stabilized finite element formulation for elastic plastic finite deformations abstract this paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic plastic solids such as metals an updated lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation this formulation circumvents the restriction on the displacement and pressure fields due to the babuska brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit typical in metal forming applications moreover it facilitates the use of low order simplex elements i e p1 p1 reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary linearization of the weak form is derived for implementation into a finite element code numerical experiments with p1 p1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis

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

a novel approach to the solution of the tensor equation ax xa h a systematic approach to the solution of the tensor equation ax xa h where a is symmetric is presented it is based upon the reformulation of the original equation in the form ax h where a a 1 1 a is the fourth order tensor obtained from the square tensor product of the second order tensors a and 1 it is shown that the solution x which is known to be an isotropic function of a and h can be effectively obtained either by providing explicit formulas for a 1 or by reconverting to the format ax h the well known representation formulas for tensor valued isotropic functions the final form of the solution can thus be established a priori by suitably choosing a set of independent generators for a 1 the coefficients of the expansion of a 1 with respect to the assigned generators are then obtained by means of basic composition rules for square tensor products in this way it is possible to provide new expressions of the solution as well as to derive the existing ones in a simpler way both three dimensional and two dimensional cases are addressed in detail

bounds on list decoding of rank metric codes so far there is no polynomial time list decoding algorithm beyond half the minimum distance for gabidulin codes these codes can be seen as the rank metric equivalent of reed solomon codes in this paper we provide bounds on the list size of rank metric codes in order to understand whether polynomial time list decoding is possible or whether it works only with exponential time complexity three bounds on the list size are proven the first one is a lower exponential bound for gabidulin codes and shows that for these codes no polynomial time list decoding beyond the johnson radius exists second an exponential upper bound is derived which holds for any rank metric code of length n and minimum rank distance d the third bound proves that there exists a rank metric code over bbfqm of length n m such that the list size is exponential in the length for any radius greater than half the minimum rank distance this implies that there cannot exist a polynomial upper bound depending only on n and d similar to the johnson bound in hamming metric all three rank metric bounds reveal significant differences to bounds for codes in hamming metric

skew feedback shift register synthesis and decoding interleaved gabidulin codes an efficient algorithm which synthesizes all shortest skew feedback shift registers defined in the paper generating l sequences of varying length over a field is derived and its correctness is proved it generalizes the berlekamp massey algorithm and some other algorithms and has time complexity o ln2 where n is the length of a longest sequence the proposed algorithm can be applied for efficiently solving the key equation when decoding interleaved or direct sum of gabidulin codes beyond half minimum distance those codes have many applications and as shown by kotter and kschischang can be used for random network coding

maximum rank array codes and their application to crisscross error correction a mu n n k array code c over a field f is a k dimensional linear space of n n matrices over f such that every nonzero matrix in c has rank or mu it is first shown that the dimension of such array codes must satisfy the singleton like bound k

conjugate stresses of the seth hill strain tensors abstract in this paper conjugate stresses of the seth hill strain measures are treated by means of the definition of energy conjugacy and hill s principal axis method this approach results in relations between components of two different conjugate stress tensors the results are valid for distinct as well as coalescent principal stretches illustrative examples are solved to demonstrate the simplicity and usefulness of the derived relations between conjugate stress tensors

numerical study of consistency of rate constitutive equations with elasticity at finite deformation the present work is concerned with the numerical study of the elasticity consistency of the spatial rate equations using the conventional oldroyd truesdell cotter rivlin jaumann and green naghdi rates and the three novel co rotational e and l based logarithmic rates and of the rotated material rate equation describing the relationship between the material time derivatives of the rotated kirchhoff stress and material logarithmic strain to this end three integration procedures for updating stress are presented the stress responses of several typical deformation processes are simulated according to the numerical results we know that among the spatial rate equations only the logarithmic rate equation is consistent with elasticity under constant material parameters integrating the other spatial rate equations will provide path dependent stress response these numerical conclusions support the arguments in h xiao et al acta mechanica 1999 138 31 50 the reasons leading to elasticity inconsistency of spatial rate equations are analysed if the material parameters are assumed to be strain dependent the logarithmic rate equation loses also its elasticity consistent property the numerical results prove also that the spatial logarithmic and rotated material rate equations are equivalent to each other copyright 2002 john wiley sons ltd

