An implicit dual-based approach to couple peridynamics with classical continuum mechanics

A general local/nonlocal implicit coupling technique called the dual-based approach is proposed to couple peridynamics (PD) with classical continuum mechanics. In the present method, physical information is transmitted mutually from local to nonlocal regions through the coupling elements; no transition region is introduced. For different mesh discretizations, two coupling methods are achieved with simplicity and effectivity. To obtain the stiffness matrix of the coupled model, without loss of generality, the implicit dual-horizon ordinary state-based peridynamic model is proposed, in which the linearization of dual-horizon ordinary state-based PD is derived and the dual assembly algorithm of the peridynamic stiffness matrix is developed. It will be seen that the implicit dual-based coupling approach provides a new implicit coupling method that is easy to implement and makes full use of the internal connection between PD and classical continuum mechanics. Several numerical examples involving static crack propagation are investigated, and the satisfactory results show both quantitative and qualitative agreement with either the analytic solution or the available experiment.     

Discretized peridynamics for linear elastic solids

Peridynamics is a theory of continuum mechanics employing a nonlocal model that can simulate fractures and discontinuities (Askari et?al. J Phys 125:012–078, 2008; Silling J Mech Phys Solids 48(1):175–209, 2000). It reformulates continuum mechanics in forms of integral equations rather than partial differential equations to calculate the force on a material point. A connection between bond forces and the stress in the classical (local) theory is established for the calculation of peridynamic stress, which is calculated by summing up bond forces passing through or ending at the cross section of a node. The peridynamic stress and the constitutive law in elasticity are used for the derivation of one- and three-dimensional numerical micromoduli. For three-dimensional discretized peridynamics, the numerical micromodulus is larger than the analytical micromodulus, and converges to the analytical value as the horizon to grid spacing ratio increases. A comparison of material responses in a three-dimensional discretized peridynamic model using numerical and analytical micromoduli, respectively, is performed for different horizons. As the horizon increases, the boundary effect is more conspicuous, and the errors increase in the back-calculated Young’s modulus and strains. For the simulation of materials of Poisson’s ratios other than 1/4, a pairwise compensation scheme for discretized peridynamics is proposed. Compared with classical (local) elasticity solutions, the computational results by applying the proposed scheme show good agreement in the strain, the resultant Young’s modulus and Poisson’s ratio.     

Discretized peridynamics for brittle and ductile solids

Peridynamics is a theory of continuum mechanics expressed in forms of integral equations rather than partial differential equations. In this paper, a peridynamics code is implemented using a graphics processing unit for highly parallel computation, and numerical studies are conducted to investigate the responses of brittle and ductile material models. Stress–strain behavior with different grid sizes and horizons is studied for a brittle material model. A comparison of stresses and strains between finite element analysis (FEA) and peridynamic solutions is performed for a ductile material. By applying the proposed procedure to bridge the material model defined for peridynamic bonds and the corresponding macroscale material model for FEA, peridynamics and FEA show good agreements as regards the stresses and strains. Copyright © 2011 John Wiley & Sons, Ltd.     

A peridynamic Reissner-Mindlin shell theory

In this work, we have developed a state-based peridynamics theory for nonlinear Reissener-Mindlin shells to model and predict large deformation of shell structures with thick wall. The nonlocal peridynamic theory of solids offers an integral formulation that is an alternative to traditional local continuum mechanics models based on partial differential equations. This formulation is applicable for solving the material failure problems involved in discontinuous displacement fields. The governing equations of the state-based peridynamic shell theory are derived based on the nonlocal balance laws by adopting the kinematic assumption of the Reissner and Mindlin plate and shell theories. In the numerical calculations, the stress points are employed to ensure the numerical stability. Several numerical examples are conducted to validate the nonlocal structure mechanics model and to verify the accuracy as well as the convergence of the proposed shell theory.     

