Three-dimensional Elasticity Equations Research Articles

Three-dimensional Elasticity Solution for Uniformly Loaded Cross-ply Laminates and Sandwich Plates

This article establishes the bending problem of cross-ply laminated plates using the three-dimensional elasticity equations as well as the technique based on the state space concept. It presents a wide variety of results for the symmetric and antisymmetric analyses of rectangular multilayer plates subjected to a sinusoidally/uniformly distributed load (SDL/UDL). In addition, this study provides a strong mathematical tool allowing one to determine, in an exact and unified manner, the state of stress and displacement of cross-ply laminated composites and sandwich plates. The well-known results (given in Pagano ((1970), Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates, Journal of Composite Materials, 4(1): 20—24.) and Pagano and Hatfield ((1972), Elastic Behaviour of Multilayered Bidirectional Composites, AIAA Journal, 10(12): 931—933.) due to the exact three-dimensional elasticity solution are in fact special cases of the present technique.

Stress singularities in bimaterial bodies of revolution

An eigenfunction expansion solution is first developed for determining the stress singularities of bimaterial bodies of revolution by directly solving the equilibrium equations of three-dimensional elasticity in terms of displacement functions. The characteristic equations are explicitly given for determining the stress singularities in the vicinity of the interface corner of two intersecting bodies of revolution having a sharp corner with free boundary conditions along the corner. The characteristic equations are found to be equivalent to a combination of the characteristic equations for plane elasticity problems and St. Venant torsion problems. The strength of stress singularities varying with the interface angles is also investigated. The first known asymptotic solutions for the displacement and stress fields are also explicitly shown for some interface angles. The present results will be useful not only for understanding the singularity behaviors of stresses in the vicinity of a revolution interface corner, but also for developing accurate numerical solutions with fast convergence for stress or vibration analysis of a body of revolution having an interface corner.

Heat capacity of suspended phonon cavities

We present a detailed analysis of the vibrational spectrum and heat capacity of suspended mesoscopic dielectric plates, for various thickness-to-side ratios at sub-Kelvin temperatures. The vibrational modes of the suspended cavity are accurately obtained from the three-dimensional (3D) elastic equations in the small strain limit and their frequencies assigned to the cavity phonon modes. The calculations demonstrate that the heat capacity of realistic quasi-2D phonon cavities approach the linear dependence on T at sub-Kelvin temperatures. The behavior is more pronounced for the thinnest cavities, but takes place also for moderately thick structures, with thickness-to-side ratios $\gamma$=0.1 to 0.2. It is also demonstrated that the heat capacity of the suspended phonon cavities is invariant under the product of the temperature (T) with a characteristic lateral dimension (L) of the sample. The present results establish a lower bound for the heat capacity of suspended mesoscopic structures and indicate the emergence of the quantum mechanical regime in the dynamics of bounded phonon cavities.

A Novel Finite-Element– Numerical-Integration Model for Composite Laminates Supported on Opposite Edges

Abstract An attempt is made here to devise a new methodology for an integrated stress analysis of laminated composite plates wherein both in-plane and transverse stresses are evaluated simultaneously. The method is based on the governing three-dimensional (3D) partial differential equations (PDEs) of elasticity. A systematic procedure is developed for a case when one of the two in-plane dimensions of the laminate is considered infinitely long (y direction) with no changes in loading and boundary conditions in that direction. The laminate could then be considered in a two-dimensional (2D) state of plane strain in x-z plane. It is here that the governing 2D PDEs are transformed into a coupled system of first-order ordinary differential equations (ODEs) in transverse z direction by introducing partial discretization in the finite inplane direction x. The mathematical model thus reduces to solution of a boundary value problem (BVP) in the transverse z direction in ODEs. This BVP is then transformed into a set of initial value problems (IVPs) so as to use the available efficient and effective numerical integrators for them. Through thickness displacement and stress fields at the finite element discrete nodes are observed to be in excellent agreement with the elasticity solution. A few new results for cross-ply laminates under clamped support conditions are also presented for future reference and also to show the generality of the formulation.

Lamb wave propagation in magnetoelectroelastic plates

Based on the three-dimensional linear elastic equations and magnetoelectroelastic constitutive relations, propagation of symmetric and antisymmetric Lamb waves in an infinite magnetoelectroelastic plate is investigated. The coupled differential equations of motion are solved, and the phase velocity equations of symmetric and antisymmetric modes are obtained for both electrically and magnetically open and shorted cases. The dispersive characteristic of wave propogation is explored. The mechanical, electric and magnetic responses of the lowest symmetric and antisymmetric Lamb wave modes are discussed in detailed. Obtained results are valuable for the analysis and design of broadband magnetoelectric transducer using composite materials.

