Inverse problem of material distribution for desired fracture characteristics in a thick-walled functionally graded material cylinder with two diametrically-opposed edge cracks

This study concerns the inverse problem of evaluating the optimum material distribution for desired fracture characteristics in a thick-walled functionally graded material (FGM) cylinder containing two diametrically-opposed edge cracks emanating from the inner surface of the cylinder. The thermal eigenstrain developed in the cylinder material due to nonuniform coefficient of thermal expansion as a result of cooling from sintering temperature is taken into account. Based on a generalized method of evaluating stress intensity factors developed in a previous study, an inverse method is developed to optimize material distribution intending to realize prescribed apparent fracture toughness in the FGM cylinder. To present some numerical results, a TiC/Al₂O₃ FGM cylinder is considered and the inverse problems are solved to evaluate material distributions for two examples of prescribed apparent fracture toughness. The effect of cylinder wall thickness on the material distribution and comparison of material distributions corresponding to a single and two cracks are also discussed. The numerical results reveal that the apparent fracture toughness of FGM cylinders can be controlled by choosing the material distributions properly.     

Crack Spacing Effect on the Brittle Fracture Characteristics of Semi-infinite Functionally Graded Materials with Periodic Edge Cracks

The effect of crack spacing on the brittle fracture characteristics of a semi-infinite functionally graded material (FGM) with periodic edge cracks is discussed. The incompatible eigenstrain induced in the material due to mismatch in the coefficients of thermal expansion is considered in the analysis. The nonhomogeneity of the material is simulated by an equivalent eigenstrain, whereby the problem is reduced to that of a cracked homogeneous material with incompatible and equivalent eigenstrains. A method is then formulated to calculate the stress intensity factor of periodic edge cracks in such a semi-infinite homogeneous medium and applied to calculate apparent fracture toughness of a semi-infinite FGM from its prescribed composition profile. Inverse calculation is also carried out to compute composition profile from prescribed apparent fracture toughness of the semi-infinite FGM. Numerical calculations are carried out for semi-infinite TiC/Al₂O₃ FGM and the results are shown in the figures.     

Stress intensity factors of two diametrically opposed edge cracks in a thick-walled functionally graded material cylinder

A method is developed to evaluate stress intensity factors for two diametrically-opposed edge cracks emanating from the inner surface of a thick-walled functionally graded material (FGM) cylinder. The crack and the cylinder inner surfaces are subjected to an internal pressure. The thermal eigenstrain induced in the cylinder material due to nonuniform coefficient of thermal expansion after cooling from the sintering temperature is taken into account. First, the FGM cylinder is homogenized by simulating its nonhomogeneous material properties by an equivalent eigenstrain, whereby the problem is reduced to the solution of a cracked homogenized cylinder with an induced thermal and an equivalent eigenstrains and under an internal pressure. Then, representing the cracks by a continuous distribution of edge dislocations and using their complex potential functions, generalized formulations are developed to calculate stress intensity factors for the cracks in the homogenized cylinder. The stress intensity factors calculated for the cracks in homogenized cylinder represents the stress intensity factors for the same cracks in the FGM cylinder. The application of the formulations are demonstrated for a thick-walled TiC/Al₂O₃ FGM cylinder and some numerical results of stress intensity factors are presented for different profiles of material distribution in the FGM cylinder.     

Analysis of apparent fracture toughness of a thick-walled FGM cylinder with two diametrically opposed edge cracks

This study focuses on the analysis of apparent fracture toughness of a thick-walled functionally graded material (FGM) cylinder with two diametrically opposed edge cracks emanating from the inner surface of the cylinder. The crack surfaces and the inner surface of the cylinder are subjected to an internal pressure. The incompatible eigenstrain developed in the cylinder due to non-uniform coefficient of thermal expansion after cooling from sintering temperature is taken into account. Based on a generalized method of evaluating stress-intensity factor developed in our previous study, an approach is presented to evaluate apparent fracture toughness. To demonstrate the approach, some numerical results of apparent fracture toughness are presented for a TiC/Al₂O₃ FGM cylinder. The effects of material distribution, cylinder wall thickness, application temperature and number of cracks on apparent fracture toughness are investigated in details. It is found that all these factors play an important role in controlling apparent fracture toughness of a thick-walled FGM cylinder.     

