Buckling and fracture behaviour of cracked thin plates under shear loading

Thin-walled structural components are widely used in several engineering applications such as in aerospace, naval, nuclear power plant, pressure vessel, mechanical and civil fields. Since they are frequently characterised by a high slenderness, the safety assessment of such structural components requires to carefully consider the buckling collapse which can heavily limit their allowable bearing capacity. For very thin plates, buckling collapse can occur under compression, shear, or even under tension. In the present paper, the buckling and fracture collapse mechanisms in an elastic rectangular thin-plate with a central straight crack under shear loading are analysed. Different boundary conditions, crack length and orientation are considered. Through a parametric finite elements (FE) numerical analysis, the crack sensitivity of the collapse load of such a structural component is examined. The obtained results are discussed, and some interesting and useful conclusions are drawn. The collapse mechanism occurring earlier (buckling or fracture) is found by varying the fracture toughness of the material, and some failure-type maps depending on the geometrical parameters of the crack are determined.     

含裂纹损伤结构强度的多源数据融合方法研究

工程结构在制造工艺过程中或使用期间会产生裂纹,对结构断裂路径的预测和研究是防治工程安全问题发生的重要手段。在考虑裂纹尖端应力场常数项T应力的基础上对传统的最大周向应力准则(Maximum tangential stress criterion, MTS)和最小应变能密度因子准则(Minimum strain energy density criterion, SED)进行修正,采用Python语言对ABAQUS的前、后处理和有限元计算模块进行二次开发,通过计算最优解的粒子群算法(Particle swarm optimization, PSO)将修正后的准则编入裂纹自动扩展程序脚本中。利用上述二次开发程序对初始纯I型裂纹的扩展路径进行模拟,结果表明：采用ABAQUS脚本程序模拟结果与相关文献实验结果吻合,表明了程序的有效性,进而实现考虑T应力的多种断裂准则对裂纹扩展路径的预测;当T应力值处于一定范围内时,修正的MTS准则无法预测裂纹发生的偏转现象,扩展路径呈直线,此时可采用修正的SED准则进行预测。

     

Finite strip with a central crack under tension

In this study, a symmetrical finite strip with a length of 2L and a width of 2h, containing a transverse symmetrical crack of width 2a at the midplane is considered. Two rigid plates are bonded to the ends of the strip through which uniformly distributed axial tensile load of magnitude 2hp₀ is applied. The material of the strip is assumed to be linearly elastic and isotropic. Both edges of the strip are free of stresses. Solution for this finite strip problem is obtained by means of an infinite strip of width 2h which contains a crack of width 2a at y = 0 and two rigid inclusions of width 2c at y = ±L and which is subjected to uniformly distributed axial tensile load of magnitude 2hp₀ at y = ±∞. When the width of the rigid inclusions approach the width of the strip, i.e., when c → h, the portion of the infinite strip between the inclusions becomes identical with the finite strip problem. Fourier transform technique is used to solve the governing equations which are reduced to a system of three singular integral equations. By using the Gauss–Jacobi and the Gauss–Lobatto integration formulas, these integral equations are converted to a system of linear algebraic equations which is solved numerically. Normal and shearing stress distributions and the stress intensity factors at the edges of the crack and at the corners of the finite strip are calculated. Results are presented in graphical and tabular forms.     

Interaction of a penny-shaped crack with an elliptic crack under shear loading

The problem of interaction between equal coplanar elliptic cracks embedded in a homogeneous isotropic elastic medium and subjected to shear loading was solved analytically by Saha et al. (1999) International Journal of Solids and Structures 36, 619–637, using an integral equation method. In the present study the same integral equation method has been used to solve the title problem. Analytical expression for the two tangential crack opening displacement potentials have been obtained as series in terms of the crack separation parameter _i up to the order _i⁵,(i=1,2) for both the elliptic as well as penny-shaped crack. Expressions for modes II and III stress intensity factors have been given for both the cracks. The present solution may be treated as benchmark to solutions of similar problems obtained by various numerical methods developed recently. The analytical results may be used to obtain solutions for interaction between macro elliptic crack and micro penny-shaped crack or vice-versa when the cracks are subjected to shear loading and are not too close. Numerical results of the stress-intensity magnification factor has been illustrated graphically for different aspect ratios, crack sizes, crack separations, Poisson ratios and loading angles. Also the present results have been compared with the existing results of Kachanov and Laures (1989) International Journal of Fracture 41, 289–313, for equal penny-shaped cracks and illustrations have been given also for the special case of interaction between unequal penny-shaped cracks subjected to uniform shear loading.     

Energy partitioning for a crack under remote shear and compression

The true nature and characteristics of crack growth mechanisms in geologic materials have not been adequately described and are poorly understood. The process by which deformation energy is converted to slipping and growing cracks under compressive stresses is complex and difficult to measure. A hybrid technique employing moiré interferometry as an experimental boundary condition to a finite element method (FEM) was employed for through-cracked polycarbonate plates under remote shear and compression. Cohesive end zone and dislocation slip models are used to approximate experimentally observed displacement characteristics. Shear-driven linear elastic fracture mechanics displacement predictions are shown to be inadequate for initial displacement progression. Moiré displacement fields of relative crack face slip reveal a near tip cohesive zone. The pre-slip moiré-FEM stress fields reveal that the maximum crack tip tensile stress occurs at approximately 45 degrees and further infers cohesive zone presence. A J integral formulation uses moiré displacement data and accounts for stored energy along the crack before and after shear driven crack face slip. These energy-partitioning results track the transfer of stored energy along the crack face to the crack tip until the entire crack is actively slipping. These laboratory-scale experiments capture basic mechanical behavior and simulate thousands of years of large-scale geologic feature displacement history in just a few hours.     

Explicit local buckling analysis of rotationally-restrained orthotropic plates under uniform shear

The explicit closed-form local buckling solution of in-plane shear-loaded orthotropic plates with two opposite edges simply supported and other two opposite edges either both rotationally restrained or one rotationally restrained and the other free is presented. Based on the boundary condition of the other two opposite edges, two types of plates are considered: the RR (both the edges rotationally restrained) and RF (one edge restrained and the other free) plate elements. Different plate buckled shape functions are proposed, and the approximate explicit expressions for the buckling loads are derived using the Rayleigh–Ritz method for the plate with the generic rotationally-restrained (R) boundary conditions which can be reduced to two extreme cases, i.e., simply supported (S) and clamped (C). The accuracy of the derived explicit solutions is verified by comparing the predictions with the existing solutions and numerical finite element analysis, and excellent agreements are obtained. The effects of material and boundary restraining parameters on the local shear buckling behavior of the plate elements are discussed. The derived explicit formulas for the shear buckling loads are straightforward, efficient and reliable for preliminary engineering design and analysis of composite structures under primarily shear-dominant loading conditions.     

Influence of the loading path on fatigue crack growth under mixed-mode loading

Fatigue crack growth tests were performed under various mixed-mode loading paths, on maraging steel. The effective loading paths were computed by finite element simulations, in which asperity-induced crack closure and friction were modelled. Application of fatigue criteria for tension or shear-dominated failure after elastic–plastic computations of stresses and strains, ahead of the crack tip, yielded predictions of the crack paths, assuming that the crack would propagate in the direction which maximises its growth rate. This approach appears successful in most cases considered herein.     

Buckling behavior of a central cracked thin plate under tension

The buckling characteristics of cracked plates subject to uniaxial tensile loads are analysed by the aid of the finite element method. Owing to the fact that crack buckling behavior is affected by the in-plane stress distribution around a crack, to get more accurate results, pre-buckling in-plane stress fields are analysed by the finite element method. For the critical loads calculation, the finite element approach adopted is based on Von Karman's linearize theory for buckling of plates subjected to pre-buckling state of plane stress. Several singular elements based on the Willian series are used in this plate bending approach. In this study, the effect of crack length, the effect of boundary condition, the effect of Poison's ratio and the effect of biaxial force on critical loads are analysed and discussed. Furthermore, the effect of initial imperfection is also discussed. There is a good agreement between other researcher's work and present results.     

The kinking behaviour of a bimaterial interface crack under indentation loading

The aims of this paper are twofold. The first is to evaluate the applicability of the formula for the crack kink angle—based on the maximum principle stress criterion—for predicting the interface kink angle in a bimaterial sample undergoing indentation loading. This formula was developed for cracks in homogenous materials but in this paper, it is used to predict the kink angle using the mode mixity at the tip of a crack lying on a bimaterial interface. The second aim is to examine the behaviour of the system, in terms of the crack kink angle and contact radius, for various coating thickness', crack lengths and combinations of properties of the coating and substrate. The system that is analysed consists of a planar bimaterial sample undergoing indentation with a tungsten-carbide spherical indenter. Two-dimensional, axisymmetric models are created to represent the system, with subdomains used for modelling the cracks. In order to determine the applicability of the kink angle formula, the angle predicted is compared to the angle that is directly calculated using boundary element method models that establish the angle of the kink which yields the maximum mechanical energy release rate. The second aim of the paper is achieved by varying the material property combinations and coating thickness of the bimaterial sample and observing the effect on the kink angle of the interface crack and the contact radius. The methodologies employed are initially verified on homogenous samples with known solutions.     

Large deformation of thin plates under localised impulsive loading

The response of thin clamped plates to a localised impulsive load imparted over a central circular region is investigated experimentally and theoretically to determine the location of tearing failure and the critical impulse to failure. Over 40 tests were performed with four different loading radii and increasing values of the applied impulse. Typical dish like deflection profiles were produced and measured. It was observed that smaller loading radii led to a more localised bulging near the plate centre. The resulting plate failure took the form of a disc. A more distributed load produced tearing failure at the clamped boundary.

In the theoretical development, the plate was modelled as a rigid-plastic membrane. The resulting initial-boundary value problem for the wave equation in the polar co-ordinate system was solved by the eigenfunction expansion method up to the point of first unloading. An approximate closed-form solution was also obtained by combining the wave form solution for early motion with the mode solution for late motion. A good correlation was obtained between the predicted and measured normalised deflection profile. The analysis of the theoretical solution gave strong indication that the initial velocity profile must have a form of a continuous and smooth function near the edge of the burning zone. Precise determination of the deflected shape is a necessary step for strain calculations and the prediction of the onset of tensile necking and failure of plates.     

Experimental investigations on buckling of cylindrical shells under axial compression and transverse shear

This paper presents experimental studies on buckling of cylindrical shell models under axial and transverse shear loads. Tests are carried out using an experimental facility specially designed, fabricated and installed, with provision forin-situ measurement of the initial geometric imperfections. The shell models are made by rolling and seam welding process and hence are expected to have imperfections more or less of a kind similar to that of real shell structures. The present work thus differs from most of the earlier investigations. The measured maximum imperfections δ_max are of the order of ±3t (t = thickness). The buckling loads obtained experimentally are compared with the numerical buckling values obtained through finite element method (FEM). In the case of axial buckling, the imperfect geometry is obtained in four ways and in the case of transverse shear buckling, the FE modelling of imperfect geometry is done in two ways. The initial geometric imperfections affect the load carrying capacity. The load reduction is considerable in the case of axial compression and is marginal in the case of transverse shear buckling. Comparisons between experimental buckling loads under axial compression, reveal that the extent of imperfection, rather than its maximum value, in a specimen influences the failure load. Buckling tests under transverse shear are conducted with and without axial constraints. While differences in experimental loads are seen to exist between the two conditions, the numerical values are almost equal. The buckling modes are different, and the experimentally observed and numerically predicted values are in complete disagreement.     

Linear buckling analysis of orthotropic inhomogeneous rectangular plates under uniform in-plane compression

Summary A linear buckling analysis is carried out for orthotropic inhomogeneous rectangular plates under uniform in-plane compression. It is assumed that material inhomogeneities of Young's modulus and shear modulus of elasticity are continuously changed in the thickness direction with the power law of the coordinate variable, while Poisson's ratio is assumed to be constant. The buckling equation can be successfully constructed as the linearized von Kármán plate model by introducing the newly defined position of the reference plane which enables us to easily deal with the problem. The critical buckling loads of the simply supported rectangular plate are presented using the derived fundamental relations. Effects of material inhomogeneity, material orthotropy, aspect ratio, width-to-thickness ratio and load ratio are discussed.     

Considerations in straight fold analysis of thin tubes under axial compression

A straight fold model for the analysis of the tubes under axial compression is reconsidered. Based on experiments, the tube is assumed to fold partly inside and partly outside the mean diameter of tube. The model considers the variation of circumferential strain during the formation of a fold, change in thickness of the tube during the deformation, and the influence of change in the yield strength of material in tension and compression. The first fold ends at the mean diameter and all subsequent folds initiate and end at the mean diameter. Some parameters have been introduced for making the expressions valid for first as well as the subsequent folds. The size of fold, load–compression curves, variation of inside and outside fold lengths and mean crushing load have been computed. All the variables involved in the analysis such as, the size of fold, folding parameter, mean crushing load, and the peak load have been obtained analytically. The model developed has been validated with the experiments.     

Influence of woven ply degradation on fatigue crack growth in thin notched composites under tensile loading

This paper deals with the fatigue of the through-the-thickness crack propagation in thin notched composite laminates made of two glass woven plies. It highlights the different crack growths between warp and weft directions of the woven ply. Experimental results show a decrease of the crack growth rate per cycle with the increase of the crack initiation time. Moreover, it has been shown that it is necessary to take into account the fatigue damage of the woven plies in term of loss of rigidity in the initiation phase. The fatigue crack growth rates are then quantified using Paris law type equations and linear elastic fracture mechanics (LEFM).     

Influence of loading path on fatigue crack growth under multiaxial loading condition

In this study, the specimens made of carbon steel S45 with an initial surface straight edge notch were subjected to combined cyclic axial‐torsion loading at room temperature. The fatigue life, surface crack extension direction and crack length were experimentally investigated. The effects of loading path, stress amplitude ratio and phase angle on the crack growth behaviour were also discussed. The results showed that, under the combination of cyclic axial and torsion loading, the tension stress amplitude had more effect on the initial crack growth path than the latter. The shear stress amplitude contributed mainly to the latter crack extension. The crack extension path was mainly determined by the stress amplitudes and the ratio of the normal stress to the shear stress, and almost independent of the mean stresses. The increase of the tension stress amplitude and shear stress amplitude would both accelerate the crack growth rate.     

Surface effects on time-dependent fracture parameter of subsurface crack in plates under tension

Based on the comprehensive finite element (FE) creep analyses, the influence of free surface on the time dependent fracture mechanics parameter of a crack near the free surface in plates under tension has been investigated. It is found that the time dependent fracture parameter C* increases as the crack tip closes to the free surface. Such an increment is related not only to the crack configurations but also to the material properties, especially the creep exponent n of power creep law. In addition, more pronounced interaction is observed between the C* of subsurface crack and that of a single isolated crack compared to that denoted by SIF under the linear elastic fracture condition. Under the framework of reference stress method, we also developed a closed form solution for creep interaction factor. Overall good agreement is achieved between the proposed method for the C* of subsurface crack and the FE results which provides us confidence in practical application.     

Radial stretching of a thin hollow membrane: biaxial tension, tension field and buckling domains

The radial stretching of a hollow thin membrane without compressive strength is considered within the framework of the small strain tension field theory. For each type of the uniform boundary conditions, the loading plane is partitioned into the domains of biaxial tension, tension field and buckling. The extent of these domains critically depends on the value of the Poisson??s coefficient and on the aspect ratio of the membrane. The stress and displacement fields are determined at an arbitrary stage of loading, when the outer biaxially stressed (taut) annulus surrounds the inner (tension field) portion of the membrane, characterized by continuously distributed infinitesimal wrinkles. The growth of the tension field as the loading increases is analyzed. It is shown that, depending on the Poisson??s coefficient and the aspect ratio of the membrane, the tension field may or may not spread throughout the whole membrane. For the fixed outer boundary, and the applied tension or a negative displacement at the inner boundary, and for a particular combination of the material and geometric parameters, the tension field instantly spreads to a specific depth within the membrane, dependent on the Poisson??s ratio and the outer radius of the membrane, remaining constant during further increase of the loading.     

The crack growth resistance of thin steel sheets under eccentric tension

The stable crack growth in thin steel sheets is the topic of this paper. The crack opening was observed using a videoextensometry system, allowing the crack extension determination. J_R-curve and δ_R-curve were established from obtained data. The ductile tearing properties of different thin sheets of steel were determined, including the impact of the specimen orientation, from test performed on compact tension specimens loaded under two conditions. The effect of the material, the rolling direction, and loading rate on the crack growth resistance of thin steel sheets was analyzed. In addition to the crack growth resistance, J-integral values for crack initiation were also estimated. The relation between J _i and J_0.2 was assessed using the basic mathematical and statistical methods. This relation was described by a linear regression model.     

Dynamic buckling of functionally graded cylindrical thin shells under non-periodic impulsive loading

Summary.  This paper is concerned with the study of the amplitude of the steady-state vibration in a right finite cylinder made of a mixture consisting of three components: an elastic solid, a viscous fluid and a gas. An exponential decay estimate of Saint-Venant type in terms of the distance from one end of the cylinder is obtained from a first-order differential inequality for a cross-sectional area measure associated with the amplitude of the steady-state vibration. The decay constant depends on the excitation frequency, the constitutive coefficients and the first positive eigenvalue for the fixed membrane problem for the cross-sectional geometry. The paper also indicates how to extend the results to a semi-infinite cylinder. Received October 24, 2002 Published online: April 17, 2003