Uniformity of stresses inside an elliptic inclusion in finite plane elastostatics

We consider an elastic inclusion embedded in a particular class of harmonic materials subjected to uniform remote stress. Using complex variable techniques, we show that if the Piola stress within the inclusion is uniform, the inclusion is necessarily an ellipse except in the special case when the (uniform) remote stress assumes a particular form. In addition, we obtain the complete solution for an elliptic inclusion with uniform interior stress for any uniform remote stress distribution.     

几种金属材料断裂条件的试验研究

考察了几种金属材料在常规破坏试验与拉-扭双轴破坏试验中，随受力条件不同，材料不同形式断裂破坏变化规律和相应的断裂条件；利用几种韧性材料复合型断裂试验结果，分析了随应力三维度变化，材料中孔洞成核形状与聚合方式的变化规律，分析了不同断裂形式时启裂点、启裂方向变化规律及主要影响因素。研究表明，从材料断裂物理机制来看，裂纹体与无裂纹体断裂破坏实质是相同的，随材料塑性因素与应力三维度变化的影响。材料内孔洞成核形状由椭球形逐渐变化为细长形，材料断裂由正拉断转向剪断，裂纹体与无裂体材料的断裂形式、断裂危险点、危险点上断裂方向等宏观量也有着相同的变化规律；区分不同物理机制建立断裂条件，可能适合解决金属材料不同形式的断裂力学问题。     

纵向剪切三相共焦点椭圆模型的精确解及其应用

推广夹杂问题的三相同心圆模型,提出了三相共焦点椭圆模型,从而计及了夹杂形状变化的重要因素。利用保角变换结合罗朗级数展开技术获得了纵向剪切载荷下的封闭形式解。本模型和解答对夹杂、界面层附近的细观应力场分析、对复合材料和含缺陷材料的有效弹性模量预测有重要实用价值,获得了比经典公式更精确的结果。对特殊模型,可以得到许多有意义的问题（如二相模型）的解答。     

A MIXED MODE FRACTURE CRITERION BASED ON THE MAXIMUM TANGENTIAL STRESS IN BRITTLE INCLUSION

A closed-form solution for predicting the tangential stress of an inclusion located in mixed mode Ⅰ and Ⅱ crack tip field was developed based on the Eshelby equivalent inclusion theory. Then a mixed mode fracture criterion, including the fracture direction and the critical load, was established based on the maximum tangential stress in the inclusion for brittle inclusioninduced fracture materials. The proposed fracture criterion is a function of the inclusion fracture stress, its size and volume fraction, as well as the elastic constants of the inclusion and the matrix material. The present criterion will reduce to the conventional one as the inclusion having the same elastic behavior as the matrix material. The proposed solutions are in good agreement with detailed finite element analysis and measurement.     

The null field approach to two-dimensional elastostatics

The vector basis functions, necessary for solving two-dimensional inclusion problems in an elastic solid under time independent conditions by means of the null field approach (T-matrix method), are obtained as a zero frequency limit of the corresponding basis functions commonly used in elastodynamics. The expansion of the fields appearing in the surface integral representation of the static displacement can thus be achieved, leading to the T-matrix equations of 2d-elastostatics. We specialize the problem to the simple boundary condition case of a single cavity and develop the analytical expressions as much as possible before numerical implementation. A numerical test for the ellipse and some examples for the superellipse, with applied static pressure or shear stress at infinity, are given.     