Static and dynamic analysis of elastohydrodynamic elliptical journal bearing with micropolar lubricant

In this paper the effect of deformation of the bearing liner on the static and dynamic performance characteristics of an elliptical (two-lobe) journal bearing operating with micropolar lubricant is presented. Lubricating oil containing additives and contaminants is modeled as micropolar fluid. A generalized form of Reynold's equation is derived from the fluid flow and diffusion equations. Finite element technique is used to solve the modified Reynold's equation governing the flow of micropolar lubricant in the clearance space of the journal bearing and the three-dimensional elasticity equations governing the displacement field in the bearing shell. The static and dynamic characteristics of the bearing are computed for a wide range of deformation coefficient which takes into accountant the flexibility of bearing liner by treating operating lubricant as (i) Newtonian and (ii) micropolar. The computed results show that the increasing volume concentration of additives and mass transfer of additives produce significant changes on the performance characteristics.

Bending Solutions for Inhomogeneous and Laminated Elastic Plates

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a thick plate of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the homogeneous material. Recently this theory has been formulated in terms of functions of a complex variable. It was shown that the displacement and stress fields in the inhomogeneous material could be expressed in terms of four complex potentials that are analytic functions of the complex variable ζ = x + iy in the mid-plane of the plate. However, the analysis performed so far applies only to the case of a plate with traction-free upper and lower faces. The present paper extends these solutions to the case where the plate is bent by a pressure distribution applied to a face.

Complex Variable Solutions for Inhomogeneous and Laminated Elastic Plates

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we formulate this theory in terms of complex potentials. It is shown that the displacement and stress in the inhomogeneous material can be expressed in terms of four complex potentials that are analytic functions of ζ = x + iy. Expressions for stress and moment resultants are derived in terms of these complex potentials. As examples we solve the problems of biaxial stress in an infinite plate and of an in-plane point force in an infinite plate and at the boundary of a semiinfinite plate. We also formulate the general boundary-value problem for a half-plane and develop a procedure to determine the complex potentials for an inhomogeneous plate in terms of the complex potentials for the corresponding plane-strain problem.

Optimal Stacking Sequence of Laminated Cylindrical Shells Using Genetic Algorithm

Stacking sequence of a laminated cylindrical shell is optimized based on natural frequency. In the optimization problem, it is desired to have the maximum possible natural frequency. Laminated thickness is fixed and orientations are changed in a set of angles. A laminated cylindrical shell is considered anisotropic with finite length and simply supported edges. Free-vibration analysis is based on fully three-dimensional elasticity equations. The highly coupled partial differential equations are reduced to ordinary differential equations with variable coefficients by means of trigonometric function expansion in the axial dimension. The resulting differential equations are solved exactly. In this solution, the continuity conditions between any two layers are satisfied. A genetic algorithm is devised for designing composite laminates composed of only one material. The genetic algorithm procedure, referred to as GA-I, will be used for maximization of the lowest natural frequency of a laminated cylindrical shell with strength and ply contiguity constraints. Results are presented for stacking sequence optimization of three- and five-layer cylindrical shells.

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.

Dispersion analysis of a nonconforming finite element method for the three-dimensional scalar and elastic wave equations

We investigate the dispersive properties of a nonconforming finite element method (NC-method) to solve the three-dimensional scalar and elastic wave equations. The study is performed by constructing and analyzing the dispersion relations, and by evaluating derived quantities such as dimensionless phase and group velocities. The behaviour of the present algorithm is compared with that of a conforming finite element method (C-method) with the same order of spatial approximation. It is observed that the NC-method introduces less numerical anisotropy and dispersion than the C-method, allowing to work with fewer than 10 points per wavelength with reasonably small relative errors. On the other hand, the computational cost of the NC-method is higher than that of the C-method. However, the present NC-method is suitable for domain decomposed parallel algorithms; therefore the fact that the NC-method needs more computing time may be partially compensated by its lower transmission load.

