some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

on rank one convex functions that are homogeneous of degree one we show that positively 1 homogeneous rank one convex functions are convex at 0 and at matrices of rank one the result is a special case of an abstract convexity result that we establish for positively 1 homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space from these results we derive a number of consequences including various generalizations of the ornstein l1 non inequalities most of the results were announced in c r acad sci paris ser i 349 407 409 2011

invariant properties of the stress in plane elasticity and equivalence classes of composites attention is drawn to the invariance of the stress field in a two dimensional body loaded at the boundary by fixed forces when the compliance tensor scr g chi is shifted uniformly by ell text i lambda lambda where lambda is an arbitrary constant and scr g text i kappa mu is the compliance tensor of a isotropic material with two dimensional bulk and shear moduli kappa and mu this invariance is explained from two simple observations first that in two dimensions the tensor scr g text i frac 1 2 frac 1 2 acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain for composite materials the invariance of the stress field implies that the effective compliance tensor ell ast also gets shifted by scr g text i lambda lambda when the constituent moduli are each shifted by ell text i lambda lambda this imposes constraints on the functional dependence of ell ast on the material moduli of the components applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1 lambda and the inverse shear modulus is shifted by 1 lambda then the inverse effective bulk and shear moduli undergo precisely the same shifts in particular it explains why the effective young s modulus of a two dimensional media with holes does not depend on the poisson s ratio of the matrix material

the simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation we study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type mathcal p 432 to mathcal p 222 the free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank one connected we consider the energy minimization problem with dirichlet boundary data compatible with an arbitrary but fixed simple laminate we first show that for all but a few isolated values of transformation strains this problem has a unique young measure solution solely characterized by the boundary data that represents the simply laminated microstructure we then present a theory of stability for such a microstructure and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers

optimal bounds on the effective bulk modulus of polycrystals the classical problem of bounding the effective bulk modulus of isotropic polycrystalline aggregates built from a single anisotropic phase is studied for three dimensional composites we find that the voigt and reuss bounds are each attained by multi layered laminate geometries and hence are optimal for two dimensional composites the reuss lower bound is proved to be optimal while a new optimal upper bound is derived

second order analysis for thin structures the eective behavior of second order strain energy densities is obtained using relaxation and i convergence techniques the cosserat theory see e g 13 14 and 2 is recovered within a dimension reduction analysis for 3d non linear elastic thin domains with varying profiles homogeneous and inhomogeneous plate models with periodic profiles are treated

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

lower semicontinuity of multiple integrals and the biting lemma weak lower semicontinuity theorems in the sense of chacon s biting lemma are proved for multiple integrals of the calculus of variations a general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions the biting weak limit of the corresponding integrands is characterised via the young measure and related to the weak limit in the sense of measures finally an example is given which shows that the young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case

association schemes in which the thin residue is an elementary abelian p group of rank 2 abstract in this article we investigate the existence and schurity problem of association schemes whose thin residues are isomorphic to an elementary abelian p group of rank 2

global equivalence for deformable theormoeleastic bodies we introduce the notion of global physical equivalence between two deformable thermoelastic bodies and this equivalence is understood in the sense that a global thermokinetic process for is admissible in the sense of coleman noll if and only if its counterpart for also is admissible according to a suitable bijective correspondence hat k between and here we prove a general theorem which establishes some conditions on the response functions of two deformable thermoelastic bodies that are necessary and sufficient for these bodies to be globally equivalent in the absence of heat sources

energy minimizers in nonlinear elastostatics and the implicit function theorem the purpose of this paper is to give conditions under which the equilibria obtained by two different approaches are the same these two approaches are the calculus of variation and the theory of polyconvex functions and one using the implicit function theorem in infinite dimensional space

