Modeling dynamic brittle behavior of materials with circular flaws or pores

Compressive failure of brittle materials is driven primarily by crack growth from pre-existing flaws in the material. These flaws, such as grain boundaries, pores, preexisting cracks, inclusions and missing grains, are randomly spaced and have a range of possible shapes and sizes. The current work proposes a micromechanics-based model for compressive dynamic failure of brittle materials with circular pore flaws, which incorporates both the number density and the size distribution of flaws. Results show that the distribution of flaw sizes is very important, particularly at moderate strain rate, since analyses based solely on the mean flaw size overpredict strength. Therefore, in order to increase dynamic strength at low to moderate strain rates, it is most effective to control the presence of large flaws. At very high strain rates, however, crack growth is activated even in small flaws and therefore controlling the total number density rather than the size of the flaws is effective for increasing dynamic strength. Finally, the model shows that neglecting very small flaws in the pore population may not have significant effects on the results in many cases, suggesting that the model is a useful tool for identifying a minimum resolution required for experimental characterization of microstructure.     

Experimental study on cracking behaviour of moulded gypsum containing two non-parallel overlapping flaws under uniaxial compression

Failure of rock mass that is subjected to compres-sive loads occurs from initiation, propagation, and linkage of new cracks from preexisting fissures. Our research inves-tigates the cracking behaviour and coalescence process in a brittle material with two non-parallel overlapping flaws using a high-speed camera. The coalescence tensile crack and tensile wing cracks were the first cracks to occur from the pre-existing flaws. The initiation stresses of the primary cracks at the two tips of each flaw were simultaneous and decreased with reduced flaw inclination angle. The following types of coalescence cracks were identified between the flaws: pri-mary tensile coalescence crack, tensile crack linkage, shear crack linkage, mixed tensile-shear crack, and indirect crack coalescence. Coalescence through tensile linkage occurred mostly at pre-peak stress. In contrast, coalescence through shear or mixed tensile-shear cracks occurred at higher stress. Overall, this study indicates that the geometry of preexisting flaws affect crack initiation and coalescence behaviour.     

An experimental and numerical study of fracture coalescence in pre-cracked specimens under uniaxial compression

This study presents crack initiation, propagation and coalescence at or near pre-existing open cracks or flaws in a specimen under uniaxial compression. The flaw geometry in the specimen was a combination of a horizontal flaw and an inclined flaw underneath. This flaw geometry is different from those reported in the previous studies, where a pair of parallel flaws was used. Three materials were used, PMMA (Poly Methyl MethAcrylate), Diastone (types of molded gypsum), and Hwangdeung granite. Crack initiation and propagation showed similar and different patterns depending on the material. In PMMA, tensile cracks initiated at the flaw tips and propagated to the tip of the other flaw in the bridge area. The cracks then coalesced at a point of the inclined flaw, which is affected by the flaw inclination angle. For Diastone and Hwangdeung granite, tensile cracks were observed followed by the initiation of shear cracks. Coalescence occurred mainly through the tensile cracks or tensile and shear cracks. Crack coalescence was classified according to the crack coalescence types of parallel flaws for overlapping flaw geometry in the past works. In addition, crack initiation and coalescence stresses in the double-flawed specimens were analyzed and compared with those in the single-flawed specimen. Numerical simulations using PFC^2D (Particle Flow Code in two dimensions) based on the DEM (Discrete Element Method) were carried out and showed a good agreement with the experimental results in the coalescence characteristics in Hwangdeung granite. These experimental and numerical results are expected to improve the understanding of the characteristics of cracking and crack coalescence and can be used to analyze the stability of rock and rock structures, such as the excavated underground openings or slopes, tunneling construction, where pre-existing cracks or fractures play a crucial role in the overall integrity of such structures.     

An interacting micro-crack damage model for failure of brittle materials under compression

A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.     

