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.     

On the Green's Functions and Boundary Integral Formulation of Elastic Planes with Cutouts

ABSTRACT

The problem of an infinite elastic plane that contains a hole of arbitrary shape and is subjected to a concentrated unit load is considered. The Green's function (influence function) for the problem is formulated by means of two complex potential functions. This is accomplished by mapping the region that is exterior to the hole onto a unit circle. A class of closed contour hole shapes is analyzed. Green's functions for an elliptical hole and a class of triangular holes are determined. Green's functions for a class of rectangular holes are also discussed. In order to determine stress and displacement fields for the finite plane problem, Green's function is employed and an indirect boundary integral equation is formulated, with the integrand of the integral equation incorporating the effect of the hole. The contour of the hole is no longer considered a part of the boundary and only the contour of the region that is exterior to the hole is subdivided into boundary elements. Examples for elliptical and triangular holes are solved.     

Complex variable function method fob hole shape optimization in an elastic plane

In this paper, a complex variable function method for solving the hole shape optimization problem in an elastic plane is presented. In this method, the stresses in hole problems are analysed by taking advantage of the efficiency of the complex variable function method. To optimize the hole shape, the coeffecients in conformal mapping functions are taken as design variables, and the sensitivity analysis and gradient methods are used to reduce the largest circumferential stress in absolute value and at the same time to make the second largest circumferential stress in absolute value not to exceed the largest one (in fact, these two stresses are the stationary values of the circumferential stresses). The coefficients in conformal mapping function are revised by iteration step by step until the largest circumferential stress in absolute value is reduced to the second largest stress. This method guarantees the continuity, differentiability and accuracy of the stress solution along the boundary, and it is evident that this method is better than either the difference method or the finite element method.     

A HYBRID METHOD FOR ANALYZING THE DYNAMIC RESPONSES OF CAVITIES OR SHELLS BURIED IN AN ELASTIC HALF-PLANE

In this paper, based on a variational formalism which originally proposed by Mei [1] for infinite elastic medium and extended by Yeh, et al. [2,3] for elastic half-plane, a hybrid method which combines the finite element and series expansion method is implemented to solve the diffraction of plane waves by a cavity buried in an elastic half-plane. The finite domain which encloses all inhomogeneities including the cavity can be easily formulated by finite element methods. The unknown boundary data obtained by subtracting the known free fields from the total fields which include the boundary nodal displacements and tractions at the interface between the finite domain and the surrounding elastic half-plane are not independent of each other and can be correlated through aseries repre sentation. Due to the continuity condition at the interface, the same series representation is still valid for the exterior elastic half-plane to represents the scattered wave. The unknown coefficients of this series are treated as generalized coordinates and can be easily formulated by the same variational principle. The expansion function of the series is composed of basis function. Each basis function is constructed from the basis function for an infinite plane by superimposing an additional homogeneous reflective term to satisfy both traction free conditions at ground surface and radiation conditions at infinity. The numerical results are made against those obtained by boundary element methods, and good agreements are found.     

Harmonic elastic inclusions in the presence of point moment

We employ conformal mapping techniques to design harmonic elastic inclusions when the surrounding matrix is simultaneously subjected to remote uniform stresses and a point moment located at an arbitrary position in the matrix. Our analysis indicates that the uniform and hydrostatic stress field inside the inclusion as well as the constant hoop stress along the entire inclusion–matrix interface (on the matrix side) are independent of the action of the point moment. In contrast, the non-elliptical shape of the harmonic inclusion depends on both the remote uniform stresses and the point moment.     

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.     

含有非圆形双孔的无限平板中应力的解析解研究

无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.      

