Refined solution of the timoshenko problem for an orthotropic beam on a rigid base

We have obtained the solution of the problem of a composite (orthotropic) beam partly supported on a perfectly rigid base under a uniformly distributed load. For calculations, we use the refined model of beams that takes into account the strains of transverse shear and compression. We have derived an equation for evaluating the size of the contact area and relations for calculating the contact pressure of the rigid base on the external surface of the beam. Numerical results obtained by the refined model for isotropic and orthotropic materials are compared with those corresponding to the classical theory, the shear model of beams, and the plane problem of the theory of elasticity.     

On stress concentrations for isotropic/orthotropic plates and cylinders with a circular hole

Scale factors (SFs) are widely used in engineering applications to describe the stress concentration factor (SCF) of a finite width isotropic plate with a circular hole and under uniaxial loading. In this paper, these SFs were also found to be valid in an isotropic plate with biaxial loading and an isotropic cylinder with uniaxial loading or internal pressure, if a suitable hole to structure dimension ratio was chosen. The study was further expanded to consider orthotropic plates and cylinders with a center hole and under uniaxial loading. The applicable range of the SFs was given based on the orthotropic material parameters. The influence of the structural dimension on the SCF was also studied. An empirical calculation method for the stress concentrations for isotropic/orthotropic plates and cylinders with a circular hole was proposed and the results agreed well with the FEM simulations. This research work may provide structure engineers a simple and efficient way to estimate the hole effect on plate structures or pressure vessels made of isotropic or orthotropic materials.     

Propagation of Gaussian Schell-model Beams in Dispersive and Absorbing Media

Abstract

The spectral properties and the coherence properties of Gaussian Schell-model beams, propagating in dispersive and absorbing media, are discussed. Unlike in free space, the ratio of the transverse spectral correlation length to the beam width is found to increase on propagation. Consequently upon propagation the beam becomes spatially more coherent at each frequency. Numerical results for the spectrum and for the degrees of spectral and of temporal coherence of the field are presented for some selected values of the beam parameters and for several values of the propagation distance. Propagation in a gain medium is also briefly discussed.     

The Spatial Coherence Properties of Gaussian Schell-model Beams

The transverse and longitudinal spatial coherence properties of the light beams generated by planar gaussian Schell-model sources are discussed. It is found that for all gaussian Schell-model beams the ratio of the transverse coherence length to the beam width remains invariant upon propagation. An examination of the longitudinal coherence for both on-axis and off-axis pairs of points indicates that the longitudinal coherence will not, in general, die out as the separation between the points is increased. Rather, the degree of longitudinal coherence will approach a finite (non-zero) value as long as the source contains a finite coherence area, regardless of how small this area may be. Gaussian quasihomogeneous beams are studied as a limiting case. The relation of the present work to the analysis of speckle size is briefly discussed.     

Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory

This article presents the buckling analysis of orthotropic nanoplates such as graphene using the two-variable refined plate theory and nonlocal small-scale effects. The two-variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Nonlocal governing equations of motion for the monolayer graphene are derived from the principle of virtual displacements. The closed-form solution for buckling load of a simply supported rectangular orthotropic nanoplate subjected to in-plane loading has been obtained by using the Navier’s method. Numerical results obtained by the present theory are compared with first-order shear deformation theory for various shear correction factors. It has been proven that the nondimensional buckling load of the orthotropic nanoplate is always smaller than that of the isotropic nanoplate. It is also shown that small-scale effects contribute significantly to the mechanical behavior of orthotropic graphene sheets and cannot be neglected. Further, buckling load decreases with the increase of the nonlocal scale parameter value. The effects of the mode number, compression ratio and aspect ratio on the buckling load of the orthotropic nanoplate are also captured and discussed in detail. The results presented in this work may provide useful guidance for design and development of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic nanoplates.     

