Experimental and numerical investigation of a cracked transversely graded plate subjected to in plane bending

The variation of stress intensity factor along the thickness in a cracked transversely graded plate subjected to in plane bending is investigated in this study. A transversely graded plate having elastic modulus varying continuously along the thickness was prepared by embedding glass beads in epoxy resin. An edge crack in this plate was subjected to in plane bending and the crack tip displacement field on the surfaces of the plate was measured using digital image correlation (DIC). Using the recently reported asymptotic displacement fields for cracked transversely graded plates (Wadgaonkar, S.C., Parameswaran, V., 2009. Structure of near tip stress field and variation of stress intensity factor for a crack in a transversely graded material, Journal of Applied Mechanics 76 (1), 011014), the stress intensity factor (SIF) on the surfaces of the plate was calculated from the experimental data. The results of this part of the study indicated that the extent of variation of the SIF across the plate thickness is nearly the same as that of the elastic modulus. An expression to calculate the variation of the SIF through the plate thickness was developed assuming simple bending of the plate. The predicted variation of SIF was verified through finite element calculations. Further, the behavior of the SIF near the intersection of the crack front and the plate surfaces, the extent of dominance of the corner singular field and the exponent of the corner singularity were also investigated in detail. Finally, the effect of gradation strength and gradation type on the SIF was also studied.     

有限厚度板穿透裂纹前缘附近三维弹性应力场分析

通过三维有限元计算来研究有限宽度、有限厚度含有穿透裂纹板的裂纹前缘应力场，从中找出应力强度因子与板的厚度、裂纹长度之间的关系，同时还分析了裂尖的三维约束程度和三维约束区的大小。分析结果表明：应力强度因子沿厚度的分布是不均匀的，应力强度因子的最大值及其位置与厚度有关；有限厚度板中面应力强度因子(KI)m-p及最大应力强度因子(KI)max均大于平面应力或平面应变的应力强度因子。对有限厚度裂纹问题，按平面应力或平面应变来考虑是不安全的；板中面的应力强度因子(KI)m-p及最大应力强度因子(KI)max是厚度B／a的函数；板的中面离面约束系数Tx最大，自由面(z=B)Tx=0。沿厚度方向裂尖附近的离面约束系数Tx也是z／B和B／a的函数，随着厚度的增加离面约束系数Tx增大，离中面越近离面约束系数Tx越大。Tx随着x的增大急剧减小，三维约束影响区域大小大约为板厚的一半，且裂纹长度a／W对应力强度因子沿厚度变化规律及Tx影响区域大小影响较小。     

含单个Ⅰ型裂纹的高温超导块材应力强度因子修正及计算

高温超导块材在外磁场下降过程中会受到拉伸应力的作用,可能导致其断裂,且其在受到拉伸应力作用时裂纹尖端产生小范围屈服,此时用线弹性理论计算应力强度因子将不准确。本文通过统一强度理论对裂纹尖端塑性区的应力强度因子计算公式进行修正。随后基于Bean模型分别讨论了外磁场变化速率和外磁场大小对应力强度因子的影响。结果表明高温超导块材充磁过程中,外磁场以较小速率下降时主要考虑外磁场下降速率对裂纹扩展的影响,当外磁场较大时,主要考虑外磁场大小对裂纹扩展的影响。本文结果对高温超导块材充磁过程中的裂纹扩展和应力强度因子的计算具有一定的参考价值。

     

差异沉降下的拱形门径向裂纹应力强度因子解析解研究

拱形门结构的裂纹应力强度因子是表征径向裂纹尖端应力场强弱的重要参数,但是差异沉降下拱形门结构的径向裂纹应力强度因子面临着难以模式化计算的问题。为此,本文推导了差异沉降作用下的拱形门径向裂纹应力强度因子解析解公式。首先,根据拱形门的对称性,将其分解为正、反对称状态的叠加形式,利用力法对拱形门内力分布进行分析;然后,依据内力情况采用截面法对拱形门径向裂纹截面上的应力分布沿厚度方向进行表达;最后,利用权函数法将权重函数和裂纹表面应力分布的乘积进行积分,得到差异沉降下的拱形门径向裂纹应力强度因子解析解。解析解推导结果与ANSYS数值仿真结果一致,即Ⅰ型、Ⅱ型应力强度因子对差异沉降值与径向裂纹倾角均有显著的线性响应趋势。实验结果表明,本文解析解可以准确有效地将差异沉降下的拱形门径向裂纹应力强度因子进行模式化计算,为裂纹的扩展延伸提供参考。

