Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms

A variant of the refined geometric nonlinear theory is suggested for nonshallow shells with a transversely soft core of medium thickness with regard to modifications of metric characteristics across the core thickness. The kinematic relations for the core are derived by sequential integration of the initial three-dimensional equations of elasticity theory along the transverse coordinate. The equations are preliminarily simplified by the assumption that the tangential stress components are equal to zero. With the example of sandwich plates, it is shown that these equations allow us to investigate synphasic, antiphasic, mixed flexural, and mixed flexural-shear buckling forms of load-bearing layers and the core depending on the precritical stress-strain state. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 95–108, January–February, 2000.     

Transient response of sandwich viscoelastic beams, plates, and shells under impulse loading

The transient response of sandwich beams, plates, and shells with viscoelastic layers under impulse loading is studied using the finite element method. The viscoelastic material behavior is represented by a complex modulus model. An efficient method using the fast Fourier transform is proposed. This method is based on the trigonometric representation of the input signals and the matrix of the transfer functions. The present approach makes it possible to preserve exactly the frequency dependence of the storage and loss moduli of viscoelastic materials. The logarithmic decrements are determined using the steady state vibrations of sandwich structures to characterize their damping properties. Test problems and numerical examples are given to demonstrate the validity and application of the approach suggested in this paper. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 367–378, March–April, 2000.     

Application of the technical theory to sandwich shells with a solid polymeric core

The possibilities of using the technical theory for analyzing cylindrical sandwich shells with a core of low-modulus polymeric material are considered. It is shown to be necessary to make assumptions concerning the distribution of the deformations over the elements of the three-layer section and to take account of the shear strains in the core, the flexural rigidity in the longitudinal direction, and the Poisson ratio in determining the forces and moments. The theoretical conclusions have been experimentally confirmed by static tests on a model.All-Union Structural Engineering Correspondence Institute, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 298–304, March–April, 1973.     

Refined stability theory for sandwich shells with transversally stiff core. 1. Nonlinear equilibrium equations

A refined version of geometrically nonlinear relationships is proposed for the static thermoelastic response of sandwich shells with face sheets made of composite or homogeneous materials and a transversally stiff core. This theory has primary importance for studying mixed forms of buckling of the bearing sheets, which are mainly realized in the zones of a momentary stress-deformed state of the shell on the whole. An iteration procedure was developed for construction of the model. In the first step, assuming that the core is transversally soft, expressions are derived for the components of the displacement vector after integration of the three-dimensional equilibrium equations. In the second step, the tangential stresses are determined assuming a transversally stiff core to obtain the in-plane stresses and highly accurate transverse normal stresses. The proposed model admits a formal changeover to the model of a shell with a transversely soft core.Center for the Study of Dynamics and Stability. A. N. Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 513–524, July–August, 1996.     

Refined stability theory for sandwich shells with transversally stiff core. 3. Simplest one-dimensional problems

The applicability and accuracy of stability equations of the refined theory for sandwich shells with a transversally stiff core proposed in [1] are investigated. The model problem of calculating the critical loads and stress fields in the core at mixed forms of the loss of stability is solved for an infinitely wide sandwich plate with an orthotropic core and composite load-carrying layers subjected to in-plane edge loads. The case of pure bending of the plate is considered in detail. The results obtained by variation of the physical-mechanical parameters are compared with the solutions of the three-dimensional theory for the core [2]. It is shown that the version of the refined theory [1] is more accurate than the other two-dimensional theories.For Pt. 2 see [1].Center for Study of Dynamics and Stability, Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 57–65, January–Feburary, 1998.     

Stress-strain state and stability of composite sandwich shells with a scaling zone between the core and facings

A finite element model is presented for analyzing the strength and stability of sandwich shells of arbitrary configuration with an adhesion failure zone between the core and one of the facings. The model is based on the assumptions that both facings are laminated Timoshenko-type composite shells, only transverse shear stresses in the core and normal stresses in the thickness direction have nonzero values, a free slip in the tangential plane in the adhesion failure zone and unilateral contact along the normal are possible, and the prebuckling state in the stability problem is linear. Biquadratic nine-node approximations for all functions and numerical integration were used. The displacements and rotation angles of the normals toward the facings as well as stresses in the core are taken as global degrees of freedom. The algebraic problem is solved using a special step-by-step procedure of determining the contact area in the scaling zone and employing unilateral constraints for some of the unknowns. Numerical examples are also given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 29, No. 5, pp. 640–652, September–October, 1993.     

