CALCULATION OF THE STRESS INTENSITY FACTOR FOR A PARTIAL CIRCUMFERENTIALLY CRACKED TUBE LOADED IN BENDING BY USING THE SHELL LINE-SPRING MODEL

Abstract— In this paper the line-spring model (LSM) developed by Rice and Levy is used to obtain an approximate solution of the stress intensity factor for a partial circumferential, externally cracked tube under axial tension and four point bending. The calculation is based on the work done by Delale and Erdogan for cylindrical shells containing a circumferential or an axial semi-elliptical, part-through crack. The range of utility of their analysis is enlarged to thicker wall tubes with nonelliptic and longer part-through circumferential cracks. Values of K ₁ calculated by the LSM are compared with those from a finite element analysis for remote tensile loading and bending cases, which shows fairly good agreement. The calculations are also applied to a fatigue crack growth test in a tube in four point bending to correlate the d a /d N vs λ K ₁ data.     

Energy theory of linear fracture mechanics of shells with crack-cuts

A variational approach to calculating shells with crack-cuts based on a nonclassical Timoshenko-type theory is proposed and realized. Compliance and virtual crack growth methods are proposed and developed for calculating the stress intensity factor, which is the main characteristic of the local stress field at the crack tip. These methods are realized by the finite-element method in the displacement and finite-difference variant. Analytical and mathematical models of the crack-cut are proposed. A spherical shell with one and two through cracks rectilinear in plan and a square cylindrical panel with a crack located in the center of the diagonal were calculated.Translated from Problemy Prochnosti, No. 7, pp. 59–67, July, 1995.     

Numerical and analytical study of severity of cracks in cylindrical and spherical shells

The present study deals with the severity of cracks in pressure equipments, where such defects are often involved. Our work is particularly concerned with the problem of cylindrical shells and also the little well-known problem of spherical shells, including all sorts of practical defects, namely axisymmetric or semi-elliptic, both internal and external cracks. The stress intensity factor in the linear elastic domain and the J integral in the elastoplastic range are performed using the finite element method and compared to the results provided by the application of the semi-analytical A16 or R6 simplified criteria, depending on a limit load calculation. The nocivity of the defects depends on the crack shape and size and other structural geometrical parameters. Use is made of a polynomial decomposition of the stress field in the vicinity of the crack in order to cover all industrial loadings. All the numerical results, for a wide range of shell and crack geometries, are depicted using appropriate tables and curves in order to check the fracture criteria more easily.     

The elastic analysis of arbitrary thin shells having part-through cracks, using the integrated line spring and the semiloof shell elements

This work describes the Integrated Line Spring Element, which was specially designed to be associated to Semiloof shell elements for the simulation of the structural behaviour of the remaining ligament in a part-through crack in a thin shell. In its formulation, it follows the “line spring model” introduced by Rice and Levy (1972) for virtually infinite plates of small thickness. This element was used to determine the stress intensity factor for internal and external semielliptical cracks in arbitrary shells under mode I, mode II and mode III load conditions.     

Tension of a cylindrical bar having an infinite row of circumferential cracks

In this paper, the crack problems in the case of a cylindrical bar having a circumferential crack and a cylindrical bar having an infinite row of circumferential cracks under tension are analyzed by the body force method. The stress field for a periodic array of ring forces in an infinite body is used to solve the problems. The solution is obtained by superposing the stress fields of ring forces in order to satisfy a given boundary condition. The stress intensity factors are calculated for various geometrical conditions. The obtained values of stress intensity factor of a single circumferential crack are considered to be more reliable than the results of other paper's. As the crack becomes very shallow, the stress intensity factor of a row of circumferential cracks approaches the value corresponding to that of a row of edge cracks in a semi-infinite plate under tension. As the crack becomes very deep, it approaches the values corresponding to that of a single deep circumferential crack.     

Fatigue growth of surface cracks in bending

A simple procedure for calculating the fatigue crack propagation lifetimes of surface-initiated part-through cracks in bending is discussed and agreement is obtained with previously reported experimental data. It is shown that a realistic fatigue analysis of these cracks must include an explicit determination of crack depth as a function of surface length and that, in general, the depth-to-length ratio varies with crack length, crack nucleus geometry and with the stress intensity factor exponent in the Paris growth rate relationship. Numerical results are presented for some typical situations.     

