Peridynamic modeling of membranes and fibers

The peridynamic theory of continuum mechanics allows damage, fracture, and long-range forces to be treated as natural components of the deformation of a material. In this paper, the peridynamic approach is applied to small thickness two- and one-dimensional structures. For membranes, a constitutive model is described appropriate for rubbery sheets that can form cracks. This model is used to perform numerical simulations of the stretching and dynamic tearing of membranes. A similar approach is applied to one-dimensional string like structures that undergrow stretching, bending, and failure. Long-range forces similar to van der Waals interactions at the nanoscale influence the equilibrium configurations of these structures, how they deform, and possibly self-assembly.     

Linearized Theory of Peridynamic States

A state-based peridynamic material model describes internal forces acting on a point in terms of the collective deformation of all the material within a neighborhood of the point. In this paper, the response of a state-based peridynamic material is investigated for a small deformation superposed on a large deformation. The appropriate notion of a small deformation restricts the relative displacement between points, but it does not involve the deformation gradient (which would be undefined on a crack). The material properties that govern the linearized material response are expressed in terms of a new quantity called the modulus state. This determines the force in each bond resulting from an incremental deformation of itself or of other bonds. Conditions are derived for a linearized material model to be elastic, objective, and to satisfy balance of angular momentum. If the material is elastic, then the modulus state is obtainable from the second Fréchet derivative of the strain energy density function. The equation of equilibrium with a linearized material model is a linear Fredholm integral equation of the second kind. An analogue of Poincaré’s theorem is proved that applies to the infinite dimensional space of all peridynamic vector states, providing a condition similar to irrotationality in vector calculus.     

Peridynamic modeling of pitting corrosion damage

In this paper we introduce a peridynamic model for the evolution of damage from pitting corrosion capable of capturing subsurface damage. We model the anodic reaction in corrosion processes (in which electroplating is negligible) as an effective peridynamic diffusion process in the electrolyte/solid system coupled with a phase-change mechanism that allows for autonomous evolution of the moving interface. In order to simulate creation of subsurface damage, we introduce a corrosion damage model based on a stochastic relationship that connects the concentration in the metal to the damage of peridynamic mechanical-bonds that are superposed onto diffusion-bonds. We study convergence of this formulation for diffusion-dominated stage. The model leads to formation of a subsurface damage layer, seen in experiments. We validate results against experiments on pit growth rate and polarization data for pitting corrosion. We extend the 1D model to the 2D and 3D, and introduce a new damage-dependent corrosion model to account for broken mechanical bonds that enhance the corrosion rate. This coupled model can predict the pit shape and damage profile in materials with microstructural heterogeneities, such as defects, interfaces, inclusions, and grain boundaries.     

Peridynamic States and Constitutive Modeling

A generalization of the original peridynamic framework for solid mechanics is proposed. This generalization permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. This extends the types of material response that can be reproduced by peridynamic theory to include an explicit dependence on such collectively determined quantities as volume change or shear angle. To accomplish this generalization, a mathematical object called a deformation state is defined, a function that maps any bond onto its image under the deformation. A similar object called a force state is defined, which contains the forces within bonds of all lengths and orientation. The relation between the deformation state and force state is the constitutive model for the material. In addition to providing a more general capability for reproducing material response, the new framework provides a means to incorporate a constitutive model from the conventional theory of solid mechanics directly into a peridynamic model. It also allows the condition of plastic incompressibility to be enforced in a peridynamic material model for permanent deformation analogous to conventional plasticity theory.      

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.     

A Peridynamic Model of Fracture Mechanics with Bond-Breaking

Wave propagation in elastic dielectrics with flexoelectricity, micro-inertia and strain gradient elasticity is investigated in this paper. Dispersion phenomenon, which does not exist in classical elastic dielectric theory, is observed in the flexoelectric microstructured solids. Analytical solutions for the phase velocity \(C_{p}\), group velocity \(C_{g}\) and their ratio \(\gamma = C_{g} / C_{p}\) are calculated for the case of harmonic decomposition. The magnitudes of the phase velocity and group velocity changed with the increasing of the wave number, while they are constant in the classical elastic dielectric theory. It is shown that the flexoelectricity, micro-inertia and microstructural effects are significant to predict the real behavior of longitudinal wave propagating in flexoelectric microstructured solids. Microstructural effects are not sufficient for dealing with realistic dispersion curves in flexoelectric solids, the micro-inertia and flexoelectricity are needed to obtain a physically acceptable value of the phase and group velocities.     

