Strain of a flexible orthotropic cylindrical shell as a function of orthotropy parameters

A procedure is proposed for calculating the stress-strain state of flexible orthotropic cylindrical shells of constant thickness with unsymtnetric load and nonhomogeneous boundary conditions. The system of nonlinear partial differential equations is solved by the method of lines. The system of nonlinear ordinary differential equations is reduced by linearization to a sequence of linear systems. The sequence of linear boundary-value problems is solved by the discrete orthogonalization method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 57–61, 1986.     

Second-approximation shear theory for shallow layered shells and plates

Analysis of a second-approximation refined shear model for shallow layered composite shells and plates with a substantially inhomogeneous structure over the thickness is presented. The tangential displacements and corresponding normal stresses are expressed in the form of a polynomial of the fith degree in the transverse coordinate and contain squared rigidity characteristics. In this way, the accuracy of results and practical coincidence with the 3D solutions is ensured. Based on the refined model, a theory of shallow layered shells is developed. A system of resolving equations of sixteenth power together with appropriate boundary conditions was obtained and solved analytically. It is shown that the area of application of the formed model is extended as compared with the model of the first approximation. The model proposed allows us to examine the stress-strain state of layered composite structures of substantially different thickness and physical-mechanical characteristics of the layers, including the possibility of simulating relatively large shear deformations of rigid layers separated by a low-modulus thin interlayer pliable to transverse shear.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 363–370, May–June, 1998.     

Equations of thermoelasticity for glass-plastic shells

The differential equations of thermoelasticity are derived for nonuniformly heated glass-plastic shells of arbitrary shape subjected to a normal surface load. It is assumed that the elastic properties of the shell material depend upon temperature and are orthotropic.     

Stress State of Multilayer Thin-Walled Elements with Interfacial Defects of Structure

Based on the discrete-structural theory of thin plates and shells, a calculation model for thin-walled elements consisting of a number of rigid anisotropic layers is put forward. It is assumed that the transverse shear and compression stresses are equal on the interfaces. Elastic slippage is allowed over the interfaces between adjacent layers. The solution to the problem is obtained in a geometrically nonlinear statement with account of the influence of transverse shear and compression strains. The stress-strain state of circular two-layer transversely isotropic plates, both without defects and with a local area of adhesion failure at their center, is investigated numerically and experimentally. It is found that the kinematic and static contact conditions on the interfaces of layered thin-walled structural members greatly affect the magnitude of stresses and strains. With the use of three variants of calculation models, in the cases of perfect and weakened contact conditions between layers, the calculation results for circular plates are compared. It is revealed that the variant suggested in this paper adequately reflects the behavior of layered thin-walled structural elements under large deformations. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 761–772, November–December, 2005.     

Hamiltonian approach to studying thin shells of revolution made of composite materials

A method for constructing defining relations of the linear theory of shells of revolution in complex Hamiltonian form has been proposed. Based on the Lagrange variational principle, we have constructed a mathematical model of a multilayer orthotropic shell of revolution. We have obtained explicit expressions for the coefficients and right-hand sides of the Hamiltonian complex system of equations describing the statics of shells of revolution in terms of their rigid characteristics and acting loads. The Hamiltonian resolving system of linear differential equations, formulated in the axially symmetric case, has some specific properties facilitating both analytical studies and numerical procedures of their solution.     

正交各向异性旋转扁壳的非线性振动*

本文提出一种时间模态假设,由此导出描述圆柱正交各向异性薄扁球壳和锥壳非线性轴对称自由振动的非线性耦合的代数和微分特征值方程组.我们求出了该方程组的近似解析解,并获得壳体振动的幅频响应关系的渐近展开式.文中还讨论了壳体的几何及材料参量对其振动性态的影响.     

The theory of orthotropic layered cylindrical shells

The author examines orthotropic layered cylindrical shells for which the moduli of elasticity of the load-carrying layers substantially exceed the shear modulus between layers. This class of structure includes, in particular, shells made of orthotropic glass-reinforced plastic. In this case the classical theory based on the Kirchhoff-Love hypotheses requires refinement; the corresponding equations obtained as a result of approximating the distribution of shear stresses or displacements over the thickness of the shell by a certain known function are presented in [4, 7, 8]. In this paper equations that make it possible to construct the stress distribution over the shell thickness are obtained within the framework of the engineering theory on the basis of the hypothesis of the incompressibility of a normal element.Mekhanika Polimerov, Vol. 4, No. 1, pp. 136–144, 1968     

Long-time stability of orthotropic cylindrical shells under external pressure

The problem is solved using a refined theory of shells that takes shear strains into account. The shell deformations are described by means of the relations for an orthotropic material, it being assumed that creep strains develop only as a result of shear forces. The geometrically linear problem is considered. For the sake of comparison, the long-time critical load is calculated on a Minsk-22 computer using the Kirchhoff-Love and refined models. It is shown that when shears are taken into account, in certain cases the critical load may be reduced by 30%.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 315–320, March-April, 1969.     

