Abstracts and Talk Materials
IMA Symposium: Prospects for Mathematics and Mechanics upon the 80th Birthday of Jerry Ericksen

November 5-6, 2004

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Kaushik Bhattacharya (Department of Mechanics and Materials Science, California Institute of Technology) bhatta@caltech.edu http://www.mechmat.caltech.edu

Ferroelectric ceramics

This talk will outline recent progress in understanding ferroelectric perovskite ceramics with an emphasis on their electromechanical behavior.

Howard Brenner (Department of Chemical Engineering, Massachusetts Institute of Technology) hbrenner@MIT.EDU

Navier-Stokes-Fourier revisited

This talk builds upon the pioneering work of Dan Joseph and co-workers in clarifying the notion of what is meant by an "incompressible" fluid when density gradients are present. In particular, we introduce the notion of volume as an extensive transportable physical property of a fluid continuum in both liquids and gases. Of special interest is the kinematical notion of the "diffuse" transport of volume, above and beyond the conventional (albeit implicit) view of convective volume transport, the latter being simply and inseparably linked to mass transport through the agency of the fluid's density; that is, in the presence of density gradients, volume can be transported through space without a concomitant movement of mass. Beyond the purely kinematical aspects of volume transport reflected in the work of Joseph et al., the diffuse transport of volume is accompanied by both momentum and energy transport in amounts above and beyond the amounts heretofore considered in standard continuum-mechanical theories of diffuse momentum and energy transport. This leads to constitutive revisions of both Newton's law of viscosity governing the diffuse transport of momentum and Fourier's law of heat conduction governing the diffuse transport of energy (the latter when a clear distinction is drawn between the respective fluxes of heat and internal energy). Experimental evidence based upon the phenomena of thermophoresis and thermal transpiration in single-component gases undergoing heat transfer, together with replacement of the no-slip mass-velocity condition by a comparable no-slip volume-velocity condition, is used to quantitatively support the proposed constitutive revisions to Newton's and Fourier's laws.

Jim Casey (Department of Mechanical Engineering, University of California - Berkeley) jcasey@me.berkeley.edu

A theory of pseudo-rigid bodies

Bodies that are somehow capable of keeping their deformation fields homogeneous have been studied extensively in the literature. Such "pseudo-rigid" bodies, or "Cosserat points," have been used successfully in a variety of applications. The main question addressed in this lecture is: How, in principle, can a 3-dimensional continuum, subjected to arbitrary applied loads, keep its deformation field homogeneous? The homogeneity condition is regarded as a "global constraint," and a system of indeterminate reactive stresses is introduced. The remaining part of the stress tensor is specified by a constitutive equation. The reactive stresses play the same role as in rigid body dynamics. It is also shown how a pseudo-rigid body can be represented by a point moving in a 12-dimensional Euclidean space, the metric of which is determined by the radius of gyration of the body. In the presence of holonomic constraints, the configuration manifold is Riemannian, and a set of Lagrange's equations emerge as the covariant components of the governing balance equation.

Patricia E. Cladis (Advanced Liquid Crystal Technologies, Inc.) cladis@alct.com

Hedgehog-antihedghog annihilation to a static soliton
Slides:  pdf

Nematic liquid crystals are liquids with orientational order in a preferred direction denoted by n, a unit pseudo-vector i.e. with neither head nor tail. We call one of its point defects a hedgehog (H) because, in that case, n radiates from a point so is reminiscent of a hedgehog's quills when in a defensive posture. The other point defect is then, by default, an antihedgehog (.H), because it annihilates with a hedgehog to leave behind a static soliton [1].

Hedgehogs and antihedghogs are a direct consequence of the intrinsic nonlinear elastic properties of nematics in a cylindrical geometry to break spontaneously the symmetry imposed by the boundary condition. This was first pointed out by me and Maurice Kléman in 1972 [2] then subsequently dubbed "escape into the third dimension" [3]. While the statics of nematic point defects is relatively well understood (or so we thought), their surprising dynamics is of general interest in fields extending beyond liquid crystals.

As Jerry asked (based on preliminary observations Mayola Walters and I found at Bell Labs in the late '70's using my first computer controlled imaging system}[4]: Why is H.H annihilation dynamics so nonlinear? Indeed, why do point defects move at all when their range of interactions is limited by the cylinder radius, R, so that they should be "asymptotically free" when separated by many times R?