     

考虑夹杂相互作用的复合陶瓷夹杂界面的断裂分析

复合材料中夹杂含量较高时,夹杂间的相互作用能显著改变材料细观应力应变场分布,基体和夹杂中的平均应力应变水平也会发生较大变化,导致复合材料强度等力学性能发生显著变化. 为修正单一夹杂模型运用在实际材料中的误差,基于相互作用直推估计法,建立一种考虑含夹杂相互作用的夹杂界面裂纹开裂模型. 首先根据相互作用直推估计法,得到残余应力和外载应力共同作用下夹杂中的平均应力,再计算无限大基体中相同的夹杂达到相同应力场时的等效加载应力,将此加载应力作为含界面裂纹夹杂的等效应力边界条件,在此边界条件下求得界面裂纹尖端的应力强度因子,进而得到界面裂纹开裂的极限加载条件,并分析了夹杂弹性性能、含量、热残余应力、夹杂尺寸等因素对界面裂纹开裂条件的影响. 结果表明,方法能够有效修正单夹杂模型运用在实际材料中的误差,较大的残余应力对界面裂纹开裂有重要的影响,夹杂刚度的影响并非单调且比较复杂;在残余应力较小时,降低柔性夹杂刚度或者增大刚性夹杂刚度都有利于提高材料强度;扩大夹杂尺寸将导致裂纹开裂极限应力显著降低,从而降低材料强度.      

A study on caustic method in cracked plate under symmetric bending

Based on Chun-Tu Liu's result for Reissner's plate, an equation of caustics in cracked plate under symmetric bending was obtained. The shape and the distribution of light intensity of the caustic curve were discussed in detail. The relation between the stress intensity factor (SIF) and the size of the caustic curve was found by numerical method, and a simple and accurate experimental method for evaluating SIF in plate under bending was proposed.     

Stress intensities for nozzle cracks in reactor vessels

A series of six frozen stress photoelastic tests was conducted to investigate the distribution of stress-intensity factor (SIF) along a crack which occurred at the juncture of a pipe (nozzle) with a cylindrical pressure vessel. Typical photoelastic-fringe patterns are shown for slices which were taken mutually orthogonal to the flaw border and the flaw surface. A typical plot of normalized apparent SIF vs. square root of normalized distance from the crack tip is presented. The variation in SIF along the flaw border is given for all six different crack geometries and, also, the variation of SIF with varyinga/T is presented.     

Crack kinking from an interface crack with initial contact between the crack lips

The authors recently theoretically studied crack kinking and opening from an initially closed crack (without friction) in some homogeneous medium. The same problem, but for an interface crack, is considered here. Comninou has shown that the asymptotic stress field prior to kinking is governed by a single, mode II stress intensity factor (SIF). Using this result, plus a homogeneity property of the problems of elastic fracture mechanics with unilateral contact envisaged, a change of scale, and two reasonable hypotheses, we establish the expression of the SIF at the tip of the small, open crack extension. It is shown that whatever the geometry of the external boundary and the crack and whatever the loading, these SIF depend solely upon the initial (mode II) SIF (in a linear way), the kink angle and Dundurs' parameters α and β. Using this result and Goldstein and Salganik's “principle of local symmetry” to predict the kink angle, one finds that it is independent of the loading but does depend upon Dundurs' parameters α and β. This contrasts with the case of an ordinary (initially closed) crack in some homogeneous medium, for which the kink angle was not only independent of the loading but an absolute constant. However, it is numerically found that the influence of the mismatch of elastic properties upon the kink angle is rather weak.     

A piezoelectric-elastic body with inhomogeneities and a crack under plane electrical and anti-plane mechanical loads

Summary  In this paper, we study a two-dimensional electroelastic problem of an infinite piezoelectric body with two circular piezoelectric inhomogeneities, one of which contains a crack. We formulate the stress intensity factor (SIF) analytically and investigate it numerically. The problem is solved based on Bueckner's principle, and is reduced to a problem of a singular integral equation of the first kind with respect to the distribution function of screw dislocation. The effect of interaction between the two inhomogeneities and the crack on the electroelastic field as well as the control of the SIF by electrical loads is investigated. Received 18 April 2000; accepted for publication 24 October 2000     