Load and frequency interaction in dynamic buckling of soft-core sandwich plates and shells

The present study is concerned with an analysis of load-frequency interaction effects in the dynamic buckling response of soft-core sandwich plates and shells. Based on a higher-order geometrically non-linear sandwich shell theory in conjunction with an extended Galerkin procedure, a mathematical model together with analytical solution for simply supported sandwich plates and shells with with rectangular projection is obtained. By linearization of the non-linear solution with respect to the oscillating parts, the interaction of the natural frequencies with static preloads can be analyzed. The results reveal that the transverse compressibility of soft sandwich cores might have distinct effects on the natural frequencies. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Refined stability theory for sandwich shells with transversally stiff core. 2. Linearized equations of neutral equilibrium

Neutral equilibrium equations of the refined theory of stability for sandwich shells with a transversally stiff core are constructed and used for studying local mixed forms of stability loss (FSL), as well as admitting different variants of simplification, depending on the type of precritical state and realized FSL. The generalized Reissner variational principle used for deriving the stability equations allows us to refine transverse shear stresses in the core as compared to [1]. A method for a highly accurate definition of these stresses is proposed. Namely, after the integration of three-dimensional equilibrium equations over the transverse coordinate, the number of free constants and the number of static conditions to be satisfied are equalized according to the actual stress distribution across the thickness.Science and Technology Center for Study of Dynamics and Strength. Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 786–795, November–December, 1997.     

Nonlinear free dynamic response of laminated compressible cylindrical shell panels

The governing equations for a free dynamic response of a symmetrically laminated composite shell are used to analyze a nonlinear differential panel. The FEM and the Lindstedt–Poincare perturbation technique are invoked to construct a uniform asymptotic expansion of the solution to a nonlinear differential equation ofmotion. A comparison between numerical and finite-element methods for analyzing a symmetrically laminated graphite/epoxy shell panel is performed to show that the nonlinearities are of hardening type and are more repeated for smaller opening angles. It is also shown that large-amplitude motions are dominated by lower modes.     

Transient response of a transversely isotropic multilayered half-space due to a vertical loading

This paper investigates the transient response of a transversely isotropic multilayered half-space under vertical loadings. With the aid of a Laplace–Hankel transform, the global stiffness matrix for a multilayered half-space is acquired by assembling the analytical layer-element of each layer medium. The solutions for the displacements in the time domain are obtained by using the global stiffness matrix equations and a numerical inversion procedure. The accuracy of the proposed method is verified through comparisons with existing solutions for displacements induced by a step and rectangular pulse loading. In addition, selected numerical results for displacements induced by the buried loading are presented to illustrate the effect of transient loading type and material anisotropy on the transient response.     

Stability of composite shells with an elastic core

The stability in axial compression of orthotropic cylindrical shells with an elastic core is investigated with allowance for the following factors: transverse shear strains in the shell material, precritical state of stress of the core, and the moments due to the surface forces exerted on the shell by the core. The numerical results obtained are compared with the results of approximate methods in which the above-mentioned factors are disregarded.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 544–547, May–June, 1976.     

Elastoplastic stability of sandwich plates with a light core

The elastoplastic stability of sandwich plates with a light core is theoretically investigated. Certain specific problems are considered.Tashkent Lenin State University. Translated from Mekhanika Polimerov, No. 3, pp. 568–571, May–June, 1971.     

Parametric vibrations of cylindrical shells with a viscoelastic core

The method of calculating the axisymmetric and nonaxisymmetric parametric vibrations of a cylindrical shell bonded to an elastic core [2] is extended to the case of hollow and solid viscoelastic cores by substituting for the material moduli in the equations of motion of the core integral operators with kernels in the form of an exponential and a sum of exponentials. Expressions are given for the reaction of the viscoelastic core, together with the equation of the boundaries of the spectrum of principal regions of dynamic instability. The effect of relaxation time and the long-term modulus of elasticity of the core on the shape and location of the regions of dynamic instability is analyzed.     

Transient response of inhomogeneous thermoelastic media to a dynamic input

A problem involving a half-space or an infinite space with a cylindrical or spherical cavity subjected to a uniform dynamic input on its surface is studied. The material is elastic and has an inhomogeneity in the direction perpendicular to the boundary surface for the half-space problem and an inhomogeneity in the radial direction for the problem with a cylindrical or spherical cavity. The numerical results, obtained when the dynamic input is a step pressure, indicate that the wave profiles are influenced by two kinds of dispersions: one caused by thermal effects and the other by inhomogeneity. The influence of thermal dispersion is observed more pronouncedly along the initial portion of wave profiles. Compared to the homogeneous case the inhomogeneities in which Lamé's constants increase and decrease with the distance measured from the boundary have respectively amplifying and attenuating effects on both the axial stress and velocity.

