saint venant end effects in multilayered piezoelectric laminates the saint venant end effects and stress decay in multilayered laminates of piezoelectric materials are studied a state space approach is developed in the context of generalized plane strain by means of matrix algebra and transfer matrix the characteristics of the stress decay in 2 d piezoelectric strips and laminates are examined their elastic counterparts are included as special cases evaluation of the characteristic decay length for typical materials shows that the saint venant end effects are significant it also indicates that the analysis based on the plane strain assumption is misleading

on the index of convergence of an irreducible boolean matrix abstract first we give a new proof of the sharp upper bound for the indices of convergence of n x n irreducible boolean or nonnegative matrices with period p then we characterize the matrices with the largest index

deconstructing plane anisotropic elasticity part i the latent structure of lekhnitskii s formalism abstract general solutions of the stress and displacements in two dimensional anisotropic elasticity may be represented by eigenvectors and analytic functions of the complex variables x iy but the representation takes different forms for five distinct types of materials as determined by the elastic compliance matrix in this paper explicit expressions of the general solutions are derived for each type of anisotropic materials in terms of the eigenvalues i and the elements of it is shown that for degenerate and extra degenerate materials the generalized eigenvectors and associated eigensolutions may be obtained by the derivative rule the barnett lothe tensors are defined in terms of unnormalized eigenvectors by the same set of relations regardless of material degeneracy explicit expressions of these tensors are given in concise forms depending only on the multiplicity of the eigenvalues the six dimensional matrix formalism and normalization of the eigenvectors are found to be neither essential nor expedient for the analysis except as a device for abridged expressions of matrix identities

numerical analysis of end effects in laminated piezoelectric circular cylinders abstract a method to quantify saint venant s principle for laminated piezoelectric circular cylinders is described an algebraic eigensystem is constructed based on homogeneous equations of equilibrium with displacement and electric potential voltage fields stated in exponential form the end effects for displacements and voltages as well as stresses and electric displacements of self equilibrated states are represented by the eigendata extracted from the eigensystem the real parts of the eigenvalues convey information on the inverse decay lengths and their corresponding eigenvectors are displacement and voltage distributions of the self equilibrated states stress and electric displacement eigenvectors are formed by appropriate differentiation of the eigenvectors for the displacement and voltage fields and through the coupled electromechanical constitutive laws the right and left eigenvectors which are obtained from the eigensystem and its adjoint are related through bi orthogonality relationships the end effects due to arbitrary displacements voltages tractions or electric displacements prescribed at the cylinder s ends are obtained by means of an expansion theorem based on these bi orthogonality relationships or by a least squares method a number of verification examples are provided to demonstrate that the present results compare well with known analytical solutions and numerical results obtained via three dimensional finite element analyses

semi analytical solutions for free vibration of anisotropic laminated plates in cylindrical bending elasticity solutions for free vibration of angle ply laminates subjected to cylindrical bending are obtained using a newly developed semi analytical approach the thickness domain is solved analytically using the transfer matrix method based on the state space concept while the in plane domain is solved approximately via the technique of differential quadrature the present method is applicable to arbitrarily thick laminates and for treating arbitrary edge conditions the method is verified by comparisons with the exact solutions of pagano s problem effects of variation of ply angle on the vibration properties of laminates are investigated mode shape switching is observed when ply angle varies numerical results for fully clamped thick laminates are presented for future references

modal analysis of magneto electro elastic plates using the state vector approach the state vector approach is proposed to analyze the free vibration of magneto electro elastic laminate plates the extended displacements and stresses can be divided into the so called in plane and out of plane variables once the state equation for the out of plane variables is obtained a complex boundary value problem is converted into an equivalent simple initial value problem through the state equation the propagator matrix between the top and bottom interfaces of every layer can be easily derived the global propagator matrix can also be assembled using the continuity conditions it is obvious that the order of global propagator matrix is not related to the number of layers consequently this approach possesses certain virtues including simple formulation less expensive computation etc to test the formulation the developed solution is then applied to a simply supported multilayered plate constructed of piezoelectric and or piezomagnetic materials the natural frequencies and corresponding mode shapes are computed and compared with existing results furthermore the fundamental modes along with a couple of other higher modes which have never been reported in previous literature are presented therefore this completed set of frequencies and mode shapes can be used as benchmarks for future research in this field it is also believed that the approach could be useful in the analysis and design of smart structures constructed from piezoelectric piezomagnetic composites

