Microscopic analysis of crack propagation for multiple cracks, inclusions and voids

The elastic crack interaction with internal defects, such as microcracks, voids and rigid inclusions, is investigated in this study for the purpose of analyzing crack propagation. The elastic stress field is obtained using linear theory of elasticity for isotropic materials. The cracks are modeled as pile-ups of edge dislocations resulting into a coupled set of integral equations, whose kernels are those of a dislocation in a medium with or without an inclusion or void. The numerical solution of these equations gives the stress intensity factors and the complete stress field in the given domain. The solution is valid for a general solid, however the propagation analysis is valid mostly for brittle materials. Among different propagation models the ones based on maximum circumferential stress and minimum strain energy density theories, are employed. A special emphasis is given to the estimation of the crack propagation direction that defines the direction of crack branching or kinking. Once a propagation direction is determined, an improved model dealing with kinked cracks must be employed to follow the propagation behavior.     

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.     

Combined analysis of fracture under stresses acting along cracks

A joint approach to the study of two non-classical fracture mechanisms, namely fracture of cracked materials with initial (residual) stresses acting along the crack planes and fracture under compression along parallel cracks, is considered in the framework of three-dimensional linearized solid mechanics. Mathematical statements of problems for pre-stressed solids that contain interacting circular cracks are given. Problems for an infinite solid containing two parallel co-axial cracks and for a space with the periodical set of co-axial parallel cracks as well as for a half-space with near-the-surface crack are solved. Several patterns of loading on the crack faces (normal loading, radial shear and torsion) are considered. The effects of initial stresses on stress intensity factors are analyzed for highly elastic materials with some types of elastic potentials. Formulation of fracture criteria accounting effect of initial (residual) stresses is given. Critical parameters of fracture of solids containing interacting cracks under compression along the cracks are calculated. The influence of geometrical parameters of the problems as well as physical and mechanical properties of materials on these critical parameters is analyzed.     

带裂纹的椭圆孔口问题的应力分析

断裂现象与材料和结构中的孔洞、缺口或裂纹等缺陷密切相关,这是因为缺陷附近的应力集中明显.该文利用复变方法,通过保角映射研究了带裂纹的椭圆孔洞的平面弹性问题,给出了应力强度因子的解析解.并由此计算了两互相垂直的裂纹问题.     

裂纹体分析的权函数理论与应用: 回顾和展望

断裂力学是工程材料和结构的疲劳与断裂分析、损伤容限设计和结构完整性评定的理论基础. 应力强度因子作为线弹性裂纹尖端奇异场的单一表征参量和裂纹扩展驱动力, 在裂纹体的断裂力学分析中发挥着关键作用. 权函数法为复杂受载裂纹体的应力强度因子求解计算提供了强有力的解析工具, 不但具有远高于各类数值解法的计算效率, 而且精度可靠, 使用方便. 本文结合笔者团队在权函数法方面的长期研究工作, 对该方法自20世纪70年代初提出至今半个世纪以来, 国际断裂界在二维和三维权函数理论与应用方面的主要研究进展作了回顾和评述, 并对其未来发展提出了展望. 主要内容涵盖: 当前国际断裂界广泛应用的3种二维裂纹解析权函数法简介和以格林函数为基准的验证评价; 三维裂纹问题的片条合成权函数法和点载荷权函数法; 权函数法在复杂受载裂纹体的应力强度因子和裂纹张开位移等关键力学参量计算、内聚力/桥连等裂纹模型分析、共线多裂纹权函数理论及其在剩余强度预测等方面的应用, 以及复杂裂纹几何的工程化权函数分析和权函数法的反向应用问题.      

玄武岩中裂隙分布形式对声波传播的影响

复杂岩体含有大量的裂隙,这些裂隙尺寸及其分布形式等对弹性波传播都有很大的影响.本文加工了含单个裂隙、双裂隙和三个裂隙的玄武岩岩样单元对其进行组合,进行了25kHz、 50kHz、 400kHz、 600kHz和1000kHz 等5种频率的声波测试.通过考虑垂直或平行波传播方向的裂隙长度,来探索裂隙分布形式和不同裂隙长度对弹性波传播的影响,研究玄武岩的频散效应和波的衰减.结果表明:裂隙方向与波传播方向夹角对弹性波传播有很大的影响.当裂隙方向与波传播方向垂直时,散射效应最大;而当裂隙方向与波传播方向平行时,影响最小.上述结果可为理论模型和数值分析提供依据.     