Effect of FGM coating thickness on apparent fracture toughness of a thick-walled cylinder

This study describes the effect of FGM coating thickness on apparent fracture toughness (AFT) of a thick-walled cylinder containing two diametrically-opposed edge cracks emanating from the inner surface of the cylinder. The incompatible eigenstrain developed in the cylinder due to nonuniform coefficient of thermal expansion because of cooling from sintering temperature is taken into account. Based on a generalized method of evaluating stress intensity factor (SIF) addressed in a previous study, an approach is developed to calculate AFT. Some numerical results of AFT are presented for a TiC/Al₂O₃ FGM coating at the inner surface of an Al₂O₃ thick-walled cylinder.     

含圆孔缺陷三点弯曲梁的裂纹扩展特性实验研究

采用动态焦散线实验系统,对含圆孔缺陷的PMMA材料进行冲击断裂力学实验,研究三点弯曲梁中不同直径和位置的圆孔缺陷对扩展裂纹的影响。实验结果表明:扩展裂纹与圆孔缺陷相互作用前,呈现Ⅰ型拉伸断裂,扩展路径平直;与圆孔缺陷相互作用后,贯通萌生的次裂纹沿直线继续扩展,未贯通的裂纹偏移扩展。扩展裂纹与路径上圆孔缺陷贯通过程中,裂纹扩展速度和动态应力强度因子快速降为零,裂纹的扩展受到抑制,且圆孔直径越大、距离越近,抑制作用越显著;贯通萌生次生裂纹的起裂速度和起裂韧度,随着圆孔缺陷直径的增大而变大。扩展裂纹与偏置圆孔缺陷相互作用过程中,当圆孔缺陷直径越大、偏置距离越小,裂纹起偏距离越短,最大偏移量越大,并且扩展裂纹动态应力强度因子和扩展速度局部小幅增大。研究结果为分析动态裂纹扩展特征和材料破坏模式提供借鉴。     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Experimental Investigations on Deformation and Fracture Behavior of Glass Sphere Filled Epoxy Functionally Graded Materials

In this paper, deformation and fracture behavior of glass sphere filled epoxy functionally graded materials (FGM) are numerically evaluated and experimentally studied. The fabrication of the FGM is described in detail, and the spatial gradation of elastic modulus and the microscopic structure in FGM are measured and analyzed. The deformation and fracture characterization of the FGM specimen with a crack oriented along the direction of the elastic gradient under three point bend are studied by the experimental and the finite element method. The influences of crack location at both the stiff and the compliant sides of the FGM specimen on crack initiation, deformation field and stress intensity factor are analyzed. The results are: (a) The neutral-axis in the FGM specimen under three-point-bending will shift toward the stiffer side; (b) The initial fracture load increases with the increase of elastic modulus at the crack tip; (c) The elastic gradients shield a crack on the compliant side and lower the stress intensity factor when compared to the one with crack on the stiff side. These results will be useful for better design and reliable evaluation of FGM.     

梯度复合材料裂纹扩展路径和起裂载荷的有限元分析

为了模拟功能梯度材料(FGM)在工程应用中可能会出现的断裂问题并计算相应的开裂载荷,通过编写用户自定义UEL子程序将梯度扩展单元嵌入到ABAQUS软件中模拟功能梯度材料的物理场,并编写交互能量积分后处理子程序计算裂纹尖端的混合模式应力强度因子(SIF),采用最大周向应力准则编写子程序计算裂纹的偏转角,并模拟了裂纹扩展路径,计算了裂纹的起裂载荷。讨论了材料梯度参数对裂纹扩展路径以及起裂载荷的影响规律。通过与均匀材料的对比,验证了功能梯度材料断裂性能的优越性。研究表明:外载平行于梯度方向时,垂直梯度方向的初始裂纹朝着等效弹性模量小的方向扩展,且偏转角在梯度指数线性时出现峰值,并随着组分弹性模量比的增加而变大;当外载和初始裂纹均平行于梯度方向时,材料等效弹性模量和断裂韧性的增加或者梯度指数的减小都导致起裂载荷变大。     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