Most recently, using observations from my latest imaging system at ALCT, Helmut Brand and I found a new, unprecedented result: during annihilation, the hedgehog always moved faster than the antihedgehog. This gave us an idea that, after many referee battles, we finally published [5]. This talk is a tribute to Jerry's long standing affection and profound appreciation for all hedgehogs big and small.

References

Masao Doi (Department of Applied Physics, University of Tokyo) doi@rheo.t.u-tokyo.ac.jp

A variational principle in dissipative systems

In constructing kinetic equations which describe the motion of complex fluids (fluid mixtures, liquid crystals, polymer solutions and gels), thermodynamics gives us an important guide. The way how one uses the principle of thermodynamics, however, seems to have various versions. In this talk, I would like to discuss a way which I found useful. It is based on Rayleigh's extension of the Lagrangean mechanics to dissipative systems. In this talk, I explain the method and discuss its applications to several systems (two fluid model, electrolyte solutions, gels etc).

Richard D. James (Department of Aerospace Engineering and Mechanics, University of Minnesota) james@aem.umn.edu http://www.aem.umn.edu/people/faculty/bio/james.shtml

Hysteresis and geometry: a way to search for new materials with "unlikely" physical properties

These thoughts begin with the observation by physicists, probing new phenomena through the use of first principles' studies, that the simultaneous occurrence of ferromagnetism and ferroelectricity is unlikely. While these studies do not consider the possibility of a phase transformation, there is a lot of indirect evidence that, if the lattice parameters are allowed to change a little, then one might have co-existence of "incompatible properties" like ferromagnetism and ferroelectricity. Thus, one could try the following: seek a reversible first order phase transformation, necessarily also involving a distortion, from, say, ferroelectric to ferromagnetic phases. If it were highly reversible, there would be the interesting additional possibility of controlling the volume fraction of phases with fields or stress. Thus, one could imagine having a strong magnet; apply stress to it and it becomes a strong ferroelectric. The key point is reversibility.

Even big first order phase changes can be highly reversible (liquid water to ice, some shape memory materials), and we argue that it is the nature of the shape change that is critical. We suggest, based on a close examination of measured hysteresis loops in various martensitic systems, that reversibility is governed by the presence of certain special relations among lattice parameters. While these relations are naively geometric, their fundamental status is not clear, but they likely relate to a concept of metastability for an energy functional that includes both interfacial and bulk energy. Fundamentally, we lack the ability to formulate the appropriate concept of metastability because we do not really understand how to model interfacial energy, as we explain.

Acknowledgment: John Ball, Karin Rabe, Jerry Zhang.

David Kinderlehrer (Department of Mathematical Sciences, Carnegie Mellon University ) davidk@andrew.cmu.edu http://www.math.cmu.edu/people/fac/kinderlehrer.html

Issues for interfaces in polycrystals

Nearly all technologically useful materials are polycrystalline. Their ability to meet system level specifications of performance and reliability is influenced by the types of grain boundaries present and their connectivity. We explore the role of mesoscale theory and experiment designed to establish predictive models of material behavior. Traditionally we have studied geometry-based statistics, like relative area statistics or distributions of numbers of sides of grains. With the advent of automated data acquisition we now have the possibility of obtaining large quantities of both geometric and crystallographic information. In particular we shall discuss two new results which lead to the astonishing conclusion that a polycrystal may leave its "footprint" in a microsopic scan: simple analysis of the scan reveals the identity of the material.

This is part of the CMU MRSEC project.

Fanghua Lin (Courant Institute of Mathematical Sciences, New York University) linf@courant.nyu.edu

A mathematical journey on defects motions

In this lecture,I recollect a few personel interactions with Jerry Ericksen on a few mathematical issues concerning liquid crystals. In particular,I shall describe briefly some thoughts and works related to the problem of the defect motion law for flows of liquid crystal.

Gianni Royer-Carfagni (Civil-Environmental Engineering and Architecture, Universita di Parma (CNR)) royer@aem.umn.edu

The role of stress on chemical transformations in an elastic bar with nonconvex chemomechanical free energy

The equilibrium of an elastic bar capable of undergoing chemomechanical transformations and in contact with a chemically aggressive environment is considered. In the proposed model, stable equilibrium states are identified with minimizers of a specific free energy functional, which depends upon the axial strain of the bar and the extent of reaction with an external agent, which is dispersed in a surrounding vapor or liquid solution with assigned chemical potential. In general, the corresponding minimization problem is nonconvex and, therefore, it predicts the coexistence of equilibrium phases. This work is related to the now classical problem of Ericksen for an elastic bar stretched in either a "hard" or a "soft" testing machine. However, here, the presence of an additional internal variable, which represents the extent of reaction, allows for phase transformations which are stress-induced and/or driven by changes in the chemical composition of the surrounding environment. We discuss a characterization of "hard" or "soft" environmental chemical boundary conditions. The model is germane to the description of several phenomena, such as the swelling of ionic gels under chemomechanical actions, or the formation of expanding crusts in stone monuments due to acid rain or an otherwise polluted atmosphere.

Anja Schlömerkemper (Institut fuer Analysis, Dynamik und Modellierung, Universität Stuttgart) schloeme@mathematik.uni-stuttgart.de http://www.mathematik.uni-stuttgart.de/~schloeme

About magnetic force formulae

The formula for the magnetic force that is exerted by a magnetic field on a single magnetic dipole is well accepted. On the other hand, the formulation of magnetic force formulae for macroscopic magnetic systems has been under debate for a long time. A final answer to this question is of interest in the context of deformable magnetic bodies as for instance of ferromagnetic shape memory alloys.

In the first part of the talk, a brief overview of Brown's [1] approach is given and related work that was initiated by Brown's approach is mentioned.

Secondly, we focus on the formula for the magnetic force between rigid magnetic bodies which was derived from a lattice of magnetic dipoles in a continuum limit [2]. The main ideas of this approach and of the mathematical proof are presented. In addition to the classical magnetic force formulae, one obtains a surface force density which depends on the underlying lattice structure and includes short range contributions of the magnetic interaction at interfaces.

In the final part of the talk we address the question of whether this magnetic force formula describes nature well and compare it with Brown's formula.

[1] Brown, W.F., Magnetoelastic Interactions, Springer-Verlag, Berlin, 1966

[2] Schlömerkemper, A., Mathematical derivation of the continuum limit of the magnetic force between two parts of a rigid crystalline material, accepted for publ. in Arch. Rational Mech. Anal.

Lev Truskinovsky (Laboratory of Solid Mechanics, École Polytechnique) trusk@lms.polytechnique.fr

Thermodynamics of rate independent plasticity

We show that the singular dissipative potential of the phenomenological rate independent plasticity can be obtained by homogenization of a micro-model with quadratic dissipation. The essential ingredient making this reduction possible is a rugged energy landscape at the micro-scale, generating under external loading a regular cascade of subcritical bifurcations. Such landscape may appear as a result of a sufficiently strong pinning or jamming of defects, leading to elastic micro-metastability. The rate independent plastic deformation emerges in this description as a continuous succession of infinitesimal viscous events; the limiting procedure presumes the elimination of small time and length scales. We present an explicit example of a simple viscoelastic mass-spring system whose macroscopic dissipative behavior is plastic rate independent.

Anna Vainchtein (Department of Mathematics, University of Pittsburgh) annav@euler.math.pitt.edu http://www.math.pitt.edu/~annav/

Kinetics of lattice phase transitions

Understanding the origin of energy dissipation and the associated kinetics of phase boundaries remains an important open problem in modeling lattice phase transitions in martensites. Following the pioneering work of Ericksen [1975], it has become common to model these materials by an up-down-up stress-strain relation in the framework of continuum elasticity theory. The corresponding dynamic problem changes type and is ill-posed; however, it may be regularized by prescribing an additional kinetic relation between the driving force and the velocity of a phase boundary. This relation is usually either postulated or derived from a phenomenological model accounting for dispersive and dissipative effects.

In this talk we will describe how one can avoid introducing additional phenomenological parameters and instead obtain a kinetic relation by replacing the continuum model with its natural discrete analog. We consider a lattice model of martensitic phase transition which takes into account long-range interactions of an arbitrary range. Although the model is Hamiltonian at the microscale, it generates a nontrivial macroscopic kinetic relation. The apparent dissipation is due to the induced radiation of lattice waves carrying energy away from the front.

This is joint work with Lev Truskinovsky (Ecole Polytechnique, France).

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