Three-layer Shell Research Articles

Analysis of temperature-dependent layered shells subjected to thermomechanical loading

An analytical method is proposed for the analysis of simply supported temperature-dependent layered shells subjected to thermomechanical loading. The method is developed by introducing a sublayer model: assuming each layer of the shell comprises numerous fictitious sublayers, and the material properties of each sublayer are uniform. Based on this model, an iterative procedure is used to predict the radial temperature distribution; the transfer-matrix technique is used to predict the displacement and stress distributions based on the three-dimensional thermoelasticity theory. Some examples are provided to study the effects of surface temperature and material properties on the thermomechanical performances of a three-layer shell.

Methodology for the Design of Combustible Structures of Separating Launch Vehicle Parts

The presence of parts separating from the launch vehicle (LV) during of the active part of the flight, such as the tail and interstage compartments, and payload fairing calls for the solution of the problem associated with the allocation of significant impact areas. This is a common problem in all spaceships, including the cases when recoverable and reusable first stages are used, for example, Falcon 9 launch vehicles. This paper considers one of the possible solutions to this problem, based on the combustion of separating parts during the descent trajectory flight. The study subject is a combustible demonstrator (CD), a small element of the three-layer shell of the payload fairing. The layers of the shell are the sheathings and the filler charge placed between them. The filler charge is a pyrotechnic composition (PC) initiated by the command, and the sheathing is a combustible polymer composite material (PCM). The material for the structures of the separating parts is produced as follows. First, a CD is created due to the chosen PCs and sheathings. Then, the required strength of the CD is ensured. Finally, the demonstrator structure is guaranteed to burn up or be fragmented finely for the given time.

Efficient Approximate Analytic Solution for the Problem of Stability of a Three-Layer Conic Shell Under Combined Loading

We propose a solution of the linear problem of behavior of an elastic three-layer conic shell under the combined action of several external force factors (uniform pressure, axial compression, and torque), which may lead to the loss of stability of the analyzed structure. The problem is reduced to the integration of the resolving singular sixth-order ordinary differential equation with variable coefficients. The solutions obtained by the methods of phase integrals (WKB method), hybrid WKB–Galerkin method, and finite-difference method are compared. We demonstrate the advantages of the asymptotic hybrid approach to the solution of the analyzed class of equations. The rational relationships between the thicknesses and elasticity moduli of layers of the shell required for attainment of the highest stability of the investigated structure under given external loads are analyzed. The problem of construction of the boundary surfaces separating the regions of stability and instability of the shell is discussed. The efficiency of three-layer shells as load-bearing elements aimed at improving stability is confirmed.

Eigenwaves propagation in three-layer cylindrical viscoelastic shells with a filler non-uniform in thickness

The paper provides the statement of the problem, the development of a calculation method and an algorithm to solve the problems of propagation and absorption of non-axisymmetric natural waves in layered dissipative-inhomogeneous viscoelastic three-layer cylindrical bodies. A detailed analysis of well-known publications devoted to this problem is given. The paper posed the problem of non-axisymmetric natural wave propagation in three-layer cylindrical shells with fillers, taking into account the theory of hereditary viscoelasticity. Dispersion equations were obtained and the dependence of the phase velocity on the wavenumber for the three-layer shells was constructed. A calculation method was developed based on the Muller, Gauss, and orthogonal sweep methods. The program (in C++ language) was compiled based on the developed algorithm. Numerical results were obtained for the complex phase velocity depending on various wavenumbers and parameters of an axisymmetric cylindrical mechanical system for the Kirchhoff-Love and Timoshenko hypotheses. The change in the real and imaginary parts of the complex phase velocity under various parameters of the system was investigated for structurally homogeneous and inhomogeneous mechanical systems. The results obtained by the Kirchhoff-Love and Timoshenko hypotheses are in good agreement but differ in the wave absorbing abilities. It was found that a decrease in filler thickness at a rigid inner shell leads to a rapid increase in the real and imaginary parts of the vibration frequencies.

Analysis of the dynamic characteristics of three-layer shells with discrete filler

Analysis of the dynamic characteristics of three-layer shells with discrete filler

Mathematical modelling of torsional vibrations of the three-layer cylindrical viscoelastic shell

The paper considers a circular cylindrical three-layer shell of arbitrary thickness from a viscoelastic material. It is believed that it consists of two outermost bearing layers and a middle layer between them, the materials of which are generally different. The problem of unsteady torsional vibrations of such a shell with rigid contact between the layers is formulated. Proceeding from the assumption that there is a rigid contact between the layers, the dynamic and kinematic contact conditions of the problem are formulated. On the basis of exact solutions in transformations of the three-dimensional problem of the linear theory of viscoelasticity for a circular cylindrical three-layer shell, a mathematical model of its unsteady torsional vibrations has been developed. The proposed model includes the derivation of the general equations of torsional vibrations of the shell with respect to two auxiliary functions, which are the main parts of the torsional displacement of the points of some intermediate surface of the middle layer of the shell. Along with the equations, an algorithm for calculating was created that allows, based on the results of solving the equations of vibration, to unambiguously determine the stress-strain state of the shell and its layers in their arbitrary sections.