Buckling stability of multilayer plates and shells with interfacial structural defects under axial compression

Based on the discrete-structural theory of thin plates and shells, a variant of the equations of buckling stability, containing a parameter of critical loading, is put forward for the thin-walled elements of a layered structure with a weakened interfacial contact. It is assumed that the transverse shear and compression stresses are equal on the interfaces. Elastic slippage is allowed over the interfaces between adjacent layers. The stability equations include the components of geometrically nonlinear moment subcritical buckling conditions for the compressed thin-walled elements. The buckling of two-layer transversely isotropic plates and cylinders under axial compression is investigated numerically and experimentally. It is found that variations in the kinematic and static contact conditions on the interfaces of layered thin-walled structural members greatly affect the magnitude of critical stresses. In solving test problems, a comparative analysis of the results of stability calculations for anisotropic plates and shells is performed with account of both perfect and weakened contacts between adjacent layers. It is found that the model variant suggested adequately reflects the behavior of layered thin-walled structural elements in calculating their buckling stability. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 513–530, July–August, 2007.     

Application of the Newton Method for Calculating the Axisymmetric Thermoelastoplastic State of Flexible Laminar Branched Shells Using the Shear Model

A method for calculating the thermoelastoplastic geometrically nonlinear state of branched laminar shells is elaborated. The method is based on the shear kinematic model for the whole package of layers and on the theory of simple loading processes. The linearization of geometrically nonlinear equations is realized using the Newton method.     

Investigation of the Stress State of Pressure Vessels of Inhomogeneous Structure Made of Composite Materials

The problem on the stress-strain state of layered cylindrical shells with bottoms of intricate shape under the action of internal pressure is considered. The elastic system examined is formed by spiral-circular winding. Two variants of the shell bottom structure are investigated. In the first variant, one spiral layer is installed, which leads to great variations in the bottom thickness along the meridian. In the second one, the bottoms are formed according to the zone-winding scheme. The stress state of the shell constructions of the classes considered is determined by solving boundary-value problems for systems of ordinary differential equations. The solution results for cylindrical shells with elliptic bottoms for the two types of winding are given. It is shown that the zone winding leads to smaller deflections and stresses than the conventional ways of reinforcing shell bottoms.     

Analyzing the harmonic wave propagation in the layered thick tubes

In this study, the propagation of time harmonic waves in prestressed, anisotropic elastic tubes filled with viscous fluid is studied. The fluid is assumed to be incompressible and Newtonian. A two layered hyperelastic anisotropic structural model is used for the compliant arterial wall. The tube is subjected to a static inner pressure P_i and an axial stretch λ. The governing differential equations of tube are obtained in cylindrical coordinates, utilizing the theory of “Superposing small deformations on large initial static deformations”. The analytical solutions of the equations of motion for the fluid have been obtained. Due to variability of the coefficients of the resulting equations for the solid body they are solved numerically. The dispersion relation is obtained as a function of the stretch and material parameters.     

Effect of random irregularities of the creep of reinforced layered plastics

The authors investigate the creep of inhomogeneous materials consisting of a large number of stiff orthotropic elastic layers alternating with layers of linear isotropic viscoelastic material. The elastic layers are assumed to be almost plane; the functions describing the irregularities (curvature) form a random field. The averaged characteristics of the medium are found together with the variation of the averaged displacements and strains in time. An analogous problem was previously considered in [1, 6] on the assumption that the binder layers are elastic. The present paper is based on the equations of [1] and the elastic-viscoelastic correspondence principle [4]. When the correlation scales of the irregularities are small as compared with the dimensions of the body and the characteristic distances over which the averaged parameters of the stress-strain state vary appreciably is considered in detail. A relation is established between the creep functions for simple cases of the state of stress and the parameters characterizing the properties of the components, the properties of the random field of initial irregularities, etc. The development of perturbations with different wave numbers is investigated. The theory is used to describe the creep of reinforced layered plastics.Mekhanika Polimerov, Vol. 2, No. 5, pp. 755–762, 1966     

Soft elastic shells undergoing large deformations

Soft shells made of elastomers and undergoing large deformations under load are studied. The inverse design problem, non-linear under large deformations, is solved. The results obtained are illustrated on a two-parameter shell of revolution fabricated from a two-constant material. The problems of coupling the biaxial and uniaxial zones of the shell and of designing the composite shell are clarified. Amongst the papers dealing with the theory of soft shells and, generally, under small deformations, /1–7/ merit attention.     

Buckling and the initial postbuckling behavior of cylindrical shells made of composites with one plane of symmetry

A method for calculating the buckling stability of layered cylindrical shells made of composite materials with one plane of symmetry of mechanical characteristics is worked out. As a special case, shells made of fibrous materials by winding in directions not coinciding with coordinate axes are considered. An analysis of stability of shells under an axial compression, external pressure, and torsion is carried out. It is shown that, at a great number of layers and appropriate reinforcing angles, the shells can be considered orthotropic. The solution to the problem of the initial postbuckling behavior of shells made of composites with one plane of symmetry is also obtained. It is found that shells of this type can be less sensitive to geometrical imperfections. This fact is important from the practical point of view. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 213–236, March–April, 2007.     

Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

In this study, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow under magneto-thermo-mechanical loadings using modified couple stress theory are investigated. It is assumed that the cylindrical shells with foam core rested in an orthotropic elastic medium and the face sheets are made of composites with temperature-dependent material properties. Also, the Lorentz functions are applied to simulation of magnetic field in the thickness direction of each face sheets. Then, the governing equations of motions are obtained using Hamilton's principle. Moreover, the generalized differential quadrature method is used to discretize the equations of motions and solve them. There are a good agreement between the obtained results from this method and the previous studies. Numerical results are presented to predict the effects of size-dependent length scale parameter, third order shear deformation theory, magnetic intensity, length-to-radius and thickness ratios, Knudsen number, orthotropic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of cylindrical shells. Also, it is demonstrated that the magnetic intensity, temperature changes and carbon nanotubes volume fraction have important effects on the behavior of micro composite sandwich cylindrical shells. So that, increasing the magnetic intensity, volume fraction and Winkler spring constant lead to increase the dimensionless natural frequency and stability of micro shells, while this parameter reduce by increasing the temperature changes. It is noted that sandwich structures conveying fluid flow are used as sensors and actuators in smart devices and aerospace industries. Moreover, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In fact, the present study can be provided a valuable background for more research and further experimental investigation.     

The Effect of Inhomogeneity of Elastic Properties on the Stress State in Composite Cylindrical Panels

The effect of inhomogeneity of elastic properties in the circumferential direction on the distribution of stress and displacement fields in orthotropic cylindrical panels is studied. The mechanical properties of the panels and the load acting on them are constant in the axial direction, which makes it possible to neglect the influence of the curvilinear ends. From the initial relations of a three-dimensional problem of the elasticity theory of inhomogeneous anisotropic bodies, a resolving system of partial differential equations is obtained, whose solution is presented in the form of truncated Fourier series, so that the conditions of free support of the rectilinear ends are satisfied. This allows us to separate the variables and to get a system of ordinary high-order differential equations, which is integrated by a stable numerical method. The problem on the stress-strain state of an orthotropic composite panel with a varying relative volume content of reinforcing elements in the circumferential direction is solved. The effect of the change in the reinforcement density on the stresses and displacements of the panel is studied.     

Initial imperfections and buckling modes of polymeric cylindrical shells under long-term loading

The problem of buckling of polymeric cylindrical shells under long-term loading is considered. The results of compressive and external-pressure tests on shells [10, 11, 12, 17] are discussed and their carrying capacity is estimated [16]. The buckling problem is solved theoretically on the basis of an investigation of the development of axisymmetric modes and their conversion to nonaxisymmetric modes [15]. The equations of the nonlinear theory of viscoelasticity are used as the physical relations. The theoretical model gives qualitative agreement with the experimental behavior of the shell under load.     

The shell theory including transverse shear deformations and thickness change

A variational method for refining the theory of shells based on power series expansion of displacements has been described. The particular case of a cubic approximation for the tangential displacements and a quadratic approximation for the deflections is considered in detail. A constitutive system of differential equations in the canonical form for the axisymmetrical deformation of cyclindrical shells is derived. As an example, axisymmetrical deformations of a cylindrical shell made of an orthotropic composite material are discussed.Martin Luther Universität Halle-Wittenberg, Fachbereich Werkstoffwissenschaften. Germany. Kharkov State Polytechnical University, Department of Dynamics and Strength of Machines. Ukraine. Published in Mekhanika Kompozimykh Materialov, No. 6, pp. 768–780. November–December, 1997.     

Nonlinear nonaxisymmetric deformation of composite shells of revolution

We discuss the results of the determination of the stress and displacement fields in nonaxisymmetrically loaded nonlinear-elastic shells of revolution. The original nonlinear system of equations is linearized in accordance with the method of variation of elastic parameters. The two-dimensional linear boundary-value problem is reduced to a sequence of one-dimensional problems, which are solved using a numerical method. We carry out an analysis of the stress-strain state of a conical shell made of a composite material of granular structure. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 80–83.