Numerical analysis of quasi-static crack branching in brittle solids by a modified displacement discontinuity method

Mechanism of quasi-static crack branching in brittle solids has been analyzed by a modified displacement discontinuity method. It has been assumed that the pre-existing cracks in brittle solids may propagate at the crack tips due to the initiation and propagation of the kink (or wing) cracks. The originated wing cracks will act as new cracks and can be further propagated from their tips according to the linear elastic fracture mechanics (LEFM) theory. The kink displacement discontinuity formulations (considering the linear and quadratic interpolation functions) are specially developed to calculate the displacement discontinuities for the left and right sides of a kink point so that the first and second mode kink stress intensity factors can be estimated. The crack tips are also treated by boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip. The propagating direction of the secondary cracks can be predicted by using the maximum tangential stress criterion. An iterative algorithm is used to predict the crack propagating path assuming an incremental increase of the crack length in the predicted direction (straight and curved cracks have been treated). The same approach has been used for estimating the crack propagating direction and path of the original and wing cracks considering the special crack tip elements. Some example problems are numerically solved assuming quasi-static conditions. These results are compared with the corresponding experimental and numerical results given in the literature. This comparison validates the accuracy and applicability of the proposed method.     

Edge cracks in plastically deforming surface grains

A selection of surface crack problems is presented to provide insights into Stage I and early Stage II fatigue crack growth. Edge cracks at 45^o and 90^o to the surface are considered for cracks growing in single crystals. Both single crystal slip and conventional plasticity are employed as constitutive models. Edge cracks at 45^o to the surface are considered that either (i) kink in the direction perpendicular to the surface, or (ii) approach a grain boundary across which only elastic deformations occur.     

Dynamic fracture of idealized fiber-reinforced materials

Dynamic plane stress of sheets composed of two orthogonal families of inextensible fibers, with infinitesimal elastic shearing stress response, is considered. The fibers through the tip of a propagating tear or crack carry finite forces. Fracture criteria that can be expressed in terms of these tip forces are discussed. In a particular example it is shown that the maximum energy release rate criterion leads to a circular crack trajectory, while the so-called critical force and critical stress criteria imply that the crack is L-shaped, like cracks or tears in real fibrous materials.     

Growth predictions of two interacting cracks

The elastic fracture behavior of a plate subjected to uniform stress surrounding two equal cracks inclined at an angle is investigated. The orientation of the crack plane with applied stress can be varied. Among the cases are: (1) two cracks inclined symmetrically with respect to the vertical and horizontal applied stress, and (2) one crack is horizontal while the other is inclined to the vertical applied stress. The strain energy density criterion is used for determining the combined crack and load arrangement that correspond to the lowest critical load at global instability. The direction of crack initiation is also determined. Quantitative results pertaining to the fracture characteristics are given in graphical forms.     

Stable growth of a central crack

The elastic fracture behavior of a plate subjected to uniform stress surrounding two equal cracks inclined at an angle is investigated. The orientation of the crack plane with applied stress can be varied. Among the cases are: (1) two cracks inclined symmetrically with respect to the vertical and horizontal applied stress, and (2) one crack is horizontal while the other is inclined to the vertical applied stress. The strain energy density criterion is used for determining the combined crack and load arrangement that correspond to the lowest critical load at global instability. The direction of crack initiation is also determined. Quantitative results pertaining to the fracture characteristics are given in graphical forms.     

Antiplane mechanical and inplane electric time-dependent load applied to two coplanar cracks in piezoelectric ceramic material

This work is concerned with the dynamic response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric time-dependent load. The cracks are assumed to act either as an insulator or as a conductor. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain. A numerical Laplace inversion algorithm is used to determine the dynamic stress and electric displacement factors that depend on time and geometry. A normalized equivalent parameter describing the ratio of the equivalent magnitude of electric load to that of mechanical load is introduced in the numerical computation of the dynamic stress intensity factor (DSIF) which has a similar trend as that for the pure elastic material. The results show that the dynamic electric field will impede or enhance crack propagation in a piezoelectric ceramic material at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the crack length to the ligament between the cracks. The stress and electric displacement intensity factor can be combined by the energy density factor or function to address the fracture of piezoelectric materials under the combined influence of electromechanical loading.     