saint venant end effects for plane deformations of linear piezoelectric solids the recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity in this paper we investigate the decay of saint venant end effects for plane deformations of a piezoelectric semi infinite strip first of all we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid just as in the mechanical case not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non trivial state of plane deformation the governing system of four second order partial differential equations for the two in plane displacements and electric potential are overdetermined in general sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation the resulting traction boundary value problem with prescribed surface charge is an oblique derivative boundary value problem for a coupled elliptic system of three second order partial differential equations the special case of a piezoelectric material transversely isotropic about the poling axis is then considered thus the results are valid for the hexagonal crystal class 6mm the geometry is then specialized to be a two dimensional semi infinite strip and the poling axis is the axis transverse to the longitudinal direction we consider such a strip with sides traction free subject to zero surface charge and self equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity a formulation of the problem in terms of an airy type stress function and an induction function is adopted the governing partial differential equations are a coupled system of a fourth and third order equation for these two functions on seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second order ordinary differential equations this problem is the piezoelectric analog of the well known eigenvalue problem arising in the case of an anisotropic elastic strip it is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti symmetric deformations and electric fields assuming completeness of the eigenfunctions the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part numerical results are given for pzt 5h pzt 5 pzt 4 and ceramic b it is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials

a state space formalism for anisotropic elasticity part ii cylindrical anisotropy abstract a state space formalism for thermoelastic analysis of a cylindrically anisotropic elastic body is developed by proper grouping of the stress and taking r ij instead of ij as the stress variables the three dimensional equations of elasticity in the cylindrical coordinates are concisely formulated into a state equation and an output equation using matrix notations the general solution for the generalized plane problems is derived in a simple manner effects of extension torsion bending temperature change and body force are accounted for systematically the eigen relation arising from the solution process and the degenerate cases of repeated eigenvalues are examined to demonstrate the power of the formalism exact solutions to extension torsion bending and thermo mechanical loading of a general cylindrically anisotropic circular tube or bar are obtained the surface tractions and temperature field may vary in but not in z displacement as well as traction boundary conditions are considered it appears that many problems of anisotropic elasticity are best viewed and treated in the state space framework

a sextic formalism for three dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials elasticity of arbitrary cylindrically anisotropic radially inhomogeneous media is considered the system of six first order ordinary differential equations describing time harmonic cylindrical waves propagating along the cylinder axis is obtained the matrix of the system coefficients has a specific algebraic symmetry underlying the structure of the formalism the matricant solution in the form of the peano expansion is discussed the wentzel kramers brillouin approximation is derived algebraic features of the solution are analysed and their physical content is elucidated the dispersion equation for waves in a hollow cylinder with free or clamped lateral surfaces is formulated in terms of the impedance admittance matrices their analytical properties are established which are useful for analysis of the resulting frequency spectrum the theory is extended to anisotropic radially inhomogeneous piezoelectric cylinders for which the octet generalization of the formalism is developed and studied

alternative state space formulations for magnetoelectric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates abstract by introducing two displacement functions and two stress functions the governing equations of the linear theory of magneto electro thermo elasticity with transverse isotropy are simplified on selecting certain physical quantities as the basic unknowns two new state equations are established each of them is order reduced when compared with the one reported recently in literature leading to a higher numerical efficiency the material inhomogeneity along the axis of symmetry z direction can be taken into account and an approximate laminate model is employed to facilitate deriving analytical solutions the validity of new formulations is examined by considering a laminated magneto electro elastic rectangular plate and good agreement is obtained with existent results a plate with a functionally graded property is then analyzed the effect of magnetoelectric coupling in a batio3 cofe2o4 composite predicted from the micromechanics simulation is studied quantitatively