some characterizations of operators satisfying a browder s theorem abstract we characterize the bounded linear operators t defined on banach spaces satisfying a browder s theorem or a weyl s theorem by means of the discontinuity of some maps defined on certain subsets of c several other characterizations are given in terms of localized svep as well as by means of the quasi nilpotent part the hyper kernel or the analytic core of i t

operators which do not have the single valued extension property abstract in this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator t l x on a banach space x and some quantities associated with t which play an important role in fredholm theory in particular we shall consider some conditions for which t does not have the single valued extension property at a point o c

on the dual variable of the logarithmic strain tensor the dual variable of the cauchy stress tensor and related issues the paper discusses relations related to the dual variables of the cauchy stress tensor and the logarithmic strain tensor we give a new proof that the logarithmic strain is dual to the cauchy stress tensor in the isotropic hyperelastic case corresponding relations are given in the case of multiplicative finite strain elasto viscoplasticity for the material eshelby like stress tensor and the rotated material stress tensor as well the general case of anisotropic response is discussed the dual variable of the logarithmic strain tensor is derived and some new expressions are given specifically it is proven that the skew symmetric part of the eshelby tensor is determined by the commutator of the logarithmic strain tensor and its conjugate variable the question of the existence of dual variables for the cauchy stress tensor and its material counterparts is rigorously answered where it is shown that in the general case a dual variable does not exist

a generalization of the berlekamp massey algorithm for multisequence shift register synthesis with applications to decoding cyclic codes a generalization of the berlekamp massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences a more general problem is first considered that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field which includes the multisequence problem as a special case a simple iterative algorithm the fundamental iterative algorithm fia is presented for solving this problem the generalized algorithm is then derived through a refinement of the fia application of this generalized algorithm to decoding cyclic codes up to the hartmann tzeng ht bound and roos bound making use of multiple syndrome sequences is considered conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error locator polynomial are determined with respect to decoding up to the ht bound and special cases of the roos bound

single valued extension property at the points of the approximate point spectrum abstract a localized version of the single valued extension property is studied at the points which are not limit points of the approximate point spectrum as well as of the surjectivity spectrum in particular we shall characterize the single valued extension property at a point o c in the case that oi t is of kato type from this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum

the computation of the exponential and logarithmic mappings and their first and second linearizations we describe two simple methods for the evaluation of the exponential and logarithmic mappings and their first and second linearizations based on the taylor expansion and the spectral representation we also provide guidelines for switching between those representations on the basis of the size of the argument the first and second linearizations of the exponential and logarithmic mappings provided here are based directly on the exponential formula for the solutions of systems of linear ordinary differential equations this representation does not require the use of perturbation formulae for eigenvalues and eigenvectors our approach leads to workable and straightforward expressions for the first and second linearizations of the exponential and logarithmic mappings regardless of degeneracies in the spectral decomposition of the argument

hencky s logarithmic strain and dual stress strain and strain stress relations in isotropic finite hyperelasticity abstract it has been known that the kirchhoff stress tensor and hencky s logarithmic strain tensor h may be useful in formulations of isotropic finite elasticity and elastoplasticity in this work a straightforward proof is presented to demonstrate that for an isotropic hyperelastic solid the just mentioned stress strain pair and h are derivable from two dual scalar potentials with respect to each other these results establish a simple explicit dual formulation of isotropic finite hyperelasticity as a result they supply a complete solution to the problem of finding out the inverted stress strain relation for isotropic hyperelastic solids raised by j a blume int j non linear mech 27 1992 413 moreover an explicit form of such an inverted hyperelastic stress strain relation is derived in terms of the powers i and 2

the stress conjugate to logarithmic strain abstract a stress s is said to be conjugate to a strain measure e if the inner product s e is the power per unit volume the logarithmic strain in u with u the right stretch tensor has been considered an interesting strain measure because of the relationship of its material time derivative ln u with the stretching tensor d in a previous article int j solids structures 22 1019 1032 1986 a formula for ln u was obtained in direct notation for the cases where the principal stretches are repeated as well as for the case where they are all distinct here the formula for ln u and the definition of conjugate stress are used to derive an explicit properly invariant expression for the stress conjugate to the logarithmic strain