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     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

Magnetoelastic fracture of soft ferromagnetic materials

Effects of magnetic field on fracture toughness of soft ferromagnetic materials were studied using experimental techniques and theoretical models. The manganese–zinc ferrite with a single-edge-notch-beam (SENB) were chosen to be the specimen and the Vickers’ indentation specimen subjected to a magnetic field were chosen to be the specimens. Results indicate that there is no significant variations of the measured fracture toughness of the manganese–zinc ferrite ceramic in the presence of the magnetic field. The theoretical model involves an anti-plane shear crack with finite length in an infinite magnetostrictive body where an in-plane magnetic field prevails at infinity. Magnetoelasticity is used. The crack-tip elastic field is different from that of the classical mode III fracture problem. Furthermore, the magnetoelastic fracture of the soft ferromagnetic material was studied by solving the stress field for a soft ferromagnetic plane with a center-through elliptical crack. The stress field at the tip of a slender elliptical crack is obtained for which only external magnetic field normal to the major axis of the ellipse is applied at infinity. The results indicate that the near field stresses are governed by the magnetostriction and permeability of the soft ferromagnetic material. The induction magnetostrictive modulus is a key parameter for finding whether magnetostriction or magnetic-force-induced deformation is dominant near the front an elliptically-shaped crack. The influence of the magnetic field on the apparent toughness of a soft ferromagnetic material with a crack-like flaw can be regarded approximately in two ways: one possesses a large induction magnetostrictive modulus and the other has a small modulus. Finally, a small-scale magnetic-yielding model was developed on the basis of linear magnetization to interpret the experimental results related to the fracture of the manganese–zinc ferrite ceramics under magnetic field. Studied also is the fracture test of the soft ferromagnetic steel with compact tension specimens published in the existing literature.     

Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion

We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lamé equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion. Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions.     

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。     

Effect of the parameters of the biaxial field of rock pressure on the shape of the fracture zone formed by an explosion of a cylindrical charge in a brittle medium

The effect of the biaxial field of external rock pressure on the deformation of the fracture zone formed by radial cracks in an elastic-brittle medium is studied. We consider cylindrical charges that are rather thin compared to the diameter of the explosive borehole. This allows one to exclude the grinding zone from consideration. At the initial moment of time, the system of emerging cracks originating at the boundary of the circular hole is assumed to be symmetric. To solve the problem, we use singular integral equations and the fracture criterion σ_ϑ. The propagation trajectories of the system cracks are calculated in the quasistatic approximation in a step-by-step manner in relation to the parameters of the external compressive stress field. Two ideal variants of loading of the crack system are analyzed. In the first variant, gaseous detonation products penetrate into cracks, and the pressure in the explosion cavity and the cracks instantaneously equalizes. In the second variant, gases do not penetrate into the cracks of the system. The fracture zone is shown to become an ellipse whose long axis is oriented in the direction of the largest compressive stress in magnitude acting at infinity. The effect of the variants of loading of the crack system on the shape and dimensions of the deformed fracture zone is evaluated. Mining Institute, Siberian Division, Russian Academy of Sciences, Novosibirsk 630091. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 129–137, January–February, 1998.     

Crack buckling in soft gels under compression

Recent interest in designing soft gels with high fracture toughness has called for simple and robust methods to test fracture behavior. The conventional method of applying tension to a gel sample suffers from a difficulty of sample gripping. In this paper, we study a possible fracture mechanism of soft gels under uni-axial compression. We show that the surfaces of a pre-existing crack, oriented parallel to the loading axis, can buckle at a critical compressive stress. This buckling instability can open the crack surfaces and create highly concentrated stress fields near the crack tip, which can lead to crack growth. We show that the onset of crack buckling can be deduced by a dimensional argument com- bined with an analysis to determine the critical compression needed to induce surface instabilities of an elastic half space. The critical compression for buckling was verified for a neo-Hookean material model using finite element simulations.     

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.     

Formation of a contact zone during compression of a plate with an elliptic hole

The problem of compression of a thin plate with an elliptic hole is considered. It is assumed that increasing the distant compressive load can lead to contact of opposite regions of the boundaries of the ellipse. The problem is solved within the framework of a modified Leonov-Panasyuk-Dugdale model and an elastoplastic analog of the Griffith problem for an ellipse using the Goodier and Kanninen model. The critical fracture parameters providing an equilibrium configuration of the system are determined from a sufficient strength criterion representing a system of two equations, one of which specifies the absence of partial overlapping of the upper and lower surfaces of the contact zone, and the other is a deformation criterion of critical opening of the ellipse. The compression-induced deformation of the boundaries of ellipses with various curvature radii at the top is shown by the example of annealed copper having nanostructure.     