焊趾表面裂纹的形态发展曲线与疲劳寿命预测

以作者建立的焊地椭圆表面裂纹应力强度因子数据库以及复杂应力场中焊践半随圆表面裂纹前缘应力强度因子分布计算的基本模式法为基础上,给出了复杂应力场中焊践表面表纹在疲劳扩展过程中形态变化规律及寿命的工程分析方法。     

SINGULARITIES AND EXPLICIT SOLUTIONS OF COLLINEAR CRACK PROBLEM IN ANISOTROPIC MEDIA

The singularities of collinear cracks both in anisotropic single mediumand at the interface of anisotropie bimaterials are studied by combining Stroh formalismand the analytic function method.The formulae for calculating the field potential andstress intensity factor(SIF)are obtained.It is found that the field potentials areexplicitly related to material matrix L and the in-plane and anti-plane fields can beseparately calculated when orthotropic bimaterials are considered.     

Why traction-free? Piezoelectric crack and Coulombic traction

The non-zero traction condition is introduced in piezoelectric crack problems with the unknown Coulombic traction acting on the crack surfaces. An analytical solution under this condition is obtained by means of the generalized Stroh formalism and by accounting for the permittivity of medium inside the crack gap. As the crack in such materials can be thought of as a low-capacitance medium carrying a potential drop, the Coulombic traction always pulls the two opposite surfaces of the crack together. It is proved that under relatively larger mechanical loading and relatively smaller electrical field, the Coulombic traction may be negligible and the previous investigations under the traction-free crack condition may be accepted in a tolerant way, otherwise the Coulombic traction may lead to some erroneous results with over 10% relative errors. It is also shown that, unlike the traction-free crack condition, the applied electric field does change the Mode I stress intensity factor (SIF) for a central crack in an infinite plane piezoelectric material, and in this way may significantly influence piezoelectric fracture. It is also concluded that the variable tendencies of the normalized SIF and the ERR against the applied electric field depend on the mechanical loading levels. This load-dependence feature may lead to a transformation of the normalized SIF and the ERR from an even functional dependence to an odd functional dependence on the applied electric field.     

WEIGHT FUNCTION FOR STRESS INTENSITY FACTORS IN ROTATING THICK-WALLED CYLINDER

The equation of stress intensity factors(SIF) of internally pressurized thick- walled cylinder was used as the reference case.SIF equation of rotating thick-walled cylinder containing a radial crack along the internal bore was presented in weight function method.The weight fumction formulas were worked out and can be used for all kinds of depth of cracks,rotating speed,material,size of thick-walled cylinder to calculate the stress intensity factors.The results indicated the validity and effectiveness of these formulas.Meanwhile,the rules of the stress intensity factors in rotating thick-walled cylinder with the change of crack depths and the ratio of outer radius to inner radius were studied.The studies are valuable to engineering application.     

STRESS INTENSITY FACTOR OF AN ANTI-PLANE CRACK PARALLEL TO THE WEAK/MICRO-DISCONTINUOUS INTERFACE IN A BI-FGM COMPOSITE

The problem considered is a mode Ⅲ crack lying parallel to the interface of an exponential-type functional graded material （FGM） strip bonded to a linear-type FGM substrate with infinite thickness. By applying the Fourier integral transform, the problem was reduced as a Cauchy singular integral equation with an unknown dislocation density function. The collocation method based on Chebyshev polynomials proposed by Erdogan and Gupta was used to solve the singular integral equation numerically. With the numerical solution, the effects of the geometrical and physical parameters on the stress intensity factor （SIF） were analyzed and the following conclusions were drawn： （a） The region affected by the interface or free surface varies with the material rigidity, and higher material rigidity will lead to bigger affected region. （b） The SIF of the crack in the affected region and parallel to the micro-discontinuous interface is lower than those of the weak discontinuous cases. Reducing the weak-discontinuity of the interface will be beneficial to decrease the SIF of the interface-parallel crack in the region affected by the interface. （c） The effect of the free surface on SIF is more remarkable than that of the interface, and the latter is still more notable than that of the material rigidity. When the effects of the interface and free surface are fixed, increase of the material rigidity will enhance the value of SIF.     

