a refined quasi 3d isogeometric analysis for functionally graded microplates based on the modified couple stress theory the isogeometric analysis associated with a novel quasi 3d shear deformation theory is proposed to investigate size dependent behaviours of functionally graded microplates the modified couple stress theory with only one material length scale parameter is employed to effectively capture the size dependent effects within the microplates meanwhile the quasi 3d theory which is constructed from a novel seventh order shear deformation refined plate theory with four unknowns is able to consider both shear deformations and thickness stretching effect without requiring shear correction factors the nurbs based isogeometric analysis is integrated to exactly describe the geometry and approximately calculate the unknown fields with higher order derivative and continuity requirements the proposed approach is successfully applied to study the static bending free vibration and buckling responses of rectangular and circular functionally graded microplates with various types of boundary conditions in which some benchmark numerical examples are presented a number of investigations are also conducted to illustrate the effects of the material length scale material index and aspect ratios on the responses of the microplates

energy pairs in the micropolar continuum in this paper the concept of energy pairs in the micropolar continuum is introduced a brief review of the micropolar continuum theory is presented for using in the subsequent derivations a mathematical lagrangian strain and a wryness tensor for the micropolar continuum are introduced using the first law of thermodynamics and for isothermal processes the power of deformation is obtained and the energy pairs in the eulerian and lagrangian descriptions are defined also the micropolar stress and couple stress tensors which are energy pairs to the micropolar lagrangian strain and wryness measures are determined

vibration of two dimensional imperfect functionally graded 2d fg porous nano micro beams abstract this study presents analysis on the vibration behavior of the two dimensional functionally graded 2d fg nano and microbeams which are made of two kinds of porous materials for the first time based on timoshenko beam theory the material of the nano and microbeams is modeled as 2d fgms according to the power law the eringen s nonlocal elasticity and the modified couple stress theories are used respectively in case of nano and microbeams the boundary conditions are considered as clamped cc simply supported ss clamped simply supported cs and cantilever cf the governing equations are solved using the generalized differential quadrature method gdqm the effects of fg power indexes along length and thickness are studied along each other and also exclusively the parametric studies are presented for functionally graded along thickness fg functionally graded along length afg and 2d fg porous and perfect micro nanobeams under different boundary conditions

free vibration analysis for micro structures used in mems considering surface effects based on state space method a three dimensional 3 d approach is proposed to study the size dependent dynamical properties of micro structures considering surface effects the structure is modeled as a laminate composed of a bulk bounded with upper and bottom surface layers which are allowed to have different material properties from the bulk layer on the basis of 3 d fundamental elasticity the state equations including the surface properties of the structure can be established to analyze the size dependent dynamic responses of the plate like thin film structures used in mems to show the feasibility of the proposed approach a simply supported plate like thin structure including the surface layers is considered an algorithm strategy is proposed for the calculation of the state equations obtained to ensure that the numerical results can reveal the surface effects clearly even for extremely thin surface layers comparing with the two dimensional plate theories based size dependent models for the thin film structures in literature the present 3 d approach is exact which can provide benchmark results to assess the accuracy of various 2 d plate theories and numerical methods numerical tests prepared for the prediction of natural frequency are carried out to illustrate the surface effects some discussions and conclusions are presented based on the results of the numerical examples

uniqueness for plane crack problems in dipolar gradient elasticity and in couple stress elasticity the present work deals with the uniqueness theorem for plane crack problems in solids characterized by dipolar gradient elasticity the theory of gradient elasticity derives from considerations of microstructure in elastic continua mindlin r d 1964 micro structure in linear elasticity arch ration mech anal 16 51 78 and is appropriate to model materials with periodic structure according to this theory the strain energy density assumes the form of a positive definite function of the strain as in classical elasticity and the second gradient of the displacement additional term specific cases of the general theory employed here are the well known theory of couple stress elasticity and the recently popularized theory of strain gradient elasticity these cases are also treated in the present study we consider an anisotropic material response of the cracked plane body within the linear version of gradient elasticity and conditions of plane strain or anti plane strain it is emphasized that for crack problems in general a uniqueness theorem more extended than the standard kirchhoff theorem is needed because of the singular behavior of the solutions at the crack tips such a theorem will necessarily impose certain restrictions on the behavior of the fields in the vicinity of crack tips in standard elasticity a theorem was indeed established by knowles and pucik knowles j k pucik t a 1973 uniqueness for plane crack problems in linear elastostatics j elast 3 155 160 who showed that the necessary conditions for solution uniqueness are a bounded displacement field and a bounded body force field in our study we show that the additional to the two previous conditions requirement of a bounded displacement gradient field in the vicinity of the crack tips guarantees uniqueness within the general form of the theory of dipolar gradient elasticity in the specific cases of couple stress elasticity and pure strain gradient elasticity the additional requirement is less stringent this only involves a bounded rotation field for the first case and a bounded strain field for the second case

triangular mindlin microplate element in this article based on the most general form of strain gradient theory mgsgt a novel extended triangular mindlin plate element is proposed to accomplish this aim first the quadratic form of energy functional is obtained by vectorizing the higher order tensors of energy pairs from which the stiffness and mass matrices of the element are readily derived in comparison with the standard mindlin plate element the new element needs three additional nodal degrees of freedom dof including derivatives of lateral deflection and rotations which means a total of nine dofs per node also as compared to the standard mindlin plate element which requires only c0 shape functions the present one requires c1 continuous smooth shape functions due to second derivatives of deflection and rotations hence cubic polynomials are used to interpolate the displacement components the new element can be reduced to that based on the modified strain gradient theory msgt and the modified couple stress theory mcst moreover the standard mindlin plate element is recovered when the gradient based material parameters tend to zero the mindlin microplates with different boundary conditions are considered as the problem under study whose free vibration and bending are analyzed the results are compared with the exact solutions and excellent agreement is achieved

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

a boundary element method for solving 2 d and 3 d static gradient elastic problems part i integral formulation abstract a boundary element formulation is developed for the static analysis of two and three dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects the simple gradient elastic theory of aifantis expressed in the framework of mindlin s general theory is used to model this material behaviour a variational statement is established to determine all possible classical and non classical due to gradient terms boundary conditions of the general boundary value problem the gradient elastic fundamental solution for both two and three dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here it is found that for a well posed boundary value problem in addition to a boundary integral representation for the displacement a second boundary integral representation for its normal derivative is also necessary explicit expressions for interior displacements and stresses in integral form are also presented all the kernels in the integral equations are explicitly provided

