on the structure of h sup infinity control systems and related extensions an existence condition of a h sup infinity controller which achieves a prescribed norm bound of the closed loop transfer function is derived in the frequency domain based on a generalization of the notion of j lossless systems this condition is regarded as a frequency domain representation of the well known existence condition in the state space represented in terms of the two algebraic riccati equations a notion of j orthogonal complement which is introduced as a generalization of the usual orthogonal complement play an important role in clarifying the fundamental frequency domain structure of the model matching problem and simplifying the computation of controllers the results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank it is shown that the proper controller achieving the prescribed norm bound exists even in this case in nonstandard cases the controller order can be smaller than the plant order

convergences for sequences of sets and linear mappings let e and f be nontrivial real normed vector spaces and e f be the class of continuous linear operators from e into f let a a n n n e f we shall suppose that a lim a n in a certain sense strong convergence uniform convergence

effective moduli of an anisotropic material with elliptical holes of arbitrary orientational distribution abstract effective moduli of a two dimensional anisotropic solid with elliptical holes having an arbitrary non random orientational distribution are given in closed form the results are derived in the non interacting approximation besides being rigorous at small defect densities this approximation constitutes the basic building block for various approximate schemes proper tensorial parameters of defect density dependent on ellipses eccentricity and their orientations relative to the matrix anisotropy axes are identified when derived in terms of such parameters expressions for the effective moduli cover in a unified way mixtures of holes of diverse eccentricities and arbitrary orientational distribution a number of special cases circles cracks of various orientational distributions are discussed if the field of defects is geometrically isotropic holes of the circular shapes or randomly oriented cracks it reduces the matrix anisotropy

a reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials abstract the general isotropic strain gradient elasticity theory with five higher order elastic constants is reformulated by introducing two different orthogonal decompositions of the strain gradient tensor just applying the mathematical reformulations no extra conditions needed the constitutive relations equilibrium equation and boundary conditions are reformulated in the reformulated theory the number of independent higher order elastic constants is proved to be three for isotropic materials which indicates that the five higher order elastic constants in the general isotropic strain gradient elasticity theory are dependent with each other therefore the general strain gradient elasticity theory contains only three independent material length scale parameters for isotropic materials in addition to the lame constants the new theory is different from the existed strain gradient elasticity theory with one or three material length scale parameters which introduces extra conditions during deriving process moreover the reformulated theory can be directly reduced to that of incompressible materials by assuming the terms associated with hydrostatic strains to be zero some examples such as torsion of cylindrical bars shearing of fixed end layers and pure bending of thin beams are performed to reveal the necessity of including multi length scale parameters in the strain gradient elasticity theory to predict size effects at micron scale

establishment of strain gradient constitutive relations by using asymptotic analysis and the finite element method for complex periodic microstructures abstract in this study we present a method of numerical homogenization combining asymptotic analysis and finite element modelling to establish constitutive laws of heterogeneous materials with periodic microstructures tacking into account high order strain gradients by performing asymptotic analysis the problem of local homogenization was split up into differential equations of different orders which were solved using finite element modelling this approach allows researcher to accurately calculate high order stiffness tensors from representative volume elements rves with complicated microstructures and geometrical forms the efficiency and accuracy of this approach were verified by means of numerical examples the mechanical implications consistency and strain energy convexity of the strain gradient constitutive laws obtained using the proposed approach are analyzed on the basis of the numerical results

