Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock

A finite element formulation for stress analysis of functionally graded material (FGM) sandwich plates and shell panels under thermal shock is presented in this work. A higher-order layerwise theory in conjunction with Sanders’ approximation for shells is used to develop the finite element formulation for transient stress analysis of FGM sandwich panels. The top and the bottom surfaces of FGM sandwich panels are made of pure ceramic and metal, respectively, and core of the sandwich is assumed to be made of FGM. The temperature profile in the thickness direction of the panels is considered to be varying as per the Fourier’s law of heat conduction equation for unsteady state. The heat conduction equations are solved using the central difference method in conjunction with the Crank–Nicolson approach. Transient thermal displacements of the sandwich panels are obtained using Newmark average acceleration method and the transient thermal stresses are obtained using stress–strain relations, subsequently. Results obtained from the present layerwise finite element formulations are first validated with available solutions in literature. Parametric studies are taken up to study the effects of volume fraction index, temperature dependency of material properties, core thickness, panel configuration, geometric and thermal boundary conditions on transient thermal stresses of FGM sandwich plates and shells.     

Three-Dimensional Pyroelectric Analysis of a Functionally Graded Piezoelectric Hollow Sphere

A three-dimensional analytical piezothermoelastic solution is presented for a functionally graded piezoelectric spherical shell subjected to various thermal boundary conditions applied on the inner and outer surfaces. Material properties are assumed to vary along the radius, r, obeying a power law. Both the thermal field and the pyroelectric responses are resolved using the state space method. On introducing three displacement and two stress functions, two independent kinds of state equations are derived from the basic equations of piezoelectricity. It is interesting that the first kind is a homogeneous equation only related to the purely elastic behavior of the sphere, yet the second has an inhomogeneous term associated with the thermal effect to determine the pyroelectric responses. The state equations are solved by expanding the field variables into series of spherical harmonic functions. Numerical examples are performed to investigate the influence of material inhomogeneity on the pyroelectric responses of the spherical shell.     

Static and vibration analysis of axi-symmetric angle-ply laminated cylindrical shell using state space differential quadrature method

In this study, static and free vibration characteristics of anisotropic laminated cylindrical shell with various end conditions are considered by making the use of differential quadrature method (DQM). Equations of motion are derived based on three-dimensional theory of elasticity. Applying the state space in conjunction with DQM to the governing differential equations and to the edges boundary conditions in term of displacements, new state equations at discrete points are derived. By solving the obtained state equations, static and frequency behavior of laminated shell are evaluated. To ensure the accuracy of the present approach, comparisons are made with those for the shell with simply supported edges which can be solved analytically. Finally, the effect of edges condition on the static and vibration behaviour of shell is investigated.     

Buckling of circular cylindrical shells by the Differential Quadrature Method

A study is conducted of the linear elastic buckling of circular cylindrical shells by the new Differential Quadrature Method (DQM). To date this numerical method in the area of buckling analysis has been applied only to rectangular plates. The Fluegge shell stability equations serve as the basis of the analysis. By assuming the form of the buckling modes in the circumferential direction the stability equations are transformed into ones dependent only on the axial coordinate of the shell. The resulting ordinary differential equations are then solved using the one-dimensional DQM approach. Results are first given for shells with simple or clamped boundary conditions, and these are compared with previosly published results. Finally, new results are presented for shells with clamped-free boundary conditions, which have relevance to the buckling analysis of liquid storage tanks.     

Nonlinear free vibration of spherical shell panel using higher order shear deformation theory – A finite element approach

A nonlinear finite element model for geometrically large amplitude free vibration analysis of doubly curved composite spherical shell panel is presented using higher order shear deformation theory (HSDT). The nonlinearity is introduced in the Green–Lagrange sense. The governing equations of the vibrated shell panel are derived using the Variational approach. Frequency ratios (nonlinear frequency to linear frequency) of the spherical panels are determined as a function of shell amplitude ratio. The results are computed for different orthotropicity ratios, stacking sequences, thickness ratios, amplitude ratios and boundary conditions and also compared with those available in literature.     

