双轴动载作用下脆性裂纹扩展问题的近场动力学建模与分析

在双参数微极近场动力学弹脆性模型基础上,引入反映长程力尺寸效应的核函数修正项以提高定量计算精度和收敛稳定性。通过定量变形计算和双轴动载作用下含中心裂纹准脆性板的裂纹扩展过程模拟,验证了本文模型与算法。应用本文模型能准确反映动载作用下的裂纹起裂、扩展、分叉与二次分叉等现象。进一步分析了双轴动载作用下含不同倾角、不同间距平行双裂纹脆性板的破坏模式、起裂时间与承载能力,以及含不同倾角多裂纹脆性板的破坏机制。结果表明:初始裂纹倾角与间距对脆性板承载能力有显著影响,在平行双裂纹倾角一定时,间距越大脆性板承载能力越强;双轴动载作用下多裂纹扩展时,裂纹之间首先相互连接且无分叉,在初始裂纹连接贯通后向边界扩展时出现分叉及二次分叉。     

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。     

基于键基近场动力学理论的单裂纹圆孔板冲击破坏研究

基于键基近场动力学理论,建立分离式霍普金森杆冲击单裂纹圆孔板动力学模型,其中霍普金森杆用一维PD模型、单裂纹圆孔板用二维PD模型描述,采用短程斥力模型描述碰撞过程,模拟冲击压缩条件下单裂纹圆孔板动态破坏行为。通过试样端面受力分析得到端面载荷的V形分布规律,解决了传统实验-数值研究法在端面加载上的局限性。研究不同入射速度下试件裂纹的扩展过程和破坏模式,准确捕获了裂纹起裂、止裂及二次起裂时间。根据键基近场动力学理论下材料动态应力强度因子计算方法,求得裂纹的起裂韧度,为材料动态断裂韧度计算提供了新的途径。     

含单边缺口混凝土梁冲击破坏的近场动力学模拟

改进了近场动力学（peridynamics,PD）微观弹脆性（prototype microelastic brittle,PMB）模型中微观模量的计算方法,解决了PMB模型的“边界效应”问题。利用改进的PMB模型计算了冲击荷载作用下单边缺口混凝土梁的破坏全过程,得到了混合型裂纹扩展的角度、路径以及裂尖最大速度,并与试验和其他数值方法模拟结果进行了对比分析,验证了改进的PMB模型的有效性。研究结果表明,近场动力学方法在模拟破坏问题时不存在网格依赖性,其本构模型自然包含了损伤与断裂的描述,是求解结构冲击破坏问题的一种有效方法。     

单向复合材料随机裂纹核扩展过程概率分析

以纤维呈六边形分布的单向复合材料为研究对象,结合局部载荷分配法则,提出了随机裂纹核理想扩展过程,给出了基于理想扩展过程的随机裂纹核扩展概率算法,并对随机裂纹核断裂纤维由1根扩展到多根的概率进行了算例分析。通过与基于Markov过程的计算结果比较,表明基于理想扩展过程的随机裂纹核扩展概率算法具有较高的精度。该算法化繁为简,便于考虑裂纹扩展过程中多根纤维同时断裂这一因素。计算表明：忽略多根纤维同时断裂的算法会使随机裂纹核扩展概率计算结果产生较大的误差,而考虑多根纤维同时断裂的算法可以提高裂纹扩展概率的计算精度,从而有利于提高复合材料强度的预测精度。     