Numerical modeling of a double-walled spherical reservoir

Extensive and important class of multilayer shell structures is three-layer structures. In a three-layer structure, a rigid filler plays an important role, due to which the bearing layers are spaced that gives the layer stack high rigidity and durability with a relatively low weight. By combining the thicknesses of the bearing layers and the filler, the desired properties of a three-layer shell structure can be achieved. Compared with traditional single-walled, three-layer construction has increased rigidity and durability, which allows reducing the thickness and weight of the shells.
 In order to reduce the metal content of the spherical reservoir for storing liquefied gases, this work considers the design of a double-walled reservoir, in which the inter-wall space is filled with reinforced polyurethane. Numerical modeling made it possible to determine the parameters of the stress-strain state of the structure with an error of no more than 5 %. It has been established on the example of a reservoir with a volume of 4000 m3 that the spatial structure of the spherical reservoir wall can reduce the metal content up to 19 %. Field of application for the research results is the assessment of the stress-strain state of spherical reservoirs at their designing.
 Method for building the structure of a double-walled spherical reservoir in the SCAD software has been developed, which allows calculating the stress-strain state (SSS) by the finite element method.
 Numerical model of a double-walled spherical reservoir has been developed. It was found that to obtain calculation results with an error of P ≤ 5 % the size of the final element should not exceed 300×300×δ mm.
 Design of a double-walled spherical reservoir was investigated. Design parameters have been established to ensure the operational reliability of the structure with a decrease in metal content in comparison with a single-wall reservoir by 19 %.

Stress-strain state of a three-layer cylindrical shell under internal axisymmetric pulse load

The problem of dynamic deformation of a three-layer cylindrical shell under non-stationary loads in the case of rigid clamping of the shell ends is considered. The article presents the results of assessing the stress-strain state of a three-layer cylindrical shell, taking into account its structural feature, the ratio of the sheathing thickness and the physical and mechanical characteristics of a one-piece polymer filler. Calculations were performed by software complex Nastran. The values of displacements and stresses were calculated by the algorithm of direct transient dynamic process. The step duration of the time interval was 0.0000025 s, and the total number of steps was 200. The choice of the type of three-dimensional finite element was due to obtaining more detailed and accurate calculation results. The finite element model included 19000 three-dimensional finite elements and numbered 20800 nodes.
 The influence of geometrical parameters of shell layers with different physical and mechanical properties of one-piece filler on the stress-strain state under axisymmetric internal impulse load is investigated. Numerical results on the dynamics of the three-layer structure, obtained by the finite element method, allow to characterize the stress-strain state of the three-layer elastic structure of the cylindrical type at any time in the studied time interval. Optimization of the shell design is recommended. Changing the ratio of the thickness of the internal and external shells of the shell significantly affects the stress-strain state of the shell and its performance. Increasing the thickness of the internal layer of the shell significantly contributes to the increase of the latter. Comparison of the given results with materials of other similar researches and positions, testify to objectivity of the made approach.

Design and manufacture of 3D technology for printing a three-layer spherical shell with discrete filler

Three-layer construction with a discrete filler structure, which are distinguished by high performance indicators, are investigated. An algorithm for constructing a mathematical model of this construction and the process of manufacturing a sample using 3D printing are presented. Keywords three-layer construction, spherical shell, discrete filler, 3D printing, model, sample. aa-zotov@inbox.ru

Calculation of three-layer rotation shells for an axisymmetric load in conditions of nonlinear creep

The article is devoted to the calculation of three-layer shells of rotation with a light filler under the creep conditions. Unlike previous works of the authors, the hypotheses of the theory of shallow shells are not used. General geometric and physical equations are presented, as well as a system of resolving equations for axisymmetrically loaded structures. An example of calculating a spherical dome is given. The features of changes in the stress-strain state of the considered structure during creep are revealed.

