An analytical solution for the elastic fields near spheroidal nano-inclusions

When the size of an inclusion shrinks to nanometers, interface energy plays an important role in the deformation around it. In the present paper, we consider the effect of interface energy on the elastic fields near a spheroidal nanoinclusion embedded in an elastic medium on the basis of surface elasticity theory. Using Boussinesq-Sadowsky potential function method, we obtain the deformation field near the inclusion subjected to a uniformly uniaxial loading at infinity. The results show that the elastic fields near the nano-inclusion depend strongly on the interface properties, the size and shape of inclusion. These new characteristics may be helpful to understand various relevant mechanical performances of nanosized inhomogeneities.     

SURFACE EFFECTS ON ELASTIC FIELDS AROUND SURFACE DEFECTS

<正>There are always severe stress concentrations around surface defects like grooves or bugles,which might induce the failure of solid materials and structures.In the present paper,we consider the elastic fields around nanosized bugles and grooves on solid surfaces.The influence of surface elasticity on the elastic deformation is addressed through a finite element method.It is found that when the size of defects shrinks to nanometer,the stress fields around such defects will be affected significantly by surface effects.     

Stress concentration around a nano-scale spherical cavity in elastic media: effect of surface stress

Influence of surface effect on stress concentration around a spherical cavity in a linearly isotropic elastic medium is studied on the basis of continuum surface elasticity. Following Goodier's work, a close form solution of the elastic field created by biaxial uniform load is presented. The stress concentration factors under different load combinations are obtained. It is concluded that consideration of surface effect leads to dependence of stress concentration factors on cavity size. Besides, numerical result indicates that stress concentration factors around the cavity are mainly affected by residual surface tension. The result is significant in the understanding of relevant mechanical phenomena in solids with nano-sized cavities.     

Analysis of stress concentration around a spheroidal cavity under asymmetric dynamic loading

The fracture and fatigue properties of porous materials are strongly influenced by stress concentrations around the pores. In addition, failure of structural components initiates at locations of high stress concentration which is often caused by holes, inclusions or other discontinuities. In view of this, the stress concentration around a spheroidal cavity embedded in an elastic medium is studied under dynamic loading conditions. While solutions abound for static loads, only limited solutions exist for dynamic loads. The stress field around a spheroidal cavity is determined by using a hybrid methodology that combines the finite element technique with a spherical wave function expansion method. The stress concentrations within the matrix are found to be dependent on the frequency of excitation, aspect ratio of the cavity and the Poisson’s ratio of the matrix. The study reveals that dynamic stress concentrations can reach much higher values than those encountered under static loading.     

Stresses in an elastic circular cylinder with a prolate spheroidal cavity under torsion

This paper contains an exact solution for the stresses arising from torsion of an elastic circular cylinder with a prolate spheroidal cavity. The solution, which is represented as a combination of a solution that is regular outside the cavity and a solution regular in the solid infinite circular cylinder, is deduced with the aid of a harmonic displacement potential. The two harmonic functions needed are given by simple expressions referred to cylindrical and prolate spheroidal coordinates. The boundary conditions on the surfaces of the cylinder and of the cavity are satisfied by using the relations between cylindrical and prolate spheroidal harmonics. Numerical results are presented for different shapes and sizes of the cavity, and the ensuing stress distribution in the neighborhood of the cavity is illustrated graphically.     

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.     

Analysis of the nonstationary spatial propagation of elastic waves from cavities subject to mechanical and thermal loads

A numerical analytic method is proposed to solve nonstationary coupled problems of thermoelasticity with regard to the finite velocity of thermal waves. The method is used to analyze the nonstationary spatial propagation of elastic waves from a cavity subjected on its surface to mechanical and thermal loads. The ray theory of propagation of wavefield discontinuities is used. To determine the time dependence of the field parameters behind the wavefront and to account for the relationship between the mechanical and thermal fields with prescribed accuracy, a numerical iterative procedure that employs the properties of characteristics is used. Plots are presented for the nonstationary stresses and temperature near a prolate spheroidal cavity subject to step mechanical loading and near an elliptical cylindrical cavity subject to thermal shock __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 79–88, August 2006.     