Predicting variability in the dynamic failure strength of brittle materials considering pre-existing flaws

We perform two-dimensional dynamic fracture simulations of a specimen in biaxial tension, incorporating various distributions of pre-existing microcracks. The simulations consider the spatial distribution of flaws while modeling the discrete failure processes of crack interactions and coalescence, and predict the macroscopic variability in failure strength. The model quantitatively predicts the effect (on the dynamic failure strength) of different shapes of the flaw size distribution function, the random spatial distribution of flaws, and the random local resistance to crack growth (i.e. strength) associated with each flaw. The effect of changing material volumes on the variability in failure strengths is also examined in relation to the flaw size distribution. The effect of loading rate on the variability in failure strengths is presented in a form that will enable improved constitutive modeling using non-local formulations at the continuum scale.     

Influence of flaw inclination angle and loading condition on crack initiation and propagation

With reference to the experimental observation of crack initiation and propagation from pre-existing flaws in rock specimens under compression, the influences of pre-existing flaw inclination angle on the cracking processes were analyzed by means of finite element method (FEM) and non-linear dynamics method. FEM analysis on the stress field distribution induced by the presence of a pre-existing flaw provided better understanding for the influence of flaw inclination angle on the initiation position and initiation angle of the potential cracks. Numerical analysis based on the non-linear dynamics method was performed to simulate the cracking processes. The resultant crack types, crack initiation sequences and the overall crack pattern were different under different loading conditions. Under a relatively low loading rate or a small magnitude of maximum loading pressure, tensile cracks would tend to initiate prior to shear cracks. In contrast, under a relatively high loading rate and a large magnitude of maximum loading pressure, shear cracks would tend to initiate prior to tensile cracks instead.     

Finite Element study of fracture initiation in flaws subject to internal fluid pressure and vertical stress

Hydraulic fracturing is a method used routinely in oil and gas exploitation and in engineered geothermal systems. While used frequently, there are many aspects of hydraulic fracturing, such as the direction of propagation of the newly-created fractures, which are not very well understood. Even though it is known that the local stress field plays a fundamental role in the orientation of the new fractures, there may be other factors, such as the geometry of the existing fractures and the magnitude of the hydraulic pressure applied, that may play a major role in the path that a new fracture follows when pressurized.The main goal of this study is to numerically analyze the effect of the ratio between a vertical load, or stress, and the hydraulic pressure applied in existing flaws on the stress field in the vicinity of the flaw tips. For that purpose, a double flaw geometry 2a-30-30 was modeled in the Finite Element code ABAQUS, and different vertical loads and internal flaw pressures were applied to the model. The variation of the maximum principal stresses and maximum shear stresses around the flaw tips were analyzed and related to fracture initiation.The study showed that the ratio between the water pressure applied in the flaws and the vertical load/stress (WP/VL) plays a crucial role in the magnitude and shape of the stress field around a flaw tip, and therefore in the location of tensile and shear fracture initiation. As WP/VL increases, the location of initiation of new tensile fractures shifts from the upper face of the studied flaw towards the region right ahead of the flaw tip; simultaneously, the location of initiation of new shear fractures shifts from the region ahead of the flaw tip to the upper face of the analyzed tip.     

Fracture energy based constitutive models for tensile fracture of metal powder compacts

Diametral compression test or the Brazilian disc test is commonly used to characterise the tensile strength of brittle materials. A general fracture model based on energy assumptions is proposed for simulation of the discrete and localised tensile fracturing process in metal powder. The characteristics of the tensile fracture development of the central crack in diametral tested specimen is numerically studied. The softening rate of the model is obtained from the corresponding rate of the dissipated energy. Finite element simulations of the diametral compression test are performed with the proposed tensile fracture model used in conjunction with a Cap model for the deformation of the powder material. The results agree reasonably with experiments.     