岩体多裂纹扩展演化过程数值流形方法研究

岩体中含有大量节理、裂隙、断层等各类结构面,结构面在应力作用下的扩展与贯通是导致岩体破坏的重要原因。数值流形方法 (NMM)可以有效模拟连续和非连续问题,然而,其在多裂纹动态扩展的模拟方面仍处于探索阶段。该文以线弹性断裂力学原理为基础,提出了一种基于高阶数值流形方法的多裂纹扩展模拟算法。通过在基函数中增加关键项来考虑裂纹尖端位移场的奇异性;裂纹尖端的应力强度因子则采用了J积分来计算;Ⅰ型-Ⅱ型混合裂纹的开裂和扩展方向依据最大周向拉应力准则来判断;采用假设-修正的多裂纹扩展算法解决了多裂纹的扩展问题。根据强化后的基函数,对于不符合单纯形积分形式的被积函数,采用了泰勒级数展开式计算近似解。通过多个静态裂纹扩展的经典问题的数值模拟对计算方法的合理性和计算精度及进行了验证。     

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。     

近场动力学方法研究复合材料失效的进展

复合材料与单一材料相比具有更高的比刚度和比模量,是重要的工程结构材料,但复合材料失效产生和扩展的机理非常复杂。基于传统连续介质力学理论的方法(如有限元法)求解复合材料的静态和准静态问题时,其理论解和标准试验结果一致。但在求解动态问题时,则需要对连续介质理论进行修改,而且需要额外的失效判断标准和附加函数。尽管如此,传统方法仍无法准确模拟三维裂纹和群裂纹等复杂裂纹的扩展。近场动力学理论(简称PD理论)将传统连续介质力学本构方程中的微分项用积分项代替,避免了由裂纹造成的导数求解奇异性。PD理论应用于失效扩展的模拟具有三大优势:(1)不需要额外的失效判断标准,自发模拟裂纹的产生和扩展;(2)更改本构力函数能对不同尺度的问题进行建模;(3)同一计算体系框架下,能够同时处理多条裂纹的产生和扩展,并考虑它们之间的相互作用。复合材料的不均匀性及其力学性能的各向异性,使得PD模型中的点对力函数无法全面地描述复合材料的各向异性行为,构建理想的数学模型较为困难。PD理论的实质是将模型离散为一系列点,计算在一个点近场范围内所有其他点对该点作用力的合力,这导致PD方法的计算量非常大。因此,近几年PD方法应用于复合材料失效的研究主要集中于理论模型和计算体系的不断发展完善,并取得一系列成果。目前,已发展出多种复合材料的PD模型,开发出新的算法和求解策略,能够较好地模拟复合材料的多种失效模式,并提高计算效率。成功模拟复合材料多种失效模式的PD模型包括:基于纤维键和基体键的模型与基于法向键和剪切键的模型。基于纤维键和基体键的模型是最早建立的复合材料PD模型,通过在材料点对的本构力函数中增加适当的修改项来描述材料的本构信息。基于法向键和剪切键构建的模型,本构力函数中变形量的求解形式类似于传统连续介质力学中应变的表达,能直接在失效结果图中显示力学参量的变化。动态自适应松弛技术、并行算法等已经应用于PD方法并成功提高了计算效率,此外,针对PD方法的计算体系开发了快速算法和转化方程;求解策略上,成功将PD模型和有限元模型进行耦合,将PD模型布置在核心(失效扩展)区域,有限元模型布置在其他区域,在保证求解精度和正确性的基础上提高计算效率。本文归纳了PD方法研究复合材料失效的进展,分别对PD方法的理论框架、复合材料的PD模型、新的求解算法和求解策略以及PD方法在复合材料失效方面的应用等进行介绍,分析了PD方法在研究复合材料失效中存在的问题并展望其前景,以期为PD方法在复合材料失效机理研究中的进一步应用提供参考。     