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

a variational approximation scheme for three dimensional elastodynamics with polyconvex energy we construct a variational approximation scheme for the equations of three dimensional elastodynamics with polyconvex stored energy the scheme is motivated by some recently discovered geometric identities qin 18 for the null lagrangians the determinant and cofactor matrix and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy the scheme decreases the energy and its solvability is reduced to the solution of a constrained convex minimization problem we prove that the approximating process admits regular weak solutions which in the limit produce a measure valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities this latter property is related to the weak continuity properties of minors of jacobian matrices here exploited in a time dependent setting

on p schemes of order p 3 abstract let x s be a p scheme of order p 3 and t the thin residue of s assuming that t has valency p 2 one of the following holds i t p 2 and t c p 2 ii t p 2 and t c p c p iii t p 2 it is known that x s is schurian if i holds in case ii we define a partial linear space on x t and prove that the sizes of its lines coincide with the character degrees of x s in case iii we give a construction of non schurian p schemes which are algebraically isomorphic to a schurian p scheme of order p 3

on the justification of the nonlinear inextensional plate model we consider a cylindrical three dimensional body made of a saint venant kirchhoff material and we let its thickness go to zero for a specific order of magnitude for the applied loads and under appropriate restrictions on the set of admissible deformations we show that the almost minimizers of the total energy converge toward deformations that minimize the nonlinear bending energy obtained by fox raoult and simo using formal asymptotic expansions our result is obtained by convergence arguments

a note on meyers theorem in wk 1 lower semicontinuity properties of multiple integrals u w k 1 r d f x u x k u x dx are studied when f may grow linearly with respect to the highest order derivative k u and admissible w k 1 r d sequences converge strongly in w k 1 1 r d it is shown that under certain continuity assumptions on f convexity 1 quasiconvexity or k polyconvexity of f x 0 u x 0 k 1 u x 0 ensures lower semicontinuity the case where f x 0 u x 0 k 1 u x 0 is k quasiconvex remains open except in some very particular cases such as when f x u x k u x h x g k u x

a global method for relaxation in w1 p and in sbvp an integral representation formula for a class of functionals defined on and in the space of special functions of bounded variation is obtained without requiring the regularity conditions usually imposed in the literature the approach is based on the general results of 10 and on a poincare wirtinger type inequality introduced by de giorgi carriero leaci 25 applications to relaxation problems and dimension reduction problems in brittle thin films are presented

on the optimality of certain sobolev exponents for the weak continuity of determinants keywords optimality of sobolev exponents weak continuity of determinant radon measure reference caa article 1992 005doi 10 1016 0022 1236 92 90071 pview record in web of science record created on 2008 11 25 modified on 2017 05 12

asymptotic analysis of linearly elastic shells iii justification of koiter s shell equations we consider as in part i a family of linearly elastic shells of thickness 2 all having the same middle surfaces r3 where r2 is a bounded and connected open set with a lipschitz continuous boundary and l3 r3 the shells are clamped on a portion of their lateral face whose middle line is 0 where 0 is any portion of withlength 0 0 we make an essential geometrical assumption on the middle surfaces and on the set 0 which states that the space of inextensional displacements begin gathered v f omega eta eta i in h 1 omega times h 1 omega times h 2 omega hfill eta i partial v eta 3 0 on gamma 0 gamma alpha beta eta 0 in omega hfill end gathered where gamma alpha beta are the components of the linearized change is metric tensor ofs contains non zero functions this assumption is satisfied in particular ifs is a portion of cylinder and 0 is contained in a generatrix ofs

a hopf algebraic approach to the picard vessiot theory abstract this paper shows what is really happening in picard vessiot theory our hopf algebraic approach simplifies and generalizes the theory among other things the generalization gives a picard vessiot theory for positive characteristic for fields with not necessarily commuting derivations and even for fields with a set of higher derivations

on structures of quasi thin table algebras and applications to association schemes abstract in this paper we develop a structure theory for a class of table algebras i e the so called quasi thin table algebras they are direct generalizations of quasi thin association schemes although this class of table algebras is rather close to finite groups its structural properties are quite different from those of finite groups the thin radical and thin residue give rise to two group structures for every table algebra they play the central role in our discussions applications to association schemes are also discussed and some new results about quasi thin association schemes are obtained in particular we will give necessary and sufficient conditions under which a commutative quasi thin association scheme has multiplicities 1 or 2 we will need these results in the study of the characterization and classification of double quasi thin association schemes in 22