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.     

破片群作用下复合材料层合板近场动力学损伤模拟

采用一种新兴的无网格法——近场动力学理论,模拟复合材料结构在破片群载荷作用下的损伤情况。根据复合材料结构受到载荷的特性,总结破片群冲击作用下复合材料结构损伤特性,分析其破坏过程,研究破片群增强效应,并对破片速度、破片数量、破片群间距对侵彻能力增强效应的影响进行分析。结果表明:层合板结构在高速破片群侵彻作用下损伤模式多样,与破片数量、速度、间距相关;破片数量的增加,对破片群侵彻能力增强效应明显;破片间距与破片群侵彻能力增强效应负相关,破片间距减小,破片群损伤效应提高;破片速度直接决定穿透时间,破片速度的提高使得穿透时间缩短,应力波的叠加效应不足以影响破片群的侵彻能力。     

动载作用下复合型裂纹扩展的近场动力学模拟

构建改进的非局部近场动力学模型和粒子系统动力加载算法,开展复合型裂纹动力扩展过程模拟。在常规近场动力学微极模型中引入能反映非局部长程力尺寸效应的核函数修正项,以提高计算精度和收敛稳定性。通过模拟动载作用下含复合型裂纹混凝土三点弯梁的裂纹扩展与破坏过程并与试验结果对比,验证了本文模型和算法的可靠性。在此基础上,通过分组模拟分析了动载作用下,初始裂纹的位置、长度和方向对含复合型裂纹三点弯梁破坏模式、起裂时间与承载能力的影响规律。     

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.     

层合板冲击损伤的PD率效应本构模型分析

从近场动力学(简称PD)理论的PMB材料模型出发,结合Kelvin-Voigt粘弹性固体模型,建立PD率效应本构模型。采用LAMMPS软件模拟了环氧树脂板、纤维增强复合材料单向层板和多向铺层层合板受冲击的情况。通过分析各板的冲击损伤,探索纤维对板的增强作用。同时,分析了不同冲击速度下层合板上下表面的损伤程度,初步探讨了从低速碰撞到高速冲击过程中复合材料层合板的破坏机理及规律。     

复合材料层合板冲击损伤近场动力学模型与分析

近场动力学(简称PD)理论通过域内积分建立物质基本运动方程。不同于传统理论中通过微分建立运动方程的方法,该理论对场函数没有连续性的要求,因而适合求解各类不连续问题。基于此,本文建立了正交各向异性单层板PD理论模型,进而引入单层板层间作用,发展了正交各向异性层合板PD模型及其损伤模型,并模拟了各向同性与各向异性层合板冲击损伤;通过对比分析,对模型的有效性进行了验证。     

破片冲击作用下舰船复合材料结构损伤的近场动力学模拟

基于近场动力学方法,综合分析了破片的速度、层合板的铺层方式、加筋板的筋条尺寸和破片相对筋条的冲击位置对结构损伤模式和破片剩余速度的影响。结果显示：高速破片冲击作用下,层合板会发生侵彻和穿透现象,层合板的损伤模式以基体损伤为主,且随着破片冲击速度的增大,板上下表面的损伤区域呈现出一种先增大后减小的趋势;高速破片冲击作用下,层合的板损伤扩展方向和纤维铺设方向有关,对于纤维铺层方向相同的层合板,其上下表面的损伤扩展方向一般与纤维方向相同;加筋板通过增加少量质量可以获得比层合板更好的抗破片冲击性能,且加筋板的筋条尺寸和破片相对筋条的冲击位置对加筋板的损伤具有明显影响。

     

Peridynamic modelling of impact damage in three-point bending beam with offset notch

