Strong Ellipticity Research Articles

Remarks on an overdetermined boundary value problem

We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragala et al. (Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006).

A Note on Uniqueness in Linear Elastostatics

We obtain a uniqueness theorem for linear elastostatics, for a homogeneous isotropic body, under the condition of strong ellipticity, in the case of “mixed–mixed” boundary conditions. The theorem applies to bodies of restricted local concavity and convexity. Some example domains are illustrated.

On constitutive inequalities in nonlinear theory of elastic shells

AbstractConstitutive inequalities in general static and dynamic theory of elastic shells undergoing finite deformation are discussed. Constitutive inequalities are well known in continuum mechanics. They express physical or mathematical restrictions for constitutive equations of 3D elastic materials. In this paper we discuss the analogs of the strong ellipticity, Hadamard and Coleman‐Noll (GCN‐condition) inequalities for nonlinear elastic shells. It is shown that the GCN‐condition implies the strong elipticity for shell theory whereas the strong ellipticity is equivalent to the existence conditions of acceleration waves in shell.

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: II—Results

In Part I of this paper, we developed a homogenization-based constitutive model for the effective behavior of isotropic porous elastomers subjected to finite deformations. In this part, we make use of the proposed model to predict the overall response of porous elastomers with (compressible and incompressible) Gent matrix phases under a wide variety of loading conditions and initial values of porosity. The results indicate that the evolution of the underlying microstructure—which results from the finite changes in geometry that are induced by the applied loading—has a significant effect on the overall behavior of porous elastomers. Further, the model is in very good agreement with the exact and numerical results available from the literature for special loading conditions and generally improves on existing models for more general conditions. More specifically, we find that, in spite of the fact that Gent elastomers are strongly elliptic materials, the constitutive models for the porous elastomers are found to lose strong ellipticity at sufficiently large compressive deformations, corresponding to the possible onset of “macroscopic” ( shear band-type) instabilities. This capability of the proposed model appears to be unique among theoretical models to date and is in agreement with numerical simulations and physical experience. The resulting elliptic and non-elliptic domains, which serve to define the macroscopic “failure surfaces” of these materials, are presented and discussed in both strain and stress space.

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: I—Analysis

The purpose of this paper is to provide homogenization-based constitutive models for the overall, finite-deformation response of isotropic porous rubbers with random microstructures. The proposed model is generated by means of the “second-order” homogenization method, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” The constitutive model takes into account the evolution of the size, shape, orientation, and distribution of the underlying pores in the material, resulting from the finite changes in geometry that are induced by the applied loading. This point is key, as the evolution of the microstructure provides geometric softening/stiffening mechanisms that can have a very significant effect on the overall behavior and stability of porous rubbers. In this work, explicit results are generated for porous elastomers with isotropic, (in)compressible, strongly elliptic matrix phases. In spite of the strong ellipticity of the matrix phases, the derived constitutive model may lose strong ellipticity, indicating the possible development of shear/compaction band-type instabilities. The general model developed in this paper will be applied in Part II of this work to a special, but representative, class of isotropic porous elastomers with the objective of exploring the complex interplay between geometric and constitutive softening/stiffening in these materials.

Sensitivity analysis framework for micromagnetism with application to the optimal shape design of magnetic random access memories

The paper concerns the Landau–Lifshitz (LL) equation with space variable gyromagnetic factor γ ⩾ 0. This allows for a flexible description of the magnetic properties of materials. The area of a domain where the material exhibits ferromagnetic behaviour is simply compact support of γ. Outside this region, material is non-magnetic (γ = 0). First, a stability analysis of the LL equation and the corresponding sensitivity equation is carried out. We have to deal with a loss of strong ellipticity due to the fact that we allow γ to vanish in a part of a domain. Therefore we introduce a regularized problem where γ > ϵ for a constant positive ϵ and we investigate the limit process ϵ → 0. Next, we apply previous theoretical results to the optimal shape design of a magnetic random access memory (MRAM) core. A primal-dual approach is employed to tackle this problem. In the last section, numerical results for the standard model of MRAM are presented.

