An efficient and accurate iterative stress solution for an infinite elastic plate around two elliptic holes,subjected to uniform loads on the hole boundaries and at infinity

Using the Schwarz's alternating method and the Muskhelishvili's complex variable function techniques, an efficient and accurate stress solution for an infinite elastic plate around two elliptic holes, subjected to uniform loads on the hole boundaries and at infinity, is presented in this paper. The present algorithm can be used to compute the stress concentration factors (SCF), i.e., the ratio of the maximum tangential hoop stress to the applied uniform load, on the boundaries of the two elliptical holes of different sizes and layouts under different loading conditions, as illustrated in two numerical cases.     

Surface tension-induced stress concentration around a nanosized hole of arbitrary shape in an elastic half-plane

This paper studies surface tension-induced stress concentration around a nanosized hole of arbitrary shape inside an elastic half-plane. Of particular interest is the maximum hoop stress on the hole’s boundary with relation to the point of maximum curvature and the distance between the hole and the free surface of the half-plane. The shape of the hole is characterized by a conformal mapping which maps the exterior of the hole onto the exterior of the unit circle in the image plane. On using the technique of conformal mapping and analytic continuation, the complex potentials of the half-plane are expressed in a series form with unknown coefficients to be determined by Fourier expansion method. Detailed numerical results are shown for elliptical, triangular, square and rectangular holes. Two basic conclusions are that the hoop stress increases with decreasing hole size and the maximum hoop stress generally appears nearby but not exactly at the point of maximum curvature. In addition, it is shown that the hoop stress nearby the point of maximum curvature on the hole’s boundary increases rapidly with decreasing distance between the hole and the free surface of the half-plane. On the other hand, if the distance between the hole and the free surface is more than three times the hole size, the effect of the free surface on the stress concentration around the hole is ignorable and the elastic half-plane can be treated approximately as an elastic whole plane.     

弯曲波在含非圆孔洞无限压电薄板中的散射

基于线性压电动力学理论,采用波函数展开法、保角映射以及复变函数,对含非圆孔洞无限大压电薄板弹性波的散射及动应力集中问题进行了分析,给出了其动弯矩集中系数（DMCF）的解析表达式。为说明问题,以PZT-4为例,讨论了外加电场、椭圆孔长短半轴比、椭圆孔倾角以及入射波频率对含圆孔和椭圆孔无限大压电薄板弹性波散射的影响,并分别给出了无限压电薄板开圆孔和椭圆孔动弯矩集中系数的数值结果。     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

A coated rigid elliptical inclusion loaded by a couple in the presence of uniform interfacial and hoop stresses

We consider a confocally coated rigid elliptical inclusion, loaded by a couple and introduced into a remote uniform stress field. We show that uniform interfacial and hoop stresses along the inclusion–coating interface can be achieved when the two remote normal stresses and the remote shear stress each satisfy certain conditions. Our analysis indicates that: (i) the uniform interfacial tangential stress depends only on the area of the inclusion and the moment of the couple; (ii) the rigid-body rotation of the rigid inclusion depends only on the area of the inclusion, the coating thickness, the shear moduli of the composite and the moment of the couple; (iii) for given remote normal stresses and material parameters, the coating thickness and the aspect ratio of the inclusion are required to satisfy a particular relationship; (iv) for prescribed remote shear stress, moment and given material parameters, the coating thickness, the size and aspect ratio of the inclusion are also related. Finally, a harmonic rigid inclusion emerges as a special case if the coating and the matrix have identical elastic properties.     

Surface tension-induced stress concentration around an elliptical hole in an anisotropic half-plane

We examine the surface tension-induced stress concentration around an elliptical hole inside an anisotropic half-plane with traction-free surface. Using conformal mapping techniques, the corresponding complex potential in the half-plane is expressed in a series whose unknown coefficients are determined numerically. Our results indicate that the maximum hoop stress around the hole (which appears in the vicinity of the point of maximum curvature) increases rapidly with decreasing distance between the hole and the free surface. In particular, for an elliptical or even circular hole in an anisotropic half-plane we find that, with decreasing distance between the hole and the free surface, the hoop stress can switch from compressive to tensile at certain points on the hole's boundary and from tensile to compressive at others. This phenomenon is absent in the case of an elliptical or even circular hole in the corresponding case of an isotropic half-plane.     