Free vibration of symmetrically laminated plates using a higher-order theory with finite element technique

A C⁰ finite element formulation of the higher-order theory is used to determine the natural frequencies of isotropic, orthotropic and layered anisotropic composite and sandwich plates. The material properties that are typical of high modulus fibre reinforced composites are used to show the parametric effects of plate aspect ratio, length-to-thickness ratio, degree of orthotropy, number of layers and lamination angle/scheme. The present theory is based on a higher-order displacement model and the three-dimensional Hooke's laws for plate material. The theory represents a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strains through the plate thickness. A special mass matrix diagonalization scheme is adopted which conserves the total mass of the element and includes the effects due to rotary inertia terms. The results presented should be useful in obtaining better correlation between theory and experiment, and to numerical analysts in verifying their results.     

Size dependent buckling analysis of nano sandwich beams by two schemes

Abstract

Within the framework of Timoshenko beam theory, the buckling of nano sandwich beams is developed. The material properties are assumed to vary arbitrarily in both axial and thickness directions. These types of beams are referred to as bi-directional functionally graded (BDFG) beams. Two types of nano sandwich beams with different material distribution patterns and immovable supports are considered. Since the size effects play a significant role in mechanical behavior of nanostructures, the small-scale effects are captured by Eringen’s nonlocal theory of elasticity. The governing equations are derived using the variational formulation. Symmetric smoothed particle hydrodynamics (SSPH) and the Galerkin method are adopted as numerical solution approaches. As a truly meshless method, the convergence of the SSPH technique mainly depends on the smoothing length value and distribution of particles in the compact support domain of the kernel function. The Revised Super Gauss Function is used as the kernel function and an optimum value for the smoothing length that bears the fastest convergence rate is obtained. The solution methods are verified through benchmark problems found in the literature. Numerical and illustrative results show that various parameters, including the aspect ratio, nonlocal parameter, gradient indexes, and cross-sectional types have significant effects on the buckling responses of BDFG nano sandwich beams.     

Locking-free finite elements for shear deformable orthotropic thin-walled beams

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are presented. The assumed kinematical model is based on Timoshenko–Reissner theory so as to take shear strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms arise between bending in the two principal planes and between bending and torsion. The adopted model holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared with the available numerical examples. In particular, some shape functions are obtained from ‘modified’ Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections are performed to verify convergence rate and accuracy of the proposed formulations, especially in the presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models are compared with results given by plate-shell models. Copyright © 2007 John Wiley & Sons, Ltd.     

Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers

This paper presents analytical solution for functionally graded material (FGM) beams integrated with piezoelectric actuator and sensor under an applied electric field and thermo-mechanical load. In FGM host properties is assumed to vary exponentially in thickness direction and the Poisson’s ratio is held constant. The hybrid beam is in a state of plane stress and the piezoelectric is composed of orthotropic materials. The beam is simply supported with the bottom surface traction free and zero temperature. By using of state-space method in thickness direction and Fourier series in longitudinal direction, the solution can be made. To verify the accuracy of the present formulation, numerical results for the simple case is compared with results obtained in the published literature. Finally, effects of FGM index, electromechanical coupling, thickness ratio and thermo-mechanical surface boundary condition on the bending behaviour of beam are investigated.     

Deformation analysis of functionally graded beams by the direct approach

In this paper we employ the direct approach to the theory of rods and beams, which is based on the deformable curve model with a triad of rotating directors attached to each point. We show that this model (also called directed curve) is an efficient approach for analyzing the deformation of elastic beams with a complex material structure. Thus, we consider non-homogeneous, composite and functionally graded beams made of isotropic or orthotropic materials and we determine the effective stiffness properties in terms of the three-dimensional elasticity constants. We present general analytical expressions of the effective stiffness coefficients, valid for beams of arbitrary cross-section shape. Finally, we apply this method for FGM beams made of metal foams and compare our analytical results with the numerical results obtained by a finite element analysis.     

Improved engineering theory for uniform beams

Summary A small static symmetric bending deformation of isotropic linear elastic beams under arbitrary transverse loading varying slowly with the axial coordinate is considered. An asymptotic analysis of the three-dimensional variational equation — in which the small parameter is the ratio of maximum cross-sectional dimension to beam length — gives Timoshenko type governing equations, corresponding boundary conditions, improved formulae for the displacements, and, unlike known beam theories, for all stresses a plane problem in the cross-sectional domain to be solved. Predictions of the theory for beams of narrow rectangular and circular cross-sections are compared with explicit elasticity solutions.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

Constitutive models for ductile solids reinforced by rigid spheroidal inclusions

We study the effective constitutive response of composite materials made of rigid spheroidal inclusions dispersed in a ductile matrix phase. Given a general convex potential characterizing the plastic “in the context of J₂-deformation theory” behavior of the isotropic matrix, we derive expressions for the corresponding effective potentials of the rigidly reinforced composites, under general loading conditions. The derivation of the effective potentials for the nonlinear composites is based on a variational procedure developed recently by Ponte Castaneda (1991a, J. Mech. Phys. Solids 39, 45–71). We consider two classes of composites. In the first class, the spheroidal inclusions are aligned, resulting in overall transversely isotropic symmetry for the composite. In the second class, the inclusions are randomly oriented, and thus the composite is macroscopically isotropic. The effective response of composites with aligned inclusions depends on both the orientation of the loading relative to the inclusions and on the inclusion concentration and shape. Comparing the strengthening effects of rigid oblate and prolate spheroids, we find that prolate spheroids give rise to stiffer effective response under axisymmetric “relative to the axis of transverse isotropy” loading, while oblate spheroids provide greater reinforcement for materials loaded in transverse shear. On the other hand, nearly spherical “slightly prolaterd spheroids are most effective in strengthening the composite under longitudinal shear. Thus, the optimal shape for strengthening composites with aligned inclusions depends strongly on the loading mode. Alternatively, the properties of composites with randomly oriented spheroidal inclusions, being isotropic, depend only on the concentration and shape of the inclusions. We find that both oblate and prolate inclusions lead to significant strengthening for this class of composites.     

Micromechanics determination of electroelastic properties of piezoelectric materials containing voids

This paper examines the electroelastic properties of piezoelectric materials that contain voids. A unified micromechanics approach is adopted for determining the properties. Voids are treated as spheroidal inclusions with zero elastic moduli. The surrounding material is assumed to be linearly piezoelastic and transversely isotropic. The electroelastic Eshelby tensors for spheroidal inclusions have been evaluated numerically for different aspect ratios. Utilizing these tensors and applying the Mori–Tanaka mean field theory that accounts for the interaction between inclusions and matrix, the effective electroelastic properties of the materials are obtained. Numerical examples are given based on PZT-5H and BaTiO₃. Influences of the volume fraction and aspect ratio of voids on the material properties have been studied. Emphasis has been placed on the piezoelastic coupling effect of the material. For both materials, the piezoelastic coupling provides a stiffening effect on the materials, and the influence is more pronounced when void volume increases and when the aspect ratio of voids becomes shorter.     

Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration

This paper initiates the theoretical analysis of nonlinear microbeams and investigates the static bending, postbuckling and free vibration. The nonlinear model is conducted within the context of non-classical continuum mechanics, by introducing a material length scale parameter. The nonlinear equation of motion, in which the nonlinear term is associated with the mean axial extension of the beam, is derived by using a combination of the modified couple stress theory and Hamilton’s principle. Based on this newly developed model, calculations have been performed for microbeams simply supported between two immobile supports. The static deflections of a bending beam subjected to transverse force, the critical buckling loads and buckled configurations of an axially loaded beam, and the nonlinear frequencies of a beam with initial lateral displacement are discussed. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio. Our results also indicate that the nonlinearity has a great effect on the static and dynamic behaviors of microscale beams. To attain accurate and reliable characterization of the static and dynamic properties of microscale beams, therefore, both the microstructure-dependent parameters and the nonlinearities have to be incorporated in the design of microscale beam devices and systems.     

On universal relations in orthotropic material subclasses

Universal relations that hold true for specific classes or subclasses of elastic materials form an integral part of the characterisation process for any material that can deform elastically. On the other hand, it has become understood that any class of hyper-elastic materials possessing anisotropic properties more complicated to those of transverse isotropy obeys no universal relations. However, there are met in practice special cases (subclasses) of anisotropic hyper-elastic materials that, under certain circumstances, may still obey a certain number of universal relations. Material characterisation purposes suggest therefore that there is a practical purpose in identifying and classifying different subclasses of anisotropic hyper-elastic materials that may obey any kind of universal relations and, hence, to derive and gradually build up catalogues or tables of different universal relations that such material subclasses obey when subjected to different deformation patterns. This paper heads towards this research direction by initially considering the orthotropic material class which is alternatively identified as the hyper-elastic crystal class of rhombic symmetry [A.E. Green, J.E. Adkins, Large Elastic Deformations, Clarendon Press, Oxford, 1960]. Then, for a certain set of appropriately chosen orthotropic and transverse isotropic hyper-elastic material subclasses, it develops and presents a considerable number of relevant universal relations, most of which are new in the literature.     

A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading

Due to the variation in material properties through the thickness, bifurcation buckling cannot generally occur for plates or beams made of functionally graded materials (FGM) with simply supported edges. Further investigation in this paper indicates that FGM beams subjected to an in-plane thermal loading do exhibit some unique and interesting characteristics in both static and dynamic behaviors, particularly when effects of transverse shear deformation and the temperature-dependent material properties are simultaneously taken into account. In the analysis, based on the nonlinear first-order shear deformation beam theory (FBT) and the physical neutral surface concept, governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform in-plane thermal loading are derived. Then, a shooting method is employed to numerically solve the resulting equations. The material properties of the beams are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and to be temperature-dependent. The effects of material constants, transverse shear deformation, temperature-dependent material properties, in-plane loading and boundary conditions on the nonlinear behavior of FGM beams are discussed in detail.     

End corrections for orthotropic DCB specimens

An extension of the beam on an elastic foundation model for estimating the end rotation correction for orthotropic double cantilever beam specimens is described. The solutions are compared with finite element results and a small empirical modification to the analytical result gives a simple formula for calculating an effective crack length correction of the form (a + χh), where h is the thickness. Values of χ for orthotropic materials are about 2·5 compared with 0·67 for isotropic materials. Such large corrections are likely to be important in data analysis.     

The Effect of Load and Geometry on the Failure Modes of Sandwich Beams

Facing compressive failure, facing wrinkling and core shear failure are the most commonly encountered failure modes in sandwich beams with facings made of composite materials. The occurrence and sequence of these failure modes depends on the geometrical dimensions, the form of loading and type of support of the beam. In this paper the above three failure modes in sandwich beams with facings made of carbon/epoxy composites and cores made of aluminum honeycomb and two types of foam have been investigated. Two types of beams, the simply supported and the cantilever have been considered. Loading included concentrated, uniform and triangular. It was found that in beams with foam core facing wrinkling and core shear failure occur, whereas in beams with honeycomb core facing compressive failure and core shear crimping take place. Results were obtained for the dependence of failure mode on the geometry of the beam and the type of loading. The critical beam spans for failure mode transition from core shear to wrinkling failure were established. It was found that initiation of a particular failure mode depends on the properties of the facing and core materials, the geometrical configuration, the type of support and loading of sandwich beams.     

Transverse Shear and Normal Deformation Theory for Bending Analysis of Laminated and Sandwich Elastic Beams

The influence of transverse normal strain on bending analysis of cross-ply laminated and sandwich beams is presented. A higher-order shear deformation beam theory is developed. Euler-Bernoulli classical, Timoshenko first-order and simple higher-order theories have been also used in the analysis. The governing equations for a beam composed of orthotropic layers and subjected to any given mechanical load distribution are derived. Making use of Navier-like approach, exact solutions are obtained for cross-ply laminated and sandwich beams subjected to arbitrary loadings. Numerical results for beams with the simply-supported boundary conditions are presented. The effects due to transverse normal strain, transverse shear deformation and number of layers on the static response of the beams are investigated.