Magneto-rotation-fibre-reinforced thermoelastic with gravity and energy dissipation

In this article, two theories of the generalized thermoelasticity Green-Naghdi theory (of type II and III) are applied, as well as the coupled theory to study the effect of magnetic field and rotation under influence of gravity on 2D problem of a fibre-reinforced thermoelastic. The normal mode analysis is used to obtain the expressions for the temperature, displacement components and the thermal stresses distributions. The resulting formulation is applied for two different concrete problems. The first concerns the case of a punch moving across the surface of semi-infinite thermoelastic half-space subjected to appropriate boundary conditions. The second deals with a thick plate subjected to a time-dependent heat source on each face. Numerical results are illustrated graphically for each problem considered. A comparison is made with the results predicted obtained by the two theories in the presence and absence of magnetic field, rotation and gravity field.     

A two-dimensional problem of a mode-I crack in a rotating fibre-reinforced isotropic thermoelastic medium under dual-phase-lag model

In the present work, a general model of the equations of generalized thermoelasticity for a homogeneous isotropic elastic half-space solid whose surface is subjected to a mode-I crack problem under the effect of rotation is investigated. The normal mode analyses are used to obtain the expressions for the temperature distribution, the displacement component and thermal stresses in the context of the dual-phase-lag theory of thermoelasticity proposed by Tzou. The boundary of the crack is subjected to a prescribed stress distribution and temperature. Some particular cases are also discussed in the context of the problem. The numerical values of the temperature distribution, the displacement components and thermal stresses are also computed for a suitable material and the results are presented graphically. The effects of rotation, reinforcement and the phase lags parameters are discussed in detail in the light of earlier works.     

Dependence of a crack growth path on the elastic moduli of an anisotropic solid

One of the ways to increase the resistance of a structure to catastrophic fracture is to force a main line crack to deviate from its path. For this reason the influence of the elastic moduli of an anisotropic material on crack rotation are studied. In particular a linear elastic problem for a straight Mode I crack, located on a symmetry axis of an orthotropic plane is considered. The strength properties of the material are assumed to be isotropic. Several crack models are considered for studying the direction of a crack growth path. It is shown that a crack modeled as a thin, elongated, elliptical hole leads to more plausible results concerning crack rotation conditions than an ideal cut model. The maximal tensile stresses are taken as a crack growth criterion. It is shown that for a class of orthotropic materials a crack deviates from the straight path just after it starts to grow, even in the conditions of uniaxial normal tension. The problem of the stability of a straight crack path under Mode I loadings is also considered. This problem is reduced to the problem of the fracture direction determination for thin, elongated, elliptical cavities slightly inclined to the initial direction. The conditions of instability are obtained within the framework of the proposed approach. It is shown that for a class of orthotropic materials a straight crack path is unstable in the conditions of uniaxial normal tension. This class of materials is larger than the one for which a crack deviates from the straight crack path just after its start.     

The rotation of a rigid spheroidal inclusion embedded in a transversely isotropic medium

The exact three-dimensional elasticity solutions are given for two problems related to a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium. The first is of axisymmetric nature in which the inclusion is given a constant rotation about its axis of revolution which coincides with the axis of symmetry of the material. The second problem is asymmetric where the spheroidal inclusion is given a constant rotation about a direction that is perpendicular to the axis of elastic symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. In both cases, the moment-rotation relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of the aspect ratio of the spheroid on the rotational stiffness.     

