Shell Equations Research Articles

Modeling and Vibration Analysis of a Porous Rotational Shell Based on Biot Theory

Combining the Biot theory and classical elastic theory for thin shells, a new dynamic model of a thin fluid-saturated porous rotational shell is proposed. First-order ordinary differential control equations of the porous rotational shell are derived in the frequency domain. These equations are then solved by using the precise element method. The accuracy of this model has been verified by comparing with a vibration experiment. Moreover, the comparisons between the present model and two equivalent property models are carried out. Because the present approach considers the fluid-solid coupling effect and makes no assumptions for the fluid displacements, it is more accurate in the high-frequency range. Lastly, the dynamic characteristics of porous rotational shells are demonstrated by the proposed method.

NONLINEAR DYNAMIC EQUATIONS FOR ELASTIC MICROMORPHIC SOLIDS AND SHELLS. PART I

The present paper develops a general approach to deriving nonlinear equations of motion for solids whose material points possess additional degrees of freedom. The essential characteristic of this approach is theaccount of incompatible deformations that may occur in the body due to distributed defects or in the result of the some kind of process like growth or remodelling. The mathematical formalism is based on least action principle and Noether symmetries. The peculiarity of such formalism is in formal description of reference shape of the body, which in the case of incompatible deformations has to be regarded either as a continual family of shapes or some shape embedded into non-Euclidean space. Although the general approach yields equations for Cosserat-type solids, micromorphic bodies and shells, the latter differ significantly in the formal description of enhanced geometric structures upon which the action integral has to be defined. Detailed discussion of this disparity is given.

Buckling Analysis of Functionally Graded Shells under Mixed ‎Boundary ‎Conditions Subjected to Uniform Lateral Pressure

In this study, the buckling problem of shells consisting of functionally graded ‎materials (FGMs) under uniform compressive lateral pressure is solved at mixed ‎boundary conditions. After creating the FGM models, the basic differential equations ‎of FGM shells under compressive lateral pressure are derived within the scope of ‎classical shell theory (CST). The basic differential equations are solved with the help ‎of Galerkin method and the formula for the lateral buckling pressure is obtained. The ‎minimum values of the lateral buckling pressure are found numerically by minimizing ‎the obtained expression according to the numbers of transverse and longitudinal ‎waves. The accuracy is confirmed by comparing the numerical values for the lateral ‎buckling pressure of homogeneous and FGM shells with the results in the literature. ‎The influences of FGM profiles and shell characteristics on the magnitudes of lateral ‎buckling pressure are investigated in detail by performing specific numerical analyzes.

Free vibration analysis of circular cylindrical shell on elastic foundation using the Rayleigh-Ritz method

A general approach is presented for the free vibration analysis of circular cylindrical shell resting on elastic foundation and subjected to classical boundary conditions of any type. The Winkler/Pasternak model is utilized to simulate the elastic foundation imposed on the cylindrical shell, and then it is easily to derive the potential energy of the elastic foundation. Based on the Flugge shell theory, explicit expressions for the mass and stiffness matrices are obtained. By taking the characteristic beam modal functions as the admissible functions, the Rayleigh-Ritz method is employed to derive the frequency equations of circular cylindrical shell with all the classical boundary conditions and resting on elastic foundation. Once the frequency equation has been determined, the frequencies can be calculated numerically. The excellent accuracy and validity of the present approach are demonstrated by numerical examples and comparisons with the results available in the literature. Finally, some further numerical results are given to illustrate the comprehensive effect of geometric properties and foundation coefficients on the frequencies of circular cylindrical shell in contact with elastic foundation.

A Symplectic Approach for Buckling Analysis of Natural Fiber Reinforced Composite Shells Under Hygrothermal Aging

A buckling analysis of natural fiber reinforced (NFRC) cylindrical shells is performed based on Reissner's shell theory and symplectic approach. The governing buckling equations of axially-compressed NFRC cylindrical shells are established in the Hamilton system. Therefore, the original problem is reduced into a symplectic eigen-problem in the symplectic space. Accurate critical loads and analytical buckling mode shapes are directly obtained from the symplectic eigenvalue and eigensolutions. In numerical examples, the effect of hygroscopic aging on the expressions of eigensolutions is investigated. In addition, the influences of ageing time, fiber content and geometric parameters on the buckling behavior of NFRC cylindrical shell is discussed in detail.

