Estimation of the error of approximation of the analytic solution of problems for nonlinear viscoelastic materials

Estimates are given for the error of approximation of the solution of problems of nonlinear viscoelastic media, for which the nonlinear elastic solution depends analytically on the material characteristics, the body and surface forces, and on the boundary displacements. From the integral estimates particular estimates are derived for viscoelastic materials in cases when the nonlinear elastic solution is majorized by a geometric progression or the sum of products of geometric progressions.Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, No. 5, pp. 827–834, September–October, 1971.     

Methods of finite element analysis of arbitrary buckling forms of sandwich plates and shells

Two statements of the problem of arbitrary buckling forms (BFs) (including synphasic, antiphasic, mixed flexural, flexural-shear, and shear forms in the tangential directions) of general-form sandwich shells and two schemes of its solution by the FEM are given. The first of the schemes is based on the use of refined linear equations for determination of the precritical stress-strain state and linearized equations of neutral equilibrium with all parametric addends necessary to determine the critical loads and reveal the possible BFs. The second one uses the general geometrically nonlinear relations of elasticity theory for investigation of the whole deformation process up to buckling in terms of a modified incremental (stepwise) statement of the problem. Examples of solution of particular problems are given.Center for Study of Dynamics and Stability, Tupolev Kazan' State Technical University, Kazan', Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 473–486, July–August, 2000.     

Local buckling of compressed elements of layered viscoelastic material

The problem of local surface buckling of compressed elements of layered material is solved using the equations of the theory of multilayer plates [1, 2, 5]. The local buckling deformations are described by means of the solutions of differential-difference equations damping into the interior of the medium. The effect of compressive forces, buckling wavelength, and the elastic and viscoelastic constants of the material on the behavior of the deformations and on the thickness of the layer to which significant buckling extends is investigated. Numerical estimates are presented for the buckling parameters of compressed laminated-plastic elements.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 816–821, September–October, 1968.     

Effect of the stiffness of the elastic core on the buckling mode and the critical load for glass-reinforced plastic cylindrical shells in axial compression

The effect of the stiffness of the core, treated as a three-dimensional elastic body, on the buckling mode and the critical load is investigated for orthotropic cylindrical shells in axial compression. Parametric equations that permit the easy determination of the critical loads are obtained together with an expression for the stiffness parameter at which the buckling mode changes. The effect on the critical load of a central opening in the core is considered.Translated from Mekhanika Polimerov, No. 5, pp. 931–939, September–October, 1971.     

Polarized Hessian covariant: Contribution to pattern formation in the Föppl–von Kármán shell equations

We analyze the structure of the Föppl–von Kármán shell equations of linear elastic shell theory using surface geometry and classical invariant theory. This equation describes the buckling of a thin shell subjected to a compressive load. In particular, we analyze the role of polarized Hessian covariant, also known as second transvectant, in linear elastic shell theory and its connection to minimal surfaces. We show how the terms of the Föppl–von Kármán equations related to in-plane stretching can be linearized using the hodograph transform and relate this result to the integrability of the classical membrane equations. Finally, we study the effect of the nonlinear second transvectant term in the Föppl–von Kármán equations on the buckling configurations of cylinders.     

Determination of the mechanical indices of composite materials by testing multilayered samples

The mechanical indices of materials such as composites reinforced by braids and ribbons are difficult to determine by standard methods. Moreover, the indices of such materials may be greatly altered when they are converted into various structures. It has been suggested that these indices be determined by analyzing the structure testing data. The determination of linear and nonlinear elastic, plastic, and viscoelastic parameters of reinforced materials is discussed. A very simple structure, namely, cylindrical shells made by symmetric winding, is studied as an example. Equilibrium conditions are used to obtain resolving equations relative to the mechanical indices. Convergence of iteration methods is examined. The effect of the scatter of experimental data on the calculated results is analyzed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 607–615, September–October, 1995.     