Homogenization estimates for fiber-reinforced elastomers with periodic microstructures

This work presents a homogenization-based constitutive model for the mechanical behavior of elastomers reinforced with aligned cylindrical fibers subjected to finite deformations. The proposed model is derived by making use of the second-order homogenization method [Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—theory. J. Mech. Phys. Solids 54, 807–830], which is based on suitably designed variational principles utilizing the idea of a “linear comparison composite.” Specific results are generated for the case when the matrix and fiber materials are characterized by generalized Neo-Hookean solids, and the distribution of fibers is periodic. In particular, model predictions are provided and analyzed for fiber-reinforced elastomers with Gent phases and square and hexagonal fiber distributions, subjected to a wide variety of three-dimensional loading conditions. It is found that for compressive loadings in the fiber direction, the derived constitutive model may lose strong ellipticity, indicating the possible development of macroscopic instabilities that may lead to kink band formation. The onset of shear band-type instabilities is also detected for certain in-plane modes of deformation. Furthermore, the subtle influence of the distribution, volume fraction, and stiffness of the fibers on the effective behavior and onset of macroscopic instabilities in these materials is investigated thoroughly.

Effective action and heat kernel in a toy model of brane-induced gravity

We apply a recently suggested technique of the Neumann-Dirichlet reduction to a toy model of brane-induced gravity for the calculation of its quantum one-loop effective action. This model is represented by a massive scalar field in the (d+1)-dimensional flat bulk supplied with the d-dimensional kinetic term localized on a flat brane and mimicking the brane Einstein term of the Dvali-Gabadadze-Porrati (DGP) model. We obtain the inverse mass expansion of the effective action and its ultraviolet divergences which turn out to be nonvanishing for both even and odd spacetime dimensionality d. For the massless case, which corresponds to a limit of the toy DGP model, we obtain the Coleman-Weinberg type effective potential of the system. We also obtain the proper-time expansion of the heat kernel in this model associated with the generalized Neumann boundary conditions containing second-order tangential derivatives. We show that in addition to the usual integer and half-integer powers of the proper time this expansion exhibits, depending on the dimension d, either logarithmic terms or powers multiple of one quarter. This property is considered in the context of strong ellipticity of the boundary value problem, which can be violated when the Euclidean action of the theory is not positivemore » definite.« less

On the Strong Ellipticity of the Anisotropic Linearly Elastic Materials

In this paper we derive necessary and sufficient conditions for strong ellipticity in several classes of anisotropic linearly elastic materials. Our results cover all classes in the rhombic system (nine elasticities), four classes of the tetragonal system (six elasticities) and all classes in the cubic system (three elasticities). As a special case we recover necessary and sufficient conditions for strong ellipticity in transversely isotropic materials. The central result shows that for the rhombic system strong ellipticity restricts some appropriate combinations of elasticities to take values inside a domain whose boundary is the third order algebraic surface defined by x2+y2+z2−2xyz−1=0 situated in the cube \(\left\vert x\right\vert <1\), \(\left\vert y\right\vert <1\), \( \left\vert z\right\vert <1\). For more symmetric situations, the general analysis restricts combinations of elasticities to range inside either a plane domain (for four classes in the tetragonal system) or in an one-dimensional interval (for the hexagonal systems, transverse isotropy and cubic system). The proof involves only the basic statement of the strong ellipticity condition.

The Vishik–Lyusternik method in elliptic problems with a small parameter

We consider boundary value problems where the operator defined in a domain and the boundary operators depend on a small parameter. Elliptic and properly elliptic problems with a small parameter are defined. It is proved that small parameter ellipticity is a necessary and sufficient condition for the existence of a priori estimates that are uniform with respect to the parameter. The proof of uniform estimates is based on the construction of the exponential boundary layer introduced in the classical paper by Vishik and Lyusternik. 0. Introduction The problem we study in this paper can be formulated as follows. On a manifold M with a smooth boundary ∂M we consider the following problem for an elliptic operator of order 2m: A(x,D, e)u(x) = f(x), x ∈ M, (0.1) Bj(x, D, e)u(x′) = gj(x), x′ ∈ ∂M, j = 1, . . . ,m, (0.2) where the operators in (0.1) and (0.2) depend polynomially on a small parameter e: A(x,D, e) := eA2m(x,D) + eA2m−1(x,D) + · · ·+A2μ(x,D), (0.3) Bj(x, D, e) := ejjBj,bj (x ′, D) + ejjBj,bj−1(x ′, D) + · · ·+Bj,βj (x′, D). (0.4) Here by A2m−j , j = 0, . . . , 2m − 2μ (μ < m) and Bj,bj−k, k = 0, . . . , bj − βj (βj < bj) we denote differential operators of orders 2m − j and bj − k, respectively, with leading homogeneous terms A2m−j and B 0 j,bj−k. Replacing in (0.3) A2m−j with A2m−j we obtain the leading term A (x,D, e) of the operator A(x,D, e). Similarly, replacing in (0.4) Bj,bj−k with B 0 j,bj−k we obtain the leading term B j (x ′, D, e) of the boundary operator (0.4). 0.1. Although specific problems of the type (0.1), (0.2) with operators of the form (0.3), (0.4) did appear quite some time ago in various areas of mathematical physics (especially in elasticity) and were well known for quite some time, a comprehensive theory of such operators had started up with a remarkable paper (essentially a monograph) by M. I. Vishik and L. A. Lyusternik [1]. In this paper the main condition on the dependence of the operator (0.1) on a small parameter was formulated and the asymptotics as e → 0 of the solution of the Dirichlet problem was constructed. In the proof of this asymptotics the condition of strong ellipticity of the operator (0.3) was used. The main feature of the problem (0.1), (0.2) is that at e = 0 we obtain an elliptic equation of order 2μ < 2m, which requires only μ boundary conditions. Therefore 2000 Mathematics Subject Classification. Primary 35B40. c ©2006 American Mathematical Society 87 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

