Limit velocity of fracture front and dynamic strength of brittle solids

The theories of propagation of brittle fracture fronts in solid materials are compared with experimental data. Instead of the well-known theory of the limit fracture stress the theory of limit velocity of fracture front is developed. Accordingly between the moving boundary at which the static strength is attained and the front of fracture the material can stand essential dynamic over-loadings. The experimental data on contained explosions in optically transparent intact blocks show that the limit velocity of brittle cracks front takes place immediately after the separation of the shock front and the front of brittle fracture. The hypothesis of the existence of limit front velocity leads to the conclusion that in the two-front structure of plane shock waves the amplitude of elastic precursors, known as “the Hugoniot elastic limit”, exceeds the value of ultimate static strength of a solid material and has to increase with increasing of a finite shock pressure. This effect is justified by a number of experiments with brittle materials. The analogue with the plane problem of a self-supporting brittle burst is shown. The explanation of exceeding of the ultimate static strength and of “the delay time” of fracture under the spall condition is given. The increasing of internal fractures, which is described by the dilatancy loosening of materials is discussed. The well-known laws of “the geometrical similarity” of contained explosions are in accordance with expression of the strength in terms of the ultimate stress but not in terms of Griffith's energy for creating of new cracks. The possibility of the regime of a limit front velocity of fracture at explosion motions in real rocks, for which the dilatancy has place, is discussed.     

Integral estimates to the fast brittle fracture mechanics

In the terms of a general approach the problem of the fast crack propagation in elastic bodies is considered. The time dependence of, so called domain of possible fractures due to the fast crack propagation, is obtained. The dynamic domain of possible fractures is the upper estimate of the zone of real fractures and thereby it takes into account such important features of the fast brittle fracture as the instability of the fast crack propagation, branching, waving, the influence of boundary conditions and so on. The problem of dynamic crack initiation is considered in some details. Within the linear fracture mechanics the dynamic criterion of crack initiation is offered.     

The influence of the cohesive process zone in hydraulic fracturing modelling

This paper studies the importance of the cohesive zone in the modelling of a fluid driven fracture under plain strain conditions. The fracture is driven by pumping of an incompressible viscous fluid at the fracture inlet. Rock deformation is modeled for linear elastic and poroelastic solids. Fluid flow in the fracture is modeled by lubrication theory. The cohesive zone approach is used as the fracture propagation criterion. Finite element analysis was used to compute the solution for the crack length, the fracture opening and propagation pressure as a function of the time and distance from the wellbore. It is demonstrated that the crack profiles and the propagation pressures are larger in the case of elastic-softening cohesive model compared to the results of the rigid-softening cohesive model for both elastic and poroelastic cohesive solids. It is found that the results are affected by the slope of the loading branch of the cohesive model and they are nearly unaffected from the exact form of the softening branch. Furthermore, the size of the process zone, the fracture geometry and the propagation pressure increase with increasing confining stresses. These results may explain partially the discrepancies in net-pressures between field measurements and conventional model predictions.     

The cleavage energy at initiation of (110) silicon

We evaluated the cleavage energy at initiation of the (110)[1 $\bar{{1}}$ 0] low energy cleavage system of silicon crystal under pure Mode I, at room conditions and under inert argon gas at atmospheric pressure. The results revealed significant reduction of the room cleavage energy, presumably due to the effect of environmental molecules at the crack front. We also show that misalignment between the precrack and the maximum $G_{I}$ plane yields large variations in the cleavage energy, which may explain the large discrepancy appearing in the literature. Comparison of our results with those existing in the literature, enabled conclusion regarding the lower limit of the cleavage energy at initiation of brittle crystals and comments on other physical effects associated with crack initiation in these materials. We describe in details the experimental method that was used to evaluate the cleavage energy at initiation. This method is aimed at cleaving brittle materials in a controllable and noiseless manner and to generate high resolution fracture experiments. It consists in gluing a thin, precracked, rectangular brittle specimen in a rectangular aluminum frame, using thin layers of epoxy resin. Being very complaint, these layers enable to reduce the energy flow to the crack tip and to manipulate crack speed for accurate evaluation of materials properties. Crack initiation, propagation, and arrest, when needed, occur upon heating the assembly by only a few $^{\circ }\hbox {C}$ , due to the coefficients of thermal expansion mismatch between the specimen and the aluminum frame. Among other advantages, it facilitates accurate evaluation of fracture properties of brittle crystals, through utilizing the assembly as a boundary value problem appropriate for finite element analysis without requiring any assumptions regarding boundary conditions.     

Loading Rates and the Dynamic Initiation Toughness in Brittle Solids

The experimentally determined marked rise of the stress intensity factor required to initiate crack propagation in brittle solids under variably high loading rates, is analyzed. This problem of fracture initiation at the tip of a crack is considered in terms of activating a flaw at some distance away from the tip. By using a semi-infinite crack in an unbounded two-dimensional solid subjected to spatially uniform but temporally varying crack-face pressure, we consider the evolution of stress at the failure initiation site. Fracture initiation is assumed synonymous with attaining a critical stress at the fracture site. The results conform to typical experimental data of dynamic crack initiation in brittle solids.     

Soil models as a basis for modelling the behaviour of geophysical materials

Summary The aim of this paper is to show that the conceptual structure of constitutive models for granular materials, such as soils, may be taken as a starting point to model the mechanical behaviour of materials of geophysical interest. Clearly some modifications are necessary to cope with the range of pressures, time and temperature which are relevant in geophysics.To start with, it will be shown that an elastic plastic strainhardening model with an elastic nucleus may describe the brittle ductile transition of rocks due to increasing confining pressure. This will be done by using the cam clay model, well known in soil mechanics, with slight modifications. A further extension of the model will allow to describe the influence of inherent anisotropy on strength and deformability.The effect of temperature on yielding may be taken into account by means of the Prager's consistency rule. It is shown that when temperature increases a shrinking of the initial yield function occurs, which is the primary reason for ductile behaviour under high temperatures.Finally the modelling of creep of rocks under constant loading is considered.With 10 Figures     

Crack propagation in brittle heterogeneous solids: Material disorder and crack dynamics

Crack propagation in a linear elastic material with weakly inhomogeneous failure properties is analyzed. An equation of motion for the crack is derived in the limit of slow velocity. Predictions of this equation on both the average crack growth velocity and its fluctuations are compared with recent experimental results performed on brittle heterogeneous materials (Ponson in Phys Rev Lett, 103, 055501; Måløy et al. in Phys Rev Lett, 96, 045501). They are found to reproduce quantitatively the main features of crack propagation in disordered systems. This theoretical framework provides new tools to predict life time and fracture energy of materials from their properties at the micro-scale.     

梯度复合材料裂纹扩展路径和起裂载荷的有限元分析

为了模拟功能梯度材料(FGM)在工程应用中可能会出现的断裂问题并计算相应的开裂载荷,通过编写用户自定义UEL子程序将梯度扩展单元嵌入到ABAQUS软件中模拟功能梯度材料的物理场,并编写交互能量积分后处理子程序计算裂纹尖端的混合模式应力强度因子(SIF),采用最大周向应力准则编写子程序计算裂纹的偏转角,并模拟了裂纹扩展路径,计算了裂纹的起裂载荷。讨论了材料梯度参数对裂纹扩展路径以及起裂载荷的影响规律。通过与均匀材料的对比,验证了功能梯度材料断裂性能的优越性。研究表明:外载平行于梯度方向时,垂直梯度方向的初始裂纹朝着等效弹性模量小的方向扩展,且偏转角在梯度指数线性时出现峰值,并随着组分弹性模量比的增加而变大;当外载和初始裂纹均平行于梯度方向时,材料等效弹性模量和断裂韧性的增加或者梯度指数的减小都导致起裂载荷变大。     

Aspects of energy propagation in highly anisotropic elastic solids

Summary Anomalies in the theory of wave propagation in constrained materials may be reconciled with the standard theory of wave propagation in unconstrained materials by relaxing the constraint slightly and then taking the limit as the constraint is obeyed exactly. In this paper the same method is employed in an attempt to reconcile anomalies in the propagation of energy in a constrained material with the known propagation propertics for unconstrained materials. On relaxing the constraint in a singly constrained material, it is found that the energetics associated with two of the three propagating waves tend to the appropriate known forms for the corresponding constrained material in the limit where the constraint holds exactly. The third wave has no counterpart in the constrained theory and it is conjectured that both the total energy density and the energy flux vector tend to zero as the constrained limit is approached. This conjecture is shown to be true for two simple boundary value problems involving incompressible, and inextensible, elastic half-spaces.     

A boundary integral method for a dynamic, transient mode I crack problem with viscoelastic cohesive zone

We consider the problem of the dynamic, transient propagation of a semi-infinite, mode I crack in an infinite elastic body with a nonlinear, viscoelastic cohesize zone. Our problem formulation includes boundary conditions that preclude crack face interpenetration, in contrast to the usual mode I boundary conditions that assume all unloaded crack faces are stress-free. The nonlinear viscoelastic cohesive zone behavior is motivated by dynamic fracture in brittle polymers in which crack propagation is preceeded by significant crazing in a thin region surrounding the crack tip. We present a combined analytical/numerical solution method that involves reducing the problem to a Dirichlet-to-Neumann map along the crack face plane, resulting in a differo-integral equation relating the displacement and stress along the crack faces and within the cohesive zone.     

Experimental research on dynamic mechanical properties of PZT ceramic under hydrostatic pressure

An experimental technique for initially applied hydrostatic pressure in specimens subjected to axial impact has been developed to study the dynamic mechanical properties of materials. The technique was employed for the purpose of examining the dynamic mechanical properties of lead zirconate titanate (PZT) at zero to 15 MPa hydrostatic pressures. Experimental results unambiguously exhibit the ductile behavior of PZT when hydrostatic pressure is involved. The compressive strength is demonstrated sensitive to the initial hydrostatic pressure and the strain-rate. The fracture modes are analyzed by means of scanning electron microscopy (SEM). Moreover, a failure criterion based on Mohr-Coulomb failure theory is suggested to explain the brittle and ductile failure of PZT.     

Damage of concrete in a very high stress state: experimental investigation

This study is intended to characterize the evolution in triaxial behavior of a standard concrete subjected to confining pressures varying from 0 to 600 MPa. Hydrostatic and triaxial tests, with several unloading–reloading cycles, are carried out on concrete samples using a high-capacity triaxial press. These tests serve to identify the evolution of the elastic unloading characteristics of concrete, depending on both confining pressure and axial strain. A number of optical observations are also provided to allow visualizing the evolution in concrete damage mode in the middle of the sample. Experimental results indicate a sizable change in concrete behavior with confining pressure. At low pressure values, Young’s modulus decreases and Poisson’s ratio rises sharply with axial strain. The concrete exhibits brittle behavior with failure caused by a localized damage mechanism. In contrast, at high confining pressures, the concrete becomes a ductile material, and the evolution in its unloading characteristics is negligible. Failure is thus associated with diffuse material damage. The concrete behaves like a granular material controlled by plasticity, meaning that the damage phenomenon observed at low confinement is completely inhibited.     

Numerical simulation with experimental verification of the fracture behavior in granite under confining pressures based on the tension-softening model

The use of the tension-softening model for analyzing fracture processes of rock is examined with special reference to the effect of confining pressure on the fracture extension. Tension-softening curves are measured by means of the J-based technique from unconfined tests performed on compact tension (CT) specimens of granite. On the basis of the determined tension-softening relation, numerical analyses are executed using a boundary element method (BEM) to simulate fracture of the granite under confining pressures. Numerical results are compared to the experimental results of two series of tests for which CT specimens and thick-walled cylindrical specimens were loaded to failure under confining pressures ranging from 0 to 26.5 MPa. It is shown that the BEM analyses can predict the observed fracture behavior. Based on the results, it is demonstrated that the tension-softening relation provides a suitable model to analyze the fracture process in the rock. The source mechanism for the pressure sensitive fracture is discussed by examining the growth of the fracture process zone.     

Numerical simulation with experimental verification of the fracture behavior in granite under confining pressures based on the tension-softening model

The use of the tension-softening model for analyzing fracture processes of rock is examined with special reference to the effect of confining pressure on the fracture extension. Tension-softening curves are measured by means of theJ-based technique from unconfined tests performed on compact tension (CT) specimens of granite. On the basis of the determined tension-softening relation, numerical analyses are executed using a boundary element method (BEM) to simulate fracture of the granite under confining pressures. Numerical results are compared to the experimental results of two series of tests for which CT specimens and thick-walled cylindrical specimens were loaded to failure under confining pressures ranging from 0 to 26.5 MPa. It is shown that the BEM analyses can predict the observed fracture behavior. Based on the results, it is demonstrated that the tension-softening relation provides a suitable model to analyze the fracture process in the rock. The source mechanism for the pressure sensitive fracture is discussed by examining the growth of the fracture process zone.     

陶瓷薄基板材料裂纹预制与断裂韧性评价

预制长度可控的裂纹以及原位观察裂纹扩展是研究陶瓷薄基板抗断裂行为的两大重点.本研究提出应变诱导法,通过将基板与黄铜梁粘结形成复合体,利用黄铜梁弯曲变形带动侧面陶瓷薄板受拉侧拉伸变形产生可控裂纹.在工具显微镜下对复合体进行四点弯曲,原位观察样品的裂纹扩展情况,通过调节黄铜梁宽度来控制初始裂纹长度,在初始裂纹萌发后继续加载...     

The methods of complex potentials, singular integral equations and integral transformations in a series of problems for the reinforcement of cracked plates

We study the initiation and propagation of a vertical crack in an elastic semi-infinite plate, reinforced on its boundary by an infinite discontinuous stringer within the limits of the theory of brittle failure. The plate is subjected to uniform distributed tensile forces at infinity, as well as to contact stresses due to application of forces to the stringer. We find the appropriate loading of the coherent stringer, and consequently we consider a problem where the stringer is cracked and a vertical crack has developed within the plate. We deduce the exact analytical solution for the principal singular integral equation for this case; hence the stringer is perfectly rigid and we calculate characteristic parameters of the problem. The results show that the crack tip has a logarithmic singularity, and the tangential contact stresses under the stringer at that end point are finite and generally differ from zero.