nucleation barriers for the cubic to tetragonal phase transformation we are interested in the phase transformation from austenite to martensite this transformation is typically accompanied by the generation and growth of small inclusions of martensite we consider a model from geometrically linear elasticity with sharp energy penalization for phase boundaries focusing on a cubicto tetragonal phase transformation we show that the minimal energy for an inclusion of martensite scales like maxfv 2 3 v 9 11 g in terms of the volume v moreover our arguments illustrate the importance of self accommodation for achieving the minimal scaling of the energy the analysis is based on fourier representation of the elastic energy 2012 wiley periodicals inc

bounds on the micromagnetic energy of a uniaxial ferromagnet we identify the optimal scaling law for a nonconvex nonlocal variational problem representing the magnetic energy of a uniaxial ferromagnet our analysis is restricted to a certain parameter regime in which the surface tension is sufficiently small relative to the other parameters of the problem 1998 john wiley sons inc

piecewise affine stress free martensitic inclusions in planar nonlinear elasticity we consider a partial differential inclusion problem which models stress free martensitic inclusions in an austenitic matrix based on the standard geometrically nonlinear elasticity theory we show that for specific parameter choices there exist piecewise affine continuous solutions for the square to oblique and the hexagonal to oblique phase transitions this suggests that for specific crystallographic parameters the hysteresis of the phase transformation will be particularly small

a resolution of paz s conjecture in the presence of a nonderogatory matrix abstract let m n f be the algebra of n n matrices over the field f and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space it is a long standing conjecture of paz that the length of any generating set of m n f cannot exceed 2 n 2 we prove this conjecture under the assumption that the generating set s contains a nonderogatory matrix in addition we find linear bounds for the length of generating sets that include a matrix with some conditions on its jordan canonical form finally we investigate cases when the length 2 n 2 is achieved

cylinder buckling the mountain pass as an organizing center we revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell by examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections key to obtaining this estimate is the existence of a mountain pass point for the system we prove the existence on bounded domains of such solutions for almost all loads and then numerically compute example mountain pass solutions numerically the mountain pass solution with lowest energy has the form of a single dimple we interpret these results and validate the lower bound against some experimental results available in the literature

the matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells using the notion of convergence we discuss the limiting behavior of the three dimensional nonlinear elastic energy for thin elliptic shells as their thickness h converges to zero under the assumption that the elastic energy of deformations scales like h with 2 4 we establish that for the given scaling regime the limiting theory reduces to linear pure bending two major ingredients of the proofs are the density of smooth infinitesimal isometries in the space of w2 2 first order infinitesimal isometries and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces

boundary layer analysis of the ridge singularity in a thin plate large deformations of thin elastic plates and shells present a formidable problem in continuum mechanics which is generally intractable except by numerical methods conventional approaches break down in the limit of small plate thickness due to appearance of discontinuities in the solution which require boundary layer treatment we examine a simple case of a plate bent by forces exerted along its boundary so as to create a sharp crease in the limit of infinitely small thickness we find a separable boundary layer solution of the von karman plate equations which is valid along the ridge line we confirm a scaling argument t a witten and hao li europhys lett 23 51 1993 that asserts that the ridge possesses a characteristic radius of curvature r given by the thickness of the sheet h and the length of the ridge x it viz r sim h 1 3 x 2 3 the elastic energy of the ridge scales as e sim kappa x h 1 3 where kappa is the bending modulus of the sheet we determine the dependence of these quantities on the dihedral angle of the ridge pi 2 alpha for all angles r sim alpha 4 3 and e sim alpha 7 3 the framework developed in this paper is suitable for determination of other properties of ridges such as their interaction or behavior under various types of loading we expect these results to have broad importance in describing forced crumpling of thin sheets