optimal use of the translation method and relaxations of variational problems the translation method has been used with great success in bounding the effective moduli of composite materials we consider here the analogous method for bounding the relaxations of variational problems we optimize the bound over the set of all available translations our method is to cast this in the form of a minmax problem using techniques of nonsmooth analysis we are able to identify the optimal translation bound meanwhile proving the existence of at least one optimal combination rank one convex quadratic and null lagrangian translation the optimal translation bound proves to be a better general lower bound on relaxations of variational problems than is the polyconvexification in three dimensions in two dimensions we discuss the negative result that the optimal translation bound is exactly the polyconvexification several examples of optimal applications of translation bounds to non convex nonlinear variational problems are given

a hierarchy of plate models derived from nonlinear elasticity by gamma convergence we derive a hierarchy of plate models from three dimensional nonlinear elasticity by convergence what distinguishes the different limit models is the scaling of the elastic energy per unit volume h where h is the thickness of the plate this is in turn related to the strength of the applied force h membrane theory derived earlier by le dret and raoult corresponds to 0 nonlinear bending theory to 2 von karman theory to 3 4 and linearized vk theory to 3 intermediate values of lead to certain theories with constraints a key ingredient in the proof is a generalization to higher derivatives of our rigidity result 29 which states that for maps v 0 1 3 3 the l2 distance of v from a single rotation is bounded by a multiple of the l2 distance from the set so 3 of all rotations

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

a global method for relaxation a new method for the identification of the integral representation of a class of functionals defined on bv omega bbb r d times cal a omega where cal a omega represents the family of open subsets of is presented applications are derived such as the integral representation of the relaxed energy in bv omega bbb r d corresponding to a functional defined in w 1 1 omega bbb r d with a discontinuous integrand with linear growth relaxation and homogenization results in sbv omega bbb r d are recovered in the case where bulk and surface energies are present

schurity and separability of quasiregular coherent configurations abstract a permutation group is said to be quasiregular if each of its transitive constituents is regular and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group the transitive constituents are replaced by the homogeneous components in this paper we are interested in the question when the configuration is schurian i e formed by the orbitals of a permutation group or and separable i e uniquely determined by the intersection numbers in these terms an old result of hanna neumann is in a sense dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian in the present paper we a establish the duality in a precise form and b generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with the homogeneous components have distributive lattices of normal subgroups

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

determinantal ideals and the capelli identities abstract recent work in the formal calculus of variations and also in riemannian geometry and general relativity have relied upon certain important previously established properties of determinant ideals in characteristic zero this note presents direct elementary proofs of these properties by utilizing the capelli identities of classical invariant theory

a new route to finding bounds on the generalized spectrum of many physical operators here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depending linearly on a vector of parameters z z1 z2 zn we obtain constraints on z that ensure that green s function exists and hence which provide bounds on the generalized spectrum here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depend

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

a framework for nonlinear shells based on generalized stress and strain measures a very general class of shell theories is established based on a unified variational principle and on expansions with respect to biorthogonal function systems the shell models derived are able to accommodate the flow theory of large strain thermoelastoplasticity including the effects due to unloading the specific case of a compressible neo hookean material with a von mises yield condition is discussed in detail

higher order quasiconvexity reduces to quasiconvexity in this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions precisely it is proved that a smooth strictly 2 quasiconvex function with p growth at infinity p 1 is the restriction to symmetric matrices of a 1 quasiconvex function with the same growth as a consequence lower semicontinuity results for second order variational problems are deduced as corollaries of well known first order theorems

the intricate global structure of buckled states of compressible columns singularity theory and global multiparameter bifurcation theory are combined to show that nonbifurcating buckled states of compressible nonlinearly elastic columns lie on branches having detailed qualitative properties these properties distinguish the branches globally together with a delicate asymptotic analysis they are used to show that the global disposition of solution branches both bifurcating and not is strikingly different from that for incompressible columns

addendum to sharp inequalities that generalize the divergence theorem an extension of the notion of quasi convexity the paper sharp inequalities that generalize the divergence theorem an extension of the notion of quasi convexity published in proc r soc a 2013 469 20130075 doi 10 1098 rspa 2013 0075 is c

