Deformation of a Peridynamic Bar

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venant's principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Green's function for general loading problems.     

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅰ)──MICROPOLAR THERMOELASTICITY

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Motion of gas bubbles in a liquid under the influence of a temperature gradient

This study deals with the motion of a gas bubble developing under the influence of surface-tension forces in an imponderable viscous liquid with a temperature gradient. A theory of steady-state motion of a bubble in a field with constant temperature gradient is given for the case of small Reynolds numbers. Experimental results that show qualitative agreement with the theory are presented.The authors wish to thank M. A. Lavrent'ev for formulating the problem and giving constant attention to their work.     

The self-similarly expanding Eshelby ellipsoidal inclusion: I. Field solution

The solution of a self-similarly (subsonically) dynamically expanding ellipsoidal inclusion with general spatially uniform transformation strain temporally constant is obtained by the use of the Radon transform and the satisfaction of the zero initial conditions and the radiation condition at infinity. It constitutes the self-similar evolution of the inclusion singularity (jump discontinuity at the inclusion boundary) starting from zero dimension. The field solutions for the displacement gradient and particle velocity are presented. Due to the fact that for a self-similarly expanding subsonic motion the hyperbolic system of the partial differential equations of motion becomes elliptic (as proved in Ni and Markenscoff, 2015), it is shown here explicitly that the solution for the displacement gradient in the interior domain of the expanding ellipsoid is constant, thus extending the Eshelby property to the self-similarly expanding ellipsoids as pointed out by Burridge and Willis (1969). Also, the particle velocity is shown to be zero in the interior domain (lacuna) as the waves emitted by the self-similarly expanding inclusion cancel each other due to the symmetries of geometry and motion.     

Restudy of coupled field theories for micropolar continua (I)—Micropolar thermoelasticity

Problems of micropolar thermoelasticity have been presented and discussed by some authors in the traditional framework of micropolar continuum field theory. In this paper the theory of micropolar thermoelasticity is restudied. The reason why it was restricted to a linear one is analyzed. The rather general principle of virtual work and the new formulation for the virtual work of internal forces as well as the rather complete Hamilton principle in micropolar thermoelasticity are established. From this new Hamilton principle not only the equations of motion, the balance equation of entropy, the boundary conditions of stress, couple stress and heat, but also the boundary conditions of displacement, microrotation and temperature are simultaneously derived. Contributed by DAI Tian-min Foundation item: the National Natural Science Foundation of China (10072024); the International Cooperation Project of the NSFC (10011130235) and the DFG (51520001); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)     

Two-Dimensional Electroelastic Problem for a Multiply Connected Piezoelectric Body

In this paper, we present the basic relationships for the complex potentials of a two-dimensional electroelastic problem, their general representations for a multiply connected domain, expressions for stress, displacement, electrostatic field intensity and induction, and potential. A closed solution is found for a body with one elliptic cavity or one elliptic crack under the action at infinity of a constant electroelastic field or concentrated forces and charges     

Axisymmetric contact problem of cubic quasicrystalline materials

The axisymmetric elasticity theory of cubic quasicrystal was developed in Ref. [1]. The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function, based on which, the exact analytic solutions for the elastic field of an axisymmetric contact problem of cubic quasicrystalline materials are obtained for universal contact stress or contact displacement. The result shows that if the contact stress has order −1/2 singularity on the edge of the contact domain, the contact displacement is a constant in the contact domain. Conversely, if the contact displacement is a constant, the contact stress must have order −1/2 singularity on the edge of the contact domain. Project supported by the National Natural Science Foundation of China (No. 19972011).     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.     

两微极磁热弹性固体分界面上波的反射与折射

给出了磁场、热场和弹性场多场耦合作用下微极广义热弹性固体的一般控制方程.该方 程既包含了磁场、热场和弹性场的耦合作用,又在其广义热传导方程中涵盖了耦合热弹理论 (C-D)及其5类推广(L-S理论,G-L理论,G-N(II,III)理论和C-T理论).运用该微极广义磁热 弹性控制方程,研究了在定常磁场作用下, 具有均匀初始温度的两理想接触微极弹性介质平面分界面上磁热弹性波的反射和折射现象.给出了分别在缺少磁场、热场作用或不同广义热传 导理论下反射或折射热波、纵向位移波、耦合横向和微旋转波与入射纵向位移波的振幅比随 入射角变化的关系曲线.对缺少磁、热和微极性以及热松弛时间时对应的反射、折射系数进 行了对比.结果表明磁、热和微极性以及热松弛时间对振幅比均有不同程度的影 响,与磁、热和微极性一样,热松弛时间对不同类型波的影响能力差别明显,但对同 一类型的反射波和折射波的影响相似.     