在受轴向周期扰动作用下双壁碳纳米管动力屈曲的研究

对双壁碳纳米管受轴向周期扰动的动力响应进行了研究．采用连续体模型研究双壁碳纳米管的动力屈曲问题,考虑了壁间van der Waals力和周围弹性介质对轴向动力屈曲的影响．给出了受轴向周期扰动的屈曲模型及临界应变和临界频率．发现双壁碳纳米管由于壁间van der Waals力的作用较单壁碳纳米管具有较低的临界应变．van der Waals力和周围弹性介质将影响双壁碳纳米管不稳定区,van der Waals力使受轴向周期性扰动的双壁碳纳米管的临界频率增大,周围弹性介质对双壁碳纳米管的临界频率影响不大．     

Stability of hydrostatic-pressured FGM thick rings with material nonlinearity

Buckling behaviors of elastoplastic ceramic/metallic functionally graded material (FGM) rings are investigated by using the first order shear deformation theory. The hydrostatic-pressured rings are assumed to be in both the plane-stress case and the plane-strain case, which lead respectively to a uniaxial and a biaxial elastoplastic stress states in prebuckling stage. A uniform strain hypothesis helps to deal with the elastoplastic stress states. By introducing in the graded material properties, the constitutive model of FGMs is formulated under the framework of J₂ deformation theory. By considering the kinetic relations of von-Kárman type and employing the principle of virtual displacement, the equilibrium equations and the buckling governing equations of FGM circular rings are formulated, and the analytical solution of the anisotropic rings is obtained. Finally, the elastoplastic buckling problem is numerically solved through a semi-analytical method, which is proposed to seek the real circumferential strain of FGM rings at the buckling point and determinate the elastoplastic buckling critical hydrostatic pressure. The effects of the inhomogeneous and geometrical parameters on the buckling critical load and the position of the elastoplastic interface are discussed. Results show that, in both the plane-stress and the plane-strain cases, the elastoplastic critical loads are generally lower than their elastic counterparts due to material flow, and the plane-strain critical load is generally larger than the plane-stress one. The elastoplastic critical load does not always decrease monotonously with the increase of the inhomogeneous parameters, which is quite different from their elastic counterparts.     

New mathematical model for nonuniform deformation of composites

A new mathematical model is suggested for nonuniform deformation of composite materials valid for arbitrary external load gradients. As a basis stochastic equations are suggested for the statics of an elastic microinhomogeneous two-component material with volumetric forces differing from zero. Expressions are obtained for all coefficients in terms of elastic constants of the components and geometric parameters of the structure.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October, 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 310–318, May–June, 1995.     

Mechanical stability of functionally graded stiffened cylindrical shells

This paper is concerned with the elastic buckling of stiffened cylindrical shells by rings and stringers made of functionally graded materials subjected to axial compression loading. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations, the equilibrium and stability equations are derived using the Sander’s assumption. Resulting equations are employed to obtain the closed-form solution for the critical buckling loads. The results show that the inhomogeneity parameter and geometry of shell significantly affect the critical buckling loads. The analytical results are compared and validated using the finite element method.     

Limiting state of a multilayered nonlinearly elastic long cylindrical shell under the action of nonuniform external pressure

The buckling of a long multilayered nonlinearly elastic shell made of different materials and subject to the action of external pressure is investigated. The load is not hydrostatic and greatly varies in value and direction. Neglecting the effect of end fastening of the shell, the problem is reduced to an analysis of the loss of load-carrying ability of a ring of unit width separated from the shell. The solution is based on a variational method of mixed type formulated for heterogeneous nonlinearly elastic bodies, taking into account the geometrical nonlinearity, in a combination with the Rayleigh–Ritz method. The initial analysis is reduced to solving the Cauchy problem for a nonlinear ordinary differential equation resolved for the derivative. Numerically, using the Runge–Kutta method, the effect of the number of layers and of the parameter of nonuniformity of the external pressure on the critical buckling force is revealed. The urgency and importance of the problem are connected with the research of reserves in the saving of materials with a simultaneous possibility of increasing the load-carrying ability of a structure.     

Generalized Kojima–Functions and Lipschitz Stability of Critical Points

In this paper we consider systems of equations which are defined by nonsmooth functions of a special structure. Functions of this type are adapted from Kojima's form of the Karush–Kuhn–Tucker conditions for C²—optimization problems. We shall show that such systems often represent conditions for critical points of variational problems (nonlinear programs, complementarity problems, generalized equations, equilibrium problems and others). Our main purpose is to point out how different concepts of generalized derivatives lead to characterizations of different Lipschitz properties of the critical point or the stationary solution set maps.     