FREE VIBRATIONS OF A CONICAL SHELL WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES

This article presents an analysis of the free vibrations of a truncated conical thin shell subjected to thermal gradients. The governing equations of the shell are based on the Donnell-Mushtari theory of thin shells. Simply supported and clamped boundary conditions are considered at both ends of truncated conical shell. Temperature loading due to supersonic flow is assumed to vary along the meridian and across the thickness of the shell Hamilton's principle is used to derive the appropriate governing equations of a conical shell with temperature-dependent material properties. The shell material has a kind of inhomogeneity due to the varying temperature load and temperature dependency of material properties. The resulting differential equations are solved numerically using the collocation method. The results are compared with certain earlier results. The influence of temperature load on the vibration characteristics is examined for the conical shells with various geometrical properties.     

Investigation of combined heat and mass transfer by Lie group analysis with variable diffusivity taking into account hydrodynamic slip and thermal convective boundary conditions

The present paper investigates heat and mass transfer over a moving porous plate with hydrodynamic slip and thermal convective boundary conditions and concentration dependent diffusivity. The similarity representation of the system of partial differential equations of the problem is obtained through Lie group analysis. The resulting equations are solved numerically by Maple with Runge–Kutta–Fehlberg fourth–fifth order method. A representative set of results for the physical problem is displayed to illustrate the influence of parameters (velocity slip parameter, convective heat transfer parameter, concentration diffusivity parameter, Prandtl number and Schmidt number) on the dimensionless axial velocity, temperature and concentration field as well as the wall shear stress, the rate of heat transfer and the rate of mass transfer. The analytical solutions for velocity and temperature are obtained. Very good agreements are found between the analytical and numerical results of the present paper with published results.     

Thermoelastic analysis of laminated composite and sandwich shells considering the effects of transverse shear and normal deformations

Abstract

In the present study, thermoelastic analysis of laminated composite and sandwich shells (cylindrical/spherical) is presented using fifth-order shear and normal deformation theory. The significant characteristic of the present theory is that it includes the effects of both transverse shear and normal deformations. The mathematical formulation uses the principle of virtual work to derive the variationally consistent governing equations and traction free boundary conditions. To obtain the static solution, these governing equations are solved by employing Navier’s solution technique. The shell is subjected to a mechanical/thermal load sinusoidally distributed over the top surface of the shell. The thermal load linearly varies across the thickness of the shell. The present results are compared with other higher-order models and 3D elasticity solution wherever possible. Thermal stresses presented in this study will act as a benchmark for the future work.     

Application of two-steps perturbation technique to geometrically nonlinear analysis of long FGM cylindrical panels on elastic foundation under thermal load

Present research deals with the geometrically nonlinear bending of a long cylindrical panel made of a through-the-thickness functionally graded material subjected to thermal load. A panel under the action of uniform temperature rise loading is considered. Formulation of the shell is based on the third-order shear deformation shell theory, where the first-order shear deformation and classical shell theory may be extracted as special cases. Thermomechanical properties of the shell are assumed to be temperature dependent and are estimated according to a power law function across the shell thickness. Also, it is assumed that shell is in contact with an elastic foundation which acts in tension as well as in compression. The nonlinear governing equations of the shell are obtained using the von Kármán type of geometrical nonlinearity. The obtained governing equations are solved for two cases, i.e., simply supported shells and clamped shells. The developed equations are solved using a two-step perturbation technique. Accurate closed-form expressions are provided to obtain the mid-span deflection of the shell as a function of temperature elevation. Numerical results are provided to analyze the effects of power law exponent, boundary conditions, temperature dependency, side to radius ratio, and side to thickness ratio.     

Asymmetric thermal buckling of annular plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behavior of isotropic homogeneous annular plates resting on a partial Winkler-type elastic foundation under uniform temperature elevation is investigated. First-order shear deformation plate theory is used to obtain the governing equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained under the solution of a plane stress formulation, neglecting the rotations and lateral deflection. Applying the adjacent equilibrium criterion, the linearized stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region, and the second set, which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalized differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved to obtain the critical conditions of the plate and the associated circumferential mode number. Benchmark results are given in tabular and graphical presentations for combinations of simply supported and clamped types of boundary conditions. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and the boundary conditions.     

