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.

基于近场动力学理论的裂纹模型波场数值模拟及频散特性分析

近场动力学理论采用积分形式的运动方程,能够有效避免空间偏导数不存在的问题,在模拟不同材料发生裂纹扩展、断裂破坏等不连续问题方面有着广泛应用,但是对近场动力学理论中波的传播问题和频散特性研究较少。为了研究近场动力学理论中应力波的传播规律,考虑平面应力情况,建立积分形式的运动方程,在均匀各向同性模型及含水平裂纹模型下进行了波场数值模拟,并与经典连续介质理论中的纵波和横波波场进行对比分析。基于常规态型近场动力学理论的运动方程,详细推导了一维和二维情况下的频散关系,并探讨了邻域内物质点数、邻域半径和形状参数等近场动力学参数对频散曲线的影响。在波场数值模拟中,线性近场动力学理论中固体与经典连续介质理论中线弹性介质的波场数值模拟结果具有良好的一致性,介质内裂纹的存在会阻碍波的传播;在频散特性分析中,所选邻域半径越小、形状参数越大,线性近场动力学理论中的频散特性越接近经典连续介质力学理论中的频散特性。

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.

一种近场动力学非普通状态理论零能模式控制方法

近场动力学非普通状态理论在采用节点积分时将引起零能模式,造成位移场、应力应变场的数值不稳定性,影响计算精度甚至会导致完全错误的结果,因此必须对其进行控制.目前国际上还没有十分有效的零能模式控制方法.本文针对零能模式问题,提出了一种通用的、高效的控制方法.根据近场动力学线性键理论,确定非均匀变形对应弹性张量的具体形式,考虑了微模量随不同作用键的变化.通过最小位能原理推导出非均匀变形引起的力状态,结合近场动力学力状态,得到稳定力状态表达式.从而建立起基于线性键理论的稳定关联材料模型,并应用于含圆孔平板、三点弯试件线弹性变形和损伤破坏过程模拟.数值结果表明,本文模型能有效抑制近场动力学非普通状态理论中的零能模式现象.与已有零能模式控制方法相比,其物理意义明确,不包含控制参数,避免了复杂的零能模式参数调节过程,提高了计算效率.

冰柱冲击问题的数值仿真分析

为研究近场动力学(PD)方法在冰力学行为领域的应用特性及在冰破坏数值预报过程中的参数敏感性,采用常规状态型近场动力方法系统地计算分析了冰柱冲击破坏过程,并进行了参数敏感分析.结果表明:该方法模拟冰冲击过程与试验对比结果基本一致,在选定的时间步长和粒子间距下本文的计算结果收敛.海冰冲击速度、泊松比、弹性模量对海冰冲击过程影响明显,而海冰尺寸和断裂韧度对冲击过程影响较小.文章创新之处在于将状态型PD方法应用于海冰冲击问题中,弥补了键型PD方法限制海冰泊松比的缺点.

非常规态型近场动力学热黏塑性模型及其应用

在非常规态型近场动力学(non-ordinary state-based peridynamics,NOSB-PD)理论框架下构建了考虑应变率效应、塑性硬化、热软化效应和材料断裂特征的非局部三维热黏塑性固体本构模型以及相应的非局部空间积分型数值算法,并应用于金属类材料和构件在冲击载荷作用等工况下的高应变率热黏塑性变形与破坏分析.通过对经典含初始裂纹Kalthoff-Winkler板冲击试验进行三维近场动力学模拟,可得到裂纹的起裂角度、扩展路径、扩展速度以及裂纹扩展过程中靶板等效应力和温度分布,所得结果与已有试验结果和其他数值方法结果吻合较好.在此基础上,应用该模型分析了不同冲击速度作用下金属靶板的变形与裂纹扩展过程,结果表明:该模型能较好地模拟不同冲击速度(应变率)情况下靶板的变形与破坏全过程.随着冲击速度变化,初始裂纹的起裂时间、扩展方向和扩展速度呈一定规律变化.冲击速度越低,起裂时间越晚(直至冲击速度低于某值时初始裂纹不扩展),裂纹扩展速度峰值越低,冲击过程中靶板温度峰值越低,完全扩展所需时间越长.

基于近场动力学的二维疲劳裂纹扩展模型

为模拟二维疲劳裂纹的扩展,建立了一种基于普通态基近场动力学的疲劳裂纹扩展模型.首先,在普通态基近场动力学模型的基础上,利用动态松弛算法得到最大循环载荷下键的应变值,并通过计算物质点的平均键应变值确定裂纹尖端位置及损伤区域;然后,在模拟过程中设置循环断键数,当损伤区域内的断键数达到循环断键数时,更新键的应变值并进行下一轮运算;最后,分别通过模拟紧凑拉伸试样和中心斜裂纹板的疲劳裂纹扩展来验证模型的有效性.结果表明,设置合适的循环断键数后,模型计算效率得到提高,且能获得精细的裂纹扩展路径和准确的裂纹扩展速率,所得裂纹扩展路径及a-N曲线均与试验结果一致.

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.