Strain fields caused by doughnut-like and tubular inclusions with uniform eigenstrains

Inclusion problems for elongated toroidal inclusions are solved to discuss the strain fields caused by eigenstrained doughnut-like and tubular inclusions. For an infinitely long tubular inclusion, uniform eigenstrains do not generate elastic strains in the matrix region inside the tube. Effects of tube length on the elastic strains are shown for the purely dilatational eigenstrain ^*. For the tubular inclusion with the length 2b, the elastic strain at the center of the matrix region inside the tube becomes the maximum as much as ≈(a/b)^* when the tubular inclusion has the inner diameter ≈2(b−a) and the outer diameter ≈2(b+a).     

Morphological evolution during diffusive healing of internal cracks within grains of α-iron

Micron-sized internal cracks were introduced into rounded bars of pure iron by low cycle fatigue, and the cracks had irregular penny-shaped morphology with the critical diameter of about 30μm and the thickness of 0.5–1.5μm. The initiation and propagation of the cracks were investigated quantitatively as well as their location and geometry. After vacuum annealing of the samples fatigued, the morphology in a two-dimensional longitudinal section of cracks within grains had evolved from initially elliptical one into arrays of spherical voids controlled by surface diffusion. Furthermore, a typical morphology for a broken crack with a center spherical void surrounded by outer doughnut-like cavities was observed along a perpendicular section of the specimen. Subsequently the spherical voids shrink and diminish gradually dominated by bulk diffusion. A physical model to heal an internal microcrack was proposed, in particular for the various healing stages controlled by the related dominant diffusion mechanism and their dependencies upon the morphology and geometry of an original micro-crack in materials. The project supported by the National Outstanding Young Investigator Grant of China (59925104) and the National Natural Science Foundation of China (59889101)     

ANTI-PLANE ANALYSIS FOR ELLIPTICAL INCLUSION IN MAGNETOELECTROELASTIC MATERIALS

This paper considers the multi-field coupling in magneroelectroelastic composite materials consisting of the inclusion and the matrix are magnetoelectroelastic materials. The mechanical,electric and magnetic fields around an elliptical cylinder inclusion are formulated by complex potentials. Inside the inclusion,the strain,electric and magnetic fields are found to be uniform and vary with the shape of the ellipse. When the inclusion is reduced to a crack,along the interface,the strain,electric field strength and magnetic field strength equal the corresponding remote ones,which can be used as the boundary condition. Special cases,such as a rigid and permeable inclusion,a soft and impermeable inclusion,a line inclusion and a crack problem are discussed in detail.     

Special properties of Eshelby tensor for a regular polygonal inclusion

When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor: Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.The project supported by the National Natural Science Foundation of China (10172003 and 10372003) The English text was polished by Keren Wang.     

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.     

Green's function of the displacement boundary value problem for a heat source in an infinite plane with an arbitrary shaped rigid inclusion

Summary Green's functions of the displacement boundary value problem are derived within two-dimensional thermoelasticity for a heat source in an infinite plane with an arbitrary shaped rigid inclusion. The following two cases are considered: either rigid-body displacements and rigid-body rotations of the inclusion are allowed or no rigid-body displacements and no rigid-body rotations of the inclusion are possible. To solve these problems, fundamental solutions are developed for a point heat source, for rigid-body rotations of the inclusion, and for concentrated loads acting on the inclusion. Complex stress functions, temperature function, a rational mapping function and the thermal dislocation method are used for the analysis. In analytical examples, distributions of stresses are developed for an infinite plane with a rectangular rigid inclusion. Received 5 August 1998; accepted for publication 1 December 1998     

Effect of interface stress on the fracture behavior of a nanoscale linear inclusion along the interface of bimaterials

This paper investigates the influence of residual interface tension on the fracture behavior of a nanoscale linear interface inclusion in a bimaterial matrix. Solutions to the inclusion opening displacement and the energy release rate are obtained. The results show that the interface effect on the inclusion deformation and inclusion tip field are prominent at nanoscale. Especially, the residual interface stress has a dramatic influence on the energy release rate. It is also found that the importance of the interface effect depends on the size of the inclusion, the shear modulus ratios of the bimaterial. The inclusion opening displacement and the energy release rate can be reduced considerably by decreasing the inclusion length at nanoscale.     

The Arithmetic Mean Theorem of Eshelby Tensor for a Rotational Symmetrical Inclusion

In 1997, H. Nozaki and M. Taya found numerically that for any regular polygonal inclusion except for a square, both the Eshelby tensor at the center and the average Eshelby tensor over the inclusion domain are equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Then in 2001, these remarkable properties were mathematically justified by Kawashita and Nozaki. In this paper, a more radical property is presented for a rotational symmetrical inclusion: For any N-fold (N is an integer greater than 2 and unequal to 4) rotational symmetrical inclusion, the arithmetic mean of the Eshelby tensors at N rotational symmetrical points in the inclusion is the same as the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. It follows that the Eshelby tensor at the center and the average Eshelby tensor over the rotational symmetrical inclusion domain are identical to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion as well. This paper shows that although the Eshelby property does not hold for non-ellipsoidal inclusions, the Eshelby tensor for a rotational symmetrical inclusion satisfies the arithmetic mean property. Mathematics Subject Classifications (2000) 73C02.     

