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.      

Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation

In this paper, we carry out further mathematical studies of nonlocal constrained value problems for a peridynamic Navier equation derived from linear state-based peridynamic models. Given the nonlocal interactions effected in the model, constraints on the solution over a volume of nonzero measure are natural conditions to impose. We generalize previous well-posedness results that were formulated for very special kernels of nonlocal interactions. We also give a more rigorous treatment to the convergence of solutions to nonlocal peridynamic models to the solution of the conventional Navier equation of linear elasticity as the horizon parameter goes to zero. The results are valid for arbitrary Poisson ratio, which is a characteristic of the state-based peridynamic model.     

近场动力学研究进展与岩石破裂过程模拟

近场动力学（Peridynamics，PD）作为一种新兴的非局部性理论，在非连续处不需要任何处理，能够很好表述模型从连续到非连续的过程。首先，在PD基本理论简介的基础上，系统回顾了PD的国内外研究现状。其次，采用键型PD理论对非均匀性的圆孔岩板单轴拉伸破裂过程进行了二维数值模拟，采用态型PD理论对单轴、常规三轴以及真三轴等不同压缩条件下的岩石破裂过程进行了三维数值模拟，并以加拿大Mine-by隧洞为例对现场岩体破裂过程进行了模拟，结果表明PD在岩石破裂过程模拟上具有较强适用性。最后，指出当前PD在岩石破裂过程模拟中存在的主要问题和未来值得开展的若干研究课题。     

近场动力学在断裂力学领域的研究进展

近场动力学采用非局部积分计算节点内力, 利用统一数学框架描述空间连续与非连续, 避免了非连续区局部空间导数引起的应力奇异, 数值上具有无网格属性, 可自然模拟材料结构的断裂问题. 本文概述了近场动力学的弹性本构力模型, 系统介绍了近场动力学临界伸长率、临界能量密度以及材料强度相关的键失效准则. 详细介绍了近场动力学在断裂力学领域的研究进展, 包括断裂参数能量释放率与应力强度因子的求解、J积分、混合型裂纹、弹塑性断裂、黏聚力模型、动态断裂、材料界面断裂以及疲劳裂纹扩展等. 最后讨论了断裂问题近场动力学研究的发展方向.      

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.     

非局部弹性理论概述及在当代材料背景下的一些进展

局部作用原理在发展经典连续介质力学的本构关系中起着重要的作用，由此导出的简单物质理论得到了广泛的应用．然而，随着科技的发展，各种具有微结构的新材料不断涌现，理论和实验表明，非局部理论可以更好地刻画这些材料的宏观力学行为．本文简要介绍了一些传统的非局部弹性理论，包括Eringen 理论、Kunin 理论、Mindlin 理论；阐述了针对复合材料发展的，具有时间-空间非局部特征的Willis 方程、最新的时间-空间耦合非局部弹性动力学理论以及近场动力学理论．时间-空间非局部理论反映了复合材料宏观性能固有的非局部特征，而具有空间非局部特征的近场动力学理论便于处理具有不连续性的问题．最后，本文讨论了非局部理论的发展中值得关注的一些问题.     

基于近场动力学微分算子的含裂纹正交各向异性板热传导分析方法

采用近场动力学微分算子(Peridynamic Differential Operator, PDDO)理论建立正交各向异性板热传导的非局部模型。通过构造近场动力学函数,将边界条件和热传导方程由局部微分形式转化为非局部积分形式,引入Lagrange乘子,采用变分分析对含裂纹正交各向异性板温度及裂纹尖端的热通量分布进行求解。通过对比算例,验证了该模型具有较好的收敛性和有效性。分析了正交各向因子、材料铺设角、裂纹倾角及间距对裂纹尖端热通量的影响。结果表明,基于PDDO建立的含裂纹正交各向异性板热传导模型,考虑了热传导问题中的非局部性,能有效提高计算精度,预测含裂纹板中裂纹尖端出现的奇异性。     

Adaptive coupling between damage mechanics and peridynamics: A route for objective simulation of material degradation up to complete failure