The mechanically-based approach to 3D non-local linear elasticity theory: Long-range central interactions

This paper presents the generalization to a three-dimensional (3D) case of a mechanically-based approach to non-local elasticity theory, recently proposed by the authors in a one-dimensional (1D) case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range forces exerted by non-adjacent elements. Specifically, the long-range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements, measured along the line connecting the centroids. Further, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decaying function and to the products of the interacting volumes. Consistently with the modelling of the long-range forces as central body forces, the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces.The proposed 3D formulation is developed both in a mechanical and in a variational context. For this the elastic energy functionals of the solid with long-range interactions are introduced, based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model. Numerical applications are reported for 2D solids under plane stress conditions.     

A self-consistent solution for nonlinear theory of the positive column in a magnetic field

Only the electron and ion gases were taken into account in all previous theories of the positive column of intermediately-low-pressure arc discharge with or without the longitudinal magnetic field, while the motion of neutral gas was neglected. In 1982, the authors^[1] presented a nonlinear theory of a positive column which indicated that the rotating velocities of neutral gas and ion gas were nearly equal, and the motion of neutral gas could not be ignored. They further discussed the problem of validity of Bohm's criterion. However, some of the parameters with which the computation was worked out in Ref. [1] were not correlated to the initial discharge parameters. In the present paper, two integral relations are supplemented, so that a complete mathematical formation of the problem is given. A convergent numerical solution is obtained by iteration and the solution of Ref. [1] turns out to be the first iteration approximation. It is shown that both functions and parameters obtained by self-consistent solution differ significantly from those obtained in the first iteration approximation. According to this paper the computation can be conducted when the initial discharge parameters are given, so this method could have certain practical applications.     

Boundary layer on the moving surface of a cylinder

A study is made of the problem of the boundary layer on a cylinder with a moving surface when the cylinder moves with constant velocity in an incompressible fluid. Expressions are obtained for the distributions of the frictional stress on the surface of the cylinder and the coordinate of the singular point in the solution of the boundary layer equations that indicates the appearance of a region of reverse flow for different values of the relative velocity of the motion of the surface of the cylinder. Numerical calculations have been made of the work of the force of friction associated with displacement of the cylinder, the work expended on the motion of its surface, and, in the case of flow separation, the work of the pressure forces (it being assumed here that the pressure and friction on the wall behind the singular point are constant and equal to the pressure and friction at the singular point).     

Inclined entry of a blunt profile into an ideal fluid

A study is made of the two-dimensional unsteady motion of an ideal incompressible fluid due to the entry into it of a blunt profile at a given angle of attack. In the initial stage of the process, when the penetration depth is relatively small, the problem can be investigated by the methods of asymptotic analysis. The dimensionless time t plays the part of the small parameter. It is shown that to 0(t²) as t 0 the displacement field of the fluid particles does not depend on the angle of attack and is determined by the solution to the problem of vertical entry. The asymptotic behaviors of the principal vector and principal moment of the forces exerted on the profile by the fluid at short times are found. The asymptotic behavior of the principal moment of the forces is proportional to the distance traversed by the body along the surface of the fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 145–150, May–June, 1988.     

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

A quasi-three dimensional model is proposed for the vibration analysis of functionally graded(FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.     

Magnetohydrodynamic lubrication theory for a cylindrical bearing

Liquid metal, which is a conductor of electric current, may be used as a lubricant at high temperatures. In recent years considerable attention has been devoted to various problems on the motion of an electrically conducting liquid lubricant in magnetic and electric fields (magnetohydrodynamic theory of lubrication), Thus, for example, references [1–3] study the flow of a conducting lubricating fluid between two plane walls located in a magnetic field. An electrically conducting lubricating layer in a magnetohydrodynamic bearing with cylindrical surfaces is considered in [4–8] and elsewhere.The present work is concerned with the solution of the plane magnetohydrodynamic problem on the pressure distribution of a viscous eletrically conducting liquid in the lubricating layer of a cylindrical bearing along whose axis there is directed a constant magnetic field, while a potential difference from an external source is applied between the journal and the bearing. The radial gap in the bearing is not assumed small, and the problem reduces to two-dimensional system of magnetohydrodynamic equations.An expression is obtained for the additional pressure in the lubricating layer resulting from the electromagnetic forces. In the particular case of a very thin layer the result reported in [4–8] is obtained. SI units are used.     

Principle of equal effects of general relativity and invariance of classical mechanics equations

This paper gives the proof that the“inerial forces”in a noninertial system arenot fabricated forces,but potential forces which actually act on the objects in motionin the acceleration field,according to the equivalent principle between gravitation andinertial forces in the theory general relativity.Further,the invariance of kineticalequation is illuminated.