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels

Based on the three-dimensional elasticity theory, free vibration analysis of functionally graded (FG) curved thick panels under various boundary conditions is studied. Panel with two opposite edges simply supported and arbitrary boundary conditions at the other edges are considered. Two different models of material properties variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential distribution of the material properties through the thickness are considered. Differential quadrature method in conjunction with the trigonometric functions is used to discretize the governing equations. With a continuous material properties variation assumption through the thickness of the curved panel, differential quadrature method is efficiently used to discretize the governing equations and to implement the related boundary conditions at the top and bottom surfaces of the curved panel and in strong form. The convergence of the method is demonstrated and to validate the results, comparisons are made with the solutions for isotropic and FG curved panels. By examining the results of thick FG curved panels for various geometrical and material parameters and subjected to different boundary conditions, the influence of these parameters and in particular, those due to functionally graded material parameters are studied.     

Approximate Analytical Solution of Stagnation Point Flow and Heat Transfer over an Exponential Stretching Sheet with Convective Boundary Condition

This study is concerned with the stagnation point flow and heat transfer over an exponential stretching sheet via an approximate analytical method known as optimal homotopy asymptotic method (OHAM). The governing partial differential equations are converted into ordinary nonlinear differential equations using similarity transformations available in the literature. The heat transfer problem is modeled using two‐point convective boundary condition. These equations are then solved using the OHAM approach. The effects of controlling parameters on the dimensionless velocity, temperature, friction factor, and heat transfer rate are analyzed and discussed through graphs and tables. It is found that the OHAM results match well with numerical results obtained by Runge–Kutta Fehlberg fourth‐fifth order method for different assigned values of parameters. The rate of heat transfer increases with the stretching parameter. It is also found that the stretching parameter reduces the hydrodynamic boundary layer thickness whereas the Prandtl number reduces the thermal boundary layer thickness.     

BEM/FDM conjugate heat transfer analysis of a two-dimensional air-cooled turbine blade boundary layer

A coupled boundary element method （BEM） and finite difference method （FDM） are applied to solve conjugate heat transfer problem of a two-dimensional air-cooled turbine blade boundary layer. A loosely coupled strategy is adopted, in which each set of field equations is solved to provide boundary conditions for the other. The Navier-Stokes equations are solved by HIT-NS code. In this code, the FDM is adopted and is used to resolve the convective heat transfer in the fluid region. The BEM code is used to resolve the conduction heat transfer in the solid region. An iterated convergence criterion is the continuity of temperature and heat flux at the fluid-solid interface. The numerical results from the BEM adopted in this paper are in good agreement with the results of analytical solution and the results of commercial code, such as Fluent 6.2. The BEM avoids the complicated mesh needed in other computation method and saves the computation time. The results prove that the BEM adopted in this paper can give the same precision in numerical results with less boundary points. Comparing the conjugate results with the numerical results of an adiabatic wall flow solution, it reveals a significant difference in the distribution of metal temperatures. The results from conjugate heat transfer analysis are more accurate and they are closer to realistic thermal environment of turbines.     

Lie Group Analysis and Numerical Solutions for Magnetoconvective Slip Flow along a Moving Chemically Reacting Radiating Plate in Porous Media with Variable Mass Diffusivity

A numerical investigation of magnetoconvective boundary layer slip flow along a nonisothermal continuously moving permeable nonlinear radiating plate in Darcian porous media is reported. The concentration dependent mass diffusivity, viscous dissipation, Joule heating, and chemical reaction are taken into account. A Lie group of transformation is applied to the governing transport equations and boundary condition to find the corresponding similarity equations. Furthermore, the similarity equations with the relevant boundary conditions are solved numerically using the Runge‐Kutta‐Fehlberg fourth‐fifth order numerical method. Numerical results for the dimensionless velocity, temperature, and concentration distributions as well as friction factor, local Nusselt, and local Sherwood numbers are discussed for various controlling parameters. It is found that that the dimensionless concentration increases whilst the rate of mass transfer decreases with the mass diffusivity parameter. An excellent correlation is found between the present results and published results. The study finds applications in the polymer industry and metallurgy.     

Effect of material parameter on mixed convective fully developed micropolar fluid flow in a vertical channel

In this study, the effect of material parameter on the mixed convective fully developed micropolar fluid flow in a vertical channel has been analyzed. By considering appropriate boundary and interface conditions, the coupled nonlinear equations are solved analytically. The analytical results are plotted for various important parameters. It is found that an increase in the material parameter enhances the microrotation velocity and decreases the fluid velocity, and the results are shown graphically.     

