Theoretical analysis of crack front instability in mode I+III

This paper focusses on the theoretical prediction of the widely observed crack front instability in mode I+III, that causes both the crack surface and crack front to deviate from planar and straight shapes, respectively. This problem is addressed within the classical framework of fracture mechanics, where the crack front evolution is governed by conditions of constant energy-release-rate (Griffith criterion) and vanishing stress intensity factor of mode II (principle of local symmetry) along the front. The formulation of the linear stability problem for the evolution of small perturbations of the crack front exploits previous results of Movchan et al. (1998) (suitably extended) and Gao and Rice (1986), which are used to derive expressions for the variations of the stress intensity factors along the front resulting from both in-plane and out-of-plane perturbations. We find exact eigenmode solutions to this problem, which correspond to perturbations of the crack front that are shaped as elliptic helices with their axis coinciding with the unperturbed straight front and an amplitude exponentially growing or decaying along the propagation direction. Exponential growth corresponding to unstable propagation occurs when the ratio of the unperturbed mode III to mode I stress intensity factors exceeds some “threshold” depending on Poisson's ratio. Moreover, the growth rate of helical perturbations is inversely proportional to their wavelength along the front. This growth rate therefore diverges when this wavelength goes to zero, which emphasizes the need for some “regularization” of crack propagation laws at very short scales. This divergence also reveals an interesting similarity between crack front instability in mode I+III and well-known growth front instabilities of interfaces governed by a Laplacian or diffusion field.     

Intergranular strain evolution near fatigue crack tips in polycrystalline metals

The deformation field near a steady fatigue crack includes a plastic zone in front of the crack tip and a plastic wake behind it, and the magnitude, distribution, and history of the residual strain along the crack path depend on the stress multiaxiality, material properties, and history of stress intensity factor and crack growth rate. An in situ, full-field, non-destructive measurement of lattice strain (which relies on the intergranular interactions of the inhomogeneous deformation fields in neighboring grains) by neutron diffraction techniques has been performed for the fatigue test of a Ni-based superalloy compact tension specimen. These microscopic grain level measurements provided unprecedented information on the fatigue growth mechanisms. A two-scale model is developed to predict the lattice strain evolution near fatigue crack tips in polycrystalline materials. An irreversible, hysteretic cohesive interface model is adopted to simulate a steady fatigue crack, which allows us to generate the stress/strain distribution and history near the fatigue crack tip. The continuum deformation history is used as inputs for the micromechanical analysis of lattice strain evolution using the slip-based crystal plasticity model, thus making a mechanistic connection between macro- and micro-strains. Predictions from perfect grain-boundary simulations exhibit the same lattice strain distributions as in neutron diffraction measurements, except for discrepancies near the crack tip within about one-tenth of the plastic zone size. By considering the intergranular damage, which leads to vanishing intergranular strains as damage proceeds, we find a significantly improved agreement between predicted and measured lattice strains inside the fatigue process zone. Consequently, the intergranular damage near fatigue crack tip is concluded to be responsible for fatigue crack growth.     

Modeling of fatigue crack growth from a notch

When a fatigue crack is nucleated and propagates into the vicinity of the notch, the crack growth rate is generally higher than that can be expected by using the stress intensity factor concept. The current study attempted to describe the crack growth at notches quantitatively with a detailed consideration of the cyclic plasticity of the material. An elastic–plastic finite element analysis was conducted to obtain the stress and strain histories of the notched component. A single multiaxial fatigue criterion was used to determine the crack initiation from the notch and the subsequent crack growth. Round compact specimens made of 1070 steel were subjected to Mode I cyclic loading with different R-ratios at room temperature. The approach developed was able to quantitatively capture the crack growth behavior near the notch. When the R-ratio was positive, the crack growth near a notch was mainly influenced by the plasticity created by the notch and the resulted fatigue damage during crack initiation. When the R-ratio was negative, the contact of the cracked surfaces during a part of a loading cycle reduced the cyclic plasticity of the material near the crack tip. The combined effect of notch plasticity and possible contact of cracked surface were responsible for the observed crack growth phenomenon near a notch.     

Molecular-dynamics simulation-based cohesive zone representation of intergranular fracture processes in aluminum

A traction-displacement relationship that may be embedded into a cohesive zone model for microscale problems of intergranular fracture is extracted from atomistic molecular-dynamics (MD) simulations. An MD model for crack propagation under steady-state conditions is developed to analyze intergranular fracture along a flat Σ99 [1 1 0] symmetric tilt grain boundary in aluminum. Under hydrostatic tensile load, the simulation reveals asymmetric crack propagation in the two opposite directions along the grain boundary. In one direction, the crack propagates in a brittle manner by cleavage with very little or no dislocation emission, and in the other direction, the propagation is ductile through the mechanism of deformation twinning. This behavior is consistent with the Rice criterion for cleavage vs. dislocation blunting transition at the crack tip. The preference for twinning to dislocation slip is in agreement with the predictions of the Tadmor and Hai criterion. A comparison with finite element calculations shows that while the stress field around the brittle crack tip follows the expected elastic solution for the given boundary conditions of the model, the stress field around the twinning crack tip has a strong plastic contribution. Through the definition of a Cohesive-Zone-Volume-Element—an atomistic analog to a continuum cohesive zone model element—the results from the MD simulation are recast to obtain an average continuum traction-displacement relationship to represent cohesive zone interaction along a characteristic length of the grain boundary interface for the cases of ductile and brittle decohesion.     

Stress intensity factor analysis of a three-dimensional interfacial corner between anisotropic bimaterials under thermal stress

A numerical method using a path-independent H-integral based on the conservation integral was developed to analyze the singular stress field of a three-dimensional interfacial corner between anisotropic bimaterials under thermal stress. In the present method, the shape of the corner front is smooth. According to the theory of linear elasticity, asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch of the materials’ elastic constants. The eigenvalues and the eigenfunctions are obtained using the Williams eigenfunction method, which depends on the anisotropic materials’ properties and the geometry of an interfacial corner. The order of the singularity related to the eigenvalue is real, complex or power-logarithmic. The amplitudes of the singular stress terms can be calculated using the H-integral. The stress and displacement around an interfacial corner for the H-integral are obtained using finite element analysis. In this study, a proposed definition of the stress intensity factors of an interfacial corner, which includes those of an interfacial crack and a homogeneous crack, is used to evaluate the singular stress fields. Asymptotic solutions of stress and displacement around an interfacial corner front are uniquely obtained using these stress intensity factors. To prove the accuracy of the present method, several different kinds of examples are shown such as interfacial corners or cracks in three-dimensional structures.     

On the higher order asymptotic analysis of a non-uniformly propagating dynamic crack along an arbitrary path

Transient mixed-mode elastodynamic crack growth along arbitrary smoothly varying paths is considered. Asymptotically, the crack tip stress field is square root singular with the angular variation of the singular term depending weakly on the instantaneous values of the crack tip speed and on the mode-I and mode-II stress intensity factors. However, for a material particle at a small distance away from the moving crack tip, the local stress field will depend not only on the instantaneous values of the crack tip speed and stress intensity factors, but also on the past history of these time dependent quantities. In addition, for cracks propagating along curved paths the stress field is also expected to depend on the nature of the curved crack path. Here, a representation of the crack tip fields in the form of an expansion about the crack tip is obtained in powers of radial distance from the tip. The higher order coefficients of this expansion are found to depend on the time derivative of crack tip speed, the time derivatives of the two stress intensity factors as well as on the instantaneous value of the local curvature of the crack path. It is also demonstrated that even if cracks follow a curved path dictated by the criterion K ₁₁ ^d =0, the stress field may still retain higher order asymmetric components related to non-zero local curvature of the crack path.     

扩展有限元裂尖场精度研究

论述了扩展有限元方法和基本原理,研究了单元类型(四边形单元和三角形单元、线性单元和二次单元)、网格密度、J积分区域半径等因素对裂尖局部应力场(应力强度因子)计算精度的影响。研究发现,上述因素对裂尖应力强度因子计算的收敛速度与稳定性影响不大,证实了XFEM可以用较少的节点获得较高的裂尖场精度,并提出了通过固定裂尖附加区半径可以进一步改善XFEM的收敛速度。     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

Controlling parameter of the stress intensity factors for a planar interfacial crack in three-dimensional bimaterials

Numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three-dimensional bimaterials subjected to tension. The problem is formulated as a system of singular integral equations on the basis of the body force method. In the numerical analysis, unknown body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental density functions are chosen to express singular behavior along the crack front of the interface crack exactly. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary. The stress intensity factors are given with varying the material combination and aspect ratio of the crack. It is found that the stress intensity factors K_I and K_II are determined by the bimaterial constant ε alone, independent of elastic modulus ratio and Poisson’s ratio.     

Effect of grain size on slow fatigue crack propagation and plastic deformation near crack tip

Fatigue crack growth and its threshold are investigated at a stress ratio of 0.5 for the three-point bend specimen made of Austenitic stainless steel. The effect of grain size on the crack tip plastic deformation is investigated. The results show that the threshold value Δk_th increases linearly with the square root of grain size d and the growth rate is slower for materials with larger grain size. The plastic zone size and ratio for different grain sizes are different at the threshold. The maximum stress intensity factor is k_max and σ_ys is the yield strength. At the same time, the characteristics of the plastic deformation development is discontinuous and anti-symmetric as the growth rate is increased from 2·10^—8 to 10⁻⁷ mm/cycle.A dimensionless relation of the form for collating fatigue crack starting growth data is proposed in which Δk_th represents the stress intensity factor range at the threshold. Based on experimental results, this relation attains the value of 0.6 for a fatigue crack to start growth in the Austenitic stainless steel investigated in this work. Metallurgical examinations were also carried out to show a transgranular shear mode of cyclic cleavage and plastic shear.     

Effect of rising elastic modulus in surface layer on stress intensity and gradient

The relaxation element method is applied to obtain the stress field around a crack under normal tension. A surface layer is assumed to surround the crack periphery taken to be in the shape of a narrow ellipse. The elastic modulus within this layer increases from zero to the bulk value of the medium outside. Calculations show that the stresses are finite at the crack tip; they reach a maximum in the layer and then decay to the well known solution of Griffith outside the layer. The influence of plastic deformation on the crack front stresses can also be simulated by the surface layer model. Stress concentration at the crack front is found to be lower when plastic deformation takes place. Sharp decay of stress next to the crack is accompanied by increase of local stress gradients. Severity of the local stress fluctuation depends on the width of the crack surface layer.     

Investigation on 3D fatigue crack propagation in surface-cracked specimens

In the present work the fatigue crack growth in AISI304 specimens is investigated experimentally. In 3D finite element analysis the virtual crack closure technique is applied to calculate distributions and variations of the stress intensity factor along the surface crack front. It is confirmed that the stress intensity factor along the surface crack front varies non-uniformly with crack growth. Crack growth rate is proportional to the stress intensity factor distribution in the 3D cracked specimen. The fatigue crack growth in surface cracked specimens can be described by the Forman model identified in conventional compact tension specimens. For crack growth in the free specimen surface the arc length seems more suitable to quantify crack progress. Geometry and loading configuration of the surface cracked specimen seem to not affect the fatigue crack growth substantially.     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

A technique for studying interaction between a screw dislocation dipole or a concentrated load and a mode III crack crossing an interface

A method of potentially wide application is developed for deriving analytical expressions of the elastic interaction between a screw dislocation dipole or a concentrated force and a crack cutting perpendicularly across the interface of a bimaterial. The cross line composed of the interface and the crack is mapped into a line, and then the complex potentials are educed. The Muskhelishvili method is extended by creating a Plemelj function that matches the singularity of the real crack tips, and eliminates the pseudo tips’ singularity induced by the conformal mapping. The stress field is obtained after solving the Riemann–Hilbert boundary value problem. Based on the stress field expressions, crack tip stress intensity factors, dislocation dipole image forces and image torque are formulated. Numerical curves show that both the translation and rotation must be considered in the static equilibrium of the dipole system. The crack tip stress intensity factor induced by the dipole may rise or drop and the crack may attract or reject the dipole. These trends depend not only on the crack length, but also on the dipole location, the length and the angle of the dipole span. Generally, the horizontal image force exerted at the center of the dislocation dipole is much smaller than the vertical one. Whether the dipole subjected to clockwise torque or anticlockwise torque is determined by whether the Burgers vector of the crack-nearby dislocation of the dipole is positive or negative. A concentrated load induces no singularity to crack tip stress fields as the load is located at the crack line. However, as the concentrated force is not located on the crack line but approaches the crack tip, the nearby crack tip stress intensity factor K_IIIu increases steeply to infinity.     

An electrically conducting interface crack with a contact zone in a piezoelectric bimaterial

A plane problem for an electrically conducting interface crack in a piezoelectric bimaterial is studied. The bimaterial is polarized in the direction orthogonal to the crack faces and loaded by remote tension and shear forces and an electrical field parallel to the crack faces. All fields are assumed to be independent of the coordinate co-directed with the crack front. Using special presentations of electromechanical quantities via sectionally-analytic functions, a combined Dirichlet–Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical and electrical parameters are derived. Also, a contact zone solution is obtained as a particular case. For the determination of the contact zone length, a simple transcendental equation is derived. Stress and electric field intensity factors and, also, the contact zone length are found for various material combinations and different loadings. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Electrically permeable conditions in the crack region are considered as well and matching of different crack models has been performed.     

Crack paths in three-dimensional elastic solids. ii: three-term expansion of the stress intensity factors—applications and perspectives

This work continues the calculation of the stress intensity factors, as a function of position s along the front of an arbitrary (kinked and curved) infinitesimal extension of some arbitrary crack on some three-dimensional body. More precisely, ε denoting a small parameter which the crack extension length is proportional to, what is studied here is the third term, proportional to εfn2 = ε and noted K ⁽¹⁾ (s) ε, of the expansion of these stress intensity factors at the point s of the crack front in powers of ε. The novelties with respect to previous works due to Gao and Rice on the one hand and Nazarov on the other hand, are that both the original crack and its extension need not necessarily be planar, and that a kink (discontinuity of the tangent plane to the crack) can occur all along the original crack front. Two expressions of K ⁽¹⁾ (s) are obtained; the difference is that the first one is more synthetic whereas the second one makes the influence of the kink angle (which can vary along the original crack front) more explicit. Application of some criterion then allows to obtain the apriori unknown geometric parameters of the small crack extension (length, kink angle, curvature parameters). The small scale segmentation of the crack front which is observed experimentally in the presence of mode III is disregarded here because a large scale point of view is adopted; this phenomenon will be discussed in a separate paper. It is shown how these results can be used to numerically predict crack paths over arbitrary distances in three dimensions. Simple applications to problems of configurational stability and bifurcation of the crack front are finally presented.     

Numerical simulations of crack deflection at a twist-misoriented grain boundary between two ideally brittle crystals

Finite-element simulations are used to model crack propagation across twist-misoriented grain boundaries, which are an important source of toughness in lamellar microstructures such as TiAl. We consider a twist grain boundary (GB) between two adjacent grains, and assume that each grain has a single cleavage orientation. The cleavage planes and GB are modeled as a set of cohesive surfaces, and the crack path and effective toughness of the system are simulated using a dynamic finite-element method (FEM). As the crack approaches the GB under remote mode I loading, it is allowed to either deflect along the GB and/or induce the nucleation of a periodic array of cracks in the adjacent grain. The simulations predict (i) a critical toughness ratio between the GB and the cleavage planes for the crack to propagate into the adjacent grain; (ii) an array of cracks in the GB and the twisted grain; (iii) the macroscopic mode I toughness of the solid as a function of a generalized measure of crack length; and (iv) the influence of GB toughness and twist misorientation on the effective toughness of the solid.     

基于数字散斑相关方法测定Ⅰ型裂纹应力强度因子

提出了一种通过数字散斑相关方法测定金属材料Ⅰ型裂纹尖端位置和应力强度因子的实验方法.实验采用疲劳试验机对含Ⅰ型缺口的Cr12MoV钢试件预制裂纹,通过数字散斑相关方法测试试件在三点弯曲加载条件下裂纹的扩展过程及裂尖区域的位移场.将位移场数据代入裂尖位移场方程组,采用牛顿-拉普森方法求解含未知参量的裂尖非线性位移场方程组,计算裂尖位置和应力强度因子.实验结果表明,采用该方法可以准确地测定金属材料Ⅰ型裂纹应力强度因子、裂尖位置及裂纹扩展长度,解决了以往研究中因不能准确测定裂纹尖端位置,而无法准确计算Ⅰ型裂纹裂尖断裂参数的难题,揭示了金属材料裂纹扩展过程中应力强度因子演化特征.     

Crack front penetration in transgranular cleavage cracking in an iron-silicon alloy

Based on the analysis of the crack trapping effect of cleavage ridges and penetration of crack front segments in an iron-silicon alloy, the distance between break-through points along a high-angle grain boundary is investigated. If the break-through points are close to each other, the crack trapping effect is dominant; otherwise, the grain boundary separation in break-through windows governs the front transmission process. The relationship between the overall grain boundary fracture resistance and the break-through-point distance is quite complex. The minimum grain boundary resistance is achieved when the break-through points are about 2–3 μm apart, which is also influenced by the grain boundary shear strength, the crystallographic toughness and orientation, as well as the crack front profile.     

A Long Crack Penetrating a Circular Inhomogeneity

This paper presents the effects of elastic mismatch and crack-tip position on the stress intensity factors of a long crack penetrating a circular inhomogeneity. The analysis relies on closed-form solutions, derived using complex variable techniques, for the stresses and displacements produced by dislocations positioned inside and outside the inhomogeneity. Dislocation distributions are introduced to express the traction boundary condition along the crack surfaces as a system of singular integral equations, whose solution is obtained through a numerical procedure. It is shown that if the elastic mismatch is interpreted correctly, then the stress intensity factors of this micromechanical model are very good approximations to those computed using a Monte Carlo finite element model of a long crack in a polycrystalline plate with compliant grain boundaries.