冲击载荷下两裂纹间的连通规律

脆性材料内部含有大量裂纹,当某一裂纹扩展时,其他裂纹会对扩展裂纹产生影响。为了研究冲击载荷下,脆性材料内两裂纹的相互影响、连通规律及裂纹尖端应力强度因子的变化规律,利用有机玻璃板制作了含非平行双裂纹的实验试件,利用落板冲击设备进行了中低速冲击实验,结合有限元分析软件ABAQUS计算出裂纹尖端应力强度因子,利用有限差分软件AUTODYN进行了动态数值模拟研究,并将其模拟结果与实验结果进行对比分析。实验及模拟结果表明:裂纹破坏形态与AUTODYN数值模拟破坏形态基本一致;试件的断裂形态随着两裂纹间距不同而不同;裂纹间的相互影响程度随着裂纹间间距增大而减小;裂纹尖端应力强度因子K_I随着裂纹间距的增大而减小,而K_II随着裂纹间距增大而增大。     

脆性材料在双向应力下的断裂实验与理论分析

研究了脆性材料在双向应力下的断裂特性和失效机理，特别是在平行于裂纹的应力对临界断裂参数的影响方面进行了实验上和理论上的研究．采用玻璃、陶瓷等脆性材料进行了平面双向拉伸和单向拉伸试验，并对实验结果进行比较．观测直通裂纹的启裂和扩展过程，证明了双向应力对裂纹驱动力有明显影响，讨论了裂纹扩展的应变准则．     

Statistical characterization of meso-scale uniaxial compressive strength in brittle materials with randomly occurring flaws

Increasingly fine spatial resolution in numerical models of brittle materials promises to improve prediction and characterization of dynamic failure in these materials. However, as the resolution of these numerical models begins to approach the material micro-scale, the associated discretization requires a definitive connection to the microstructure. In many cases a numerical model (e.g., a finite element mesh) that explicitly resolves each flaw within the material is not feasible for macro-scale analyses. As an alternative, each element can be treated as a meso-scale continuum with constitutive properties that reflect the characteristics of the underlying microstructure. Small scale elements will exhibit random variations in the constitutive properties as a result of the random variations in the number and types of flaws and the flaw sizes contained within each element. The present paper proposes a technique for assigning probability distributions to these element properties, which can be thought of as the meso-scale constitutive properties. In particular, the strain-rate dependent compressive uniaxial strength of a ceramic is modeled using a two-dimensional analytical model developed by Paliwal and Ramesh (2008). The effect on the probability distribution of meso-scale (or element-level) strength from flaw density, flaw size distribution, flaw clustering, and strain rate are studied. Higher strain rates, more flaw clustering, and decreasing element size all contribute to greater scatter in uniaxial compressive strength. Variations in flaw size increase the scatter in the strength more for low strain rate loadings and less clustered microstructures. The results provide interesting comparisons to the classical assumption of a two-parameter Weibull-distributed strength, showing that a three-parameter Weibull distribution and even a lognormal distribution fit better with the simulated strength data.     

基于近场动力学理论的准脆性材料的本构力函数的构建

近场动力学理论(PD)是基于非局部思想的连续介质力学新理论,用于研究材料破坏问题。根据准脆性材料破坏的线性和非线性的力学行为,在初始微观弹脆性材料(PMB)的本构力函数中引入了键的损伤模型,将键的断裂过程分成了线性的弹性变形阶段和非线性的损伤变形阶段,以此构建了准脆性材料的本构力函数的基本形式。以典型的准脆性材料为例构建了其本构力函数,通过在压缩载荷下对含预制不同角度单裂纹缺陷的类岩材料的裂纹扩展进行PD数值模拟仿真,裂纹起裂位置和扩展方向与试样试验结果在一定程度上保持了一致,证明了该基于近场动力学理论的典型准脆性材料的本构力函数可用于该类材料的破坏分析。     

Estimation of Fracture Toughness of Metallic Materials Using Instrumented Indentation: Critical Indentation Stress and Strain Model

We propose a generalized approach based on fracture mechanics and contact mechanics to estimate the fracture toughness in metallic materials from instrumented indentation testing. Models were developed for brittle and ductile fracture. Different criteria were applied to each model to determine the critical fracture point during indentation. For brittle fracture, the critical fracture point was defined in terms of the critical mean pressure; for ductile fracture, the critical fracture point was derived from fracture strain and critical plastic zone size. Each fracture criterion was used to determine the indentation fracture energy corresponding to the fracture energy required for crack extension. The fracture toughness was estimated for various metallic materials using each model and compared with standard fracture toughness tests.     