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

singular perturbation of an elastic energy with a singular weight abstract we study the singular perturbation of an elastic energy with a singular weight the minimization of this energy results in a multi scale pattern formation we derive an energy scaling law in terms of the perturbation parameter and prove that although one cannot expect periodicity of minimizers the energy of a minimizer is uniformly distributed across the sample finally following the approach developed by alberti and muller 2001 we prove that a sequence of minimizers of the perturbed energies converges to a young measure supported on piecewise linear periodic functions of slope 1 whose period depends on the location in the domain and the weights in the energy

optimal fine scale structures in compliance minimization for a uniaxial load we consider the optimization of the topology and geometry of an elastic structure o r2 subjected to a fixed boundary load i e we aim to minimize a weighted sum of material volume vol o structur

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

dislocation microstructures and the effective behavior of single crystals we consider single crystal plasticity in the limiting case of infinite latent hardening which signifies that the crystal must deform in single slip at all material points this requirement introduces a nonconvex constraint and thereby induces the formation of fine scale structures we restrict attention throughout to linearized kinematics and deformation theory of plasticity which is appropriate for monotonic proportional loading and confers the boundary value problem of plasticity a well defined variational structure analogous to elasticity we first study a scale invariant local problem we show that by developing microstructures in the form of sequential laminates of finite depth crystals can beat the single slip constraint i e the macroscopic relaxed constitutive behavior is indistinguishable from multislip ideal plasticity in a second step we include dislocation line energies and hence a length scale into the model different regimes lead to several possible types of microstructure patterns we present constructions which achieve the various optimal scaling laws and discuss the relation with experimentally known scalings such as the hall petch law

branched microstructures scaling and asymptotic self similarity we address some properties of a scalar two dimensional model that has been proposed to describe microstructure in martensitic phase transformations consisting of minimizing the bulk energy where uy 1 a e and u 0 0 kohn and muller r v kohn and s muller comm pure and appl math 47 1994 405 proved the existence of a minimizer for 0 and obtained bounds on the total energy that suggested self similarity of the minimizer building upon their work we derive a local upper bound on the energy and on the minimizer itself and show that the minimizer u is asymptotically self similar in the sense that the sequence 0 1 has a strongly converging subsequence in w1 2 2000 john wiley sons inc

froth like minimizers of a non local free energy functional with competing interactions we investigate the ground and low energy states of a one dimensional non local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions in particular the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one the competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction in this sense the optimal or quasi optimal profiles are froth like if seen on the scale of the antiferromagnetic potential they look neutral but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other in this paper we prove the validity of this picture we compute the oscillation scale of the quasi optimal profiles and we quantify their distance in norm from a reference periodic profile the proof consists of two main steps we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale in this way we reduce the original functional to an effective sharp interface one next we study the latter by reflection positivity methods which require as a key ingredient the exact locality of the short range term our proof has the conceptual interest of combining coarse graining with reflection positivity methods an idea that is presumably useful in much more general contexts than the one studied here

on locally complex algebras and low dimensional cayley dickson algebras the paper begins with short proofs of classical theorems by frobenius and resp zorn on associative and resp alternative real division algebras these theorems characterize the first three resp four cayley dickson algebras then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to c we call them locally complex algebras in particular we describe all such algebras that have dimension at most 4 our main motivation however for introducing locally complex algebras is that this concept makes it possible for us to extend frobenius and zorn s theorems in a way that it also involves the fifth cayley dickson algebra the sedenions

regularity properties of isometric immersions we show that an isometric immersion y from a two dimensional domain s with c1 boundary to 3 which belongs to the critical sobolev space w2 2 is c1 up to the boundary more generally c1 regularity up to the boundary holds for all scalar functions v w2 2 s which satisfy det 2v 0 if s has only lipschitz boundary we show such v can be approximated in w2 2 by functions vk w1 w2 2 with det 2vk 0