НЕЛИНЕЙНЫЕ МЕТОДЫ МЕХАНИКИ ДЕРЕВЯННЫХ ПОЛОГИХ ОБОЛОЧЕК НА ПРЯМОУГОЛЬНОМ ПЛАНЕ

The coatings construction of buildings and structures in the form of wooden shallow shells on a rectangular plan are considered. The orthotropic of the material and the geometric nonlinearity of the thin-walled structure taken into account in the equations. The Bubnov-Galerkin method is used to solve differential equations of wooden shallow shells with different supports. The influences of the ratio of elasticity in mutually perpendicular directions, shape and thickness of a structure on the value of stresses, critical load and low frequencies of small values of vibrations are investigated. The investigations results are given in dimensionless form and graphs are shown. This makes them useful in engineering calculations. Recommendations for adjusting the shape and thickness of the structure of coatings in the form of shallow shells to increase their bearing capacity or reduce material consumption are given.

Nonlinear buckling analysis of FGP shallow spherical shells under thermomechanical condition

In this study, an analytical method is presented to investigate the nonlinear buckling analysis of functionally graded porous (FGP) shallow spherical shells exposed to external excitation in a thermal environment resting on elastic foundations. The elastic foundations of the FGP shell are consist of a two-term Winkler-Pasternak's model. Using the classical theory of shells and considering nonlinear von Karman-Donnell relations and Hook law, the governing equations of thin FG porous spherical shells are extracted. Galerkin's method is utilized to discretize the governing equation of shallow spherical shell. Regarding the discretized equations of motion, the explicit expressions for dynamic and static critical buckling loading are determined under thermomechanical effect. Two types of distributions for FG porous, including the uniform and symmetric porosity, are considered. To investigate the dynamic buckling exposed to external pressure and thermal effect, the equations of motion of FGP shallow spherical shells are examined via a numerical method named Runge-Kutta approach, and then with a procedure presented by Budiansky-Roth, the critical load for the nonlinear dynamic buckling is obtained. The influence of thermal environment changes, porosity coefficients, various porosity distributions, elastic foundations, and geometrical characteristics on the nonlinear static and dynamic buckling response of FGP shallow spherical shell is examined.

Finite element simulation of multi‐component vesicle morphologies undergoing phase separation

AbstractMorphological changes in lipid bilayer vesicles are due to phase transitions and surface deformations occurring in unison. We present a dynamic chemo‐mechanical finite element model to study such changes. This is achieved by coupling two fourth order partial differential equations (PDEs): The Cahn‐Hilliard [2] mass balance equation based on irreversible thermodynamics, and the Kirchhoff‐Love [1] rotation free thin shell equation. The Helmholtz free energy consists of Helfrich energy to model elastic bending, and also includes in‐plane elastic energy with a finite shear modulus for the purpose of regularization. The geometry is discretized by C1‐continuous NURBS shape functions. An implicit second order accurate generalized‐α scheme is used for time integration. Newton‐Raphson iterations are utilized to solve the resulting linearized weak form monolithically. The proposed formulation is illustrated by a numerical example.

A New Analytical Method for Spherical Thin Shells’ Axisymmetric Vibrations

In order to improve the spherical thin shells’ vibrations analysis, we introduce a new analytical method. In this method, we take into consideration the terms of the inertial couples in the stress couples’ differential equations of motion. These inertial couples are omitted in the theories provided by Naghdi–Kalnins and Kunieda. The results show that the current method can solve the axisymmetric vibrations’ equations of elastic thin spherical shells. In this paper, we focus on verifying the current method, particularly for free vibrations with free edge and clamped edge boundary conditions. To check the validity and accuracy of the current analytical method, the natural frequencies determined by this method are compared with those available in the literature and those obtained by a finite element calculation.