Numerical solutions of three dimensional dynamic crack problems and simulation of dynamic fracture phenomena by a “non-standard” finite difference method

A brief presentation of the novel features of a non-standard finite difference method (HEMP and HEMP 3D computer codes) is given. To demonstrate the capability of the HEMP 3D code with the extrapolation sub-routine for solving dynamic problems in linear fracture mechanics, through-thc-thickness cracks and semielliptic surface cracks of a section of cylindrical pipe under Heaviside-function time dependence pressure applied to the interior surface are considered and their dynamic stress intensity factors are calculated. The dynamic fracture phenomenon caused by the penetration of a rigid cylindrical punch into the center of a cylindrical wafer of linear-elastic brittle material is simulated numerically by using the HEMP code with the incorporation of the method of equivalent free-surface boundary conditions and the method of artificial velocity. A simple physical model that crack will initiate and propagate in any region where a critical stress has been reached and the crack will propagate in a direction perpendicular to the maximum principal stress is chosen here.     

Prediction of part-through crack growth under cyclic loading

A comparative analysis is performed of the applicability of the local, averaged and effective stress intensity factor ranges as part-through crack growth criteria under cyclic loading. The effective stress intensity factor range is found to be preferable for the purpose of part-through crack growth prediction. The surface and corner crack shape variations under cyclic loading are predicted and fatigue lives of cracked specimens are estimated. Part-through crack growth prediction results are compared with experimental data.     

Experimental study of pressurized,foam reinforced,cracked cylindrical aluminum shells

Cracks in pressurized cylindrical shells are subject to stress fields peculiar to the shell geometry and loading conditions which makes them more critical than cracks in flat plates under similar conditions. In order to mitigate the stress field around the crack tips a foam layer is applied to the shell as reinforcement. The effect of this solution on crack behavior is evaluated experimentally. Pre-cracked cylindrical aluminum shells with layers of rigid foam applied to their inner side are tested by subjecting them to cyclic internal pressure. A test bench built especially for the purpose of cyclic testing is described as well as the testing procedure. Results indicate that crack growth rates are significantly reduced and fatigue life is extended by as much as 161%. The application of a foam layer to may help slow down the growth of existing cracks. This solution may prove useful especially in ageing aircraft fuselage structures.     

Stress intensity factors for circumferential through‐wall cracks in thin‐walled cylindrical shells subjected to tension and torsion

In order to assess the structural integrity of tubular members or pipes containing circumferential through‐wall cracks, their stress intensity factor solutions are required. While stress intensity factors for tension and bending are available, few solutions exist for the case of torsion, even though these components may also be subjected to torque. In this paper, the finite element method is used to compute the stress intensity factors for this geometry under tension and torsion. Shell elements are employed to compute the results for thin shells by the means of the displacement extrapolation technique. The computed results indicate that the available analytical solution for torsional loading, which is based on shallow shell theory, is nonconservative for long cracks in thin shells. Shallow shell theory is in general not applicable to long cracks, and the present work is therefore able to provide solutions for a wider range of crack lengths than what is currently available.     

Stress-strain state in a closed cylindrical shell containing a system of part-through cracks

Summary A study has been made on the state of stress and strain in a closed cylindrical shell weakened by a system of part-through longitudinal surface cracks. The line-spring model has been used in the examination.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 4, pp. 89–92, July–August, 1989.     

A [90/0]s composite laminate with part-through matrix cracks in adjacent plies

A [90/0]_s orthotropic composite laminate with part-through matrix cracks is considered. Stress intensity factors are determined for the cracks using a linear-elastic analysis. These matrix cracks run along the fiber direction of the individual plies. The crack-geometry considered here is one where the matrix cracks in adjacent plies form a cross-like pattern in the plan view of the laminate. The plies are assumed bonded by thin resin-rich adhesive layers. These adhesive layers are modeled as distributed shear springs. Each ply of the laminate is modeled as a thin elastic orthotropic layer under plane stress. The laminate is subject to both tensile and shear loading. The mathematical model for the stresses and displacements in the layers reduces to a pair of Fredholm integral equations which are solved numerically. The stress intensity factors show a strong dependence on crack-sizes and nature of loading. In particular, the magnitudes of the stress intensity factors for the matrix crack in the 0° layer are increased significantly by the crack in the adjacent 90° layer.     

Numerical investigation of the dynamic strength of thick-walled cylindrical shells with cracklike technological defects

A numerical method is proposed for the evaluation of the stress intensity factors in elastic thick-walled cylindrical shells with cracks subjected to dynamic loading. The method is based on the two-dimensional axially symmetric Wilkins algorithm and the equations of mechanics of brittle fracture. The strength of thick-walled cylinders loaded by pulses of internal pressure and containing technological defects (similar to mathematical cuts) formed at the sites of fixing of their ends by welding is investigated.Translated from Problemy Prochnosti, No. 1, pp. 76–87, January–February, 2005.     