derivation of von karman plate theory in the framework of three dimensional viscoelasticity we apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite strain setting in kelvin s voigt s rheology to derive a viscoelastic plate model of von karman type we start from time discrete solutions to a model of three dimensional viscoelasticity considered in friedrich and kru ik siam j math anal 50 4426 4456 2018 where the viscosity stress tensor complies with the principle of time continuous frame indifference combining the derivation of nonlinear plate theory by friesecke james and muller commun pure appl math 55 1461 1506 2002 arch ration mech anal 180 183 236 2006 and the abstract theory of gradient flows in metric spaces by sandier and serfaty commun pure appl math 57 1627 1672 2004 we perform a dimension reduction from three dimensions to two dimensions and identify weak solutions of viscoelastic form of von karman plates

optimal fine scale structures in compliance minimization for a shear load we return to a classic problem of structural optimization whose solution requires microstructure it is well known that perimeter penalization assures the existence of an optimal design we are interested in the regime where the perimeter penalization is weak i e in the effect of perimeter as a selection mechanism in structural optimization to explore this topic in a simple yet challenging example we focus on a two dimensional elastic shape optimization problem involving the optimal removal of material from a rectangular region loaded in shear we consider the minimization of a weighted sum of volume perimeter and compliance i e the work done by the load focusing on the behavior as the weight of the perimeter term tends to 0 our main result concerns the scaling of the optimal value with respect to our analysis combines an upper bound and a lower bound the upper bound is proved by finding a near optimal structure which resembles a rank 2 laminate except that the approximate interfaces are replaced by branching constructions the lower bound which shows that no other microstructure can be much better uses arguments based on the hashin shtrikman variational principle the regime being considered here is particularly difficult to explore numerically due to the intrinsic nonconvexity of structural optimization and the spatial complexity of the optimal structures while perimeter has been considered as a selection mechanism in other problems involving microstructure the example considered here is novel because optimality seems to require the use of two well separated length scales 2015 wiley periodicals inc

convergence theorems for hl integrable vector valued functions with applications abstract in this paper we prove dominated and monotone convergence theorems for hl integrable banach valued functions these results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of fredholm type in ordered banach spaces involving kurzweil integrals or improper integrals results are used also to solve concrete second order functional boundary value problems involving discontinuities and singularities

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

on the sobolev space of isometric immersions we prove that every w 2 2 isometric immersion from a convex regular domain of r 2 into r3 can be approximated in w 2 2 norm by smooth isometric immersions from the same domain into r3

limitations on the smooth confinement of an unstretchable manifold we prove that an m dimensional unit ball dm in the euclidean space rm cannot be isometrically embedded into a higher dimensional euclidean ball brd rd of radius r 1 2 unless one of two conditions is met 1 the embedding manifold has dimension d 2m 2 the embedding is not smooth the proof uses differential geometry to show that if d 2m and the embedding is smooth and isometric we can construct a line from the center of dm to the boundary that is geodesic in both dm and in the embedding manifold rd since such a line has length 1 the diameter of the embedding ball must exceed 1

differential and integral equations with henstock kurzweil integrable functions abstract in this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing henstock kurzweil integrable functions

quantitative immersability of riemann metrics and the infinite hierarchy of prestrained shell models we propose results that relate the following two contexts for each n we identify conditions for the validity of the scaling h 2n in terms of the vanishing of riemann curvatures of g up to appropriate orders and in terms of the matched isometry expansions we also establish the asymptotic behaviour of the minimizing immersions as h rightarrow 0

wrinkles as the result of compressive stresses in an annular thin film it is well known that an elastic sheet loaded in tension will wrinkle and that the length scale of the wrinkles tends to 0 with vanishing thickness of the sheet cerda and mahadevan phys rev lett 90 074302 2003 we give the first mathematically rigorous analysis of such a problem since our methods require an explicit understanding of the underlying convex relaxed problem we focus on the wrinkling of an annular sheet loaded in the radial direction davidovitch et al proc natl acad sci 108 2011 18227 our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to 0 this requires proving an upper bound and a lower bound that scale the same way we prove both bounds first in a simplified kirchhoff love setting and then in the nonlinear three dimensional setting to obtain the optimal upper bound we need to adjust a naive construction one family of wrinkles superimposed on a planar deformation by introducing a cascade of wrinkles the lower bound is more subtle since it must be ansatz free 2014 wiley periodicals inc

a density result in two dimensional linearized elasticity and applications we show that in a two dimensional bounded open set whose complement has a finite number of connected components the vector fields u h 1 2 are dense in the space of fields whose symmetrized gradient e u is in l 2 4 this allows us to show the continuity of some linearized elasticity problems with respect to variations of the set with applications to shape optimization or the study of crack evolution

a compactness result in the gradient theory of phase transitions in the plane as 0 this functional favours jraj 1 and penalizes singularities where jrraj concentrates our main result is a compactness theorem if fe a g 0 is uniformly bounded then fra g 0 is compact in l thus in the limit 0 a solves the eikonal equation jraj 1 almost everywhere our analysis uses entropy relations and the div curl lemma adopting tartar s approach to the interaction of linear di erential equations and nonlinear algebraic relations

on the n dimensional dirichlet problem for isometric maps we exhibit explicit lipschitz maps from r n to r n which have almost everywhere orthogonal gradient and are equal to zero on the boundary of a cube we solve the problem by induction on the dimension n

upper bounds for the length of non associative algebras abstract we introduce the notion of length for non associative finite dimensional unitary algebras and obtain a sharp upper bound for the lengths of algebras belonging to this class we also put forward a new method of characteristic sequences based on linear algebra technique which provides an efficient tool for computing the length function in non associative case then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra in the last case the length is bounded in terms of the fibonacci sequence we present concrete examples demonstrating the sharpness of our bounds

gaussian curvature as an identifier of shell rigidity in the paper we deal with shells with non zero gaussian curvature we derive sharp korn s first linear geometric rigidity estimate and second inequalities on that kind of shell for zero or periodic dirichlet neumann and robin type boundary conditions we prove that if the gaussian curvature is positive then the optimal constant in the first korn inequality scales like h and if the gaussian curvature is negative then the korn constant scales like h 4 3 where h is the thickness of the shell these results have a classical flavour in continuum mechanics in particular shell theory the korn first inequalities are the linear version of the famous geometric rigidity estimate by friesecke et al for plates in arch ration mech anal 180 2 183 236 2006 where they show that the korn constant in the nonlinear korn s first inequality scales like h 2 extended to shells with nonzero curvature we also recover the uniform korn poincare inequality proven for boundary less shells by lewicka and muller in annales de l institute henri poincare c non linear anal 28 3 443 469 2011 in the setting of our problem the new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in plane loads as well as to derive energy scaling laws in the pre buckled regime the exponents 1 and 4 3 in the present work appear for the first time in any sharp geometric rigidity estimate

a rigidity result for a perturbation of the geometrically linear three well problem we study a three dimensional model for alloys that undergo a cubic to tetragonal phase transition in the martensitic phase any pair of the three martensitic variants can form a stress free laminate however this laminate is only compatible on average with the remaining variant the resulting local stresses favor a microstructure if all three variants are present for instance because of an externally imposed average strain next to the linearized elastic energy the variational model features an interfacial energy between the three variants this introduces a material length scale which together with the sample size which we mimic by periodic boundary conditions gives rise to a nondimensional parameter we rigorously establish the scaling of the minimal energy e per volume in case of externally imposed volume fractions of the martensitic variants in more precisely we show e 2 3 the upper bound construction is achieved by a few patches of branched laminates the lower bound relies on suitable interpolation inequalities this is in the spirit of a celebrated work by kohn and muller and relies on techniques developed for domain branching in micromagnetics we also prove a rigidity result in the sense that if the energy per volume e of a configuration is much smaller than 2 3 the configuration is approximately a simple unbranched laminate in particular one of the three volume fractions has to be small this is related to similar rigidity results by dolzmann and muller and by kirchheim 2009 wiley periodicals inc

confining thin elastic sheets and folding paper crumpling a sheet of paper leads to the formation of complex folding patterns over several length scales this can be understood on the basis of the interplay of a nonconvex elastic energy which favors locally isometric deformations and a small singular perturbation which penalizes high curvature based on three dimensional nonlinear elasticity and by using a combination of explicit constructions and general results from differential geometry we prove that in agreement with previous heuristic results in the physics literature the total energy per unit thickness of such folding patterns scales at most as the thickness of the sheet to the power 5 3 for the case of a single fold we also obtain a corresponding lower bound

energy estimates for the von karman model of thin film blistering we consider the behavior of buckling driven thin film blisterings using von karman s plate theory our focus is on the setting where the blistered region is the unit square with clamped boundary conditions at the vertical sides and periodic ones along the horizontal sides in this setting we prove rigorous upper and lower bounds for the elastic energy which are of the same order as the film thickness we also present a convincing argument for the necessity of branching of folds near the boundary as has been observed in experiments

length realizability for pairs of quasi commuting matrices abstract for the pairs of quasi commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators

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

domain branching in uniaxial ferromagnets a scaling law for the minimum energy we address the branching of magnetic domains in a uniaxial ferromagnet our thesis is that branching is required by energy minimization to show this we consider the nonlocal nonconvex variational problem of micromagnetics we identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already known upper bound we further show that any domain pattern achieving this scaling law must have average width of order l 2 3 where l is the length of the magnet in the easy direction finally we argue that branching is required by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction its scaling law is different

an upper bound for the length of a finite dimensional algebra abstract letfbe a field and letabe a finite dimensionalf algebra writed dimf a and letebe the largest degree of the minimal polynomial for anya a define the function f d e e 2d e 1 1 4 e 2 2 we prove that ifsis any finite generating set foraas anf algebra the words insof length less thanf d e spanaas anf vector space in the special case ofn by nmatrices this bound becomes f n 2 n n 2n 2 n 1 1 4 n 2 2 o n 3 2 this is a substantial improvement over previous bounds which have all beeno n2 we also prove that for particular setssof matrices the bound can be sharpened to one that is linear inn as an application of these results we reprove a theorem of small stafford and warfield about semiprime affinef algebras

optimal design in elasticity and plasticity a direct weight minimization subject to compliance constraints or plastic yielding constraints leads to a non convex variational problem both the theoretical and the numerical analysis are unsatisfactory the minimum weight is not achieved by any design and the approximate designs oscillate as the element mesh is refined we look for equivalent relaxed problems with the same minima they come from allowing composite materials constructed in an optimal way from the original materials the constructions are different for elasticity and plasticity but surprisingly the final relaxed problems are in some cases the same

blister patterns and energy minimization in compressed thin films on compliant substrates this paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick compliant substrates these patterns are often induced by an elastic misfit that compresses the film blistering permits the film to expand locally reducing the elastic energy of the system it is therefore natural to ask what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate this is a variational problem involving both the elastic deformation of the film and substrate and the geometry of the blistered region it involves three small parameters the nondimensionalized thickness of the film the compliance ratio of the film substrate pair and the mismatch strain in formulating the problem we use a small slope foppl von karman approximation for the elastic energy of the film and a local approximation for the elastic energy of the substrate for a one dimensional version of the problem we obtain matching upper and lower bounds on the minimum energy in the sense that both bounds have the same scaling behavior with respect to the small parameters the upper bound is straightforward and familiar it is achieved by periodic blistering on a specific length scale the lower bound is more subtle since it must be proved without any assumption on the geometry of the blistered region for a two dimensional version of the problem our results are less complete our upper and lower bounds only match in their scaling with respect to the nondimensionalized thickness not in the dependence on the compliance ratio and the mismatch strain the lower bound is an easy consequence of our one dimensional analysis the upper bound considers a two dimensional lattice of blisters and uses ideas from the literature on the folding or crumpling of a confined elastic sheet our main two dimensional result is that in a certain parameter regime the elastic energy of this lattice is significantly lower than that of a few large blisters 2015 wiley periodicals inc