Viscoplasticity using peridynamics

Peridynamics is a continuum reformulation of the standard theory of solid mechanics. Unlike the partial differential equations of the standard theory, the basic equations of peridynamics are applicable even when cracks and other singularities appear in the deformation field. The assumptions in the original peridynamic theory resulted in severe restrictions on the types of material response that could be modeled, including a limitation on the Poisson ratio. Recent theoretical developments have shown promise for overcoming these limitations, but have not previously incorporated rate dependence and have not been demonstrated in realistic applications. In this paper, a new method for implementing a rate‐dependent plastic material within a peridynamic numerical model is proposed and demonstrated. The resulting material model implementation is fitted to rate‐dependent test data on 6061‐T6 aluminum alloy. It is shown that with this material model, the peridynamic method accurately reproduces the experimental results for Taylor impact tests over a wide range of impact velocities. The resulting model retains the advantages of the peridynamic formulation regarding discontinuities while allowing greater generality in material response than was previously possible. Copyright © 2009 John Wiley & Sons, Ltd.     

Capturing nanoscale effects by peridynamics

Molecular dynamic simulations inevitably demand large computational resources for structures of liner measures even as small as a few tens or hundreds of nanometers. Thus, a computationally efficient method to simulate larger structures and, at the same time, retain the properties and the mechanical response at the atomic scale is in demand. One such approach is peridynamics, which is a nonlocal extension of continuum mechanics. In this study, we investigate the possibility to efficiently reproduce results from molecular dynamic (MD) simulations by calibration of two parameters inherent in peridynamics: the length scale parameter and the interparticle bond strength. The free-ware LAMMPS supports both numerical approaches, and thus LAMMPS has been used as the common framework. Beams of single-crystal fcc copper of various sizes and under tension along the crystallographic [100]- and [110]-directions act as the modeling example. The force–displacement curves and the elastic–plastic transitions have been compared between the approaches. The conclusion is that proper calibration of the peridynamic two parameters to MD simulations results in proper reproduction of the molecular dynamic results. This in turn allows for geometrical upscaling or simulation of geometrically more complicated structures, without loss of features derived from the atomic scale but to a much lower computational cost.     

Kinematic constraints in the state-based peridynamics with mixed local/nonlocal gradient approximations

In contrast to the partial differential equation in the classical continuum mechanics, the equation of motion in standard state-based peridynamics utilizes an integral form and follows an anti-symmetric relationship for the pairwise particle forces. As a consequence, the kinematic constraints such as the boundary displacements and the coupling with other numerical methods in state-based peridynamics cannot be prescribed directly on the geometric boundary for solid mechanics applications. In this paper, an enhanced variant of the state-based peridynamics for the numerical simulation of continuum mechanics problems is presented. The method is first devised based on a convex kernel approximation to localize the influence function on the boundary. A mixed local/nonlocal gradient approximation is introduced to the computation of particle equation of motion and allows a direct imposition of kinematic constraint in the analysis model. The new formulation is shown to retain the conservation nature of state-based peridynamics. Three numerical benchmarks are studied in this paper to demonstrate the effectiveness and accuracy of the proposed method.     

基于近场动力学方法的混凝土板侵彻问题研究

由于近场动力学(Peridynamics)用统一空间积分-时间微分方程描述物体连续或不连续区域,改进常用微观弹脆性模型对势本构力函数,给出刚性体与变形体冲击问题接触算法;编制计算程序,验证经典简支梁变形及Kalthoff-Winkler试验;数值模拟刚性弹丸侵彻混凝土矩形板破坏过程,揭示损伤累积及裂纹扩展全过程与最终破坏形态。结果表明,改进的近场动力学模型及算法合理、可靠,能有效模拟混凝土结构冲击破坏及侵彻问题。     

基于非局部LSM优化的近场动力学及脆性材料变形模拟

在经典近场动力学模型的基础上，通过小变形假定将近场动力学中的微模量与经典理论中的弹性常数建立联系，引入可以反映非局部作用特性的核函数提高计算精度，利用刚度等效的方式建立有关微模量的线性方程组，并通过寻求不定线性方程组最小二乘(LSM)最小范数解的方式对近场动力学中微模量进行优化，根据二次规划得到最优非负模量。利用优化后的方法对二维平板在单轴和双轴荷载作用下的变形及含预制裂纹脆性材料在荷载下的裂纹扩展进行了模拟并将结果与理论经典近场动力学方法结果对比。结果表明:优化后的方法可以较好的反映结构在荷载条件下的变形与破坏特性，与经典方法相比材料变形模拟在最大误差及误差范围具有良好的改善，并且模拟裂纹扩展过程在同等计算成本下具有更优的收敛速度及收敛结果，进一步验证了所提出方法的有效性，有着较为广泛的应用前景。     

Analysis of the plastic zone near the crack tips under the uniaxial tension using ordinary state‐based peridynamics

In this paper, the plastic model of ordinary state‐based peridynamics is established. The size and shape of plastic zone around crack tips with the different inclination angles are simulated using ordinary state‐based peridynamics. Comparison of the size and shape of plastic zone around the crack tips obtained from peridynamic solution and analytic solution is made. It is found that the relative errors between the analytical and peridynamic solution are very little. Therefore, it is feasible to predict the plastic zone around crack tips using ordinary state‐based peridynamics.     

Predicting crack propagation with peridynamics: a comparative study

The fidelity of the peridynamic theory in predicting fracture is investigated through a comparative study. Peridynamic predictions for fracture propagation paths and speeds are compared against various experimental observations. Furthermore, these predictions are compared to the previous predictions from extended finite elements (XFEM) and the cohesive zone model (CZM). Three different fracture experiments are modeled using peridynamics: two experimental benchmark dynamic fracture problems and one experimental crack growth study involving the impact of a matrix plate with a stiff embedded inclusion. In all cases, it is found that the peridynamic simulations capture fracture paths, including branching and microbranching that are in agreement with experimental observations. Crack speeds computed from the peridynamic simulation are on the same order as those of XFEM and CZM simulations. It is concluded that the peridynamic theory is a suitable analysis method for dynamic fracture problems involving multiple cracks with complex branching patterns.     

The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture

We introduce a framework that adapts local and non-local continuum models to simulate static fracture problems. Non-local models based on the peridynamic theory are promising for the simulation of fracture, as they allow discontinuities in the displacement field. However, they remain computationally expensive. As an alternative, we develop an adaptive coupling technique based on the morphing method to restrict the non-local model adaptively during the evolution of the fracture. The rest of the structure is described by local continuum mechanics. We conduct all simulations in three dimensions, using the relevant discretization scheme in each domain, i.e., the discontinuous Galerkin finite element method in the peridynamic domain and the continuous finite element method in the local continuum mechanics domain.     

基于非局部微分算子的近场动力学及固体材料应力数值模拟

在经典近场动力学模型的基础上引入非局部微分算子求解理论,建立近场动力学微弹性应力分析模型。在近场动力学模型物质点处进行泰勒级数展开,利用正交非局部函数构建微分算子的数值积分方程并且根据矩阵正交性求解函数未知系数,最终由平衡方程等价性建立近场动力学应力求解模型。采用所提出的方法对固体材料变形破坏过程中的应力进行模拟,并将计算结果与理论解对比以验证方法有效性,同时对粒子离散间距、泰勒项数及权函数的数值收敛性进行分析。结果表明:该文提出的方法可以较准确的反映完整及非完整固体脆性材料在荷载作用下的应力分布,并且离散间距及权函数对数值收敛结果具有显著影响,可为使用近场动力学方法模拟变形破坏时提供新的应力分析思路,有着较为广泛的应用前景。     

Influence of surface energy on the nanoindentation response of elastically-layered viscoelastic materials

In the context of integrated nonlinear viscoelastic contact mechanics, a nonlinear finite element model is developed to predict and analyze the quasistatic response of nanoindentation problems of an elastically-layered viscoelastic materials considering the surface elasticity effects. Effects of surface energy are accounted for by employing the Gurtin–Murdoch continuum model for surface elasticity. The linear viscoelastic response is modeled by the Schapery’s creep model with a Prony’s series to express the transient component in the creep compliance. The viscoelastic constitutive equations are cast into a recursive form that needs only the previous time increment rather than the entire strain history. To satisfy the contact constraints exactly, the Lagrange multiplier method is adopted to enforce the contact conditions into the system. The equilibrium indentation configuration is obtained through the Newton–Raphson iterative procedure. The developed model is verified then applied to investigate the quasistatic nanoindentation response of two different indentation problems with different geometry and loading conditions. Results show the significant effects of surface energy and viscoelasticity on the quasistatic nanoindentation response.     

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond‐based peridynamic model with a local elasticity model or a linearized peridynamic model away from the fracture set. The nonlocal model treated here is characterized by a double‐well potential and is a smooth version of the peridynamic model introduced in the work of Silling. The nonlinear peridynamic evolutions are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale ε of nonlocal interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation, the consistency error for the numerical approximation is found to depend on the ratio between mesh size h and ε. More generally, for local Lagrange interpolation of order p≥1, the consistency error is of order h^p/ε. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size h is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.     

A coupled thermomechanical nonordinary state‐based peridynamics for thermally induced cracking of rocks

A novel coupled thermo‐mechanical nonordinary state‐based peridynamics is proposed to study thermally induced damage in rocks. The thermal expansion characteristics of solid material are introduced into the coupled thermomechanical model to consider the influence of temperature. The deformation gradient tensor is obtained by the temperature fields, which is solved by peridynamic heat conduction theory. By introducing the deformation gradient tensor into the force state function of the nonordinary state‐based peridynamics, the coupling of thermal and mechanical is realized. A failure criterion is developed to investigate the thermally induced cracking of rocks. Then, the validity of the coupled thermo‐mechanical model is demonstrated by a numerical simulation. The correctness of the coupled model is validated by a benchmark example with analytic solution. Moreover, the thermal cracking progress in rocks is simulated using the proposed coupled nonordinary state‐based peridynamic model, and it is found that the numerical results are in good agreement with the previous experimental observations.     

A peridynamic failure analysis of fiber-reinforced composite laminates using finite element discontinuous Galerkin approximations

This paper presents a discontinuous Galerkin weak form for bond-based peridynamic models to predict the damage of fiber-reinforced composite laminates. To represent the anisotropy of a laminate in a peridynamic model, a lamina is simplified as a transversely isotropic medium under a plane stress condition. The laminated structure is modeled by stacking the surface mesh layers along the thickness direction according to the laminate sequence. To avoid a mesh dependence on either the fiber orientation or the discretization, the spherical harmonic expansion theory is employed to construct a function for the micro-elastic modulus in terms of the bond-fiber angle. The laminate material is decomposed into an isotropic matrix material part and a transversely isotropic material part. The bond stiffness can be evaluated using the engineering material constants, based on the equivalence between the elastic energy density in the peridynamic theory and the elastic energy density in the classic continuum mechanics theory. Benchmark tests are conducted to verify the proposed model. Numerical results illustrate that the convergence of simulations with different horizon sizes and meshes can be achieved. In terms of damage analysis, the proposed model can capture the dynamic process of the complex coupling of the inner-layer and delamination damage modes.     

Linearized state‐based peridynamics for 2‐D problems

In this work, the authors formulate a 2‐D linearized ordinary state‐based peridynamic model of elastic deformations and compute the stiffness matrix for 2‐D plane stress/strain conditions. This model is then verified by testing the recovery of elastic properties for given Poisson's ratios in the range 0.1–0.45. The convergence behavior of peridynamic solutions in terms of the size of the nonlocal region by comparison with the classical (local) mechanics model is also discussed. The degree to which the peridynamic surface effect influences the recovery of elastic properties is examined, and stress/strain recovery values are found to have a definite influence on the results. The technique used here can provide the basis for applying 2‐D peridynamic models to the study of fatigue failure and quasi‐static fracture problems. Copyright © 2016 John Wiley & Sons, Ltd.     

基于改进型近场动力学方法的多裂纹扩展分析

在非局部键型近场动力学理论基础上,提出了能够反映混凝土、岩石类材料力学特性和非局部长程力尺寸效应的改进型近场动力学微极模型,弥补常规微观弹脆性（Prototype Microelastic Brittle,PMB）键型近场动力学本构模型的应用范围限制和定量计算误差大等缺陷,并构建了相应的适合于模拟脆性多裂纹扩展问题的近场动力学算法体系。通过对不同核函数修正项对应的近场动力学定量计算结果进行比较,验证了改进型近场动力学模型和数值算法的计算精度并确定了精度最高的核函数修正项;模拟双裂纹脆性板受压和随机多裂纹脆性板受拉的裂纹扩展全过程并与已有结果对比,进一步验证了模型和算法在模拟多裂纹扩展问题时的可靠性。分析了含多裂纹三点弯梁的起裂和裂纹失稳扩展过程,并研究了裂纹初始倾角、初始长度等因素对构件破坏形式和破坏荷载的影响规律。