Using a nonlinear theory of sandwich shells with a transversely soft core, approximate analytical solutions were found to the one-dimensional linearized stability problem for a sandwich beam in the ...case of axial compression of its outer layer. The equations applied are based on the introduction into consideration unknown contact forces of interaction of the outer layers with core, of outer layers and core with stiffening elements at all points of their interfaces. A numerical solution of the nonlinear problem formulated was obtained invoking the method of finite sums (integrating matrices) by reducing the original problem in a differential form to a system of integroalgebraic equations.
For a semi-infinite strip beam with fixed section of finite size on one of the front faces, it is shown that, in order to study static and dynamic deformation processes, the corresponding problems of ...mechanics should take into account the transformation of the types of stress–strain state at the interface between the free and fixed sections. Within the classical Kirchhoff–Love model, it is impossible to take into account the deformability of the fixed section. When using the Timoshenko simplest refined shear model, its transformation is possible when the section is fixed only on one of the front faces. Within these models and their combinations, kinematic and force conditions for conjugating the fixed and free sections are formulated. Based on the relations derived, an exact analytical solution was found for the simplest linear problem of the transverse bending of a beam with its cantilever fixation. It is shown that, taking into account the deformability of the fixed section of a finite length is especially important for thin-walled structural elements made of composite materials.
The brief review and analysis devoted to the problem of the degradation processes of materials, including fiber reinforced plastics, is carried out. As a specific object, a unidirectional carbon ...fiber reinforced plastic with ±45
2s
lay-up under cyclic loading was selected. In the theoretical description of this process, it was assumed that the strain includes the elastic, viscoelastic, and viscoplastic components and the strain, formed as a result of the microdamages accumulation in the material. Modeling the degradation process is based on a phenomenological approach; moreover, the kinetic equation for the degradation parameter contains as the arguments both physical time and number of cycles transformed into a continuous variable. When determining the parameters from the experimental results, that are included in the constitutive relations for the strain components, the hypothesis is used, that with a small number of cycles, the strain caused by degradation is much less the strain caused by the rheological properties of the material. In addition, a number of hypotheses were introduced (a generalization of the Kachanov hypothesis, as well as the assumption that the rates of various inelastic strain cannot always be of the same order at all times of loading). It makes possible to simplify the problem of mechanical characteristics identification, as well as to reduce the variety and amount of experiments. The results of experiments and the problems solved for determining the parameters included in the physical relationships proposed were presented, and their good agreement was obtained.
For an analysis of internal and external buckling modes of a monolayer inside or at the periphery of a layered composite, refined geometrically nonlinear equations are constructed. They are based on ...modeling the monolayer as a thin plate interacting with binder layers at the points of boundary surfaces. The binder layer is modeled as a transversely soft foundation. It is assumed the foundations, previously compressed in the transverse direction (the first loading stage), have zero displacements of its external boundary surfaces at the second loading stage, but the contact interaction of the plate with foundations occurs without slippage or delamination. The deformation of the plate at a medium flexure is described by geometrically nonlinear relations of the classical plate theory based on the Kirchhoff–Love hypothesis (the first variant) or the refined Timoshenko model with account of the transverse shear and compression (the second variant). The foundation is described by linearized 3D equations of elasticity theory, which are simplified within the framework of the model of a transversely soft layer. Integrating the linearized equations along the transverse coordinate and satisfying the kinematic joining conditions of the plate with foundations, with account of their initial compression in the thickness direction, a system of 2D geometrically nonlinear equations and appropriate boundary conditions are derived. These equations describe the contact interaction between elements of the deformable system. The relations obtained are simplified for the case of a symmetric stacking sequence.
A series of tests to identify the physical-mechanical properties of a unidirectional carbon-fiber-reinforced composite based on an ELUR-P carbon fibers and an XT-118 epoxy binder were performed. The ...form of the stress–strain diagrams of specimens loaded in tension in the longitudinal, transverse, and ±45° directions and in compression in the longitudinal and ±45° directions were examined. Tensile diagrams were also determined for the XT-118 binder alone. The relation between the tangential shear modulus and shear strains of the composite was highly nonlinear from the very beginning of loading and depended on the loading type. Such a nonlinear response of the carbon-fiber-reinforced composite in shear cannot be the result of plastic deformation of binder, but can be explained only by structural changes caused by the inner buckling instability of the composite at micro- and mesolevels..