压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向刚性压电材料中受拉伸和电载荷作用的平片裂纹Ⅰ型断裂力学问题．使用自限部分概念，从二维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意．     

Three-dimensional elasticity problems related to cracks: Exact solution by inversion transformation

Prior to determining the conditions of brittle and quasi-brittle fracture of elastic solids with cracks, it is necessary to solve the corresponding three-dimensional elasticity problem. Since analytical solutions are known only for certain simple configurations such as an infinite space containing a plane, penny-shaped, or elliptical crack, and a half-plane or a strip crack [1–3], numerical procedure may have to be used.This paper is concerned with a tensile crack (or Mode I crack) on a plane domain in infinite elastic medium. A technique is proposed for constructing exact solutions for crack configurations that are obtainable by inversion transformation from the canonical contours such as a circle, ellipse or half-plane, for which exact solutions are known. Simple formulae are derived which yield the stress intensity factor distribution over the transformed crack boundary with loads appropriately adjusted according to the initial crack region. Hence, it is not necessary to find the complete solution. Solutions obtained by this method are presented for cracks bounded by convex as well as nonconvex contour, i.e., oval and sickle-shape contours.     

A finite difference method for elastic wave scattering by a planar crack with contacting faces

This paper presents a finite difference time-domain technique for 2D problems of elastic wave scattering by cracks with interacting faces. The proposed technique introduces cracks into the finite difference model using a set of split computational nodes. The split-node pair is bound together when the crack is closed while the nodes move freely when open, thereby a unilateral contact condition is considered. The development of the open/close status is determined by solving the equation of motion so as to yield a non-negative crack opening displacement. To check validity of the proposed scheme, 1D and 2D scattering problems for which exact solutions are known are solved numerically. The 1D problem demonstrates accuracy and stability of the scheme in the presence of the crack-face interaction. The 2D problem, in which the crack-face interaction is not considered, shows that the proposed scheme can properly reproduce the stress singularity at the tip of the crack. Finally, scattered fields from cracks with interacting faces are investigated assuming a stick and a frictionless contact conditions. In particular, the directivity and higher-harmonics are investigated in conjunction with the pre-stress since those are the basic information required for a successful ultrasonic testing of closed cracks.     

Evolving crack patterns in thin films with the extended finite element method

This paper develops the extended finite element method (XFEM) to evolve patterns of multiple cracks, in a brittle thin film bonded to an elastic substrate, with a relatively coarse mesh, and without remeshing during evolution. A shear lag model describes the deformation in three dimensions with approximate field equations in two-dimensions. The film is susceptible to subcritical cracking, obeying a kinetic law that relates the velocity of each crack to its energy release rate. At a given time, the XFEM solves the field equations and calculates the energy release rate of every crack. For a small time step, each crack is extended in the direction of maximal hoop stress, and by a length set by the kinetic law. To confirm the accuracy of the XFEM, we compare our simulation to the exiting solutions for several simple crack patterns, such as a single crack and a set of parallel cracks. We then simulate the evolution of multiple cracks, initially in a small region of the film but of different lengths, showing curved crack propagation and crack tip shielding. Starting with multiple small cracks throughout the film, the XFEM can generate the well-known mud crack pattern.     

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     

Longitudinal and Bending Stresses in a Beam with Saw Cut Edge Cracks∗

Abstract

The Griffith-Irwin theory of brittle fracture of elastic solids predicts the propagation of cracks on the basis of the energy release rate. This depends upon the stress intensity factors for a given crack configuration. The present paper provides these informations for the problem of an infinite number of periodic, non-coplanar, parallel edge cracks in a strip. Two types of crack configurations, namely, periodic cracks of equal length starting from one edge and a set of two coplanar symmetrical edge cracks of equal length are solved for constant and linearly varying pressure distributions. These problems arise naturally in structural mechanics while investigating stresses in extension and bending of cracked strips. Final results are obtained from the numerical solution of certain Fredholm integral equations of the second kind derived from a dual series of Papkovich-Fadle eigenfunctions     

Crack-tip stress fields in functionally graded materials with linearly varying properties

Crack-tip stress fields for a stationary crack along or inclined to the direction of property gradation in functionally graded materials (FGMs) are obtained through an asymptotic analysis coupled with Westergaard’s stress function approach. The elastic modulus of the FGM is assumed to vary linearly along the gradation direction. The first six terms for a crack along the direction of property variation and first four terms for a crack inclined to the direction of property variation in the expansion of the stress field are derived to explicitly bring out the influence of nonhomogeneity on the structure of the stress field. Using these stress fields, contours of constant maximum shear stress and constant out of plane displacement are generated and the effect of inclination of property gradation direction on these contours is discussed. The strain energy density criterion is applied to obtain critical conditions for crack initiation and the effect of property gradation is discussed. It is shown that the materials with varying properties can offer more resistance to crack propagation and will suppress crack growth in some situations.