The objective (mesh-independent) simulation of evolving discontinuities, such as cracks, remains a challenge. Current techniques are highly complex or involve intractable computational costs, making simulations up to complete failure difficult. We propose a framework as a new route toward solving this problem that adaptively couples local-continuum damage mechanics with peridynamics to objectively simulate all the steps that lead to material failure: damage nucleation, crack formation and propagation. Local-continuum damage mechanics successfully describes the degradation related to dispersed microdefects before the formation of a macrocrack. However, when damage localizes, it suffers spurious mesh dependency, making the simulation of macrocracks challenging. On the other hand, the peridynamic theory is promising for the simulation of fractures, as it naturally allows discontinuities in the displacement field. Here, we present a hybrid local-continuum damage/peridynamic model. Local-continuum damage mechanics is used to describe “volume” damage before localization. Once localization is detected at a point, the remaining part of the energy is dissipated through an adaptive peridynamic model capable of the transition to a “surface” degradation, typically a crack. We believe that this framework, which actually mimics the real physical process of crack formation, is the first bridge between continuum damage theories and peridynamics. Two-dimensional numerical examples are used to illustrate that an objective simulation of material failure can be achieved by this method.     

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.     

Peridynamic beams: A non-ordinary,state-based model

This paper develops a new peridynamic state based model to represent the bending of an Euler–Bernoulli beam. This model is non-ordinary and derived from the concept of a rotational spring between bonds. While multiple peridynamic material models capture the behavior of solid materials, this is the first 1D state based peridynamic model to resist bending. For sufficiently homogeneous and differentiable displacements, the model is shown to be equivalent to Eringen’s nonlocal elasticity. As the peridynamic horizon approaches 0, it reduces to the classical Euler–Bernoulli beam equations. Simple test cases demonstrate the model’s performance.     

含初始裂纹巴西圆盘劈裂问题的非局部近 场动力学建模

巴西圆盘劈裂是弹性力学及岩石力学与工程中的经典问题。在非局部键型近场动力 学理论的基础上,引入物质点对的转动自由度构建双参数微观弹脆性近场动力学本构力模型 以突破常规模型的应用范围限制,并考虑岩石混凝土类材料的宏观拉压异性和断裂特征。引 入动态松弛、粒子系统力边界条件和系统平衡弛豫等算法,实现了含不同倾角中心裂纹巴西 圆盘受压劈裂破坏全过程的近场动力学数值模拟,裂纹扩展路径及破坏形式均与试验结果高 度吻合,为裂纹扩展和断裂破坏问题的数值模拟提供了新的选择。     

Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.      

Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod

Peridynamics is a nonlocal theory of continuum mechanics, which was developed by Silling (2000). Since then peridynamics has been applied to a variety of solid mechanics problems ranging from fracture, damage, failure to wave propagation, buckling, and detonation physics. Since the governing equation of peridynamics is an integro-differential equation, most of the treatment in the literature is often numerical. However, the analytical treatment is very important for the development of the peridynamic theory, which is continually developing at the present time. In this paper, peristatic and peridynamic problems for a 1D infinite rod are analytically investigated. We have developed a method to obtain a valid analytical solution starting from a formal analytical solution, which may be divergent. The primary contribution of the present paper is a systematic analytical treatment of peristatic and peridynamic problems for a 1D infinite rod. Additionally, dispersion curves and group velocities for the materials with three different micromoduli are also studied. It is found from the study that some peridynamic materials can have negative group velocities in certain regions of wavenumber. This indicates that peridynamics can be used for modeling certain types of dispersive media with anomalous dispersion such as the one discussed by Mobley (2007).     