Diagrams of critical equilibrium for brittle bodies with sharp flaws

Certain forms of the stress-intensity factors close to the tips of sharp flaws (plane problem) are used as the basis of a method for plotting critical equilibrium diagrams for brittle bodies with flaws in the form of pointed cavity-cracks [5]. Concrete examples are discussed, mainly in the context of such diagrams, for a brittle body weakened by a circular cavity flaw with a crack leaving the edge of the flaw. Determination of the stress-intensity factors for this problem is based on approximate solution of an integral equation by the method of collocations. Plots of some familiar diagrams are also analyzed.Translated from Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 98–104, May–June, 1970.     

Fracture in metal powder compaction

This paper presents a preliminary assessment and qualitative analysis on fracture criterion and crack growth in metal powder compact during the cold compaction process. Based on the fracture criterion of granular materials in compression, a displacement based finite element model has been developed to analyse fracture initiation and crack growth in metal powder compact. Approximate estimation of fracture toughness variation with relative density is established in order to provide the fracture parameter as compaction proceed. A single crack initiated from the boundary of a multi-level component made of iron powder is considered in this work. The finite element simulation of the crack propagation indicates that shear crack grows during the compaction process and propagates in the direction of higher shear stress and higher relative density. This also implies that the crack grows in the direction where the compaction pressure is much higher, which is in line with the conclusion made by previous researchers on shear crack growth in materials under compression. In agreement with reported work by previous researchers, high stress concentration and high density gradient at the inner corner in multi-level component results in fracture of the component during preparation.     

压剪条件下预埋椭圆裂纹三维扩展实验研究

试制出两种不同性质的脆性类岩石材料：透明的低温树脂材料和非透明的砂浆类材料。实验研究了这两种材料内置椭圆形裂隙的三维始裂、扩展过程及其破裂过程中声发射信号特征,并采用三维断裂理论确定了内部裂纹的三维始裂状态。实验结果表明,在单轴压缩荷载作用下,两种材料中内置裂隙初期破裂面形态基本一致,均为一对反对称的包裹状翼裂纹,但初始起裂位置有所不同。透明树脂试样中,初始的新生裂纹几乎在预制裂纹长轴端部附近同时起裂,但起裂点并不在裂隙长轴端部,而在砂浆类材料内的起裂位置则靠近短轴端部,亦不在短轴端部。相同条件下,两种材料的最终破裂状态明显不同,透明树脂试样是以宏观劈拉破裂为主,而砂浆类试样则以宏观剪切破裂为主,这可能与材料性质和三维裂隙的起裂位置不同有关。三维断裂分析结果与实验结果一致。二维简化模型仅为三维模型的主法平面上的特征解,无法反映出III型断裂模式在三维裂隙的起裂和扩展中的作用。三维断裂机制分析应加强对III型断裂模式及其相关的复合模式的深入分析。     

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。     

Numerical study of failure behaviour of pre-cracked rock specimens under conventional triaxial compression

Macroscopic pre-existing flaws play an important role in evaluating the strength and the failure modes of a heterogeneous rock mass. Crack initiation, propagation and coalescence from macroscopic pre-existing flaws are considered in a 3-D numerical model (RFPA3D) to investigate their effects on the underlying failure modes of rock. A feature of the code RFPA3D is that it can numerically simulate the evolution of cracks in three-dimensional space, as well as the heterogeneity of the rock mass. Three types of flaw geometries were evaluated numerically against experimental results: Type A for intact specimen, and Types B and C for flawed cylindrical specimens with different macroscopic pre-existing flaws, respectively. The effect of confining pressure on the fracture evolution was also considered. Numerical results showed that both the ligament angle and the flaw angle of two pre-existing cracks can affect the uniaxial compressive strength of the specimen and the mechanism of fracture evolution. In addition, both the uniaxial compressive strength and the accumulated acoustic emission increase with increasing heterogeneity.     

The brittle fracture criterion based on the maximum tensile stress on the surface of blunt crack tip

The brittle fracture criterion is developed for a blunted crack. The curvature radius of the blunt crack tip is suggested as a characteristic length for brittle materials, and then the fracture toughness of the brittle materials can be determined from the cohesion strength and the characteristic length of the materials.     

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。