Exact analytic solutions to the problem on plane buckling modes of rectangular orthotropic plates with free edges under biaxial loading

A two-dimensional linearized problem on plane buckling modes (BMs) of a rectangular plate with free edges, made of an elastic orthotropic material, underbiaxial tension-compression is considered. With the use of double trigonometric basis functions, displacement functions exactly satisfying all static boundary condition on plate edges are constructed. It is shown that the exact analytic solutions found describe only the pure shear BMs, and if the normal stress in one direction is assumed equal to zero, an analog of the solution given by the kinematic Timoshenko model can be obtained. Upon performing the limit passage to the zero harmonic in the displacement functions of one of the directions, the solution to the problem of biaxial compression can be obtained by equating the Poisson ratio to zero; in the case of uniaxial compression, this solution exactly agrees with that following from the classical Bernoulli-Euler model. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 149–170, March–April, 2007.     

Correlation between the properties of eigenfrequencies and eigenmodes in a chain of rigid bodies with torque connections

A chain of N rigid bodies with elastic connections transmitting a torque interaction is considered. Similar individual inertial elements of the chain have one degree of freedom. An exact analytical solution of the problem of oscillation eigenfrequencies and eigenmodes is constructed for an arbitrary value of N. The properties of frequencies and modes of such a chain are compared with those of a canonical Newton chain. It is established that the correlation between eigenfrequencies and the properties of the corresponding eigenmodes fundamentally differs from that for a canonical Newton chain. Lower eigenfrequencies of a chain of rigid bodies with inertialess torque connections correspond to modes with a larger number of nodes, while higher eigenfrequencies correspond to the smoother modes. The nontypical correlation between the oscillation eigenfrequencies and eigenmodes discovered based on the exact analytical solution to the problem of free oscillations of a chain of N rotating rigid bodies contradicts the ideas underlying theoretical studies in the field of solid state physics devoted to simulation of mechanical and thermal dynamic processes in crystalline lattices.     

各向异性矩形板自由振动的一般解析解法

根据各向异性矩形薄板自由振动横向位移函数的微分方程建立了一般性的解析解．该一般解包括三角函数和双曲线函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式和双正弦级数解,它能满足4个角的边界条件问题．因此,这一解析解可用于精确地求解具有任意边界条件的各向异性矩形卞的振动问题．解中的积分常数可由4边和4角的边界条件来确定．由此得出的齐次线性代数方程系数矩阵行列式等于零可以求得各阶固有频率及其振型,以四边平夹的对称角铺设复合材料迭层板为例进行了计算和讨论．     

Trapped modes of oscillation of a liquid for surface-piercing bodies in oblique waves

The linear problem of the harmonic oscillations of an ideal incompressible heavy liquid with a free surface in the presence of two and more infinitely long partially submerged cylindrical bodies of arbitrary cross-section is considered. It is proved that there are configurations of the bodies which provide examples of the non-uniqueness of the boundary-value problem in the case of an arbitrary frequency of the oscillations and an arbitrary non-zero angle between the generatrix of the cylinders and the direction of propagation of the surface waves. In the case of these configurations, the homogeneous boundary-value problem has non-trivial solutions with a finite energy integral, which describe trapped modes of oscillation of the liquid.     

The small free oscillations of a strip

The refined equations of the free oscillations of a rod-strip, constructed previously in a first approximation by reducing the two-dimensional equations to one-dimensional equations by using trigonometric basis functions and satisfying the static boundary conditions on the boundary surfaces are analysed. These equations, the solutions of which are obtained for the case of hinge-supported end sections of the rod, are split into two independent systems of equations. The first of these describe non-classical fixed longitudinal-transverse forms of free oscillations, which are accompanied by a distortion of the plane form of the cross section. It is shown that the oscillation frequencies corresponding to them depend considerably on Poisson's ratio and the modulus of elasticity in the transverse direction, while for a rod of average thickness for the same value of the frequency parameter (the tone) they may be considerably lower than the frequencies corresponding to the classical longitudinal forms of free oscillations, which are performed while preserving the plane form of the cross sections. The second system of equations describes transverse flexural-shear forms of free oscillations, whose frequencies decrease as the transverse shear modulus decreases. They are practically equivalent in quality and content to the similar equations of well-known versions of the refined theories, but, unlike them, when the number of the tone increases and the relative thickness parameter decreases they lead to the solutions of the classical theory of rods.     