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

a reciprocity theorem in linear gradient elasticity and the corresponding saint venant principle in many practical applications of nanotechnology and in microelectromechanical devices typical structural components are in the form of beams plates shells and membranes when the scale of such components is very small the material microstructural lengths become important and strain gradient elasticity can provide useful material modelling in addition small scale beams and bars can be used as test specimens for measuring the lengths that enter the constitutive equations of gradient elasticity it is then useful to be able to apply approximate solutions for the extension shear and flexure of slender bodies such approach requires the existence of some form of the saint venant principle the present work presents a statement of the saint venant principle in the context of linear strain gradient elasticity a reciprocity theorem analogous to betti s theorem in classic elasticity is provided first together with necessary restrictions on the constitutive equations and the body forces it is shown that the order of magnitude of displacements are in accord with the sternberg s statement of the saint venant principle the cases of stretching shearing and bending of a beam were examined in detail using two dimensional finite elements the numerical examples confirmed the theoretical results 2005 elsevier ltd all rights reserved

conservative numerical schemes for the vlasov equation abstract a new scheme for solving the vlasov equation using a phase space grid is proposed the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity several numerical results are presented in two and four dimensional phase space and the scheme is compared with the semilagrangian method this method is almost as accurate as the semilagrangian one and the local reconstruction technique is well suited for parallel computation

mechanical analysis of higher order gradient nanobeams this paper figures out effective of a new proposed beam element to handle mechanical behaviors of size dependent nanobeams on the basis of the higher order gradient model a higher order strain gradient model is derived from the nonlocal eringen differential equation the kinematic assumption of nanobeam is proposed by euler bernoulli theory galerkin finite element technique is employed for discretizing a beam domain and solves the equation of motion numerically the output results are compared and verified with those previously published works static buckling and dynamic behaviors of nonlocal and higher order gradient beams are investigated and discussed also the effectiveness of a higher order gradient model to incorporate the size dependent of nanobeams with different boundary conditions is presented the applicability of proposed model to analyze a carbon nanotubes is illustrated

uniqueness of initial boundary value problems in nonlocal elasticity abstract in this study we have dealt with the uniqueness of the solutions of a class of initial boundary value problems in linear isotropic homogeneous nonlocal elasticity the proof of uniqueness is based on the positive definiteness of total strain energy after giving the sufficient conditions under which total strain energy is positive definite it has been shown that the solutions of the initial boundary value problems which are considered in this study are unique existence of the solutions of these problems is an a priori assumption

long range cohesive interactions of non local continuum faced by fractional calculus abstract a non local continuum model including long range forces between non adjacent volume elements has been studied in this paper the proposed continuum model has been obtained as limit case of two fully equivalent mechanical models i a volume element model including contact forces between adjacent volumes as well as long range interactions distance decaying between non adjacent elements ii a discrete point spring model with local springs between adjacent points and non local springs with distance decaying stiffness connecting non adjacent points under the assumption of fractional distance decaying interactions between non adjacent elements a fractional differential equation involving marchaud type fractional derivatives has been obtained for unbounded domains it is shown that for unbounded domains the two mechanical models revert to lazopoulos and eringen model with fractional distance decaying functions it has also been shown that for a confined bar the stress strain relation is substantially different from that obtained simply using the truncated marchaud derivatives since a double integral instead of convolution integral appears moreover in the analysis of bounded domains the governing equations turn out to an integro differential equation including only the integral part of marchaud fractional derivatives on finite interval the mechanical boundary condition for the proposed model has been introduced consistently on the basis of mechanical considerations and the constitutive law of the proposed continuum model has been reported by mathematical induction several numerical applications have been reported to show verify and assess the concepts listed in this paper

nonlinear size dependent dynamics of an imperfect shear deformable microplate abstract the nonlinear size dependent dynamics of a geometrically imperfect shear deformable microplate is investigated based on the modified couple stress theory an initial imperfection in the out of plane direction is taken into account the third order shear deformation theory is employed to model the system by retaining in plane displacements and inertia the governing equations of motion of the system are obtained employing an energy method through use of the lagrange equations which upon employing an assumed mode technique yields a set of second order nonlinear ordinary differential equations with coupled terms a high dimensional discretised system is considered and made double dimensional via a change of variables in order to obtain a set of first order nonlinear ordinary differential equations the resulting equations are solved using a direct time integration technique resulting in time dependent generalised coordinates for the in plane and out of plane displacements and two rotations from these generalised coordinates phase plane portraits and fast fourier transforms ffts are also obtained moreover the frequency response and force response curves of the system are obtained using a continuation technique stability analysis is conducted via the floquet theory the effect of the initial imperfection as well as the length scale parameter on the system response is also examined

free vibration of size dependent mindlin microplates based on the modified couple stress theory this paper develops a mindlin microplate model based on the modified couple stress theory for the free vibration analysis of microplates this non classical plate model contains an internal material length scale parameter related to the material microstructures and is capable of interpreting the size effect that the classical mindlin plate model is unable to describe the higher order governing equations of motion and boundary conditions are derived using the hamilton principle the p version ritz method is employed to determine the natural frequencies of the microplate with different boundary conditions a detailed parametric study is conducted to study the influences of the length scale parameter side to thickness ratio and aspect ratio on the free vibration characteristics of the microplate it is found that the size effect is significant when the thickness of microplate is close to the material length scale parameter

classification construction and similitudes of quadratic forms we present a theory of classifying quadratic forms over an algebraic number field which is a reformulation of eichler s methods and different from the traditional theory of hasse as applications we determine all possible values of the discriminant of an integer valued quadratic form over q with a given dimension and index and also the number of genera of such forms we also investigate purely algebraic problems on the clifford groups and similitudes of quadratic forms

mixed finite element formulations of strain gradient elasticity problems theories on intrinsic or material length scales find applications in the modeling of size dependent phenomena in elasticity length scales enter the constitutive equations through the elastic strain energy function which in this case depends not only on the strain tensor but also on gradients of the rotation and strain tensors in the present paper the strain gradient elasticity theories developed by mindlin and co workers in the 1960s are treated in detail in such theories when the problem is formulated in terms of displacements the governing partial differential equation is of fourth order if traditional finite elements are used for the numerical solution of such problems then c 1 displacement continuity is required an alternative mixed finite element formulation is developed in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an integralsense a variational formulation is developed which can be used for both linear and non linear strain gradient elasticity theories the resulting finite elements require only c 0 continuity and are simple to formulate the proposed technique is applied to a number of problems and comparisons with available exact solutions are made 2002 elsevier science b v all rights reserved

three dimensional free vibration of arbitrarily shaped laminated micro plates with sliding interfaces within couple stress theory abstract free vibration of laminated micro plates with arbitrary geometry and boundary conditions consisting of several micro layers with free sliding frictional sliding perfect interfaces is of interest the inter layer bond in the direction normal to the interfaces is perfect and thus the corresponding displacement component is continuous across the interfaces the mentioned in plane interface conditions may be realized by varying the stiffness of the sliding inter layer spring of vanishing thickness for free sliding and perfect interface conditions the stiffness 0 and respectively couple stress theory as a useful higher order continuum theory is utilized to formulate the problem subsequently the corresponding hamiltonian is presented the three components of the displacement field in each layer are given by three distinct expressions each expression consists of the product of a base function and three dimensional polynomials with unknown coefficients the base functions are chosen according to the geometries of the system and the corresponding layer and to satisfy the homogeneous essential boundary conditions for the enforcement of the displacement continuity in the direction normal to the interfaces an appropriate series with unknown coefficients pertinent to each layer is added the influences of the level of inter layer imperfection and couple stress on the dynamical characteristics including the angular frequencies of free vibration and the corresponding mode shapes are addressed

a strain difference based nonlocal elasticity model abstract a two component local nonlocal constitutive model for macroscopically inhomogeneous linear elastic materials but constant internal length is proposed in which the stress is the sum of the local stress and a nonlocal type stress expressed in terms of the strain difference field hence identically vanishing in the case of uniform strain attention is focused upon the particular case of piecewise homogeneous material the proposed model is thermodynamically consistent with a suitable free energy potential it constitutes an improved form of the vermeer and brinkgreve a new effective nonlocal strain measure for softening plasticity in chambon r desrues j vardulakis i eds localization and bifurcation theory for soils and rocks balkema rotterdam 1994 pp 89 100 model and can also be considered derivable from the eringen nonlocal elasticity model through a suitable enhancement technique based on the concept of redistribution of the local stress the concept of equivalent distance is introduced to macroscopically account for the further attenuation effects produced by the inhomogeneity upon the long distance interaction forces with the aid of a piecewise homogeneous bar in tension a portion of which degrades progressively till failure it is shown that under a suitable choice of a material constant the solution procedure exhibits no pathological features numerical instability mesh sensitivity in every degraded bar condition including the limit idealized stress free condition of the failed bar

benchmarks in nonlocal elasticity defined by eringen s integral model abstract in this paper we present low residual approximate solutions for nonlocal 1d and 2d elasticity problems defined according to eringen s integral model the benchmarks in the 1d cases are defined by prescribing the stress field while the unknown fields are the strains or the displacements for the 2d cases we define problems with equilibrated tractions and evaluate the approximate displacement field meanwhile a fourier series as well as a set of chebyshev polynomials are used as the basis functions for the main unknown fields we increase the number of the approximation functions to decrease the norm of the residuals and repeat the procedure until reasonable accuracy is obtained for the final solution since the procedure is very time consuming in some benchmark problems we present the calculated coefficients and in some other we give some point wise values for further use the results presented in this paper are particularly useful for the validation and convergence studies when numerical methods are to be used for the solution of the nonlocal elasticity problems

size dependent nonlinear analysis and damping responses of fg cntrc micro plates abstract this paper presents a nonlinear numerical model which is based on the modified couple stress theory mcst and trigonometric shear deformation theory coupled with isogeometric analysis the present approach captures the small scale effects on the geometrically nonlinear behaviors of functionally graded carbon nanotube reinforced composite fg cntrc micro plate with four patterns distribution the equations of motion are established based on a galerkin weak form associated with von karman nonlinear strains the mcst utilizes only one material length scale parameter to predict the size effect in fg cntrc micro plate for which its material properties are derived from an extended rule of mixture the solutions of nonlinear static equation are obtained by using the newton raphson technique and the newmark time iteration procedure in association with picard method is assigned to get responses of the nonlinear dynamic problems in addition the rayleigh damping is applied to consider the influence of damping characteristic on the oscillation of fg cntrc micro plates comparisons are performed to verify the proposed approach afterward the numerical examples are used to show the effects of the distribution of carbon nanotubes cnt their volume fraction the material length scale parameter and the boundary conditions on the nonlinear static and dynamic behaviors of fg cntrc micro plates

mathematical model of a three layer micro and nano beams based on the hypotheses of the grigolyuk chulkov and the modified couple stress theory abstract the mathematical model of three layered beams developed based on the hypothesis of the grigolyuk chulkov and the modified couple stress theory and the size depended equations governing the layers motions on the micro and nano scales is constructed the hamilton s principle yields the novel equations of motion as well as the boundary initial conditions regarding beams displacement the latter ones clearly exhibit the size dependent dynamics of the studied micro and nano beams and the introduced theory overlaps with the classical beam equations for large enough layer thickness in particular a three layer beam with the micro layer thickness has been investigated with respect to the classical theory of grigolyuk chulkov the derived boundary problem is of sixth order and can be solved analytically in the case of statics the carried out numerical experiments allowed to detect and explain size dependent effects exhibited by the micro beams the beam deflections and stress yielded by the employed couple stress model are less than those predicted by the classical grigolyuk chulkov theory while the estimated eigen frequencies are higher respectively it has been shown that the proposed model can be reduced to the classical three layer grigolyuk chulkov beam through increase of the layers thickness which validates our approach

couple stress theory for solids by relying on the definition of admissible boundary conditions the principle of virtual work and some kinematical considerations we establish the skew symmetric character of the couple stress tensor in size dependent continuum representations of matter this fundamental result which is independent of the material behavior resolves all difficulties in developing a consistent couple stress theory we then develop the corresponding size dependent theory of small deformations in elastic bodies including the energy and constitutive relations displacement formulations the uniqueness theorem for the corresponding boundary value problem and the reciprocal theorem for linear elasticity theory next we consider the more restrictive case of isotropic materials and present general solutions for two dimensional problems based on stress functions and for problems of anti plane deformation finally we examine several boundary value problems within this consistent size dependent theory of elasticity

the finite element method for the mechanically based model of non local continuum summary in this paper the finite element method fem for the mechanically based non local elastic continuum model is proposed in such a model each volume element of the domain is considered mutually interacting with the others beside classical interactions involved by the cauchy stress field by means of central body forces that are monotonically decreasing with their inter distance and proportional to the product of the interacting volume elements the constitutive relations of the long range interactions involve the product of the relative displacement of the centroids of volume elements by a proper distance decaying function which accounts for the decrement of the long range interactions as long as distance increases in this study the elastic problem involving long range central interactions for isotropic elastic continuum will be solved with the aid of the fem the accuracy of the solution obtained with the proposed fem code is compared with other solutions obtained with galerkins approximation as well as with finite difference method moreover a parametric study regarding the effect of the material length scale in the mechanically based model and in the kroner eringen non local elasticity has been investigated for a plane elasticity problem copyright 2011 john wiley sons ltd

a non classical mindlin plate model incorporating microstructure surface energy and foundation effects a non classical model for a mindlin plate resting on an elastic foundation is developed in a general form using a modified couple stress theory a surface elasticity theory and a two parameter winkler pasternak foundation model it includes all five kinematic variables possible for a mindlin plate the equations of motion and the complete boundary conditions are obtained simultaneously through a variational formulation based on hamilton s principle and the microstructure surface energy and foundation effects are treated in a unified manner the newly developed model contains one material length scale parameter to describe the microstructure effect three surface elastic constants to account for the surface energy effect and two foundation parameters to capture the foundation effect the current non classical plate model reduces to its classical elasticity based counterpart when the microstructure surface energy and foundation effects are all suppressed in addition the new model includes the mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases recovers the kirchhoff plate model incorporating the microstructure surface energy and foundation effects and degenerates to the timoshenko beam model including the microstructure effect to illustrate the new mindlin plate model the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived

subgrid interaction and micro randomness novel invariance requirements in infinitesimal gradient elasticity abstract we present a micromechanically motivated form of the curvature energy in infinitesimal isotropic gradient elasticity the basis is a homogenization averaging scheme using a micro randomness assumption imposed on a directional higher gradient interaction term these directional interaction terms are matrix valued allowing to apply the standard orthogonal cartan lie algebra decomposition averaging over all subgrid directions leads to three quadratic curvature terms which are conformally invariant when neglecting volumetric effects restricted to rotational inhomogeneities we motivate thereby a symmetric couple stress tensor in the infinitesimal indeterminate couple stress model of koiter mindlin toupin type relations are established to a novel conformally invariant linear cosserat model

