Layers Of Isotropic Materials Research Articles

Analytical and numerical (FEA) solution for steady-state heat transfer in generic FGM cylinder coated with two layers of isotropic material under convective-radiative boundary conditions

An investigation was carried out in order to develop an accurate analytical solution and a numerical (FEA) solution for steady-state heat transfer in a circular sandwich structure incorporated with convective-radiative boundary conditions. The dimensional governing equations and boundary conditions were developed in the form of a 4th order algebraic equation, and then the solution was obtained using Ferrari's method. By solving for the roots of the quartic equation, we were able to determine the dimensionless temperature fields of the FG sandwich composite. The findings obtained utilizing the exact analytical solution for the FG sandwich composite under thermal loads were satisfactorily validated against those data obtained using the Galerkin finite element approximation. The impact of geometric and thermo-physical characteristics, such as Biot number (Bii=1,2), Inner and outer surface thickness ratio (ri=1,2Ro), ambient temperature ratio (θd), radiation-conduction parameter (Nr), and thermal conductivity ratio (λ3λ1) on the efficiency of heat transfer, has also been studied. This study reveals the distinct effect of Biot number on the inner and outer layers of the composite cylinder. It shows that Bi1 has a negligent effect on temperature distribution; on the other hand, the outer surface (Bi2≤1) minimizes temperature variation. However, for design consideration, a thicker inner face sheet is not recommended in high thermal load, as Nr>4 has an insignificant impact on inner surface thickness on top surface temperature. Moreover, the outer surface temperature appears to be more sensitive to θd than the radiation-convection side. Furthermore, the given analytical solution is adequately verified against the proposed FEA method, having an error of less than 1.5%.

Eigenproblem Versus the Load-Carrying Capacity of Hybrid Thin-Walled Columns with Open Cross-Sections in the Elastic Range.

The phenomena that occur during compression of hybrid thin-walled columns with open cross-sections in the elastic range are discussed. Nonlinear buckling problems were solved within Koiter’s approximation theory. A multimodal approach was assumed to investigate an effect of symmetrical and anti-symmetrical buckling modes on the ultimate load-carrying capacity. Detailed simulations were carried out for freely supported columns with a C-section and a top-hat type section of medium lengths. The columns under analysis were made of two layers of isotropic materials characterized by various mechanical properties. The results attained were verified with the finite element method (FEM). The boundary conditions applied in the FEM allowed us to confirm the eigensolutions obtained within Koiter’s theory with very high accuracy. Nonlinear solutions comply within these two approaches for low and medium overloads. To trace the correctness of the solutions, the Riks algorithm, which allows for investigating unsteady paths, was used in the FEM. The results for the ultimate load-carrying capacity obtained within the FEM are higher than those attained with Koiter’s approximation method, but the leap takes place on the identical equilibrium path as the one determined from Koiter’s theory.

In Situ Observation of Fouling Behavior under Thermal Cracking Conditions: Hue, Saturation, and Intensity Image Analyses

Thermal cracking reactions of a variety of heavy petroleum feeds were analyzed by in situ cross-polarized microscopy using a reactor equipped with a sapphire window. Cross-polarized microscopy is a powerful technique for detecting the formation of anisotropic domains of carbonaceous material (mesophase coke). The present study, however, focused on the characterization of the chemical and physical events prior to the formation of new phases by evaluating the changes in image properties with reaction time. Results showed that changes in image brightness were strongly correlated to the conversion of 524+ °C material but could not provide much insight regarding the stability of reacting oils. Color analyses of the cross-polarized micrographs, however, revealed a consistent red-to-blue color shift around the point of instability. This color shift corresponded to either the formation of a fouling layer of isotropic material or a homogeneous color change of the reacting medium, indicating the formation of CS2-in...