On localised vibrations in incompressible pre-stressed transversely isotropic elastic solids

This paper is concerned with 2D localised vibration in incompressible pre-stressed fibre-reinforced elastic solids and the closely related problem of surface wave propagation in such materials. An appropriate constitutive model is derived and its stability discussed within the context of the strong ellipticity condition. Surface wave propagation in an associated half-space is considered, with the particular cases of propagation along a principal direction of primary deformation and that of almost inextensible fibres also investigated. The problems of free and forced edge vibration of a semi-infinite strip are analysed, revealing a link between the natural edge frequencies and the associated Rayleigh surface wave speed.

A unique 2-D hollow sheet structure and magnetic behavior of a cyanide- and triamine-bridged Mn IICr III ferrimagnet

A 2-D cyanide- and triamine-bridged Mn IICr III ferrimagnet, [Mn 3(dien) 2(H 2O) 2][Cr(CN) 6] 2 · 4H 2O ( 1), has been prepared by the combination of Mn 2+, diethylenetriamine (dien) co-ligand and [Cr(CN) 6] 3−. This compound forms a unique 2-D hollow sheet structure constructed by 1-D ribbon networks on the basis of triamine (dien)-bridged trinuclear Mn II units. Compound 1 readily looses all lattice water molecules and irreversibly changes to a dehydrated form, [Mn 3(dien) 2(H 2O) 2][Cr(CN) 6] 2 ( 1a), in the air. Cryomagnetic studies of 1 and 1a reveal an antiferromagnetic interaction between Cr III and Mn II ions, and an unusual long-range ferrimagnetic ordering below 30 K ( 1) and 40 K ( 1a) with multiple magnetic phase changes below T C. MCD spectra of 1a show a strong Faraday ellipticity associated with the LMCT band of the Cr–CN below 300 nm. Faraday ellipticity is remarkably enhanced below T C in line with the characteristics long-range ferrimagnetic ordering.

Negative incremental bulk modulus in foams

Negative incremental stiffness is known to occur in structures such as post-buckled flexible tubes and single-cell models. A single foam cell under uniaxial loading buckles and exhibits a non-monotonic S-shaped deformation curve, which is indicative of negative incremental stiffness. Negative stiffness is not observed in bulk materials. For example, individual foam cells display negative stiffness but foams tested in uniaxial compression exhibit a plateau in the stress–strain curve because the buckled cells localize in bands. This behaviour is consistent with the continuum view in which strong ellipticity and, hence, a positive shear modulus G and positive C 11 modulus are required for stability, even for a constrained object. It is hypothesized that a solid with negative bulk modulus can be stabilized by control of the surface displacement. Experimentally, foams were hydrostatically compressed by controlled injections of small volumes of water into a plastic chamber, causing volumetric deformation. A negative incremental bulk modulus was observed in a foam with 0.4-mm cell size beyond about 20% volumetric strain. A foam with large cells, 2.5–4 mm in size, was anisotropic and did not exhibit the cell buckling required for negative modulus.

On the equivalence of strong ellipticity in the material and spatial settings of finite elasticity

It is known from Shield’s theorem that the equilibrium equations for the inverse deformation in finite elasticity are equivalent to those for the primary deformation. We review this result for compressible elastic solids and then use it to provide a simple proof that strong ellipticity in the material setting is equivalent to that in the spatial setting. The result is then extended to incompressible materials.