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Zusammenfassung Das Problem der gleichmässigen dynamischen Belastung an den Grenzen eines halbunendlichen Körpers oder eines unendlichen Körpers mit einem zylindrischen oder sphärischen Halbraum wurde untersucht.Der Körper ist elastisch und enthält Inhomogenitäten, im halbunendlichen Körper in der Richtung senkrecht zur Grenzebene und im unendlichen Körper mit Hohlräumen in radialer Richtung. Die numerischen Ergebnisse, die für eine Belastung nach einer Schritt-Funktion erzielt wurden, zeigen dass die Profile der Ausbreitungswellen von zwei Streuungen beeinflusst sind: die erste ist eine Folge der thermischen Effekte, die zweite eine solche der Inhomogenitäten. Der Effekt der thermischen Zerstreuung ist im anfänglichen Bereich der Profile stärker. Im Vergleich zum homogenen Fall hat die Zu-beziehungsweise die Abnahme der Lamé'schen Konstanten mit dem Abstand von der Grenze erhöhenden beziehungsweise vermindernden Einfluss sowohl auf die axiale Spannung als auch auf die Geschwindigkeit.

     

On the stability of a compressible Rivlin's cube made of transversely isotropic material

^** Email: kostas.soldatos{at}nottingham.ac.uk This paper considers a unit cube made of a compressible, transverselyisotropic elastic material, with the direction of transverseisotropy being aligned normal to one pair of the cube's faces,and investigates the stability of a dilatation equilibrium stateof that cube, with respect to superposed pure homogeneous deformationswith principal directions parallel to the cube edges. This dilatation,intermediate equilibrium state (state I) of the cube is assumedattainable in two different ways. Accordingly, in what is termedas the ‘principal stability problem’ under investigation,state I is considered to be that of uniform dilatation, whichis attained upon loading normally and uniformly the oppositefaces of the unit cube with certain pairs of equal and oppositelydirected forces having appropriately selected magnitudes. Inwhat is termed as the ‘modified stability problem’under investigation, the same compressible, transversely isotropicunit cube is loaded uniformly by three identical pairs of equaland oppositely directed forces acting normally to its faces,and, hence, it attains in state I the shape of a certain rectangularparallelepiped. The necessary and sufficient conditions forstability of state I of the cube deformation are obtained inthe form of three inequalities, which are found to hold regardlessof whether the intermediate equilibrium state I is that of uniformdilatation or that of the aforementioned rectangular parallelepiped(non-uniform dilatation). These, however, lead to quite differentspecific results and conclusions when applied in connectionwith, first, the principal and, then, the modified stabilityproblem of a unit cube made of a particular type of a transverselyisotropic extension of the Blatz–Ko (isotropic) material.     

Optimization of the thickness of layers of sandwich conical shells

We deal with an optimal control problem for variational inequalities, where the linear operators as well as the convex sets of possible states depend on the control parameter. The existence theorem for optimal control is applied to optimal design problems for sandwich conical shells where a variable thickness of given layers appears as a control variable.     

Closed-form elasticity solution for smart curved sandwich panels with soft core

Due to the rapid development of intelligent engineering structure, the demand for high performance smart structures has been increased in recent years. In this paper, an exact elasticity solution for bending analysis of sandwich panel with generally orthotropic facings and core is developed. Two electro-orthotropic piezoelectric layers in the top and bottom surfaces of sandwich panel as sensor and actuator are considered. From practical point of view, an initially curved sectorial geometry is considered for smart sandwich panel. In this regard, the constitutive relations and governing equations are considered in polar coordinate and coupled Euler–Cauchy Equations are derived. The characteristic equations are determined and closed-form basis functions of displacements and stresses are achieved for various material and geometrical conditions.Furthermore, based on classical and first-order shell theories, the governing equations of smart curved sandwich panels are derived. The governing equations are solved analytically and compared with exact elasticity solution.Several parametric studies are performed on both material and geometrical properties such as angular span, facing and core thickness, and external electrical voltage.     

Stability of a layered cylindrical shell with a transversely isotropic core in axial compression

The problem of the stability of a cylindrical shell with a transversely isotropic core in axial compression is considered. A solution is obtained for the axisymmetric case.