Natural frequency characteristics of thin-walled homogeneous and manifold layered cylindrical shells under pressure using energy method

Energy method for the vibration of two types of cylindrical shells, namely thin-walled homogeneous isotropic and manifold layered isotropic cylindrical shells under uniform external lateral pressure is presented. The study is carried out based on strain-displacement relationship from Love’s shell theory with beam functions as axial modal function. A manifold layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The homogeneous cylindrical shell is made-up of isotropic one layer with stainless steel. The governing equations with uniform external lateral pressure for homogeneous isotropic and manifold layered isotropic cylindrical shells are obtained using energy functional by the Lagrangian function with Rayleigh-Ritz method. The boundary conditions that are presented at the end conditions of the cylindrical shell are simply supported-simply supported, clamped-clamped and free-free. The influences of uniform external lateral pressure and symmetrical boundary conditions on the natural frequency characteristics for both homogeneous and manifold layered isotropic cylindrical shells are examined. For all boundary conditions considered, the natural frequency of both cylindrical shells with symmetric uniform lateral pressure increases as h/R ratio increases and those considering natural frequency of the both cylindrical shells with symmetric uniform lateral pressure decrease as L/R ratio increases.

Frequency analysis of multiple layered cylindrical shells under lateral pressure with asymmetric boundary conditions

Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The multiple layered shell equations with lateral pressure are established based on Love’s shell theory. The governing equations of motion with lateral pressure are employed by using energy functional and applying the Ritz method. The boundary conditions represented by end conditions of the multiple layered cylindrical shell are simply supported-clamped(SS-C), free-clamped(F-C) and simply supported-free(SS-F). The influence of different lateral pressures, different thickness to radius ratios, different length to radius ratios and effect of the asymmetric boundary conditions on natural frequency characteristics are studied. It is shown that the lateral pressure has effect on the natural frequency of multiple layered cylindrical shell and causes the natural frequency to increase. The natural frequency of the developed multilayered cylindrical shell is validated by comparing with those in the literature. The proposed research provides an effective approach for vibration analysis shell structures subjected to lateral pressure with an energy method.

Effects of Ring Support and Internal Pressure on the Vibration Behavior of Multiple Layered Cylindrical Shells

This paper presents the study on natural frequency characteristics of a multiple layered cylindrical shell with ring support under internal pressure. The multiple layered cylindrical shell configuration is formed by three layers of isotropic materials where the inner and outer layers are stainless steel and the middle layer is aluminum. The isotropic multiple layered shell equations with ring support and internal pressure are established based on first order shear deformation theory (FSDT). The governing equations of motion were employed by using energy functional and by applying the Ritz method. The boundary conditions represented by end conditions of the multiple cylindrical shell are simply supported-simply supported (SS-SS), clamped-clamped (C-C), free-free (F-F), clamped-free (C-F), clamped-simply supported (C-SS), and free-simply supported (F-SS). The influences of internal pressure and ring support and the effect of the different boundary conditions on natural frequencies characteristics are studied. The results are validated by comparing them with those in the literature.

Vibration of bilayered cylindrical shells with layers of different materials

In this analysis, a comparative study for natural frequencies of two-layered cylindrical shells was presented with one layer composed of functionally graded material and the other layer of isotropic material. Love’s thin shell theory was exploited for the strain-displacement and curvature-displacement relationships. For governing frequency equations, the Rayleigh-Ritz method was utilized to minimize the Lagrangian functional in the form of an eigenvalue problem. Frequency spectra were computed for long, short, thick, and thin cylindrical shells by varying the nondimensional geometrical parameters, length-to-radius and thickness-to-radius ratios for a simply supported end condition. Influence of different configurations of cylindrical shells on the shell frequencies was studied. For validity, the results obtained were compared with some results of isotropic and single-layered functionally graded cylindrical shells from the literature.

Simulating the junctions of basic machine-tool components

In developing models of machine-tool supporting systems, it is important to determine the initial data. For the basic parts, handbook data are available; it is more difficult to determine the initial data for the junctions. For the example of special samples, the simulation of junctions between the basic parts in the form of a thin layer of isotropic material was described in [1]. It was shown that, on the basis of experimental data on the junction deformation in specified contact conditions, a model of the junction may be developed, and the elastic moduli E and G of the material simulating the junction may be determined. However, the handbook and specialized data refer to the rigidity of the junction and the damping there, but not to the deformation. Therefore, we need to determine the initial data for a model of the junction (values of E and G of the material modeling the junction) from the known rigidity and damping of the junction. Given the benefits of shell finite elements, we use a shell model in solving this problem. As an example, consider a chuck‐column junction in a 654 noncantilever milling machine. On the basis of the dimensions and configuration of the junction (Fig. 1a), a geometric model may be developed, in the form of two identical boxes of cellular structure (Fig.1b). The dimensions of the model in the XOZ plane conform to those of the junction. The box height z in the model and its wall thickness s are assumed to be 1 and 0.02 mm, respectively [1]. In this model, the junction between the chuck and column is oriented so that its rear face is connected to the guide surface of the column and its front face to the guide surface of the chuck. Given that it is very difficult to determine the initial data for junction calculation from the known rigidity and damping, we will first consider an elementary volume dV of the material simulating the junction. To determine E and G , we analyze the compression (Fig. 2a) and shear (Fig. 2b) of volume dV . In the compression of volume dV , the displacement of its points along the OX , OY , and OZ axes will be denoted by dm , dn , and dw , respectively. Only the deformation dw of volume dV (height dz , cross-sectional area dS xy ) under the action of force dF z in the direction of the OZ axis is shown in Fig. 2a. The deformation in the direction of the OX and OY axes is analogous.

Slow decay of end effects in layered structures with an imperfect interface

An asymptotic analysis of a layered structure with an imperfect interface subject to an anti-plane shear deformation and non-homogeneous Dirichlet end conditions is presented in this paper. Two layers of isotropic materials are bonded via a middle interface layer (adhesive joint), which is thin and soft; effectively, this can be described as a discontinuity surface for the displacement. Model fields are constructed to compensate for the error produced by the asymptotic solution for the case when the layered structure is subject to non-homogeneous Dirichlet end conditions. Numerical examples and analytical estimates are presented to illustrate the slow decay of the 'boundary-layer' fields.

Acoustic Radiation by Point- or Line-Excited Laminated Plates

This paper presents an elasticity theory solution for computation of acoustic radiation by a point- or line-excited fluid-loaded laminated plate, which may consist of a stack of an arbitrary number of different isotropic material layers. A one-side water-loaded three-layer sandwich plate, which consists of a hard rubber core sandwiched between two steel plates of equal thickness, was used as an example of the laminated plates. The approximated equivalent sandwich plate solutions were compared with the elasticity theory solutions. These results show that the approximated solutions are, as expected, valid only at frequencies much lower than the coincidence frequency. The numerical result also shows that, even at about one-tenth of the coincidence frequency, the approximated solutions suffer substantial error. The differences between the dry-side- and the wet-side-excited radiated fields of a single-layer uniform plate and a sandwich plate were investigated and compared, and found to be significantly different at frequencies above the coincidence frequency. [S0739-3717(00)01803-1]

Bending and extension of a multilayered plate with body forces and thermal loading

A multilayered plate composed of thin layers of isotropic materials is analyzed. The problem for the multilayered plate with body forces is formulated by using the lamination theory in which displacement fields are expressed in terms of in-plane displacements on a main plane and transverse displacement. Placing the main plane at an appropriate distance from the lower surface of the plate, a set of equilibrium equations is shown to be written in uncoupled forms, which are identical to those for an uncoupled plate such as a single layer plate. It is proved that the complete solutions of the multilayered plates subject to the specified in-plane resultant tractions or in-plane displacements on its whole boundary can be obtained from the sum of solutions for uncoupled plates. Closed form solutions are obtained for a circular laminate clamped or simply supported on its the boundary as well as for a rotating disk with a constant angular velocity. The calculations of thermoelastic stresses and displacements in multilayered plates are also discussed. Closed form solutions are obtained for a circular laminate with distributed temperature varying in the radial direction and through the thickness.