Dynamic instability characteristics of advanced grid stiffened conical shell with laminated composite skins

Dynamical instability characteristics of sandwich truncated conical shell are investigated. The three-layered shell is composed of advanced grid stiffened core and laminated composite skins. The core maybe made of three different fiber paths. The conical shell with simply-supported ends is subjected to two different types of time-dependent axial compressions. The equations of motion and compatibility are derived by considering Kirchhoff-Love assumptions and von Karman relations. The solution procedure is divided to two steps. First, the terms consisting of spatial derivatives are eliminated by applying a stress function and following the Galerkin method. Second, the terms with temporal derivatives are solved with Runge-Kutta method. A new condition of Budiansky-Roth criterion is offered for detection of dynamic buckling load. The effects of different fiber paths and lay-up configurations are investigated.

Impact behavior of a stiffened shell structure with optimized GFRP corrugated sandwich panel skins

The aim of this work is to provide a virtual testing for the three-layered shells made of the optimized corrugated core sandwich panels. Starting from the unit cells level, we define their geometry found from the preliminary optimization and ensured the requirements of strength, stability and effective thermal conductivity. Effective stiffness properties of the core are evaluated then by using asymptotic homogenization method. The mechanical performance of the non-axisymmetric stiffened shell made of the considered sandwiches is studied numerically for the low-velocity impact. Solid-shell model with homogenized properties of the core is used for the skin material of the shell. Multi-scale approach is involved in the evaluation of the global and local stress state of the panels using stress concentration tensor estimated during asymptotic homogenization. As result, we check the robustness of the design approaches for the unit cells and its validity in the case of the complex loading conditions in the three-dimensional structure.

Phylogenetic constraint and phenotypic plasticity in the shell microstructure of vent and seep pectinodontid limpets

Pectinodontid limpets of the genus Bathyacmaea are endemic to hot vents and cold seeps and exhibit greatly variable shell and radular macro-morphologies, rendering reliable species-level identification challenging. Here, we analyzed shell microstructures of western Pacific vent/seep Bathyacmaea limpets using scanning electron microscopy and Raman spectrophotometry to test its usefulness in providing phylogenetic signals. Bathyacmaea shells comprised of two forms of calcitic microstructure including irregular spherulitic prismatic type-A (ISP type-A) and semi-foliated (SF), as well as the aragonitic crossed lamellar (CL) microstructure. Despite marked differences in macroscopic shell morphologies once leading them to be classified into different species or even genera, six morphotypes of Bathyacmaea nipponica from different chemosynthetic localities and substrates shared an outermost ISP-A layer and alternating layers of SF and CL structures in their outer and inner shell layers. A genetically divergent lineage recovered from the South Chamorro Seamount, however, differed in having a simple three-layered shell composition consisting of ISP-A, SF, and CL structures, in that order, from the outside, and an unusually thin inner shell layer consisting of only CL structure. Moreover, the ratio of aragonite and calcite varied with habitat conditions, with calcite dominating in vents and aragonite dominating in seeps. These results suggest that the shell microstructure of pectinodontids is under phylogenetic constraints and provides useful taxonomic signals, while the mineral polymorphism in aragonite/calcite ratio varies according to environmental factors. Furthermore, microstructures of two ‘species’ from Cretaceous seeps confirmed the same patterns in fossil lineages.

Effective Model of Load-Bearing Layers for Layer-by-Layer Analysis of the Stress-Strain State of Three-Layer Cylindrical Irregular Shells of Revolution

The use of effective approximations that increase the rate of convergence of numerical results when constructing a finite element model of bearing layers for a more accurate layer-by-layer analysis of the stress-strain state of three-layer irregular cylindrical shells is considered. It is believed that the carrier layers are sufficiently thin and rigid, and two-dimensional finite elements of natural curvature, constructed on the basis of the classical theory of moment shells, are used to simulate the stress-strain state. It is assumed that the aggregate layer can be modeled in thickness by the required number of three-dimensional finite elements, which allows one to take into account the change in geometric and physical-mechanical characteristics, as well as the parameters of the stress-strain state, not only along the meridional and circumferential coordinates, but also along the thickness of the shell and the aggregate layer. The finite element approximations considered allow one to reduce the order of systems of equations, i.e. to reduce the dimensionality of the problems being solved in comparison with the traditionally used approximations, which is especially important for layer-by-layer analysis of layered-heterogeneous structures. The high convergence rate of the numerical results obtained using the considered finite element model of the bearing layers is confirmed by a comparison with other known finite elements.

Model for Layer-by-Layer Analysis of the Stress-Strain State of Three-Layer Irregular Shells of Revolution of Double Curvature

Based on the layer-by-layer analysis approach, the construction of a model of two types of finite elements (FE) of natural curvature (two-dimensional FE of the momentary bearing layers and three-dimensional FE of the filler) is considered for refined investigation of the stress-strain state (SSS) in layers of three-layer generally irregular shells of revolution of double curvature. The presented model-building algorithm allows one to avoid errors due to the discontinuity of generalized displacements on the docking surfaces of the FE of the bearing layers and the aggregate, to analyze the SSS for all three coordinates to which the shell is assigned, and to obtain a solution in an updated statement for various shell shapes and boundary conditions of the layers, as well as violations of their continuity. At the same time, the aggregate layer can be modeled in thickness by the required number of finite elements, which allows one to take into account the change in geometric characteristics, physical and mechanical properties, and SSS parameters not only in the meridional and circumferential coordinates, but also in the thickness of the shell and aggregate layer. The proposed approach significantly expands the range of problems solved in the refined formulation and allows us to carry out the calculation with a high degree of accuracy and detail. As an example, the calculation of the SSS of a three-layer animated casing with cut-outs is given. The influence of the size of the cut-outs on the stress – strain state in the layers of the three-layer shell of rotation of double curvature is studied.

Математическая модель нелинейного деформирования трехслойных оболочек

The article presents the study results of geometrically nonlinear deformation of elastic shells of arbitrary type with consideration of transverse shifts. There is constructed a new mathematical model of nonlinear deformation of thin-walled elastic isotropic three-layer shells. Each layer of the shell is made of different materials, but with similar shear modules. The thickness of the layers can be different. Averaging of all three layers becomes possible, and deformation of a three-layer shell as a single-layer shell with the given characteristics of the modulus of elasticity and the Poisson's ratio can be considered.

Dynamic Design of Compound Shell Structures of Revolution Under Nonstationary Loads*

A technique for solving dynamic problems of the behavior of compound shell structures (solving the equations of motion subject to appropriate boundary and initial conditions) is developed. The structurally orthotropic model of a three-layer shell structure with a cellular core is used, for which the integral values of the elastic modulus and Poisson’s ratios are determined experimentally. Numerical algorithms are developed, and the corresponding problems of mathematical theory of elasticity are solved. The numerical results are analyzed.

Investigation of the Influence of the Cutout Dimensions on the Stress-strain State of Three-layer Shells with Load-bearing Layers of Composite Materials

The description of the model for investigation of the influence of rectangular dimensions in terms of cutouts on the stress-strain state (SSS) in the layers of three-layer conical shells is given. The finite element model is based on the layer-by-layer analysis approach and allows to perform calculations of SSS in shell layers in a wide range of changes in the physical and mechanical characteristics of the layers in the presence of cut-outs. The equations of the classical theory of shells at reception of effective approximations (leading to high speed of convergence of received results) and construction of finite-element model of moment bearing layers are used. The approach with application of the elasticity theory relations for a three-dimensional body in curvilinear coordinates and obtained approximations for construction of a finite element model of the filler layer is described. Influence of the cut-out mortar angle on the parameters of the three-layer shell with load-bearing layers of composite materials is investigated.

Improved lithium and sodium ion storage properties of WS2 anode with three-layer shell structure

A three-layer shell structured tungsten disulfide-based material for high capacity anode is prepared by a hydrothermal method. The three layers consist of a stable porous carbon shell (PCS), a tungsten disulfide nanosheet (WS2) and a highly conductive nitrogen-doped graphene (NG). While the nanostructured WS2 and NG ensured the fast lithium/sodium ions (Li+/Na+) insertion/extraction, the stable PCS shell provides structural stability during cycling, resulted in the improved rate capability and long cycle life of the PCS/WS2/NG anode material. For lithium-ion batteries (LIBs), the capacity of PCS/WS2/NG anode is 324.7 mAh g−1 after 510 cycles at a high current density of 2 A g−1. The discharge capacities of PCS/WS2/NG, PCS/WS2, WS2 and PCS anodes are 272.9, 105.1, 8.1 and 6.7 mAh g−1, respectively, at a high current density of 4 A g−1. When the current density rebound to 0.1 A g−1, their capacity retentions are 57.8%, 38.3%, 17.6% and 26.4%, respectively. The capacity retention of PCS/WS2/NG anode is 69.6%, higher than that of the pristine WS2 anode. For sodium-ion batteries (SIBs), PCS/WS2/NG anode exhibits a specific capacity of 205.0 mAh g−1 after the 900th cycle at the current density of 0.5 A g−1. The excellent electrochemical performance of the three-layer shell structured PCS/WS2/NG nanomaterial is a promising anode for high performance LIBs and SIBs.

Erratum to: Layer-by-Layer Analysis of the Stress-Strain State of Three-Layer Shells with Cutouts

There was a mistake in the affiliations. The right affiliation is “Institute of Applied Mechanics of the Russian Academy of Sciences, Leningradskii pr. 7, Moscow, 125040 Russia.”