Bifurcations in the Regularized Ericksen Bar Model

We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Müller’s conjecture (Müller, Calc. Var. 1:169–204, 1993) concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of Müller’s conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a “loop” structure that characterizes (k,3k) bifurcations.      

GENERALIZATION OF STEINER FORMULAFOR THE HOMOTHETIC MOTIONS ONTHE PLANAR KINEMATICS

The Steiner formula and the mixture area formula given by M(U|¨)ller were expressed under the one-parameter closed planar homothetic motions in the complex sense . Also, using the generalization of Steiner formula, the result of Holditch theorem for homothetic motions is got. In the case of the homothetic scale h≡1 the results given by M(U|¨)ller are obtained as a special case.     

Response of a Thin Cylindrical Shell Excited by a Turbulent Internal Flow

This paper presents results obtained within the scope of a collaboration between the “Laboratoire de Mécanique et d'Acoustique” (Marseille, France), the “Laboratoire de Mécanique des Fluides et d'Acoustique” of the Ecole Centrale (Lyon, France) and the “Laboratoire de Vibrations et d'Acoustique” of the Institut National des Sciences Appliquées (Lyon, France). The main aspect of this collaboration is to establish a comparison between a measured and a model vibro-acoustics response of a thin cylindrical pipe excited by a turbulent internal flow. After a brief review of the literature, a model of the response of the shell, based on a matched asymptotic expansion, is given. Some numerical results are also given. The spectral density of the acceleration of the shell is compared with experimental results. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Justification of the Nonlinear Inextensional Plate Model

We consider a cylindrical three-dimensional body, made of a Saint Venant-Kirchhoff material, and we let its thickness go to zero. For a specific order of magnitude for the applied loads and under appropriate restrictions on the set of admissible deformations, we show that the almost-minimizers of the total energy converge toward deformations that minimize the nonlinear bending energy obtained by Fox, Raoult and Simo using formal asymptotic expansions. Our result is obtained by Γ-convergence arguments. (Accepted September 18, 2002) Published online January 15, 2003 Communicated by S. Müller     

Zur hydrodynamischen Stoffübergangshemmung durch Adsorptionsschichten an flüssig/flüssig-Phasengrenzen

Ohne Zusammenfassung

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The hydrodynamic effect of adsorption layers on the mass transfer in liquid-liquid systems

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Auszug aus der Dissertation v. K.D. Heck, München 1974.

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Die verfasser danken der Deutschen Forschungsgemeinschaft für die Unterstfitzung im Rahmen des Sonderforschungsbereiches 153 “Reaktions- und Austauschtechnik disperser Zweiphasensysteme”.     

A rheological approach to drill-in fluids optimization

The aim of this work is to propose design criteria, based on rheological characterisation for improving drill-in fluids performance. In particular, it reports an example in which rheological approaches helped improve drill-in fluids resistance to temperature. As a starting system a commercial drill-in fluid containing xanthan gum and calcium carbonate was chosen and evaluated. Different samples were then prepared by changing the initial formulation in order to increase the system's stability to temperature. Drill-in fluids' performance have been compared by considering their “damaging potential”, filtration properties and, “cakes”. All drill-in fluids have been tested before and after aging at a given temperature with “hot rolling tests”. The systems' gel structure was characterized by measuring dynamic moduli (G′ and G′′) in the linear viscoelastic range and all samples were compared by evaluating their “melting” temperature and gel network strength during time cure tests. The results obtained from this work suggest that the rheological tests carried out on the whole drill-in fluid can provide insights into fluids' damaging potential and “cake” structure. In particular, rheology proved to be able to provide quantitative information about gel strength and temperature stability that permitted one to improve drill-in fluids' formulation in order to preclude formation damage and to meet industrial requirements. Received: 6 February 2000 Accepted: 15 November 2000     

Symmetry classification of quasi-linear PDE’s containing arbitrary functions

We consider the problem of performing the preliminary “symmetry classification” of a class of quasi-linear PDE’s containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial Lie point symmetries be admitted, and a “geometrical” characterization of the relevant system of equations determining these symmetries. Two detailed examples will elucidate the idea and the procedure: the first one concerns a nonlinear Laplace-type equation, the second a generalization of an equation (the Grad–Schlüter–Shafranov equation) which is used in magnetohydrodynamics.     

Invertibility and Non-Invertibility in Thin Elastic Structures

The nonlinear elastic energy of a thin film of thickness h is given by a functional E ^h . Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h ^−α E ^h for α ≧ 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.     

Addendum to: Symmetry classification of quasi-linear PDE’s. II: an exceptional case

This short note completes the symmetry analysis of a class of quasi-linear partial differential equations considered in the previous paper (Nonlinear Dynamics 51: 309–316, 2008): it deals with an “exceptional” Lie point symmetry which is admitted only if the involved parameters are fixed by precise relationships. The peculiarity of this symmetry is enhanced by the fact that, combined with the presence of a conditional symmetry of “weak” type, it leads to a family of solutions which include, as a particular case, a relevant solution of the Grad–Schlüter–Shafranov equation, well known in plasma physics.     

Entropy Flux Relation for Viscoelastic Bodies

Thermodynamic restrictions of elastic materials in general are well-known based on the Clausius–Duhem inequality by employing the simple Coleman–Noll procedure. One of the basic assumptions in this entropy inequality is that the entropy flux is defined as the heat flux divided by the absolute temperature. To avoid this unnecessary and possibly too restrictive assumption, the general entropy inequality has been proposed and its thermodynamic consequences exploited following the Müller–Liu procedure in which supply-free bodies are considered and Lagrange Multipliers are introduced. In this new thermodynamic theory, the entropy flux and heat flux relation identical to the above assumption has not been proved for elastic bodies in general. For isotropic elastic bodies, it was proved by Müller in 1971, using explicit isotropic representations for constitutive functions. Unfortunately, the procedure contains a flaw which was later pointed out, but can not be easily resolved. Although it was shown later that it can be proved by Müller–Liu procedure, it has not been available in the literature. In this paper, we shall establish this result, providing the missing details in the previous proof. The analysis will be carried out for isotropic viscoelastic materials and the case of elastic materials follows as a special case.      

