On One Two-Level Model in the Mesomechanics of Compression Fracture of Cracked Composites

The article is devoted to a two-level model in the mesomechanics of compression fracture of composites with cracks. Consideration is given to fiber-laminated panels with an opening compressed along the fibers so that two cracks propagate from the opening edge in a direction perpendicular to the loading axis. The previous results obtained by the author in the three-dimensional theory of stability of deformable bodies and nonclassical fracture mechanics are used to create the above-mentioned two-level model     

混凝土双K断裂参数计算的半解析有限元法

混凝土裂缝扩展的双$K$断裂准则，用于描述混凝土结构裂缝的起裂、稳定扩 展和失稳断裂. 其相应的双$K$断裂参数(起裂断裂韧度$K_{\rm IC}^{\rm ini} $和失 稳断裂韧度$K_{\rm IC}^{\rm un}$)一般通过简便的试验和基于虚拟裂缝扩展粘 聚力的解析方法确定. 利用平面扇形域哈 密顿体系的方程，通过分离变量法及共轭辛本征函数向量展开法，以解析的方法推导出基于混 凝土虚拟裂缝扩展线性粘聚力模型的平面裂缝解析元列式. 将该解析元与有限元相结合，构成 半解析的有限元法，可求解任意结构几何形状的混凝土平面裂缝双$K$断裂参数的计算问题. 数值计算结果表明半解析有限元法对该类问题的求解是十分有效的.     

Slope stability analysis using fracture mechanics approach

The analysis of slope failure is complicated due to its mechanism as well as the geological history of the slope. In classical slope stability analysis, the slope failure is assessed using the basic continuum mechanics or the limit equilibrium approach. This analysis, however, must be slightly modified when tension cracks exist at the upper edge of the slope. From post-mortem analyses of slope failures in the past, it was found that tension cracks had some considerable importance affecting the failure mechanism of the slope.Some attempts had also been directed in the past towards simple vertical cut slopes with tension cracks. Terzaghi [40]. Using an elasto-plastic model and a few analytical assumptions, Terzaghi [40] obtained an expression for a critical height of a vertical cut slope. However, since the stress distribution is changing during crack propagation, the failure mechanism cannot in principle conjecture from a static (undisturbed) stress field alone. In this report, a unified numerical approach combining Finite Element Method (FEM), fracture mechanics and remeshing technique is used to model the failure analysis of vertical cut slopes with tension cracks. Using this approach, the crack can be extended incrementally under a certain energy based failure criterion (Strain Energy Density criterion - SED) while taking into account the existing stress singularity field in the vicinity of the crack-tip.The use of fracture mechanics in soils, however, poses some difficulties in obtaining its relevant parameters due to the granular structure of soil. Because of its granular structure, the shear strength of soil depends on cohesion and the applied confining pressure making the determination of fracture parameters in soil difficult. These relevant parameters are not yet available in literature and hence, certain assumptions and interpolations had to be made in obtaining these parameters. Future research direction in this area is badly needed. Using a certain set of soil properties and crack geometry, a series of curves relating parameters S (Strain Energy Density Factor) and a non-dimensional variable N ( = H/C - slope height vs. crack distance from edge ratio) for vertical cut slopes can be constructed through numerous parametric studies. These curves can serve as a quick guide to obtain the critical height for a given vertical cut slope geometry with tension cracks.     

Effects of static electric field on the fracture behavior of piezoelectric ceramics

The paper gives an overview on experimental observations of the failure behavior of electrically insulating and conducting cracks in piezoelectric ceramics. The experiments include the indentation fracture test, the bending test on smooth samples, and the fracture test on pre-notched (or pre-cracked) compact tension samples. For electrically insulating cracks, the experimental results show a complicated fracture behavior under electrical and mechanical loading. Fracture data are much scattered when a static electric field is applied. A statistically based fracture criterion is required. For electrically conducting cracks, the experimental results demonstrate that static electric fields can fracture poled and depoled lead zirconate titanate ceramics and that the concepts of fracture mechanics can be used to measure the electrical fracture toughness. Furthermore, the electrical fracture toughness is much higher than the mechanical fracture toughness. The highly electrical fracture toughness arises from the greater energy dissipation around the conductive crack tip under purely electric loading, which is impossible under mechanical loading in the brittle ceramics. The project supported by an RGC grant from the Research Grant Council of the Hong Kong Special Administrative Region, China     