Study on dynamic instability characteristics of functionally graded material sandwich conical shells with arbitrary boundary conditions

This paper presents analytical studies on the dynamic instability of functionally graded material (FGM) sandwich conical shell subjected to time dependent periodic parametric axial and lateral load. In the analysis, four kinds of sandwich distributions, including FGM core and isotropic skins, and isotropic core and FGM skins are considered and the power law distribution is employed to estimate the volume of the constituents. The arbitrary boundary conditions of the FGM sandwich conical shell are achieved by using the artificial spring boundary. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton’s principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin’s method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of sandwich distribution types, gradient indexes, skin-core-skin ratio and load forms on the dynamic stability of FGM sandwich conical shell have been investigated.

Ultimate Limit State Equations and Plasticity of Tubular Conical Transitions

AbstractConical transitions have wide applications in wind turbine foundation as well as oil and gas jacket type of structures. The junctions where tubular and cone meet experience a sharp stress rise from shell edge effects. Like all structures experiencing sharp stress rises, fatigue considerations are critical. In addition to fatigue, the existing offshore structural design standards also require ultimate limit state checks. It is known from the lower bound theorem of plasticity limit analysis that the junction local edge effects do not impact the global capacity. Designing for the local junction ultimate limit state contains wide variations among existing design standards. In this paper, the design practices from API RP-2A, NORSOK N-004, and ISO 19902:2020 draft are assessed. They are compared to the shell plastic yield criteria of Hodge and Ilyushin. In addition, this paper provides a semi-analytical plasticity solution to determine junction plastic deformations. The formulation is based on cylindrical shell equations coupled with deformation plasticity theory. It is found that the growth of the junction plasticity zone is limited, which is consistent with the anticipation from the lower bound limit analysis theorem. The observations made in this paper show that the local junction plasticity is a secondary issue compared to other design considerations. Its ultimate limit state design equation can afford to be more lenient if chooses for future standards’ development.

Vibration control of a smart shell reinforced by graphene nanoplatelets under external load: Semi-numerical and finite element modeling

In this article, smart control and frequency characteristics of a graphene nanoplatelets composite (GPLRC) cylindrical shell surrounded by piezoelectric layers as sensor and actuator (PLSA) are presented. The current structure is under an external load. For the semi-numerical method, the strain-stress relations can be determined through the first-order shear deformable theory (FSDT). For access to various mass densities as well as the Poisson ratio, the rule of the mixture is applied, although the modified Halpin-Tsai theory for obtaining the module of elasticity. The external voltage is applied to the sensor layer, while a Proportional-Derivative (PD) controller has been utilized for controlling the output of the sensor. The boundary conditions are derived through governing equations of the GPLRC cylindrical shell surrounded by PLSA using an energy method known as Hamilton's principle and finally are solved using a generalized differential quadrature method (GDQM). Apart from a semi-numerical solution, a finite element model was presented using the finite element package to simulate the response of the smart GPLRC cylindrical shell. The results created from a finite element simulation illustrates a close agreement with the semi-numerical method results. The outcomes show that the PD controller, viscoelastic foundation, slenderness factor (L/R), external voltage, and GPL's weight fraction have a considerable impact on the amplitude and vibration behavior of a GPLRC smart cylindrical shell. As an applicable result in related industries, the parameter and consideration of the PD controller have a positive effect on the static and dynamic behaviors of the structure subjected to an external load.

Existence and stability of static spherical fluid shells in a Schwarzschild-Rindler–anti–de Sitter metric

