Stress intensity factors for a crack in planar disking

Disking is a relatively new manufacturing process for cutting/slicing brittle plates and rods. In the planar disking configuration, a pre-cracked plate is placed against an elastic plate and the two are squeezed together by fluid pressure. At a critical pressure the crack runs across the thickness of the brittle plate producing a clean cut. In this paper a fracture criterion is developed for the process using linear elastic fracture mechanics. The geometry of the process is modeled here as two perfectly bonded, infinite elastic layers with a crack perpendicular to the interface. The problem is formulated in terms of a singular integral equation with the derivative of the crack surface displacement (dislocation density) as the unknown function. Numerical quadrature is used to determine the stress intensity factors as a function of the parameters of the problem.     

Fracture energy of epoxy resin under plane strain conditions

The fracture behaviour of an epoxy resin has been studied by a method which involves the pressurization of an internal circular crack. The method can be used to study both cohesive fracture and the adhesive failure of an interface. Plane strain conditions are assured because the crack does not intersect a free surface and (for adhesive failure) shrinkage stresses are eliminated as a crack driving force. Using high speed photography, the dependence of crack speed on critical pressure and specimen geometry was determined. An elastic analysis permits the derivation of fracture energy as a function of crack velocity. Fracture energy values lay between 100 and 200 Jm^–3 at 35° C with a peak at a crack velocity of 37 m sec^–1.     

Explicit relations between elastic and conductive properties of materials containing annular cracks

The impact of annular cracks on the effective elastic and conductive properties of a material is analysed. The compliance contribution tensor of an annular crack - the quantity that determines the increase in compliance of a solid due to introduction of such a crack - is derived analytically. The resistivity contribution tensor of an annular crack is calculated numerically. It is shown that an effective circular crack, i.e. a crack which yields the same change in elastic/conductive properties of a material as the given annular crack, can be chosen to match both of these tensors. Using this result, the explicit relation between elastic and conductive properties of a material containing annular cracks is obtained. The relation is derived using a non-interaction approximation. Applicability of the derived formulae to real materials (to plasma-sprayed coatings, in particular) is discussed.     

Cohesive zone modelling of interface fracture near flaws in adhesive joints

A cohesive zone model is suggested for modelling of interface fracture near flaws in adhesive joints. A shear-loaded adhesive joint bonded with a planar circular bond region is modelled using both the cohesive zone model and a fracture mechanical model. Results from the models show good agreement of crack propagation on the location and shape of the crack front and on the initial joint strength. Subsequently, the cohesive zone model is used to model interface fracture through a planar adhesive layer containing a periodic array of elliptical flaws. The effects of flaw shape are investigated, as well as the significance of fracture process parameters. The results from simulations of fracture in a bond containing circular flaws show that localization of crack propagation in the vicinity of a flaw has significant effect on the joint strength and crack front shape. The localization effects are highly dependent on the fracture process zone width relative to the flaw dimensions. It is also seen that with increasing fracture process zone width, the strength variation with the flaw shape decreases, however, the strength is effected over a wider range of propagation.     

The use of interlayers in modeling interface crack propagation

Delaminations are a common mode of failure at interfaces between two material layers which have dissimilar elastic constants. There is a well-known oscillatory nature to the singularity in the stress fields at the crack tips in these bimaterial delaminations, which creates a lack of convergence in the modewise energy release rates. This makes constructing fracture criteria somewhat difficult. An approach used to overcome this is to artificially insert a thin, homogeneous, isotropic layer (the interlayer) at the interface. The crack is positioned in the middle of this homogeneous interlayer, thus modifying the original ‘bare’ interface crack problem into a companion ‘interlayer’ crack problem. Individual modes I and II energy release rates are convergent and calculable for the companion problem and can be used in the construction of a fracture criterion or locus. However, the choices of interlayer elastic and geometric properties are not obvious. Moreover, a sound, consistent, and comprehensive methodology does not exist for utilizing interlayers in the construction and application of mixed-mode fracture criteria in interface fracture mechanics. These issues are addressed here. The role of interlayer elastic modulus and thickness is examined in the context of a standard interface fracture test specimen. With the help of a previously published analytical relation that relates the bare interface crack stress intensity factor to the corresponding interlayer crack stress intensity factor, a suitable thickness and elastic modulus are identified for the interlayer in a bimaterial four-point bend test specimen geometry. Interlayer properties are chosen to make the interlayer fracture problem equivalent to the bare interface fracture problem. A suitable mixed-mode phase angle and a form for the fracture criterion for interlayer-based interface fracture are defined. A scheme is outlined for the use of interlayers for predicting interface fracture in bimaterial systems such as laminated composites. Finally, a simple procedure is presented for converting existing bare interface crack fracture loci/criteria into corresponding interlayer crack fracture loci.     

A calibrated frature process zone model for thin film blistering

The purpose of this work was to examine the feasibility of using fracture process zone models for extracting the adhesive fracture energy of thin films on a thick substrate from circular blister experiments that involve a substantial amount of inelastic deformation in the thin film. The interface produced by vapor depositing polyimide on aluminum formed the basis of the experiments that were conducted. The experiments were conducted in volume control while the pressure history and the corresponding three dimensional blister shape were measured. The analysis accounted for the nonlinear kinematics and material behavior of the polyimide film and included a traction-separation law for the interface. The traction-separation law for the interface was calibrated in an iterative manner by comparing measured pressure-volume responses and crack opening displacements. The adhesive fracture energy obtained from the selected traction-separation law was reasonable considering that fracture occurred in an interphase region. It was bounded by values that were obtained from elastic analyses (with updated kinematics) of the type performed by Gent and Lewandowski (1987) and Chu et al. (1992 a, b). This revised version was published online in July 2006 with corrections to the Cover Date.     

Conditions for matrix crack deflection at an interface in ceramic matrix composites

This paper describes a numerical approach developed to simulate the mechanism of matrix crack deflection at the fibre/matrix interface in brittle matrix composites. For this purpose, the fracture behaviour of a unit cell (microcomposite) consisting of a single fibre surrounded by a cylindrical tube of matrix was studied with the help of a finite element model. A fracture mechanics approach was used to design a criterion for deflection at the fibre/matrix interface of an annular crack present in the matrix. The analysis of the fracture behaviour of SiC/SiC and SiC/glass ceramics microcomposites shows that the introduction of a low modulus and low toughness interfacial layer at the fibre/matrix interface (e.g. a carbon coating) greatly favours matrix crack deflection at the interphase/fibre interface.     

Diffraction of torsional waves by a flat annular crack at the interface of two bonded dissimilar elastic solids

The paper deals with the problem of finding the stress distribution near an annular crack located at the interface of two bonded dissimilar elastic solids. The crack is opened by the interaction of a torsional wave incident normally on the annular crack. The problem is reduced to the solution of three simultaneous Fredholm integral equations. The numerical solution of these simultaneous integral equations has been obtained. The solution is used to calculate the stress-intensity factors at the tips of the crack.     

Primary fracture propagation from circular cavities loaded in compression

When a brittle elastic material containing a cavity is loaded in uniaxial compression, fractures may form in three basic positions around the cavity; at the tensile stress concentration (primary fracture), at positions inside the material remote from the perimeter of the cavity, and at the compressive stress concentration. Granite blocks containing a circular cavity of radius between 2.5 mm and 50 mm were tested in uniaxial compression to collect data on primary fracture propagation. The laboratory results indicate that primary crack propagation is a stable process at small scales but approaches instability at large scales. A finite width crack model is presented which is able to capture this scale dependent behavior. The model illustrates that both tensile and compressive stresses play an important role in the primary fracture process.     

Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media

This study is concerned with the inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in functionally graded material (FGM) coatings around a circular hole in infinite elastic media. The incompatible eigenstrain induced in the FGM coatings after cooling from the sintering temperature, due to mismatch in the coefficients of thermal expansion, is taken into consideration. An approximation method of determining stress intensity factors is introduced for a crack in the FGM coatings in which the FGM coatings are homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case of a uniform pressure applied to the surfaces of the hole and the crack. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress intensity factors for the same crack in the FGM coatings, and are used in the inverse problem of calculating material distributions in the FGM coatings intending to realize prescribed apparent fracture toughness in the coatings. Numerical results are obtained for a TiC/Al₂O₃ FGM coating, which reveal that the apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media can be controlled within possible limits by choosing an appropriate material distribution profile in the coatings.     

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to the crack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts the symmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity. This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeled using a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structures made of quasi-brittle materials failing at concrete-concrete interfaces. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.     

A FINITE ELEMENT TECHNIQUE TO SIMULATE THE STABLE SHAPE EVOLUTION OF PLANAR CRACKS WITH AN APPLICATION TO A SEMI-ELLIPTICAL SURFACE CRACK IN A BIMATERIAL FINITE SOLID

This paper describes a numerical procedure to model the crack front evolution of initially arbitrary shaped planar cracks in a three-dimensional solid. The influence of a bimaterial interface on the fracture path of a semi-elliptical surface crack in a three-dimensional structure is examined. The analysis is based on the assumption that fracture is controlled by small-scale yielding and linear elastic fracture mechanics. The finite element method and the crack-tip contour J-integral in a volume domain representation are utilized to calculate the crack front energy release rate. The computed values of the energy release rate are used with a crack-tip velocity growth law to model crack growth increment. The progress of the crack growth evolution is brought forward by successive iterations. Examples of computed crack evolution are given for an embedded circular crack, a semi-elliptical surface crack in a finite plate, and a configuration that defines an isotropic homogeneous material layer with a surface crack located between two material layers. © 1997 by John Wiley & Sons, Ltd.     

Effect of anisotropic plasticity on mixed mode interface crack growth

Crack growth along an interface between a solid with plastic anisotropy and an elastic substrate is modelled by representing the fracture process in terms of a traction-separation law specified on a crack plane. A phenomenological elastic-viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy. Conditions of small-scale yielding are assumed, and due to the mismatch of elastic properties across the interface the corresponding oscillating stress singularity field is applied as boundary conditions on the outer edge of the region analyzed. Crack growth resistance curves are calculated numerically, and based on these results the dependence of the steady-state fracture toughness on the near-tip mode mixity is determined. Different initial orientations of the principal axes relative to the interface are considered and it is found that the steady-state fracture toughness is quite sensitive to this orientation of the anisotropy.     

Bimodular behaviour and crack closure in compression in a brittle material

A vibrating beam method was used to determine the elastic modulus of graphite rods. The frequency and apparent modulus were determined as a function of compressive end-loading. Following fracture of the rod, the frequency and apparent modulus were decreased. At a compressive end-loading of about 0.83 MPa (120 p.s.i.), crack closure was sufficient for the fractured rod to behave similarly in vibration to the unfractured rod. Thus, the fractured material behaves in a bimodular fashion and crack closure can be achieved to enable unimpeded stress transfer across the fracture surface during vibration.     

Stress intensity factors for cracks of arbitrary shape near an interfacial boundary

Modes I, II and III stress intensity factors for a crack of arbitrary planar shape near a bimaterial interface are calculated. The solution utilizes the body-force method and requires Green's functions for perfectly bonded elastic half-spaces. The formulation leads to a system of two-dimensional singular integral equations whose solutions represent the three modes of crack opening displacement. Numerical examples of a semicircular or semielliptical crack terminating at the interface and circular or elliptical cracks contained in one material are given for both internal pressure and farfield tension.     