Application of the finite-element method to improve the quasi-exact reinforcement theory of fibrous polymeric composites

In the paper, the WL quasi-exact reinforcement theory of fibrous polymeric composites is improved. An optimum compatibility condition related to the transverse shear problem for a unit cell, which brings solutions closest to reality, is derived. This condition is formulated in the form of a linear combination of maximum radial and circumferential displacements. Optimum coefficients of this combination are determined by comparing analytical and numerical solutions for a test specimen in the form of a rectangular thin plate, which is in a plane strain state and is subject to selected loading schemes. The analytic solutions are obtained for a homogenized material by using the WL reinforcement theory. The numerical solutions are found for an actual heterogeneous composite material by using the finite-element method, and they verify the WL reinforcement theory, in particular, the admissibility of Hills assumption. An analysis performed for two composite materials shows that the improved WL reinforcement theory gives adequate displacement fields.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 79–92, January–Febrauary, 2005.     

Free bending vibrations of an isotropic plate with free holes

A solution of the problem of the free bending vibrations of a thin finite isotropic plate with stiffly clamped outer edge is obtained by using functions of a complex variable. Results of numerical investigations are presented for an elliptic plate with acircular hole.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 67–70, 1989.     

Limiting equilibrium of a plane with a circular hole

The elastoplastic state of a compressible isotropic plane with a circular hole is studied by the method of a small parameter. An unknown boundary separating the domain of limiting equilibrium and the elastic domain is determined. We construct the complex Kolosov-Muskhelishvili functions that describe the elastic state of a plane and compare these with the solutions of Galin's problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 980–981. July, 1993.     

压电梁的多项式解（Ⅱ）——典型问题解析解

从正交各向异性压电介质平面问题,对于材料3个特征根互不相等情况下,以3个拟调和位移函数表达位移、电势、应力和电位移的通解出发,利用调和多项式的显式表达式,结合试凑法,给出了平面压电梁的若干典型问题的解析解,包括悬臂压电梁自由端作用横向集中力和点电荷,悬臂压电梁表面作用线性电势和均布载荷,以及两端简支压电梁作用均布载荷等的解析解．     

趋化模型中的一个自由边界问题

本文就一个常见的生物趋化问题建立了相应的自由边界模型,对上述模型得到了其局部弱解的存在性.     

微极热弹性无限板的轴对称自由振动

研究了在应力自由和刚性固定边界条件下,无能量耗散的均匀、各向同性微极热弹性无限板的轴对称自由振动波的传播,导出了相应的对称和斜对称模态波传播的闭合式特征方程和不同区域的特征方程.对短波的情况,应力自由热绝缘和等温板中对称和斜对称模态波传播的特征方程退化为Rayleigh表面波频率方程.根据导出的特征方程得到了热弹性、微极弹性和弹性板的结果.在对称和斜对称运动中计算了板的位移分量幅值、微转动幅值和温度分布,给出了对称和斜对称模式的频散曲线,并示出了位移分量和微转动幅值和温度分布的曲线.能够发现理论分析和数值结论是非常一致的.     

Exact analytical and numerical solutions of stability problems for a straight composite bar subjected to axial compression and torsion

Based on linearized equations of the theory of elastic stability of straight composite bars with a low shear rigidity, which are constructed using the consistent geometrically nonlinear equations of elasticity theory for small deformations and arbitrary displacements and a kinematic model of Timoshenko type, exact analytical solutions of nonclassical stability problems are obtained for a bar subjected to axial compression and torsion for various modes of end fixation. It is shown that the problem of direct determination of the critical parameter of the compressive load at a given torque parameter leads to transcendental characteristic equations that are solvable only if bar ends have cylindrical hinges. At the same time, we succeeded in obtaining solutions to these equations in terms of wave formation parameters of the bar; these parameters, in turn, enabled us to find the parameter of the critical load at any boundary conditions. Also, an algorithm for numerical solution of the problems stated is proposed, which is based on reducing the problems to systems of integroalgebraic equations with Volterra-type operators and on solving these equations by the method of mechanical quadratures (finite sums). It is demonstrated that such numerical solutions exist only for certain ranges of parameters of the bar and of the parameter of torque. In the general case, they can not be obtained by the numerical method used. It is also shown that the well-known solutions of the stability problem for a bar subjected to torsion or to compression with torsion are in correct. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 167–200, March–April, 2009.     