The Brinkman model for boundary layer regime in a rectangular cavity with uniform heat flux from the side

This paper summarizes an analytical and numerical study of natural convection in a rectangular porous layer subjected to uniform heat fluxes along its vertical boundaries. In the formulation of the problem, use is made of the Brinkman-extended Darcy model which allows the no-slip boundary condition to be satisfied. The boundary layer equations are solved using a modified Oseen linearization method. It is found that the boundary effects have a non-negligible influence on the flow field and heat transfer. These effects are more pronounced in high porosity media where the flow rate and heat transfer are significantly reduced. For low porosity media the results obtained on the basis of a pure Darcy's law model are recovered as a limiting case of the present theory. Numerical results are reported in the range 20 ≤ R ≤ 1000, 10 ⁻⁷ ≤ Da ≤ 10 and 2 ≤ A ≤ 4. The boundary layer analytical solution is shown to agree well with the numerical results.     

Orthotropic cylindrical pressure vessels under line load

The work carried out earlier in this field is reviewed. Equations of equilibrium in terms of displacement components are derived for an orthotropic thin circular cylindrical shell subjected to a load that is not symmetric about the axis of the shell. Solutions that satisfy the boundary conditions are assumed in the form of exponential and trigonometric terms. Line loads along generator, as well as, circumference are considered. Eighth order differential equations in terms of parameter ‘p’ are obtained. These equations may be solved using suitable numerical methods.     

THERMOELASTIC BUCKLING OF THIN SPHERICAL SHELLS

Thermoelastic stability of thin perfect spherical shells based on deep and shallow shell theories is presented. To derive the equilibrium and stability equations according to deep shell theory, Sanders's nonlinear kinematic relations are substituted into the total potential energy function of the shell and the results are extremized by the Euler equations in the calculus of variation. The same equations are also derived based on quasi-shallow shell theory. An improvement is obtained for equilibrium and stability equations related to the deep shell theory in comparison with the same equations related to shallow shell theory. Approximate one-term solutions that satisfy the boundary conditions are assumed for the displacement components. The Galerkin-Bubnov method is used to minimize the errors due to this approximation. The eigenvalue solution of the stability equations is obtained using computer programs. For several thermal loads it is found that the deep shell theory results are slightly more stable as compared to the shallow shell theory results under the same thermal loads. The results are compared with the Algor finite element program and other known data in the literature.     

MHD radiative ohmic heating nanofluid flow of a stretching penetrable wedge: A numerical analysis

The present numerical investigation has focused on the magnetohydrodynamics flow of a viscous nanofluid over a stretching wedge with the boundary convective conditions, thermal radiation, and ohmic heating. Buongiorno's two-component nonhomogeneous nanoscale model was used and a dilute nanofluid with spherical type particles is considered. Similarity transformations are used to render the system of governing partial differential equations into a system of coupled similarity equations. The transformed equations are solved numerically with the BVP4C method. Validation of solutions with previous studies based on special cases of the general model is included. The salient features of fluid velocity profile, temperature as well as concentration profiles are discussed in a graphical manner for various values of selected governing factors. The skin friction coefficient, mass, and heat transfer rates are calculated and summarized. It is worthwhile noticing that the validation results exhibit an excellent agreement with already existing reports. The modeling of the present problem is useful in the thermal processing of sheet-like substances that is a necessary operation in paper procurement, wire drawing, drawing of plastic films, polymeric sheets, and metal spinning.     

CONVECTIVE HEAT TRANSFER IN AN ANNULAR POROUS LAYER WITH CENTRIFUGAL FORCE FIELD

The present study deals with natural convection in an annular porous layer under the influence of a centrifugal force field. It is assumed that the outer boundary is heated by a constant heat flux, while the inner boundary is perfectly insulated. The problem is formulated in terms of Darcy-Boussinesq equations and solved using analytical and numerical techniques. An analytical solution for the flaw and heat transfer variables, based on a concentric flow assumption, is obtained in terms of the Rayleigh number and the radius ratio. Finite amplitude results are verified by a numerical approach. Predicted thresholds in terms of critical Rayleigh numbers are verified by a linear stability analysis. Results obtained from the numerical approach indicate the existence of multiple solutions differing by the number of cells involved.