chirality in isotropic linear gradient elasticity abstract chirality is generally speaking the property of an object that can be classified as left or right handed though it plays an important role in many branches of science chirality is encountered less often in continuum mechanics so most classical material models do not account for it in the context of elasticity for example classical elasticity is not chiral leading different authors to use cosserat elasticity to allow modelling of chiral behaviour gradient elasticity can also model chiral behaviour however this has received much less attention than its cosserat counterpart this paper shows how in the case of isotropic linear gradient elasticity a single additional parameter can be introduced that describes chiral behaviour this additional parameter directly linked to three dimensional deformation can be either negative or positive with its sign indicating a discrimination between the two opposite directions of torsion two simple examples are presented to show the practical effects of the chiral behaviour

nonlocal optimal design a new perspective about the approximation of solutions in optimal design abstract it is well known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations in the present work a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem the new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals the purpose of this paper is to prove existence of optimal design for the new model this work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional see 14

capturing the effect of thickness on size dependent behavior of plates with nonlocal theory abstract the effective elastic properties of nano structures are shown to be strongly size dependent in this paper using a three dimensional strong nonlocal elasticity we have presented a formulation to capture the size dependent behavior of plate structures as a function of their thickness this paper discusses some new aspects of employing a three dimensional nonlocal formulation for analysis of plates namely the confining of the nonlocal kernel in the near boundary regions at the two surfaces of the plate to address this aspect we have studied two different types of nonlocal kernels one bounded in a finite domain of the structure and the other non bounded this study shows that the influence of the plate s thickness on its bending stiffness can be captured within the nonlocal elasticity framework and this influence highly depends on the bounding of the nonlocal kernel particularly for a uniformly deformed plate with a homogeneous isotropic material using the nonlocal formulation with the bounded domain reflects the physics of the problem better

characterisation of the weak lower semicontinuity for a type of nonlocal integral functional the n dimensional scalar case abstract in this work we are going to prove the functional j defined by j u w u x u y d x d y is weakly lower semicontinuous in w 1 p if and only if w is separately convex we assume that is an open set in r n and w is a real valued continuous function fulfilling standard growth and coerciveness conditions the key to state this equivalence is a variational result established in terms of young measures

on the transformation formulas of theta series here s is a nondegenerate rational symmetric matrix of signature q s s 3 x iy is a variable in the siegel half space hn of degree n u is a variable in the space cn of all complex n x q matrices runs over the set qq of all rational q x n matrices a is a c valued function on qn that is a finite linear combination of characteristic functions of cosets of qq modulo a z lattice a is a real matrix such that s ta diag lq 1 a and e c exp 2iric for c e c if q 1 s a 1 and a is suitably chosen the series g becomes a classical theta function of jacobi riemann if s is positive definite n 1 u 0 and q is even g is an elliptic modular form of weight q 2 studied by hecke also siegel introduced the series with u 0 in his theory of quadratic forms in si now g satisfies a transformation formula of the following type for every

a new kirchhoff plate model based on a modified couple stress theory in this paper a new kirchhoff plate model is developed for the static analysis of isotropic micro plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect the proposed model is capable of handling plates with complex geometries and boundary conditions from a detailed variational procedure the governing equilibrium equation of the micro plate and the most general boundary conditions are derived in terms of the deflection using the principle of minimum potential energy the resulting boundary value problem is of the fourth order instead of existing gradient theories which is of the sixth order and it is solved using the method of fundamental solutions mfs which is a boundary type meshless method several plates of various shapes aspect and poisson s ratios are analyzed to illustrate the applicability of the developed micro plate model and to reveal the differences between the current model and the classical plate model moreover useful conclusions are drawn from the micron scale response of this new kirchhoff plate model

an efficient computational approach for size dependent analysis of functionally graded nanoplates abstract in this paper an efficient computational approach based on refined plate theory rpt including the thickness stretching effect namely quasi 3d theory in conjunction with isogeometric formulation iga is proposed for the size dependent bending free vibration and buckling analysis of functionally graded nanoplate structures the present novel quasi 3d theory not only possesses 4 variables as refined plate theory but also accounts for both shear deformation and stretching effect without any requirement of shear correction factors scfs the size dependent effect is taken into account by nonlocal elasticity theory isogeometric analysis shows a great advantage in dealing with the high continuity and high order derivative requirements of the displacement fields used in quasi 3d and nonlocal theory the reliability and accuracy of the present method are ascertained by comparing the obtained results with other published ones numerical examples are also performed to show the significance of nonlocal effect material distribution profile aspect ratios and boundary conditions on the behaviour of fgm nanoplates

geometric material analogy for multiscale modelling of twisted plates abstract it is well known that the macroscopic behaviour of many engineering materials is strongly affected by the role of underlying microstructure currently though mathematical expressions linking behaviour of large scale structures to the geometry of their microscopic structure are largely lacking in this respect establishing quantitative links across different material lengthscales may offer new pathways for engineering design in the present work an analogy between cross sectional geometrical properties representing macrostructure and a material length parameter representing microstructure is presented the analogy is established through the study of a thin plate subject to axial loading undergoing finite displacements from two alternative perspectives first we consider a thin elastic plate with a pretwist about the loading axis where a warping term is introduced accounting for the out of plane deformation of the cross section the coupled governing differential equations and the corresponding coupled boundary conditions are explicitly derived employing a classical structural mechanics approach utilising an energy variational statement secondly an axially loaded thin flat plate i e with no pretwist is studied with strain gradient elasticity theory incorporating only one material length parameter representing the microstructure in addition to the two classical lame stiffness constants the ensuing analogy emerges by comparison of the governing equations of the two formulations which shows a mathematical expression can be identified which incorporates both geometric and material length variables that formalises the link between microscale and macroscale this mathematical expression which constitutes the kernel of the proposed multiscale approach admits a twofold interpretation depending on the assumed independent variable on the one hand the proposed multiscale modelling approach suggests that a plate with complex global geometry can be substituted by a structurally equivalent flat plate with constitutive relations given by a non local strain gradient theory on the other hand the material length parameter can be interpreted on a physical basis because for the first time it has been identified as a known function of geometrical features of the structure through simple algebraic relationships for various cross sectional profiles

a deterministic solver for a hybrid quantum classical transport model in nanomosfets we model a nanomosfet by a mesoscopic time dependent coupled quantum classical system based on a sub band decomposition and a simple scattering operator we first compute the sub band decomposition and electrostatic force field described by a schrodinger poisson coupled system solved by a newton raphson iteration using the eigenvalue eigenfunction decomposition the transport in the classical direction for each sub band modeled by semiclassical boltzmann type equations is solved by conservative semi lagrangian characteristic based methods numerical results are shown for both the thermodynamical equilibrium and time dependent simulations in typical nowadays nanomosfets

