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     

Wave front shape for a constrained cylindrically anisotropic elastic solid

The two-dimensional wave front shape caused by a point impulse excitation in a cylindrically anisotropic elastic solid is considered. The elastic parameters of the solid are constrained such that E = G This constraint allows the parametric equations of the wave front to be expressed exactly in terms of elementary transcendental functions. The precise location of double and cusp points on the front is treated in detail. Time histories of several wave front patterns are presented and an interesting feature of the front is generalized to the unconstrained solid.     

A compatible‐incompatible decomposition of symmetric tensors in Lp with application to elasticity

In this paper, we prove the Saint‐Venant compatibility conditions in L^p for p∈(1,+∞), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in L^p are provided. We also use the Helmholtz decomposition in L^p to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence‐free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami‐type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < p < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case (p = 2), to the full range p∈(1,+∞). Copyright © 2015 John Wiley & Sons, Ltd.     

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).      

Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity

We show that the averaged response of random isotropic Cauchy elastic material can be described analytically. It leads to a higher gradient model with explicit expressions for the dependence on the second derivatives of the mean field. A subsequent penalty formulation coincides with a linear elastic micro-stretch model with specific choice of constitutive parameters, depending only on the average cut-off length (the internal characteristic length scale L_c > 0). Thus the microstretch displacement field can be interpreted as an approximated mean field response for these parameter ranges. The mean field free energy in this micro-stretch formulation is not uniformly pointwise positive, nevertheless, the model is well posed.      

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.     

Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities

In this paper, we study the model of Groma and Balogh [I. Groma, P. Balogh, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater. 47 (1999) 3647–3654] describing the dynamics of dislocation densities. This is a two-dimensional model where the dislocation densities satisfy a system of two transport equations. The velocity vector field is the shear stress in the material solving the equations of elasticity. This shear stress can be related to Riesz transforms of the dislocation densities. Basing on some commutator estimates type, we show that this model has a unique local-in-time solution corresponding to any initial datum in the space Cr(R²)∩Lp(R²)$C^r({\mathbb{R}}^2)\cap L^p({\mathbb{R}}^2)$ for r>1$r>1$ and 1<p<+∞$1<p<+\infty$, where Cr(R²)$C^r({\mathbb{R}}^2)$ is the Hölder–Zygmund space.     

Compactness for maps minimizing the n-energy under a free boundary constraint

Let X denote a compact subset of ℝ ⁿ and B the unit ball in ℝ ⁿ . In this paper we investigate analytical and topological compactness properties of minimizing sequences for the n-energy in the class of maps , the homotopy class . Received: 5 June 2000     

Eigenvalue distributions from impacts on a ring

We consider the collision dynamics produced by three beads with masses (m ₁, m ₂, m ₃) sliding without friction on a ring, where the masses are scaled so that m ₁ = 1/ɛ, m ₂ = 1, m ₃ = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. 0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ ² ∼ αɛ ^β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ ² ∼ log(ɛ ^β).     

Γ-convergence for incompressible elastic plates

We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by Friesecke et al. (Comm Pure Appl Math 55:1461–1506, 2002). We extend their result to the case of p growth at infinity with p ϵ [1, 2), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in W ^2,2 isometries, which was obtained by Pakzad (J Differ Geom 66:47–69, 2004) for convex domains and by Hornung (Comptes Rendus Mathematique 346:189–192, 2008) for piecewise C ¹ domains.     

Classification of Material <Emphasis Type="Italic">G</Emphasis>-Structures

A geometrical interpretation of the G-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained.     

Parabolic Type Decay Rates for 1-D-Thermoelastic Systems with Time-Dependent Coefficients

 The purpose of this paper is to derive L ^p − L ^q decay estimates for linear thermoelastic systems with time-dependent coefficient in one space variable. When all coefficients in the system have the same growth speed with small oscillations, we obtain a parabolic type decay estimate. For the system with time-dependent coefficients, we need to investigate the delicate asymptotic behaviour of characteristic roots and the remainder of diagonalization, which will be treated by dividing the phase space into three regions. Received September 15, 2001; in revised form April 20, 2002     

Besov regularity and new error estimates for finite volume approximations of the p-laplacian

Summary. In [1], we have constructed a family of finite volume schemes on rectangular meshes for the p-laplacian and we proved error estimates in case the exact solution lies in W^2,p. Actually, W^2,p is not a natural space for solutions of the p-laplacian in the case p>2. Indeed, for general L^p data it can be shown that the solution only belongs to the Besov space In this paper, we prove Besov kind a priori estimates on the approximate solution for any data in L^p. We then obtain new error estimates for such solutions in the case of uniform meshes     

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)     

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.     

Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density

This paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one-dimensional thermoviscoelasticity with a non-convex free energy density. The constitutive assumptions for the Helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys. To describe physically phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u$u$ belongs to L^∞$L^{\infty }$. New approaches are introduced and more delicate estimates are derived to establish the crucial L^∞$L^{\infty }$-estimate of strain u$u$ in deriving the compactness of the orbit of the semiflow and the existence of an absorbing set.     

The exceptional set for the sum of a prime and a square

Let E(X) denote the number of natural numbers not exceeding X which cannot be written as a sum of a prime and a square. In this paper we show that for sufficiently large X we have E(X)<< X^0.982. This revised version was published online in June 2006 with corrections to the Cover Date.     

TheN-membranes problem

We study the equilibrium position ofN elastic membranes attached to rigid supports and submitted to the action of forces. They are constrained because they cannot pass through each other. As in the case of the obstacle problem, the solution fails to beC ² and thus fails to be classical, so we provide some new regularity results in different larger spaces using an iterative penalization technique.On leave from Departement de Mathematiques, Universite de Metz, Ile du Saulcy, 57045 Metz-Cedex, FranceOn leave from Dipartamento di Matematica, Universita di Pisa, 56100 Pisa, Italy     

Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle

In this article we consider the self-adjoint operator governing the propagation of elastic waves in a perturbed isotropic half space with a free boundary condition. We prove the limiting absorption principle in appropriate Hilbert spaces for this operator. We also prove decreasing properties for the eigenfunctions associated with strictly positive eigenvalues of this operator. The proofs are based on the limiting absorption principle for the self-adjoint operator governing the propagation of elastic waves in a homogeneous isotropic half space with a free boundary and on the so called division theorem for it. Both perturbations of R ₊² ={(x₁, x₂) ? R ²; x₂ > 0} and R ₊² = {(x₁, x₂, x₃) ? R ³; x₃ > 0} are considered.     

The number of exceptional orbits of a pseudofree circle action on S 5

Let M be a closed Riemannian manifold of dimension 5 which admits a Riemannian metric of nonnegative sectional curvature. The aim of this short paper is to show that under certain lower bound of the orders of isotropy subgroups, every pseudofree and isometric S ¹-action on M cannot have more than five exceptional circle orbits. As a consequence, we conclude that a pseudofree and isometric S ¹-action on a 5-sphere S ⁵ with a Riemannian metric of nonnegative sectional curvature cannot have more than five exceptional circle orbits. This gives a result related to the Montgomery–Yang problem. In addition, we also give some further related result about nonnegatively curved manifolds of dimension 5 with an isometric but not necessarily pseudofree circle action.