We demonstrate the existence of static stable spherical fluid shells in the Schwarzschild-Rindler-anti-de Sitter (SRAdS) spacetime where $ds^2 = f(r)dt^{2} -\frac{dr^{2}}{f(r)}-r^{2}(d\theta ^2 +\sin ^2 \theta d\phi ^2)$ with $f(r) = 1 -\frac{2Gm}{r} + 2 b r -\frac{\Lambda}{3}r^2$. This is an alternative to the well known gravastar geometry where the stability emerges due to the combination of the repulsive forces of the interior de Sitter space with the attractive forces of the exterior Schwarzschild spacetime. In the SRAdS spacetime the repulsion that leads to stability of the shell comes from a negative Rindler term while the Schwarzschild and anti-de Sitter terms are attractive. We demonstrate the existence of such stable spherical shells for three shell fluid equations of state: vacuum shell ($p=-\sigma$), stiff matter shell ($p=\sigma$) and dust shell ($p=0$) where $p$ is the shell pressure and $\sigma$ is the shell surface density. We also identify the metric parameter conditions that need to be satisfied for shell stability in each case. The vacuum stable shell solution in the SRAdS spacetime is consistent with previous studies by two of the authors that demonstrated the existence sf stable spherical scalar field domain walls in the SRAdS spacetime.

An adaptive shell element for explicit dynamic analysis of failure in laminated composites Part 2: Progressive failure and model validation

To enable modelling of the progressive failure of large, laminated composite components under crash or impact loading, it is key to have a numerical methodology that is both efficient and numerically robust. A potential way is to adopt an adaptive method where the structure is initially represented by an equivalent single-layer shell model, which during the analysis is adaptively transformed to a high-resolution layer-wise model in areas where higher accuracy is required. Such a method was recently developed and implemented in the commercial finite element solver LS-DYNA, aiming at explicit crash analysis (Främby, Fagerström and Karlsson: An adaptive shell element for explicit dynamic analysis of failure in laminated composites - Part 1: Adaptive kinematics and numerical implementation, 2020). In the current work, the method is extended to the case of interacting inter- and intralaminar damage evolution. As a key part, we demonstrate the importance of properly regularising the intralaminar failure described by a smeared-crack model, and show that neglecting to account for the crack-versus-mesh orientation may lead to significant errors in the predicted energy dissipation. We also validate the adaptive approach against a four-point beam bending test with matrix-induced delamination growth, and simultaneously show the capability of the proposed method to – at lower computational expense – replicate the results from a refined, non-adaptive model.

Length-scale effect in stability problems for thin biperiodic cylindrical shells: extended tolerance modelling

Thin linearly elastic Kirchhoff–Love-type circular cylindrical shells of periodically micro-inhomogeneous structure in circumferential and axial directions (biperiodic shells) are investigated. The aim of this contribution is to formulate and discuss a new averaged nonasymptotic model for the analysis of selected stability problems for these shells. This, so-called, general nonasymptotic tolerance model is derived by applying a certain extended version of the known tolerance modelling procedure. Contrary to the starting exact shell equations with highly oscillating, noncontinuous and periodic coefficients, governing equations of the tolerance model have constant coefficients depending also on a cell size. Hence, the model makes it possible to investigate the effect of a microstructure size on the global shell stability (the length-scale effect).

Nonlinear analysis of thermal-mechanical coupling bending of FGP infinite length cylindrical panels based on PNS and NSGT

The present paper investigates the nonlinear static bending behavior of infinite length cylindrical panels made of functionally graded porous (FGP) materials. The shell with clamped edges is subjected to uniform temperature rise and transverse pressure loading. Thermomechanical properties of the shell with even distributed porosities are temperature-dependent and are graded along the thickness. The principle of virtual displacement and nonlocal strain gradient theory (NSGT) are employed to derive the equilibrium equations of the shallow shell resting on nonlinear elastic foundation. The concept of physical neutral surface (PNS) and higher-order shear deformation theory are also included into the formulation. Using the uncoupled thermoelasticity and Donnell kinematic assumptions, three differential equations of the shell under thermomechanical loading are established. The nonlinear system of governing equations is solved for the shell with infinite length which is clamped on both straight edges and free at other curved edges. The two-step perturbation technique and Galerkin procedure are employed to derive analytical solutions for the thermal, mechanical, and thermomechanical responses of cylindrical panels. The important parameters governing the bending behavior of shells under thermal-mechanical coupling load are identified and discussed. Parametric studies are given to show the influences of nonlocal/length scale parameter, porosity coefficient, foundation stiffness, geometrical parameter and functionally graded pattern. It is shown that the geometrical parameters have an important role on the bending behavior of cylindrical panels. Also, the temperature dependence of material properties results in higher deflection of the heated shells.