Scattering of SH-wave by cracks originating at an elliptic hole and dynamic stress intensity factor

The method of complex function and the method of Green‘s function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamicstress intensity factor at the crack tip was given. A Green‘s function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed.     

The plane problem of an elliptically reinforced circular hole in an anisotropic plate or laminate

Summary  Within the scope of linear elasticity, the in-plane problem of an anisotropic plate or laminate with a circular hole and an elliptical hole reinforcement is considered. Arbitrary anisotropic elastic stiffnesses are allowed for the base plate and the reinforcement material, and for the reinforcement there is no restriction for its elliptical shape and size. The analysis of the problem is performed by the complex potential method with appropriately chosen series representations inside and outside the reinforcement region. The derived closed-form solution provides all resultant in-plane stresses and deformations within and around the hole reinforcement with little computational effort and at high accuracy. The determined solution allows a proper and effective assessment of hole reinforcements for many technical applications. Received 26 June 2000; accepted for publication 26 September 2000     

In-plane and out-of-plane constraint effects under pressurized thermal shocks

Transferability of fracture toughness data obtained on small scale specimens to a full-scale cracked structure is one of the key issues in integrity assessment of engineering structures. In order to transfer fracture toughness under different constraints, both in-plane and out-of-plane constraint effect should be considered for the specimens and structures. In this paper both in-plane and out-of-plane constraint effects of a crack in a reference reactor pressure vessel (RPV) subjected to pressurized thermal shocks (PTSs) are analyzed by two-parameter and three-parameter methods. The comparison between elastic and elastic–plastic analysis shows that the constraint effect varies with the material property. T₁₁ (the second term of William’s extension acting parallel to the crack plane) generally displays a reversed relation to the stress intensity factor (SIF) with the transient time, which indicates that the loading (SIF) plays an important role on the in-plane constraint effect. The thickness at the crack tip contributes more than the loading to the out-of-plane constraint, such that T₃₃ (the second term of William’s extension acting along the thickness) displays a similar relation to ε₃₃ (strain along the thickness direction) and a different relation to T₁₁ during the transient. The results demonstrate that both in-plane and out-of-plane constraint effect should be analyzed separately in order to describe precisely the stress distribution ahead of the crack tip.     

含椭圆孔的变角度复合材料层合板的屈曲性能研究

本文利用变角度复合材料的纤维方向角可沿平面位置任意连续变化的特点,提出在孔附近采用与孔同心的椭圆曲线作为纤维铺设路径的层合板铺层方案,以改善含椭圆孔的层合板的孔边应力集中,进而提高层合板的抗屈曲性能．主要研究内容有：利用ABAQUS软件分析本文提出的孔边特殊铺层方式下变角度复合材料层合板的面内应力分布及屈曲性能,通过与传统直线铺层方式以及线性变角度铺层方式进行比较,说明了本文提出的新铺层方式的优越性,并详细分析了椭圆孔的离心率、开孔尺寸及开孔方位对层合板的屈曲临界荷载的影响．研究结果可为含椭圆孔的变角度复合材料层合板的结构设计和优化提供一定的参考．     

Edge waves in asymmetrically laminated plates

We study edge waves propagating along the edge of an asymmetrically laminated elastic plate for which the out-of-plane component of displacement is coupled with the in-plane components. A Stroh-like formulation is used to show that such a plate can support at most two edge waves. An efficient method for computing the edge-wave speeds is proposed and explained through examples.     

In-plane/out-of-plane concentrated forces and moments on composite laminates with elliptical elastic inclusions

The problems of composite laminates containing elliptical elastic inclusions subjected to concentrated forces and moments are considered in this paper. By employing Stroh-like formalism for the coupled stretching–bending analysis, analytical closed form solutions are obtained explicitly. The generality of the solutions provided in this paper can be shown as follows: (1) The laminates include any kinds of laminate lay-ups, symmetric or unsymmetric, which allow the stretching and bending deformations couple each other. (2) The concentrated forces and moments can be applied in in-plane and/or out-of-plane directions, located inside and/or outside the inclusions. (3) The elliptical elastic inclusions can be any kinds of elastic materials including the limiting cases such as holes, rigid inclusions, cracks, line inclusions, etc. Since no such general solution has been found in the literature, the solutions are checked and verified by the special cases that no inclusions are embedded in the laminates, and that the inclusions are replaced by holes. Moreover, with various hardness ratios of inclusion and matrix some numerical examples showing the stress resultants along the interface are presented. Like the Green’s functions for the infinite laminates and those containing holes/cracks, the present solutions associated with the in-plane concentrated forces and out-of-plane concentrated moments have exactly the same mathematical form as those of the corresponding two-dimensional problems, in which the only difference is the contents of the symbols. While for the other loading cases, new types of solutions are obtained explicitly.     

Path Independent Integral for an Elliptical Hole in a Plate under Tension for Plane Stress Deformation Theory

An exact expression is derived for a path independent integral surrounding an elliptical hole in an infinite plate with tensile tractions at infinity for plane stress loading conditions. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under proportional loading and small strain assumptions. This problem may serve as a simple classroom example for the derivation of a relationship between crack opening displacement and path independent integral for a nonlinear elastic material satisfying the Tresca yield condition.      

CONSTRAINTS ON THE CRACK TIP PLASTIC FIELDS UNDER LARGE SCALE YIELD

Plane strain elastic-plastic finite element analyses are used to study the stress, strain fields around a straight crack in power hardening plastic material. Center crack panel (CCP), single edge crack panel (SECP) and double edge crack panel (DECP) tension specimens are analyzed with various crack lengths. Two local constraint parameters, i.e. in-plane stress ratioT _x and out-of-plane constraintT are analyzed, which are defined as tangential stress dividing normal (open) stress and out-of-plane stress dividing the sum of tangential stress and normal stress respectively. Numerical analyses indicate that the two local constraint parameters are nearly independent of the specimen geometry and the loading level in the plastic zone. Methods of estimating two constraint parameters and ways to define the boundary of the blunt zone are given. By using these parameters, the stress distributions in the plastic zone, especially in the blunt zone can be accurately predicted. Project supported by the National Foundation of Distinguished Young Scientists of China (No. 59625510).     

The effect of out-of-plane restraint on plane-stress solutions near straight boundaries

The existence of out-of-plane displacement restraint where thin photoelastic plates are bonded to relaively rigid boundaries or inclusions will induce transverse stress components whose presence can cause the in-plane stresses to deviate significantly from the plane-stress solution. The extended generalized planestress formulation, which was developed in a previous paper to study the effect of out-of-plane restraint on the through-the-thickness averages of the in-plane stresses and displacements in thin plates, is applied here to obtain correction factors by which the photoelastically determined in-plane stress components at a straight restrained boundary can be multiplied to recover the desired twodimensional solution. Cases of mechanical and thermal loadings which vary sinusoidally in the direction parallel to the restrained boundary are treated. The stress-field alterations due to out-of-lane restraint are shown to depend strongly on Poisson's ratiov and the ratioL/H of the half-wavelength of the load variation to the plate thickness, and can be minimized by choosing a photoelastic material with the smallest values ofv andH possible. The simple asymptotic form of the stress-field alterations for large values ofL/H has enabled us to prescribe simple correction factors in this range. Finally, it is observed that, although the normal stress component which is parallel to the restrained boundary is altered to a greater degree than are the other in-plane stress components, the effect on this stress component penetrates into the plate a distance of only about one plate thickness, whereas the effect on the other components penetrates a distance of approximately one half-wavelength of the load variation.     