convergence and performance of the h and p extensions with mixed finite element c0 continuity formulations for tension and buckling of a gradient elastic beam abstract mixed formulations with c 0 continuity basis functions are employed for the solution of some types of one dimensional fourth and sixth order equations resulting from axial tension and buckling of gradient elastic beams respectively a basic characteristic of gradient elasticity type equations is the appearance of boundary layers in the higher order derivatives of the displacements e g in the stress fields this is due to the small parameters related to the size of the microstructure entering the governing equations the proposed mixed formulations are based on generalizations of the well known ciarlet raviart mixed method where the new main variables are related to second order or fourth order for the buckling problem derivatives of the displacement field the continuous and discrete babuska brezzi inf sup conditions are established the mixed formulations are numerically tested for both the uniform h and p extensions with regard to the axial tension problem the standard quasi optimal rates of convergence are numerically verified in all cases i e algebraic rate of convergence for the h extension and exponential rate for the p extension on the other hand the h extension observed convergence rates of the critical buckling load for the second model problem are slightly higher than the theoretical ones found in the literature especially for polynomial order p 1 the respective observed rates of convergence of the buckling load for the p extension are still exponential

locking free variational formulations and isogeometric analysis for the timoshenko beam models of strain gradient and classical elasticity abstract the timoshenko beam bending problem is formulated in the context of strain gradient elasticity for both static and dynamic analysis two non standard variational formulations in the sobolev space framework are presented in order to avoid the numerical shear locking effect pronounced in the strain gradient context both formulations are shown to be reducible to their locking free counterparts of classical elasticity conforming galerkin discretizations for numerical results are obtained by an isogeometric c p 1 continuous approach with b spline basis functions of order p 2 convergence analyses cover both statics and free vibrations as well as both strain gradient and classical elasticity parameter studies for the thickness and gradient parameters including micro inertia terms demonstrate the capability of the beam model in capturing size effects finally a model comparison between the gradient elastic timoshenko and euler bernoulli beam models justifies the relevance of the former confirmed by experimental results on nano beams from literature

a different approach to eringen s nonlocal integral stress model with applications for beams abstract the nonlocal continuum theory either integral or differential form has been formulated evolved and widely used in our era to explain size effect phenomena in micro and nano structures in the case of euler bernoulli beam theory ebbt the nonlocal integral form produces energy consistent formulas unlike the nonlocal differential form revealing inconsistencies and giving rise to paradoxes in this work our overall research objective mainly focuses on nonlocal integral elasticity analysis of beams by employing a normalized symmetric kernel this kernel corresponds to a finite domain and will be called modified kernel from now on that satisfies all the properties of a probability density function as well as successfully handles the physical inconsistencies of classic types of kernel our concern is to investigate the static response of a beam with various types of loading and boundary conditions bcs by making use the modified kernel and the kernel corresponding to the two phase nonlocal integral tpni model carrying out numerical methods to our problems the deducing results appear more flexible behavior than those of classic local and the common nonlocal differential respectively what is more a comparison is made between the results of the aforementioned kernels to demonstrate the advantages of the modified kernel unlike the common nonlocal differential form the nonlocal integral forms do not give rise to paradoxes moreover it is critical that the tpni model does not raise paradoxes as has been observed in other publications presented in the literature

a galerkin method for distributed systems with non local damping in this paper a non local damping model including time and spatial hysteresis effects is used for the dynamic analysis of structures consisting of euler bernoulli beams and kirchoff plates unlike ordinary local damping models the damping force in a non local model is obtained as a weighted average of the velocity field over the spatial domain determined by a kernel function based on distance measures the resulting equation of motion for the beam or plate structures is an integro partial differential equation rather than the partial differential equation obtained for a local damping model approximate solutions for the complex eigenvalues and modes with non local damping are obtained using the galerkin method numerical examples demonstrate the efficiency of the proposed method for beam and plate structures with simple boundary conditions for non local and non viscous damping models and different kernel functions

nonlinear thermo elastic bending behavior of graphene sheets embedded in an elastic medium based on nonlocal elasticity theory this paper investigates the large deflection behavior of orthotropic single layered graphene sheet slgs embedded in a winkler pasternak elastic medium under a uniform transverse load in thermal environments using the nonlocal differential constitutive relations of eringen the slgs is modeled as a nonlocal orthotropic plate using the principle of virtual work the coupled nonlinear equilibrium equations are obtained based on first order shear deformation theory fsdt and the von karman geometrical model the differential quadrature dq discretized form of the governing equations with clamped and simply supported boundary conditions is derived the newton raphson iterative scheme is used to solve the resulting system of nonlinear algebraic equations effects of small scale parameter thermal environment width to length elasticity ratio aspect ratio thickness elastic foundation load value and boundary conditions are considered in detail the results show that unlike the simply supported boundary conditions increase of small scale parameter plays a decreasing role in effect of thermal environment on the deflections of nanoplates with clamped edges it is also observed that increasing the plate thickness thermal load effects decline noticeably and depending on the small scale value the thermal loads do not have any significant effect on the results beyond a specified thickness

classification of integer valued symmetric forms we classify all bf z valued symmetric forms that are reduced in the sense that they cannot be represented nontrivially by another bf z valued symmetric form we determine the possible values of the discriminant and signature of each such form and also the number of genera of the forms with the given discriminant and signature

variational analysis of gradient elastic flexural plates under static loading gradient elastic flexural kirchhoff plates under static loading are considered their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case a variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions classical and non classical ones two circular gradient elastic plates clamped or simply supported at their boundaries are analyzed analytically and the gradient effect on their static response is assessed in detail a rectangular gradient elastic plate simply supported at its boundaries is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors finally a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple stress elasticity theories

atomistic viewpoint of the applicability of microcontinuum theories abstract microcontinuum field theories including micromorphic theory microstructure theory micropolar theory cosserat theory nonlocal theory and couple stress theory are the extensions of the classical field theories for the applications in microscopic space and time scales they have been expected to overlap atomic model at microscale and encompass classical continuum mechanics at macroscale this work provides an atomic viewpoint to examine the physical foundations of those well established microcontinuum theories and to justify their applicability through lattice dynamics and molecular dynamics

bending and stability analysis of gradient elastic beams abstract the problems of bending and stability of bernoulli euler beams are solved analytically on the basis of a simple linear theory of gradient elasticity with surface energy the governing equations of equilibrium are obtained by both a combination of the basic equations and a variational statement the additional boundary conditions are obtained by both variational and weighted residual approaches two boundary value problems one for bending and one for stability are solved and the gradient elasticity effect on the beam bending response and its critical buckling load is assessed for both cases it is found that beam deflections decrease and buckling load increases for increasing values of the gradient coefficient while the surface energy effect is small and insignificant for bending and buckling respectively

free vibration analysis of functionally graded size dependent nanobeams abstract this paper presents free vibration analysis of functionally graded fg size dependent nanobeams using finite element method the size dependent fg nanobeam is investigated on the basis of the nonlocal continuum model the nonlocal elastic behavior is described by the differential constitutive model of eringen which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators the material properties of fg nanobeams are assumed to vary through the thickness according to a power law the nanobeam is modeled according to euler bernoulli beam theory and its equations of motion are derived using hamilton s principle the finite element method is used to discretize the model and obtain a numerical approximation of the equation of motion the model is validated by comparing the obtained results with benchmark results numerical results are presented to show the significance of the material distribution profile nonlocal effect and boundary conditions on the dynamic characteristics of nanobeams