Dynamic instability analysis of porous FGM conical shells subjected to parametric excitation in thermal environment within FSDT

The conical shell structure is prone to dynamic instability when subjected to time dependent periodic axial loads and it can cause structural damage. Based on that this paper presents an accurate and semi-analytical method for investigation the dynamic instability of porous functionally graded material (PFGM) conical shell in thermal environment. In the analysis, three common types of PFGM conical shells, namely, uniform, symmetric and asymmetric distribution are considered assuming that material properties are related to temperature. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton's principle considering first order shear deformation theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin's method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of porosity distribution type, gradient index, radius-to-thickness ratio, porosity volume fraction and temperature fields on the dynamic stability of PFGM conical shell have been investigated, wherein for different porosity distribution types, the UPD type is more sensitive to gradient index as compared to other three types, while the SPD has the minimum relative width.

Dynamic instability analysis of FG-CNTRC laminated conical shells surrounded by elastic foundations within FSDT

A comprehensive understanding of the dynamic instability of shell structure is critical to avoid resonance damage. On the basis of that, an accurate and analytical method for investigation the dynamic instability of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell surrounded by the elastic foundations is presented in this work based on the first-order shear deformation theory. In the analysis, uniform or functionally graded distributions of reinforcements across the shell thickness are considered and the extended Voigt model is employed to estimate the CNTRC material properties. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton's principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin's method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of volume fractions and types of CNTs, lamination angle, elastic foundations stiffness and lamination angle on the dynamic stability of laminated FG-CNTRC conical shell have been investigated.

The Effect of Porosity on Elastic Stability of Toroidal Shell Segments Made of Saturated Porous Functionally Graded Materials.

This research deals with the stability analysis of shallow segments of the toroidal shell made of saturated porous functionally graded (FG) material. The nonhomogeneous material properties of porous shell are assumed to be functionally graded as a function of the thickness and porosity parameters. The porous toroidal shell segments with positive and negative Gaussian curvatures and nonuniform distributed porosity are considered. The nonlinear equilibrium equations of the porous shell are derived via the total potential energy of the system. The governing equations are obtained on the basis of classical thin shell theory and the assumptions of Biot's poroelasticity theory. The equations are a set of the coupled partial differential equations. The analytical method including the Airy stress function is used to solve the stability equations of porous shell under mechanical loads in three cases. Porous toroidal shell segments subjected to lateral pressure, axial compression, and hydrostatic pressure loads are analytically analyzed. Closed-form solutions are expressed for the elastic buckling behavior of the convex and concave porous toroidal shell segments. The effects of porosity distribution and geometrical parameters of the shell on the critical buckling loads of porous toroidal shell segments are studied.

The study of the strength of dentures with different surface reliefs under the action of static loading

The paper presents modeling of denture specimens with different surface reliefs under the action of static loading using modern numerical methods - finite and boundary elements. Based on the analysis of literature, it is shown that the most effective approaches for calculating the stress-strain state (SSS) of dentures is the finite element method (FEM), which allows you to create 3D models with any complexity of geometry and surface, as well as the boundary element method (BEM), which allows in some cases to obtain more accurate calculation results than finite element method. The algorithm and procedure for exact integration of differential equations of hollow shells according to the algorithm of Kantorovich-Vlasov variational method are presented for the formation of the calculated finite element method ratios. The analytical expressions for the parameters of the hollow shells used for calculations of the state of dentures are given. Finite element method is represented in the work by the universal package SolidWorks. Different stages of solid-state modeling of prototypes of dentures with different surface reliefs are shown in detail. Dental stress-strain state calculations were performed by two methods. The results of the calculations are in good agreement with each other, which proves the reliability of both models developed and the results of the stress-strain state received. It is shown that the smallest values of stresses occur in dentures with a rhombic lattice; they are 5.9% less than prosthesis with a smooth surface, and 18.78% less than in prosthesis with a square lattice. Equivalent displacement of a rhombic lath prosthesis is less by 3,864% than that of a smooth surface prosthesis and 8 52% less than a square lattice prosthesis.