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     

Stress Concentration in a Transversally Isotropic Piezoceramic Cylinder Near a Hyperboloidal Neck

The coupled electric and elastic fields near a hyperboloidal neck in a piezoceramic cylinder under static loading are studied. Uniform tensile forces and potential difference are applied to its ends. The dependences of the stress components and the projections of the electric-displacement vector near the neck on the load and the neck curvature are plotted     

The Stress State of a Transversally Isotropic Piezoceramic Body with a Parabolic Crack in a Uniform Heat Flow

The explicit solution is constructed for a static thermoelastic problem for an infinite transversally isotropic piezoceramic body containing a heat-insulated parabolic crack in the isotropy plane. The crack surface is assumed free of forces. The body is under a uniform heat flow, which is perpendicular to the crack surface and is far from the crack itself. The problem is solved for two cases of electric conditions on the crack surface. In the first case, an electric potential is absent on the crack surface and, in the second case, the normal component of the electric-displacement vector is equal to zero. The intensity factors, which depend on the heat flow, crack geometry, and the thermoelectroelastic properties of the piezoceramic body, are determined for the force field and electric displacement near the crack tip     

Elastic–plastic near field solution of an eccentric crack under shear in a finite width plate

The near crack line analysis method is used to investigate an eccentric crack loaded by shear forces in a finite width plate, and the analytical solution is obtained in this paper. The solution includes: the unit normal vector of the elastic–plastic boundary near the crack line, the elastic–plastic stress fields near crack line, variations of the length of the plastic zone along the crack line with an external loads, and the bearing capacity of a finite plate with a centric crack loaded by shear stress in the far field. The results obtained in this paper are sufficiently precise near the crack line because the assumptions of small scale yielding theory have not been made and no other assumptions have been taken. Subsequently, the present results are compared with the traditional line elastic fracture mechanical solutions and elastoplastic near field solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

偏心裂纹板在裂纹面受一对集中拉应力作用时弹塑性解析解

利用裂纹线场方法对理想弹塑性材料偏心裂纹板在裂纹面受一对集中拉力问题进行了弹塑性分析,并且获得了理论解．这个解包括：裂纹线附近弹塑性边界上的单位法向矢量,裂纹线附近的弹塑性解析解、最大塑性区长度、裂纹线上的塑性区长度随荷载的变化规律及其承载力．该分析不受小范围屈服假设的限制,并且不附加假使条件．结果在裂纹线附近足够精确．     

Elastoplastic solution for an eccentric crack loaded by two pairs of point tensile forces

The strain energy density theory and the near crack line analysis method are applied to investigate an eccentric crack loaded by two pairs of tensile point forces in a finite plate. The minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. Obtained is the elastic-plastic solution near the crack line of an eccentric crack loaded by two pairs of point tensile forces under large scale yielding condition. More specifically, the near field solution contains the unit normal vector of the elastic-plastic boundary and the elastic-plastic stress field. The length of the plastic zone along the crack line is found to vary with the external load and the bearing capacity of a finite plate with an eccentric crack loaded by two pairs of tensile point forces. Compared with small scale yielding condition, the normalized load obtained is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Perturbed magnetic field of an infinite plate with a centered crack

Deforming a cracked magnetoelastic body in a magnetic field induces a perturbed magnetic field around the crack. The quantitative relationship between this perturbed field and the stress around the crack is crucial in developing a new generation of magnetism-based nondestructive testing technologies. In this paper, an analytical expression of the perturbed magnetic field induced by structural deforma- tion of an infinite ferromagnetic elastic plate containing a centered crack in a weak external magnetic field is obtained by using the linearized magnetoelastic theory and Fourier transform methods. The main finding is that the perturbed magnetic field intensity is proportional to the applied tensile stress, and is dominated by the displacement gradient on the boundary of the magnetoelastic solid. The tangential component of the perturbed magnetic-field intensity near the crack exhibits an antisymmetric distribution along the crack that reverses its direction sharply across its two faces, while the normal component shows a symmetric distribution along the crack with singular points at the crack tips.     

Wall-modeling for large-eddy simulation of flows around an axisymmetric body using the diffuse-interface immersed boundary method

A novel method is proposed to combine the wall-modeled large-eddy simulation(LES) with the diffuse-interface direct-forcing immersed boundary(IB) method.The new developments in this method include:(i) the momentum equation is integrated along the wall-normal direction to link the tangential component of the effective body force for the IB method to the wall shear stress predicted by the wall model;(ii) a set of Lagrangian points near the wall are introduced to compute the normal component of the effective body force for the IB method by reconstructing the normal component of the velocity. This novel method will be a classical direct-forcing IB method if the grid is fine enough to resolve the flow near the wall. The method is used to simulate the flows around the DARPA SUBOFF model. The results obtained are well comparable to the measured experimental data and wall-resolved LES results.     

The Stress State of a Ferromagnetic with an Elliptic Crack in a Homogeneous Magnetic Field

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing an elliptic crack is considered. It is assumed that the body is under an external magnetic field perpendicular to the crack plane. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the elliptic crack are studied. Formulas are given for the stress intensity factors for the force and magnetic fields near the crack tip     

Exact Analytical Solutions of Static Electroelastic and Thermoelectroelastic Problems for a Transversely Isotropic Body in Curvilinear Coordinate Systems

An approach to the solution of three-dimensional static problems for a transversely isotropic (rectilinear anisotropy) body is expounded and the solutions for piezoceramic canonical bodies are systematized. The result of the study is explicit analytical solutions of three-dimensional problems. Bodies are examined whose boundary surface is the coordinate surfaces in coordinate systems that permit the separation of the variables in the three-dimensional Laplace equation. The stress concentration in bodies near necks, cavities, inclusions, and cracks is investigated. The stress intensity factors of the force field and electric induction near elliptic and parabolic cracks are determined. The contact interaction of a piezoceramic half-space with elliptic and parabolic dies is studied. The bodies are under various mechanical, thermal, and electric loads     