The nonlocal peridynamic theory has been proven to be a promising method for the material failure and damage analyses in solid mechanics.Based upon the integrodifferential equations,peridynamics enables predicting the complex fracture phenomena such as spontaneous crack nucleation and crack branching,curving,and arrest.In this paper,the bond-based peridynamic approach is used to study the impact damage in a beam with an offset notch,which is widely used to investigate the mixed I-II crack propagation in brittle materials.The predictions from the peridynamic analysis agree well with available experimental observations.The numerical results show that the dynamic fracture behaviors of the beam under the impact load,such as crack initiation,curving,and branching,rely on the location of the offset notch and the impact speed of the drop hammer.     

基于显微CT图像的近场动力学建模与复合材料微结构破坏模拟

随着纤维增强复合材料应用领域的不断扩展且用量激增,亟需理清复合材料微观结构损伤对宏观力学性能影响的内在机制。因此,发展针对纤维增强复合材料微结构破坏过程的建模与高效模拟方法就显得十分重要。本文借助显微CT(Micro-computed Tomography)扫描技术,提出了一种基于显微CT图像中像素点离散的近场动力学建模与模拟方法。一方面,近场动力学作为一种由积分方程建模的非局部理论,便于采用基于空间点离散的数值计算方法,相比传统的连续介质力学能够更有效地模拟材料从连续变形到裂纹萌生与扩展(非连续变形)的全过程。另一方面,对显微CT图像使用像素点灰度阈值分割处理技术,能够快速建立含有复合材料原位微结构信息的空间点离散模型。该离散模型可以直接用于微结构破坏过程的近场动力学模拟,从而避免了传统的数值模拟技术需要依据像素点先建立光滑的几何模型、再划分成有限单元网格的复杂前处理过程,并且极大地保留了复合材料的原位组分分布信息。数值模拟结果表明,基于显微CT图像的近场动力学建模方法能够精确捕捉到复合材料微结构信息,并能准确模拟纤维增强复合材料的微结构破坏过程。     

混凝土梁三点弯曲断裂模式转变的近场动力学模拟

混凝土结构的宏观损伤开裂与其非均质微观结构紧密相关。底部带切口的混凝土梁在进行三点弯曲破坏时,随着切口的位置由梁中向梁边转移,裂纹由从切口处萌生并生长转变为从梁的中部萌生。本文采用半均质化近场动力学（IH-PD）模型和全均质化近场动力学（FH-PD）模型,分别对混凝土梁三点弯断裂问题进行模拟研究。IH-PD模型根据混凝土中骨料体积分数随机生成不同键的组合方式,将微观尺度的非均质性引入模型,无需详细描绘骨料形状和分布即可考虑混凝土非均质性。本文将IH-PD与FH-PD模型得到的断裂模式随切口位置的变化关系,与实验结果对比,分析微观结构对混凝土梁开裂的影响;基于非均质材料特征尺寸与IH-PD模型网格参数的相关性,模拟骨料大小对混凝土梁断裂模式的影响;另外,通过在IH-PD模型中设置预损伤的方式引入随机分布的孔隙,探讨孔隙率对混凝土断裂模式的影响。     

近场动力学框架下钢轨疲劳裂纹萌生预测的数值方法研究

经典连续介质力学在求解由裂纹引起的不连续问题时,会出现数学构架失效的情况.为克服这一难题,基于近场动力学理论,构建铁路钢轨疲劳裂纹萌生的数值预测方法,可实现钢轨疲劳裂纹萌生寿命与位置的预测.当未出现疲劳裂纹时,通过与经典连续介质力学模型的结果对比,验证近场动力学模型的正确性和适用性.分析了车轮全滑动、黏着-滑动和无摩擦三种状态对钢轨疲劳裂纹萌生的影响规律,结果表明:车轮由全滑动向无摩擦转变的过程中,裂纹萌生位置由钢轨表层转移到内部,裂纹萌生所需的荷载循环次数由0.45×10⁷次增至2.05×10⁷次,可见车轮滚滑状态会影响裂纹的萌生位置,并且较大的切向接触应力会显著降低钢轨的疲劳裂纹萌生寿命.