脆性断裂的非局部力学理论

本文提出一种脆性材料断裂的非局部力学理论,内容包括:Ⅰ、Ⅱ、Ⅲ型Griffith裂纹的非局部弹性应力场,裂纹尖端邻域非局部弹性应力场的渐近形式,脆性开裂的最大拉应力准则。文中给出了这种理论应用于三种基本型裂纹和Ⅰ-Ⅱ、Ⅰ-Ⅲ复合型裂纹临界开裂条件的计算结果,并把它们与一些试验资料和最小应变能密度因子理论进行了对比。     

Mechanics of crack propagation in materials with initial (residual) stresses (review)

Major results on the mechanics of crack propagation in materials with initial (residual) stresses are analyzed. The case of straight cracks of constant width that propagate at a constant speed in a material with initial (residual) stresses acting along the cracks is examined. The results were obtained, based on linearized solid mechanics, in a universal form for isotropic and orthotropic, compressible and incompressible elastic materials with an arbitrary elastic potential in the cases of finite (large) and small initial strains. The stresses and displacements in the linearized theory are expressed in terms of analytical functions of complex variables when solving dynamic plane and antiplane problems. These complex variables depend on the crack propagation rate and the material properties. The exact solutions analyzed were obtained for growing (mode I, II, III) cracks and the case of wedging by using methods of complex variable theory, such as Riemann–Hilbert problem methods and the Keldysh–Sedov formula. As the initial (residual) stresses tend to zero, these exact solutions of linearized solid mechanics transform into the respective exact solutions of classical linear solid mechanics based on the Muskhelishvili, Lekhnitskii, and Galin complex representations. New mechanical effects in the dynamic problems under consideration are analyzed. The influence of initial (residual) stresses and crack propagation rate is established. In addition, the following two related problems are briefly analyzed within the framework of linearized solid mechanics: growing cracks at the interface of two materials with initial (residual) stresses and brittle fracture under compression along cracks     

Perfectly plastic stress field at a stationary crack tip

Under the hypothesis that all the perfectly plastic stress components at a orach tip are the functions of θ only, making use of yield conditions and equilibrium equations. we derive the generally analytical expressions of the perfectly plastic stress field at a crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the tips of Mode Ⅰ Mode Ⅱ, Mode Ⅲ and Mixed Mode Ⅰ-Ⅱ cracks are obtained.     

Bond-based peridynamic modelling of singular and nonsingular crack-tip fields

A static meshfree implementation of the bond-based peridynamics formulation for linearly elastic solids is applied to the study of the transition from local to nonlocal behavior of the stress and displacement fields in the vicinity of a crack front and other sources of stress concentration. The long-range nature of the interactions between material points that is intrinsic to and can be modulated within peridynamics enables the smooth transition from the square-root singular stress fields predicted by the classical (local) linear theory of elasticity, to the nonsingular fields associated with nonlocal theories. The accuracy of the peridynamics scheme and the transition from local to nonlocal behavior, which are dictated by the lattice spacing and micromodulus function, are assessed by performing an analysis of the boundary layer that surrounds the front of a two dimensional crack subjected to mode-I loading and of a cracked plate subjected to far-field tension.     

Experimental analysis of crack tip fields in rubber materials under large deformation

A three-nested-deformation model is proposed to describe crack-tip fields in rubber-like materials with large deformation.The model is inspired by the distribution of the measured in-plane and out-of-plane deformation.The inplane displacement of crack-tip fields under both Mode I and mixed-mode(Mode I-II) fracture conditions is measured by using the digital Moire’ method.The deformation characteristics and experimental sector division mode are investigated by comparing the measured displacement fields under different fracture modes.The out-of-plane displacement field near the crack tip is measured using the three-dimensional digital speckle correlation method.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

Anisotropic plastic stress field near a singular point

On condition that any perfectly plastic stress component near a singular point is nothing but the function of θ only, making use of equilibrium equations and Hill anisotropic yield condition, we derive the general analytical expressions of the anisotropic plastic stress field near a singular point in both the cases of anti-plane and in-plane strains. Applying these general analytical expressions to the concrete cracks and the plane-strain bodies with a singular point, the anisotropic plastic stress fields at the tips of Mode Ⅰ, Mode Ⅱ, Mode Ⅲ and mixed mode Ⅰ-Ⅱ cracks, and the limit loads of anisotropic plastic plane-strain bodies with a singular point are obtained.     