The electroelasticity problem for piezoceramic media with bilateral hyperbolic tunnel cavities

An explicit solution of the static problem of electroelasticity is obtained for a transversally isotropic medium that contains bilateral hyperbolic tunnel cavities. It is assumed that the plane of isotropy of the medium coincides with the plane of symmetry of the medium, and also that the surfaces of the cavities are free of mechanical forces and that the normal component of the electric induction vector is equal to zero on the cavities. A uniform tensile force and difference in the electric potentials are specified at a sufficient distance from the cavities in a direction perpendicular to the plane of isotropy of the medium. A solution of the corresponding problem for a piezoceramic medium containing external bilateral rectilinear cracks is obtained as a special case. S. P. Timoshenko Institute of Mechanics. National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 54–60, August, 1999.     

Elastoplasticity of crack loaded by two pairs of isolated shear forces in finite width plate

The failure behavior of an elastic-perfectly plastic body with a crack loaded by two pairs of concentrated shear forces is discussed. The analytical solutions of an eccentric crack in a finite plate loaded by two pairs of point shear forces are obtained. It includes the unit normal vector of the elastic-plastic boundary near the crack line, the elastic-plastic stress fields near crack line and the law of the plastic zone along the crack line with external loads. The solutions of this paper are sufficiently precise near the crack line in elastic-perfectly plastic materials. Subsequently, the present results are compared with solutions based on the minimum strain energy density theory and elastic-plastic solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

The Stress State of a Ferromagnetic with a Spheroidal Inclusion in a Homogeneous Magnetic Field

The stress–strain state of an infinite isotropic magnetically soft ferromagnetic body with a spheroidal inclusion is analyzed. It is assumed that the body is in an external magnetic field. The basic stress–strain characteristics and the induced magnetic field near and inside the inclusion are analyzed. The plots and the table presented show how the total magnetoelastic and Maxwell stresses near and inside the inclusion depend on the ratio of the spheroid axes, the latitude angle, and the magnetic induction when the medium and the inclusion are dissimilar materials.     

The Stress State of a Transversely Isotropic Ferromagnetic with a Parabolic Crack in a Homogeneous Magnetic Field

The magnetoelastic problem for a transversely isotropic ferromagnetic body with a parabolic crack in the plane of isotropy is solved explicitly. The body is in an external magnetic field, which is perpendicular to the plane of isotropy. The field induces elastic strains and a magnetic field in the body. The characteristics of the stress–strain distribution and induced magnetic field are determined; and their singularities in the neighborhood of the crack are analyzed. Formulas for the stress intensity factors of the mechanical and magnetic fields near the crack tip are presented     

Axisymmetric Deformation of a Transversely Isotropic Cylindrical Body: A Hamiltonian State-Space Approach

A state-space approach for exact analysis of axisymmetric deformation and stress distribution in a circular cylindrical body of transversely isotropic material is developed. By means of Hamiltonian variational formulation via Legendre’s transformation, the basic equations in cylindrical coordinates are formulated into a state-space framework in which the unknown state vector comprises the displacements and associated stress components as the dual variables and the system matrix possesses the symplectic characteristics of a Hamiltonian system. Upon delineating the symplecticity of the formulation, a viable solution approach using eigenfunction expansion is developed. For illustration, an exact analysis of a finite thick-walled circular cylinder under internal and external pressures is presented, with emphasis on the end effects.     

Exact Analytical Solutions of Three-Dimensional Static Thermoelastic Problems for a Transversally Isotropic Body in Curvilinear Coordinate Systems

An approach to the solution of three-dimensional static thermoelastic problems for a transversally isotropic (the case of rectangular anisotropy) body is proposed. The results of construction of the general analytic solutions to thermoelastic problems for canonical bodies are systematized. The exact analytic solutions of three-dimensional problems are obtained. It is assumed that the bodies under consideration are thermoelastic and their boundary surface corresponds to the coordinate surfaces in coordinate systems that allow separating the variables in the three-dimensional Laplace equation. The stress concentration near cavities and inclusions is studied. The stress intensity factors near elliptic and hyperbolic cracks are determined. Formulas are presented for the stress intensity factors on the surface of a rigid elliptic inclusion and inside the body near a homogeneity under various thermal effects     

Three-dimensional local stress analysis on grain boundaries in polycrystalline material

In order to understand the initiation behavior of microstructurally small cracks in a stress corrosion cracking condition, it is important to know the tensile normal stress acting on the grain boundary (normal GB stress). The local stress in a polycrystalline body is enhanced by the inhomogeneity which stems from the shape and orientation of each grain. The stress in a three-dimensional polycrystalline body consisting of 100 grains with random orientation, under a remote uniform tensile stress condition, is evaluated by the finite element method. It was revealed that the local stress on the polycrystalline body is inhomogeneous under uniform applied stress and becomes large at those grain boundaries that are perpendicular to the load axis, though there is large fluctuation. It was also shown that the normal GB stress tends to be large near the triple points due to the deformation constraint caused by adjacent grains. Finally, the maximum stress on the surface of a large component caused by the inhomogeneity was evaluated by using Gumbel statistics.     

A Magnetoelastic Field in a Ferromagnetic Medium with a Spherical Cavity

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing a spherical cavity is considered. It is assumed that the body is under an external magnetic field. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the cavity are studied. Graphs are presented for the total magnitoelastic and Maxwell stresses as functions of the magnetic induction, the angle of dip, and the mechanical and magnetic properties of the material     

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.