Variants of sandwich structural elements in the form of plates and shells with a transversely soft core are analyzed. Their outer, load-carrying layers are reinforced along their outer contour with ...elastic bars to ensure the transfer of loads to the layers during their interaction with other structural elements. For such structures, at small strains and moderate displacements, a refined geometrically nonlinear a theory is constructed that allows one to describe their subcritical deformation and reveal all possible buckling modes (cophasal, antiphasal, mixed flexural, mixed shear-flexural, and arbitrary modes including all the listed ones) of the load-carrying layers and the reinforcing elements (flexural, shear-flexural, and pure shear ones at various subcritical stressstrain states). This theory is based on considering the interaction forces of outer layers with the core and of the layers and core with the reinforcing bars as unknowns. To derive the basic equations of static equilibrium, boundary conditions for the shell and stiffening bars, the conditions of kinematic conjunction of outer layers with the core and of the outer layers and core with the reinforcing bars, and the generalized Lagrange variational principle proposed earlier are utilized. The theory suggested differs from all known variants by a high degree of accuracy and maningfulness at a minimum number of unknown two-dimensional functions for the shells, one-dimensional functions for the reinforcing bars, and one and two-dimensional contact forces of interaction between structural elements.
An improved mathematical model was constructed to describe the geometrically and physically nonlinear deformation of specimens of a fiber-reinforced plastic with a rectangular cross-section. The ...specimens had thin elastic side tabs on the clamping ends in the test fixture to transfer the external load to the specimens in kinematic loadings (by the friction forces that arise between the tabs and rigid elements of the test fixture). The specimens had the form of a three-layered rod. For the tabs, the S. P. Timoshenko shear model taking into account the transverse compression was used. For the middle layer across the thickness, a linear approximation for the transverse displacement and a cubic approximation for the axial displacement were accepted. The kinematic relations and equilibrium equations of the theory were obtained based on geometrically nonlinear relations of elasticity theory in a simplified quadratic approximation. They contained geometrically nonlinear terms that, having the necessary degree of accuracy and content, make it possible to identify the classical bending and nonclassical transverse shear buckling modes of the specimens during their compression tests. For unidirectional fiber-reinforced plastics, the physical nonlinearity was taken into account only in the relationship between the transverse shear stress and the corresponding shear strain. When compressing a ±45 fiber-reinforced plastic, the physical nonlinearity was also taken into account in the relation between the normal stress in the specimen cross-section and the corresponding axial strain.
In the first part of the article 1, a physically and geometrically nonlinear boundary-value problem, that describes the compression of a fiber-reinforced plastic rod with 0
s
layup, was formulated. ...The rod had a rectangular cross-section and thin elastic side tabs. The boundary-value problem was reduced to a system of integral-algebraic equilibrium equations containing Volterra integral operators of the second type. To find its numerical solution, the method of finite sums in the variant of integrating matrices was used. The advantage of the method is the possibility of a strong local thickening of the computational grid in the region of large gradients of solutions. Based on the algorithm constructed, an application software package was developed. The results of computational experiments showed that the test specimens under compression according to one of the most commonly used test schemes predominantly failed when the localized transverse shear stresses reached their ultimate values. Failure was also possible according to the shear buckling mode in stress concentration zones. The identification of such modes was possible by using a proposed refined geometrically and physically nonlinear deformation model built in the quadratic approximation with account of transverse shear strains and transverse compression. To verify the numerical method developed, physical experiments were carried out on unidirectional carbon-fiber-reinforced specimens with 0
s
layup. They showed a good agreement between the theoretical and experimental results of the research.
The problem of forced bending vibrations of a plane rod with a finite-length fastening section under the action of an external transverse force at its free end was solved. The classical ...Kirchhoff–Love model in the classical geometrically nonlinear approximation was used to describe the deformation process of the free part of the rod. The deformation of its fixed part was described by the Timoshenko refined shear model that takes into account transverse strains. The conditions of kinematic conjugation of the free and fixed parts of the rod were formulated. The equations of motion, the corresponding boundary conditions, and the force conditions of conjugation of the rod parts were obtained using the Hamilton–Ostrogradsky variational principle. An exact analytical solution of the problem of forced vibrations of a rod under the action of a harmonic transverse force at the free end of the unfastened part of the rod was deduced. Numerical experiments were carried out to study the resonant vibrations of rods made of unidirectional fiber composite. The effect of a noticeable increase of the amplitudes of transverse vibrations of the ends of the cantilever parts of the rods studied due to transverse contraction of the fixed section was revealed. Taking into account the transverse contraction caused an almost twofold reduction of the maximum transverse shear stresses in the fixed part of the duralumin rod.
New refined geometrically nonlinear equations of motion of composite elongated rod-type plates in plane stress-strain state are derived for the case when the axes of the selected coordinate system ...coincide with the axes of the orthotropy of the plate material. The equations are based on the previously proposed relations of a consistent version of the geometrically nonlinear theory of elasticity under small deformations and on the refined shear model of S.P. Timoshenko. Such equations describe the high-frequency torsional vibrations in elongated rod-type plates that can occur during low-frequency flexural vibrations. By limit transition to the classical model of the theory of rods, the derived equations simplified to a system of equations of a lower order. For the case of the same deformation models for a composite plate with inclined reinforcement, similar equations of motion in a linear approximation are obtained. It is shown that they describe the coupled flexural-torsional vibrations in the case of small displacements.