平面幂律塑性Ⅰ型裂纹尖端应力场全解

在航空航天、船舶、石油管道和核电等领域,服役结构或部件在长期极端条件下运行,不可避免地会产生裂纹,因此,为研究含裂纹结构的准静态断裂行为,必须了解裂纹尖端附近区域的应力应变场特点.对于幂律材料裂纹构元,研究平面应变和平面应力条件下Ⅰ型裂纹尖端应力场的解析分布.基于能量密度等效和量纲分析,推导了能量密度中值点代表性体积单元(representative volume element, RVE)的等效应力解析方程,并定义其为应力因子,进而针对有限平面应变和平面应力紧凑拉伸(compact tension, CT)试样和单边裂纹弯曲(single edge bend, SEB)试样,以应力因子作为应力特征量,并构造用于表征裂尖等效应力等值线的蝶翅轮廓式和扇贝轮廓式三角特殊函数,提出描述幂律塑性条件下平面I型裂纹尖端应力场的半解析模型.该半解析模型形式简单,对CT和SEB试样的裂尖应力场的预测结果与有限元分析的结果比较表明,两者之间均密切吻合,模型公式可直接用于预测Ⅰ型裂纹尖端应力分布,方便于断裂安全评价和理论发展.     

含初始裂纹巴西圆盘劈裂问题的非局部近 场动力学建模

巴西圆盘劈裂是弹性力学及岩石力学与工程中的经典问题。在非局部键型近场动力 学理论的基础上,引入物质点对的转动自由度构建双参数微观弹脆性近场动力学本构力模型 以突破常规模型的应用范围限制,并考虑岩石混凝土类材料的宏观拉压异性和断裂特征。引 入动态松弛、粒子系统力边界条件和系统平衡弛豫等算法,实现了含不同倾角中心裂纹巴西 圆盘受压劈裂破坏全过程的近场动力学数值模拟,裂纹扩展路径及破坏形式均与试验结果高 度吻合,为裂纹扩展和断裂破坏问题的数值模拟提供了新的选择。     

Influence of the root radius of crack-like notches on the fracture load of brittle components

Narrow notches often cause damage that can lead to the destruction of components. The stress field in the vicinity of such crack-like notches in two-dimensional (2D) structures is similar to the stress field around equivalent cracks. Therefore similar investigations are necessary to predict the fracture load for components with cracks or narrow notches. Thus, the asymptotical stress field for a narrow notch with a rounded notch root is deduced from an Airy’s stress function. Based on this stress field a fracture criterion is developed. Comparing the theoretical fracture limit curves derived from the fracture criterion with experimental results it can be shown that for brittle material the local stress state at the fracture initiation point is the same for mode I, mixed-mode and mode II loading.     

Nonlocal theory solution of two collinear cracks in the functionally graded materials

In this paper, the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti-plane shear loading is investigated by means of nonlocal theory. The traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near the crack tips. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. The magnitude of the finite stress field depends on the crack length, the distance between two cracks, the parameter describing the functionally graded materials and the lattice parameter of the materials.     

The mixed mode brittle fracture criteria in siliding mode fracture

It is well-known that the present mixed mode brittle fracture criteria are all theopening mode fracture criterion.We consider that mixed mode brittle fracture of slidingmode fracture exists too.Hence we propose three criteria of mixed mode brittle fracture ofsliding mode fracture;:the radial shearing stress criterion,the maximum shearing stresscriterion and the distortional strain-energy-density criterion.Thus,we can overall explainthe phenomena of brittle fracture in the structural elements with cracks.     