On a finite element model for the analysis of through cracks in laminated anisotropic cylindrical shells

A finite element model for the analysis of laminated composite cylindrical shells with through cracks is presented. The analysis takes into account anisotropic elastic behaviour, bending-extensional coupling and transverse shear deformation effects. The proposed finite element model is based on the approach of dividing a cracked configuration into triangular shaped singular elements around the crack tip with adjoining quadrilateral shaped regular elements. The parabolic isoparametric cylindrical shell elements (both singular and regular) used in this model employ independent displacement and rotation interpolation in the shell middle surface. The numerical comparisons show the evidence to the conclusion that the proposed model will yield accurate stress intensity factors from a relatively coarse mesh. Through the analysis of a pressurised fibre composite cylindrical shell with an axial crack, the effect of material orthotropy on the crack tip stress intensity factors is shown to be quite significant.     

A method for stress intensity factor calculation of infinite plate containing multiple hole-edge cracks

Multiple site damage is the occurrence of small fatigue cracks at several sites within aging aircraft structures. Focusing on this typical structure, an analytical method for calculating the stress intensity factor of an infinite plate containing multiple hole-edge cracks was introduced in this paper. The properties of complex variable functions are used to evaluate the stress function. The approximate superposition method is applied to solve stress intensity factor problems on multiple holes. The equivalent crack is introduced to modify the method. Some numerical examples of an infinite plate containing two hole-edge cracks are examined by the method. By comparing the analytical and finite element analysis results it was realized that the analytical results are accurate and reliable. This modified analytical method is easier to apply than some traditional analytical methods and can provide stress intensity factor solutions for an infinite plate containing a random distribution of multiple hole-edge cracks.     

基于非傅里叶热传导对热冲击下带裂纹厚壁圆筒的分析

随着科学技术、工业水平的发展,传统的傅里叶导热在极端条件下不再适用。基于双曲型单相延迟非傅里叶热传导方程,推导了热冲击下有限元方程,编写了有限元算法程序,研究了在热冲击载荷下含裂纹厚壁圆筒结构的热力学响应,计算出厚壁圆筒在非经典传热条件下的温度场、位移场和裂纹尖端应力强度因子的数值解,分析不同热冲击载荷、不同裂纹长度、不同相位延迟下非傅里叶热传导的波动性效应以及温度应力强度因子的变化,得到相应的结论。为非经典工程条件下,带裂纹厚壁圆筒构件的可靠性以及构件的优化设计提供了数值上的参考。     

Stress and Failure Analysis of Brittle Matrix Composites. Part II: Failure Analysis

A linear elastic fracture mechanics analysis of a cylindrical element of matrix with a single fiber and two matrix annular cracks perpendicular to the fiber direction under longitudinal tensile load was undertaken. The order of singularity and the angular dependence of the stress field in the neighborhood of the crack periphery were determined by using the stress function approach proposed by Zak and Williams. The stress intensity factor was evaluated by combining the results of the local stress solution with a finite element analysis. The case of fiber debonding originating from the periphery of the annular cracks was also studied. For that problem both opening-mode and sliding-mode stress intensity factors and the strain energy release rate were determined. These results help to understand the various failure mechanisms including matrix cracking, debonding along interfaces and kinking of interface cracks into fibers in brittle matrix composites. This revised version was published online in July 2006 with corrections to the Cover Date.     

Shear stress intensity factor near semi-infinite cylindrical crack edges under their shock loading

The discontinuous solution of the torsional vibration equation for an elastic medium with a flaw in the form of a semi-infinite cylindrical crack is constructed. The method of solving the integro-differential equation describing the distribution of shear stresses along the edges of a cylindrical crack is presented. The evaluation procedure for a stress intensity factor and its numerical calculation for the case of short times under the shock loading of cylindrical crack edges are given. It is established that the magnitude of a dynamic stress intensity factor can be used to determine the condition of shock wave interactions with structural heterogeneities at the high-rate deformation of treated surfaces containing flaws in the form of cylindrical cracks. Translated from Problemy Prochnosti, No. 3, pp 63–72, May–June, 1999.     

Weight function calculation for surface cracks using the line-spring model

A numerical method for calculating weight functions for surface cracks in plates and shells is proposed. Thick-shell finite elements are used to create the discrete model of a body with a through-wall flaw. Line-spring elements transform the through-wall flaw into a surface crack. A quadratic line-spring element is presented. Weight functions for some semielliptical surface cracks in a plate have been calculated. The weight functions obtained may be used for computing stress intensity factors related to two-dimensional stress fields at the crack surface.     

椭圆形半露头裂纹的线弹簧模型

建立了无限平板椭圆形半露头裂纹线弹簧模型,由积分变换方法推导了问题的控制方程和应力强度因子表达式,给出了相应于表面裂纹及内埋裂纹的计算结果,通过与现有有限元解或交替迭代解比较,表明了解的良好性。由此说明半露头裂纹的线弹簧模型解也将是合理的可靠的。