coherent configurations associated with ti subgroups let x be a coherent configuration associated with a transitive group g in terms of the intersection numbers of x a necessary condition for the point stabilizer of g to be a ti subgroup is established furthermore under this condition x is determined up to isomorphism by the intersection numbers it is also proved that asymptotically this condition is also sufficient more precisely an arbitrary homogeneous coherent configuration satisfying this condition is associated with a transitive group the point stabilizer of which is a ti subgroup as a byproduct of the developed theory recent results on pseudocyclic and quasi thin association schemes are generalized and improved in particular it is shown that any scheme of prime degree p and valency k is associated with a transitive group whenever p 1 6k k 1 2

sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations let omega subset of r n be a bounded domain and let f omega x r n x r nxn r consider the functional i u f omega f x u du dx over the class of sobolev functions w 1 w q omega r n 1 less than or equal to q less than or equal to infinity for which the integral on the right is well defined in this paper we establish sufficient conditions on a given function u 0 and f to ensure that u 0 provides an l r local minimizer for i where 1 less than or equal to r less than or equal to infinity the case r infinity is somewhat known and there is a considerable literature on the subject treating the case min n n 1 mostly based on the field theory of the calculus of variations the main contribution here is to present a set of sufficient conditions for the case 1 less than or equal to r less than or equal to infinity our proof is based on an indirect approach and is largely motivated by an argument of hestenes relying on the concept of directional convergence

terwilliger algebras of wreath products by quasi thin schemes abstract the structure of terwilliger algebras of wreath products by thin schemes or one class schemes was studied in a hanaki k kim y maekawa terwilliger algebras of direct and wreath products of association schemes j algebra 343 2011 195 200 in this paper we will consider the structure of terwilliger algebras of wreath products by quasi thin schemes this gives a generalization of their result

heterogeneous thin films of martensitic materials we study the effective behavior of heterogeneous thin films with three competing length scales the film thickness and the length scales of heterogeneity and material microstructure we start with three dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der waals type and derive the effective energy density as all length scales tend to zero with given limiting ratios we do not require any a priori selection of asymptotic expansion or ansatz in deriving our results depending on the dominating length scale the effective energy density can be identified by three procedures averaging homogenization and thin film limit we apply our theory to martensitic materials with multi well energy density and use a model example to show that the shape memory behavior can crucially depend on the ratios of these length scales we comment on the effective conductivity of linear composites and also on multilayers made of shape memory and elastic materials

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

justification of the nonlinear kirchhoff love theory of plates as the application of a new singular inverse method in the framework of isotropic homogeneous nonlinear elasticity for a st venant kirchhoff material we consider a three dimensional plate of thickness and periodic in the two other directions using a new method that we call the singular inverse method we prove the existence of a solution rescaled uniformly in for small forces and at the same time we prove the rigorous convergence of this rescaled solution to the solution of the nonlinear kirchhoff love plate model we also state a 3d 2d error estimate

an invariant infinitesimal theory of motions superposed on a given motion abstract the development is applicable to any material but special attention is given to elastic solids included as a special case is an infinitesimal theory of elasticity with the following properties 1 it is properly invariant under arbitrary not necessarily infinitesimal superposed rigid body motions 2 it reduces by specialization to the theory of rigid bodies undergoing finite motion and 3 it can be brought into correspondence with the classical linear elasticity through a suitable reinterpretation of the symbols in the constitutive equation of the latter author

geometrically nonlinear flexural vibrations of plates in plane boundary conditions and some symmetry properties abstract this study is devoted to the derivation of some properties of the von karman equations for geometrically nonlinear models of plates with a boundary of arbitrary shape for applications to nonlinear vibration and buckling an intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an airy stress function as unknowns is provided classical homogeneous boundary conditions with vanishing prescribed forces and displacements are derived in terms of the airy stress function in the case of a boundary of arbitrary geometry a special property of this operator crucial for some energy conserving numerical schemes and called triple self adjointness is derived in the case of an edge of arbitrary shape it is shown that this property takes a simple form for some classical boundary conditions so that the calculations in some practical cases are also simplified the applications of this work are either semi analytical methods of solution using an expansion of the solution onto an eigenmode basis of the associated linear problem or special energy conserving numerical methods