state space approach for stress decay in laminates abstract stress decay in laminates due to edge boundary effects are studied through a state space formulation a self equilibrium eigenstress field accounting for the multilayer construction of the laminate is derived using the state variables and the transfer matrix method the eigenvalue determination requires only the solution of 6 6 determinants irrespective of the number of laminae through combinations of the eigenstress field and the interior stress field a complete solution valid in the boundary layer as well as in the interior region of the laminate can be obtained for verification the formulation is first applied to determining the eigenstress in a homogeneous anisotropic layer and then the free edge stress decay in laminates under uniform extension is examined

three dimensional asymptotic finite element method for anisotropic inhomogeneous and laminated plates abstract an asymptotic finite element model for anisotropic inhomogeneous and laminated plates is developed within the framework of three dimensional elasticity the formulation begins with a hellinger reissner functional in which the displacements and transverse stresses are taken to be the functions subject to variation by means of asymptotic expansion the h r functional for the problem is decomposed into functionals of various orders from which the asymptotic finite element equations are derived in the multilevel computations the transverse stresses and displacements may be interpolated independently and the midplane displacements are the only unknown nodal degree of freedoms dof in the system equations thus the total dof at each level is less than that of a homogeneous kirchhoff plate the stiffness matrix remains unchanged the one generated at the leading order level is always used at subsequent levels the formulation is three dimensional yet requires only two dimensional finite element discretization with no need of interpolation in the thickness direction the through thickness effect can be accounted for in a consistent and hierarchical manner numerical comparisons with the benchmark solutions show that the method is effective in modeling of multilayered composite plates

connection between static electromagnetoelasticity and anisotropic elasticity and its applications connection between electromagnetoelasticity and anisotropic elasticity is explored in the state space setting in the absence of electric charges and currents the basic equations of static electromagnetoelasticity are formulated into a state equation and an output equation which bear a remarkable resemblance to the corresponding equations of elasticity accordingly the solutions for various steady state problems of electromagnetoelasticity can be determined in parallel to their elastic counterparts for illustration the generalized plane problems are treated within the context exact solutions for the electromagnetoelastic fields in a half space subjected to line loads and in an infinite plate with an elliptic notch under extension are determined in a simple way

deconstructing plane anisotropic elasticity part ii stroh s formalism sans frills abstract eigensolutions for all types of anisotropic elastic materials are obtained in terms of the eigenvalues and the anisotropic elastic stiffness the generalized eigenvectors and eigensolutions in the degenerate and extra degenerate cases are obtained by the derivative rule a complete set of unnormlized eigenvectors now given in terms of the elastic moduli define the barnett lothe tensors by the same expressions irrespective of material degeneracy explicit expressions of the barnett lothe tensors are obtained in various forms depending on the multiplicity of eigenvalues these expressions complement the alternative expressions of part i in terms of the elastic compliances a new family of extra degenerate materials is found suggesting the superabundance of such materials a concise proof of the equivalence of the eigensystems of the compliance based and elasticity based formalisms is given eigenrelations applicable to all cases of material degeneracy are presented in both three dimensional and six dimensional matrix formalisms

new solutions to pressuring shearing torsion and extension of a cylindrically anisotropic elastic circular tube or bar the problem of a cylindrically anisotropic elastic circular tube or bar subjected to a uniform pressure shearing torsion or extension has been studied by ting the stroh formalism for twodimensio

torsion of cylindrically orthotropic elastic circular bars with radial inhomogeneity some exact solutions and end effects abstract torsion of elastic circular bars of radially inhomogeneous cylindrically orthotropic materials is studied with emphasis on the end effects to examine the conjecture of saint venant s torsion we consider torsion of circular bars with one end fixed and the other end free on which tractions that results in a pure torque are prescribed arbitrarily over the free end surface exact solutions that satisfy the prescribed boundary conditions point by point over the entire boundary surfaces are derived in a unified manner for cylindrically orthotropic bars with or without radial inhomogeneity and for their counterparts of saint venant s torsion stress diffusion due to the end effect is examined in the light of the exact solutions

a state space formalism for piezothermoelasticity a state space formalism for electrothermoelastic analysis of a linear piezoelectric body is developed a novel feature of the formalism is that by proper grouping of the field variables and using matrix notations the three dimensional equations of piezothermoelasticity are concisely formulated into a state equation and an output equation which bear a striking resemblance to their elastic counterparts the formalism is remarkably simple with which one deals with only three vector quantities with the 13 independent electromechanical variables as their components and six submatrices that represent all the material constants for a piezoelectric material of the most general kind in this work emphasis is placed on the state space formulation and solution to the generalized plane problem exact solutions for a piezoelectric half space under a line of electromechanical loading and an infinite piezoelectric plate with an elliptical notch subjected to inplane loads are determined with relative ease in many cases piezoelectric solutions can be obtained directly from the corresponding elastic solutions by a simple replacement of the corresponding matrices on the basis of the formalism

asymptotic splitting in the three dimensional problem of elasticity for non homogeneous piezoelectric plates a novel asymptotic approach to the theory of non homogeneous anisotropic plates is suggested for the problem of linear static deformations we consider solutions which are slowly varying in the plane of the plate in comparison to the thickness direction a small parameter is introduced in the general equations of the theory of elasticity according to the procedure of asymptotic splitting the principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms three dimensional conditions of compatibility make the analysis more efficient and straightforward we obtain the system of equations of classical kirchhoff s plate theory including the balance equations compatibility conditions elastic relations and kinematic relations between the displacements and strain measures subsequent analysis of the edge layer near the contour of the plate is required in order to satisfy the remaining boundary conditions of the three dimensional problem matching of the asymptotic expansions of the solution in the edge layer and inside the domain provides four classical plate boundary conditions additional effects like electromechanical coupling for piezoelectric plates can easily be incorporated into the model due to the modular structure of the analysis the results of the paper constitute a sound basis to the equations of the theory of classical plates with piezoelectric effects and provide a trustworthy algorithm for computation of the stressed state in the three dimensional problem numerical and analytical studies of a sample electromechanical problem demonstrate the asymptotic nature of the present theory

laminated composite tubes under extension torsion bending shearing and pressuring a state space approach we present a state space approach to extension torsion bending shearing and pressuring of laminated composite tubes one of the novel features is that we have formulated the basic equations of anisotropic elasticity in the cylindrical coordinate system into a state equation by a judicious arrangement of the displacement and stress variables so that the system matrix is independent of r the formulation suggests a systematic way using matrix algebra and the transfer matrix to determine the stress and deformation in a multilayered cylindrically anisotropic tube under applied loads that do not vary in the axial direction an exact analysis of the tube subjected to uniform surface tractions an axial force a torque and bending moments is presented the solution consists of an axisymmetric state due to extension torsion uniform pressuring and shearing and an asymmetric state due to bending the formalism indicates that extension torsion and pressuring interact uniform shearing causes pure shears in the laminated tube regardless of the number of layers these deformations are uncoupled with bending of the tube

an exact correspondence between plane piezoelectricity and generalized plane strain in elasticity we consider an anisotropic body bounded by a cylindrical surface which is infinitely long in the axial direction suppose the body is loaded in such a way that the field variables do not vary along the generators an exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity in particular we show that by setting a linkage between the two sets of material constants any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa the assertation is true for rectilinearly as well as for cylindrically anisotropic solids the equivalence is found for the most general anisotropic case which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic triclinic elastic solid a few degenerate systems are also identified in addition these correspondences can be extended to inhomogeneous media applied to composite materials or polycrystalline aggregates they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors and vice versa we also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by cherkaev lurie and milton in illustration we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity lekhnitskii9s complex potential approach together with the conformal mapping technique are employed general solutions of the fields inside the inhomogeneity and the matrix are obtained the results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem

a bound on the exponent of a primitive matrix using boolean rank we present a bound on the exponent exp a of an n x n primitive matrix a in terms of its boolean rank b b a namely exp a b 1j2 2 further we show that for each 2 q b q n 1 there is an n x n primitive matrix a with b a b such that exp a b 1 2 2 and we explicitly describe all such matrices the new bound is compared with a well known bound of dulmage and mendelsohn and with a conjectured bound of hartwig and neumann several open problems are posed

imprimitive matrices of zeros and ones with extremal numbers of ones abstract in x zhan extremal numbers of positive entries of imprimitive nonnegative matrices linear algebra appl 424 2007 132 138 zhan determined the maximum and minimum numbers of positive entries of irreducible nonnegative matrices in this paper we characterize the irreducible 0 1 matrices with the maximum and minimum numbers of positive entries

the boundary layer phenomena in two dimensional transversely isotropic piezoelectric media by exact symplectic expansion a separable variable method is introduced to find the exact homogeneous solutions of a two dimensional transversely isotropic piezoelectric media to handle general boundary conditions the usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives introducing dual variables of stresses we obtain a set of first order hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions the solutions consist of two parts the first part is the derogative zero eigenvalue solutions of the saint venant type together with all their jordan chains the second part is the decaying non zero eigenvalue solutions describing the boundary layer effects the classical solutions are actually the zero eigenvalue solutions representing the simple extension bending equipotential field and the uniform electric displacement on the other hand the non zero eigenvalue solutions represent the localized solutions which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries some rate of decay curves of the newly found non zero eigenvalue solutions are shown by numerical examples finally the complete boundary layer effects are quantified for the first time

matrices permutation equivalent to primitive matrices we give a necessary and sufficient condition for an n n 0 1 matrix or more generally an n n nonnegative matrix to be permutation equivalent to a primitive matrix more precisely except for two simple permutation equivalent classes of n n 0 1 matrices each n n 0 1 matrix having no zero row or zero column is permutation equivalent to some primitive matrix as an application we use this result to characterize the subsemigroup of bn bn is the multiplicative semigroup of n n boolean matrices generated by all the primitive matrices and permutation matrices we also consider a more general problem and give a necessary and sufficient condition for an n n nonnegative matrix to be permutation equivalent to an irreducible matrix with given imprimitive index

a modified lekhnitskii formalism a la stroh for anisotropic elasticity and classifications of the 6 6 matrix n the stroh formalism for two dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation the vector b is then determined from a depending on the number of repeated eigenvalues p and the number of independent eigenvectors a b the system has it can be classified into six groups the lekhnitskii formalism has no eigenrelation to speak of we present a modified lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation the vector a is then determined from b thus the modified lekhnitskii formalism is a dual to the stroh formalism not only does the modified formalism enable us to do the classifications it is much simpler than using the stroh formalism the six groups are the sp ss d1 d2 ed and es groups the es group does not exist for a real material the ss group that has p 1 p 2 p 3 and the d2 group that has p 1 p 2 p 3 can be identified without computing the eigenvalues p and the eigenvectors a b we show that the repeated eigenvalue p 1 p 2 in the ss and d2 groups is simply a root of the quadratic equation l 2 0 we present an explicit expression of p 3 for the ss group the ed group that has three identical p can also be identified without computing p and a b however we do present an explicit expression of p we show that monoclinic materials with the symmetry plane at x 3 0 cannot belong to the ed group the identification of the sp and d1 groups is the only one that requires computation of p but not a b for special classes of materials however they can be identified without computing p in all cases the eigenvectors and the generalized eigenvectors are obtained explicitly