水力劈裂问题的态型近场动力学建模

将态型近场动力学理论引入水力劈裂问题的模拟。构建了能反映岩土类材料准脆性断裂特征的态型近场动力学本构模型,并在物质点间相互作用力模型中加入等效水压力项,以实现在新生裂纹面上跟踪施加水压力。同时,考虑裂纹面间的接触,引入物质点间的短程排斥力作用,并设计了相应的接触算法。通过自编程序将模型和算法应用于含初始裂纹、不含初始裂纹以及含坝基软弱结构面的混凝土重力坝在高水头作用下的水力劈裂过程模拟,并与扩展有限元等模拟结果对比,验证了本文模型和算法的可行性和准确性。     

Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations

Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations, including crack propagation and strain localization is developed. By this approach, a peridynamic model capable of effectively treating strong and weak discontinuities is superimposed in the critical regions over an underlying finite element mesh placed over the entire problem domain. A rigorous variational framework of coupling local finite element and nonlocal peridynamics approximations that is free of blending parameters is developed. Several numerical examples involving mixed-model fracture, three-dimensional adaptive crack propagation and strain localization induced ductile failure demonstrate the rational and efficiency of the proposed superposition-based coupling approach.

     

Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics

We consider the nonlocal formulation of continuum mechanics described by peridynamics. We provide a link between peridynamic evolution and brittle fracture evolution for a broad class of peridynamic potentials associated with unstable peridynamic constitutive laws. Distinguished limits of peridynamic evolutions are identified that correspond to vanishing peridynamic horizon. The limit evolution has both bounded linear elastic energy and Griffith surface energy. The limit evolution corresponds to the simultaneous evolution of elastic displacement and fracture. For points in spacetime not on the crack set the displacement field evolves according to the linear elastic wave equation. The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media. The elastic moduli, wave speed and energy release rate for the evolution are explicitly determined by moments of the peridynamic influence function and the peridynamic potential energy.     

Deformation of a Peridynamic Bar

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venant's principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Green's function for general loading problems.     

A nonlocal diffuse interface model for microstructure evolution of tin-lead solder

Microstructural length scales are relatively large in typical soldered connections. A microstructure which is continuously evolving is known to have a strong influence on damage initiation and propagation in solder materials. In order to make accurate lifetime predictions by numerical simulations, it is therefore necessary to take the microstructural evolution into account. In this work this is accomplished by using a diffuse interface model incorporating a strongly nonlocal variable. It is presented as an extension of the Cahn-Hilliard model, which is weakly nonlocal since it depends on higher order gradients which are by definition confined to the infinitesimal neighbourhood of the considered material point. Next to introducing a truly nonlocal measure in the free energy, this nonlocal formulation has the advantage that it is numerically more efficient. Additionally, the model is extended to include the elastically stored energy as a driving force for diffusion after which the entire system is solved using the finite element approach. The model results in a computational efficient algorithm which is capable of simulating the phase separation and coarsening of a solder material caused by combined thermal and mechanical loading.     

Multiscale Dynamics of Heterogeneous Media in?the?Peridynamic Formulation

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodology is a novel two-scale evolution equation. The rescaled solution of this equation is shown to provide a strong approximation to the actual deformation inside the peridynamic material. The two scale evolution can be split into a microscopic component tracking the dynamics at the length scale of the heterogeneities and a macroscopic component tracking the volume averaged (homogenized) dynamics. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations. The equations show that the forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation.     

考虑相变的近场动力学热?力耦合模型及多孔介质冻结破坏模拟

多孔介质的传热传质现象广泛存在于自然界和工业领域中. 低温条件可能导致多孔介质中的组分发生相变, 并由此诱发材料损伤, 甚至导致结构失效破坏. 对这类破坏现象的预测需要精细化建模, 以能够反映物质的相变过程和材料的破坏特征. 本文采用热焓法改写经典的热传导方程, 在近场动力学框架下, 建立了一种考虑物质相变的热?力耦合模型, 发展了交错显式求解的数值计算方法, 进行了方板角冻结、热致变形和多孔介质冻结破坏等问题的模拟, 得到了方板的冻结特征、温度场和变形场的分布规律以及多孔介质的冻结破坏过程, 与试验和其他数值方法的结果具有较好的一致性. 研究表明, 本文所建立的考虑物质相变的近场动力学热?力耦合模型能够反映材料的非局部效应和物质相变潜热的影响, 准确捕捉相变过程中液固界面的演化特征, 再现多孔介质中材料相变、基质热致变形和冻结破坏过程, 突破了传统连续性模型求解这类破坏问题时面临的瓶颈, 为深入研究多孔介质冻融破坏过程和破坏机理提供了有效途径.