V. A. Polyakov

DEFORMATION OF SANDWICH PANELS CYLINDRICALLY BENT BY POINT FORCES. 2. DEVELOPMENT OF PROCEDURE

Our proposed method [1] is extended for the cylindrical bending of an asymmetrical composite panel upon piecewise constant loading and point forces, with and without allowance for transversal stiffness (perfect compliance) in shear or tension/compression along the normal. A set of fundamental functions was obtained for all the cases examined. The properties of functions were studied taking account of the discontinuous nature of the surface loading. The set of functions was normalized for initial values of the variable coordinate. Integral relationships required for analysis were derived and an identical expression of the unit function was represented in terms of the fundamental function set. The boundary problem of a panel supported along the surface of its lower face layer with free ends is reduced to the Cauchy problem. The solution is greatly simplified for a panel symmetrical relative to its mean plane. Asymptotic formulas were obtained for the case of infinite panel length. Relationships are give for the stresses and layer deflections, which permit consideration of all the features of the stress state in addition to simplified calculations for actual panel design.

A refined model for three-point bending of sandwich panels. 2. Transverse stresses

According to the relationships derived in [1], transverse normal and tangential stresses in a sandwich panel have been analyzed. Asymptotic formulas for the stress concentration area in the vicinity of point forces are derived. Analytical estimates of a normal stress at the central and end sections of the panel are deduced. The Saint-Venant effect of the degeneration of a panel of finite length into an infinite strip is studied. For the estimation of the concentration of the transverse tangential stress, the possibility of a superposition of the solution of the slippage problem of the face layers and the classical solution allowing for shear is substantiated. It is shown that the local Reissner-type effects are specified by reducing the concentration of the tangential stress in the face layers along the longitudinal coordinate and transition to the steady tangential stress state in the filler layer. The concentration coefficients of the tangential stress are derived as functions of the dimensional parameters of the panel section.

Bending features of a sandwich panel with asymmetric structure under local loads

The bending characteristics of a composite panel with asymmetric layered structure under local surface loads are obtained. A refined version of the applied theory is developed using the analytical solution of the bending problem of a sandwich plate with arbitrary asymmetric structure under a point load. Local effects are investigated within the limits of a discrete model allowing for the specific character of elastic properties of a soft filler. The advantages of the solution are expressions of bending characteristics — layer curvatures, displacements, and stresses — in a closed form. It is shown that these characteristics can vary several times depending on the asymmetry parameters of the structure. Degeneration peculiarities of the solution, stemming from the slipping of layers or, otherwise, their rigid linking by the Kirchoff—Love hypothesis, as well as from account of the transverse shear and compression of the normal, are examined in line with the degeneration of geometric and physical parameters of the discrete model adopted. The results obtained are illustrated by curves and surfaces for the characteristics studied.

A refined model for three-point bending of sandwich panels. 1. Deflections and bending stresses

Design formulas for the flexural characteristics of sandwich panels under three-point loading by point forces, taking into account local effects, have been derived. Transverse deformation of the normal in the modified model is deduced in terms of the difference between deflections of face layers. It is considered that the rotation of the normal depends also on shear of the filler. The deflections, local curvatures, and bending stresses, dependent on the face-layer thicknesses and transverse characteristics of the filler, are studied. The danger of initial failure caused by the local moment stresses at the central panel section is shown. Comparative estimates refining the conventional designs are established.

Analysis of Compression Stress of a Filler and Bending of a Sandwich Panel under Multipoint Loading

The distribution of transverse stresses in the midlayer of a composite sandwich panel under multipoint loading is investigated. The stresses averaged across the thickness of a soft filler are estimated using a discrete model. Finite expressions for the compression of the filler along the length of the panel are derived by means of superposition of the local effects from the bending of face layers under an infinite system of transverse point forces constant across the panel thickness. The effects of compression and transverse extension of the filler, in the case of a high distribution frequency of these forces, i.e., when the distance between the forces is comparable to the panel thickness, are revealed. Compression of the panel by two systems of forces applied symmetrically or nonsymmetrically to the upper and lower faces is considered. The bending characteristics in the cases of loading with point forces and piecewise distributed loads are compared. The formulas obtained are used to determine the length of a small region on the panel surface for which the local effects from the distributed pressure and the point force are equivalent. The corresponding estimates are obtained in a closed form. The analysis, carried out with varied parameters of the structure, allows us to elucidate the peculiarities of the effect of discontinuous loads on the design characteristics in the local zones, using finite expressions derived by the operational method.

Waves of transverse stresses under a plane impact on the surface of a sandwich panel (acoustic approximation)

The wave process arising in a sandwich panel with a free back surface under the action of a short-term dynamic load on the front surface of the upper layer (plane deformation) is investigated. The calculation procedure for displacements, rates, and stresses under a rectangular short-time pulse, whose duration does not exceed the double time of wave travel within a layer, is based on the representation of the solution to the one-dimensional wave equation in terms of characteristics. The transmission and reflection coefficients of the pressure pulses on the contact surfaces of layers with different physical properties are determined. The expressions for tensile stresses in the panel face layers and filler, which are responsible for the material failure by spalling, are presented. The stresses in relation to the geometry and dynamic parameters of the sandwich structure are analyzed. In the case of a symmetric panel structure, the stress pattern in the midlayer and on its contact boundaries is given, which takes into account the branching and superposition of pulses.

Limiting stresses in spatially reinforced composites with components-having different structural geometries nd elastic properties

A model which is proposed for calculating structural stresses in spatially reinforced composites and an invariant-polynomial criterion for evaluating their limiting values are used to predict the effect of the elastic and strength properties of the components and their relative content on the limiting stress-strain state of composites of different structures. Emphasis is given to tri-orthogonal and 4D cubic structures, in addition to structures with hexagonal and angle-ply fiber reinforcement schemes in the plane and perpendicular to it. The types of composite loading typical of standard tests are examined in separate numerical experiments for shear, tension, compression, and their proportional combination. A computational variant of a criterional estimate of the limiting stresses is substantiated for an anisotropic composite of variable strength. The limiting-stress surface is obtained along with contour maps showing stress isolines as a function of the properties of the components and the geometry of the structure. The maps are suitable for practical use. The cases of maximum resistance to shear, tension, compression, and combination loading of 3D and 4D composites are compared to the analogous cases for two-dimensional structures.

Structural model analysis of ultimate stresses in spatially reinforced 4D composites

Conclusions1.The approximate structural model for calculating the stressed state of 4D composites and the tensor-polynomial criterion, accounting for the anisotropy and matrix of unlike strengths upon tension-compression of a material permits us to elucidate the change in the ultimate stresses of the composite relative to the reinforcement angles and load application direction.2.Variation of the fiber packing directions in two orthogonal planes was used to show that isotropy in the composite strength determined by the ultimate stresses upon failure of its weakest link cannot be achieved. The least deviation from such isotropy is obtained at the spatial reinforcement angle θ = 54.7 °. For this angle, we find a very low percentage of the ultimate tensile stress (about 7%), while there is a significant enhancement of the ultimate shear stress (to 220% in comparison with the corresponding indices of the unidirectional material along the fiber packing direction).3.In the case of symmetrical spatial arrangement of straight fibers at an angle θ = π/4 to the direction of the tensile force, the value of δx* is twice that for planar fiber packing (±ϕ) when ϕ = θ. There is no significant difference in the θx* values for these structures in the case of low reinforcement angles. When θ > 65 °, the values of the ultimate stress θx* are constant and minimal when evaluated according to the polynomial criterion, though they are twice as large when evaluated according to the criterion for greatest deformation transverse to the fibers.4.The ultimate shear stress between the two planes of a spatially reinforced composite with arbitrary fiber directions parallel to these planes is independent of the reinforcement angles and close in magnitude to the shear strength along the fibers of a unidirectional composite with identical properties of the fiber and matrix and same reinforcement coefficient.

Evaluation of the tensile and shear strengths of composites formed by a system of three fibers

1, Composites formed by a system of three fibers have a high resistance to shear, transverse rupture, and low-rate impact loading [1-3], which makes them promising materials for use in different areas of technology. However, the effective use of such composites in the load-bearing elements of structures is impossible without reliable methods of evaluating their load-carrying capacity. At present, there is no information on this subject in the literature. Thus, one of the main goals in this study is to develop an approach that will make it possible to use the elastic and strength properties of the initial components and the structural parameters of the composite to estimate its load-carrying capacity for a given type of load. The essence of the proposed method is as follows. The structure of the composite is represented as a set of individual structural elements [3] forming a unidirectional monotropic material. The conditions adopted to characterize the deformation of the elements are such as to rigidly connect them with one another. This then allows us to use an assigned strain tensor to determine the stresses in the elements. The relationship between the mean stresses of the composite aij and the stresses of the structural elements aij ~ is determined by a fourth-rank tensor. In the case when the direction of all of the elements is the same or the material is isotropic, this tensor degenerates into a transformation tensor with rotation of the coordinate axes. The limiting stresses of the composite a* i can be expressed through the limiting stresses of a structural element Rj, the elastic properties of the element Bij ~ and the composite Bij, and the parameter characterizing the orientation of the fibers of the element 0:

Evaluation of stresses in the edge effect zone in tension of composite laminates

i. A specific feature in the loading of rectangular specimens made of laminate materials is the occurrence of high stress peaks along the line of intersection of the lateral side with the interlaminar plane. It is accepted practice in the mechanics of composite laminates to call edge effects the effects connected with large stress gradients localized near . this line [i]. Among the edge effects are premature delamination of material before the load bearing capacity of the specimen is exhausted [i] and the characteristic broadening of the lateral side (swelling) under tensile loading [2]. The onset of delamination was noted at certain levels of tensile strain [3, 4] depending on the nature of the stacking. A change of tensile strength in dependence on the stacking sequence was noted in [i, 5].