saint venant end effects in piezoceramic materials the stress decay rate and characteristic decay length for anisotropic materials and fiber composites have been intensively investigated however those for piezoelectric materials have not been studied in this paper we examine the saint venant end effects of piezoceramic materials by considering the problem of a semi infinite piezoceramic strip polarized in the thickness direction and with applied voltages on the upper and lower surfaces it is assumed that the gradient of electric potential in the axial direction is much smaller than that in the thickness direction thus the governing equations in terms of the airy stress function and electric potential function can be uncoupled the governing equation in terms of the airy stress function involves only two non dimensional parameters after non dimensionalization finally the stress decay rates and characteristic decay lengths for a variety of piezoceramic materials are computed numerically and their variation with the two non dimensional parameters is presented

a revisit of a cylindrically anisotropic tube subjected to pressuring shearing torsion extension and a uniform temperature change abstract the problem of a cylindrically anisotropic tube or bar was seemed to be first examined by lekhnitskii 1981 lekhnitskii s g 1981 theory of elasticity of an anisotropic body trans from the revised 1977 russian edition mir moscow recently a thorough investigation of the subject was performed by ting 1996 ting t c t 1996 pressuring shearing torsion and extension of a circular tube or bar of cylindrically anisotropic material proc roy soc lond a452 2397 2421 in which a formulation akin to that of stroh s formalism is employed to resolve the boundary value problem subjected to a uniform pressure shearing torsion and uniform extension in a continuing paper ting 1999 ting t c t 1999 new solutions to pressuring shearing torsion and extension of a cylindrically anisotropic elastic circular tube or bar proc roy soc lond to appear rederived the solutions based on a modified formalism of lekhnitskii in which the solutions are in terms of elastic compliances reduced elastic compliances as well as doubly reduced compliance the results are much more compact and simpler than those of the earlier one independently in this work we construct the governing system also under the lekhnitskii s framework nevertheless the present work and ting s formulation 1999 are not alike besides the loads considered in ting 1996 1999 we add the effect of a uniform temperature change in the formulation the assumption that the stresses depend only on r makes it possible to incorporate the various loading cases considered in addition to the explicit forms of admissible stresses we derive the admissible displacements which are ensured to be single valued for a multiply connected domain in contrast to the ting s works 1996 1999 which often require superpositions of two or more basic solutions the present solutions offer complete forms of solutions ready for direct calculations we also report that as in rectilinearly anisotropic solids an entire analogy is observed between the fields of a uniform axial extension and a uniform temperature change in cylindrically anisotropic solids

pressuring shearing torsion and extension of a circular tube or bar of cylindrically anisotropic material one of the novel features of the present paper is that we have written the equation of equilibrium and the stress strain law of an inhomogeneous anisotropic linear elastic material in a compact form for cylindrical coordinate system using matrix notation for a two dimensional deformation the result resembles stroh s sextic formalism in a rectangular coordinate system we then consider the material to be cylindrically anisotropic it means that the elastic stiffnesses referred to a cylindrical coordinate system are constants the problem of a circular tube subjected to a uniform normal stress and shearing stresses at the inner and outer surfaces of the tube is studied also studied are the axial extension and torsion of the tube unlike isotropic materials for which the applied normal stress or shear stress induces only the normal or shear stress all three displacement components and most of the six stress components are nonzero for general anisotropic materials this is particularly interesting for the uniform axial extension of the tube for an isotropic material the stress 33 is the only non zero and uniform stress inside the tube for a cylindrically anisotropic material the stresses rr and 3 are also non zero moreover they depend on r and are not uniform a solid cylinder or a cylinder with a pin hole is a special case of a tube it is shown that for the loads mentioned above including the axial extension the stress may be unbounded at the pinhole

strongly regular elements of noetherian rings any regular element of a commutative ring remains regular modulo every annihilator ideal of the ring in general this is not the case if the assumption of commutativity is dropped although certain elements for example the units stay regular modulo every right or left annihilator ideal this may not be the case for other regular elements thus the multiplicatively closed set y o of all those regular elements which are well behaved in this sense deserves some special attention its elements are called strolzgly regular this set is particularly interesting for questions concerning quotient rings for it turns out that in a right noetherian ring any regular right ore set is contained in y o proposition 1 6 thus the equality of i o and g o the set of regular elements becomes a necessary condition for the existence of a full right quotient ring for a left and right noetherian ring r it turns out that y o n g p where p runs through the set of all prime middle annihilator ideals theorem 2 1 a similar description of g o in terms of finitely many affiliated primes obtained by l small and j t stafford 8 thus makes the comparison of the two sets relatively easy although the inequality of y o and y o provides a criterion for ruling out a full right quotient ring equality of the two sets by no means guarantees its existence example 2 6

a state space formalism for anisotropic elasticity part i rectilinear anisotropy a state space formalism for anisotropic elasticity including the thermal effect is developed a salient feature of the formalism is that it bridges the compliance based and stiffness based formalisms in a natural way the displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation yet it is simple and clear this is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three dimensional equations of anisotropic elasticity into a compact state equation and an output equation the homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of stroh extension twisting bending temperature change and body forces are accounted for through the particular solution based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner

on plate theories and saint venant s principle abstract it is generally known hat the classical germain kirchhoff plate theory is the leading term of the outer asymptotic expansion or interior solution in a small thickness parameter for the linear elastostatics of thin flat isotropic bodies this leading term or the actual outer solution alone cannot satisfy arbitrarily prescribed admissible edge data on the other hand the complementary inner asymptotic expansion or boundary layer solution is determined by a sequence of boundary value problems which are nearly as difficult to solve as the original problem for stress edge data saint venant s principle has been invoked to generate a set of stress boundary conditions for the classical plate theory as well as for higher order terms in the outer expansion giving various thick plate theories without any reference to the inner solution attempts to derive the corresponding boundary conditions for displacement and other types of edge data in the literature for general shape plates have not been successful the present study applies a general method developed by the authors to derive the correct set of boundary conditions for arbitrarily prescribed admissible edge data without an explicit solution of the inner or boundary layer solution for a number of special cases of general interest including cases with displacement edge data our general results also show that to be strictly correct saint venant s principle should be applied only to the leading term outer solution i e the classical plate theory

a refined state space formalism for piezothermoelasticity abstract the state space formalism for piezothermoelasticity tarn j q 2002c a state space formalism for piezothermoelasticity international journal of solids and structures 39 5173 5184 is refined by introducing the generalized displacement vector and generalized stress vectors as the fundamental variables in which appropriate electrical variables are included the basic equations of piezoelectricity with temperature change are formulated neatly into a state equation and an output equation in terms of the generalized displacement vector and generalized stress vectors the formalism bears a remarkable resemblance to its elastic counterpart various problems of piezothermoelasticity can be solved by simple extension of the corresponding solutions of anisotropic elasticity for illustration some fundamental problems are studied within the context and exact solutions are obtained in a systematic and self contained manner

a circular elastic cylinder under its own weight an exact analysis of deformation and stress field in a finite circular elastic cylinder under its own weight is presented with emphasis on the end effect the problem is formulated on the basis of the state space formalism for axisymmetric deformation of a transversely isotropic body upon delineating the hamiltonian characteristics of the formulation a rigorous solution which satisfies the end conditions is determined by using eigenfunction expansion the results show that the end effect is significant but confined to a local region near the base where the displacement and stress distributions are remarkably different from those according to the simplified solution that gives a uniaxial stress state it is more pronounced in the cylinder with the bottom plane being perfectly bonded than in smooth contact with a rigid base