conjugate stress and tensor equation r 1mum 1xur 1 c abstract in this paper we seek an explicit tensorial expression which for each integer m 0 gives the stress t m conjugate to the strain e m in the seth hill class the preceding problem leads us to find an intrinsic i e coordinate free representation of the solution of the tensor equation displayed in the title we obtain the requisite representation by using hill s principal axis method and the representation theorem of isotropic tensor functions we illustrate the general procedure to obtain an explicit lensorial expression for t m by working out the instance m 3 in detail

conjugate stresses to two point deformation tensors in this paper some expressions for stresses conjugate to two point deformation tensors are derived these expressions are offered in both the component and basis free forms although the material time rate of a two point deformation tensor is not an objective quantity the stress tensor conjugate to it may be determined the component form expressions are obtained by using the notion of conjugacy together with the objectivity of the stress power the component form expressions are then extended to a unified basis free form using a theorem established for this purpose the specific results are provided for all different cases of distinct and coalescent principal stretches in a three dimensional inner product space

a family of metric strains and conjugate stresses prolonging usual material laws from small to large transformations abstract a new family of simple generalized strains and conjugate stresses based on the material metric right cauchy green tensor is proposed it includes an interesting quasilinear pair it is a close approximation of the seth hill family with the advantage of being easier to calculate it extends the realm of application of the classical theories of linear elasticity and perfect plasticity from small to large transformations for isotropic and anisotropic materials without any modification

on the theory of fourth order tensors and their applications in computational mechanics many problems concerned with the mathematical treatment of fourth order tensors still remain open in the literature in the present paper they will be considered in the framework of a complete theory involving a set of notations and definitions a tensor operation algebra differentiation rules eigenvalue problems applications of fourth order tensors to isotropic tensor functions and some other relevant aspects a tensor is understood to be an invariant quantity with respect to any co ordinate system transformation which justifies the use of absolute notation preferred in this work as a most important application field of fourth order tensors elastic moduli are formulated in a material and a spatial description and given for various hyperelastic material models

limitations to the large strain theory in the formulation of a large strain problem the euler cauchy stokes decomposition and the eulerian objective stress and strain rates used for the formulation of the problem are actually applicable only for some special cases and are not suitable for general large strain and rotation rates in this paper the limitations and applicabilities of the conventional large strain formulation under various deformation cases will be studied

some basis free expressions for stresses conjugate to hill s strains through solving the tensor equation ax xa c abstract in the present paper some new and compact basis free expressions are derived for representing stresses conjugate to the lagrangean hill s strain measures solving a tensor equation in the form of ax xa c a rather simple general expression is obtained for the conjugate stresses in the case of distinct principal stretches the result decomposes the conjugate stresses additively into two parts such that one part is coaxial with the right stretch tensor while another part is orthogonal to it an expression for the case of double coalescent principal stretches is also obtained as an example of the general results simple expressions are obtained for the stress conjugate to the logarithmic strain

the material time derivative of logarithmic strain abstract in this article we derive explicit formulas for the time rate of change of the logarithmic strains ln u and ln v where u and v are the right and left stretch tensors respectively results are obtained in both two and three dimensions for the cases where the principal stretches are repeated as well as for the case where they are distinct the formulas are displayed in terms of v the stretching tensor d the rotation r and the principal stretches finally we consider the corotational and jaumann derivatives of ln v and establish conditions under which these logarithmic strain rates are equal to the stretching tensor

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

a new two point deformation tensor and its relation to the classical kinematical framework and the stress concept abstract starting from the issue of what is the correct form for a legendre transformation of the strain energy in terms of eulerian and two point tensor variables we introduce a new two point deformation tensor namely h f f t 2 as a possible deformation measure involving points in two distinct configurations the lie derivative of h is work conjugate to the first piola kirchhoff stress tensor p the deformation measure h leads to straightforward manipulations within a two point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation the manipulations are analogous to those used for the lagrangian and eulerian frameworks it is also shown that the legendre transformation in terms of two point tensors and spatial tensors require lie derivatives as an illustrative example we propose a simple saint venant kirchhoff type of a strain energy function in terms of h the constitutive model leads to physically meaningful results also for the large compressive strain domain which is not the case for the classical saint venant kirchhoff material