the von karman equations the stress function and elastic ridges in high dimensions the elastic energy functional of a thin elastic rod or sheet is generalized to the case of an m dimensional manifold in n dimensional space we derive potentials for the stress field and curvatures and find the generalized von karman equations for a manifold in elastic equilibrium we perform a scaling analysis of an m 1 dimensional ridge in an m n 1 dimensional manifold a ridge of linear size x in a manifold with thickness h x has a width w h1 3x2 3 and a total energy e hm x h m 5 3 where is a stretching modulus we also prove that the total bending energy of the ridge is exactly five times the total stretching energy these results match those of a lobkovsky phys rev e 53 3750 1996 for the case of a bent plate in three dimensions

convergence of equilibria of thin elastic plates under physical growth conditions for the energy density abstract the asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied as the thickness h of the plate goes to zero more precisely it is shown that critical points of the nonlinear elastic functional e h whose energies per unit thickness are bounded by ch 4 converge to critical points of the limit of h 4 e h this is proved under the physical assumption that the energy density w f blows up as det f 0

rim curvature anomaly in thin conical sheets revisited this paper revisits one of the puzzling behaviors in a developable cone d cone the shape obtained by pushing a thin sheet into a circular container of radius r by a distance the mean curvature was reported to vanish at the rim where the d cone is supported we investigate the ratio of the two principal curvatures versus sheet thickness h over a wider dynamic range than was used previously holding r and fixed instead of tending toward 1 as suggested by previous work the ratio scales as h r 1 3 thus the mean curvature does not vanish for very thin sheets as previously claimed moreover we find that the normalized rim profile of radial curvature in a d cone is identical to that in a c cone which is made by pushing a regular cone into a circular container in both c cones and d cones the ratio of the principal curvatures at the rim scales as r h 5 2 f yr 2 where f is the pushing force and y is the young s modulus scaling arguments and analytical solutions confirm the numerical results

well posedness results for differential and integral equations containing henstock kurzweil integrable vector valued functions abstract in this paper we prove existence uniqueness and comparison results for solutions of differential and integral equations in banach spaces containing henstock kurzweil integrable functions from a compact real interval to an ordered banach space

coarsening of folds in hanging drapes we consider the elastic energy of a hanging drape a thin elastic sheet pulled down by the force of gravity with fine scale folding at the top that achieves approximately uniform confinement this example of energy driven pattern formation in a thin elastic sheet is of particular interest because the length scale of folding varies with height we focus on how the minimum elastic energy depends on the physical parameters as the sheet thickness vanishes the limiting energy is due to the gravitational force and is relatively easy to understand our main accomplishment is to identify the scaling law of the correction due to positive thickness we do this by i proving an upper bound by considering the energies of several constructions and taking the best and ii proving an ansatz free lower bound which agrees with the upper bound up to a parameter independent prefactor the coarsening of folds in hanging drapes has also been considered in the recent physics literature by using a self similar construction whose basic cell has been called a wrinklon our results complement and extend that work by showing that self similar coarsening achieves the optimal scaling law in a certain parameter regime and by showing that other constructions involving lateral spreading of the sheet do better in other regions of parameter space our analysis uses a geometrically linear foppl von karman model for the elastic energy and is restricted to the case when poisson s ratio is 0 2016 wiley periodicals inc

axial compression of a thin elastic cylinder bounds on the minimum energy scaling law we consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core treating the core as an obstacle we prove upper and lower bounds on the minimum energy of the cylinder that depend on its relative thickness and the magnitude of axial compression we focus exclusively on the setting where the radius of the core is greater than or equal to the natural radius of the cylinder we consider two cases the large mandrel case where the radius of the core exceeds that of the cylinder and the neutral mandrel case where the radii of the core and cylinder are the same in the large mandrel case our upper and lower bounds match in their scaling with respect to thickness compression and the magnitude of pre strain induced by the core we construct three types of axisymmetric wrinkling patterns whose energy scales as the minimum in different parameter regimes corresponding to the presence of many wrinkles few wrinkles or no wrinkles at all in the neutral mandrel case our upper and lower bounds match in a certain regime in which the compression is small as compared to the thickness in this regime the minimum energy scales as that of the unbuckled configuration we achieve these results for both the von karman donnell model and a geometrically nonlinear model of elasticity 2017 wiley periodicals inc

monotone convergence theorems for henstock kurzweil integrable functions and applications abstract in this paper we prove monotone convergence theorems for henstock kurzweil integrable functions from a compact real interval to an ordered banach space these theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing henstock kurzweil integrable functions

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

commutative matrix subalgebras and length function it is proved that if the length of a commutative matrix subalgebra is maximal then this subalgebra is maximal under inclusion the examples are given showing that the converse does not hold to establish this result we prove several fundamental properties of the length function

existence and comparison results for operator and differential equations in abstract spaces in this paper we apply the chain methods developed in heikkila and lakshmikantham 1994 to obtain new fixed point theorems and new existence and comparison results for operator equations in partially ordered sets these results are then applied to discontinuous implicit functional differential equations in ordered banach spaces

a cubic hermite finite element continuation method for numerical solutions of the von karman equations abstract we study a cubic hermite finite element method for numerical solutions of the von karman equations defined in a rectangular domain with totally clamped boundary conditions a novel iterative method combined with a predictor corrector continuation algorithm is exploited to trace solution curves of the von karman equations the fourth order finite element approximations compute accurate numerical solutions for the deformation and the airy stress function as well as their first order partial derivatives and the mixed second order partial derivatives in this regard the classical predictor corrector continuation method is interpreted in a different way our numerical results show that the bifurcation scenario of the von karman equations with totally clamped boundary conditions is different from those with simply supported and partially clamped boundary conditions

energy scaling of compressed elastic films three dimensional elasticity and reduced theories we derive an optimal scaling law for the energy of thin elastic films under isotropic compression starting from three dimensional nonlinear elasticity as a consequence we show that any deformation with optimal energy scaling must exhibit fine scale oscillations along the boundary which coarsen in the interior this agrees with experimental observations of folds which refine as they approach the boundary

closure properties for integral problems driven by regulated functions via convergence results abstract in this paper we give necessary and sufficient conditions for the convergence of kurzweil stieltjes integrals with respect to regulated functions using the notion of asymptotical equiintegrability one thus generalizes several well known convergence theorems as applications we provide existence and closure results for integral problems driven by regulated functions both in single and set valued cases in the particular setting of bounded variation functions driving the equations we get features of the solution set of measure integrals problems

on lower semicontinuity of a defect energy obtained by a singular limit of the ginzburg landau type energy for gradient fields a defect energy j which measures jump discontinuities of a unit length gradient field is studied the number indicates the power of the jumps of the gradient fields that appear in the density of j it is shown that j for 3 is lower semicontinuous on the space of unit gradient fields belonging to bv in l 1 convergence of gradient fields a similar result holds for the modified energy which measures only a particular type of defect the result turns out to be very subtle since with 3 is not lower semicontinuous as is shown in this paper the key idea behind semicontinuity is a duality representation for j 3 and the duality representation is also important for obtaining a lower bound by using j 3 for the relaxation limit of the ginzburg landau type energy for gradient fields the lower bound obtained here agrees with the conjectured value of the relaxation limit

existence results for operator equations in abstract spaces and an application in this paper we apply the recursion method presented in heikkila and lakshmikantham 1994 to derive fixed point theorems and existence results for operator equations in partially ordered sets the obtained results combined with related results of heikkila 2002 are then applied to operator equations in ordered hilbert spaces and to a functional elliptic boundary value problem