Determination of crack surface displacements for cracks emanating from a circular hole using weight function method

Cracks emanating from a circular hole are of significant engineering importance, especially in aerospace industry. Accurate determination of key fracture mechanics parameters is essential for damage tolerance design and fatigue life predictions. The purpose of this paper is to provide an efficient and accurate closed‐form weight function approach to the calculation of crack surface displacements for radial crack(s) emanating from a circular hole in an infinite and finite‐width plate. Results were presented for two loading conditions: remote applied stress and uniform stress segment applied to crack surfaces, and extensively compared to recent studies using other methods in the literature. Both single and double radial cracks were considered, and also the effect of finite plate width on crack surface displacements has been investigated. A brief assessment was made on an engineering estimation of displacements based on a correction of stress intensity factor ratio. It has been demonstrated that the Wu‐Carlsson closed‐form weight functions are very efficient, accurate and easy‐to‐use for calculating crack surface displacements for arbitrary load conditions. The method will facilitate fatigue crack closure and other fracture mechanics analyses where accurate crack surface displacements are required.     

Dynamic fracture of a functionally gradient material having discrete property variation

A functionally gradient material (FGM) with discrete property variation is prepared, and the dynamic fracture in this material is studied using the technique of photoelasticity combined with high-speed photography. Transparent sheets required for the study are made by casting a polyester resin mixed with varying amounts of plasticizer. The mechanical (quasi-static and dynamic) and optical properties of the material are evaluated as a function of the plasticizer content. Results of material characterization show that the fracture toughness increases with increasing plasticizer content, whereas the Young's modulus decreases. The material fringe constant and the dynamic modulus are observed to be relatively insensitive to plasticizer content. The FGM is then prepared by casting together thin strips having different plasticizer content. The dynamic crack propagation phenomenon is studied for four different property variations along the crack propagation direction, and the effects of these property variations on crack speed, crack jump distance and dynamic stress intensity factor are investigated. Results of this investigation show that increasing the toughness in the direction of crack growth reduces the crack jump distance as compared to on increasing-decreasing toughness variation for the same initial energy.     

Ductile–brittle fatigue and fracture behaviour of aluminium/PMMA bimaterial 3PB specimens

Fracture toughness and fatigue crack growth tests and numerical simulations on 3PB specimens were carried out to study the behaviour of a crack lying perpendicular to the interface in a ductile/brittle bimaterial. Polymethylmethacrylate acrylic (PMMA) and aluminium alloy 2024 T531 were joined together using epoxy resin. A precrack was introduced into the ductile material and tests were carried out to obtain fracture toughness and fatigue properties. The body force method and elastic–plastic finite-element analyses were used to simulate the experimental stress intensity K_I and cracking behaviour under monotonic and cyclic loads. It was found that the bimaterial fatigue crack growth rate is higher than that for monolithic aluminium 2024 but lower than the rate for a monolithic PMMA. This agreed with the trend for the fracture toughness values and was consistent with the numerical method results. The initial Mode I stable ductile cracking in the aluminium appears to ‘jump’ the interface and continues under mixed fracture Mode (I and II) in the PMMA material up to the final failure. A consistency between the simulation methods has indicated that the bimaterial fatigue crack growth is dominantly elastic with a small plastic zone near the crack tip.     