Blind-hole residual stress determination using optical interferometry

The in-plane method and the out-of-plane method are used to analyze blind-hole residual stress as measured by optical interferometry. The in-plane method, which constructs a relation between the in-plane displacement field and the residual stress released from blind-hole drilling, is applicable when the sensitivity vector of the interferometer used in the measuring system is parallel to the object surface. Three in-plane displacements obtained from one interference pattern are sufficient to determine the residual stress. The out-of-plane method, which establishes a new relation between the out-of-plane displacement field and the released residual stress, is suggested when the sensitivity vector is perpendicular to the object surface. Two relative out-of-plane displacements extracted from one interference pattern are sufficient to determine the residual stress. With the adoption of these two methods, interpolating calculation is not needed to determine the fringe order of each data point, since the selections of the required data points are flexible using these two methods. Two experiments, one for the in-plane method and the other for the out-of-plane method, were carried out to illustrate the applicability and usefulness of these two methods.     

Elasto-plastic buckling analysis for perforated steel plates subject to uniform compression

This study investigates the elasto-plastic buckling behaviour of simply supported square and rectangular thin steel plates having elliptic cut-outs by means of finite element method. Plates with simply supported in the out-of-plane direction are applied uniform compression in long-edge direction. A50 steel was used in the analysis and the focus was on the effect of plate aspect ratio, elliptical hole size, elliptical hole angle, elliptical hole location and slenderness ratio on buckling behaviour. It was found in the study that as the plate slenderness ratio increases, the critical buckling stress decreases for all the perforated plates.     

Nonlinear analysis of a composite panel with a cutout repaired by a bonded tapered composite patch

This study presents a solution method to analyze the geometrically nonlinear response of a patch-repaired flat panel (skin) with a cutout under general loading conditions. The effect of induced stiffening due to tensile loading on the in-plane and, particularly, the out-of-plane behaviors of the patch-repaired skin are investigated. The damage to the skin is represented in the form of a cutout under the patch. The patch with tapered edges is free of external tractions. The skin is subjected to general boundary and loading conditions along its external edge. The solution method provides the transverse shear and normal stresses in the adhesive between the skin and the patch, and in-plane and bending stresses in the patch and skin. Both the patch and skin are made of linearly elastic composite laminates, and the adhesive between them is homogeneous and isotropic, exhibiting a bi-linear elastic behavior. Modified Green’s strain–displacement relations in conjunction with von Karman assumptions are employed in determining the in-plane strains in the skin and patch; however, the transverse shear strains in the adhesive are determined based on the shear-lag theory. The present solution method utilizes the principle of virtual work in conjunction with complex potential functions.     

Singular full-field stresses in composite laminates with open holes

A method of the superposition of a hybrid and displacement approximation was developed to provide the accurate stress fields in a multilayered composite laminate, including the singular neighborhood of the ply interface and the hole edge. Asymptotic analysis was used to derive the hybrid stress functions. The displacement approximation is based on the polynomial B-spline functions. The method provides the determination of the coefficient of the singular term along with convergent stress components including the singular regions. Reissner’s variational principle was employed. Simple [45/−45]_s and quasi-isotropic IM7/5250 [45/90/−45/0]_s laminates were analyzed. Uniaxial loading and residual stress calculation (quasi-isotropic laminate) were considered. A convergence study showed that accurate values of the coefficient of the singular term of the asymptotic stress expansion could be obtained with coarse out-of-plane and in-plane subdivisions. The interaction between the singular terms on the neighboring interfaces was found to be important for the convergence with coarse subdivisions. Converged transverse interlaminar stress components as a function of the distance from the hole edge, were shown for all examples.     

Nonlinear Oscillations of a Nonresonant Cable under In-Plane Excitation with a Longitudinal Control

The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.