夹杂对两相材料界面裂纹的影响

根据界面上应力和位移的连续条件,得到了单向拉伸状态下,含有椭圆夹杂的无限大双材料组合板的复势解。进一步通过求解Hilbert问题,得到了含有夹杂和半无限界面裂纹的无限大板的应力场,并由此给出了裂尖的应力强度因子K。计算了夹杂的形状、夹杂的位置、夹杂的材料选取以及上、下半平面材料与夹杂材料的不同组合对裂尖应力强度的影响。计算结果表明夹杂到裂尖的距离和夹杂材料的性质对K影响较大,对于不同材料组合,该影响有较大差异。夹杂距裂尖较近时,会对K产生明显屏蔽作用,随着夹杂远离裂尖,对K的影响也逐渐减小。另外,软夹杂对K有屏蔽作用,硬夹杂对K有反屏蔽作用,而夹杂形状对K几乎没有影响。     

Flexural vibrations of a thin circular elastic inclusion in an unbounded body under the action of a plane harmonic wave

The interaction of a plane harmonic longitudinal wave with a thin circular elastic inclusion is considered. The wave front is assumed to be parallel to the inclusion plane. Since the inclusion is thin, the matrix-inclusion interface conditions (perfect bonding) are formulated on the mid-plane of the inclusion. The bending displacements of the inclusion are determined from the bending equation for a thin plate. The problem is solved using discontinuous Lamé solutions for harmonic vibrations. Therefore, the problem can be reduced to the Fredholm equation of the second kind for a function associated with the discontinuity of normal stresses on the inclusion. The equation obtained is solved by the method of mechanical quadratures using Gaussian quadrature formulas. Approximate formulas for the stress intensity factors are derived. Results from a numerical analysis of the dependence of the SIFs on the dimensionless wave number and the stiffness of the inclusion are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 16–21, May 2008.     

Exact solution for elliptical inclusion in magnetoelectroelastic materials

This paper considers the magneto-electro-mechanical coupling between an inclusion and matrix, which are both of magnetoelectroelastic materials. The general cases including the mode I, mode II and mode III are studied. Analytical solutions for an elliptical cylinder inclusion inside an infinite magnetoelectroelastic medium under combined mechanical–electrical–magnetic loadings are formulated via the Stroh formalism. Crack problem is also investigated and the stress, electric and magnetic fields in the vicinity of the crack tip are determined by a complex vector of intensity factors. Various special cases, including an impermeable inclusion, a permeable crack, a rigid and permeable inclusion, a rigid and permeable line inclusion, a permeable cavity, and an impermeable cavity are discussed.     

Stress concentration near a thin elastic inclusion under interaction with harmonic waves in the case of smooth contact

We solve the problem on the interaction of plane harmonic waves with a thin elastic plate-shaped inclusion. The ambient medium is assumed to be in plane strain. The smooth contact conditions are satisfied on both sides of the inclusion. The bending displacements of the inclusion are determined from the corresponding differential equation. In the statement of boundary conditions for this equation, one should take into account the transverse forces and bending moments applied to the lateral edges of the inclusion, while the boundary conditions are posed on the midplane of the inclusion. Using the discontinuous solution method, we reduce the problem to a system of two singular integral equations, which are solved numerically by the mechanical quadrature method. We obtain approximate formulas for the stress intensity coefficients near the ends of the inclusion and for the transverse forces and moments applied to the inclusion.     

Role of inclusion stiffness and interfacial strength on dynamic matrix crack growth: An experimental study

Experimental simulations of dynamic crack growth past inclusions of two different elastic moduli, stiff (glass) and compliant (polyurethane) relative to the matrix (epoxy), are carried out in a 2D setting. Full-field surface deformations are mapped in the crack–inclusion vicinity optically. The crack growth behavior as a function of inclusion–matrix interfacial strength and the inclusion location relative to the crack is studied under stress-wave loading conditions. An ultra high-speed rotating mirror-type digital camera is used to record random speckle patterns in the crack–inclusion vicinity to quantify in-plane displacement fields. The crack-tip deformation histories from the time of impact until complete fracture are mapped and fracture parameters are extracted. The crack front is arrested by the symmetrically located compliant inclusion for about half the duration needed for complete fracture event. The dynamically propagating crack is attracted and trapped by the weakly bonded inclusion interface for both stiff and compliant symmetrically located inclusion cases, whereas it is deflected away by the strongly bonded stiff inclusion and attracted by strongly bonded compliant inclusion when located eccentrically. The crack is arrested by a strongly bonded compliant inclusion for a significant fraction of the total dynamic event and is longer than the one for the weakly bonded counterpart. The compliant inclusion cases show higher fracture toughness than the stiff inclusion cases. Measured crack-tip mode-mixities correlate well with the observed crack attraction and repulsion mechanisms. Macroscopic examination of fracture surfaces reveals much higher surface roughness and ruggedness after crack–inclusion interaction for compliant inclusion than the stiff one. Implications of these observations on the dynamic fracture behavior of micron size A-glass and polyamide (PA6) particle filled epoxy is demonstrated. Filled-epoxy with 3% V_f of PA6 filler is shown to produce the same dynamic fracture toughness enhancement as the one due to 10% V_f glass.     