Modelling of circumferential waves in cylindrical thermoelastic plates with voids

The propagation of thermoelastic waves along circumferential direction in homogeneous, isotropic, cylindrical curved solid plates with voids has been investigated in the context of linear generalized theory of thermoelasticity. The plate is subjected to stress free or rigidly fixed, thermally insulated or isothermal boundary conditions. Mathematical modeling of the problem for the considered cylindrical curved plate with voids leads to a system of coupled partial differential equations. The model has been simplified by using the Helmholtz decomposition technique and the resulting equations are solved by using the method of separation of variables. The formal solution obtained by using Bessel’s functions with complex arguments is utilized to derive the secular equations which govern the wave motion in the plate with voids. The longitudinal shear motion and axially symmetric shear vibration modes get decoupled from the rest of the motion in contrast to non-axially symmetric plane strain vibrations. These modes remain unaffected due to thermal variations and presence of voids. In order to illustrate theoretical developments, numerical solutions have been carried out for a stress free, thermally insulated or isothermal magnesium plate and are presented graphically. The obtained results are also compared with those available in the literature.     

Free vibrations of a cantilevered plate with a polymer coating

The article considers some of the special characteristics of the free vibrations of a steel cantilevered plate, with the formation of a polymer coating on the plate. It compares the results of an experimental investigation of the effect of internal stresses in the coating on the vibrations and on the static bending of the plate. It demonstrates the possibility of determining the internal stresses in the coating from the change in the frequency of the natural vibrations of a plate with a coating.Institute for the Mechanics of Metal-Polymer Systems, Academy of Sciences of the Belorussian SSR, Gomel'. Translated from Mehanika Polimerov, No. 3, pp. 541–544, May–June, 1970.     

On possibilities of the method of homogeneous solutions in the mixed problem of elasticity theory for a half-strip

The mixed plane boundary value problem for an elastic half-strip under kinematic loading of its endface and lateral sides free of force loads is examined. An asymptotic is obtained for the coefficients of the series expansion of the displacement vector in homogeneous solutions. Regularization of the series in homogeneous solutions is proposed which would permit computation of the stress field on the half-strip enface.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 3–8, 1987.     

Concentration of electroelastic fields in a composite piezoceramic plate with a hole intersecting the interface of materials

A plane problem of electroelasticity is considered for an infinite compound plate with a hole located in both constituents of the plate. The corresponding boundary value problems is reduced to a system of singular integral equations of second kind, which is solved in numerical quadratures. Calculation results are presented that describe the concentration of electroelastic fields near the hole upon action at infinity of the vectors of mechanical stresses and electric field strength.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 359–366, May–June, 1999.     

Stability and control of transversal oscillations of a tethered satellite system

The tethered satellite system is characterized by weak nonlinearities but it practically works in conditions of internal resonance which produces unstable oscillations. The effect of a longitudinal control force is investigated. Since the displacement component in the orbit plane is always present in the motion due to the nonlinear coupling, the control force is assumed depending only on this component and also when a prevailing out-of-plane oscillation is considered. The harmonic balance method and numerical solutions of amplitude modulated equations are used to obtain stationary and nonstationary oscillations, respectively; the Floquet theory is followed in the stability analysis. The assumed control force is shown to be effective in reducing the primary and secondary instability regions of oscillations perturbed by internally resonant disturbance components.     

The asymmetrical mixed temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably.     

Asymmetric free vibration of circular plate in contact with incompressible fluid

Vibration of circular plates in contact with fluid has extensive applications in the industry. This paper derives added mass and frequencies for asymmetric free vibration of coupled system including clamped circular plate in contact with incompressible bounded fluid. Considering small oscillations induced by the plate vibration in the incompressible and inviscid fluid, velocity potential function is used to describe the fluid motion. Derivation uses Kirchoff’s thin plate theory. Two approaches are used to derive the free vibration frequency of the system. The solutions include an analytical solution employing Fourier–Bessel series and a variational formulation applied simultaneously on the plate and fluid. Strong correlation is found between free vibration frequencies of the two solutions. Finally the effect of fluid depth on the added mass and free vibration frequencies of the coupled system is investigated.     

The construction of exact solutions in the floating-plate problem

It is shown that it is possible to construct, by an inverse method, exact solutions of the problem of the flexural-gravitational oscillations of a floating elastic plate. The results obtained are used to check the accuracy of numerical solutions of the problem. It is shown that the numerical algorithm given in Ref. [Khabakhpasheva TI. The plane problem of an elastic floating plate. In Continuum Dynamics. Inst. Gidrodinamiki SO Ross Akad Nauk 2000;16:166–9.], predicts, with high accuracy, the values of the amplitudes of the oscillations of the plate and the distributions of the bending moments and hydrodynamic pressure for a wide frequency range.