On the stress concentration around a hole in a half-plane subject to moving contact loads

We study the stress concentration due to a pore in an elastic half-plane, subject to moving contact loading, in the entire range of possible geometrical parameters (contact area/hole diameter, hole depth/hole diameter). Since the number of cases is very large to study with FEM even with modern machines, the use of a recent simple approximate formula due to Greenwood based on the stress field in the absence of the hole is first attempted, and compared with a full FEM analysis in sample cases. To further distillate the effects of the hole distance from the free surface and of the contact area size, the limiting cases are studied of: (i) concentrated load perpendicular to the surface and aligned with the hole centre; (ii) constant unit pressure on the top surface of the half-plane and (iii) hydrostatic load. A full investigation is then conduced for the case of Hertzian load on the surface, and it is seen that the tensile stress concentration is significantly reduced with respect to that of the concentrated load, when the contact area size is of the same order of the hole radius. Results obtained with the approximate Greenwood formula are generally accurate however only if the hole distance from the surface is greater than two times the hole radius.     

基于表面弹性理论的开裂椭圆孔断裂力学研究

研究纳米尺度时开裂椭圆孔的III型断裂性能。基于表面弹性理论和保角映射技术，利用复势函数理论获得了缺陷(裂纹和椭圆孔)周围应力场和裂纹尖端应力强度因子的闭合解答。所得结果具有一般性，许多已有和新的解答可由本文退化的特殊情形得到。利用解析结果讨论了缺陷的绝对尺寸、椭圆孔的形状比以及裂纹的相对尺寸对应力强度因子的影响。结果表明：考虑表面效应且缺陷尺寸在纳米尺度时，应力强度因子具有显著的尺寸依赖效应；应力强度因子随椭圆孔形状比的变化规律受缺陷表面常数的影响；缺陷表面效应的影响取决于椭圆孔的形状比，非常大的形状比屏蔽了表面效应的影响；裂纹相对尺寸非常小时表面效应影响较弱，裂纹相对尺寸较大时表面效应较为明显。     

颗粒增强复合材料的椭圆形刚性夹杂呈双周期分布模型的反平面问题研究

在遵循复合材料中各夹杂相互影响的重要条件下，构造呈双周期分布且相互影响的椭圆形刚性夹杂模型的复应力函数，采用坐标变换和复变函数的依次保角映射方法，达到满足各个夹杂的边界条件，利用围线积分将求解方程化为线性代数方程，推导出了在无穷远双向均匀剪切，椭圆形刚性夹杂呈双周期分布的界面应力解析表达式，最后的算例分析给出了夹杂的形状对界面应力最大值(应力集中系数)的影响规律，并描绘出了曲线。     

A plane theory of inextensible transversely isotropic elastic composites

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic material, with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and traction conditions lead to standard and nonstandard boundary value problems of potential theory. Examples for a rectangular plane, half-plane and infinite plate with elliptic hole, are presented in illustration.     

Shape optimization of two equal holes in an infinite elastic plate

Abstract

The optimal design of the stress state in elastic plate structures with openings is a problem of great significance in engineering practice. Achieving proper shape of hole can reduce stress concentration around the boundaries remarkably. The optimal shape of a single hole in an infinite plate under uniform stresses has been obtained by complex variable method based on different optimal criteria. The complex variable method is particularly suitable for the hole shape optimization in infinite plate, in which the continuous hole boundary can be represented by the mapping function. It can also be used to solve the shape optimization problems of two or more holes. However, because of the difficulty of finding the mapping function for multi connected domain, the holes are mapped onto slits or separately mapped onto a circle. In this article, the two symmetrical and identical holes are mapped onto an annulus simultaneously by the newly found mapping function, which has a general form. The maximum tangential stress around the boundaries is minimized to achieve the optimal hole shape. And the coefficients of mapping function which describe the boundary are calculated by differential-evolution algorithm.     

Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

求弹性半平面问题基本解的一个新方法

本文所提到的弹性半平面问题的基本解是一个满足特殊条件的弹性半平面的应力位移解答。这些条件为:(1)半平面内一点处作用有集中力X,Y或集中力偶M;(2)半平面边界为自由或固定边。利用平面弹性的复变函数方法,文中把弹性半平面基本解的问题归结为下列问题,使一个特定解析函数和另一个解析函数的共轭值在半平面边界上相等。对上述转化后的问题,只要利用复变函数的性质,不难从基本解的第一部分推导出基本解的第二部分。其中,基本解的第一部分是弹性全平面的本基解。从而,半平面问题基本解可以方便地得到。此外,文中还首次给出了:(1)集中力偶作用于半平面内一点时的基本解;(2)当半平面边界固定情况下的基本解。     

Geometrical shape of in-plane inclusion characterized by polynomial internal stress field under uniform eigenstrains

The internal stress field of an inhomogeneous or homogeneous inclusion in an infinite elastic plane under uniform stress-free eigenstrains is studied. The study is restricted to the inclusion shapes defined by the polynomial mapping functions mapping the exterior of the inclusion onto the exterior of a unit circle. The inclusion shapes, giving a polynomial internal stress field, are determined for three types of inclusions, i.e., an inhomogeneous inclusion with an elastic modulus different from the surrounding matrix, an inhomogeneous inclusion with the same shear modulus but a different Poisson’s ratio from the surrounding matrix, and a homogeneous inclusion with the same elastic modulus as the surrounding matrix. Examples are presented, and several specific conclusions are achieved for the relation between the degree of the polynomial internal stress field and the degree of the mapping function defining the inclusion shape.     

BOUNDARY VALUE PROBLEMS OF TWO-DIMENSIONAL ANISOTROPIC BODY WITH A PARABOLIC BOUNDARY

Based upon Stroh formalism we derive a novel and convenient scheme for determiningthe elastic fields of a two-dimensional anisotropic body with a parabolic boundary subject to two kindsof boundary conditions, which are free surface and rigid surface, respectively. The correspondingGreen's functions are found by using the conformal mapping method. When the parabolic curve de-generates into a half-infinite crack or rigid inclusion, the singular stress fields near the tip of defectsare obtained. In particular, those Green's functions for a concentrated moment M_0 applied at a pointon the parabolic curve are also studied. It is easily found that arbitrary parabolic boundary value prob-lems can be solved by using these Green's functions and associate integrals.     

带裂纹的椭圆孔口问题的应力分析

断裂现象与材料和结构中的孔洞、缺口或裂纹等缺陷密切相关,这是因为缺陷附近的应力集中明显.该文利用复变方法,通过保角映射研究了带裂纹的椭圆孔洞的平面弹性问题,给出了应力强度因子的解析解.并由此计算了两互相垂直的裂纹问题.     

A unified approach to problems of stress concentration near V-shaped notches with sharp and rounded tip

The paper proposes a unified approach to problems of stress concentration near notches with sharp and rounded tip based on the method of singular integral equations. A solution for an elastic region having a V-shaped notch with rounded tip of large curvature is first found. Then, the stress intensity factor at the tip of a sharp-tipped notch is calculated by passing to the limit. Numerical results are obtained for a slit and a square hole in an elastic plane and an edge notch in a half-plane __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 2, pp. 70–87, February 2007. For the centenary of the birth of G. N. Savin.     

Complex variable function method for the scattering of plane waves by an arbitrary hole in a porous medium

Based on the complex variable function method, a new approach for solving the scattering of plane elastic waves by a hole with an arbitrary configuration embedded in an infinite poroelastic medium is developed in the paper. The poroelastic medium is described by Biot's theory. By introducing three potentials, the governing equations for Biot's theory are reduced to three Helmholtz equations for the three potentials. The series solutions of the Helmholtz equations are obtained by the wave function expansion method. Through the conformal mapping method, the arbitrary hole in the physical plane is mapped into a unit circle in the image plane. Integration of the boundary conditions along the unit circle in the image plane yields the algebraic equations for the coefficients of the series solutions. Numerical solution of the resulting algebraic equations yields the displacements, the stresses and the pore pressure for the porous medium. In order to demonstrate the proposed approach, some numerical results are given in the paper.