SH波作用下界面任意形状孔洞附近的动应力集中

采用Green函数和复变函数法求解了平面SH波在界面任意形状孔洞上的散射问题.首先,取含有任意形状凹陷的弹性半空间,在其水平表面上任意一点承受时间谐和的反平面线源荷载作用时的位移场作为Green函数.然后,按契合方式构造出界面任意形状孔洞对SH波的散射模型,利用所得Green函数按界面位移连续条件建立求解问题的定解积分方程组,求解界面孔附近的动应力集中系数.最后,给出了界面上椭圆孔和方孔边缘动应力集中系数的数值结果,并讨论了不同介质参数和孔洞形状对孔附近动应力集中系数的影响.     

Static equilibrium of an elastic orthotropic medium with an arbitrarily oriented triaxial ellipsoidal inclusion

The stress state of an elastic orthotropic medium with an arbitrarily oriented triaxial ellipsoidal inclusion is analyzed. A solution is obtained using the triple Fourier transform and the Fourier-transformed Green’s function for an infinite anisotropic medium. The high efficiency of the approach is demonstrated by solving the problem for a transversely isotropic material with a spheroidal cavity for which the exact solution is known. A numerical analysis is conducted to study the stress distribution over the surface of the inclusion with different orientations in the orthotropic space. It is revealed that in some cases the orientation of the inclusion has a strong effect on the stress concentration __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 55–61, April 2007.     

Damage in a viscoplastic material part I: Cavity growth

The exact velocity, stress and strain rate fields around a spheroidal cavity in an infinite linear viscoplastic compressible matrix are derived analytically by the ‘three function approach’. The perturbation of the velocity field due to the cavity is the superposition of three independent modes, inducing homothetic growth, pure distortion and both volume and shape changes, respectively. This solution is then used to investigate the velocity field around a spheroidal cavity in a nonlinear viscous compressible material by means of a variational principle. The behaviour of such damaged linear and nonlinear materials will be discussed in a forthcoming companion paper.The importance of the reference strain, while studying void growth in a compressible material, is emphasized. If the axial strain is chosen as a reference, void growth is found to be enhanced at low triaxiality ratios, but lowered at high triaxiality ratios in a compressible matrix relative to an incompressible one. Finally, the transition from a power law to a linear law with intercept, at increasing strain rates, is shown to reduce damage growth rate.     

SH波绕界面孔的散射

用波函数展开方法研究了SH波绕界面孔的散射问题。由入射、反射和透射波组成的自由波场与孔的散射场叠加成总波场。按照一定方式将两个半平面散射波场延拓于全平面，通过Hankel-Fourier展开方法求得了任意形状孔散射场的级数解。以椭圆形孔为例计算了孔边缘的动应力集中系数。     

Multiple scattering of electro-elastic waves from a buried cavity in a functionally graded piezoelectric material layer

This paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and the dynamic stress around a buried cavity in a functionally graded piezoelectric material layer bonded to a homogeneous piezoelectric material. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions around the cavity. The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure. According to the analytical expression of this problem, the numerical solutions of the dynamic stress concentration factor around the cavity are presented. The effects of the piezoelectric property, the position of the cavity in the layer, the incident wave number and the material properties on the dynamic stress around the cavity are analyzed. Analyses show that the piezoelectric property has great effect on the dynamic stress in the region of higher frequencies, and the effect increases with the decrease of the thickness of FGPM layer. If the material properties of the homogeneous piezoelectric material are greater than those at the surface of the structure, the dynamic stress resulting from the piezoelectric property is greater. The effect material properties at the two boundaries of FGPM layer on the distribution of dynamic stress around the cavity is also examined.     

Antiplane study on confocally elliptical inhomogeneity problem using an alternating technique

This paper presents a novel efficient procedure to analyze the two-phase confocally elliptical inclusion embedded in an unbounded matrix under antiplane loadings. The antiplane loadings considered in this paper include a point force and a screw dislocation or a far-field antiplane shear. The analytical continuation method together with an alternating technique is used to derive the general forms of the elastic fields in terms of the corresponding problem subjected to the same loadings in a homogeneous body. This approach could lead to some interesting simplifications in solution procedures, and the derived analytical solution for singularity problems could be employed as a Green's function to investigate matrix cracking in the corresponding crack problems. Several specific solutions are provided in closed form, which are verified by comparison with existing ones. Numerical results are provided to show the effect of the material mismatch, the aspect ratio, and the loading condition on the elastic field due to the presence of inhomogeneities.     