冲击荷载作用下混凝土结构破坏过程的近场动力学模拟

混凝土在冲击、侵彻等动载荷作用下产生损伤和破坏的过程,其实质是力学模型从连续体到非连续体的转变过程。传统的连续介质理论基于连续性假设并运用偏微分方程求解问题,难以直接用于计算和模拟材料及结构发生破坏的整个过程。近场动力学(Peridynamics,PD)是一种新兴的基于非局部模型描述材料特性的数值计算方法。该方法假定位于连续体内的粒子通过有限的距离与其它粒子相互作用,通过积分计算在一定近场范围(horizon)内具有一定影响域的材料点之间的相互作用力,而不论位移场的连续与否,避免了传统的局部微分方程求解在面临不连续问题时的奇异性和现有多尺度算法的复杂性。该文概述了PD方法的理论基础,描述了其建模思路及计算体系,给出了用近场动力学方法模拟结构受冲击荷载的计算格式。算例结果表明:PD方法可以很好地刻画和模拟材料及结构的损伤累积与渐进破坏过程。最后讨论了PD方法在理论、计算和应用等方面有待进一步研究的问题。     

Why do cracks branch? A peridynamic investigation of dynamic brittle fracture

In this paper we review the peridynamic model for brittle fracture and use it to investigate crack branching in brittle homogeneous and isotropic materials. The peridynamic simulations offer a possible explanation for the generation of dynamic instabilities in dynamic brittle crack growth and crack branching. We focus on two systems, glass and homalite, often used in crack branching experiments. After a brief review of theoretical and computational models on crack branching, we discuss the peridynamic model for dynamic fracture in linear elastic–brittle materials. Three loading types are used to investigate the role of stress waves interactions on crack propagation and branching. We analyze the influence of sample geometry on branching. Simulation results are compared with experimental ones in terms of crack patterns, propagation speed at branching and branching angles. The peridynamic results indicate that as stress intensity around the crack tip increases, stress waves pile-up against the material directly in front of the crack tip that moves against the advancing crack; this process “deflects” the strain energy away from the symmetry line and into the crack surfaces creating damage away from the crack line. This damage “migration”, seen as roughness on the crack surface in experiments, modifies, in turn, the strain energy landscape around the crack tip and leads to preferential crack growth directions that branch from the original crack line. We argue that nonlocality of damage growth is one key feature in modeling of the crack branching phenomenon in brittle fracture. The results show that, at least to first order, no ingredients beyond linear elasticity and a capable damage model are necessary to explain/predict crack branching in brittle homogeneous and isotropic materials.     

A state-based peridynamic model for quantitative fracture analysis

A new state-based peridynamic model is proposed to quantitatively analyze fracture behavior (crack initiation and propagation) of materials. In this model, the general relationship of the critical stretch and the critical energy release rate is for the first time obtained for the state-based peridynamic model of linear elastic brittle materials, and the released energy density is defined to quantitatively track the energy released during crack propagation. The three-dimensional (3D) and two-dimensional (2D) (for both plane stress and plane strain) cases are all considered. As illustrations, the compact tension and double cantilever beam tests are analyzed using the proposed model, which is capable of successfully capturing fracture behaviors (e.g., crack path and concentration of strain energy density) of the considered fracture tests. The characteristic parameters (i.e., critical load, critical energy release rate, etc.) are calculated and compared with available experimental and numerical data in the literature to demonstrate validity of the proposed model.     

Studies of dynamic crack propagation and crack branching with peridynamics

In this paper we discuss the peridynamic analysis of dynamic crack branching in brittle materials and show results of convergence studies under uniform grid refinement (m-convergence) and under decreasing the peridynamic horizon (δ-convergence). Comparisons with experimentally obtained values are made for the crack-tip propagation speed with three different peridynamic horizons. We also analyze the influence of the particular shape of the micro-modulus function and of different materials (Duran 50 glass and soda-lime glass) on the crack propagation behavior. We show that the peridynamic solution for this problem captures all the main features, observed experimentally, of dynamic crack propagation and branching, as well as it obtains crack propagation speeds that compare well, qualitatively and quantitatively, with experimental results published in the literature. The branching patterns also correlate remarkably well with tests published in the literature that show several branching levels at higher stress levels reached when the initial notch starts propagating. We notice the strong influence reflecting stress waves from the boundaries have on the shape and structure of the crack paths in dynamic fracture. All these computational solutions are obtained by using the minimum amount of input information: density, elastic stiffness, and constant fracture energy. No special criteria for crack propagation, crack curving, or crack branching are used: dynamic crack propagation is obtained here as part of the solution. We conclude that peridynamics is a reliable formulation for modeling dynamic crack propagation.     