Stress intensity factors and crack extension in a cracked pressurised cylinder

This paper considers the plane elastic problem corresponding to single or multiple radial cracks emanating from the internal boundary of a circular ring, under uniform external tension and internal pressure. The stress intensity factors are calculated by using the dual boundary element method with the J-integral technique. Accurate data are found for varying crack depths over a representative range of wall ratios for fracture mechanics applications to pressurised circular cylinders. The interaction of multiple cracks and crack extension are investigated in the case of an internal pressure loading condition. The analysis shows that, for a multi-cracked pressurised cylinder, it is sufficient to calculate the stress intensity of the main crack in isolation for the purposes of safety assessment.     

Matrix cracking initiated by fibre breaks in model composites

Fracture of resin in a composite material can be initiated by a tensile break in a fibre. This process has been investigated for a simple model composite, consisting of two inextensible rods placed along the axis of a cylindrical elastic block and touching in the centre. The rods represent a broken fibre. Energy release rates,G, were calculated by finite element methods for a circular crack growing outwards from the point where the rod ends separated as they were pulled apart. Results are compared with experimental observations on cracking of a silicone rubber cylinder containing two steel rods. It was found that a crack grew outwards under increasing load until its radius reached a certain size, approximately half-way to the surface of the resin cylinder. At this point,G reached a minimum value and then increased. Simultaneously, the crack accelerated and the sample broke. Forces required to propagate the crack were successfully predicted by linear elastic fracture mechanics at all stages of crack growth and for a wide range of fibre and sample radii. In particular, good agreement was obtained with the maximum force that the model system could support, i.e. the breaking load. When the sample was surrounded by a rigid tube, representing neighbouring fibres surrounding the broken one, growth of a crack required an increasing load at all stages. The sample finally fractured when the broken fibre pulled out with resin still attached to it. Application of these results to unidirectional fibre-reinforced materials is discussed.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

坝基界面在非线性水压力驱动下的非线性断裂过程模拟

基于多边形比例边界有限元法和粘聚裂缝模型提出了混凝土坝坝基界面在随缝宽非线性变化的水压力驱动下的非线性断裂数值模型。混凝土和基岩采用多边形比例边界单元模拟,界面裂缝的断裂过程区采用粘性界面单元模拟。因为界面裂缝总是处于复合断裂模态,故同时引入了法向和切向的界面单元,且考虑了裂纹面作用有法向和切向任意荷载时的应力强度因子求解。以裂尖为原点,裂尖附近的位移场和应力场在径向上解析求解,在环向具有有限元精度。因此无需在裂尖附近加密网格或采用富集技术即可求得高精度的解。对于界面断裂,可模拟出与两种材料差异性相关的非1/2奇异性。断裂过程区的水压力随缝面宽度变化,采用指数函数的形式进行表征,通过参数调整可实现不同分布的水压力的模拟。水压力与粘聚力考虑为与裂缝宽度相关的组合函数,便于非线性迭代的实现。结合多边形网格生成和重剖分技术,可方便地模拟界面裂缝在水力驱动下的扩展过程。算例研究表明了该文模型的有效性,从中也可看出考虑缝内水压及其具体分布形式对研究坝的稳定性具有重要影响。     

基于断裂力学的高强度钢材梁柱节点受力性能分析

为了研究国产Q460C高强度结构钢材梁柱节点的断裂行为,该文基于断裂力学理论,计算了Q460C高强度钢材焊缝及热影响区材料的断裂韧性,并且采用三维有限元断裂模型,以I型裂纹尖端应力强度因子K_I和J积分为定量的评价指标,分析了焊缝及热影响区不同长度的裂纹对梁柱节点断裂韧性的需求。弹性分析表明,K_I沿梁翼缘宽度方向呈W形分布,最大值出现在翼缘中心,且与名义弯曲应力呈线性关系,而焊根裂纹的断裂韧性需求比热影响区裂纹更高。弹塑性分析表明,J_I最大值出现在翼缘的边缘,热影响区裂纹的断裂韧性需求比焊根裂纹更高。研究结果表明,Q460C高强度钢材梁柱节点的断裂由焊根裂纹控制,断裂承载力与梁全截面塑性承载力相近,临界转角小于0.02rad,因此建议通过改善焊接工艺或局部构造来保证节点拥有足够的转动能力。