The Thermoelectroelastic State of a Transversally Isotropic Piezoceramic Body with a Spheroidal Cavity in a Uniform Heat Flow

A static thermoelectroelastic problem for an infinite transversally isotropic body containing a spheroidal cavity is explicitly solved. The symmetry axis of the spheroid coincides with the anisotropy axis of the body. It is assumed that at a rather large distance from the cavity the body is in a uniform heat flow directed along the anisotropy axis. Formulas are derived for the stress components and the projections of the electric displacement vector near the cavity, which depend on the heat-flow value, cavity geometry, and the thermoelectroelastic properties of the material. The solution of the problem for a body with a disk-like crack is obtained as a partial case from the solution of the problem for a piezoceramic body with a spheroidal cavity. The stress intensity factors for the force and electric fields are determined near the crack     

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。     

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.     

A magnetoelectroelastic medium with an elliptical cavity under combined mechanical–electric–magnetic loading

The solution for an elliptical cavity in an infinite two-dimensional magnetoelectroelastic medium subject to remotely uniformly applied combined mechanical–electric–magnetic loadings is obtained by using the Stroh formalism and the exact boundary conditions along the surface of the cavity. By letting the minor-axis of the cavity to zero the solution for a crack is deduced. A self-consistent method is proposed to calculate the real crack opening under the combined mechanical–electric–magnetic loadings. The method requires that the crack opening is the minor-axis of the elliptical opening profile. Beside the real crack solution, four different extreme models, i.e., the impermeable crack, permeable crack, electrically impermeable and magnetically permeable crack and electrically permeable and magnetically impermeable crack, are discussed. An expression of the strain energy density factor is derived. Numerical results of the strain energy density at the crack tip are given for a BaTiO₃–CoFe₂O₄ composite with the piezoelectric BaTiO₃ material being the inclusion and the magnetostrictive CoFe₂O₄ material being the matrix. The effects of the proportion of the two phases, permeability of the crack to electric and magnetic fields, the electric and magnetic loadings on the strain energy density factor are discussed.     

Axisymmetric indentation of curved elastic membranes by a convex rigid indenter

Motivated by applications to seed germination, we consider the transverse deflection that results from the axisymmetric indentation of an elastic membrane by a rigid body. The elastic membrane is fixed around its boundary, with or without an initial pre-stretch, and may be initially curved prior to indentation. General indenter shapes are considered, and the load-indentation curves that result for a range of spheroidal tips are obtained for both flat and curved membranes. Wrinkling may occur when the membrane is initially curved, and a relaxed strain-energy function is used to calculate the deformed profile in this case. Applications to experiments designed to measure the mechanical properties of seed endosperms are discussed.     

THE ZONAL DISINTEGRATION MECHANISM OF SURROUNDING ROCK AROUND DEEP SPHERICAL TUNNELS UNDER HYDROSTATIC PRESSURE CONDITION: A NON-EUCLIDEAN CONTINUUM DAMAGE MODEL

A new non-Euclidean continuum damage model is proposed to investigate the zonal disintegration phenomenon of the surrounding rocks around deep spherical tunnels under hydrostatic pressure condition as well as the total elastic stress field distributions.The elastic stress fields of the surrounding rocks around deep spherical tunnels under hydrostatic pressure condition are obtained.If the elastic stresses of the surrounding rocks satisfy the strength criterion of the deep rock masses,the number,size and location of fractured and nonfractured zones are determined.The efect of physico-mechanical parameters of the surrounding rocks on the zonal disintegration phenomenon is studied and numerical computation is carried out.It is found from numerical results that the number,size and location of fractured and non-fractured zones are sensitive to the physico-mechanical parameters of the surrounding rocks.     

Piezothermoelastic analysis of a piezoelectric material with an elliptic cavity under uniform heat flow

Summary The problem of a two-dimensional piezoelectric material with an elliptic cavity under a uniform heat flow is discussed, based on the modified Stroh formalism for the piezothermoelastic problem. The exact electric boundary conditions at the rim of the hole are introduced in the analysis. Expressions for the elastic and electric variables induced within and outside the cavity are derived in closed form. Hoop stress around the hole and electric fields in the hole are obtained. The limit situation when the hole is reduced to a slit crack is discussed, and the intensity factors for the problem are obtained. Received 14 April 1998; accepted for publication 25 June 1998