Calculation of deformative and thermal properties of multiphase composite materials filled with composite or hollow spherical inclusions by the effective medium method

A theoretical investigation was carried out to examine the possibilities of a structural approach to prediction of elastic constants, creep functions, and thermal properties of multiphase polymer composite materials filled with composite or hollow spherical Inclusions of several types. The problem of determining effective properties of the composite was solved by generalizing the effective medium method, a variant of the self-consistent method, for the case of a four-phase kernel-shell-matrix-equivalent homogeneous medium model. Exact analytical expressions for the bulk modulus thermal expansion coefficient, thermal conductivity coefficient, and specific heat were obtained. The solution for the shear modulus is given in the form of a nonlinear equation whose coefficients are the solution of a system of 12 linear equations.To be presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 462–472, July–August, 1995.     

The method of averaging in nonlinear dynamical problems occurring in the theory of viscoelasticity

The degenerate vibrations of an oscillator with slightly nonlinear elastic properties and a memory characteristic have been investigated with the aid of the averaging technique proposed in [2]. The vibrations have been studied in the case of fractional resonance and the principal resonance. A basis for the principle of averaging a nonlinear multidimensional system over an infinite time interval has been given.V. I. Lenin Tashkent State University. Translated from Mekhanika Polimerov, No. 1, pp. 39–44, January–February, 1972.     

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate,a viscoelastic (elastic) bond layer,and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 771–788, November–December, 2007.     

弹性圆底扁球壳在边缘均布力矩作用下的非线性稳定问题

本文对边缘简单支承在边缘均布力矩作用下的弹性圆底扁薄球壳的非线性问题进行了研究.用我们提出的修正迭代法求出决定上下临界载荷的二次和三次近似方程.结果画成曲线以供工程应用.并将此结果和胡海昌的结果进行了比较,我们还研究了临界点,即上下临界载荷相重点附近的情况,并指出二次近似的适用范围.文末还附带得出本问题的特殊情况:相同边缘支承和载荷下的圆平板大挠度问题的刚度设计公式及p=0.3的相应曲线,并与黄择言用摄动法得到的结果进行了比较.     

Asymptotic methods in the nonlinear theory of viscoelasticity

Dynamic and quasistatic problems of the nonlinear theory of viscoelasticity are described by nonlinear integrodifferential and integral equations. Methods of averaging various classes of nonlinear integrodifferential and integral equations are described and asymptotic expansions of the solutions of these equations are given.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 221–229, March–April, 1974.     

The limiting state of a multilayer nonlinearly elastic eccentric ring

The limiting state of a multilayer eccentric ring made of a nonlinearly elastic material and subjected to a uniform external pressure is investigated. The topicality and importance of the problem are connected with the search for reserves of savings in materials, with a simultaneous in crease in the load-carrying capacity of structures. Since rings often must have walls of varying thickness, their critical buckling force is determined as a function of a parameter considering this fact. In solving the problem, the geometric nonlinearity is also taken into account. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 761–770, November–December, 2007.     

Delamination buckling of a rectangular orthotropic composite plate containing a band crack

The delamination buckling problem for a rectangular plate made of an orthotropic composite material is studied. The plate contains a band crack whose faces have an initial infinitesimal imperfection. The subsequent development of this imperfection due to an external compressive load acting along the crack is studied through the use of the three-dimensional geometrically nonlinear field equations of elasticity theory for anisotropic bodies. A criterion of initial imperfection is used in determining the critical forces. The corresponding boundary-value problems are solved by employing the boundary-form perturbation technique and the FEM. Numerical results for the critical force are presented.     

Stability of transversely isotropic annular plates with low shear resistance

The stability differential equations of a cylindrically orthotropic circular plate are obtained on the assumption of an axisymmetric buckling mode with allowance for transverse shears. These equations are solved for the case of a transversely isotropic material when the inner and outer edges of the plate are identically loaded by uniformly distributed radial forces. The transcendental equations for the critical load parameter are constructed for various edge conditions. The dependence of this parameter on the boundary conditions and the relative thickness of the plate, Poisson's ratio, and the ratio of the Young's and shear moduli of the material are investigated. Certain conclusions are reached concerning the design of reinforced-plastic plates.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 872–880, September–October, 1969.