an efficient higher order zigzag theory for laminated plates subjected to thermal loading a new efficient higher order zigzag theory is presented for laminated plates under thermal loading the third order zigzag model is modified by replacing the uniform across the thickness approximation for the deflection with a layer wise variable approximation for deflection which explicitly accounts for the transverse thermal strain the thermal field is approximated as piecewise linear across the sub layers the displacement field is expressed in terms of the thermal field and only five primary displacement variables by satisfying exactly the conditions of zero transverse shear stresses at the top and the bottom and their continuity at the layer interfaces the governing equations are derived using the principle of virtual work comparison of navier solutions for simply supported rectangular test plate devised for this study and composite and sandwich plates with the exact three dimensional thermo elasticity solutions for two kinds of thermal loads establishes that the present efficient zigzag theory is generally more accurate than the existing zigzag theory

analysis of laminated plates by trigonometric theory radial basis and unified formulation a study using the radial basis function rbf collocation method to analyze static deformations free vibrations and buckling loads of square cross ply laminated plates using a sinus based shear and normal deformation theory is presented allowing for through the thickness deformations the principle of virtual displacements is used to obtain the equations of motions and boundary conditions and the governing equations in strong form for the rbf method are observed a simply supported square laminated plate of side a and thickness h composed of four equally layers oriented at 0 90 90 0 is considered it is found that the present mesh less results are in very close agreement with the mesh free results the laminated plate is supported along the edges parallel to the x axis while the other two edges may be simply supported s or clamped c an excellent agreement is achieved for all edge conditions when comparing the results obtained by the present rbf approach with those found earlier

free vibration of laminated composite and sandwich plates using global local higher order theory abstract in this paper the global local higher order theory is used to study the free vibration of laminated composite and sandwich plates this global local theory can satisfy the free surface conditions and the geometric and stress continuity conditions at interfaces and the number of unknowns is independent of the layer numbers of the laminate based on the higher order theory a refined three noded triangular element satisfying c1 weak continuity conditions is presented for general laminated composite plates results obtained from present global local higher order theory have been found in good agreement with those obtained from three dimensional elasticity theories moreover this theory is still suitable for analysis of laminated plates with arbitrary layouts and soft core sandwich plates whereas numerical results show that the global higher order and first order theory overestimate natural frequency for these special structures this theory cannot only calculate the natural frequencies but can accurately predict the modal stress distributions in the thickness direction without any smooth techniques

operator means of probability measures abstract let p be the complete metric space consisting of positive invertible operators on an infinite dimensional hilbert space with the thompson metric we introduce the notion of operator means of probability measures on p in parallel with kubo and ando s definition of two variable operator means and show that every operator mean is contractive for the wasserstein distance by means of a fixed point method we consider deformation of such operator means and show that the deformation of any operator mean becomes again an operator mean in our sense based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures

evaluation of layerwise mixed theories for laminated plates analysis the evaluation of mixed layerwise theories to calculate the in plane and out of plane responses of thick plates in two dimensional modeling of multilayered structures is made the employed models which were proposed by the author in earlier works a priori fulfill the continuity of transverse shear and normal stress components at the interface between two adjacent layers a reissner s mixed variational equation is used to derive the governing equations in terms of introduced stress and displacement variables the interface continuity conditions are imposed by writing the governing equations at a multilayered level the related standard displacement formulations are also discussed for comparison purpose closed form solutions are presented for plates made of orthotropic lamina and bent by harmonic distribution of transverse pressure symmetrically and unsymmetrically laminated as well as sandwich plates have been investigated a comparison with a three dimensional elasticity analysis shows that present mixed layerwise models furnish a better description of the in plane and out of plane response of thick plates with respect to existing layerwise and equivalent single layer theories in particular the proposed models describe with excellent accuracy the transverse shear and normal stress fields unlike available current models these fields are herein determined a priori without requiring implementation of any postprocessing procedures the distribution of the transverse displacement and transverse normal stress in the plate thickness direction are also shown for most of the problems

a variable kinematic ritz formulation for vibration study of quadrilateral plates with arbitrary thickness a new variable kinematic ritz method applied to free vibration analysis of arbitrary quadrilateral thin and thick isotropic plates is presented carrera s unified formulation and the versatile pb 2 ritz method are properly combined to build a powerful yet simple modeling and solution framework the proposed technique allows to generate arbitrarily accurate ritz solutions from a large variety of refined two dimensional plate theories by expanding so called ritz fundamental nuclei of the plate mass and stiffness matrices theoretical development of the present methodology is described in detail convergence and accuracy of the method are examined through several examples on thin moderately thick and very thick plates of rectangular skew trapezoidal and general quadrilateral shapes with an arbitrary combination of clamped free and simply supported edges present results are compared with existing three dimensional solutions from open literature maximum and average differences of various higher order plate theories and three dimensional results are also presented with the aim of providing useful guidelines on the choice of appropriate plate theory to get a desired accuracy on frequency parameters

bonded dissimilar strips with a crack perpendicular to the functionally graded interface the mode i crack problem for two dissimilar infinite strips bonded through a functionally graded interfacial zone is investigated within the framework of plane elasticity the graded interfacial zone is treated as a nonhomogeneous continuum having the continuously varying elastic modulus between the dissimilar homogeneous strips a crack is assumed to be embedded in one of the strips perpendicular to the nominal interface with the aid of the stiffness matrix approach a set of homogeneous conditions relevant to the proposed crack problem is readily satisfied subsequent application of the mixed conditions on the cracked plane then leads to a singular integral equation of the first kind which is solved numerically in consequence the variations of stress intensity factors are provided as functions of geometric and material parameters of the bonded structure the effect of the material nonhomogeneity in the interfacial zone is further addressed by measuring the degree of correspondence with the results that are obtained based on the use of the homogenized interfacial elastic property

free vibrations of multilayered doubly curved shells based on a mixed variational approach and global piecewise smooth functions abstract this paper presents the extension of a two dimensional model that recently appeared in literature deals with freely vibrating laminated plates the extension takes into account the corresponding theory describing the dynamic of freely vibrating multilayered doubly curved shells the relevant governing differential equations associated boundary conditions and constitutive equations are derived from one of reissner s mixed variational theorems both the governing differential equations and the related boundary conditions are presented in terms of resultant stresses and displacements in spite of the multi layer nature of the shell the theory is developed as if the shell were virtually made of a single layer this choice does not limit the performances of the model which are comparable to the corresponding three dimensional theory this ability is accomplished by an appropriate global expansion of the relevant kinetic and stress quantities through the thickness of the multilayered shell the mentioned expansion is realized by a novel selection of global piecewise smooth functions numerical tests illustrate the performance of the model with respect to several elements subjected to a class of simply supported boundary conditions plates circular cylindrical shells spherical and saddle shape laminates the model is first tested by comparing its resulting eigen parameters with those few existing of exact or approximate three dimensional models and finally new results are provided for several geometrical and material characteristics for plates and shells

multilayered shell finite element with interlaminar continuous shear stresses a refinement of the reissner mindlin formulation a finite element formulation for refined linear analysis of multilayered shell structures of moderate thickness is presented an underlying shell model is a direct extension of the first order shear deformation theory of reissner mindlin type a refined theory with seven unknown kinematic fields is developed i by introducing an assumption of a zig zag i e layer wise linear variation of displacement field through the thickness and ii by assuming an independent transverse shear stress fields in each layer in the framework of reissner s mixed variational principle the introduced transverse shear stress unknowns are eliminated on the cross section level at this process the interlaminar equilibrium conditions i e the interlaminar shear stress continuity conditions are imposed as a result the weak form of constitutive equations the so called weak form of hooke s law is obtained for the transverse strains transverse stress resultants relation a finite element approximation is based on the four noded isoparametric element to eliminate the shear locking effect the assumed strain variational concept is used performance of the derived finite element is illustrated with some numerical examples the results are compared with the exact three dimensional solutions as well as with the analytical and numerical solutions obtained by the classical the first order and some representative refined models copyright 2000 john wiley sons ltd

variable kinematic model for the analysis of functionally graded material plates this work addresses the static analysis of functionally graded material plates subjected to transverse mechanical loadings the unified formulation and principle of virtual displacements are employed to obtain both closed form and finite element solutions the use of the unified formulation permits a large variety of plate models with variable kinematic assumptions to be compared in the same framework these differ according to the order of the expansion in the thickness direction and the variable description the order of the expansion ranges from 1 to 4 thus covering first order as well as higher order theories the description of the unknowns can be equivalent single layer or layerwise the dependence of the material data on the thickness direction was introduced by employing thickness functions that are derived from the legendre polynomials that are used in the layerwise case the proposed approach is independent of the used transition function and as a result any continuous variations of the material properties in the thickness direction may be easily implemented the obtained solutions closed form and finite element are validated through comparison with three dimensional exact solutions and other available solutions these show the limitations of classical plate theories as well as the convenience of the use of the proposed variable kinematic models for the analysis of functionally graded material plates

free vibrations of multilayered plates based on a mixed variational approach in conjunction with global piecewise smooth functions abstract this paper presents a two dimensional model for the analysis of freely vibrating laminated plates the governing differential equations associated boundary conditions and constitutive equations are derived from reissner s mixed variational theorem both the governing differential equations and the related boundary conditions are presented in terms of resultant stresses and displacements the model is able to provide the results for the corresponding three dimensional theory such a performance is guaranteed from an appropriate expansion of relevant kinetic and stress quantities through the thickness of the multilayered plate the expansion is realized by using a novel selection of global piecewise smooth functions gpsfs the number of gpsfs can be arbitrarily increased to achieve a two dimensional plate theory which is at least as accurate as that of a full layerwise theory it is also shown that gpsfs permit to deal a multilayered plate as if it was virtually made of a single layer indeed the theory need not explicitly introduce continuity conditions for both displacements and relevant stresses the performance of the present two dimensional model in conjunction with the global piecewise smooth functions is tested and discussed by comparing its resulting eigen parameters for a class of simply supported plates with those of other two dimensional models and with those existing of the exact three dimensional theory

analysis of thick laminated anisotropic cylindrical shells using a refined shell theory bending analysis of arbitrarily laminated anisotropic panels and closed cylinders using the mixed shear deformation theory proposed by the authors is presented a set of equilibrium transverse shear compatibility and boundary conditions are obtained by using the mixed variational principle proposed by jing and liao 1989 int j numer meth engng28 2813 2827 with displacements and transverse shear as independent variables the zig zag type displacement is assumed together with piecewise parabolic transverse shear the initial curvature effect is included in the strain displacement relations stress resultants and assumed transverse shear stresses two types of shell geometry infinitely long cylindrical panels and closed cylinders of finite length are employed in the numerical study the cylindrical panels considered are subjected to a transversely sinusoidal loading while closed cylinders are under an uniform internal pressure numerical results presented here are compared with exact three dimensional elasticity solutions from these comparisons it is found that this mixed shear deformation theory can supply reasonably good results

a high order laminated plate theory with improved in plane responses in order to improve the accuracy of the in plane responses of the shear deformable laminated composite plate theories a new high order laminated plate theory was developed based upon reissner s new mixed variational principle int j num meth eng 20 1366 1984 to this end a zig zag shaped c0 function and legendre polynomials were introduced into the approximate in plane displacement distributions across the plate thickness the accuracy of the present theory was examined by applying it to the cylindrical bending problem of laminated plates which had been solved exactly by pagano j comp mat 3 398 1969 a comparison with the exact solutions obtained for several symmetric and asymmetric cross ply laminates indicates that the present theory accurately estimates in plane responses even for small span to thickness ratios

single vs multilayer plate modelings on the basis of reissner s mixed theorem the use of reissner s mixed variational theorem reissner e on a certain mixed variational theory and a proposed applications international journal for numerical methods in engineering vol 20 1984 pp 1366 1368 reissner e on a mixed variational theorem and on a shear deformable plate theory international journal of numerical methods in engineering vol 23 1986 pp 193 198 to analyze laminated plate structures is examined the two cases of single layer and multilayer models have been compared governing equilibrium and constitutive equations have been derived in a unified manner navier type closed form solutions are presented for the particular case of cross ply simply supported plates thin and thick as well as symmetrically and asymmetrically laminated plates have been investigated displacements and transverse stresses have been evaluated and compared with available mixed two dimensional results and three dimensional solutions the following have been concluded 1 reissner s mixed theorem is a very suitable tool to analyze laminated structures 2 multilayer modelings lead to an excellent agreement with exact solution for both displacement and transverse stress evaluations such an agreement which has been confirmed for very thick geometries a h 4 does not depend on laminate layouts no remarkable differences have been found for stresses evaluated a priori by the assumed model with respect to exact results 3 single layer analyses lead to an accurate description of the response of thick plates major discrepancies have been found for very thick plate geometries with exact solutions nevertheless their accuracy is very much subordinate to the order of the used expansion as well as to laminated layouts better transverse stress evaluations are obtained upon integration of three dimensional equilibrium equations a posteriori than those furnished a priori this trend has been confirmed for both thick and thin plates

geometrically nonlinear stress recovery in composite laminates a procedure is presented to compute the interlaminar stresses in laminated plates using the intralaminar stresses and cross section rotations from a plate analysis for cases in which geometric nonlinearities are important the procedure is based on integrating the geometrically nonlinear equilibrium equations from the theory of elasticity simplified for the case of moderate rotations integration is accomplished numerically by using the intralaminar stresses and cross section rotations from a finite element plate analysis finite difference representations of the derivatives of these intralaminar stresses and rotations and an iteration procedure iteration being required because of the nonlinear nature of the problem for purposes of illustration the procedure is applied to two semi infinite isotropic cross ply and angle ply plate example problems one problem considers a transverse load acting on a portion of the plate and the other considers a follower type shear load the problems are selected because they exhibit a variety of interlaminar stress responses thereby providing a test for the developed procedure the interlaminar stresses computed with the developed procedure are compared with the stresses computed using a three dimensional finite element analysis that serves as a benchmark comparisons are also made by using closedform expressions for the intralaminar stresses in the developed procedure the interlaminar stresses computed using the developed procedure agree well with the stresses as computed by both comparison approaches extensions of the procedure are discussed

a refined asymptotic theory for dynamic analysis of doubly curved laminated shells the asymptotic theory developed recently for dynamic analysis of doubly curved laminated shells is refined by including the transverse rotations as auxiliary variables the theory embraces the first order shear deformation theory fsdt and the higher order shear deformation theory hsdt as the first order approximation higher order corrections to the approximation are determined by solving the fsdt or hsdt equations in a hierarchic way the secular terms in the asymptotic solution are eliminated systematically by means of multiple scales and solvability conditions for the higher order equations the performance of the refined theory is illustrated by applying it to benchmark problems numerical comparisons are made to examine the convergence of the solutions

multilayered shell theories accounting for layerwise mixed description part 1 governing equations a reissner mixed variational equation is employed in this paper to derive the differential governing equations of multilayered double curved shells made of orthotropic laminae in linear static cases a layerwise description is referred to by assuming two independent fields in the thickness direction for the transverse stress both shear and normal components and displacement variables in each layer interlaminar values are used as the unknown variables of the introduced expansions the continuity conditions of displacements and transverse shear and normals stresses at the interfaces between two consecutive layers referred to as c 0 z requirements have been a priori fulfilled these have been used to drive the governing equations from a layer to a multilayered level classical displacement formulations and related equivalent single layer equations have been derived for comparison purposes no assumptions have been made concerning the terms of type thickness to radii shell ratio h r donnell s shallow shell type equations are given as particular cases for all of the considered theories indicial notations and arrays have been used extensively to handle the presented developments in a concise manner numerical evaluations and comparisons to exact and other available two dimensional solutions are given in a companion paper e carrera multilayered shell theories accounting for layerwise mixed description part 2 numerical evaluations aiaa journal vol 37 no 9 1999 pp 1117 1124

unitary invariance and spectral variation abstract we call a norm on operators or matrices weakly unitarily invariant if its value at operator a is not changed by replacing a by u au provided only that u is unitary this class includes such norms as the numerical radius we extend to all such norms an inequality that bounds the spectral variation when a normal operator a is replaced by another normal b in terms of the arclength of any normal path from a to b computed using the norm in question related results treat the local metric geometry of the manifold of normal operators we introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball and identify this correspondence explicitly in certain cases

general two dimensional theory of laminated cylindrical shells the theory accounts for a desired degree of approximation of the displacements through the thickness thus accounting for any discontinuities in their derivatives at the interface of laminae geometric nonlinearity in the sense of the von karman strains is also included navier type solutions of the linear theory are presented for simply supported boundary conditions

modeling method for a crack problem of functionally graded materials with arbitrary properties piecewise exponential model abstract a new model piecewise exponential model pe model is developed to investigate the crack problem of the functionally graded materials fgms with arbitrary properties in the pe model the functionally graded material is divided into some nonhomogeneous layers along the gradient direction of the properties with each layer s properties varying exponentially by this way the real material properties can be approached by a series of exponential functions since the real material properties are used on both surfaces of each nonhomogeneous layer the nature of continuously varying properties of fgms can be approached accurately the influences of the local nonhomogeneity on the crack tip fields can be fully considered by using the new model the fracture problem of a functionally graded strip with arbitrary properties and a crack vertical to the free surfaces is studied the integral transform method the theory of residues and the theory of singular integral equation are applied some representative samples with different kinds of nonhomogeneous properties are analyzed and the corresponding stress intensity factors sifs are presented it is shown that the pe mode is effective for investigating the crack problems of the fgms with arbitrary properties

on a certain mixed variational theorem and a proposed application formulation d une equation variationnelle pour les deplacements et certaines contraintes en vue d une application aux plaques elastiques stratifiees isotropes et anisotropes

variable kinematics approach for linearized buckling analysis of laminated plates and shells this work deals with the linearized buckling analysis of laminated plates and shells a variable kinematics approach with hierarchical capabilities is considered to establish the accuracy of a large variety of classical and advanced plate shell theories in order to evaluate buckling loads so called equivalent single layer as well as layerwise variable descriptions are implemented interlaminar continuity of transverse shear and normal stresses are a priori fulfilled by referring to reissner s mixed variational theorem stability equations are derived in compact form by referring to carrera s unified formulation for the most general case of doubly curved shells the eigenvalue problem is solved in the case of closed form solutions related to simply supported boundary conditions and axially loaded multilayered plates shells made of orthotropic layers the casesofaxial constant strains and constant stresses are considered and compared to available three dimensional and two dimensional results results related to love and donnell approximations are implemented for comparison purposes the accuracy of various approximations is established for significant multilayered plate and shell problems

hierarchic models for laminated composites the principles governing the formulation of hierarchic models for laminated composites are discussed the essential features of the hierarchic models described herein are a the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness and b the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity examples are presented

electromechanically coupled zigzag third order theory for thermally loaded hybrid piezoelectric plates an efficient coupled zigzag theory is presented for hybrid piezoelectric plates under thermoelectromechanical loads the thermal and potential fields are approximated as piecewise linear across sublayers the deflection is approximated as a combination of a global uniform term across the thickness and local piecewise quadratic variations across sublayers which explicitly account for the transverse normal strain resulting from thermal and electric fields the in plane displacements are approximated as a combination of a third order global variation across the thickness and a piecewise linear variation across layers the shear continuity conditions at the layer interfaces and the shear traction free conditions at the top and bottom are imposed for a general electric field to formulate the theory in terms of five primary displacement variables and potentials the field equations and boundary conditions are derived from a variational principle the accuracy of the developed theory is established by direct comparison with three dimensional exact piezothermoelasticity solutions for simply supported plates under two thermal loads for two very inhomogeneous laminate configurations and different aspect ratios

enhanced first order theory based on mixed formulation and transverse normal effect abstract an accurate prediction of displacements and stresses for laminated and sandwich plates is presented using an enhanced first order plate theory based on the mixed variational theorem efsdtm developed in this paper in the mixed formulation transverse shear stresses based on an efficient higher order plate theory ehopt developed by cho and parmerter cho m parmerter r r 1993 efficient higher order composite plate theory for general lamination configurations aiaa journal 31 1299 1306 are utilized and modified to satisfy prescribed lateral conditions and displacements are assumed to be those of a first order shear deformation theory fsdt relationships between the modified ehopt and the fsdt are systematically derived via both the mixed variational theorem and the least square approximation of difference between in plane stresses including the transverse normal stress effect it is shown that the transverse normal stress effect should be considered in predicting the in plane stresses when the poisson effect is dominant the developed efsdtm preserves the computational advantage of the classical fsdt while allowing for important local through the thickness variations of displacements and stresses through the recovery procedure the accuracy and efficiency of the present theory are assessed by comparing its results with various plate models as well as the three dimensional exact solutions for thick laminated and sandwich plates

development of a mixture model for nonlinear wave propagation in fiber reinforced composites abstract a method for constructing dispersive nonlinear mixture models for unidirectionally fiber reinforced composites is described system nonlinearities in the treated example result from nonlinear material properties of the constituents the proposed model is a nonlinear generalization of the linear model developed by murakami and hegemier journal of applied mechanics vol 53 pp 765 773 1986 for elastic constituents model construction is based upon a homogenization technique which employs multivariable asymptotic expansions in conjunction with certain weighted residual procedures the methodology furnishes the equations of motion the appropriate initial and boundary conditions and a set of consistent rate constitutive relations model validation for linear and nonlinear dynamic responses is accomplished by comparing predicted results for wave guide and wave reflect problems with available experimental data or data obtained by use of a detailed finite element fe analysis the validation studies reveal that the derived continuum model provides good simulations of complex wave phenomena and furnishes an economical alternative to detailed explicit fe models the studies performed reveal the importance of wave dispersion and attenuation phenomena in nonlinear as well as linear wave propagation in the composites

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

variation of grassman powers and spectra the operator norm of the derivative of the map which takes a finite dimensional linear operator to its kth grassman power the kth compound is evaluated this leads to a bound for the distance between the grassman powers of two operators as an important application a bound for the distance between the eigenvalues of two operators is obtained

new nonpolynomial shear deformation theories for structural behavior of laminated composite and sandwich plates in the present study new nonpolynomial shear deformation theories are proposed and implemented for structural responses of laminated composite and sandwich plates the theories assume nonlinear distribution of transverse shear stresses and also satisfy the traction free boundary conditions at the top and bottom layers of the laminates the governing differential equations are derived for a generalized shear deformation theory by implementing the dynamic version of principle of virtual work and calculus of variations a generalized closed form solution methodology of the navier type is implemented to ensure the validity and efficiency of the present theories for bending buckling and free vibration responses of the laminated composite and sandwich plates it is observed that the proposed formulation in conjunction with the solution methodology is capable of handling all existing five degree of freedom based shear deformation theories the comparison of results also shows that the adequate choice of shea

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

spectral variation of normal matrices abstract it has been a durable conjecture that the distance appropriately defined between the spectra of two normal matrices is bounded by the operator norm of their difference we report on some numerical studies that show that this conjecture is in general false in fact even in the 3 x 3 case the spectral distance may exceed the norm distance by between 1 and 2 percent we also describe the background of the conjecture and offer an analysis of extremal counterexamples

two generalized higher order theories in free vibration studies of multilayered plates this paper presents an extension of two dimensional models for the analysis of freely vibrating laminated plates the extension concerns the enlargement of higher order theories recently introduced by different authors in several forms to encompass higher order terms over the cubic one usually taken into consideration higher order effects such as rotatory inertia and transverse shear stress are naturally included without any shear correction factors namely two different models are introduced by expanding on different functional bases displacements d2d and transverse shear stresses in conjunction with displacements m2d the expansion is considered to be consistent with the traction type boundary condition on the external surfaces of the plate the governing equations and associated boundary conditions are consistently obtained by the classical hamilton s variational principle and reissner s mixed variational theorem both models are equivalent single layer type and therefore differ according to the layer wise descriptions preserve the independence of the number of unknown variables on the number of layers however this feature is presented together with intrinsic physical violations for both models model d2d violates the interlaminar stress continuity requirement and model m2d violates in a weaker from the same requirement derivatives are not piecewise continuous besides neglecting the transverse normal stress the importance of completely fulfilling the mentioned continuity is then discussed once the relevant governing equations are tailored for the cylindrical bending condition the effectiveness of the models is indicated by making numerical comparisons with the exact three dimensional theory of the elasticity for several lamination schemes angle cross ply lay ups and characteristic geometric ratios for low and higher frequencies

a modified finite particle method multi dimensional elasto statics and dynamics summary particle methods are a class of numerical methods that belong to the family of meshless methods and are not based on an underlying mesh or grid but rather on any general distribution of particles they are nowadays widely applied in many fields including for example solid mechanics fluid dynamics and thermodynamics in this paper we start from the original formulation of the so called modified finite particle method mfpm and develop a novel formulation in particular after discussing the position of the mfpm in the context of the existing literature on meshless methods and recalling and discussing the 1d formulation and its properties we introduce the novel formulation along with its extension to the approximation of 2d and 3d differential operators we then propose applications of the discussed methods to some elastostatic and elastodynamic problems the obtained results confirm the potential and the flexibility of the considered methods as well as their second order accuracy proposing mfpm as a viable alternative for the simulation of solids and structures copyright 2014 john wiley sons ltd

refined global local higher order theory and finite element for laminated plates based on completely three dimensional elasticity theory a refined global local higher order theory is presented as enhanced version of the classical global local theory proposed by li and liu int j numer meth engng 1997 40 1197 1212 in which the effect of transverse normal deformation is enhanced compared with the previous higher order theory the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature however the previous higher order theory will encounter difficulty for this problem a refined three noded triangular element satisfied the requirement of c1 weak continuity conditions in the inter element is also presented the results of numerical examples of moderately thick laminated plates and even thick plates with span thickness ratios l h 2 are given to show that in plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique in order to accurately obtain transverse normal stresses the local equilibrium equation approach in one element is employed herein copyright 2006 john wiley sons ltd

free vibration analysis of a circular cylindrical shell using the rayleigh ritz method and comparison of different shell theories abstract this study uses the rayleigh ritz method to derive a dynamic model for the free vibration analysis of a circular cylindrical shell in particular explicit expressions for the mass and stiffness matrices are obtained to easily implement a computer simulation under different shell theories and boundary conditions the dynamic model is constructed according to the donnell mushtari theory which is fully discussed herein and then dynamic models are constructed by using sanders theory love timoshenko theory reissner theory flugge theory and vlasov theory this paper also discusses the use of eigenfunctions of a uniform beam as admissible functions that produce compact expressions for the mass and stiffness matrices the numerical results indicate that the donnell mushtari theory is not sufficiently accurate to calculate the natural frequencies and that there is no discernible difference between the other shell theories considered in this study

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

on asymptotically correct linear laminated plate theory abstract the focus of this paper is the development of asymptotically correct theories for laminated plates the material properties of which vary through the thickness and for which each lamina is orthotropic this work is based on the variational asymptotical method a mathematical technique by which the three dimensional analysis of plate deformation can be split into two separate analyses a one dimensional through the thickness analysis and a two dimensional plate analysis the through the thickness analysis includes elastic constants for use in the plate theory and approximate closed form recovering relations for all three dimensional field variables expressed in terms of plate variables in general the specific type of plate theory that results from this procedure is determined by the procedure itself however in this paper only reissner like plate theories are considered often called first order shear deformation theories this paper makes three main contributions first it is shown that construction of an asymptotically correct reissner like theory for laminated plates of the type considered is not possible in general second a new point of view on the variational asymptotical method is presented leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible third numerical results from such an optimum reissner like theory are presented these results include comparisons of plate displacement as well as of three dimensional field variables and are the best of all extant reissner like theories indeed they even surpass results from theories that carry many more generalized displacement variables

analysis of thick isotropic and cross ply laminated plates by radial basis functions and a unified formulation in this paper we combine carrera s unified formulation and a radial basis function collocation technique for predicting the static deformations and free vibration behavior of thin and thick isotropic and cross ply laminated plates through numerical experiments the capability and efficiency of this collocation technique for static and vibration problems are demonstrated and the numerical accuracy and convergence are thoughtfully examined

a partially insulated interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate under heat flux supply in this paper thermal response of an orthotropic functionally graded coating substrate structure with a partially insulated interface crack under heat flux supply is considered it is assumed that there exists thermal resistance to heat conduction through the crack region the mixed boundary value problems are reduced to a system of singular integral equations and solved numerically higher order asymptotic terms for the singular integral kernels are considered to improve the accuracy and the convergence efficiency of the numerical integrals numerical results are presented to show the effects of the orthotropy parameters thermo elastic nonhomogeneity parameters and dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors sifs

guidelines and recommendations to construct theories for metallic and composite plates this work has evaluated the refinement of some classical theories such as the kirchhoff and reissner mindlin theories adding generalized displacement variables up to fourth order to the taylor type expansion in the thickness plate direction isotropic orthotropic and laminated plates have been analyzed varying the thickness ratio orthotropic ratio and stacking sequence of the layout higher order theories have been implemented according to the compact scheme known as the carrera unified formulation the results have been restricted to simply supported orthotropic plates subjected to harmonic distributions of transverse pressure for which closed form solutions are available for a given plate problem isotropic orthotropic or laminated the effectiveness of each employed generalized displacement variable has been established comparing the error obtained accounting for and removing the variable in the plate governing equations a number of theories have therefore been constructed imposing a given error with respect to the available best results guidelines and recommendations that are focused on the proper selection of the displacement variables that have to be retained in refined plate theories are then furnished it has been found that the terms that have to be used according to a given error vary from problem to problem but they also vary when the variable that has to be evaluated displacement stress components is changed diagrams errors in terms of geometrical and orthotopic ratios and graphical schemes have been built to establish the appropriate theories with respect to the data of the problem under consideration

dynamic investigation of a functionally graded layered structure with a crack crossing the interface abstract the dynamic response of a functionally graded layered structure with a crack crossing the interface is analyzed the in plane impact loading condition is considered by using the laplace and fourier integral transforms singular integral equation method and residue theory the present problem is reduced to a singular integral equation in the laplace transform domain the influences of young s modulus ratio thickness ratio and crack length and location on the dynamic stress intensity factors dsifs are investigated particularly the dsifs corresponding to different crack locations are shown in the case when the crack center moves from one layer to another layer through the interface the peak and static values and overshoot characteristics of the dsifs are analyzed it is found that these values typically exhibit kinking behavior when the crack tips arrive at the interface this study is different from previous other investigations in the following respects 1 the dynamic response of a crack crossing the interface of a functionally graded structure is studied analytically which has hardly been done in the past and 2 the present model can be reduced to some important problems such as a functionally graded coating substrate structure with a crack in the graded coating or homogeneous substrate or one intersecting the interface

a study of transverse normal stress effect on vibration of multilayered plates and shells this paper evaluates transverse normal stress zzeffect on vibration of multilayered structures to this purpose a mixed plate model initially introduced by toledano and murakami has been extended to dynamics analysis of double curved shells these models allow both continuous interlaminar transverse shear and normal stresses as well as the zigzag form of the displacement distribution in the shell thickness directions to be modelled governing equations have been derived by employing a reissner s mixed theorem classical models on the basis of standard displacement formulations have been considered for comparison purposes the evaluations of transverse stress effects have been conducted by comparing constant linear and higher order distributions of transverse displacement components in the plate thickness directions free vibrational response of layered simply supported plates cylindrical and spherical shells made of isotropic as well as orthotropic layers has been analyzed the numerical investigation carried out and comparison with earlier results has concluded that 1 the possibility of describing interlaminar continuous transverse normal stress makes the mixed theories more attractive with respect to other available modelling 2 any refinements of classical models are meaningless unless the effects of interlaminar continuous transverse shear and normal stresses are both taken into account in a multilayered shell theory

application of the variational asymptotical method to laminated composite plates this method is a technique by which the geometrically nonlinear three dimensional analysis of plate deformation can be split into a linear one dimensional through the thickness analysis and a nonlinear two dimensional plate analysis the elastic constants used in the plate analysis are obtained in closed form from the through the thickness analysis along with approximate closed form three dimensional distributions of displacement strain and stress the development of such a theory is presented herein for laminated plates in which each lamina exhibits monoclinic symmetry about its own midplane

free vibrations of laminated rectangular plates analyzed by higher order individual layer theory abstract a higher order plate theory is used in each individual layer to determine the natural frequencies and the relative stress and deflection distributions through the thickness of simply supported rectangular plates this theory approximates the in plane and normal displacements by third and second order functions of the thickness co ordinate respectively the theory satisfies the displacement compatibility and stress equilibrium conditions along the interfaces between adjacent layers numerical results are calculated for the cases of isotropic and orthotropic homogeneous plates and for symmetric and non symmetric cross ply laminates the present theory predicts more modes of free vibration than other approximate theories also the relative stress and deflection distributions obtained by the present higher order theory are in very good agreement with results by three dimensional elasticity theory

thermoelastic problem of steady state heat flows disturbed by a crack at an arbitrary angle to the graded interfacial zone in bonded materials plane thermoelasticity solutions are presented for the problem of a crack in bonded materials with a graded interfacial zone the interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermoelastic moduli between dissimilar homogeneous half planes the crack is assumed to exist in one of the half planes at an arbitrary angle to the graded interfacial zone disturbing uniform steady state heat flows the fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic thermoelasticity equations formulation of the current nonisothermal crack problem lends itself to the derivation of two sets of cauchy type singular integral equations for heat conduction and thermal stress analyses the heat flux intensity factors and the thermal stress intensity factors are defined and evaluated in order to quantify the singular characters of temperature gradients and thermal stresses respectively in the near tip region numerical results include the variations of such crack tip field intensity factors versus the crack orientation angle for various combinations of material and geometric parameters of the dissimilar media bonded through the thermoelastically graded interfacial zone the dependence of the near tip thermoelastic singular field on the degree of crack surface partial insulation is also addressed

comparative dynamic studies for symmetric cross ply circular cylindrical shells on the basis of a unified shear deformable shell theory abstract free vibrations of cross ply laminated shells subjected to different sets of edge boundary conditions are investigated on the basis of an advanced unified five degree of freedom shear deformable shell theory the theory unifies most of the shear deformable shell theories by means of a general shape function which takes into account shear deformation effects as a result most of the well known shear deformable shell theories available in the literature are obtained as particular cases the governing equations obtained are transformed into a set of first order linear ordinary differential equations and solved by use of the state space concept the presence of the general shear deformation shape function provides further open possibilities in considering the material and the geometrical constraints of the problem in a realistic manner numerical results are presented for symmetric cross ply laminated shells and compared with those of other known models

evaluation and verification of an hsdt layerwise generalized finite element formulation for adaptive piezoelectric laminated plates a formulation for the bending analysis of composite laminated plates with piezoelectric layers is implemented using the generalized finite element method this formulation is derived from a mechanical description based on higher order shear deformation theory which allows for the use of c0 continuous approximation functions on the domain on the other hand a layerwise theory is employed for interpolation of electric potential across the thickness of piezoelectric layers in such a way that the kinematical hypotheses result in a mixed model the paper presents an analysis of the approximation capability of the proposed numerical model for static analysis using c0 continuous partition of unity and polynomial enrichments to span the approximation spaces by assessment of convergence analytical solutions obtained from the same kinematical hypotheses are used as references results for relative error in the energy norm considering p and h refinements for regular and distorted meshes in addition to a pointwise evaluation of the stresses and electric field are presented the evaluations show that the numerical methodology is a very effective tool for improving the solution through the enrichment even for pointwise values across the thickness and is robust to mesh distortions moreover the results furnish insight about the physical modeling for both active and sensory modes for thick and thin plates

plane wave solutions and modal analysis in higher order shear and normal deformable plate theories abstract we use the three dimensional hellinger reissner mixed variational principle to derive a k th order k 0 1 2 shear and normal deformable plate theory the balance laws the constitutive relations and the boundary conditions for the plate theory are deduced the constitutive relations incorporate the shear and the normal tractions applied on the top and the bottom surfaces of the plate for a k th order plate theory with displacements expressed as a power series in the thickness co ordinate z with terms up to z k the transverse shear and the transverse normal stresses involve terms upto z k 2 while in plane stress components have terms up to z k the equations for the plate theory are expressed in a compact form by taking legendre polynomials as the basis functions the plate theory is used to study plane travelling waves and in particular the lengths of decay of the displacement components this allows for a rigorous ordering of the importance of the displacement descriptors in terms of decaying properties finally we study the free vibrations of a simply supported rectangular orthotropic thick plate results from the present theory are compared with an exact three dimensional solution and with other plate theories to this end a k th order compatible plate theory is also deduced the term compatible alludes to the fact that the reduction map for the stress fields is induced by the kinematical reduction map whilst in the mixed models it is postulated independently it is found that the frequencies up to the fifth mode of vibration computed with the fifth order theory and without introducing any shear correction factors match very well with the corresponding analytical solution also through the thickness distribution of all of the stress components is found to agree well with the three dimensional elasticitysolution while the stress distribution obtained from the compatible plate theories deviatesconsiderably especially for the higher modes

up to lowest 100 frequencies of rectangular plates using jacobi polynomials and tsndt abstract this paper finds up to lowest 100 free vibration frequencies of isotropic and linearly elastic plates with thickness to side length ratio varying from 0 001 to 0 5 by using the ritz method a third order shear and normal deformable plate theory tsndt and weighted jacobi polynomials as admissible functions that are mutually orthogonal and exactly satisfy essential boundary conditions the numerical method is stable even for 18th degree polynomials as basis functions it is shown that this approach requires fewer degrees of freedom than those in the traditional finite element method fem to find converged lowest 100 frequencies and the corresponding mode shapes no shear correction factor is employed in the tsndt the presently computed results agree well with those from either analytical or numerical solutions of the corresponding 3 dimensional linearly elastic problems obtained with the commercial fe software abaqus furthermore results for plates made of an incompressible material can be computed by setting poisson s ratio 0 49

proper generalized decomposition and layer wise approach for the modeling of composite plate structures abstract in the framework of the design of laminated and sandwich structures the computation of local quantities needs a layer wise approach but the computational cost of such approach increases with the number of layers in this work the introduction of the proper generalized decomposition pgd is presented for the layer wise modeling of heterogeneous structures in order to reduce the number of unknowns the displacement field is approximated as a sum of separated functions of the in plane coordinates x y and the transverse coordinate z this choice yields to an iterative process that consists of solving a 2d and 1d problem successively at each iteration in the thickness direction a fourth order expansion in each layer is considered for the in plane description classical finite element method is used after a preliminary study to show the relevance of the present approach mechanical tests for thin to thick laminated and sandwich plates with various boundary conditions are presented the results are compared with elasticity reference solutions

a new nonlinear higher order shear deformation theory with thickness variation for large amplitude vibrations of laminated doubly curved shells abstract a consistent higher order shear deformation nonlinear theory is developed for shells of generic shape allowing for thickness variation by using six variables geometric imperfections are also taken into account the geometrically nonlinear strain displacement relationships are derived retaining full nonlinear terms in the in plane displacements they are presented in curvilinear coordinates in a formulation that can be readily implemented in computer codes this new theory is applied to laminated circular cylindrical shells complete around the circumference and simply supported at the ends linear natural frequencies and geometrically nonlinear large amplitude forced response vibrations are studied by using the present theory and results are compared to those obtained by using the refined amabili reddy higher order shear deformation nonlinear shell theory which neglects thickness variations

temperature profile influence on layered plates response considering classical and advanced theories a study on the influence of the through the thickness temperature profile t z on the thermomechanical response of multilayered anisotropic thick and thin plates has been conducted the heat conduction problem is solved and the temperature variation t c z is then calculated the governing thermomechanical equations of multilayered plates are written considering a large variety of classical and advanced or zigzag theories into account the principle of virtual displacement and the reissner mixed variational theorem are employed linear up to fourth order expansions in z are retained for the assumed transverse stress and displacement fields as a result more than 20 plate theories are compared the numerical investigation is restricted to orthotropic layered plates with harmonic in plane distribution of both thermal loadings and unknown variables four sample plate problems are treated that are related to plates made of isotropic and or orthotropic layers that are loaded by different top bottom plate surface temperature conditions comparison is made to results related to a linear profile t a z which is usually assumed in open literature the following is concluded thick plates could exhibit a layerwise form temperature profile t c z t a z case is approached for thin plate geometries the use of linear temperature profile leads to large errors in tracing the response of thick plate geometries the accuracy of plate theories is affected to great extent by the form of temperature variation t z refinements of classical plate theories can be meaningless unless the calculated t c z is introduced the layerwise form of t c z would require layerwise assumptions for stresses and or displacements plate theories that neglect transverse normal strains lead to very inaccurate results in both thick and thin plates analysis at least a parabolic expansion for transverse displacement is required to capture transverse normal thermal strains that vary linearly along the plate thickness

invariant finite element model for composite structures the generalized unified formulation the generalized unified formulation presented here is a modern approach for the finite element analysis of sandwich plates and in general multilayered structures a large variety of types of theories with any combination of orders of expansion for the different displacement variables can be obtained from the expansion of six 1 x 1 arrays the kernels or fundamental nuclei of the generalized unified formulation each of the displacement variables is independently treated and different orders of expansions for the different unknowns can be chosen because infinite combinations can be freely chosen for the displacements the generalized unified formulation allows the user to write 3 higher order shear deformation theories 3 zig zag theories and 3 layerwise theories with a single invariant formulation implemented in a single finite element method code the six independent fundamental nuclei are formally invariant with respect to the order used for the expansion or with respect to the type of theory the generalized unified formulation can be also adopted for mixed variational statements in such cases the number of independent kernels would be different but their size would still be 1 x 1 arrays this paper assesses bending of sandwich structures analytical and elasticity solution are compared the effect of the zig zag form of the displacement is discussed