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

the role of non linear theories in transient dynamic analysis of flexible structures abstract transient dynamic analysis of flexible structures undergoing large motions is considered for rotating structures it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non linear at least second order beam theory use of a first order linearized linear beam theory results in a spurious loss of bending stiffness for a rotating plane beam a set of linear partial differential equations of motion that includes all inertia effects coriolis centrifugal acceleration of revolution and coupling between extensional and flexural deformations is derived from the fully non linear beam theory by consistent linearization the analysis is subsequently extended to the more general case of a plate accomodating shear deformation and undergoing a general three dimensional rotating motion the discretization process of the resulting linear equations of motion for the beam and the plate is also discussed

characterization of p valenced schemes of order p2q abstract in p h zieschang association schemes in which the thin residue is a finite cyclic group j algebra 324 2010 3572 3578 it was shown that association schemes are schurian if their thin residues are finite groups and have a distributive normal subgroup lattice however in the case that their thin residues are elementary abelian groups they do not have a distributive normal subgroup lattice actually they are not necessarily schurian due to the non schurian quasi thin scheme of order 28 in this paper we investigate association schemes such that their thin residues are elementary abelian groups of order p 2 it is shown that these schemes are schurian if the number of elements with valency p is 2p and the factor scheme over the thin residue is a cyclic group of prime order we also construct non schurian p schemes of order p 3

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results we use a variational convergence method to study the consistency of various cosserat hypotheses in shell theory with the limit nonlinear membrane model derived from three dimensional elasticity in the course of the analysis we introduce a generalization of quasiconvexity that is suitable for problems of the calculus of variations with two vectorial unknowns one of which appears through its gradient the other one through its value in a weak w 1 p l p framework

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

local minimizers and quasiconvexity the impact of topology the aim of this paper is to discuss the question of existence and multiplicity of strong local minimizers for a relatively large class of functionals open image in new window open image in new window from a purely topological point of view the basic assumptions on open image in new window are sequential lower semicontinuity with respect to w1 p weak convergence and w1 p weak coercivity and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds open image in new window and open image in new window

elastic surface substrate interactions a theory for three dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described the gurtin murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance this modification yields a model that can be used to describe equilibrium deformations in the presence of compressive surface stress fields an associated variational theory is given and material symmetry considerations are discussed the theory is illustrated by examples

on quasi thin association schemes abstract an association scheme is called quasi thin if each of its basic relations has valency 1 or 2 a quasi thin scheme is called kleinian if its thin residue is the klein four group with respect to relational multiplication it is proved that any kleinian quasi thin scheme arises from a near pencil on 3 points from an affine plane of order 2 or from a projective plane of order 2 the main result in this paper is that any non kleinian quasi thin scheme is schurian and separable we also construct an infinite family of kleinian quasi thin schemes which is neither schurian nor separable

orlicz sobolev nematic elastomers abstract we extend the existence theorems in barchiesi et al 2017 for models of nematic elastomers and magnetoelasticity to a larger class in the scale of orlicz spaces these models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration and a functional corresponding to the nematic energy or the exchange and magnetostatic energies in magnetoelasticity where the energy density is integrated over the deformed configuration in order to obtain the desired compactness and lower semicontinuity we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an orlicz class with an integrability just above the space dimension minus one we prove that the fine properties of orientation preserving maps satisfying that regularity requirement namely being weakly 1 pseudomonotone h 1 continuous a e differentiable and a e locally invertible are still valid in the orlicz sobolev setting

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

terwilliger algebras of wreath products of association schemes abstract the terwilliger algebra of an association scheme of order n introduced in 13 is a subalgebra of the matrix algebra of all n n matrices terwilliger algebras of wreath products of special association schemes are studied in several papers in this paper we study the terwilliger algebra of the wreath product t s of two arbitrary association schemes s and t we will express the terwilliger algebra of t s and its primitive central idempotents in terms of the terwilliger algebras of s and t and their primitive central idempotents the known results of hanaki kim etc cf 7 10 are special cases of our results

