On the stress problem of large elliptical cutouts and cracks in circular cylindrical shells

Numerical solutions are presented for stresses around an elliptical hole in a long, thin, circular cylindrical shell subjected to axial tension for both the symmetric orientations of the hole with respect to the shell. The method of analysis involves obtaining a series solution to the governing shell equations in terms of Mathieu functions by the method of separation of variables and satisfying the boundary conditions numerically term by term in a Fourier series formulation. Results are presented in the form of charts from which stress concentration factors can be directly read over a wide range of the two parameters, namely, axis ratio of the ellipse and a curvature parameter defining the hole size with respect to dimensions of the shell.An interesting feature of the investigation is the analysis of limiting cases of circumferential and axial cracks for axial tension and internal pressure loadings respectively. The method developed involves determining the solution completely in elliptic coordinates and then determining the singular stresses by carrying out a transformation to polar coordinates with crack tip as the origin through a Taylor series expansion. Membrane and bending stress intensity factors are computed and plotted over a sufficiently wide range of the curvature parameter extending from small to large sized cracks. As an outcome of the analysis, a “hybrid” technique has been developed by which singularity conditions at the crack tip can be handled effectively in dealing with boundary conditions in crack problems.     

拉压不同弹性模量薄板上圆孔的应力分析

采用弹性理论研究了拉压不同弹性模量薄板上圆孔的孔边应力集中问题．采用广义虎克定律推导出了拉压不同弹性模量薄板上圆孔边的应力平衡方程,并联合利用应力函数及边界条件得到了拉压不同弹性模量薄板上圆孔边的应力表达式．算例分析表明,当薄板材料的拉压弹性模量相差较大时,采用经典弹性理论研究薄板上圆孔的孔边应力是不合适的,当经典弹性理论与拉压不同弹性模量弹性理论的计算结果间的差别超过工程允许误差5%时,应该采用拉压不同弹性模量弹性理论进行计算．     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

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对具有不同拉压模量的厚壁球壳,采用双剪统一强度理论推导了其扩张问题的应力及位移的 统一解. 分析了不同模量、不同模型控制参数对厚壁球壳扩张时的扩张压力和应力场的影响. 结果表明：厚壁球壳弹性极限压力、应力场、位移场等均随着模量控制参数、模型参数的变 化而变化,在$\alpha<1$的情况下(即$E^ + < E^ - $),可以明显提高球壳的弹 性极限压力$p_e $; 厚壁球壳塑性极限压力与材料的拉压模量无关,与模型参数$\eta $有关,且随$\eta$的增加,先增大后减小. 因此若采用经典的弹性理论和单一的 模型参数对厚壁球壳进行设计计算,会带来较大的误差.     

The evolution of caustics from holes to cracks

The method of caustics was used for the study of the evolution of stress concentration around a circular hole, which progressively changes in shape and becomes an elliptic hole, tending to an internal crack. The influence of the amount of ellipticity of the holes and their orientation relative to the axis of the applied external loads at infinity on the form of caustics created around the discontinuity was studied, as the elliptic holes tended to become internal cracks. A series of experiments with tension specimens containing small elliptic holes of any ellipticity and orientation was performed. Comparison of experimentally obtained caustics with theory yielded a good agreement of both results. Finally, the use of small elliptic holes drilled all over a biaxial stress field for the determination of the individual principal stresses and the principal directions at the area of the holes was outlined.     

基于大型炉体焊接残余应力测试的盲孔法研究

为提高炉体焊接残余应力测试精度,对传统的盲孔法进行改进,即在测点打孔后多次扩孔并增加孔深;通过对SM400ZL钢焊接残余应力测试时的应变释放系数A,B进行实验标定,得出多组应变释放系数值,计算可得残余应力的估算值,并对不同状况下的残余应力值进行比较.通过大拉力标定试验得出SM400ZL钢盲孔的孔边塑性变形对测量精度的影响和塑性修正公式.结果表明,改进的盲孔法能够更准确地测得焊接残余应力值,测量结果经塑性修正,最大误差减小到2%以下.测试方法和结果可直接在相关的工程实际中应用.     

Experimental Investigation of the Stress Concentration in Axially Compressed Thick Cylindrical Shells with Rectangular Holes

The stress concentration in cylindrical shells of various thicknesses is studied experimentally. The shells are made of epoxy resin, have a hole, and are subjected to axial compression. The shell thickness and the opening angle are shown to affect the stresses near the corners of the hole. The experimental and theoretical values of the stress concentration factor are compared for a thin shell     

Application of the gradient strength criterion and the boundary-element method to a plane stress-concentration problem

A numerical algorithm for strength analysis of plane structural elements with stress concentrators is developed using the gradient strength criterion and the boundary-element method. As the first calculation test, the brittle fracture of a plate with a circular hole in tension is evaluated. To verify the algorithm and to compare the results of analysis with experimental data available in the literature, we consider symmetric and asymmetric problems of fracture of glass plates with a narrow elliptic hole in tension and compression. For all the problems, the accuracy of numerical results is estimated by comparison with analytical solutions. In comparison with the classical criteria, the use of the gradient strength criterion leads to better agreement between theoretical estimates and experimental data. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 214–221, July–August, 1999.     

Alleviation of the stress concentration with analogue reinforcement

In order to reduce the stress concentration around a hole in a plate, new, “analogue” reinforcements instead of reinforcing rings were used in this investigation. In two of these specimens, reinforcements with different volume fractions were arranged to coincide with the stress trajectories for an infinite plate with a hole under uniaxial tension. Two other specimens containing straight rectangular-grid-type reinforcements were made by using a photofabrication method. Specimens were then prepared by sandwiching these reinforcements between two epoxy-resin plates. Plane specimens, i.e., without reinforcement, were also made of the same epoxy resin for comparison. The stress concentrations at the edge of the hole under uniaxial tension were determined by photoelastic techniques. The measured stress-concentration factors were compared with well-known values for an infinite, isotropic, homogeneous plate containing a hole. Results were also compared with published data on [90/0/90/0]_s 7-ply laminated composite plates, and on plates strengthened with reinforcing rings. A definite reduction in stress concentration was observed on specimens containing analogue reinforcement.     

Normalized stresses around an elliptic hole in a finite plate of linear material subjected to large uniform in-plane loading

Stress-concentration factors associated with the large deformations of in-plane loaded plates with elliptic holes are presented in a form with which designers are familiar. An analytic expression to obtain the stress-concentration factor is included.     

拉压异性线性等向强化材料厚壁球壳极限分析

运用Mohr屈服准则对承受内压的拉压屈服强度不同的线性等向强化材料的厚壁球壳进行了极限载荷分析，得到了依赖于拉压比和强化模量的厚壁球壳极限载荷解析式和依赖于拉压比和强化模量的厚壁球壳极限载荷分析式。结果表明，材料拉压屈服强度的不同和强化特性对厚壁球壳极限载荷均有一定的影响。     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Effects of the curvature radius of a cylindrical shell on the stress intensity factors of surface cracks

The effects of the curvature radius of a cylindrical shell on stress intensity factors are investigated in circumferential (inner and outer) semielliptical surface cracks in a cylindrical shell. What is new in this paper is to have given: (1) The stress intensity factors for surface cracks in a cylindircal shell are determined by photoelastic technique. (2) By a special method photoelastic slices are handled for obtaining a clear caustic curve, and the stress intensity factors for surface cracks in a cylindrical shell are determined by the caustic method. (3) An approximate equation of curvature correction factor F_c is proposed. (4) Effects of the curvature radius R of a cylindrical shell on the stress intensity factors of surface cracks are obtained. The results of this paper are in fair agreement with already existing analytical results. The approximate equation of curvature correction factor F_c can be widely used for engineering purposes.     

Physically and geometrically nonlinear deformation of conical shells with an elliptic hole

The elastoplastic state of conical shells weakened by an elliptic hole and subjected to finite deflections is studied. The material of the shells is assumed to be isotropic and homogeneous; the load is constant internal pressure. The problem is formulated and a technique for numerical solution with allowance for physical and geometrical nonlinearities is proposed. The distribution of stresses, strains, and displacements along the hole boundary and in the zones of their concentration is studied. The solution obtained is compared with the solutions of the physically and geometrically nonlinear problems and a numerical solution of the linear elastic problem. The stress-strain state around an elliptic hole in a conical shell is analyzed considering both nonlinearities __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 69–77, February 2008.     