Relationship between fracture toughness and deformation ahead of the crack tip

An analytical relationship for the calculation of the intensity of elastoplastic strains at the front of a mode I crack is obtained on the basis of the suggested model. A strain criterion of fracture toughness has been developed which relates the stress intensity factor to the width of the stretch zone, the mechanical properties of the material, and its elastic constants. A comparison of experimental and numerical (obtained by using the proposed equations) data of the stretch zone width for 15Kh2NMFA steel in a wide temperature range demonstrates good agreement between the theory and the experiment. This proves the possibility of calculating fracture toughness via the stretch zone width and mechanical properties of the material. Translated from Problemy Prochnosti, No. 2, pp. 33–40, March–April, 1997.     

The longitudinal shear problem for an array of cracks at the edge of a circular hole in an infinite elastic solid

Summary A Mellin-type transform technique reduces the longitudinal shear problem for a set of cracks at the edge of a circular hole in an infinite elastic solid to that of solving a system of integral equations. The stress intensity factors and crack formation energy are calculated. Three special cases are considered in detail and graphical results given.     

The distribution of stress near the tip of a radial crack at the edge of a circular hole

A Mellin transform technique is used to find an integral equation the solution of which is related to the stress intensity factor and the formation energy of a crack at the edge of a circular hole in an infinite elastic solid. The biaxial loading case is considered in detail and numerical results given.     

Effect of residual stress on cleavage fracture toughness by using cohesive zone model

This study presents the effect of residual stresses on cleavage fracture toughness by using the cohesive zone model under mode I, plane stain conditions. Modified boundary layer simulations were performed with the remote boundary conditions governed by the elastic K‐field and T‐stress. The eigenstrain method was used to introduce residual stresses into the finite element model. A layer of cohesive elements was deployed ahead of the crack tip to simulate the fracture process zone. A bilinear traction–separation‐law was used to characterize the behaviour of the cohesive elements. It was assumed that the initiation of the crack occurs when the opening stress drops to zero at the first integration point of the first cohesive element ahead of the crack tip. Results show that tensile residual stresses can decrease the cleavage fracture toughness significantly. The effect of the weld zone size on cleavage fracture toughness was also investigated, and it has been found that the initiation toughness is the linear function of the size of the geometrically similar weld. Results also show that the effect of the residual stress is stronger for negative T‐stress while its effect is relatively smaller for positive T‐stress. The influence of damage parameters and material hardening was also studied.     

A three-point bend specimen with partly tapered cross-section sides

The stress intensity factor and the J-integral have been derived analytically and numerically for a modified three-point bend specimen with partly tapered sides, for various crack lengths, taper and specimen cross-section proportions, in order to allow full-thickness testing of tapered samples, common in older steel structures, to obtain a fair effective fracture toughness value for a through thickness crack in inhomogeneous materials. The stress intensity factor is obtained with the approximate analytical method of Kienzler and Herrmann, based on the concept of material forces. The J-integral is calculated numerically with a 3D finite element model for a linear elastic material and an elastic ideal-plastic material. A simple single specimen fracture toughness evaluation procedure is proposed. It is found that the effect of taper in the range encountered in practice is small, of the order of a few percent.     

基于断裂力学的钢框架梁柱节点抗震性能分析

介绍了8 个足尺节点试验的结果与连接焊缝断裂破坏的现象。采用二维有限元断裂模型,研究了抗弯钢框架梁柱节点的断裂行为。以I 型裂纹尖端应力强度因子K_I和J积分为定量评价指标,分析了焊接孔形式、初始缺陷尺寸及位置、焊接垫板、角焊缝补强等设计细节对节点材料断裂韧性需求的影响。弹性分析表明,切除垫板并以角焊缝补强的措施,能最有效地降低对材料韧性的要求。弹塑性分析表明,焊接高匹配时热影响区裂纹比界面裂纹对材料韧性的要求更高,且局部塑性应变累积使材料的断裂韧性降低;解释了节点试验中断裂易发生在梁下翼缘焊缝热影响区的现象。基于断裂分析得到的节点抗弯承载力与试验中节点的极限弯矩吻合较好,该文的研究为钢框架梁柱节点的防断设计提供了参考。