Shells In Thermal Environments Research Articles

The non-linear vibrations of rotating functionally graded cylindrical shells

This paper reports the result of an investigate on the non-linear vibrations of rotating functionally graded cylindrical shell in thermal environment, based on Hamilton’s principle, von Karman non-linear theory and the first-order shear deformation theory. The formulation includes the initial hoop tension, the centrifugal and Coriolis forces due to rotation of the shell. The effects of in-plane and rotary inertia are taken into account in the equations of motion. Galerkin’s method is utilised to convert the governing partial differential equations to non-linear ordinary differential equations. A reduction in the model is presented to investigate non-linear dynamics, including primary resonance responses, quasi-periodic and chaotic responses to harmonic transverse external forces. The modal coefficients of quadratic and cubic nonlinearities are calculated by Galerkin integration and superimposed on the linear part of equation to establish the non-linear reduction equation. To validate the approach proposed in this paper, a series of comparison are performed and the investigations demonstrate good reliability and low computational cost of the present approach.

Nonlinear stability of eccentrically stiffened S-FGM elliptical cylindrical shells in thermal environment

This paper has studied on the nonlinear buckling and postbuckling of a eccentrically stiffened functionally graded thin elliptical cylindrical shells surrounded on elastic foundations in thermal environment. Material properties are graded in the thickness direction according to a Sigmoid power law distribution (s-FGM) in terms of the volume fractions of constituents with metal - ceramic - metal layers. The elliptical cylindrical shells have stiffeners in two directions (longitudinal and transversal) on the external surface and inside surface surrounded on elastic foundations. The typical point in the study is both elliptical cylindrical shells and stiffeners having temperature-dependent properties are deformed under temperature simultaneously, leading to the equation to determine buckling thermal loads with both sides that are dependent on temperature, so the iterative algorithms are proposed to numerical technique in this case. The equilibrium and compatibility equations for the moderately elliptical cylindrical shells are derived by using the classical shell theory (CST) taking into account both geometrical nonlinearity in von Karman sense and initial geometrical imperfection. By applying the Galerkin method and using a stress function, the effects of material and geometrical properties, temperature-dependent material properties, elastic foundations and stiffeners on the buckling and postbuckling loading capacity of eccentrically stiffened FGM elliptical cylindrical shells in thermal environments are studied simultaneously for first time.

Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments

Vibration characteristics of CNT-reinforced functionally graded composite closed cylindrical shells are studied. Thermal effect is taken into account. In the structural modeling, Reddy's high-order shear deformation theory is applied. Hamilton's principle and the assumed mode method are used to formulate the equation of motion of the CNT-reinforced functionally graded cylindrical shell. Vibration properties of the cylindrical shell are analyzed through the time- and frequency-domain methods. Influences of temperature change, CNTs distribution as well as CNTs volume fractions on the natural frequency of the CNT-reinforced cylindrical shell are investigated. Vibration responses of the cylindrical shell computed by the FSDT and TSDT are compared to verify the necessity of the high-order shear deformation theory in the vibration analysis for thick CNT-reinforced structures. The effects of CNTs volume and distribution on the free and forced vibration of the cylindrical shell are studied. The influences of thermal effect on the vibration responses of the CNT-reinforced cylindrical shell are also investigated.

Nonlinear low‐velocity impact analysis of functionally graded nanotube‐reinforced composite cylindrical shells in thermal environments

This paper presents a nonlinear analysis of impact response of nanocomposites cylindrical shells reinforced by single‐walled carbon nanotubes (SWCNTs) in thermal environments. The nanotube reinforcement distributions are either uniform or functionally graded in the shell thickness direction. Micromechanical models are employed to estimate the material properties of nanotube reinforced composites. A higher‐order shear deformation theory with a von Kármán‐type of kinematic nonlinearity yields the equations of motion. By considering the nanotube reinforced composites as temperature‐dependent, the thermal effects are included. A two‐step perturbation technique is used to solve the equations of motion. A parametric study is carried out, and the effects of nanotube distribution and its volume fraction, impactor initial velocity, temperature change, geometric parameter of shell, and edge boundary condition on the impact response of the shell are discussed. The numerical results show that the central deflection of the shell subjected to impact is sinusoidal. It is found that the nanotube distribution through the thickness can alter the amplitude and the frequency of the central deflection and also the energy absorption significantly. POLYM. COMPOS., 39:730–745, 2018. © 2016 Society of Plastics Engineers

Experimental study on bistable behaviour of anti-symmetric laminated cylindrical shells in thermal environments

Anti-symmetric laminated [+α/−α]n cylindrical shell (ALCS) is a type of bistable composite structures that can deform between two stable shapes under an external loading. The maximum snap loads and shape deformations in varying thermal environments play a significant role in their bistable characteristic. This paper presents an experimental study on the bistable behaviour of ALCSs at 20°C, 40°C, 60°C, and 80°C. The load–displacement curves are obtained by using a testing machine with a thermal chamber. The effect of the temperature variation on the shell’s curvatures is also investigated by digital image processing technique. The obtained experimental results show the same trend as the analytical and numerical results. The thermal influence on the bistable behaviour in a local temperature field is also studied in detail. The experimental investigations presented in this paper is of great importance in the engineering design and applications of morphing structures manufactured from ALCSs.