on concentrated surface loads and green s functions in the toupin mindlin theory of strain gradient elasticity the two dimensional green s functions are derived for the half plane in the context of the complete toupin mindlin theory of isotropic strain gradient elasticity two types of green s functions exist for a concentrated force and a concentrated force dipole acting upon the surface of a traction free half plane our purpose is to examine the possible deviations from the predictions of classical theory of elasticity as well as from the simplified strain gradient theory which is frequently utilized in the last decade for the solution of boundary value problems of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions the boundary value problems are attacked with the aid of the fourier transform and exact full field solutions are provided our results indicate that in all cases the displacement field is bounded and continuous at the point of application of the concentrated loads the new solutions show therefore a more natural material response for the concentrated force problem both displacements and strains are found to be bounded whereas the strain gradients exhibit a logarithmic singularity thus in marked contrast with the classical elasticity solution a finite strain energy is contained within any finite portion of the body on the other hand in the case of the concentrated dipole force the strains are logarithmically singular and the strain gradients exhibit a cauchy type singularity the nature of the boundary conditions in strain gradient elasticity is highlighted through the solution of the pertinent boundary value problems finally based on our analytical solution the role of edge forces in strain gradient elasticity is elucidated employing simple equilibrium considerations

mindlin second gradient elastic properties from dilute two phase cauchy elastic composites part ii higher order constitutive properties and application cases starting from a cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first order by a certain discrepancy tensor see part i of the present article it is shown that the equivalent second gradient mindlin elastic solid i is positive definite only when the discrepancy tensor is negative defined ii the non local material symmetries are the same of the discrepancy tensor and iii the nonlocal effective behaviour is affected by the shape of the rve which does not influence the first order homogenized response furthermore explicit derivations of non local parameters from heterogeneous cauchy elastic composites are obtained in the particular cases of a circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix b n polygonal cylindrical voids in an isotropic matrix and c circular cylindrical voids in an orthortropic matrix

stress invariance in planar cosserat elasticity we extend the clm theorem cherkaev lurie milton proc r soc lond a 1992 438 519 529 to planar linear elastic materials of cosserat micropolar type that is those having microrotation as an additional degree of freedom besides the two in plane displacements more specifically it is shown that in the first planar problem of such media with smoothly varying material properties a shift in three out of four compliances is possible without changing the stress field the fourth coefficient represents a connection between the couple stress tensor and the torsion tensor while the other three represent compliances relating traction stress vector with the strains same shift holds for a locally anisotropic material whereby the shift tensor is seen to be a multiple of a rotation by a right angle a null lagrangian formulation is set up on this basis these results are obtained for materials with smooth properties with natural implications being drawn for their macroscopically effective moduli also it is shown that no shift is possible in case of a pseudo continuum a material admitting couple stresses but restricted to have the same connection between rotation and displacement gradient as in classical elasticity finally we establish that there is no shift in the second planar problem which represents a micropolar generalization of the classical out of plane elasticity

regular articleconvergences for sequences of sets and linear mappings let e f be normed vector spaces and let a an be continuous linear operators from e into f consider x xn y yn arbitrary nonempty closed subsets of e f we study the problem if x lim xn y lim yn in the sense of kuratowski or mosco or attouch wets do we have a x lim a xn a 1 y lim a 1n yn in the same sense in solving this problem we improve recent results obtained by i schochetman and r smith and published in this journal and extend some results of g beer established for the case f r

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

directional interpolation in the state space abstract this paper presents a new state space characterization of the directional interpolation based on the observation that the generalized pick matrix is regarded as a solution of a lyapunov type equation a simple representation of the set of all solutions is derived in terms of linear fractional transformations the computation of the associated j unitary matrix is simpler than the well known procedure based on the glover s scheme or the j spectral factorization of ball and ran

state space formulas for a model matching problem abstract in this paper we present a parametrization of all solutions to a model matching problem which is relevant for h control theory using an interpolation approach this was done in a specific case by h kimura in a recent paper here we shall follow the approach outlined in the book by b francis which uses a factorization approach

an interpolation theory approach to h controller degree bounds abstract we derive upper bounds for the mcmillan degree of all h optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration our analysis is confined to problems of the first kind which are characterized by the assumption that both p12 s and p21 s are square but not necessarily of the same size this paper which uses interpolation theory complements a previous paper which addresses the same problem through an approach based on approximation theory we demonstrate that the interpolation theory approach is more direct and circumvents a number of the technical difficulties in the previous method the final outcome is a much shorter proof as a by product we achieve a new result on the degree of an optimal solution of the matrix nevanlinna pick problem