Interaction of plane elastic nonstationary waves with an elastic inclusion under complete adhesion

We solve the problem on the interaction of plane elastic nonstationary waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) which in under conditions of plane strain. It is assumed that the condition of perfect adhesion between the inclusion and the matrix is satisfied. Because of the small thickness of the inclusion we assume that the bending and shear displacements at any inclusion point coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself are found from the corresponding equations of the theory of plates. The statement of the boundary conditions for these equations takes into account the forces and moments acting on the inclusion edges from the matrix. The solution method is based on representing the displacements in the space of Laplace transforms as a discontinuous solution of the Lame’ equations for the plane strain with subsequent determining the transforms of the unknown jumps from integral equations. The passage to the original functions is performed numerically by methods based on replacement of the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors for the inclusion. These formulas are used to study the time dependence of the stress intensity factors and the influence of the inclusion rigidity on their values. We also study the possibility of treating inclusions of high rigidity as absolutely rigid inclusions.     

On phase transitions in a domain of material inhomogeneity. I. Phase transitions of an inclusions in a homogeneous external field

We pose and study the problem on an inclusion experiencing a phase transition in a homogeneous external stress field transferred by a matrix. The matrix is formed by a linear-elastic material. The inclusion material admits phase transitions under strain, and the passage from one phase state into another, as well as two-phase states, is determined by the energy preference considerations and the possible existence of two-phase states. For the simplest problem we consider the problem of phase transitions in a cylindrical inclusion under homogeneous plane strain conditions. In the space of strains, we construct the domains of existence of the inclusion one-phase states and the switching surfaces between the one-phase states. We study the possibility of the inclusion two-phase states, prove the characteristic properties of axisymmetric two-phase strains, and examine their stability. We also demonstrate the scale effect, namely, the influence of the relative dimensions of the inclusion and the body on the inclusion phase state. In the second part of the paper, we study the interaction between an inclusion experiencing phase transitions and a crack.     

Effect of electric displacement saturation on the stress intensity factor for a crack in a ferroelectric ceramic

A crack in a ferroelectric ceramic with perfect saturation under electric loading is analyzed. The boundary of the electric displacement saturation zone ahead of the crack tip is assumed to be ellipse in shape. The shape and size of ferroelectric domain switching zone near a crack tip is determined based on the nonlinear electric theory. The stress intensity factor induced by ferroelectric domain switching under small-scale conditions is numerically obtained as a function of the electric saturation zone parameter and the ratio of the coercive electric field to the yield electric field. It is found that the stress intensity factor increases as the ratio of the semi-axes of the saturation ellipse increases.     

Analytical solution for motion of an elliptical inclusion in gradient stress field

We present a rigorous analytical solution for motion of an elliptical inclusion in isotropic matrix driven by gradient stress field. The interfacial diffusion is considered as the dominant mechanism for the motion. We demonstrate that normal stress gradient on the interface is the major driven force, while the strain energy density gradient is negligible. A key prediction of the solution is that for a given inclusion the motion velocity is proportional to stress gradient only, indicating that the solution is applicable for inclusion motion in nonuniform stress field of varying stress gradient, and that the inclusion tends to move towards the region of lower stress in nonuniform stressed materials.     

Growth of an arc-crack in bonded dissimilar materials

The problem of growth of a crack lying along the interface of a circular inclusion embedded in an infinite plate is studied within the framework of linear elasticity. The plate is subjected to a uniform uniaxial stress at infinity at any angle of inclination relatively to the crack. The critical load for unstable crack growth, the angle of initial crack extension and the subsequent crack path are investigated using the strain energy density fracture criterion. The combined effect of crack length and orientation on the fracture stress is considered for the case of an aluminum-epoxy composite.