a basis free formula for time rate of hill s strain tensors abstract in this paper we use the method to obtain a basis free formula for the time rate e of hill s strain tensor e u i i n ni here u is the right stretch tensor i are its eigenvalues and ni is a lagrangian triad of orthonormal eigenvectors subordinate to i is a smooth strictly increasing scalar function that satisfies 1 0 1 1 our formula is generally valid provided that u is a c1 function of time and is of class c3 until now all other basis free formulae for e in the literature have been burdened by one or other of the following deficiencies i valid only under various special circumstances ii a complete proof is wanting the formula derived herein is free of these difficulties

property w and perturbations ii abstract a bounded linear operator t l x defined on a banach space x satisfies property w a variant of weyl s theorem if the complement in the approximate point spectrum a t of the weyl essential approximate spectrum wa t coincides with the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity in this note we study the stability of property w for a bounded operator t acting on a banach space under perturbations by finite rank operators by nilpotent operator and quasi nilpotent operators commuting with t

on the relationship between the logarithmic strain rate and the stretching tensor abstract in this paper we investigate the relationship between the stretching tensor d and the logarithmic hencky strain in v with v the left stretch tensor we establish the simple formula 1 which holds for arbitrary three dimensional motions here f is the deformation gradient is the time derivative of in v measured in a coordinate system which rotates with the left principal strain axes and is the spin of the right principal strain axes we use this formula to show that or equrvatently the jaumann derivative of ln v if and only if the characteristic spaces of the right stretch tensor u are constant on any time interval in which the number of distinct principal stretches is constant finally we show that the asymptotic approximation 3 holds whenever the displacement gradient h satisfies

on consistent stress rates in solid mechanics computational implications summary in the context of general isothermal elastic processes issues related to work conjugacy and its effect on the choice of a stress rate are discussed for processes involving large strains it is shown that a unique stress rate consistent with hyperelasticity can be defined for every conjugate pair chosen through the rate form of the associated hyperelastic constitutive equation that results in non constant and generally anisotropic spatial elasticity tensors even for a linear isotropic material the anisotropic spatial elasticity tensor connected with the green naghdi rate of kirchhoff stress is evaluated which provides a basis for the spatial formulation of processes governed by the hencky s strain energy function 8949 notions of equivalent and irreducible form of stress rate are introduced using a workless component of the deformation rate the hyperelastic constitutive models for the conjugate pairs s e and t in u which are of significance for computation are stated in their explicit form exact rate forms are derived and the order of approximation of currently employed computational algorithms discussed finally computational implications of the work presented are reviewed

a rank metric approach to error control in random network coding the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of rotter and kschischang a large class of constant dimension subspace codes is investigated it is shown that codes in this class can be easily constructed from rank metric codes while preserving their distance properties moreover it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank metric codes where partial information about the error is available this partial information may be in the form of erasures knowledge of an error location but not its value and deviations knowledge of an error value but not its location taking erasures and deviations into account when they occur strictly increases the error correction capability of a code if mu erasures and delta deviations occur then errors of rank t can always be corrected provided that 2t les d 1 mu delta where d is the minimum rank distance of the code for gabidulin codes an important family of maximum rank distance codes an efficient decoding algorithm is proposed that can properly exploit erasures and deviations in a network coding application where n packets of length m over f q are transmitted the complexity of the decoding algorithm is given by o dm operations in an extension field f qn

unified explicit basis free expressions for time rate and conjugate stress of an arbitrary hill s strain abstract this paper provides simple and unified explicit basis free expressions for time rate and conjugate stress of an arbitrary hill s strain for the first time which are valid for all cases of the eigenvalues of the right stretch tensors

basis free relations for the conjugate stresses of the strains based on the right stretch tensor abstract in this paper general relations between two different stress tensors t f and t g respectively conjugate to strain measure tensors f u and g u are found the strain class f u is based on the right stretch tensor u which includes the seth hill strain tensors the method is based on the definition of energy conjugacy and hill s principal axis method the relations are derived for the cases of distinct as well as coalescent principal stretches as a special case conjugate stresses of the seth hill strain measures are then more investigated in their general form the relations are first obtained in the principal axes of the tensor u then they are used to obtain basis free tensorial equations between different conjugate stresses these basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes with a possible tensor expansion relation between the stresses with unknown coefficients the unknown coefficients to be obtained in this regard some relations are also obtained for t 0 which is the stress conjugate to the logarithmic strain tensor ln u