global inverse spectral problems for rational matrix functions abstract given rational matrix functions 1 i m c 1 i n 1 a 1 1 b 1 and 2 i m c 2 i n 2 a 2 1 b 2 which are analytic and invertible on the unit circle we characterize in terms of the operators a 1 b 1 c 1 a 2 b 2 c 2 when there exists a single rational matrix function w i m c i n a 1 b such that wh 2 m 1 h 2 m and wh 2 m 2 h 2 m when this is the case we give explicit formulae for a b c in terms of a 1 b 1 c 1 a 2 b 2 c 2 applications include wiener hopf factorization j inner outer factorization and coprime factorization the results on j inner outer factorization have application to a model reduction problem for discrete time linear systems

derivation of anisotropic matrix for bi dimensional strain gradient elasticity behavior the different forms of second order elasticity operators in mindlin s strain gradient elasticity are given for a bi dimensional physical space these different forms are obtained according to the different symmetry classes of a material media dimensional aspects are discussed together with observations made on the physical behavior of such a media

is smaller always stiffer on size effects in supposedly generalised continua abstract heterogeneous materials having constitutive behaviour described by more generalised continuum theories incorporating additional degrees of freedom such as couple stress micropolar or micromorphic elasticity are expected to exhibit size effects in which there is an apparent increase in stiffness as the size scale reduces here we briefly demonstrate that for a simple heterogeneous material the size effect predicted when loaded in bending depends on the nature of the sample surface diverse size effects may thus be exhibited by the same material we then show by detailed finite element analysis of a more representative material with regular heterogeneity that this diversity of size effects might actually be observed in practice thereby providing an explanation for the contradictory size effects that have sometimes been reported for real materials

mindlin second gradient elastic properties from dilute two phase cauchy elastic composites part i closed form expression for the effective higher order constitutive tensor abstract it is shown that second order homogenization of a cauchy elastic dilute suspension of randomly distributed inclusions yields an equivalent second gradient mindlin elastic material this result is valid for both plane and three dimensional problems and extends earlier findings by bigoni and drugan bigoni d drugan w j 2007 analytical derivation of cosserat moduli via homogenization of heterogeneous elastic materials j appl mech 74 741 753 from several points of view i the result holds for anisotropic phases with spherical or circular ellipsoid of inertia ii the displacement boundary conditions considered in the homogenization procedure is independent of the characteristics of the material iii a perfect energy match is found between heterogeneous and equivalent materials instead of an optimal bound the constitutive higher order tensor defining the equivalent mindlin solid is given in a surprisingly simple formula applications treatment of material symmetries and positive definiteness of the effective higher order constitutive tensor are deferred to part ii of the present article

a micromechanics based approach for the derivation of constitutive elastic coefficients of strain gradient media a micromechanics based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented at the local scale all phases of the composite obey the classic equations of three dimensional elasticity but since the assumption of strict separation of scales is not verified the macroscopic behavior is described by the equations of strain gradient elasticity the methodology uses the series expansions at the local scale for which higher order terms which are generally neglected in standard homogenization framework are kept in order to take into account the microstructural effects an energy based micro macro transition is then proposed for upscaling and constitutes in fact a generalization of the hill mandel lemma to the case of higher order homogenization problems the constitutive relations and the definitions for higher order elasticity tensors are retrieved by means of the state law associated to the derived macroscopic potential as an illustration purpose we derive the closed form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite for handling the problems with an arbitrary microstructure a fft based computational iterative scheme is proposed in the last part of the paper its efficiency is shown in the particular case of composites reinforced by long fibers

a complete description of bi dimensional anisotropic strain gradient elasticity in the present paper spaces of fifth order tensors involved in bidimensional strain gradient elasticity are studied as a result complete sets of matrices representing these tensors in each one of their anisotropic system are provided this paper completes and ends some previous studies on the subject providing a complete description of the anisotropic bidimensional strain gradient elasticity it is proved that this behavior is divided into 14 non equivalent anisotropic classes 8 of them being isotropic for classical elasticity the classification and matrix representations of the acoustical gyrotropic tensor are also provided these results may find interesting applications to the study of waves propagation in dispersive micro structured media