A layerwise finite element formulation for free vibration analysis of functionally graded sandwich shells

This paper presents a layerwise finite element formulation for dynamic analysis of functionally graded material (FGM) sandwich shell in thermal and non-thermal environments. The layerwise theory used in this work is based on the first-order shear deformation theory (FSDT) for each layer satisfying the displacement continuity at layer interface. Two configurations of FGM sandwich shells are considered, the first with FGM facesheet and homogenous core, and the second having homogenous facesheets and FGM core. Effective material properties of the FGM are estimated according to two micromechanical models, namely, Voigt’s rule of mixture (ROM) and Mori–Tanaka (MT) scheme. For the shells in thermal environment, a nonlinear temperature distribution in thickness direction is considered and the elastic properties are assumed to be temperature dependent. The results obtained from the proposed formulation are validated with those available in the literature. Natural frequencies obtained from Sanders’, Love’s and Donnell’s shell theories for different geometric and boundary conditions are compared to assess the performance of different shell theories for FGM sandwich shells under non-thermal environment. The effects of volume fraction index, core thickness and temperature gradient on natural frequencies of FGM sandwich shells are investigated. The present formulation is simple, accurate and computationally efficient.

Nonlinear internal resonance of functionally graded cylindrical shells using the Hamiltonian dynamics

Internal resonance in nonlinear vibration of functionally graded (FG) circular cylindrical shells in thermal environment is studied using the Hamiltonian dynamics formulation. The material properties are considered to be temperature-dependent. Based on the Kármán-Donnell's nonlinear shell theory, the kinetic and potential energy of FG cylindrical thin shells are formulated. The primary target is to investigate the two-mode internal resonance, which is triggered by geometric and material parameters of shells. Following a secular perturbation procedure, the underlying dynamic characteristics of the two-mode interactions in both exact and near resonance cases are fully discussed. It is revealed that the system will undergo a bifurcation in near resonance case, which induces the dynamic response at high energy level being distinct from the motion at low energy level. The effects of temperature and volume fractions of composition on the exact resonance condition and bifurcation characteristics of FG cylindrical shells are also investigated.

Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations

Abstract This paper studies the thermal stability of an eccentrically stiffened functionally graded truncated conical shells in thermal environment and surrounded on elastic foundations. Both of the FGM shell as well as the stiffeners are deformed under temperature. The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation. By applying Galerkin method, the closed-form expression for determining the thermal buckling load is obtained. The numerical results show that the critical thermal load in the case of the uniform temperature rise is smaller than one of the linear temperature distribution through the thickness of the shell, and the critical thermal load increases when increasing the coefficient of stiffeners and vice versa. The paper also analyzes and discussed the significant effects of material and geometrical properties, elastic foundations on the thermal buckling capacity of the eccentrically stiffened FGM truncated conical shell in thermal environment. The obtained results are validated by comparing with those in the literature.

Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment

The main aim of this paper is to investigate the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach. Shells are reinforced by closely spaced rings and stringers in which material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using Galerkin method with three-term solution of deflection, the closed form to find critical torsional load and post-buckling load–deflection curves are obtained. The effects of temperature, stiffener, foundation, material and dimensional parameters are analyzed.

Torsional postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded composite cylindrical shell reinforced by single-walled carbon nanotubes (SWCNTs) subjected to torsion in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under torsion is proposed. A singular perturbation technique along with a two-step perturbation approach is employed to determine the buckling load and postbuckling equilibrium path. The numerical illustrations concern the torsional buckling and postbuckling behavior of perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling torque as well as postbuckling strength of the shell under torsion when the reinforcement has a symmetrical distribution. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells under torsion.

Nonlinear low-velocity impact analysis of functionally graded shallow spherical shells in thermal environments

This article presents the nonlinear dynamic response of functionally graded (FG) shallow spherical shells in thermal environments subjected to low-velocity impact by an elastic ball. The material properties of a FG shallow spherical shell vary continuously through the thickness according to a power law distribution of the volume fraction of the constituents. The temperature field is considered to vary along the thickness direction due to the steady state heat transfer. Based on the higher order shear deformation theory, the governing equations of motion for the shell, which account for geometric nonlinearity is obtained using Hamilton’s principle. The contact force between the shell and the impactor is relative to local deformation and calculated using a numerical method. Then, the governing equations of motion are solved numerically by the Chebyshev collocation method and Newmark scheme. This is a complete model that can not only fully model the dynamic behavior of the shell but also fully model the impactor’s dynamic behavior. In the numerical example, the effects of material properties, temperature, initial impact velocity and mass of the impactor on the dynamic behavior of the shells, and contact force are discussed in detail.

Transient responses of functionally graded double curved shallow shells with temperature-dependent material properties in thermal environment

Abstract An analytical approach is presented to investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffened FGM double curved thin shallow shells on elastic foundation using a simple power-law distribution (P-FGM) in thermal environment. The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation. By applying Galerkin method and using stress function, explicit relations of thermal load-deflection curves for simply supported curved eccentrically stiffened FGM shells are determined. Effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners on the dynamic response and vibration of the imperfect eccentrically stiffened P-FGM double curved shallow shells in thermal environments are analyzed and discussed. The numerical results in this paper are compared with results reported in other publications.

Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on elastic foundations in thermal environments

This paper first time presents an analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened FGM double curved thin shallow shells on elastic foundation using a simple power-law distribution (P-FGM) in thermal environments. The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation. By applying Galerkin method and using stress function, explicit relations of thermal load–deflection curves for simply supported curved eccentrically stiffened FGM shells are determined. Effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners on the buckling and postbuckling loading capacity of the imperfect eccentrically stiffened P-FGM double curved shallow shells in thermal environments are analyzed and discussed.

Nonlinear resonance behavior of functionally graded cylindrical shells in thermal environments

This paper deals with the nonlinear vibrations of functionally graded cylindrical shells in thermal environments. The equivalent properties of functionally graded materials are described as a power-law distribution in the thickness direction and are considered to be temperature-dependent. A typical case with a primary resonance excitation and a 1:2 internal resonance between two modes is analyzed. The energy approach and the Lagrangian formulation are employed to derive the reduced low-dimensional nonlinear ordinary differential equations of motion based on Donnell’s nonlinear shell theory. The dynamic behaviors of system are investigated by means of the so-called multiple scale method. The amplitude–frequency curves and the bifurcation behavior of the system are analyzed using numerical continuation method. The effects of temperature and volume fractions of constituent material on the amplitude response of the system are fully discussed.

Transient response of rotating laminated functionally graded cylindrical shells in thermal environment

Based on the elasticity theory, the transient analysis of dynamically pressurized rotating multi-layered functionally graded (FG) cylindrical shells in thermal environment is presented. The variations of the field variables across the shell thickness are accurately modeled by dividing the shell into a set of co-axial mathematical layers in the radial direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The differential quadrature method and Newmark's time integration scheme are employed to discretize the obtained governing equations of each mathematical layer. After performing the convergence and comparison studies, parametric studies for two common types of FG sandwich shells, namely, the shell with homogeneous inner/outer layers and FG core and the shell with FG inner/outer layers and homogeneous core are carried out. The influences of the temperature dependence of material properties, material graded index, the convective heat transfer coefficient, the angular velocity, the boundary condition and the geometrical parameters (length and thickness to outer radius ratios) on the dynamic response of the FG shells are investigated.

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.

Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments

This paper investigates the large amplitude vibration behavior of nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) in thermal environments. The SWCNTs are assumed to be aligned and straight with a uniform layout. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC shells are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The motion equations are based on a higher-order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The equations of motion are solved by an improved perturbation technique to determine the nonlinear frequencies of the CNTRC shells. Numerical results demonstrate that in most cases the natural frequencies of the CNTRC shells are reduced but the nonlinear to linear frequency ratios of the CNTRC shells are increased as the temperature rises. It is found that the natural frequencies are increased by increasing the CNT volume fraction, whereas the CNTRC shells with intermediate CNT volume fraction do not have intermediate nonlinear to linear frequency ratios.

Thermoelastic analysis of rotating laminated functionally graded cylindrical shells using layerwise differential quadrature method

An accurate and efficient solution procedure based on the elasticity theory is employed to investigate the thermoelastic behavior of rotating laminated functionally graded (FG) cylindrical shells in thermal environment. The material properties are assumed to be temperature dependent and graded in the thickness direction. In order to accurately model the variation of the field variables across the thickness, the shell is divided into a set of mathematical layers. The differential quadrature method (DQM) is adopted to discretize the governing differential equations of each layer together with the related boundary and compatibility conditions at the interface of two adjacent layers. Using the DQM enables one to accurately and efficiently discretize the partial differential equations, especially along the graded direction, and also implement the boundary and compatibility conditions in their strong forms. After demonstrating the convergence and accuracy of the presented approach, the effects of material and geometrical parameters and also temperature dependence of material properties on the stresses and displacement components of rotating laminated FG cylindrical shells are studied.

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part II: Pressure-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to lateral or hydrostatic pressure in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under external pressure is proposed and a singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. Numerical results for pressure-loaded, perfect and imperfect, FG-CNTRC cylindrical shells are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling pressure as well as postbuckling strength of the shell under external pressure. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling pressure and postbuckling behavior of CNTRC shells.

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to axial compression in thermal environments. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially-loaded, perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling load as well as postbuckling strength of the shell under axial compression. The results reveal that the CNT volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells.