bounds for effective parameters of multicomponent media by analytic continuation recently d bergman introduced a method for obtaining bounds for the effective dielectric constant or conductivity of a two component medium this method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters we extend the method to multicomponent media using techniques of several complex variables

translation and related bounds for the response of a nonlinear composite conductor lower bounds for the overall energy functions of a class of nonlinear composite conductors are established by two different methods the translation method and the use of a comparison linear composite when the properties of the comparison linear composite are themselves estimated by the translation method the resulting comparison bound is demonstrated never to be superior to the bound obtained by direct application of the translation method to the nonlinear composite examples show however that the bounds obtained by these two methods in fact often coincide when they do the use of the comparison medium permits the development of more refined bounds by exploiting better bounds for the linear composite particular emphasis is placed on a limiting case of material behaviour for which explicit results can be obtained a composite dielectric that displays breakdown when the local electric field exceeds a critical value the formalism delivers an upper bound for the value of the mean field in the composite which induces breakdown in the composite bounds are also established for a composite with a power law relation between current and electric field

the cavity of the optimal shape under the shear stresses the problem of optimal shape of a single cavity in an infinite 2 d elastic domain is analyzed an elastic plane is subjected to a uniform load at infinity the cavity of the fixed area is said to be optimal if it provides the minimal energy change between the homogeneous plane and the plane with the cavity we show that for the case of shear loading the contour of the optimal cavity is not smooth but is shaped as a curved quadrilateral the shape is specified in terms of conformal mapping coefficients and explicit analytical representations for components of the dipole tensor associated with the cavity are employed we also find the exact values of angles at the corners of the optimal contour the applications include the problems of optimal design for dilute composites

a hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation we propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations this scheme divides the computational domain into boundary transition and inner areas the wavefields within the inner and boundary areas are computed by the wave equation and the one way wave equation respectively the wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone the results from our finite difference numerical modeling tests of the 2d acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area we obtain equally good performance using pseudospectral and finite element modeling with the same scheme our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption

sets of conductivity and elasticity tensors stable under lamination a complete characterization is obtained of any set of constitutive conductivity or elasticity tensors that is stable under lamination between any two of the elements of the set the conditions are local and are expressed in terms of the curvature and tangent plane at points on the boundary of the set they are checked to hold for several well known examples of sets stable under a more general process of homogenization a companion paper investigates the link between these conditions and quasiconvexity 1994 john wiley sons inc

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

γ convergence of power law functionals with variable exponents abstract convergence results for power law functionals with variable exponents are obtained the main motivation comes from the study of first failure dielectric breakdown some connections with the generalization of the laplace equation to the variable exponent setting are also explored

towards characterization of all 3 3 extremal quasiconvex quadratic forms given a d d quasiconvex quadratic form d 3 we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero then the form itself is an extremal quasiconvex quadratic form i e it loses its quasiconvexity whenever a convex quadratic form is subtracted from it in the special case d 3 we slightly weaken the condition namely we prove that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square then the form itself is an extremal quadratic form in the case d 3 we also prove that if the determinant of the acoustic tensor of the form is identically zero then the form is either extremal or polyconvex also if the determinant of the acoustic tensor of the form is a perfect square then the form is either extremal or polycovex or is a sum of a rank 1 form and an extremal whose acoustic tensor determinant is identically zero here we use the notion of extremality introduced by milton in 1990 2017 wiley periodicals inc

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

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

minimizers for a double well problem with affine boundary conditions this paper is concerned with the existence of minimizers for functionals having a double well integrand with affine boundary conditions such functionals are related to the so called kohn strang functional which arises in optimal shape design problems in electrostatics or elasticity they are known to be not quasi convex and therefore existence of minimizers is in general guaranteed only for their quasi convex envelopes we generalize previous results in 1 and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory we also prove by a general argument that their rank one convexifications coincide with their quasi convexifications

exact relations for effective tensors of polycrystals ii applications to elasticity and piezoelectricity an important necessary condition for an exact relation for effective moduli of polycrystals to hold is stability of that relation under lamination this requirement is so restrictive that it is possible if not always feasible to find all such relations explicitly in order to do this one needs to combine the results developed in part i of this paper and the representation theory of the rotation groups so 2 and so 3 more precisely one needs to know all rotationally invariant subspaces of the space of material moduli this paper presents an algorithm for finding all such subspaces we illustrate the workings of the algorithm on the examples of 3 dimensional elasticity where we get all the exact relations and the examples of 2 dimensional and 3 dimensional piezoelectricity where we get some possibly all of them

from microstructure independent formulas for composite materials to rank one convex non quasiconvex functions examples of non quasiconvex functions that are rank one convex are rare in this paper we construct a family of such functions by means of the algebraic methods of the theory of exact relations for polycrystalline composite materials developed to identify g closed sets of positive codimensions the algebraic methods are used to construct a set of materials of positive codimension that is closed under lamination but is not g closed the well known link between g closed sets and quasiconvex functions and sets closed under lamination and rank one convex functions is then used to construct a family of rotationally invariant nonnegative and 2 homogeneous rank one convex functions that are not quasiconvex

subrepresentation semirings and an analog of 6j symbols let v be a complex representation of the compact group g the subrepresentation semiring associated to v is the set of subrepresentations of the algebra of linear endomorphisms of v with operations induced by the matrix operations the study of these semirings has been motivated by recent advances in materials science in which the search for microstructure independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group so 3 in this case the structure constants for subrepresentation semirings can be described explicitly in terms of the 6j symbols familiar from the quantum theory of angular momentum in this paper we investigate subrepresentation semirings for the class of quasisimply reducible groups defined by mackey multiplicity free representations of finite groups pac j math 8 503 1958 we introduce a new class of symbols called twisted 6j symbols for these groups and we explicitly calculate the structure constan

variational linear comparison bounds for nonlinear composites with anisotropic phases ii crystalline materials in part i of this work bounds were derived for the effective potentials of nonlinear composites with anisotropic constituents making use of an appropriate generalization of the linear comparison variational method in this second part the special case of nonlinear composites with crystalline constituents is considered first it is shown that for this special but very important class of materials the variational bounds of part i are at least as good as an earlier version of the bounds due to debotton ponte castaneda then the relative merits of these two types of bounds are studied in the context of a model two dimensional porous composite with a power law crystalline matrix phase under anti plane loading conditions the results show that indeed the variational bounds of part i improve in general on the earlier bounds with the former becoming progressively sharper than the latter as the number of slip systems of the crystalline matrix phase increases in particular it is shown that unlike the bounds of debotton ponte castaneda the variational bounds of part i are able to recover the variational bound for composites with an isotropic matrix phase as the number of slip systems all having the same flow stress tends to infinity

minimum variational principles for time harmonic waves in a dissipative medium and associated variational principles of hashin shtrikman type minimization variational principles for linear elastodynamic acoustic or electromagnetic time harmonic waves in dissipative media were obtained by milton et al milton et al 2009 proc r soc a

multicomponent composites electrical networks and new types of continued fraction i the development of bounds on the complex effective conductivity tensor that relates the average current to the average electric field in a multicomponent composite has been hindered by lack of a suitable continued fraction representation for here a new field equation recursion method is developed which gives an expression for as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry a hierarchy of field equations is set up such that the solutions of the j 1 th order equation generate the solutions of thejth order equation consequently the effective tensor j associated with thejth order field equation is expressible as a fractional linear matrix transformation of j 1 these transformations combine to form the continued fraction expansion for 0 which is exploited in the following paper part ii to obtain bounds crude bounds on j forj 1 give narrow bounds on the continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable stieltjes functions that proved important in the development of the theory of pade approximants asymptotic analysis and the theory of orthogonal polynomials in the last century the results extend to other transport problems including conduction in polycrystalline media the viscoelasticity of composites and the response of multicomponent multiterminal linear electrical networks

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

the relaxation of two well energies with possibly unequal moduli the elastic energy of a multiphase solid is a function of its microstructure determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect microstructure evolution and optimal design mathematically the problem is to determine the relaxation under fixed phase fraction of a multiwell energy this paper addresses two such problems in the geometrically linear setting first in two dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy with arbitrary elastic moduli and transformation strains and provide a characterization of the optimal microstructures and the associated strain second in three dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy when either 1 both elastic moduli are isotropic or 2 the elastic moduli are well ordered and the smaller elastic modulus is isotropic in both cases we impose no restrictions on the transformation strains we provide a characterization of the optimal microstructures and the associated strain we also compute a lower bound that is optimal except possibly in one regime when either 1 both elastic moduli are cubic or 2 the elastic moduli are well ordered and the smaller elastic modulus is cubic for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal in all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain in both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction we also characterize optimal microstructures under applied stress

on the relation between extremal elasticity tensors with orthotropic symmetry and extremal polynomials we prove that an elasticity tensor with orthotropic symmetry is extremal if the determinant of its acoustic tensor is an extremal polynomial that is not a perfect square

a scalar inequality which bounds the effective conductivity of composites new bounds on the eigenvalues of the effective conductivity tensor in spatial dimension d 3 are proved using the compensated compactness method the inequalities are obtained using the nine quadratic null lagrangians given by the 2 times 2 minors of a jacobian matrix partial phi i partial x j it is proved that in general the bounds depend at most on four parameters

modelling the effective conductivity function of an arbitrary two dimensional polycrystal using sequential laminates the effective conductivity tensor of a two dimensional polycrystalline material depends on the conductivity tensor 0 of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal represented by a rotation field r x giving the orientation of the crystal at each point x here it is established that the dependence of on 0 in any polycrystal with r x held fixed can be mimicked exactly by a polycrystal constructed by sequential lamination it is first shown that the effective conductivity function is perturbed only slightly if we truncate the hilbert space of fields in the polycrystal to a finitedimensional space then the structure of this finite dimensional space of fields is shown to be isomorphic to the structure of the finite dimensional space of fields associated with the sequential laminate in particular there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields

homogenization of dynamic laminates abstract this paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spacial but also temporal inhomogeneities called dynamic materials the study was initiated by lurie in his pioneering work of 1997 homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation two typical examples of periodic homogenization are considered the wave equation and maxwell s system coefficients oscillating rapidly not only in space but also in time conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed in particular we examine a cell problem for periodic composites as well as the laminate formulae the effective tensors of rank one laminates for one dimensional wave equation and the full maxwell s system are computed explicitly we also note some dramatic differences between the hyperbolic and the elliptic cases

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

variational principles for complex conductivity viscoelasticity and similar problems in media with complex moduli linear processes in media with dissipation arising in conductivity optics viscoelasticity etc are considered time periodic fields in such media are described by linear differential equations for complex valued potentials the properties of the media are characterized by complex valued tensors for example by complex conductivity or complex elasticity tensors variational formulations are suggested for such problems the functionals whose euler equations coincide with the original ones are constructed four equivalent variational principles are obtained two minimax and two minimal ones the functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation the last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors

exact relations for effective tensors of composites necessary conditions and sufficient conditions typically the elastic and electrical properties of composite materials are strongly microstructure dependent so it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes they can also be regarded as fundamental invariances existing in a given physical context classic examples include hill s formulae for the effective bulk modulus of a two phase mixture when the phases have equal shear moduli levin s formulae linking the effective thermal expansion coefficient and effective bulk modulus of two phase mixtures and dykhne s result for the effective conductivity of an isotropic two dimensional polycrystalline material here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones the search for exact relations is reduced to a search for matrix subspaces having a structure of special jordan algebras one of many new exact relations is for the effective shear modulus of a class of three dimensional polycrystalline materials we present complete lists of exact relations for three dimensional thermoelectricity and for three dimensional thermopiezoelectric composites that include all exact relations for elasticity thermoelasticity and piezoelectricity as particular cases 2000 john wiley sons inc

absorbing boundary conditions for acoustic media by decomposing the acoustic wave equation into incoming and outgoing components an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation this boundary condition is characterized by a first order differential operator the differential operator or absorbing boundary operator is the basic element from which more complicated boundary conditions can be constructed the absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions by combining two or more absorption operators boundary conditions can be created which absorb plane waves propagating in any number of directions absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems

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

topology design with optimized self adaptive materials significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems we study the optimal shape design of a two dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material the macroscopic version of this problem is not well posed if no restrictions are placed on the structure topology relaxation of the optimization problem via quasiconvexification or homogenization methods is required the effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material

variational linear comparison bounds for nonlinear composites with anisotropic phases i general results this work is concerned with the development of bounds for nonlinear composites with anisotropic phases by means of an appropriate generalization of the linear comparison variational method introduced by ponte castaneda for composites with isotropic phases the bounds can be expressed in terms of a convex concave optimization problem requiring the computation of certain error functions that in turn depend on the solution of a non concave non convex optimization problem a simple formula is derived for the overall stress strain relation of the composite associated with the bound and special simpler forms are provided for power law materials as well as for ideally plastic materials where the computation of the error functions simplifies dramatically as will be seen in part ii of this work in the specific context of composites with crystalline phases e g polycrystals the new bounds have the capability of improving on earlier bounds such as the ones proposed by debotton and ponte castaneda for these specific material systems

bounds on the volume fractions of two materials in a three dimensional body from boundary measurements by the translation method using the translation method of tartar murat lurie and cherkaev bounds are derived on the volume occupied by an inclusion in a three dimensional conducting body the bounds assume that electrical impedance tomography measurements have been made for three sets of pairs of current flux and voltage measurements around the boundary additionally the conductivity of the inclusion and the conductivity of the surrounding medium are assumed to be known if the boundary data dirichlet or neumann is special i e such that the fields inside the body would be uniform were the body homogeneous then the bounds reduce to those of milton and thus when the volume fraction is small to those of capdeboscq and vogelius

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

on the effective elasticity of a two dimensional homogenised incompressible elastic composite the set of effective elasticity tensors for all two dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described in two dimensions the effective tensors are completely characterised by bounds on their eigenvalues

multiphase interchange inequalities phase interchange inequalities have been studied since the early work of keller j math phys 5 548 1964 they constrain the effective conductivity of composite materials which are obtained from each other for fixed configuration by interchanging the position of the phases optimal results in this direction for the case of a two phase composite are due to keller in spatial dimension d 2 and to avellaneda et al j appl phys 63 4989 1988 in dimension d 3 in this paper new inequalities in spatial dimension d 2 and d 3 which are valid when an arbitrary number of phases is present are proven when specialized to two phase composites they agree with those of keller in d 2 and of avellaneda et al in d 3 respectively

variational estimates for the creep behaviour of polycrystals a variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high temperature creep the procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corre sponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single crystal grains as such it constitutes an extension to locally anisotropic behaviour of the variational procedure devel oped by ponte castaneda 1991 for nonlinear heterogeneous media with locally isotropic behaviour by way of an example the procedure is applied to the de termination of bounds of the hashin shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals the bounds are computed for the special class of untextured fcc polycrystals with isotropic pure power law viscous behaviour first considered by hutchinson 1976 in the context of a calculation of the self consistent type the new bounds are found to be more restrictive than the corresponding classical taylor bishop hill bounds and also more re strictive if only slightly so than related bounds of the hashin shtrikman type by dendievel et al 1991 the new procedure has the advantage over the self consistent procedure of hutchinson 1976 that it may be applied without any essential complications to aggregates of crystals with slip systems exhibiting dif ferent creep rules with for example different power exponents and to general loading conditions however the distinctive feature of the new variational proce dure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals

absorbing boundary conditions for elastic waves absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains the boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves these conditions are based on compositions of simple first order differential operators the formulas can be applied without modification to problems in both two and three dimensions the boundary conditions are stable for all values of the ratio of p wave velocity to s wave velocity and they are effective near a free surface and in a horizontally stratified medium the boundary conditions are approximated with simple finite difference equations that use values of the solution only along grid lines perpendicular to the boundary this property facilitates implementation especially near a free surface and at other corners of the computational domain

rigid perfectly plastic two dimensional polycrystals we consider rigid perfectly plastic polycrystals in the two dimensional anti plane shear context the yield sets of the grains are identified with rectangles in the plane centred at the origin whose sides have length 2 and 2 m the limit m corresponds to the grains being rigid in one direction and ductile in the orthogonal direction we show that for large values of m there exist polycrystals whose effective yield sets are large in all directions more precisely for each value of m we construct a polycrystal whose yield set contains the set f f f f where f m o 1 we also show that the yield set of any isotropic polycrystal is contained in the ball of radius 4 m centred at the origin this bound results as an application of the div curl lemma the new component of our analysis which allowed us to obtain sharper results is that we consider simultaneously not only two but an infinite number of admissible stress fields whose averages have different directions

macroscopic properties and field fluctuations in model power law polycrystals full field solutions versus self consistent estimates this paper presents comparisons between full field numerical results and homogenization estimates for the effective behaviour and statistical fluctuations of the stress and strain rate fields in vi

a link between sets of tensors stable under lamination and quasiconvexity a link is found between quasiconvexity and the conditions for a set l of conductivity or elasticity tensors to be stable under lamination these conditions derived in the companion paper are shown here to be equivalent to the condition that for every point b on the boundary of the set l an operator tb dependent on the tangent plane and curvature of the set at b is a quasiconvex translation operator a separate class of quasiconvex translation operators is obtained which are candidates for proving that l is stable under homogenization the region stable under homogenization associated with any one of these operators shares a common boundary point and tangent plane with the set l and has curvature at that point not greater than the curvature of l the conditions under which there exists a representative subclass of these operators such that the associated regions stable under homogenization wrap around l remains unresolved it is proved that l can be characterized by minimizations of sums and dual energies in much the same way that convex sets can be characterized by their legendre transforms 1994 john wiley sons inc

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

on the effective viscoelastic moduli of two phase media iii rigorous bounds on the complex shear modulus in two dimensions the translation and the hashin shtrikman methods are used to provide bounds on the effective complex shear modulus of a two phase two dimensional viscoelastic composite they are both given by inequalities that depend on six parameters the best bounds are obtained by optimizing these parameters over the admissible set which is larger for the translation method than for the hashin shtrikman method consequently the translation method generally leads to tighter bounds than would be obtained via the standard hashin shtrikman approach equivalence classes of two dimensional viscoelastic composites directly analogous to similar classes for the pure elastic problem are found combination of the simplified versions of the translation or the hashin shtrikman type bounds and this equivalency results in simple algorithms for computing tight bounds for any choice of phase moduli and volume fractions the bounds constrain the effective shear modulus to lie inside a region of the complex plane bounded by arcs of circles the four points which correspond to the hashin shtrikman walpole bounds on the shear modulus of an elastic composite always lie inside or on the boundary of the bounding region in many cases the bounding region tends to hug the corresponding parallelogram in the complex compliance plane having these four points as vertices

optimal lower bounds on local stress inside random media a methodology is presented for bounding the higher l p norms 2 leq p leq infty of the local stress inside random media we present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites these bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads these results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media

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

an optimal absorbing boundary condition for elastic wave modeling absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries lindman 1975 clayton and engquist 1977 reynolds 1978 liao et al 1984 cerjan et al 1985 randall 1988 higdon 1991 we are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage this is practical for 3 d simulation of elastic wave propagation by a finite difference method peng and toksoz 1994 developed a method to design a class of optimal absorbing boundary conditions for a given operator length in this short note we give a brief introduction to this technique and we compare the optimal absorbing boundary conditions against those by reynolds 1978 and higdon 1991 using examples of 3 d elastic finite difference modeling on an ncube 2 parallel computer in the appendix we also give explicit formulas for computing coefficients of the optimal abso

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

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

variational estimates of the self consistent type for the effective behaviour of some model nonlinear polycrystals a new class of self consistent estimates for nonlinear polycrystals is obtained by making use of the classical self consistent estimates for the linear comparison polycrystal in the nonlinear variational procedure of debotton and ponte castaneda 1995 a distinctive feature of the new self consistent estimates is that they satisfy the recent bound of kohn and little 1997 for two dimensional ideally plastic polycrystals which is known to be significantly more restrictive than the taylor bound at large grain anisotropy in addition the new self consistent estimates are found to satisfy the taylor and hashin shtrikman bounds which are sharper than the kohn little bound at low grain anisotropy this result suggests that the new self consistent estimates may also be more accurate than other self consistent estimates like those arising from the incremental model of hill 1965 for three dimensional nonlinear polycrystals

twistor holomorphic immersions of real surfaces into kahler surfaces on etudie le lien entre la construction twisteur et l application de gauss en vue des points complexes sur des surfaces reelles

optimal bounds and microgeometries for elastic two phase composites consider two isotropic linearly elastic materials with elasticity tensors c 1 c 2 and assume c 1 c 2 we prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite rank laminates this implies that for any imposed uniform strain or stress field optimal bounds on the effective strain or stress energy per unit volume are attained by finite rank laminates explicit bounds on the strain energy with no symmetry assumptions are given in dimensions 2 and 3 these bounds are more stringent than the classical voigt reuss bounds finally explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry

optimal bounds for conduction in two dimensional multiphase polycrystalline media a complete geometrical characterization is given of the set of all possible effective conductivity tensors for two dimensional composites made of an arbitrary number of given anisotropic phases it is rigorously established that any polycrystalline composite formed from an arbitrary possibly infinite number of phases can be replaced by a composite formed from only two of the given phases without altering the effective conductivity tensor

linear comparison estimates for the effective resistivity of three dimensional nonlinear polycrystals estimates for the effective resistivity of nonlinear polycrystals are obtained using the linear comparison homogenization scheme of debotton and ponte castaneda debotton ponte castaneda 1995 proc r soc a 448 121 142 computing the effective properties of linear composites with the same microstructure as the nonlinear composite is an essential part of this scheme the classical self consistent method is employed for this purpose an important characteristic of these estimates for polycrystals with field thresholds is that they satisfy the recent bound of garroni and kohn garroni kohn 2003 proc r soc a 459 2613 2625 which dramatically improves upon the classical taylor upper bound at large crystal anisotropy in addition the estimates also satisfy the hashin shtrikman bounds which are more restrictive than the garroni kohn bound at small crystal anisotropy inte restingly the scaling exponents for the linear comparison estimates are found to be inde pendent of the constitutive nonlinearity this last observation provides an explanation for the relative weakness of an earlier linear comparison bound obtained by garroni and kohn

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

fourth order moments of nonnegative measures on s2 and applications this paper is mainly devoted to a study of the fifteen dimensional set of all fourth order moments of a nonnegative radon borel regular measure on s 2 we seek a complete characterization of the set a closed cone with the help of the hilbert decomposition theorem for nonnegative polynomials of degree four in two variables special emphasis is placed on the boundary of this set which is shown to be generated by atomic measures made of five dirac masses or fewer which are located on the intersection of s 2 with the zero set of a quadratic form as a consequence every point of the moment set is generated by atomic measures made of six dirac masses or less this study may be viewed as a contribution to the moment problem note that the analogous two dimensional case has been analyzed in avellaneda milton am potential applications for this result are however manifold especially in the field of homogenization which is of especial interest to us in the setting of linearized elasticity effective properties of fine mixtures of two phases are investigated specifically the goal is to analyze the coefficients of the linear second order elliptic system satisfied by the weak limit u of the solution fields w to a sequence of elasticity problems of the form

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

homogenisation of a class of fourth order equations with application to incompressible elasticity synopsis upon formalising an analogy between two dimensional stokes flow and two dimensional isotropic conductivity we exhibit a class of fourth order equations which behave isomorphically like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours in particular lipton s result on the g closure problem for mixtures of two incompressible elastic materials is recovered in the two dimensional case

network approximation for transport properties of high contrast materials we show that the effective complex impedance of materials with conductivity and dielectric permittivity that have high contrast can be calculated approximately by solving a suitable resistor capacitor network we use variational principles extensively for the analysis and we assess the accuracy of the network approximation by numerical computations

exact relations for effective tensors of polycrystals i necessary conditions the set of all effective moduli of a polycrystal usually has a nonempty interior when it does not we say that there is an exact relation for effective moduli this can indeed happen as evidenced by recent results 4 10 12 on polycrystals in this paper we describe a general method for finding such relations for effective moduli of laminates the method is applicable to any physical setting that can be put into the hilbert space framework developed by milton 13 the idea is to use the w function of milton 13 that transforms a lamination formula into a convex combination the method reduces the problem of finding exact relations to a problem from representation theory of so d d 2 or 3 corresponding to a particular physical setting when this last problem is solved there is a finite amount of calculation required to be done in order to answer the question completely at present each candidate relation has to be examined separately in order to confirm the stability under homogenization we apply our general theory to the settings of conductivity and two dimensional elasticity

attenuating boundary conditions for numerical modeling of acoustic wave propagation four types of boundary conditions dirichlet neumann transmitting and modified transmitting are derived by combining the damped wave equation with corresponding boundary conditions the dirichlet attenuating boundary condition is the easiest to implement for an appropriate choice of attenuation parameter it can achieve a boundary reflection coefficient of a few percent in a one wavelength wide zone the neumann attenuating boundary condition has characteristics similar to the dirichlet attenuating boundary condition but it is numerically more difficult to implement both the transmitting boundary condition and the modified transmitting boundary condition need an absorbing boundary condition at the termination of the attenuating region the modified transmitting boundary condition is the most effective in the suppression of boundary reflections for multidimensional modeling there is no perfect absorbing boundary condition and an approximate absorbing boundary condition is used