拉压性能不同材料厚壁圆筒和厚壁球壳的极限压力分析

本文用广义双剪应力强度理论对拉压性能不同的材料制成的厚壁圆筒和厚壁球壳进行了弹塑性应力分析，得出与拉压比有关的弹性极限内压力、塑性极限内压力、弹塑性区的应力以及弹塑性内压力与弹塑性半径之间的关系式．     

Physically and Geometrically Nonlinear Deformation of Spherical Shells with an Elliptic Hole

The elastoplastic state of thin spherical shells with an elliptic hole is analyzed considering that deflections are finite. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. Problems are formulated and a numerical method for their solution with regard for physical and geometrical nonlinearities is proposed. The distribution of stresses (strains or displacements) along the hole boundary and in the zone of their concentration is studied. The results obtained are compared with the solutions of problems where only physical nonlinearity (plastic deformations) or geometrical nonlinearity (finite deflections) is taken into account and with the numerical solution of the linearly elastic problem. The stress—strain state in the neighborhood of an elliptic hole in a shell is analyzed with allowance for nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 95–104, June 2005.     

Thermally conductive elliptic inhomogeneities in 2D anisotropic solids

The problem of a thermally conductive elliptic hole embedded in an anisotropic thermoelastic solid under a remote uniform heat flow is considered. For the plane problem, solutions are derived by the use of an extended thermoelastic version of the Stroh formalism. The hoop stress around the elliptic hole is obtained in an explicit real form. We analyze the influence of the interior thermal conductivity and the cavity thickness on the temperature and stress on the boundary points of the hole. By comparison with the thermally insulated cavity, we show that the consideration of the interior conductivity may lead to significantly different thermomechanical behaviors. Ana Ursescu on leave from Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie no. 13, Bucharest, Romania.     

Estimate of the strength of a plate with an elliptic hole in tension and compression

The strength of a plate with an elliptic hole under uniaxial tension or compression is estimated for arbitrary angles between the ellipse axes and the direction of loading with the use of the gradient strength criterion. The calculated critical stress agrees with the existing experimental data. Institute of Physicotechnical Problems of the North, Siberian, Division, Russian Academy of Sciences, Yakutsk 677891. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 163–168, May–June, 2000.     

Influence of the viscoelastic properties of a composite on the stress distribution around an elliptic hole in a plate

The time variation in the stresses around an elliptic hole in a composite plate is studied. Solutions that characterize the effect of the time dependence of the relaxation moduli of the composite components on stresses are obtained. The solutions in the time domain are obtained from the elastic–viscoelastic analogy and the corresponding elastic solutions for the effective moduli of the composite and the stress field around an elliptic hole in an anisotropic plate. The inverse Laplace transformation is carried out by an effective numerical method     

Tension–compression asymmetry of phosphor bronze for electronic parts and its effect on bending behavior

In-plane tension and compression experiments on copper alloy sheets (phosphor bronze) and 6000 series aluminum alloy sheets (AA6016-T4) were conducted using a specially designed testing apparatus. The apparatus is equipped with comb-type dies so that stress–strain curves of a sheet specimen subjected to tension followed by compression, and vice versa, can be measured without buckling of the specimen, as well as those for monotonic tension and compression. A difference was observed in the flow stresses between tension and compression for the as-received copper alloy, but not for the aluminum alloy. Moreover, stress reversal tests, such as tension followed by compression and compression followed by tension, were carried out in order to measure the Bauschinger effect. In the second part of the experiment, bending moment–curvature diagrams were measured for the as-received and pre-stretched specimens. The bending moment–curvature diagrams were compared with those calculated using the stress–strain curves obtained from the tension–compression tests, and were in good agreement with those calculated with the tension–compression asymmetry and the Bauschinger effect correctly reproduced.