Commutability of homogenization and linearization at identity in finite elasticity and applications

We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.     

A cusped prismatic shell-like body under the action of concentrated moments

The elastic equilibrium problem of a cusped prismatic shell-like body, when its projection is a half-plane x ₂ ≥ 0, under the action of a concentrated moment is solved in the explicit form within the framework of the zero approximation of I.Vekua’s hierarchical models of prismatic shells. The thickness of the prismatic shell-like body is proportional to the coordinate x ₂ raised to a non-negative exponent. When the exponent equals to zero, the above solution contains the well-known solution of the classical Carothers’ problem [1] in the case of an elastic half-plane (see also [2], §39).      

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.      

The (F*F) 1/4-method for the transmission problem in two-dimensional linear elasticity

In this article we present an inversion algorithm for the determination of the shape of a two-dimensional penetrable obstacle from knowledge of the elastic field generated by an incident plane compressional and shear wave. In particular, Kirsch's improved variant of the linear sampling method, the so called (F ^* F?)^1/4-method is extended to the elastic case. A mathematical analysis that reveals the compactness and normality of the far-field operator is presented. Finally, numerical results are presented showing the robustness of the (F ^* F?)^1/4-method with respect to noise.     

A weak approach to finite elasticity

On the ground of four axioms we define thekinematics of perfectly elastic bodies and in particular the notion ofweak deformations of a perfectly elastic body. Weak deformations turn out to agree withweak diffeomorphisms introduced in [10], a class of rectifiable currents which enjoys good closure and compactness properties. Defining thedynamics of perfectly elastic bodies in terms of twoconstitutive conditions on the stored energy function, we can therefore prove existence of stable equilibrium weak deformations for mixed boundary value problems, which moreover satisfy equilibrium and conservation equations.This work has been partially supported by the Ministero dell'Universitá e della Ricerca Scientifica, by C.N.R. contract n. 91.01343.CT01, by the European Research Project GADGET, and by Grant n.11957 of the zech Acad. of Sciences.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE Sp^（n＋p）（c）WITH CONSTANT SCALAR CURVATURE

Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c)， Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated.     

A variational formulation of anisotropic geometric evolution equations in higher dimensions

We present a novel variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow in , d ≥ 2. This new formulation leads to an unconditionally stable, fully discrete, parametric finite element approximation in the case d = 2 or 3. The resulting scheme has very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments for d = 3, including regularized crystalline mean curvature flow and regularized crystalline surface diffusion.     

Poincaré-Type inequalities and reconstruction of Paley-Wiener functions on manifolds

The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L ₂(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed from a set of numbers Φ_i (ƒ), i ∈ ℕwhere Φ_i is a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φ_i, i ∈ ℕ,can be a sequence of weighted Dirac measures δ_xi, x_i ∈M. It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline functions. To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya inequalities. Our approach to the problem and most of our results are new even in the one-dimensional case.     

On rectifiable curves with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounds on global curvature: self-avoidance,regularity, and minimizing knots

We discuss the analytic properties of curves γ whose global curvature function ρ _G [γ]⁻¹ is p-integrable. It turns out that the L ^p -norm is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks. We show in particular that for all p > 1 finite -energy is necessary and sufficient for W ^2,p -regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of -minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly elastic rods, where one can introduce a bound on to preclude self-intersections.     

On nonlinear universal relations in nonlinear elasticity

In the theory of nonlinear elasticity universal relations are relationships connecting the components of stress and deformation tensors that hold independently of the constitutive equation for the considered class (or sub-class) of materials. They are classified as linear or nonlinear according as the components of the stress appear linearly or nonlinearly in the relations. In this paper a general scheme is developed for the derivation of nonlinear universal relations and is applied to the constitutive law of an isotropic Cauchy elastic solid. In particular, we consider examples of quadratic and cubic universal relations. In respect of universal solutions our results confirm the general result of Pucci and Saccomandi [1] that nonlinear universal relations are necessarily generated by the linear ones. On the other hand, for non-universal solutions we develop a general method for generating nonlinear universal relations and illustrate the results in the case of cubic relations. (Received: November 9, 2005)     

Vanishing viscosity for hemivariational inequalities modeling dynamic problems in elasticity

In this paper we consider a mathematical model describing a dynamic linear elastic contact problem with nonmonotone skin effects. The subdifferential multivalued and multidimensional reaction–displacement law is employed. We treat an evolution hemivariational inequality of hyperbolic type which is a weak formulation of this mechanical problem. We establish a result on the existence of solutions to the Cauchy problem for the hemivariational inequality. This result is a consequence of an existence theorem for second order evolution inclusion. For the latter, using the parabolic regularization method, we obtain the solution as a limit when the viscosity term tends to zero.     

Boundary conditions in intrinsic nonlinear elasticity

The intrinsic formulation of the displacement-traction problem of nonlinear elasticity is a system of partial differential equations and boundary conditions whose unknown is the Cauchy–Green strain tensor field instead of the deformation as is customary. We explicitly identify here the boundary conditions satisfied by the Cauchy–Green strain tensor field appearing in such intrinsic formulations.     

Approximation order provided by refinable function vectors

In this paper we considerL _p-approximation by integer translates of a finite set of functionsϕ _v (v=0, ...,r − 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vectorϕ=(ϕ ^=0/ r−1 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates ofϕ _v, then a factorization of the refinement mask ofϕ can be given. This result is a natural generalization of the result for a single functionϕ, where the refinement mask ofϕ contains the factor ((1 +e ^−iu )/2) ^m if approximation orderm is achieved. Dedicated to Professor L. Berg on the occasion of his 65th birthday     

Generalized gluing for Einstein constraint equations

In this paper we construct a family of new (topologically distinct) solutions to the Einstein constraint equations by performing the generalized connected sum (or fiber sum) of two known compact m-dimensional constant mean curvature solutions (M ₁, g ₁, Π₁) and (M ₂, g ₂,Π₂) along a common isometrically embedded k-dimensional sub-manifold (K, g _K ). Away from the gluing locus the metric and the second fundamental form of the new solutions can be chosen as close as desired to the ones of the original solutions. The proof is essentially based on the conformal method and the geometric construction produces a polyneck between M ₁ and M ₂ whose metric is modeled fiber-wise (i. e. along the slices of the normal fiber bundle of K) around a Schwarzschild metric; for these reasons the codimension n : =  m − k of K in M ₁ and M ₂ is required to be  ≥  3. In this sense our result is a generalization of the Isenberg–Mazzeo–Pollack gluing, which works for connected sum at points and in dimension 3. The solutions we obtain for the Einstein constraint equations can be used to produce new short time vacuum solutions of the Einstein system on a Lorentzian (m + 1)-dimensional manifold, as guaranteed by a well known result of Choquet-Bruhat.     

Global curvature and self-contact of nonlinearly elastic curves and rods

Many different physical systems, e.g. super-coiled DNA molecules, have been successfully modelled as elastic curves, ribbons or rods. We will describe all such systems as framed curves, and will consider problems in which a three dimensional framed curve has an associated energy that is to be minimized subject to the constraint of there being no self-intersection. For closed curves the knot type may therefore be specified a priori. Depending on the precise form of the energy and imposed boundary conditions, local minima of both open and closed framed curves often appear to involve regions of self-contact, that is, regions in which points that are distant along the curve are close in space. While this phenomenon of self-contact is familiar through every day experience with string, rope and wire, the idea is surprisingly difficult to define in a way that is simultaneously physically reasonable, mathematically precise, and analytically tractable. Here we use the notion of global radius of curvature of a space curve in a new formulation of the self-contact constraint, and exploit our formulation to derive existence results for minimizers, in the presence of self-contact, of a range of elastic energies that define various framed curve models. As a special case we establish the existence of ideal shapes of knots. Received: 19 January 2001 / Accepted: 23 January 2001 / Published online: 23 April 2001     

Stress formulation for frictionless contact of an elastic-perfectly-plastic body

A mathematical model for frictionless contact of a deformable body with a rigid moving obstacle is analyzed. The Prandtl–Reuss elastic-perfectly-plastic constitutive law is used to describe the material's behavior, and contact is modeled with a unilateral condition imposed on the surface velocity. The problem is motivated by the process of the plowing of the ground. A variational formulation of the problem is derived in terms of the stresses and the existence of the unique weak solution is proven. The proof is based on arguments for differential inclusions obtained in A. Amassad, M. Shillor and M. Sofonea (2001). A quasistatic contact problem for an elastic perfectly plastic body with Tresca's friction. Nonlin. Anal., 35, 95–109. Finally, a study of the continuous dependence of the solution on the data is presented.     

Boundary and variational problems for a combined mathematical model of the theory of elasticity

We construct a combined mathematical model of the theory of elasticity that describes the stress-strain state of an elastic body using the equations of the theory of elasticity in one part of the body and the equations of the theory of shells of Timoshenko type in the other part. We write the resolvent equations and conditions for elastic coupling. We study the variational formulation of the boundary-value problems of the combined model.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 92–95.     

Existence of minimizers for a polyconvex energy in a crystal with dislocations

We provide existence theorems in nonlinear elasticity for minimum problems modeling the deformations of a crystal with a given dislocation. A key technical difficulty is that due to the presence of a the dislocation the elastic deformation gradient cannot be in L ². Thus one needs to consider elastic energies with slow growth, to which the original results of Ball cannot be applied directly.     

Effective transverse elastic moduli of composites at non-dilute concentration of a random field of aligned fibers

We consider a transversal loading of a linearly elastic isotropic media containing the identical isotropic aligned circular fibers at non-dilute concentration c. By the use of solution obtained by the Kolosov–Muskhelishvili complex potential method for two interacting circles subjected to three different applied stresses at infinity, and exact integral representations for both the stress and strain distributions in a microinhomogeneous medium, one estimates the effective moduli of the composite accurately to order c². Received: March 4, 2003; revised: August 8, 2003     

Least-squares mixed finite elements with applications to anisotropic elasticity and viscoplasticity

The objective of this work is to discuss a least-squares finite element method with applications to physically nonlinear and anisotropic constitutive equations at small strains. The L₂-norm minimization of the residuals of the given first order system of differential equations leads to a functional, which is a two field formulation in the displacements and the stresses, see e.g. Cai & Starke [1]. These functionals provide the foundation for the formulations of the related least-squares mixed finite elements. A main focus of the presentation lies on the extension of plane elasticity to anisotropic or nonlinear material behavior. In this context transversely isotropic elasticity and viscoplasticity is considered. Finally a numerical example is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)