Neutral spherical inhomogeneities in a twisted circular cylindrical bar

We consider the torsion problem of a circular cylindrical bar which is filled up with composite spherical inclusions. The composite inclusions consist of a core and coating both of which are spherically orthotropic with the volume fractions of the core being the same in every composite inclusion. The center points of the spherical inhomogeneities are on the axis of revolution of the circular cylinder. The neutral inhomogeneity in the considered problem of elastic equilibrium is defined as a foreign body (inclusion) which can be introduced in a host body without disturbing the elastic field (displacements, stresses) in it. The conditions of the neutral inhomogeneity for the twisted circular cylindrical bar are derived, and some special cases of inhomogeneity are analyzed. The present paper gives a new example for neutral inhomogeneity in the field of elasticity.     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

The Density Difference of Cupula and Endolymph Changes the Mechanics of Semicircular Canals

A precise balance of cupula and endolymph densities is key to the proper sensing of angular acceleration by the semicircular canals (SC). Estimates show that a density difference of cupula and endolymph (DD) as small as ~10^− 4 g/cm³ is sufficient to make the SC sensitive to gravity and centrifugal forces provided they are comparable with gravity. As a result this might cause vestibular disorders. There are conditions under which the DD may even exceed this value. One of them is a change of intra-labyrinth pressure (IP) that may take place during a spaceflight. Here, the effect of DD on SC dynamics is considered using a simplified one-dimensional toroidal mathematical model of a canal for rotation with constant or harmonically oscillating angular velocities. The DD results in: dependence of cupula dynamics on orientation of both the gravity vector relative to the SC plane and the axis of rotation, as well as on the distance between the axis of rotation and the center of SC; shift of the cupula to a new position of equilibrium that depends on both the gravity vector and the parameters of head rotation; and onset of cupula oscillations with multiple frequencies under harmonic stimulation. The DD effect may be important under conditions of artificial gravity where the directions of centrifugal forces, the values of which are comparable with Earth’s gravity, the orientations of the axis of rotation of a space station, and the axes of the SCs change during movements of the individuals and their habitat.     

The influence of shear and extensibility on the stability of a heavy rotating rod

We determine the critical rotation speed for a heavy rod, fixed at one and free at the other end, that rotates with constant angular speed about its axis. The constitutive equations of the rod are taken in the form of generalized Bernoulli-Euler theory that takes into account extensibility of the rod axis and the influence of shear stresses. The type of bifurcation pattern at the lowest eigenvalue is determined. The post-critical shape of the rod is determined by numerical integration and a new, interesting property of the equilibrium equations is discovered     

基于椭圆函数谐波平衡法的二元机翼非线性颤振

针对低阶谐波平衡法精度不高的不足,引入椭圆函数谐波平衡法解决非线性气动弹性问题。基于一阶活塞理论,建立了高速二元机翼的立方非线性颤振方程,采用椭圆函数谐波平衡法、谐波平衡法和Runge-Kutta数值计算方法进行了求解。结果表明:椭圆函数谐波平衡法的计算结果与Runge-Kutta数值计算方法的结果吻合,且与谐波平衡法相比其相对误差更小,可以有效的预测极限环振荡的幅值及其临界点。同时研究了弹性轴位置及重心位置对极限环颤振临界点的影响,随着弹性轴位置不断靠近翼弦中点,极限环振荡临界速度不断增大;而随着重心位置与弹性轴距离的增大,极限环振荡临界速度存在一个极小值点。     

Behavior of a Light Solid in a Rotating Horizontal Cylinder with Liquid Under Vibration

Dynamics of a cylindrical body in a rotating cavity is experimentally studied under transversal translational vibrations of the cavity rotation axis. Experiments are run at high rotation rate, when under the action of centrifugal force the body shifts to the rotation axis (the centrifuged state). In the absence of vibrations, the lagging rotation of the body is observed, due to the body radial shift from the axis of rotation caused by gravity. The body average rotation regime depends on the cavity rotation rate. The vibrations lead to the excitation of different regimes of body differential rotation (leading or lagging) associated with the excitation of its inertial oscillations. The dependence of the differential speed of the body rotation on the vibration frequency is investigated. The body dynamics has a complex character depending on the dimensionless vibration frequency. The analysis of body oscillation trajectory revealed that the body oscillatory motion consists of several modes, which contribute to the averaged dynamics of the body and the flows in the cavity.     

Vibrations of an elastic disc bearing a blade assembly in compound rotation

Precessional oscillations are considered for an elastic rotor whose axis of rotation turns in a plane. The rotor consists of an elastic disc fitted on a rigid shaft and a fan of blades connected at the outer edge. The compound rotation can be a source of resonant precession oscillations.Translated from Problemy Prochnosti, No. 6, pp. 71–81, November–December, 1996.     

Plane Waves in Generalized Thermo-microstretch Elastic Solid with Thermal Relaxation Using Finite Element Method

The propagation of plane waves in a thermo-microstretch elastic solid half-space as proposed by Lord?CShulman as well as the classical dynamical coupled theory are discussed. The problem has been solved numerically using a finite element method. Numerical results for the displacement components, force stresses, temperature, couple stresses, and microstress distribution are obtained. The variations of the considered variables through the horizontal distance are given and illustrated graphically. Comparisons are made with the results predicted by the theory of generalized thermoelasticity for different values of the relaxation time.     

Effect of rotation on the structure of a convecting mushy layer

Summary Nonlinear natural convection in a mushy layer during solidification of a binary alloy is investigated under a high gravity environment where the rotation axis is inclined to the high gravity vector. Asymptotic and scaling analyses are applied to convective flow within the mushy layer and in vertical chimneys. The main result is that, for some particular moderate rotation range, vertical velocity in the chimneys decreases rapidly with increasing rotation rate and appears to have opposite signs across some rotation dependent vertical level.     

Dual finite element analysis for plasticity–friction torsion of composite bar

The quasi-static problem of torsion of an elastic–plastic, prismatic, composite bar is considered in the paper. The phenomenon of slip on the interfaces between the components of the bar is taken into account. The elastic–plastic behaviour of the material is described by the Prandtl-Reuss constitutive relation. The slip on the interface is governed by the Coulomb friction law—it is assumed that there is no cohesion between components of the bar. The stresses normal to the interfaces are considered to be caused by shrinkage of the matrix of the bar or by external forces acting perpendicularly to its longitudinal axis. The problem is set in the dual variational forms and solved with the help of the finite element method. Two approximate kinematically and statically admissible solutions are obtained. The stress function is used for calculation of the second one. The iterative algorithms solving the problem and some numerical results are presented in the paper.     

Maximum circumferential stress criterion applied to orthotropic materials

The topic of the paper is the extension of the Maximum Circumferential Tensile Stress Criterion to orthotropic materials, to obtain the crack initiation angle and the fracture loci. It includes the effects of orthotropy, load biaxiality and non‐singular terms. It requires the solution of the elastostatic problem of an orthotropic plate having a central inclined crack and subjected at infinity to a uniform biaxial load. It is assumed that the crack line does not coincide with an axis of elastic symmetry of the body. The original problem has been transformed by rotation into a system of coordinates with one axis along the crack. The effects of the rotation of the reference system on the stress–strain equations as well as on the equilibrium equations have been considered. The stress field is represented in terms of two potential functions and the crack problem is solved by means of the superposition principle.     

Effect of break of the elastic axis of a rotor with transverse crack on its vibrational characteristics

We perform a numerical analysis of the effect of break of the elastic axis on the characteristics of bending vibrations of a multispanned unbalanced rotor with transverse crack mounted on elastic damping supports. The break of the elastic axis in the cross section with crack is caused by local plastic strains. The solution of the problem is sought in the form of a sum of the contributions of the forces of gravity, residual disbalance, and the crack responsible for the break of the axis. The problem is reduced to the numerical analysis of the trajectory of the shaft with its subsequent improvement by the method of iterations. Amplitudes and phases of several first harmonics of vibrational displacements are found as functions of sizes, initial orientation of the crack, and the ratio of vibrational and static displacements. In the presence of the break of the elastic axis, the magnitudes of vibrational characteristics undergo significant changes. The amplitudes of vibrations induced by the crack are comparable with the amplitudes of vibrations caused by disbalance. It is shown that the behavior of the trajectory of motion regarded as a function of principal parameters is nonlinear. Institute for Problems in Mechanical Engineering, National Academy of Sciences of Ukraine, Kharkov, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 82–89, July–August, 1997.     

Two-Temperature Effects on Plane Waves in Generalized Thermo-Microstretch Elastic Solid

In this article, the effect of two temperatures on plane waves propagating through a generalized-thermo-microstretch elastic half-space solid has been investigated. The surface of the medium is subjected to a mode-I crack, and the $z$ axis is pointing vertically into the medium. Two fascinating theories of generalized thermo-elasticity presented by Green and Naghdi and named as without energy dissipation (GN-II) and with energy dissipation (GN-III) have been used. Governing equations for each particular case are also derived, and a solution is obtained. An analytical technique of normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, the temperature, and the couple stresses distribution. The variations of the considered variables against the vertical distance are illustrated graphically. Comparisons are made with the results between type II and III in generalized-thermo-microstretch and in a particular case (without microstretch constants). Numerical work is also performed for a suitable material with the aim of illustrating the results. It is found that the maximum amplitude is obtained for the maximum value of the two temperature parametric constant.     

On the dynamic strength of a disk with a central cylinder-conic opening

The problem of nonstationary dynamic behavior of a disk weakened by a central cylinder-conic opening under the action of suddenly applied loading is considered. Determining relations are given in the form of a system of integral equations in an arbitrary oblique coordinate system. A numerical solution has been obtained by using a hybrid difference scheme constructed on the basis of the method of flow correction. Our generalized versions of the S.K. Godunov method for the case of a nonorthogonal system of coordinates were used as basic schemes. The locations of maximum stresses that precede the appearance of plasticity zones have been determined for a particular elastic body. Translated from Problemy Prochnosti, No. 2, pp. 79–87, March–April, 1997.     

Plane deformation of a body with rigid cylindrical inclusion

We study the problem of plane deformation of an infinite elastic body with thin rigid cylindrical inclusion with oval cross section. The body is loaded by biaxial uniform tensile forces at infinity. The solution of the problem is reduced to two singular integral equations with Cauchy kernels for the jumps of normal and tangential stresses on the surface of the inclusion. The solutions of these equations are obtained in the closed analytic form and, used to deduce the formulas for the concentration of stresses near the inclusion, for stresses inside the inclusion, and for the angle of rotation of the inclusion as a rigid body. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Ukrainian State University of Forestry Engineering, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 87–92, November–December, 1996.     

Stresses Near a Disk-Shaped Sharp Slot with Tangential Contact Interaction Between Its Surfaces

We propose a new statement of the problem of elastic equilibrium of a disk-shaped slot whose surfaces are loaded by an arbitrary normal load vanishing on its front according to a certain law. By the method of discontinuous Weber-Schafheitlin integrals, it is shown that, in this case, there exists a unique state of elastic equilibrium with regular distribution of stresses over the front of the slot. According to the concept of boundary layer, the jump of tangential stresses formed on the surfaces of the slot due to the continuity of the components of the vector of local rigid rotation Ω on its front is continued in the plane of the slot and guarantees the validity of the equilibrium conditions provided that the normal stresses are equal to zero (the effect of boundary layer). The numerical analysis shows that the jump of tangential stresses in the boundary layer outside the slot rapidly decreases with the distance from the front in complete agreement with the Saint-Venant principle. If a jump of tangential stresses on the surfaces of the slot is absent (and exists outside the slot), then these surfaces have a kink in the front of the slot and the volume strains possess a logarithmic singularity. This situation can be regarded as the limiting state corresponding to the onset of plastic deformation or brittle fracture. If the tangential stresses do not have jump in the entire plane of the slot, then the solution of the problem does not exist. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 3, pp. 71–84, May–June, 2005.