Generalized Baker–Ericksen Inequalities

It is well known that isotropic, nonlinearly elastic materials satisfy the Baker–Ericksen inequalities as a consequence of the strong ellipticity or rank 1 convexity. Here we present a generalization to a non-isotropic elastic material which posseses a preferred element in the symmetry group.

Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularized Fractional Derivative Kernels with Application to the Human Calcaneal Fat Pad

A viscoelastic model of the K-BKZ (Kaye, Technical Report 134, College of Aeronautics, Cranfield 1962; Bernstein et al., Trans Soc Rheol 7: 391-410, 1963) type is developed for isotropic biological tissues and applied to the fat pad of the human heel. To facilitate this pursuit, a class of elastic solids is introduced through a novel strain-energy function whose elements possess strong ellipticity, and therefore lead to stable material models. This elastic potential - via the K-BKZ hypothesis - also produces the tensorial structure of the viscoelastic model. Candidate sets of functions are proposed for the elastic and viscoelastic material functions present in the model, including two functions whose origins lie in the fractional calculus. The Akaike information criterion is used to perform multi-model inference, enabling an objective selection to be made as to the best material function from within a candidate set.

Two-dimensional motion in a Bell-constrained plate

The so-called Bell constraint has been used for several years in plasticity theory and has additionally been the subject to several investigations within an elastic context. In this paper the effects of the Bell constraint on the propagation of harmonic waves in a finitely deformed elastic plate are considered. Strong ellipticity conditions are first derived for the unbounded case, and are shown to be dependent on the scalar multiplier associated with the Bell constraint. The dispersion relation, associated with harmonic wave propagation in a plate composed of such a material with zero incremental surface traction, is derived in respect of an arbitrary strain energy function. Asymptotic expansions are then obtained for high and low wave number. These expansions, which give phase speed as a function of wave number, harmonic number and pre-stress, are shown to give excellent agreement with numerical solutions.

Global Continuation in Second-Gradient Nonlinear Elasticity

We consider three-dimensional elastic bodies characterized by a general class of stored-energy functions dependent upon the first and second gradients of the deformation. We assume that the dependence on the higher-order term ensures strong ellipticity. With only modest assumptions on the lower-order term, we use the Leray--Schauder degree to prove the existence of global solution continua to the Dirichlet problem. With additional, physically reasonable restrictions on the stored-energy function, we then demonstrate that our global solution branch is unbounded.

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II—Application to cylindrical fibers

In Part I of this paper, we presented a general homogenization framework for determining the overall behavior, the evolution of the underlying microstructure, and the possible onset of macroscopic instabilities in fiber-reinforced elastomers subjected to finite deformations. In this work, we make use of this framework to generate specific results for general plane-strain loading of elastomers reinforced with aligned, cylindrical fibers. For the special case of rigid fibers and incompressible behavior for the matrix phase, closed-form, analytical results are obtained. The results suggest that the evolution of the microstructure has a dramatic effect on the effective response of the composite. Furthermore, in spite of the fact that both the matrix and the fibers are assumed to be strongly elliptic, the homogenized behavior is found to lose strong ellipticity at sufficiently large deformations, corresponding to the possible development of macroscopic instabilities [ Geymonat, G., Müller, S., Triantafyllidis, N., 1993. Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rat. Mech. Anal. 122, 231–290]. The connection between the evolution of the microstructure and these macroscopic instabilities is put into evidence. In particular, when the reinforced elastomers are loaded in compression along the long, in-plane axis of the fibers, a certain type of “flopping” instability is detected, corresponding to the composite becoming infinitesimally soft to rotation of the fibers.

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—Theory

Abstract This work presents an analytical framework for determining the overall constitutive response of elastomers that are reinforced by rigid or compliant fibers, and are subjected to finite deformations. The framework accounts for the evolution of the underlying microstructure, including particle rotation, which results from the finite changes in geometry that are induced by the applied loading. In turn, the evolution of the microstructure can have a significant geometric softening (or hardening) effect on the overall response, leading to the possible development of macroscopic instabilities through loss of strong ellipticity of the homogenized incremental moduli. The theory is based on a recently developed “second-order” homogenization method, which makes use of information on both the first and second moments of the fields in a suitably chosen “linear comparison composite,” and generates fairly explicit estimates—linearizing properly—for the large-deformation effective response of the reinforced elastomers. More specific applications of the results developed in this paper will be presented in Part II.