explicit formulations of tangent stiffness tensors for isotropic materials a compact explicit expression for the tangent stiffness tensor is presented throughout the analysis the formulation holds for general isotropic elastic materials and does not require solving eigenvector problems on the theoretical side a very simple solution of a tensor equation is obtained then the expressions for the derivatives of general symmetric isotropic tensor functions of a symmetric tensor are developed on the computational side particular attention is given to the consideration of the special case green elastic materials in which the strain energy does not admit a closed form expression in terms of principal invariants finally a simple formulation of the tangent stiffness tensor for ogden material model is supplied copyright 2006 john wiley sons ltd

variations on weyl s theorem in this note we study the property w a variant of weyl s theorem introduced by rakocevic by means of the localized single valued extension property svep we establish for a bounded linear operator defined on a banach space several sufficient and necessary conditions for which property w holds we also relate this property with weyl s theorem and with another variant of it a weyl s theorem we show that weyl s theorem a weyl s theorem and property w for t respectively t coincide whenever t respectively t satisfies svep as a consequence of these results we obtain that several classes of commonly considered operators have property w

on the invariant representation of spin tensors with applications abstract the invariant representation of the spin tensor defined as the rotation rate or a principal triad for a symmetric and non degenerate tensor is derived on the basis of the general solution of a linear tensorial equation the result can be naturally specified to study the spin of the stretch tensors and to investigate the relations between various rotation rate tensors encountered frequently in modern continuum mechanics a remarkable formula which relates the generalized stress conjugate to the generalized strain in hill s sense to cauchy stress is obtained in invariant form through the work conjugate principle particularly a detailed discussion on the time rate of logarithmic strain and its conjugate stress is made as the principal axes ofstrain are not fixed during deformation

elastic eigenstates in finite element modelling of large anisotropic elasticity abstract fe algorithm for large anisotropic hyper elastic deformations is presented this algorithm is based on the concept of the local reference configuration the spectral decomposition of the elastic modulus tensor into eigentensors of elastic deformations has been utilized in the constitutive equations some numerical results are also presented

the logarithmic strain space description abstract the ingredients of a plasticity description in the logarithmic strain space with respect to the reference configuration are discussed within the framework of the finite deformation theory especially the features of logarithmic strain and the transformations between its time derivative the rate of deformation tensor and the work conjugate stresses the logarithmic strain space description is then illustrated using three analytical examples of homogeneous finite deformation fields

basis free expressions for the stress rate of isotropic elastic materials in the cases of coalescent principal stretches in this paper some basis free expressions for the material time derivative of lagrangian stress tensors are presented which are generally valid in all cases of coalescent principal stretches the material is assumed to be elastic and isotropic

basis free representations for the stress rate of isotopic materials in terms of the principal components of stress tensor and a scalar response function two kinds of basis free representation for the stress rate are obtained which only include the time rate of the strain tensor functions and invariants of the strain tensor the expressions for the stress rate valid in the cases that the eigenvalues of the strain are distinct doubly coalescent and triply coalescent respectively

on the derivatives of a subclass of isotropic tensor functions of a nonsymmetric tensor this paper studies a subclass of isotropic tensor valued functions of a nonsymmetric tensor which satisfy the commutative condition and their derivatives this subclass of tensor functions includes tensor power series exponential tensor function etc and is more general than those investigated before in the case of three distinct eigenvalues the derivatives of these tensor functions are constructed by solving a tensor equation which is acquired by differentiating the commutative condition by taking limits the results are extended to the cases of repeated eigenvalues

linearized shift register synthesis an efficient algorithm synthesizing all shortest q linearized feedback shift registers generating a given sequence of length n over a finite field bbfqm is derived and its correctness is proved this algorithm which is a generalization of the berlekamp massey algorithm has time complexity o ln o n2 operations in bbfqm where l is the linearized complexity of the sequence the algorithm can be applied for efficiently solving the key equation when decoding gabidulin codes