Elastic Energy Minimization and the Recoverable Strains of Polycrystalline Shape‐Memory Materials

Shape‐memory behavior is the ability of certain materials to recover, on heating, apparently plastic deformation sustained below a critical temperature. Some materials have good shape‐memory behavior as single crystals but little or none as polycrystals, while others have good shape‐memory behavior even as polycrystals. We propose a method for explaining the difference. Our approach is based on elastic energy minimization. It leads to a special class of nonlinear homogenization problems, involving integrands that are degenerate near the origin. We explore the behavior of these problems through various examples and bounds. The elementary “Taylor bound” and the newer “translation method” are central to our analysis. Accepted October 26, 1995     

Amorphous and crystalline states of ultrasoft colloids: a molecular dynamics study

In this work, we study the temperature-induced development of “dynamically arrested” states in dense suspensions of “soft colloids” (multi-arm star polymers and/or block-copolymers micelles) by means of molecular dynamics (MD) simulations. Temperature increase in marginal solvents results in “soft sphere” swelling, dynamical arrest, and eventually crystallization. However, two distinct “dynamically arrested” states were found, one almost amorphous (“glassy”) and one with a considerable degree of crystallinity, yet lower than that of the fully equilibrated crystal. It is remarkable that even that latter state permitted self-diffusion in the timescale of the simulations, an effect that underlies the importance of the “ultra-soft” nature of inter-particle potential. The “number of connections” criterion for crystallinity proved to be very successful in identifying the ultimate thermodynamic trend from the very early stages of the α-relaxation. This paper was presented at the Third Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios

Homoclinic snaking refers to the bifurcation structure of symmetric localised roll patterns that are often found to lie on two sinusoidal “snaking” bifurcation curves, which are connected by an infinite number of “rung” segments along which asymmetric localised rolls of various widths exist. The envelopes of all these structures have a unique maximum and we refer to them as symmetric or asymmetric 1-pulses. In this paper, the existence of stationary 1D patterns of symmetric 2-pulses that consist of two well-separated 1-pulses is established. Corroborating earlier numerical evidence, it is shown that symmetric 2-pulses exist along isolas in parameter space that are formed by parts of the snaking curves and the rungs mentioned above.     

On a supplementary conservation law for the governing equations in a rigid,magnetizable body with thermal and electrical conduction

Summary We consider a rigid heat conductor at rest immersed in an electromagnetic field. The interactions between thermodynamic and electromagnetic fields are decribed, in the frame work of the theory proposed by I. Müller [1]. Some properties related to the model equations proposed are pointed out. Specifically we are able to show as the governing equations may be written in symmetric and conservative form so that the Cauchy problem results well posed.

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Sommario Si considera un corpo rigido in quiete, conduttore di calore, immerso in un campo elettromagnetico esterno. Nell'ambito della teoria termodinamica proposta da I. Müller [1] si mettono in evidenza alcune proprietà del modello proposto che può essere scritto sotto forma simmetrica e conservativa assicurando, così, la buona posizione del problema di Cauchy.

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Research partially supported by C.N.R. through the G.N.F.M.     

Some properties of three-dimensional Biltrami flows

The investigation of Beltrami flows is important for the research on the mechanism of turbulent structure. In this paper the general solutions of the Beltrami flows are given, which depend explicitly on the solutions of three independent Helmholtz equations with scalar unknowns. Velocity fields of Beltrami flows can then be obtained explicitly after the application of some curl operations on the solutions of Helmholtz equations. On the basis of the exact solutions of Euler equations given above, we obtain one kind of exact solutions of non-steady Navier-Stokes equations which are also the Beltrami flows. Some interesting examples of Beltrami flows other than “ABC flows”, “Kolmogolov flows”, “Rayleigh-Bernard flows”, “Q-flows” are given. The detailed analytic results of these examples will be published in the near future.     

Is the heat flux density really non-objective? A glance back, 40?years later

Forty years ago, Ingo Müller wrote an article Müller (Arch Rat Mech Anal 45:241–250, 1972) in which a future development of material theory was predisposed. At that time, this essential article remained unnoticed by the scientific community for a long time. Also today, not everyone becomes aware of the difference between moving materials and moving observers. This distinction results in a supplementary material axiom, the “material motion dependence (MMD)”.     

Soft Neutral Elastic Inhomogeneities with Membrane-type Interface Conditions

A foreign body, called an “inhomogeneity,” when introduced in a host solid disturbs the stress field which is present in it. One can explore the possibility of modifying the contact mechanism between the inhomogeneity and the host body so as to leave the stress field in the host solid undisturbed. If such a procedure succeeds, then the inhomogeneity is called “neutral.” Modification of the contact mechanism between the inhomogeneity and the host solid can be achieved, for example, by a suitably designed thick or thin interphase between them. When the interphase is thin, it can be represented by an “imperfect interface” model. In the present study we consider “soft” inhomogeneities which are more compliant than the host body. A “membrane-type interface” which models a thin and stiff interphase is used in rendering such inhomogeneities neutral. Illustrative examples are constructed for cylindrical neutral inhomogeneities of elliptical cross section under a triaxial loading, and for spheroidal inhomogeneities subjected to an axisymmetric loading.