Characteristics of dynamic brittle fracture captured with peridynamics

Using a bond-based peridynamic model, we are able to reproduce various characteristics of dynamic brittle fracture observed in experiments; crack branching, crack-path instability, asymmetries of crack paths, successive branching, secondary cracking at right angles from existing crack surfaces, etc. We analyze the source of asymmetry in the crack path in numerical simulations with an isotropic material and symmetric coordinates about the pre-crack line. Asymmetries in the order of terms in computing the nodal forces lead to different round-off errors for symmetric nodes about the pre-crack line. This induces the observed slight asymmetries in the branched crack paths. A dramatically enhanced crack-path instability and asymmetry of the branching pattern are obtained when we use fracture energy values that change with the local damage. The peridynamic model used here captures well the experimentally observed successive branching events and secondary cracking. Secondary cracks form as a direct consequence of wave propagation and reflection from the boundaries.     

Crack initiation in brittle solids under multiaxial compression

Experiments combined with numerical simulations were used to study crack initiation in brittle materials under biaxial static compression with particular attention to the frictional resistance of the cracks. A new methodology has been developed to prepare specimens with a central crack where crack surfaces are in contact and with the desired friction coefficient: two pieces of Homalite-100 (a brittle polymer) were bonded except for a central region that served as a crack. The measured failure load for cracks with different orientation angles and surface roughness was in close agreement with theoretical predictions for compressive failure. In situ photoelastic fringes were obtained and compared with numerical results enabling the determination of stress distribution along crack faces, as well as the identification of slip and stick regions.     

An experimentally-calibrated damage mechanics model for stone fracture in shock wave lithotripsy

Instabilities in thermally-driven crack growth in thin glass plates have been observed in experiments that slowly immerse a hot, pre-notched glass slide into a cold bath. We show that a nonlocal model of thermomechanical brittle fracture with minimal input parameters can predict the entire phase diagram of fracture measured in experiments for the low immersion speed regime. Geometrical restrictions to crack growth commonly found in other approaches are absent here. We discuss a method for determining the appropriate size of the peridynamic horizon based on a data point around a separating line between crack-type zones in the experimental phase diagram. Once the nonlocal size is smaller than the length-scale introduced by the thermal gradient, the computational results show that no fracture criterion is needed beyond Griffith’s criterion to capture the observed instabilities. The combination of thermal gradients and competing contraction forces on the two sides of the crack are behind the observed crack path instabilities. Elastic (velocity) vortices of material points show how and why the cracks develop along the observed paths. Our results demonstrate that thermally-driven fracture in brittle materials can be predicted with accuracy. We anticipate that this model will lead to design protocols for controlled fracture in brittle materials relevant in materials science and advanced manufacturing.     

A damage model for crack prediction in brittle and quasi-brittle materials solved by the FFT method

In this paper, we present a damage model and its numerical solution by means of Fast Fourier Transforms (FFT). The FFT-based formulation initially proposed for linear and non-linear composite homogenization (Moulinec and Suquet in CR Acad Sci Paris Ser II 318:1417–1423 1994; Comput Methods Appl Mech Eng 157:69–94 1998) was adapted to evaluate damage growth in brittle materials. A non-local damage model based on the maximal principal stress criterion was proposed for brittle materials. This non-local model was then connected to the Griffith criterion with the aim of predicting crack growth. By using the proposed model, we carried out several numerical simulations on different specimens in order to assess the fracture process in brittle materials. From these studies, we can conclude that the present FFT-based analysis is capable of dealing with crack initiation and crack growth in brittle materials with high accuracy and efficiency.