Transverse Vibrations of Laminated Beams in Three-Dimensional Formulation

A new asymptotic method is proposed to describe the free and forced transverse vibrations of elastic laminated beams of arbitrary cross section using the three-dimensional elastic equations without additional hypotheses and constraints. For beams with layers of equal Poisson's ratio, the zero-order natural frequencies are equal to the natural frequencies predicted by classical beam theory based on the Bernoulli hypothesis. The method makes it possible to calculate the frequencies of free vibrations and amplitudes of forced vibrations with prescribed accuracy for the first natural modes

Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.

Dynamic Analysis of Orthotropic Laminated Cylindrical Panels

The dynamic response of an axisymmetric cross-ply laminated panel subjected to axisymmetric loading based on three-dimensional elasticity equations are studied. The shell panel is simply supported at four sides and has finite length. The highly coupled partial differential equations are reduced to ordinary differential equations with variable coefficients by means of a trigonometric function expansion in the circumferential and axial directions. The resulting ordinary differential equations are solved via the Galerkin finite element method. Numerical examples are presented for [0/90°], [90/0°], [0/90/0°], and [90/0/90°] laminations under dynamic loading.

Natural frequencies of orthotropic, monoclinic and hexagonal plates by a meshless method

The collocation method with multiquadrics basis functions and a first-order shear deformation theory are used to find natural flexural frequencies of a square plate with various material symmetries and subjected to different boundary conditions. Computed results are found to agree well with the literature values obtained by the solution of the three-dimensional elasticity equations using the finite element method.

Finite element analysis of elastohydrodynamic circular journal bearing with micropolar lubricants

In this paper the effect of deformation of the bearing liner on the static characteristics of a circular journal bearing operating with micropolar fluid is analysed. Lubricating oil containing additives and contaminants is modeled as micropolar fluid. A modified Reynold's equation is obtained by considering the effect of mass transfer across the fluid film in the fluid flow equations. Finite element technique has been used to solve the modified Reynold's equation governing the flow of micropolar fluid in the clearance space of the journal bearing and the three-dimensional elasticity equations governing the displacement field in the bearing shell. The static characteristics of the bearing are presented for a wide range of deformation coefficient which takes into account the flexibility of the bearing liner. The increase in volume concentration of additives in the lubricant produces significant effects in the performance characteristics of the bearing especially when the bearing operates at higher eccentricity ratio.

Benchmark solution of imperfect angle-ply laminated rectangular plates in cylindrical bending with surface piezoelectric layers as actuator and sensor

The static and dynamic responses of simply supported adaptive angle-ply laminates in cylindrical bending are considered. The interlaminar bonding of the host elastic laminate is assumed to be imperfect, described by a spring-layer model, while the bonding between the host elastic laminate and the surface piezoelectric actuator and sensor layers is perfect. The state-space approach, which is directly based on the three-dimensional exact elasticity (piezoelasticity) equations and very effective in analyzing laminated structures, is employed. The numerical results should provide a useful means of comparison in the development of simplified analyses or numerical methods.

On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies

Based on rigorous operator theory, a general solution of the three-dimensional elasticity equations of equilibrium for 1D hexagonal quasicrystals is obtained. The solution is expressed in terms of four quasiharmonic functions, which is very simple and useful. The point phonon (phason) force solution of an infinite 1D hexagonal quasicrystal body is derived, all in terms of elementary functions. They can play an important role in numerical simulations such as boundary element method.

On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies

Based on rigorous operator theory, a general solution of the three-dimensional elasticity equations of equilibrium for 1D hexagonal quasicrystals is obtained. The solution is expressed in terms of four quasiharmonic functions, which is very simple and useful. The point phonon (phason) force solution of an infinite 1D hexagonal quasicrystal body is derived, all in terms of elementary functions. They can play an important role in numerical simulations such as boundary element method .

A refined theory for thick spherical shells

A theory for thick spherical shells is presented in this paper. The equations given here do not only incorporate the effects of transverse shear deformation but also account for the initial curvature as well as the radial stress. The proposed theory presents a very good approximation for the shell constitutive equations and the nonlinear distribution of the in plane stresses across the thickness of the shell. The later is very important for thick shell analyses. The presented formulation is based on the following: (1) assumed out of plane stress components which satisfy given boundary conditions; (2) three-dimensional elasticity equations with an integral form of the equilibrium equations; (3) stress resultants and stress couples acting on the middle surface of the shell, average displacements along the normal at a point on the middle surface, and average rotations of the normal. The proposed shell equations can be conveniently used in finite element analysis. An application of this theory to the finite element analysis of spherical shells will be presented in the follow-up paper.