vibration analysis of beams with non local foundations using the finite element method in this paper a non local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method unlike local foundation models the reaction of the non local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure in the finite element analysis the interpolating shape functions of the element displacement field are identical to those of standard two node beam elements however for non local elasticity or damping nodes remote from the element do have an effect on the energy expressions and hence the damping and stiffness matrices the expressions of these direct and cross matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions alternatively numerical integration may be applied to obtain solutions numerical results for eigenvalues and associated eigenmodes of euler bernoulli beams are presented and compared where possible with results in literature using exact solutions and galerkin approximations the examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non local viscoelastic foundations copyright 2007 john wiley sons ltd

on some fundamental misunderstandings in the indeterminate couple stress model a comment on recent papers of a r hadjesfandiari and g f dargush abstract in a series of papers which are either published hadjesfandiari a dargush g f 2011a couple stress theory for solids int j solids struct 48 18 2496 2510 hadjesfandiari a dargush g f 2013 fundamental solutions for isotropic size dependent couple stress elasticity int j solids struct 50 9 1253 1265 or available as preprints hadjesfandiari a dargush g f 2010 polar continuum mechanics preprint arxiv 1009 3252 hadjesfandiari a r dargush g f 2011b couple stress theory for solids int j solids struct 48 2496 2510 hadjesfandiari a r 2013 on the skew symmetric character of the couple stress tensor preprint arxiv 1303 3569 hadjesfandiari a r dargush g f 2015a evolution of generalized couple stress continuum theories a critical analysis preprint arxiv 1501 03112 hadjesfandiari a r dargush g f 2015b foundations of consistent couple stress theory preprint arxiv 1509 06299 hadjesfandiari and dargush have reconsidered the linear indeterminate couple stress model they are postulating a certain physically plausible split in the virtual work principle based on this postulate they claim that the second order couple stress tensor must always be skew symmetric since they do not consider that the set of boundary conditions intervening in the virtual work principle is not unique their statement is not tenable and leads to some misunderstandings in the indeterminate couple stress model this is shown by specifying their development to the isotropic case however their choice of constitutive parameters is mathematically possible and we show that it still yields a well posed boundary value problem

nonlocal elasticity and related variational principles abstract the eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied the model is refined by assuming an attenuation function depending on the geodetical distance between material particles such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented a suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model the nonlocal elasticity boundary value problem for infinitesimal displacements and quasi static loads is addressed and the conditions for the solution uniqueness are established three variational principles nonlocal counterparts of classical ones i e the total potential energy the complementary energy and the mixed hu washizu principles are provided the former is used to derive a nonlocal type fem nl fem in which the symmetric global stiffness matrix reflects the nonlocality features of the problem an alternative standard fem based solution method is also provided which consists in an iterative procedure of the type local prediction nonlocal correction in which the nonlocality is simulated by an imposed like correction strain the potentialities of these analysis methods are pointed out their numerical implementations being the object of an ongoing research work

isogeometric analysis for sixth order boundary value problems of gradient elastic kirchhoff plates abstract sixth order boundary value problems of a one parameter gradient elastic kirchhoff plate model are formulated in a weak form within an h 3 sobolev space setting with the corresponding equilibrium equations and general boundary conditions the corresponding conforming galerkin method is proposed with error estimates for discretizations satisfying c 2 continuity requirements continuity coercivity and consistency of the corresponding bilinear form are utilized for proving the theoretical results numerical computations with conforming isogeometric discretizations of c p 1 continuous nurbs basis functions of order p 3 confirm the theoretical results and illustrate the features of the problem for both statics and free vibrations in particular the effects of the additional boundary conditions and parameter dependent boundary layers corresponding to the gradient elasticity theory are addressed by the numerical examples

plane stress problems in nonlocal elasticity finite element solutions with a strain difference based formulation abstract an enhanced computational version of the finite element method in the context of nonlocal strain integral elasticity of eringen type is discussed the theoretical bases of the method are illustrated focusing the attention on numerical and computational aspects as well as on the construction of the nonlocal elements matrices two numerical examples of plane stress nonlocal elasticity are presented to show the potentials and the limits of the promoted approach

nonlocal integral elasticity 2d finite element based solutions abstract a finite element based method theorized in the context of nonlocal integral elasticity and founded on a nonlocal total potential energy principle is numerically implemented for solving 2d nonlocal elastic problems the key idea of the method known as nonlocal finite element method nl fem relies on the assumption that the postulated nonlocal elastic behaviour of the material is captured by a finite element endowed with a set of cross stiffness element s matrices able to interpret the nonlocality effects induced in the element itself by the other elements in the mesh an eringen type nonlocal elastic model is assumed with a constitutive stress strain law of convolutive type which governs the nonlocal material behaviour computational issues as the construction of the nonlocal element and global stiffness matrices are treated in detail few examples are presented and the relevant numerical findings discussed both to verify the reliability of the method and to prove its effectiveness

the helicoidal modeling in computational finite elasticity part i variational formulation the finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields the rototranslation between two material particles can be described by a single complex tensorial quantity which is recognized to be orthogonal its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body after due introduction to dual tensors and rototranslations the polar description of the continuum is addressed with particular care to mixed differentiations of the rototranslation field then a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis and the three field two field and one field principles are drawn and linearized the proposed modeling is expected to be profitably exploited in non linear finite element analyses of solids undergoing finite displacements rotations and strains

couple stress based strain gradient theory for elasticity the deformation behavior of materials in the micron scale has been experimentally shown to be size dependent in the absence of stretch and dilatation gradients the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures in the couple stress theory formulation only conventional equilibrium relations of forces and moments of forces are used the couple s association with position is arbitrary in this paper an additional equilibrium relation is developed to govern the behavior of the couples the relation constrained the couple stress tensor to be symmetric and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system on the basis of this modification a linear elastic model for isotropic materials is developed the torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification

inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems abstract at small length scales and or in presence of large field gradients the implicit long wavelength assumption of classical elasticity breaks down postulating a form of second gradient elasticity with couple stresses as a suitable phenomenological model for small scale elastic phenomena we herein extend eshelby s classical formulation for inclusions and inhomogeneities while the modified size dependent eshelby s tensor and hence the complete elastic state of inclusions containing transformation strains or eigenstrains is explicitly derived the corresponding inhomogeneity problem leads to integrals equations which do not appear to have closed form solutions to that end eshelby s equivalent inclusion method is extended to the present framework in form of a perturbation series that then can be used to approximate the elastic state of inhomogeneities the approximate scheme for inhomogeneities also serves as the basis for establishing expressions for the effective properties of composites in second gradient elasticity with couple stresses the present work is expected to find application towards nano inclusions and certain types of composites in addition to being the basis for subsequent non linear homogenization schemes

a parallel deterministic solver for the schrodinger poisson boltzmann system in ultra short dg mosfets comparison with monte carlo abstract we present a parallel deterministic solver for the boltzmann schrodinger poisson system for partially confined dg mosfets our 2d model uses a dimensional coupling the electrons behave as particles along the longitudinal dimension and as waves along the transverse dimension in ben abdallah et al 2009 the authors validated in a simplified setting the solver for the eigenproblems in this paper we have focused on obtaining physically accurate results by introducing a realistic si conduction band and electron phonon interactions we have switched from cartesian to elliptic variables for the wave vector and consequently radically rebuilt the time integrator the code has been parallelized following a domain decomposition approach the results have been compared to those given by a monte carlo solver

some c0 continuous mixed formulations for general dipolar linear gradient elasticity boundary value problems and the associated energy theorems abstract the goal of this work is a systematic presentation of some classes of mixed weak formulations for general multi dimensional dipolar gradient elasticity fourth order boundary value problems the displacement field main variable is accompanied by the double stress tensor and the cauchy stress tensor case 1 or u formulation the double stress tensor alone case 2 or u formulation the double stress the cauchy stress the displacement second gradient and the standard strain field case 3 or e u formulation and the displacement first gradient along with the equilibrium stress case 4 or u formulation in all formulations the respective essential conditions are built in the structure of the solution spaces for cases 1 2 and 4 one dimensional analogues are presented for the purpose of numerical comparison moreover the standard galerkin formulation is depicted it is noted that the standard galerkin weak form demands c1 continuous conforming basis functions on the other hand up to first order derivatives appear in the bilinear forms of the current mixed formulations hence standard c0 continuous conforming basis functions may be employed in the finite element approximations the main purpose of this work is to provide a reference base for future numerical applications of this type of mixed methods in all cases the associated quadratic energy functionals are formed for the purpose of completeness

balance laws and energy release rates for cracks in dipolar gradient elasticity abstract it is the purpose of this work to derive the balance laws in the gunther knowles sternberg sense pertaining to dipolar gradient elasticity the theory of dipolar gradient or grade 2 elasticity derives from considerations of microstructure in elastic continua mindlin r d 1964 microstructure in linear elasticity arch rational mech anal 16 51 78 and is appropriate to model materials with periodic structure according to this theory the strain energy density assumes the form of a positive definite function of the strain as in classical elasticity and the gradient of both strain and rotation additional terms the balance laws are derived here through a more straightforward procedure than the one usually employed in classical elasticity i e noether s theorem indeed the pertinent balance laws are obtained through the action of the standard operators of vector calculus grad curl and div on appropriate forms of the hamiltonian of the system under consideration these laws are directly related to the energy release rates in the processes of crack translation rotation and self similar expansion under certain conditions they are identified with conservation laws and path independent integrals are obtained

energy theorems and the j integral in dipolar gradient elasticity within the framework of mindlins dipolar gradient elasticity general energy theorems are proved in this work these are the theorem of minimum potential energy the theorem of minimum complementary potential energy a variational principle analogous to that of the hellinger reissner principle in classical theory two theorems analogous to those of castigliano and engesser in classical theory a uniqueness theorem of the kirchhoff neumann type and a reciprocal theorem these results can be of importance to computational methods for analyzing practical problems in addition the j integral of fracture mechanics is derived within the same framework the new form of the j integral is identified with the energy release rate at the tip of a growing crack and its path independence is proved the theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua mindlin r d 1964 microstructure in linear elasticity arch rational mech anal 16 51 78 and is appropriate to model materials with periodic structure according to this theory the strain energy density assumes the form of a positive definite function of the strain as in classical elasticity and the second gradient of the displacement additional term specific cases of the general theory considered here are the well known theory of couple stress elasticity and the recently popularized theory of strain gradient elasticity the latter case is also treated in the present study 2005 elsevier ltd all rights reserved

closed form solution for a nonlocal elastic bar in tension a simple mechanical one dimensional problem in the context of nonlocal integral elasticity is solved analytically the nonlocal elastic material behaviour is described by the eringen model whose nonlocality features all reside in the constitutive relation this relation of integral type contains an attenuation function usually assumed symmetric aimed to capture the diffusion process of the nonlocality effects it also exhibits a convolution format the governing equation is a fredholm integral equation of second kind whose analytical treatment even for the usual choice of a symmetric kernel is not easy to develop in the present paper assuming a specific shape for the attenuation function a closed form solution in terms of strains is alternatively obtained by solving a volterra integral equation of second kind the latter can be easily solved with standard techniques at least for the adopted kernel taking also advantage from the symmetry of the solution such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity 2002 elsevier science ltd all rights reserved

theoretical analysis of a class of mixed c0 continuity formulations for general dipolar gradient elasticity boundary value problems abstract mixed weak formulations with two or three main tensor variables are stated and theoretically analyzed for general multi dimensional dipolar gradient elasticity biharmonic boundary value problems the general structure of constitutive equations is considered with and without coupling terms the mixed formulations are based on various generalizations of the so called ciarlet raviart technique hence c 0 continuity conforming basis functions may be employed in the finite element approximations or even c 1 basis functions for the cauchy stress variable all the complicated boundary conditions especially in the multi dimensional scenario are naturally considered the main variables are the displacement vector the double stress tensor and the cauchy stress tensor the latter variable may be eliminated in some of the formulations depending on the structure of the constitutive equations the standard continuous and discrete babuska brezzi inf sup conditions for the constraint equation as well as solution uniqueness for both the continuous statements and discrete approximations are established in all cases for the purpose of completeness two one dimensional mixed formulations are also analyzed the respective constitutive equations possess general structure with coupling terms for the 1 d formulations all the inf sup conditions are satisfied for both the continuous and discrete statements assuming proper selection of the polynomial spaces for the main variables hence the general babuska brezzi theory results in quasi optimality and stability for multi dimensional problems the difficulty of deducing the inf sup condition on the kernel is examined certain aspects of methodologies employed to theoretically by pass this problem are also discussed

a refined nonconforming quadrilateral element for couple stress strain gradient elasticity c0 1 patch test int j numer meth engng 2004 61 433 454 proposed by soh and chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress strain gradient elasticity the c0 1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations to pass the c0 1 patch test the element displacement functions used to calculate strains must satisfy c0 continuity or weak c0 continuity and quadratic completeness in this paper a 24 dof degrees of freedom quadrilateral element cq12 rdkq for the couple stress strain gradient elasticity is developed by combining the refined thin plate element rdkq and the nonconforming element cq12 the element rdkq which satisfies weak c1 continuity is used to calculate strain gradients whereas strains are computed by the element cq12 which is established based on an extended variational functional and satisfies weak c0 continuity and quadratic completeness numerical examples show that the element cq12 rdkq passes the c0 1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects copyright 2010 john wiley sons ltd

vibration analysis of rectangular mindlin plates resting on elastic edge supports for the first time to the authors knowledge the problem of free vibration of a moderately thick rectangular plate with edges elastically restrained against transverse and rotational displacements is considered the ritz method combined with a variational formulation and mindlin plate theory is used the admissible functions consist of polynomials and basic functions that impose the required boundary conditions on the mindlin plate the applicability of the formulation is illustrated using three examples of plates with different combinations of elastically restrained edges and classical boundary conditions numerical results are obtained to investigate the effects of elastic spring stiffness relative thickness and aspect ratio upon the natural frequencies of flexural vibration of rectangular mindlin plates

non local elastic plate theories a non local plate model is proposed based on eringen s theory of non local continuum mechanics the basic equations for the non local kirchhoff and the mindlin plate theories are derived these non local plate theories allow for the small scale effect which becomes significant when dealing with micro nanoscale plate like structures as illustrative examples the bending and free vibration problems of a rectangular plate with simply supported edges are solved and the exact non local solutions are discussed in relation to their corresponding local solutions