The Stress State of a Piezoceramic Medium with an Elliptic Inclusion under Pure Shear and Pure Bending

Explicit analytical solutions to electroelastic problems for an infinite transversely isotropic medium with a tunnel elliptic inclusion are constructed. At a sufficient distance from the inclusion, the medium is subjected to pure shear or pure bending. It is assumed that the medium and inclusion are dissimilar piezoceramic materials whose axes of symmetry coincide with each other and with the minor axis of the ellipse. The stresses and the projections of the electromagnetic induction vector acting in the medium beyond the inclusion are determined for each case of loading at infinity     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

压电复合材料中的Eshelby夹杂问题

通过采用解析延拓和共形映射技术，获得了压电复合材料中有关Eshelby夹杂几个典型问题的精确弹性解答，即横观各向同性压电介质中任意形状的Eshelby夹杂与圆柱异相夹杂间相互作用；一般各向异性压电介质中任意形状的Eshelby夹杂与双压电材料所形成界面的相互作用．成功求解这些问题的关健在于构造一个辅助函数．与Ru所采用的方法不同，所引入的辅助函数在无穷远点不存在极点，从而使得所展开的分析更加自然合理．分析结果清楚地揭示出Eshelby夹杂的存在对压电复合材料机电耦合响应将产生不容被忽视的影响．很典型的一个例于是当一个Eshelby椭圆夹杂与圆柱异相夹杂相互作用时，每个夹杂体内部的应力场和电场都将是不均匀的；另一个例于是位于界面附近的Eshelby夹杂有可能是界面发生损伤的一个重要原因．     

直角平面内弹性圆夹杂对入射平面SH波的散射

利用复变函数法、多极坐标移动技术及傅立叶级数展开求解二维直角平面内圆形弹性夹杂对稳态入射平面SH波的散射问题。首先写出直角平面内不含夹杂时的入射波场和反射波场;其次建立直角平面内含夹杂时夹杂外的散射波解和夹杂内的驻波解,并利用叠加原理写出问题的总波场,借助夹杂边界处应力和位移的连续条件建立求解散射波解和驻波解中未知系数的无穷代数方程组并求解,通过算例具体讨论了直角平面水平边界点的位移幅度比和夹杂边界处径向应力集中系数随不同无量纲波数、入射角及圆孔位置的变化情况,结果表明了算法的有效实用性。     

压电材料反平面应变状态的任意形状夹杂问题

应用复函数的Ｆａｂｅｒ级数展开方法，分析了含任意形状夹杂的压电材料反平面应变问题，给出了问题的复势函数解。利用这个解，具体讨论了椭圆形夹杂及其极限（几何方面与物理方面）问题。并给出了三角形、正方形夹杂的近似结果。其特例结果与早期工作一致     

Stress field of a coated arbitrary shape inclusion

This paper presents an effective method for the plane problem of a coated inclusion of arbitrary shape embedded in an isotropic matrix subjected to uniform stresses at infinity. Based on the complex variable method combined with the expansion of Faber series and Laurent series, the complex potentials in the matrix, the coating and the arbitrary shape inclusion are given in the form of series with unknown coefficients. The stress and displacement continuous conditions on the interfaces are then used to produce a set of linear equations containing all the coefficients. Through solving these linear equations, the complex potentials are finally obtained in the three phases. Additionally, numerical results are presented and graphically shown to investigate the influence of inclusion geometry and coating on the stress distribution along the interfaces for the cases of a coated elliptic, square and triangle inclusions, respectively. It is found that the coating has little effects on the interface stress for a hard inclusion, while it impacts greatly for a soft inclusion. Especially, it is also found that the stresses show the nature of intense fluctuations near the corner of the triangle inclusion, since the inclusion in this case is similar to a wedge.     

Timoshenko inclusions in elastic bodies crossing an external boundary at zero angle

The paper concerns an analysis of equilibrium problems for 2D elastic bodies with a thin Timoshenko inclusion crossing an external boundary at zero angle. The inclusion is assumed to be delaminated, thus forming a crack between the inclusion and the body. We consider elastic inclusions as well as rigid inclusions. To prevent a mutual penetration between the crack faces, inequality type boundary conditions are imposed at the crack faces.Theorems of existence and uniqueness are established. Passages to limits are investigated as a rigidity parameter of the elastic inclusion going to infinity.