基于临界距离点法的混凝土Ⅰ型断裂韧度的预测

混凝土的断裂韧度是极为重要的断裂力学材料参数，本文利用临界距离理论(TCD)中的点法和混凝土标准断裂梁试样裂缝尖端应力场的近似解，提出了一种对混凝土断裂问题按临界距离的点法进行分析的方法．基于已有文献的试验材料及其实验结果，分析了这种断裂研究分析方法的可行性．将临界距离点法与双K断裂准则应用于几何相似的混凝土梁的断裂韧性和断裂过程区分析，验证了TCD点法的可靠性．实验表明：临界距离理论的点法能够获得比双K断裂准则相对安全的失稳断裂韧度，利用临界距离法还可得到相应断裂过程区长度的极限估算值．     

断裂力学判据的评述

从Inglis 和Griffith 的著名论文到Irwin 和Rice 等的奠基性贡献,对断裂力学中的线弹性断裂力学的K判据,界面断裂力学的G判据,和弹塑性断裂力学的J 判据作了扼要的综述. 介绍了在界面断裂力学G判据的基础上提出的界面断裂力学的K判据,以说明断裂力学的判据存在改进的可能性. 在综述中归纳出断裂力学判据中目前还没有较好解决的几个问题. 在总结以往断裂力学研究经验的基础上,指出裂纹端应力奇异性的源是对断裂力学判据存在的问题作进一步研究的切入点. 探讨了裂纹端应变间断的奇点是裂纹端应力奇异性的源的问题,从而对裂纹端应力强度因子的物理意义进行了讨论. 最后,阐述了进行可靠的裂纹端应力场的弹塑性分析是改进弹塑性断裂力学判据的关键,而进行可靠的裂纹端应力场的弹塑性分析的前提是要通过裂纹端应力奇异性的源的研究来获得作用在裂纹端的造成裂纹端应变间断的有限值应力.      

A MOVING CRACK IN A NONHOMOGENEOUS MATERIAL STRIP

This paper considers an anti-plane moving crack in a nonhomogeneous material strip of finite thickness. The shear modulus and the mass density of the strip are considered for a class of functional forms for which the equilibrium equation has analytical solutions. The problem is solved by means of the singular integral equation technique. The stress field near the crack tip is obtained. The results are plotted to show the effect of the material non-homogeneity and crack moving velocity on the crack tip field. Crack bifurcation behaviour is also discussed. The paper points out that use of an appropriate fracture criterion is essential for studying the stability of a moving crack in nonhomogeneous materials. The prediction whether the unstable crack growth will be enhanced or retarded is strongly dependent on the type of the fracture criterion used. Based on the analysis, it seems that the maximum 'anti-plane shear' stress around the crack tip is a suitable failure criterion for moving cracks in nonhomogeneous materials.     

Damage analysis and fracture criteria for piezoelectric ceramics

Both the mechanical and the electrical damages are introduced to study fracture mechanics of piezoelectric ceramics in this paper. Two kinds of piezoelectric fracture criteria are proposed by using the damage theory combined with the well-known piezoelectric fracture experiments of Park and Sun [Fracture criteria of piezoelectric ceramics, J. Am. Ceram. Soc. 78 (1995) 1475-1480]. One is based on a critical state of the mechanical damage and the other on a critical value of a proper linear combination of both the mechanical and the electrical damage variables. It is found that the fracture load predicted, which takes the mechanical damage into account only (mode 1), has greater deviation than predicted result by considering a proper linear combination of the mechanical and the electrical damages (mode 2). And the fracture criterion corresponding to mode 2 presented is shown to be superior to mode 1. It is also demonstrated that the mechanical damage has greater effect on fracture than the electrical damage.     

Limitations of growth regimes for cracks initiated in smooth fatigue specimens

The stages of the growth of small cracks initiating at natural flaws in smooth specimens subjected to fatigue loading are characterized and the dominant propagation mechanisms and corresponding fracture paths are described. Characteristic crack lengths are introduced to assess the transition between the regimes of microstructurally small cracks, physically small cracks and long cracks. A log Δσ-log a-diagram is used to derive estimates of these crack lengths. It is shown that simple formulate can be found which relate these characteristic crack lengths to mechanical and material parameters that can be measured using standard fracture mechanics specimens and fatigue tests.     

Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.     

On the stability of a circular system subjected to nonlinear dissipative forces

A circular system is a mechanical system subjected to potential forces and positional nonconservative forces (circular forces). The latter linearly depend on the coordinates and are characterized by a skew-symmetric matrix. The influence of linear dissipative forces on the stability of a circular system is ambiguous: on the one hand, they can stabilize a stable circular system (making it asymptotically stable); on the other hand, they can destabilize it [1–4]. The action of linear dissipative forces on a circular system results in the so-called destabilization paradox: the stability threshold decreases by a finite value.A detailed survey of this phenomenon can be found in [5]. The destabilization effect is also preserved under the action of nonlinear dissipative forces. The influence of these forces on the stability of the Ziegler pendulum with a tracking force was studied in [6]. It was shown that the critical value of the tracking force decreases by a finite value. A similar effect was discovered in the analysis of a continual system in [7].In the present paper, we study how nonlinear dissipative forces affect the stability of the equilibrium of a circular mechanical system with two degrees of freedom. The stability problem is solved without any references to specific mechanical systems. The results are used to analyze the stability of a gimbal gyro with allowance for dry friction in the rotor bearings.     

A fracture damage model for Jointed rock masses and its application to the stability analysis of dam abutments

Based on continuum damage mechanics, for jointed rock masses, a fracture damage model is presented in this paper. First, the damage tensors are defined through the elastic-flexibility of intact rock and the equivalent elastic-damage flexibility for rock mass. Then, by the self-consistent principle of solid mechanics, the equivalent elastic-damage flexibility tensors involving the interaction between multicracks are deduced. The damage evolution law is proposed involving the mechanism of crack propagation process: frictional sliding, crack kinking, growing of branched tension cracks, interlinking of the microcracks near branched crack tips leading to the breakthrough of macro-cracks and finally the failure of rock mass. Thus the evolution of damage variables reasonably unified with the process of crack propagation is given. Finally, a plastic-brittle damage constitutive relation including brittle coupled strain rate, developed and applied to the stability analysis of complicated rock foundation of a dam in China, is described in this paper.     

岩石劈裂破坏过程的计算模拟

岩体的破坏主要是岩体中裂纹的扩展与贯通的结果.受压岩体的裂纹尖端,可能处于压剪状态,也可能处于拉剪状态.从断裂力学的角度,研究了地下工程围岩中的裂纹在各种受力条件下(包括拉剪、压剪两种情况)的起裂、扩展及贯通以至最终形成劈裂裂缝的特征及各个阶段的判据.针对压剪断裂,裂纹之间存在有粘结力与摩擦强度的特征,提出了岩石压剪断裂性判据.最后以受压岩体中的斜裂纹扩展过程分析为例,模拟了岩体中裂纹逐渐发展成劈裂破坏的过程,并说明了各个阶段的计算方法.     

On discrete models in fracture mechanics

In two papers published in 1969, where the foundations of the theory of brittle fracture were laid, Valentin Valentinovich Novozhilov introduced a necessary and sufficient criterion for estimating the strength of an elastic body weakened by a cut and formulated the concept of brittle fracture. The criterion is based on taking account of the body discrete structure, and the fracture process is treated as loss of stability of elastic equilibrium. These two facts, fracture discreteness and loss of stability in the deformation of the breaking bond, underlie numerous phenomena, which could not be discovered in the framework of continuum mechanics. In the present paper, the following effects of taking these facts into account are briefly discussed: an unstable leg of the stress-strain diagram and the capture of part of the fracture energy by the lattice, the role of the lattice dimension, radiation in quasi-static and dynamic crack propagation including the radiation structure, the role of the dynamic factor, irregularities in the crack growth, crack propagation under the action of high-frequency waves including steady-state modes with periodically varying speed, and cracks in continuum discrete models.     

The singularity in weakly nonlinear fracture mechanics

Material failure by crack propagation essentially involves a concentration of large displacement-gradients near a crack's tip, even at scales where no irreversible deformation and energy dissipation occurs. This physical situation provides the motivation for a systematic gradient expansion of general nonlinear elastic constitutive laws that goes beyond the first order displacement-gradient expansion that is the basis for linear elastic fracture mechanics (LEFM). A weakly nonlinear fracture mechanics theory was recently developed by considering displacement-gradients up to second order. The theory predicts that, at scales within a dynamic lengthscale ℓ from a crack's tip, significant logr displacements and 1/r displacement-gradient contributions arise. Whereas in LEFM the 1/r singularity generates an unbalanced force and must be discarded, we show that this singularity not only exists but is also necessary in the weakly nonlinear theory. The theory generates no spurious forces and is consistent with the notion of the autonomy of the near-tip nonlinear region. The J-integral in the weakly nonlinear theory is also shown to be path-independent, taking the same value as the linear elastic J-integral. Thus, the weakly nonlinear theory retains the key tenets of fracture mechanics, while providing excellent quantitative agreement with measurements near the tip of single propagating cracks. As ℓ is consistent with lengthscales that appear in crack tip instabilities, we suggest that this theory may serve as a promising starting point for resolving open questions in fracture dynamics.     

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.     

Fracture mechanics based design procedures for pressurized components having part-through cracks

This paper presents a fracture mechanics based design procedure for pressurized components having part-through cracks. Fracture critical stresses are evaluated for the assumed crack configurations and the fracture envelopes are obtained for these configurations. For small size crack configurations, Feddersen's type correction is made for proper estimation of critical stress values. Residual strength of unbroken ligament is a more appropriate criterion for the design of thin sections with parth-through cracks. A criterion based on the ligament failure is defined for cases where the critical fracture stresses are above the material yield stress value. The procedure is illustrated by presenting an example of a pressurized tankage used in aerospace industry.     

On the effect of pulse loads on dynamic rock fracture

The objective of this research was to develop an analytical model which is capable of predicting the fracture pattern in the rock surrounding a wellbore upon which a dynamic loading pulse is applied. The development of the model followed the principle of elastodynamic fracture mechanics. Specifically, inherent cracks are assumed to exist around the edge of the wellbore. Depending on the pulse shape, these cracks are activated to form various fracture patterns. The criterion for initiation of crack growth is determined by the magnitude of the dynamic stress intensity factor associated with the crack. The finite element code HONDO was used to calculate the dynamic stress intensity factors. Analytical predictions were correlated with field experimental data and good agreements were found.     

U形切口的有限断裂准则

有限断裂力学准则综合了应力和能量参数,假设裂纹或切口端部有限裂纹长度的增长．特别地,该有限裂纹的长度不是材料的基本常数,而是与构件的结构有关．基于U形切口两种形式：点方式和线方式有限断裂准则,对文献中的铝合金U形切口三点弯曲断裂实验进行了分析验证．一方面基于材料的断裂韧度和抗拉强度,预测切口件断裂载荷;另一方面根据几组不同的切口根部半径及其对应的临界切口应力强度因子,同时估算材料的断裂参数：断裂韧度和抗拉强度．将点方式和线方式两种不同形式有限断裂准则的预测结果,与平均周向应力准则、最大周向应力准则以及文献中相关结果进行了比较得出：无论是预测断裂载荷还是估算材料断裂参数,线方式有限断裂准则,与文献中相关结果比较吻合,尤其是估算的断裂韧度精度较高．     

Problems of dynamic fracture mechanics without contact of the crack faces

Conclusion In this, the first part of the survey we have discussed only certain aspects of dynamic fracture mechanics. The surveyed material has been selected with a preference for the most highly developed parts of the theory, specifically those elements which have direct bearing on the second part of the survey. We have also included information on the dynamic fracture mechanics of initially stressed materials, in the development of which one of the authors has been a major contributor.Since many problems of dynamic fracture mechanics have been overlooked in the survey, we have added supplementary references to the literature. Various aspects of the strength and fracture of materials under dynamic loading are set forth in [11, 12, 40, 57, 60, 73, 80, 83]. Criteria of the initiation, motion, branching, and arrest of cracks are discussed in [7, 9, 60, 102, 111, 113, 124]. Among the most interesting elements of dynamic fracture mechanics are the problems of crack propagation. Certain analytical results pertinent to this topic have been obtained in [43–45, 47, 67–72, 78, 87, 92, 96, 97].S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Highway Engineering University, Kharkov. Translated from Prikladnaya Mekhanika, Vol. 30, No. 10, pp. 3–28, October, 1994.