Steady growth of cracks in compressible elastic-plastic media

Based on the theoretical framework for crack growth analysis provided by Gao and Hwang, the 5-sector solution of near-tip fields of mode-I cracks growing quasi-statically and steadily in compressible elastic perfectly plastic materials is obtained. As Poisson's ratio ν tends to 1/2, the 5-sector solution degenerates to the 4-sector solution of near-tip fields of crack growth in incompressible elastic perfectly plastic materials. The project supported by the National Natural Science Foundation of China.     

Mode III fracture problem of a cracked FGPM surface layer bonded to a cracked FGPM substrate

This work deals with the mode III fracture problem of a cracked functionally graded piezoelectric surface layer bonded to a cracked functionally graded piezoelectric substrate. The cracks are normal to the interface and the electro-elastic material properties are assumed to be varied along the crack direction. Potential and flux types of boundary condition are assigned on the edge of the surface layer. The problem under the assumptions of impermeable and permeable cracks can be formulated to the standard singular integral equations, which are solved by using the Gauss–Chebyshev technique. The effects of the boundary conditions, the material properties and crack interaction on the stress and electric displacement intensity factors are discussed.     

Effect of the parameters of the biaxial field of rock pressure on the shape of the fracture zone formed by an explosion of a cylindrical charge in a brittle medium

The effect of the biaxial field of external rock pressure on the deformation of the fracture zone formed by radial cracks in an elastic-brittle medium is studied. We consider cylindrical charges that are rather thin compared to the diameter of the explosive borehole. This allows one to exclude the grinding zone from consideration. At the initial moment of time, the system of emerging cracks originating at the boundary of the circular hole is assumed to be symmetric. To solve the problem, we use singular integral equations and the fracture criterion σ_ϑ. The propagation trajectories of the system cracks are calculated in the quasistatic approximation in a step-by-step manner in relation to the parameters of the external compressive stress field. Two ideal variants of loading of the crack system are analyzed. In the first variant, gaseous detonation products penetrate into cracks, and the pressure in the explosion cavity and the cracks instantaneously equalizes. In the second variant, gases do not penetrate into the cracks of the system. The fracture zone is shown to become an ellipse whose long axis is oriented in the direction of the largest compressive stress in magnitude acting at infinity. The effect of the variants of loading of the crack system on the shape and dimensions of the deformed fracture zone is evaluated. Mining Institute, Siberian Division, Russian Academy of Sciences, Novosibirsk 630091. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 129–137, January–February, 1998.     

Effect of intrinsic stress gradient on the effective mode-I fracture toughness of amorphous diamond-like carbon films for MEMS

The influence of intrinsic stress gradient on the mode-I fracture of thin films with various thicknesses fabricated for Microelectromechanical Systems (MEMS) was investigated. The material system employed in this study was hydrogen-free tetrahedral amorphous diamond-like carbon (ta-C). Uniform gauge microscale specimens with thicknesses 0.5, 1, 2.2, and 3 μm, containing mathematically sharp edge pre-cracks were tested under mode-I loading in fixed grip configuration. The effective opening mode fracture toughness, as calculated from boundary force measurements, was 4.25±0.7 MPa√m for 0.5-μm thick specimens, 4.4±0.4 MPa√m for 1-μm specimens, 3.74±0.3 MPa√m for 2.2-μm specimens, and 3.06±0.17 MPa√m for 3-μm specimens. Thus, the apparent fracture toughness decreased with increasing film thickness. Local elastic property measurements showed no substantial change as a function of film thickness, which provided evidence for the stability of the sp²/sp³ carbon binding stoichiometry in films of different thicknesses. Detailed experiments and finite element analysis pointed out that the dependence of the effective fracture toughness on specimen thickness was due to the intrinsic stress gradient developed during fabrication and post-process annealing. This stress gradient is usually unaccounted for in mode-I fracture experiments with thin films. Thicker films, fabricated from multiple thin layers, underwent annealing for extended times, which resulted in a stress gradient across their thickness. This stress gradient caused an out-of-plane curvature upon film release from its substrate and, thus, combined bending and tensile mode-I loading at the crack tip under in-plane forces. Since the bending component cannot be isolated from the applied boundary force measurements, its contribution, becoming important for thick films, remains unaccounted for in the calculation of the critical stress intensity factor, thus resulting in reduced apparent fracture toughness that varies with film thickness and curvature. It was concluded that in the presence of a stress gradient, accounting only for the average intrinsic stresses could lead in an overestimate of the fracture resistance of a brittle film. Under these considerations the material fracture toughness of ta-C, as determined from specimens with negligible curvature, is K_IC=4.4±0.4 MPa√m.     