nonlocal elasticity defined by eringen s integral model introduction of a boundary layer method abstract in this paper we consider a nonlocal elasticity theory defined by eringen s integral model and introduce for the first time a boundary layer method by presenting the exponential basis functions ebfs for such a class of problems the ebfs playing the role of the fundamental solutions are found so that they satisfy the governing equations on an unbounded domain some insight to the theory is given by showing that the ebfs satisfying the navier equations in the classical elasticity theory also satisfy the governing equations in the nonlocal theory some additional ebfs are particularly obtained for the nonlocal theory in order to use the ebfs on bounded domains the effects of the boundary conditions are taken into account by truncating the kernel attenuation function in the constitutive equations this leads to some residuals in the governing equations which appear near the boundaries a weighted residual approach is employed to minimize the residuals near the boundaries the method presented in this paper has much in common with trefftz methods especially when the influence area of the kernel function is much smaller than the main computational domain several one two dimensional problems are solved to demonstrate the way in which the ebfs can be used through the proposed boundary layer method

a non local asymptotic theory for thin elastic plates the three dimensional dynamic non local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size the integral constitutive relations incorporating the variation of an exponential non local kernel across the thickness are adopted long wave low frequency approximations are derived for both bending and extensional motions boundary layers specific for non local behaviour are revealed near the plate faces it is established that the effect of the boundary layers leads to the first order corrections to the bending and extensional stiffness in the classical two dimensional plate equations

green s functions in non local three dimensional linear elasticity in this paper we compare small deformations in an infinite linear elastic body due to the presence of point loads within the classical local formulation to the corresponding deformations in the peridynamic non local formulation owing to the linearity of the problem the response to a point load can be used to obtain the response to general body force loading functions by superposition using laplace and fourier transforms we thus obtain an integral representation for the three dimensional peridynamic solution with the help of green s functions we illustrate this new theoretical result by dynamic and static examples in one and three dimensions in addition to this main result we also derive the non local three dimensional jump conditions as well as the weak formulation of peridynamics together with the associated finite element discretization

an advanced boundary element method for solving 2d and 3d static problems in mindlin s strain gradient theory of elasticity an advanced boundary element method bem for solving two 2d and three dimensional 3d problems in materials with microstructural effects is presented the analysis is performed in the context of mindlin s form ii gradient elastic theory the fundamental solution of the equilibrium partial differential equation is explicitly derived the integral representation of the problem consisting of two boundary integral equations one for displacements and the other for its normal derivative is developed the global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2d and 3d problems respectively representative 2d and 3d numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2d problems copyright 2010 john wiley sons ltd

continuous and discontinuous finite element methods for a peridynamics model of mechanics in contrast to classical partial differential equation models the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro differential equation that does not involve spatial derivatives of the displacement field as a result the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems the peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions based on a variational formulation continuous and discontinuous galerkin finite element methods are developed for the peridynamic model discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges through a series of simple one dimensional computational experiments we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates one issue addressed is the effect of the relative sizes of the horizon and the grid for problems with smooth solutions we find that continuous and discontinuous piecewise linear approximations result in the same accuracy as that obtained by continuous piecewise linear approximations for classical models piecewise constant approximations are less robust and require the grid size to be small with respect to the horizon we then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions in case a grid point is placed at the locations of the jump discontinuities such results are directly obtained in the general case we show that such results can be obtained through a simple automated abrupt local refinement of elements containing the discontinuity in order to reduce the number of degrees of freedom while preserving accuracy we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities

boundary element analysis of mode i and mixed mode i and ii crack problems of 2 d gradient elasticity abstract a boundary element method is developed for fracture analysis of gradient elastic planar solids under static loading a simple version of mindlin s general theory of gradient elastic materials is employed and the two required boundary integral equations one for the displacement vector and the other for its normal derivative are presented use is made of the fundamental solution of the problem and this leads to a formulation that requires only a boundary discretization two representative numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the first deals with a mode i crack while the second with a mixed mode i ii crack for the second case the proposed method is used in conjunction with the method of subregions the method is employed with regular and special near the crack tip boundary elements the gradient effect consists of modifying both the displacement and the stress field around the crack tip and resulting in a response which is more physically acceptable than the one coming from the classical theory of elasticity

variational formulation and isogeometric analysis for fourth order boundary value problems of gradient elastic bar and plane strain stress problems abstract the fourth order boundary value problems of one parameter gradient elastic bar and plane strain stress models are formulated in a variational form within an h 2 sobolev space setting for both problems the existence and uniqueness of the solution is established by proving the continuity and coercivity of the associated symmetric bilinear form for completeness the full sets of boundary conditions of the problems are derived and in particular the new types of boundary conditions featured by the gradient elastic models are given the additional attributes singly and doubly by utilizing the continuity and coercivity of the continuous problems corresponding error estimates are formulated for conforming galerkin formulations finally numerical results with isogeometric c p 1 continuous discretizations for nurbs basis functions of order p 2 confirm the theoretical results and illustrate the essentials of both static and vibration problems

nonoscillatory interpolation methods applied to vlasov based models we demonstrate the ability of nonoscillatory interpolation strategies for solving efficiently the transport phase in kinetic systems with applications to charged particle transport in plasmas and semiconductors pointwise weighted essentially nonoscillatory pweno interpolation is applied to obtain semi lagrangian and flux balance methods that together with splitting techniques form the building blocks of our numerical approach these methods do not present the restrictive cfl condition typical of finite difference methods with explicit time solvers and moreover they provide reliable results controlling parasite oscillations from classical polynomial interpolation while giving highly accurate approximations of smooth parts of the solutions we perform and compare these methods in different benchmark problems for vlasov or collisional models for charged particle transport

efficient deterministic parallel simulation of 2d semiconductor devices based on weno boltzmann schemes a flexible parallel deterministic solver of the boltzmann poisson system for 2d semiconductor device simulation on computer clusters is presented the simulator is obtained by parallelizing a previously proposed numerical scheme based on high order finite difference weighted essentially non oscillatory weno schemes although the underlying numerical scheme presents important advantages over direct simulation monte carlo methods this scheme imposes very high demands of computing power due to this the parallelization of the different calculation phases in the numerical scheme has been tackled the data subdomain which demands most

constitutive equations for micropolar hyper elastic materials in this paper the concept of hyper elasticity in the micropolar continuum theory is investigated the restrictions on the fourth order elasticity tensors are investigated using the representation theorems a general form of constitutive equations for micropolar hyper elastic isotropic materials is presented as some special cases generalizations of the neo hookean and mooney rivlin type materials to the micropolar continuum theory are presented the generalized constitutive equations reduce to those of the microplar linear elasticity theory when the deformations are infinitesimal also updated lagrangian finite element formulations for the micropolar hyper elastic materials are presented considering two planar examples it is shown that an increase in the micropolar parameter results in the reduction of the deformation of the bodies also it is shown that for a specimen with very small dimensions e g in the micron level the micropolar effects are more sensible furthermore it is shown that the influence of the micropolar parameters is dependent not only on the size of the body but also to its geometry and loading conditions for the problems in which the deformation is very close to a homogeneous state the micropolar effects are negligible