computational second order homogenization of materials with effective anisotropic strain gradient behavior abstract a computational homogenization method to determine the effective parameters of mindlin s strain gradient elasticity sge model from a local heterogeneous cauchy linear material is developed the devised method which is an extension of the classical one based on the use of quadratic boundary conditions intents to correct the well known non physical problem of persistent gradient effects when the representative volume element rve is homogeneous those spurious effects are eliminated by introducing a microstructure dependent body force field in the homogenization scheme together with alternative definitions of the localization tensors with these modifications and by a simple application of the superposition principle the higher order stiffness tensors of sge are computed from elementary numerical calculations on rve within this new framework the convergence of sge effective properties is investigated with respect to the size of the rve finally a c1 fem procedure for simulating the behavior of the effective material at the macro scale is developed we show that the proposed model is consistent with the solutions arising from asymptotic analysis and that the computed effective tensors verify the expected invariance properties for several classes of anisotropy we also point out an issue that the present model shares with asymptotic based solutions in the case of soft inclusions applications to anisotropic effective strain gradient materials are provided as well as comparisons between fully meshed structures and equivalent homogeneous models

continuum and discrete models for structures including quasi inextensible elasticae with a view to the design and modeling of composite reinforcements abstract inspired by some composite fiber reinforcements used in aeronautical engineering and by the need of conceiving new metamaterials some discrete models including quasi inextensible elasticae are considered a class of continuum models approximately describing the macroscopic mechanical behavior of introduced structures is then heuristically proposed some of these continuum models can be regarded as a special kind of second gradient elastic media in which the higher gradient elasticity is conferred by the flexural stiffnesses of elasticae constituting the microscopic lattice the discrete models are studied by means of suitably tailored numerical codes designed to avoid numerical instabilities and locking and a comparison of discrete versus continuum models is attempted the obtained results show that the theory of generalized continua may be useful in some engineering applications and it plays a relevant role in the mechanics of woven composites the introduced discrete and continuum models are used to describe the so called bias extension test on woven fabrics and it is shown that a good choice to correctly reproduce the targeted phenomenology is to use a second gradient continuum theory however as discussed throughout the paper in the context of rigorous micro macro identification procedures there still remain many open problems to be solved especially when dealing with systems subjected to particular constraints such as inextensibility

a consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non separated scales in this work the question of homogenizing linear elastic heterogeneous materials with periodic microstructures in the case of non separated scales is addressed a framework if proposed where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low pass filters on the fine scale fluctuations the present theory extends the classical linear homogenization by substituting averaging operators by integral operators and localization tensors by nonlocal operators involving appropriate green functions as a result the obtained constitutive relationship at the mesoscale appears to be nonlocal compared to nonlocal elastic models introduced from a phenomenological point of view the nonlocal behavior has been fully derived from the study of the microstructure a discrete version of the theory is presented where the mesoscopic strain field is approximated as a linear combination of basis functions it allows computing the esoscopic nonlocal operator by means of a finite number of transformation tensors which can be computed numerically on the unit cell

well posedness and stability for abstract spline problems abstract in this work well posedness and stability properties of the abstract spline problem are studied in the framework of reflexive spaces tykhonov well posedness is proved without restrictive assumptions in the context of hilbert spaces also the stronger notion of levitin polyak well posedness is established a sequence of parametric problems converging to the given abstract spline problem is considered in order to study stability under natural assumptions convergence results for sequences of solutions of the perturbed problems are obtained

effective couple stress continuum model of cellular solids and size effects analysis the purpose of this paper is to develop a homogeneous orthotropic couple stress continuum model to take the place of the periodic heterogeneous cellular solids through generalizing the definition of the characteristic length for isotropic couple stress continuum four characteristic lengths are introduced as material engineering constants for such kind of continuum in order to determine the effective moduli and the characteristic lengths of the effective couple stress continuum a representative volume element rve method is constructed the effective properties are obtained based on the response of the rve under prescribed boundary conditions and our results agree with the analytical solutions in literature in addition the influences of the relative density the topology the size and the properties of the solid material of cellular materials on the effective moduli as well as the characteristic lengths are discussed respectively furthermore the size effects in cellular solid beams are investigated using our effective couple stress continuum model the results show that the developed continuum model in this paper can precisely capture the size effects in cellular solids