Interacting cracks and ellipsoidal inhomogeneities by the equivalent inclusion method

Based on the Eshelby's equivalent inclusion method (EIM) and Hill's theorem on discontinuities of elastic fields across the interfaces, a theory for the determination of the stress intensity factors (SIFs) of arbitrarily oriented interacting cracks under non-uniform far-field applied stress (strain) is developed. As shown in this investigation the EIM proposed by Moschovidis and Mura can be extended for treatment of such problems, but their formulations are quite cumbersome and computationally inefficient. An alternative analytical approach is proposed that is computationally more efficient, and unlike the method of Moschovidis and Mura can easily handle complex problems of interacting inhomogeneities and cracks. It is seen that as the interaction between the inhomogeneities becomes stronger, this method yields results that are closer to the solutions reported in the literature than the solutions obtained using the extended EIM of Moschovidis and Mura, which is developed herein. Problems involving combinations of interacting elliptic and penny shape cracks and inhomogeneities are excellent candidates for demonstration of the accuracy and robustness of the present theory, for which the previous EIM produces less accurate results. Due to the limitations imposed on the existing methods, every reported treatment has been tailored for a certain category of problems, and only uniform far-field loadings have been remedied. In contrast, the present theory is more general than the previously reported theories and it encompasses interacting cracks having a variety of geometries subjected to non-uniform far-field applied stress (strain); moreover, it is applicable to modes, I, II, III, and mixed mode fracture.     

Crack path kinking under generalized stress state

Results of experimental investigations of three-point asymmetric bending applied to beams made of materials with substantially different structural, strength, and strain characteristics are presented. The effect of different types of fracture on the formation of crack kinks under a generalized stress state is revealed. Strength curves of the Coulomb-Mohr type are obtained for Plexiglas and fine-grained concrete, which are characterized by the quasi-brittle and brittle types of fracture, respectively. Dependences of the kink angles of the crack paths on the stress state in the case of three-point bending are described for these materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 205–213, May–June, 2009.     

Comparison of finite element techniques for 2D and 3D crack analysis under impact loading

For several technical applications the dynamic aspect in fracture mechanics cannot be neglected. When the reliability of components with macroscopic cracks has to be assessed, the consideration of dynamic effects may lead to much higher stress intensity factors than under static conditions. In this paper three different methods to calculate the dynamic stress intensity factor for the mode-I loading of stationary cracks are compared. Based on two- and three-dimensional finite element simulations, the dynamic stress intensity factor is computed with the dynamic J-integral, the modified crack closure integral and the displacement interpretation method. The theoretical fundamentals of all three methods are summarized in the paper and the numerical implementation is explained briefly. Results for different models are shown and compared to findings in the literature.     

Dynamic problems of the mechanics of the brittle fracture of materials with initial stresses for moving cracks. 1. Problem statement and general relationships

This paper studies the brittle fracture of materials with initial stresses when the stresses act only along the cracks. Dynamic problems for moving cracks are considered when the plane crack infinite in one direction moves with constant velocity. General formulas are presented for compressible and incompressible elastic bodies with an arbitrary structure of elastic potentials. The stresses and displacements are presented as analytical functions of complex variables. Some general relationships may be used in order to obtain exact information on the singularity order of the crack tip for dynamical problems under consideration in the general formulation. Translated from Prikladnaya Mechanika, Vol. 34, No. 12, pp. 3–15, December, 1998.