Natural frequencies of vibration of a class of solids composed of layers of isotropic materials

In a recent paper, the Ritz method with simple algebraic polynomials as trial functions was used to obtain an eigenvalue equation for the free vibration of a class of homogeneous solids with cavities. The method presented is here extended to the study of a class of non-homogeneous solids, in which each solid is composed of a number of isotropic layers with different material properties. The Cartesian coordinate system is used to describe the geometry of the solid which is modelled by means of a segment bounded by the yz, zx and xy orthogonal coordinate planes and by two curved surfaces which are defined by fairly general polynomial expressions in the coordinates x, y and z. The surface representing the interface between two material layers in the solid is also described by a polynomial expression in the coordinates x, y and z. In order to demonstrate the accuracy of the approach, natural frequencies are given for both a two- and three-layered spherical shell and for a homogeneous hollow cylinder, as computed using the present approach, and are compared with those obtained using an exact solution. Results are then given for a number of two- and three-layered cylinders and, to demonstrate the versatility of the approach, natural frequencies are given for a five-layered cantilevered beam with a central circular hole as well as for a number of composite solids of more general shape.

Prediction of edge stresses in layered media using the surface integral-finite element technique

A method for predicting the state of stress in a finite body consisting of two isotropic material layers, including a characterization of the singular stress state at the intersection of the interface with a stress-free boundary, is presented. The prediction of the “free edge stress” is accomplished using elastic stress intensity parameters. Arbitrary two-dimensional geometries with mechanical and thermal loading and plane strain or plane stress behavior are addressed. The approach consists of coupling the finite element solution of two unattached layers to a singular integral representation of distributed dislocations along the interface of two semi-infinite layers. Coupling of the two methods occurs along the interface, where displacement compatibility is enforced, and at the finite boundary, where correction forces are defined to eliminate the tractions from the semi-infinite dislocation solution. The stress intensity factors at the free edge interface are derived from the singular integral solution.

Dynamic Distribution of Displacement and Stress Considerations in the Ultrasonic Immersion Nondestructive Evaluation of Multilayered Plates

A unified theoretical treatment is presented for the calculation of displacements and stresses within multilayered media subjected to incident ultrasonic waves. The wave is supposed to be incident from water, at an arbitrary angle, upon a plate consisting of an arbitrary number of different isotropic material layers. In the first part of the analysis displacements and stresses are determined as functions of position within the plate while all layer interfaces are assumed to be rigidly bonded. A smooth interface is subsequently introduced to simulate debonding of two material layers. The composite plate is assumed to be bounded at the bottom by either a free surface, a fluid half-space or an elastic solid half-space. A byproduct of the analysis is the derivation of the reflection and transmission coefficients for the systems. Extensive numerical results are given in order to delineate the influence of the plate material orderings, layer thicknesses and interfacial conditions on the displacements and stresses within the plate. The model developed here will be of value in material characterization and in the nondestructive evaluation of advanced material applications.

A consistent refined theory for flexure of a symmetric laminate

Any two-dimensional plate theory is an approximation of the real three-dimensional elasticity problem. The classical laminated plate theory is based on the Kirchhoff hypothesis and ignores the effects of transverse shear deformation, normal stress, normal strain and nonlinear in-plane normal strain distribution through the plate thickness [ 1,2]. Two types of composite plates are generally identified in practice: (i) 'fibre reinforced laminates' in which layers of composite materials with high ratios of Young's-to-shear modulii are bonded together and (2) 'sandwiches' in which layers of isotropic materials with some layers having significantly lower elastic modulii than others, are bonded together. The effects of shear deformation are significant in these situations and thus the classical theory is inadequate. Exact elasticity solutions for flexure of some standard composite and sandwich plate problems have been obtained by Pagano [ 3] and Pagano and Hatfield [4]. Whitney [5] and Mau [6] have presented first-order laminate theories in which transverse shear strain is assumed constant through the thickness. This required, however, use of a transverse shear correction factor which generally varied with the lamination scheme.