静态拉伸和压缩实验获得SiC陶瓷的临界能量释放率,建立了适用于拉伸和压缩损伤分析的常规态基近场动力学模型.然后,采用Fortran语言编程实现了陶瓷壳体在线性递增外载荷作用下断裂损伤行为的数值仿真,并将所获得的仿真值与理论预测值进行对比,验证了所建模型的准确性.最后,对仿真结果进行详细分析,揭示了壳体结构的损伤扩展演化规律.仿真结果表明:初始损伤区源于壳体内壁面的压缩断裂,且沿环向呈90°均匀分布.随着载荷的逐渐增加,壳体外壁面出现拉伸断裂损伤区,内外壁面的损伤区先沿着周向扩展,再向着壳体中面缓慢延伸,并在厚度方向贯穿整个壳体.随后,在壳体轴线方向出现拉-压损伤混合区,该损伤区沿着轴向扩展,最终将壳体分成4个断裂区.所建模型能够有效地描述复杂载荷作用下SiC陶瓷结构的断裂损伤演化过程,同时弥补了态基近场动力学在断裂分析方面的不足.

基于近场动力学的复杂界面双材料结构研究

为了将近场动力学方法应用于复杂界面的双材料结构,以锯齿形界面为例,基于PD“串联”键理论开发复杂界面双材料结构PD求解程序,通过单、双材料平板进行拉伸仿真分析,利用FEM解验证PD方法及程序的准确性。结果表明:PD方法在准静态模拟中具有较高的准确性。PD程序所得位移结果:在y=0.005 m处,x位移相对误差低于5%;在x=0 m处,y位移平均相对误差低于6.08%。程序所得位移结果在小模量比、小载荷下更准确。

纤维-陶瓷复合板抗弹机理及PD理论分析

纤维—陶瓷复合靶板能够满足防护轻量化的需求,是目前军、警、航天等各类防护中轻质防护材料研究和使用的趋势。基于此,探讨防护结构研制中常用到的抗弹用陶瓷的特点、纤维类型和力学性能以及陶瓷和纤维各自在抗弹过程中的作用形式,应用近场动力学理论的微势能、耗散应变能推论复合结构的总能量耗散和损伤形成之间的关系,得到与弹靶作用相关的近场动力学模型运动轨迹,并通过对比抗弹机能的实验结果与近场动力学解析结果,证实纤维约束包覆陶瓷材料可以提升整体结构的抗弹效应,能充分发挥陶瓷的抗弹作用。

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.

尺寸相关挠曲电效应问题的PDDO方法研究

基于一致性偶应力理论描述压电介质的力学性能以及挠曲电效应的尺寸依赖行为,针对该尺寸相关挠曲电效应问题的四阶偏微分方程及其定解条件,采用近场动力学微分算子法(PDDO)转化为非局部积分形式,并开发相应的算法程序,对2个典型算例进行了数值求解。通过计算结果与解析结果对比,验证了该方法的有效性与准确性,并分析了特征长度系数及挠曲电系数对挠曲电效应的影响规律。

基于近场动力学的爆炸抛撒研究

有限元方法在处理复杂裂纹问题时存在固有缺陷,基于非局部理论的近场动力学弥补了这一短板。为了将近场动力学方法应用于爆炸抛撒研究,针对固体燃料爆炸抛撒问题,采用近场动力学键理论,在合理假设的基础上提出了基于近场动力学的二维爆炸抛撒模型。进一步基于MATLAB平台开发了数值计算程序,并验证了程序在预测抛撒药云雾分布方面的有效性。最后研究了圆形、方形中心装药结构对爆炸抛撒结果的影响。仿真结果表明:程序所得16 ms内云雾分布形状及2 ms内云雾扩散速率较为准确;不同装药结构会影响云雾扩散速度及云雾分布形状;圆形装药更有利于抛撒药的分散。

近场动力学研究进展

近场动力学(Peridynamics或PD)理论基于非局部作用思想,采用空间积分描述物质内部作用,对于从连续到非连续、微观到宏观的力学行为具有统一的表述,数值上天然具有无网格属性和不连续求解功能,在分析不连续,多尺度等问题时展现出了具有优势的适用性和可靠性.本文介绍了近场动力学的发展背景;概述了其理论基础、数值实现过程和计算体系,并在此基础上探讨了近场动力学理论和数值模型的适定性,以及与传统连续介质模型和分子动力学模型进行耦合的可行性;系统分析了近场动力学方法在各个领域上的应用发展现状和趋势,包括静态、动态破坏问题,基于近场动力学的材料模型,以及新兴的疲劳问题研究和多尺度、多物理场的耦合问题;最后对近场动力学方法目前存在的局限性和将来的研究进行了探讨.

基于近场动力学理论的层压板损伤分析方法

提出了一种基于近场动力学理论的纤维增强复合材料层压板的渐进损伤分析方法.在弹性力学和复合材料力学的基础上,推导了适用于近场动力学建模的微模量和临界伸长率等基本参量,结合经典层压板理论中的偏轴模量,构建了适用于各向异性材料的对点力函数,可分析3种形式的损伤:纤维断裂,基体开裂和分层破坏.分析了含圆孔层压板在拉伸载荷作用下的破坏过程,预测结果与试验结果吻合良好.

多物理场耦合作用分析的近场动力学理论与方法

广义来说,近场动力学(peri-dynamics, PD)是假设每个物质点在承受一定范围内的非接触相互作用下,研究整个物理系统演化过程的理论,为涉及非连续和非局部相互作用的问题提供了一个统一的数学框架,具有广泛的适用性.在简要介绍诸多工程对于多物理场模型和数值计算软件的迫切需求后,针对现有商用软件在处理结构非连续演化问题时遇到的瓶颈,引入近场动力学理论和方法.概述近场动力学固体力学模型,系统阐述近场动力学扩散模型和近场动力学多物理场耦合建模的研究现状和进展,主要涉及电子元器件、电子封装和岩土工程领域的多物理场耦合建模,包括热–力、湿–热–力、热–氧、热–力–氧、力–电、热–电、力–热–电、多孔介质的水–力流固相互作用等非耦合、半耦合与完全耦合模型,强调发展耦合方程数值解法的重要性.最后对扩散问题和多物理场耦合问题的近场动力学理论模型、数值算法和工程应用做进一步展望.

高速运动裂纹扩展和分叉现象的近场动力学数值模拟

首先介绍了近场动力学的基本理论,然后以两个实例分析了高速运动裂纹的扩展及分叉现象.分析了近场动力学参数(邻域半径、相邻节点距)及外部参数(材料的弹性模量、密度、温度改变量)等对裂纹分叉的速度和角度的影响并进行了对比分析,数值结果表明:随着邻域半径的增大,裂纹传播速度逐渐减少而裂纹分叉角度逐渐增加;随着相邻节点间距的增加,裂纹的传播速度逐渐减少而裂纹分叉角度也逐渐减少;裂纹分叉长度偏向于弹性模量小和密度大的材料;裂纹传播速度随着弹性模量差值的增大而增大,随着密度差值的减小而增大,同时随着外界温度改变量的增大而减少.近场动力学能自发地模拟裂纹扩展和分叉,不需要借助任何外部准则,不需要预先设置裂纹扩展路径,因此它具有天然的优势.