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.     

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.     

Computational micromechanics of dynamic compressive loading of a brittle polycrystalline material using a distribution of grain boundary properties

A two-dimensional finite element model is used to investigate compressive loading of a brittle ceramic. Intergranular cracking in the microstructure is captured explicitly by using a distribution of cohesive interfaces. The addition of confining stress increases the maximum strength and if high enough, can allow the effective material response to reach large strains before failure. Increasing the friction at the grain boundaries also increases the maximum strength until saturation of the strength is approached. Above a transitional strain rate, increasing the rate-of-deformation also increases the strength and as the strain rate increases, fragment sizes of the damaged specimen decrease. The effects of flaws within the specimen were investigated using a random distribution at various initial flaw densities. The model is able to capture an effective modulus change and degradation of strength as the initial flaw density increases. Effects of confinement, friction, and spatial distribution of flaws seem to depend on the crack coalescence and dilatation of the specimen, while strain-rate effects are result of inertial resistance to motion.     

A 3-D ellipsoidal flaw model for brittle fracture in compression

Engineering materials are rarely free of flaws. Mode I cracking from pre-existing flaws is the major cause of the brittle fracture in compression of materials such as concrete and rock. A 3-D ellipsoidal flaw model is used to show the significant influence of flaw geometry on crack initiation in uniform uniaxial, biaxial and triaxial compression. The model shows that the governing criterion for crack initiation may change from energy to stress with increasing crack size, and that for voids of similar size a spherical void is the most critical shape for crack initiation. The model thus provides a basis for a better understanding of both the phenomenon and the mechanism of brittle fracture in compression.     

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.     

Estimating the tensile strength of super hard brittle materials using truncated spheroidal specimens

New approaches need to be introduced to measure the tensile capacity of super hard materials since the standard methods are not effective. To pursue this objective, a series of laboratory tests were constructed to replicate the fracture mechanism of diamond-based materials. Experiments indicate that under a certain compressive test condition, stresses normal to the axisymmetric line in truncated spheroidal specimens (bullet-shaped specimens) are in tension contributing to the tensile fracture of the material. From experimental and numerical studies, it is concluded that semi-prolate spheroidal specimens can be used to determine precisely the tensile strength of brittle stiff diamond-like composites.     

Effect of adhesive thickness,adhesive type and scarf angle on the mechanical properties of scarf adhesive joints

The effects of adhesive thickness, adhesive type and scarf angle, which are determined as the main control parameters by the dimensional analysis, on the mechanical properties of a scarf adhesive joint (SJ) subjected to uniaxial tensile loading are examined using a mixed-mode cohesive zone model (CZM) with a bilinear shape to govern the interface separation. Particularly, the adhesive-dependence of the vital cohesive parameters of CZM, which mainly include initial stiffness, total fracture energy and separation strength, is introduced emphatically. The numerical results demonstrate that the ultimate tensile loading increases as the adhesive thickness decreases. Cross the ultimate tension, the joint loses the load-bearing capacity when adopting the brittle adhesive but sustains partial load-bearing capacity while selecting the ductile adhesive. In addition, for the joint with the ductile adhesive, the maximum applied displacement until the complete failure of it is directly proportional to the adhesive thickness, which is different from the case using the brittle adhesive. Taking the combination of the ultimate loading and applied displacement into account, failure energy is employed to evaluate the joint performances. The results show that the failure energy of the joint with the brittle adhesive increases as the adhesive thickness decreases. Conversely, the situation of the joint using the ductile adhesive is vice versa. Moreover, the effect of the adhesive thickness becomes more noticeable with decreasing the scarf angle owing to the variation of the proportion of each component of the mixed-mode. Furthermore, all the characteristic parameters (the ultimate tensile loading, the maximum applied displacement and the failure energy) that adopted to describe the performances of SJ increase as the scarf angle decreases. Finally, the numerical method employed in this study is validated by comparing with existing experimental results.     

随机分布共线微裂纹的强相互作用

微裂纹串接为宏观灾难性裂纹的过程取决于相邻微裂纹的强相互作用。微裂纹的分布规律影响材料的强度和韧性。本文研究简单的共线微裂纹构型，确定由于微裂纹长度和韧度尺寸的统计分布所产生的影响。研究结果预计了脆性材料的尺度效应，即对于相同密度的微裂纹分布，大尺寸构件的强度比小尺寸构件要低。计算还表明脆性体的强度随微裂纹分布函数标准方差的增加而减小。     

Porosity dependence of strength in brittle solids

The effect of pore size and pore volume fraction on strength in brittle solids is evaluated. The analysis considers that the strength degradaton of a solid containing a large number of spherical pores is due to a strong effect of porosity on Young's modulus. Each pore is assumed to possess radial or annular flaws emanating from the pore surface whose lengths are considered to be independent of pore size. The effect of stress concentration induced by the presence of the pore is included in the equation for strength through the Young's modulus dependence of porosity originally developed using the concept of crack opening displacement. It is shown that the strength of a solid containing spherical pores is controlled by the pore size, pore volume fracton and the radial (or annular) crack size to pore size ratio. Predicted variation of strength with pore volume fraction is tested against experimental data for glass and polycrystalline alumina.     

Al2O3陶瓷动静态压缩下碎片形貌与破坏机理分析

为探究Al₂O₃陶瓷的宏观力学响应与破坏机理，分别利用材料试验机和分离式霍普金森压杆对其进行准静态和动态压缩实验，同时通过原位光学成像观测试样的破坏过程，并利用同步辐射CT和扫描电镜（SEM）对回收碎片的尺寸和形状以及微观破坏模式进行表征分析。宏观强度数据表明，Al₂O₃陶瓷的抗压强度符合Weibull分布，且与加载应变率呈现指数增长关系。原位光学成像和SEM回收分析共同揭示了动静态加载下裂纹成核与扩展模式存在明显差异。准静态加载时材料微观上更易发生沿晶断裂，宏观表现为劈裂裂纹较少，且倾向于沿加载方向传播并贯穿整个试样；而动态加载时穿晶断裂占主导地位，劈裂裂纹明显增加并发生相互作用，因此在传播过程中容易分叉而形成大量次生裂纹，提高了试样内裂纹密度。这与碎片的CT表征结果一致，即碎片平均球形度和伸长、扁平指数等均随应变率对数线性增加。破坏模式的改变最终导致高应变率下陶瓷材料应变率敏感性显著增强。     

Dynamic damage constitutive relation of mesoscopic heterogenous brittle rock under rotation of principal stress axes

A micro-mechanics-based model is developed to investigate microcrack damage mechanism of four stages of brittle rock under rotation of the principal stress axes. They consist of linear elastic, non-linear hardening, rapid stress drop and strain softening. The frictional sliding crack model is applied to analyze microcracks nucleation, propagation and coalescence. The strain energy density factor approach is applied to determine the critical condition of microcrack nucleation, propagation and coalescence. The inelastic strain increments are formulated within the framework of thermodynamics with internal variables. Rotation of principal stress axes affect the dynamic damage constitutive relationship and the failure strength of brittle rock.     

Damage accumulation and crack growth in bilinear materials with softening: Application of strain energy density theory

A pseudo-elastic damage-accumulation model is developed by application of the strain energy density theory. The three-point bending specimen is analyzed to illustrate the crack growth characteristics according to a linear elastic softening constitutive law that is typical of concrete materials. Damage accumulation is accounted for by the decrease of elastic modulus and fracture toughness. Both of these effects are assessed by means of the strain energy density functions in the elements around a slowly moving crack. The rate of change of the strain energy density factor S with crack growth as expressed by the relation dS/da = constant is shown to describe the failure behavior of concrete. Results are obtained for different loading steps that yield different slopes of lines in an S versus a (crack length) plot. The lines rotate about the common intersect in an anti-clockwise direction as the load steps are increased. The intersect shifts upward according to increase in the specimen size. In this way, the combined interaction of material properties, load steps and specimen geometry and size are easily analyzed in terms of the failure mode or behavior that can change from the very brittle to the ductile involving stable crack growth. An upper limit on specimen or structural size is established beyond which stable crack growth ceases to occur and failure corresponds to unstable crack propagation or catastrophic fracture. The parameters that control the failure mode are the threshold values of the strain energy density function (dW/dV)_c and the strain energy density factor S_c.     

Strain rate effect on crack propagation in concrete

Some aspects of cracking in concrete structural elements are discussed. The flexural behaviour of precracked beams made with materials having different mechanical properties is experimentally evaluated at different deformation rates. The crack propagation in the sample is recorded with time by means of particular electric strain gauges. An interaction between ultimate tensile strength failure and brittle crack propagation is eventually pointed out.     

粗骨料粒径对混凝土动态压缩行为的影响研究

粗骨料作为混凝土材料组成最主要的部分,对混凝土力学性能和破坏模式有着很重要的影响。为了研究粗骨料平均粒径对混凝土动态力学性能的影响规律,针对不同平均粗骨料平均粒径（6、12、24 mm）的混凝土和砂浆材料进行了一系列SHPB试验,得到了不同应变率下各试件的应力-应变曲线,并对每种材料的动态增长因子（dynamic increase factor,DIF）与应变率的对数进行了线性拟合。结果表明:砂浆和混凝土材料的抗压强度具有明显的应变率效应,其动态抗压强度随着应变率的增加而逐渐增大,应力-应变曲线呈现相似的变化趋势;在相同的动态应变率条件下,平均粗骨料粒径为12 mm的混凝土的动态抗压强度最大,这与准静态条件下砂浆抗压强度最大截然不同;不同粗骨料粒径混凝土材料的应变率强化系数均大于砂浆材料,且随着粗骨料无量纲尺寸的增大,混凝土材料的应变率强化因子呈现先增大后减小的趋势。     

A 3D mechanistic model for brittle materials containing evolving flaw distributions under dynamic multiaxial loading

We present a validated fully 3D mechanism-based micromechanical constitutive model for brittle solids under dynamic multiaxial loading conditions. Flaw statistics are explicitly incorporated through a defect density, and evolving flaw distributions in both orientation and size. Interactions among cracks are modeled by means of a crack-matrix-effective-medium approach. A tensorial damage parameter is defined based upon the crack length and orientation development under local effective stress fields. At low confining stresses, the wing-cracking mechanism dominates, leading to the degradation of the modulus and peak strength of the material, whereas at high enough confining stresses, the cracking mechanism is completely shut-down and dislocation mechanisms become dominant. The model handles general multiaxial stress states, accounts for evolving internal variables in the form of evolving flaw size and orientation distributions, includes evolving anisotropic damage and irreversible damage strains in a thermodynamically consistent fashion, incorporates rate-dependence through the micromechanics, and includes dynamic bulking based on independent experimental data. Simulation results are discussed and compared with experimental results on one specific structural ceramic, aluminum nitride. We demonstrate that this 3D constitutive model is capable of capturing the general constitutive response of structural ceramics.     

综合考虑宏细观缺陷的岩体动态损伤本构模型

针对节理岩体同时含有节理、裂隙等宏观缺陷及微裂隙、微孔洞等细观缺陷的客观事实, 提出了在节理岩体动态损伤本构模型中应同时考虑宏细观缺陷的观点。为此, 首先对基于细观动态断裂机理的经典岩石动态损伤本构模型—TCK(Taylor-Chen-Kuszmaul)模型进行了阐述, 其次基于Lemaitre等效应变假设推导了综合考虑宏细观缺陷的复合损伤变量(张量), 进而在此基础上建立了相应的节理岩体动态损伤本构模型, 并利用该模型讨论了载荷应变率及节理条数对岩体动态力学特性的影响规律。结果表明, 在不同载荷应变率下试件在变形初始阶段是重合的, 而后随着应变的增加, 试件峰值强度、峰值应变及总应变均随载荷应变率的增加而增加; 随着节理条数的增加, 试件峰值强度逐渐降低, 但降低趋势逐渐变缓并趋于某一定值。上述研究结论与目前的理论及实验研究结果的基本规律是一致的, 说明了本模型的合理性。     

不同应变率下蓝宝石透明陶瓷玻璃的力学响应

蓝宝石（A1₂O₃）是透明陶瓷玻璃,它相较传统陶瓷（A1₂O₃）有优良的透光性,而且保留了陶瓷优良的力学性能。利用电子拉伸机和分离式霍普金森杆设备对试样进行准静态应变率为(10?4、10?3、10?2 s?1)和4种动态应变率(850、1 100、1 300、1 450 s?1)下的单轴压缩力学行为,用高速摄像机记录了蓝宝石透明陶瓷玻璃试样在准静态和动态压缩下的破坏过程。实验结果表明:从加载过程中的应力应变曲线是由加载段和失效段组成的,该材料是典型的脆性材料,并且有明显的应变率效应,随着应变率的提高,蓝宝石透明陶瓷玻璃的抗压强度也会提高;准静态和动态压缩下蓝宝石透明陶瓷玻璃都是在宏观裂纹扩展作用下失效破坏。通过分析不同应变率下蓝宝石透明陶瓷玻璃的破坏过程,分析得到该材料的失效是在加载的过程中,在蓝宝石透明陶瓷玻璃承载能力最低的区域出现裂纹源,然后裂纹成形并沿着加载方向扩展,然后裂纹之间相互交错,最终达到饱和状态破坏失效;在高应变率下,极短的时间内产生多处裂纹源,需要更大的能量去使裂纹成形、扩展,宏观上就表现为应变率效应。     

A micromechanical basis for partitioning the evolution of grain bridging in brittle materials

A micromechanical model is developed for grain bridging in monolithic ceramics. Specifically, bridge formation of a single, non-equiaxed grain spanning adjacent grains is addressed. A cohesive zone framework enables crack initiation and propagation along grain boundaries. The evolution of the bridge is investigated through a variance in both grain angle and aspect ratio. We propose that the bridging process can be partitioned into five distinct regimes of resistance: propagate, kink, arrest, stall, and bridge. Although crack propagation and kinking are well understood, crack arrest and subsequent “stall” have been largely overlooked. Resistance during the stall regime exposes large volumes of microstructure to stresses well in excess of the grain boundary strength. Bridging can occur through continued propagation or reinitiation ahead of the stalled crack tip. The driving force required to reinitiate is substantially greater than the driving force required to kink. In addition, the critical driving force to reinitiate is sensitive to grain aspect ratio but relatively insensitive to grain angle. The marked increase in crack resistance occurs prior to bridge formation and provides an interpretation for the rapidly rising resistance curves which govern the strength of many brittle materials at realistically small flaw sizes.     

脆性材料的分形统计强度理论

基于脆性材料中裂纹尺度的分形分面和最弱环原理的假设，导出了裂纹尺频分布的分形维数Ｄ与经典统计强度理论中Ｗｅｉｂｕｌｌ模量ｍ之间的关系，揭示了ｍ的几何实质，并且同时考虑裂纹的尺度分布和方位分布及裂纹扩展路径的不规则性对统计强度的影响，从理论上推导了复杂应力状态下脆性材料统计强度σ的一般表达式。     

Size effect upon grained materials tensile strength: The increase of the statistical dispersion at the smaller scales

The present paper provides a statistical model to the size effect on grained materials tensile strength; it is based on an Extreme Value Theory approach. Since the weakest link in grained materials is usually represented by the interface between the matrix and the aggregates, it is assumed that the flaw distribution can be represented by the aggregate distribution, expressed as a probability density function (pdf) of the grain diameters. Under the hypothesis that the strength of the material depends on the largest flaw, the tensile strength is computed as a function of the specimen size. In this way, two remarkable results are obtained: (i) a size effect for the average tensile strength that substantially agrees with the multifractal scaling law (MFSL) proposed by the first author and (ii) an increase of scatter of the tensile strength values when testing small specimens. Both these trends are confirmed by experimental data available in the literature.