ANTI-PLANE FRACTURE ANALYSIS OF FUNCTIONALLY GRADIENT MATERIAL INFINITE STRIP WITH FINITE WIDTH

The special case of a crack under mode III conditions was treated, lying parallel to the edges of an infinite strip with finite width and with the shear modulus varying exponentially perpendicular to the edges. By using Fourier transforms the problem was formulated in terms of a singular integral equation. It was numerically solved by representing the unknown dislocation density by a truncated series of Chebyshev polynomials leading to a linear system of equations. The stress intensity factor (SIF) results were discussed with respect to the influences of different geometric parameters and the strength of the non-homogeneity. It was indicated that the SIF increases with the increase of the crack length and decreases with the increase of the rigidity of the material in the vicinity of crack. The SIF of narrow strip is very sensitive to the change of the non-homogeneity parameter and its variation is complicated. With the increase of the non-homogeneity parameter, the stress intensity factor may increase, decrease or keep constant, which is mainly determined by the strip width and the relative crack location. If the crack is located at the midline of the strip or if the strip is wide, the stress intensity factor is not sensitive to the material non-homogeneity parameter.     

Interaction of plane nonstationary waves with a thin elastic inclusion under smooth contact conditions

We solve the problem of determining the stress state near a thin elastic inclusion in the form of a strip of finite width in an unbounded elastic body (matrix) with plane nonstationary waves propagating through it and with the forces exerted by the ambient medium taken into account. We assume that the matrix is in the plane strain state, and the smooth contact conditions are realized on both sides of the inclusion. The method for solving this problem consists in using the integral Laplace transform with respect to time and in representing the stress and displacement images in terms of the discontinuous solution of Lamé equations in the case of plane strain. As a result, the initial problem is reduced to a system of singular integral equations for the transforms of the unknown stress and displacement jumps. To invert the Laplace transform, we use a numerical method based on replacing the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors (SIF) for the inclusion, which are used to study the SIF time-dependence and its influence on the values of the inclusion rigidity. We also studied the possibility of considering the inclusions of higher rigidity as absolutely rigid inclusions.     

A solution for an isotropic sector under anti-plane shear loadings

In this study, the problem of an isotropic sector subjected to anti-plane shear loadings is investigated. The loadings were applied to the arc of the sector, and the radial edges of the sector were under traction-free or fixed conditions. Depending on these conditions, three problems, namely, free–free, fixed–free, and fixed–fixed edges were studied. A procedure using the finite Mellin transform combined with the Laplace transform was proposed for solving these problems. Explicit closed form solutions for the displacement and stress fields throughout the sector were obtained. The stress intensity factor (SIF) for each problem was analyzed using the obtained stress fields. It was determined that the SIF disappeared under the special condition of a fixed–fixed edge. Other special cases having anti-symmetric conditions were deduced from the derived solutions, and the results of these verified those cited in the literature as well as those obtained using finite element analysis (FEA).     

Assessing the effect of residual stresses on the fatigue behavior of integrally stiffened structures

Residual stresses emerge quite often in real structures due to the various manufacturing processes such as, welding, forming, cutting, milling, etc. In such cases, development of cracks at regions influenced by manufacturing operations demand additional attention. In the present work a numerical methodology has been developed, based on three-dimensional Finite Element Analysis, for the calculation of Stress Intensity Factors at cracks in welded components. The residual stress fields, which are used in SIF calculations, have been computed by the numerical simulation of the thermo-mechanical process. A numerical algorithm based on interpolation principles is developed, in order to introduce the three-dimensional field in the computational model of the cracked structure. The SIF calculation methodology is initially validated for the case of a welded plate by comparison of numerical results with existing analytical solutions. A cracked stiffened panel is analysed afterwards and the calculated fatigue crack propagation results are compared to experimentally measured data. Finally, the numerical procedure is applied to study the effect of more complicated residual stress fields on SIF values developing at cracks located in stiffened panels.     

估算裂纹应力强度因子的新方法

根据裂纹形状与裂纹尖端应力强度因子分布之间的固有关系,在线弹性断 裂力学条件下,提出了一种按已知I型裂纹应力强度因子分布规律求裂纹形状及相应应力强 度因子的无梯度迭代法. 通过有限厚度、有限宽度板穿透裂纹和表面裂纹的数值模拟实例验 证了所提出方法的有效性和实用性,并对不同应力强度因子分布规律对裂纹形状以及相应的 应力强度因子大小的影响进行了分析和讨论. 所提出的方法有助于提高实际扩展裂纹应 力强度因子的估算精度以及更合理地预测疲劳裂纹形状演化.     

Closed form solution for two unequal collinear semi-permeable straight cracks in a piezoelectric media

The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.