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

a micromechanically based couple stress model of an elastic two phase composite abstract the study reported in this paper concerns the determination of couple stress moduli and characteristic lengths of heterogeneous materials the study is set in the context of a planar two dimensional two phase composite with linear non couple stress classical elastic constituents with a single microstructural length scale inclusion spacing in an equilateral triangular array we use an approach which allows a replacement of this composite by an approximating couple stress continuum we determine the effective material parameters from the response of a unit cell under either displacement displacement periodic or traction boundary conditions we carry out computations of all the moduli by varying the stiffness ratio of both phases so as to cover a range of very different materials from porous solids through composites with rigid inclusions it is found that the three boundary conditions result in hierarchies of couple stress moduli in addition we observe from our numerical computations that these three boundary conditions also result in a hierarchy of characteristic lengths

strain gradient elastic homogenization of bidimensional cellular media the present paper aims at introducing an homogenization scheme for the determination of strain gradient elastic coefficients this scheme is based on a quadratic extension of homogeneous boundary condition hbc it allows computing strain elastic effective tensors this easy to handle computational procedure will then be used to construct overall honey comb like materials behavior and to verify some theoretical predictions about strain gradient elasticity

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

on optimal zeroth order bounds of linear elastic properties of multiphase materials and application in materials design abstract zeroth order bounds of elastic properties have been discussed by kroner 1977 and by nadeau and ferrari 2001 these bounds enclose the effective linear elastic properties of multiphase materials constituted of materials with arbitrary symmetry and of an arbitrary number of phases by using solely the material constants of the single materials nadeau and ferrari showed that these bounds are isotropic tensors and presented an algorithm for the determination of the upper and the lower zeroth order bound it is shown in this paper that a problem arises for the lower bound since the algorithm presented in nadeau and ferrari 2001 results in a negative compression modulus and or shear modulus although the considered stiffness is positive definite a simple analytic example for this undesirable property is given together with a short mathematica code of the algorithm in the present work the definition of the lower bound by nadeau and ferrari is modified thereby assuring its positive definiteness the mathematica code of the corrected algorithm is also given furthermore new bounds for non diagonal components are derived which give information of in principle accessible values for non diagonal stiffness components using the zeroth order bounds of the present work the practical application of zeroth order bounds for local and online material data bases of stiffness tensors is presented in order to accelerate purposes in materials design through efficient materials screening

analysis of the tilted flat punch in couple stress elasticity in the present paper we explore the response of a half plane indented by a tilted flat punch with sharp corners in the context of couple stress elasticity theory contact conditions arise in a number of modern engineering applications ranging from structural and geotechnical engineering to micro and nanotechnology as the contact scales reduce progressively the effects of the microstructure upon the macroscopic material response cannot be ignored the generalized continuum theory of couple stress elasticity introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying material microstructure the problem under investigation is interesting for three reasons firstly the indentor s geometry is simple so that benchmark results may be extracted secondly important deterioration of the macroscopic results may emerge in the case that a tilting moment is applied on the indentor inadvertently or in the case that the flat punch itself is not self aligning so that asymmetrical contact pressure distributions arise on the contact faces thirdly the voluntary application of a tilting moment on the flat punch during an experiment gives rise to potential capabilities of the flat punch for the determination of the material microstructural characteristic lengths the solution methodology is based on singular integral equations which have resulted from a treatment of the mixed boundary value problem via integral transforms and generalized functions the results show significant departure from the predictions of classical elasticity revealing that valuable information may be deducted from the indentation of a tilted punch of a microstructured solid