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

a justification of nonlinear properly invariant plate theories a single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame invariance properties of three dimensional finite elasticity by a successive scaling of the external loading on the three dimensional body the nonlinear membrane theory the nonlinear inextensional theory and the von karman equations are derived as the leading order terms in the asymptotic expansion of finite elasticity the governing equations of the nonlinear inextensional theory are of particular interest where 1 plane strain kinematics and plane stress constitutive equations are derived simultaneously from the asymptotic analysis 2 the theory can be phrased as a minimization problem over the space of isometric deformations of a surface and 3 the local equilibrium equations are identical to those arising in the one director cosserat shell model furthermore it can be concluded that with a regular single scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading order term of an asymptotic analysis

on quasi thin association schemes with odd number of points abstract let x r be an association scheme in the sense of p h zieschang 1996 an algebraic approach to association schemes lecture notes in mathematics vol 1628 springer new york berlin where x is a finite set and r is a partition of x x we say that x r is quasi thin if each element of r has a valency of at most two in this paper we focus on quasi thin association schemes with an odd number of points and obtain that x r has a regular automorphism group when n o r is square free

nonlinearly elastic shell models a formal asymptotic approach ii the flexural model this paper is the sequel of part i in which the limiting displacement field of a thin shell when its thickness approaches zero is identified as the solution of a two dimensional nonlinear membrane shell model when the geometry of the middle surface of the shell and the boundary conditions allow non zero inextensional displacements the previous membrane limit model is not relevant in this case we show how to update the assumptions on the applied forces acting on the shell so that a limiting model can be derived by an asymptotic analysis furthermore we identify this limit as the two dimensional nonlinear flexural shell model

les invariants du premier ordre de l equation d euler en dimension trois abstract we show in this article that any conservation law f u x t grad u x t dx constant of the three dimensional euler equations of an incompressible perfect fluid is a linear combination of momenta energy and helicity therefore there are no other invariants of the first order i e involving u and its first spatial derivatives than those which were already known

k quasi convexity reduces to quasi convexity the relation between quasi convexity and k quasiconvexity k greater than or equal to 2 is investigated it is shown that every smooth strictly k quasi convex integrand with p growth at infinity p 1 is the restriction to kth order symmetric tensors of a quasiconvex function with the same growth when the smoothness condition is dropped it is possible to prove an approximation result as a consequence lower semicontinuity results for kth order variational problems are deduced as corollaries of well known first order theorems this generalizes a previous work by dal maso et al in which the case where k 2 was treated

pseudo direct sums and wreath products of loose coherent algebras with applications to coherent configurations abstract we introduce the notion of a loose coherent algebra which is a special semisimple subalgebra of the matrix algebra and define two operations to obtain new loose coherent algebras from the old ones the pseudo direct sum and the wreath product for two arbitrary coherent configurations c d and their wreath product c d it is difficult to express the terwilliger algebra t x y c d in terms of the terwilliger algebras t x c and t y d by using the concept and operations of loose coherent algebras we find a very simple such expression as a direct consequence of this expression we obtain the central primitive idempotents of t x y c d in terms of the central primitive idempotents of t x c and t y d many results in 4 6 10 12 are special cases of the results in this paper

local invertibility in sobolev spaces with applications to nematic elastomers and magnetoelasticity we define a class of deformations in w 1 p omega mathbb r n p n 1 with a positive jacobian that do not exhibit cavitation we characterize that class in terms of the non negativity of the topological degree and the equality det det that the distributional determinant coincides with the pointwise determinant of the gradient maps in this class are shown to satisfy a property of weak monotonicity and as a consequence they enjoy an extra degree of regularity we also prove that these deformations are locally invertible moreover the neighbourhood of invertibility is stable along a weak convergent sequence in w 1 p and the sequence of local inverses converges to the local inverse we use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps we apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity

non lipschitz minimizers of smooth uniformly convex functionals we construct non lipschitz minimizers of smooth uniformly convex functionals of type i u f du x dx our method is based on the use of null lagrangians

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians