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Fall 1998 - Spring 1999 CNA Seminars
ABSTRACT: The influence of large inelastic deformations on the macroscopic elastic behavior is investigated for polycrystalline copper using the representative volume element technique A tensorial and a scalar quantity describing the deviation from the isotropic elastic state is formulated for polycrystals. Furthermore an isotropy condition for aggregates of cubic single crystals is expressed in terms of crystal orientations. Exact and approximate solutions are presented. The texture dependent deviation from the isotropic elastic state is estimated in terms of stiffnesses, compliances, stresses, and strains. The texture induced anisotropy is studied by an eigenvalue and eigenvector analysis of the macroscopic elasticity tensors.
ABSTRACT: Being able to accurately simulate the magnetizing (or demagnetizing) of a continuous medium would benefit many areas of the recording industry. This process can be described by Brown's equations. Formulating these equations in a discrete form that can be used to arrive at a numerical solution on a computer is a difficult problem. In this talk, I will give an introduction to an ongoing effort to cast this problem as an optimization problem. I will discuss some difficulties and some small successes encountered to date in the effort.
ABSTRACT: We study an initial/boundary-value problem for a fully nonlinear system which describes the propagation of plane shear waves in a three-dimensional soil, assuming a hypoplastic flow rule. This model was derived to study liquefaction of soils. For a periodic square-wave disturbance on the boundary, the solution is found to saturate to an asymptotic state (away from the boundary). We determine the asymptotic state in terms of the initial and boundary data. In the case of velocity boundary conditions, this asymptotic state has a fractal dependence}on the shape of the square wave! In the case of stress boundary conditions, the dependence is perfectly smooth.
ABSTRACT: This work develops a general theory of crystalline plasticity based on: 1. a collection of slip systems for dislocation glide; 2. microforces for each slip system consistent with a microforce balance; 3. a mechanical version of the second law that includes, via the microforces, work performed in the rearrangement of atoms as described by the microshear-rates on the individual slip systems; 4. a rate-independent constitutive theory that includes dependencies on plastic-strain gradients. A central result is the equivalence of the microforce balances and yield conditions for the individual slip systems. In contrast to standard theories, the yield conditions account for variations in the free energy due to slip. When the free energy is the sum of an elastic strain energy and a defect energy quadratic in the plastic-strain gradient, the resulting theory has a form identical to the standard theory of crystalline plasticity, except that the yield conditions contain an additional term involving the Laplacian of the plastic strain.
ABSTRACT: The microstructure of blood, particularly the presence of red blood cells, induces complex non-Newtonian flow behavior at macroscopic scales, the most notable of which are shear-thinning, viscoelasticity, shear-induced anisotropy, and inhomogeneous distribution of the cellular phase in the flow. This talk will discuss a new joint CMU-UPMC initiative that aims to develop models and algorithms for simulating blood flow at macroscopic, mesoscopic, and microscopic scales. The driving application is the design of a new generation artificial heart device.
ABSTRACT: A striking feature of many turbulent fluid and plasma systems is the tendency to form macroscopic organized structures while simulatneously developing intricate fluctuations on fine spatial scales. A familiar, and much-studied, example of this phenomenon is the appearance of large-scale vortices in a turbulent large Reynolds number fluid. In this talk, we will focus on another class of nonlinear pdes that exhibit the tendency to develop large-scale coherent structures in the midst of microscopic disorder. This is the class of 1-dimensional nonlinear, nonintegrable wave systems described by a nonlinear Schrodinger (NLS) equation of the form $ i \psi_t + \psi_{xx}+ f(|\psi|^2) \psi = 0 $, for various nonlinearities $f$. First, we will demonstrate through the results of direct numerical simulations that, for a wide range of initial conditions, the field obeying such a NLS equation in a bounded spatial domain relaxes in the long-time limit into a coherent soliton-like structure coupled with small-scale fluctuations, or radiation. Next, we will present a new mean-field statistical mechanics model of coherent structures for this class of NLS equations. In accord with the numerical simulations, this statistical theory predicts that the NLS dynamics approaches a long-time state consisting of a large-scale stable solitary wave coupled with small-scale local Gaussian fluctuations. Further comparisons between the simulations and the theory will be made. This is joint work with C. Josserand, University of Chicago; B. Turkington, University of Massachusetts; C. Zirbel, Bowling Green State University.
ABSTRACT: This is joint work with I. Fonseca, and also partly with Kaushik Bhattacharya and Andrea Braides. After a general discussion of the merits of various scalings in 3d-2d asymptotics, a few models are explored in a nonlinear setting. They all give rise to membrane-like behaviors whose corresponding energy densities are explicitly computable from their bulk analogues.
Dept. of Mathematics University of Pittsburgh Pittsburgh, PA ABSTRACT: In Euclidean space the standard metric induces a distance $d(x,0)=|x|$ that, except at $x=0$, it is a solution of the eikonal equation and the infinite Laplacian. In the Heisenberg group there are a variety of choices for an invariant metric. The so called Carnot-Caratheodory distance is a solution of the eikonal equation and it is also infinite harmonic, but it is not easy to compute explicitly. Other choices of metrics are not solutions of the eikonal equation, but they still are infinite-harmonic. CNA Structured Deformations WorkshopThursday, January 13, 1999 -Friday, January 15, 1999This workshop will address structured deformations as a new tool for describing both smooth and non-smooth geometrical changes at more than one length scale, including, for example, (i) the shearing of a single crystal at both macro- and mesolevels and (ii) the deformation of a bar at the macrolevel by means smooth extension and fracture. Some of the related issues to be discussed include: energetics and constitutive relationds, relaxed energies, second order structured deformations, three-level structured deformations, structured motions, single crystals, plastic deformations, hardening, P-L-C effect, defects, slip.
1:30 - 2:30 Luca Deseri Professor Universita degli Studi di Ferrara Istituto di Ingegneria Via Saragat 1 44100 Ferrara, Italy 2:30 - 3:30 Roberto Paroni Center for Nonlinear Analysis Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213
ABSTRACT: We study so-called "hybrid feedback stabilizers" for an arbitrarily general system of linear differential equations. We prove that under assumptions of controllability and observability there exists a hybrid feedback output control which makes the system asymptotically stable. The control is designed by making use of a discrete automation has infinitely many locations, but it gives rise to a "uniform" (in some sense) feedback control. The approach we propose goes back to the classical feedback control technique combined with some ideas used in the stability theory for equations with time-delay.
ABSTRACT: Using discrete decomposition techniques, multilinear operators are naturally associated with tensors. An intrinsic condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding operators on a variety of spaces, including Lebesgue, Sobolev, and Hardy.
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ABSTRACT: We consider Bernoulli bond percolation on the three dimensional lattice in the supercritical regime. We prove the existence of a macroscopic surface tension which allows us to establish a large deviation principle for the rescaled configuration. This leads to a verification of the classical Wulff construction from a microscopic point of view.
ABSTRACT: In this conference I will report on joint work with Gianni Dal Maso, Luigi Orsina and Alain Prignet. We introduce a new definition of solution (the renormalized solution) for the nonlinear monotone elliptic problem $$ -div (a(x, D u)) = \mu \; in \Omega, \quad u = 0 \; on \partial \Omega, $$ where $\mu$ is a Radon measure with bounded total variation on $\Omega \subset \rn$ and $u \mapsto -div (a(x, D u))$ is a monotone operator acting on $W^{1,p}_0(\Omega)$. We prove the existence of a renormalized solution, a stability result (the strong convergence in $W^{1,p}_0(\Omega)$ of the truncations) and partial uniqueness results.
ABSTRACT: Even simple variational models for brittle fracture problems relying on Griffith or Barenblatt approaches lead to heavy numerical computation problems and to theoretical difficulties in defining crack evolution. The main difficulty of course comes from handling the fracture site. To overcome these drawbacks, many approximations of energies depending on bulk and surface terms by energies defined on smoother functions are proposed, some of which having a mechanical interpretation, including: - discrete models with long-range interaction - singular perturbation - non-local models - approximation by damage.Some of these approximations are motivated and inspired also by their applications to Computer Vision, where energies formally similar to those in Fracture Mechanics intervene.
ABSTRACT: The explicit computation of the effective elasticity tensor of the material produced by laminating two homogeneous elastic media is used to show that, in 2-D and 3-D linear elasticity, for any isotropic material a whose elasticity tensor is strongly elliptic, but not semipositive definite, we can select very strongly elliptic materials, so that through laminations between these with material a, we can create a non-strongly elliptic media, whose existence contradicts properties concerning the propagation of elastic waves. We will also comment on the agreement between this result using only rank one laminates and the range of values of the Lame parameters for which the example of Le Dret still works.
Professor & Head Department of Mathematical Sciences Carnegie Mellon University ABSTRACT: In this talk I shall discuss various techniques for modelling solidificationand melting problems. Most of my effort will focus on the phase field equations $$\frac \partial {\partial t}(T+p)-a^2\bigtriangleup T=0\;,\;(x,y)\in \Omega \hspace{1.0in} {(\rm Con Energy)} $$ $$ \alpha \delta \frac{\partial p}{\partial t}-\lambda ^2\delta ^2\Delta p=T-p(p^2-1)\;,\;(x,y)\in \Omega \hspace{0.3in} {\rm (Ph. Field)} $$ $$ \frac{\partial T}{\partial n}=\frac{\partial p}{\partial n}=0\;,\;(x,y)\in \partial \Omega \hspace{1.0in} {\rm (BC)} $$ which may be used to model the energy balance in multiphase systems (liquid-solid or gas-liquid). I will (i) contrast and compare the behavior of solutions of the above systems to ``sharp interface'' approximating systems, (ii) present some new estimates which characterize the long-time behavior of solutions of the ``phase-field'' equations, and (iii) show some interesting computations which demonstrate these theorems.
Professor Dipartimento di Scienze e Tecnologie Avanzate Corso Borsalino 54 15100 Alessandria Italy ABSTRACT: We present some new results, obtained in collaboration with I. Fonseca, on lower semicontinuity of multiple integrals $F(u,A):= \int_A f(x,u,Du) dx$ with respect to local convergence in L^1. We are particularly interested in those cases where very weak growth and continuity (in (x,u)) conditions are imposed on the energy density f, and coercivity is not available. Our approach is based on blow-up arguments, the ``global method for relaxation'' proposed by Bouchitte' Fonseca and Mascarenhas, and approximation results due to Dal Maso and Sbordone in the scalar case, and to Kristensen in the vector-valued setting.
Postdoctoral Associate Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA ABSTRACT: An evolution problem is studied for filtration through porous media, accounting for hysteresis in the saturation versus pressure constitutive relation. A weak formulation of the problem is given, assuming that the memory effect in the constitutive relation consists not only of a rate-independent component but also of a rate-dependent one. Finally, an existence result is proved, in the case where the hysteresis operator is of Preisach-type. TITLE: "A parabolic nonlinear initial value problem with data given by measures" ABSTRACT:We consider equations of the form $(*) u_t - Laplacian(u) +|u|^{q-1}u = 0$ with $q>1$. We discuss the notion of initial trace for arbitrary positive solutions of $(*)$ in $R^n x R_+$ and the corresponding initial value problems.
Professor Universite Paris 10 Nanterre, France ABSTRACT: Taking into account the pressure dependance in the state law for the ocean's density, we show how to introduce additional pressure terms in the primitive equations for the ocean. Then, we prove an existence result for the deduced system of PDE.
Professor University of Napoli "Federico II" ABSTRACT: We prove a generalization to the vector-valued case of the Coifman-Rochberg-Weiss Theorem about commutators with BMO functions. This result will be applied to get Lp estimates for linear elliptic systems with BMO coefficients.
Professor Analyse Numerique 55-65 5 Place Jussieu Boite 187 University Paris 6 75252 France ABSTRACT: Constructive Solid Geometry (CSG) in CAD leads to Domain Decompositions which are based on primitive shapes, as briefly explained in the introduction below. A special role is played by "holes", which can be viewed in several different ways. We combine this remark with the method of Virtual Controls. In one approach the distributions resulting from the decomposition and Green's formula are thought of as virtual controls -- in another the virtual controls are introduced a priori with support in the holes, or outside the domain, or in the intersections of the domains. They are then chosen so as to (approximately) satisfy the Boundary Conditions, which is possible by virtue of approximate controlability results. These ideas are presented here on an example which is certainly not the most general one but which is sufficient to show how everything extends to very many other situations. This research is in collaboration with J.L. Lions.
Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213 ABSTRACT: We study the asymptotic behaviour of the following nonlinear problem: $$\left\{ \begin{array}{lll} -div(a( Du_h))+ \vert u_h\vert^{p-2}u_h =f\ \ {\rm in }\ \Omega_h,\\ a( Du_h)\cdot\nu = 0\ \ {\rm on }\ \partial\Omega_h, \end{array} \right. $$ with $a$ monotone, in a domain $\Omega_h$ of $R^n$ whose boundary $\partial\Omega_h$ contains an oscillating part with respect to $h$ when $h$ tends to $\infty$. The oscillating boundary is defined by a set of cylinders with axis $0x_n$ that are $h^{-1}$-periodically distributed. We prove that the limit problem in the domain corresponding to the oscillating boundary identifies with a diffusion operator with respect to $x_n$ coupled with an algebraic problem for the limit fluxes.
Professor California Institute of Technology Pasadena, CA ABSTRACT: In recent years many interesting properties of alloys (like InTl or CuAlNi) have been successfully explained by minimization of a suitable free energy functional. The mechanical properties of these materials are related to certain convex hulls of the zero set $K$ of the underlying energy density. In this talk we discuss these hulls for the simplest case of physical interest, i.e. for $K=SO(3)U_1\cup SO(3)U_2$. This is joint work with B. Kirchheim, S. Mueller and V. Sverak.
Professor Mathematical Sciences, Peking University, P.R.CHINA, currently visiting Department of Mathematics, Penn. State University, University Park, PA. ABSTRACT: Rotational transformation method, mesh transformation method, periodic transformation method and other numerical techniques for the computation of crystalline microstructures are presented. Numerical examples are given to show the effectiveness of the methods.
Professor Department of Aerospace Engineering and Mechanics University of Minnesota 107 Akerman Hall 110 Union Street SE Minneapolis, Minnesota 55455 ABSTRACT: In a typical strain test, shape memory wires exhibit a characteristic hardenning on the stage of microstructure development. In experiments, the degree of hardening is often related to the defectiveness or the inhomogeneity of the sample. We suggest a new interpretation of the hardening in martensites as the effect of entropic, rubber-like elasticity. We assume that the sample can be represented as an ensemble of ideal sub-samples and adopt a canonical distribution wich introduces a new parameter with the dimension of temperature. This parameter characterizes the degree of defectiveness of the sample. At zero effective temperature we recover the results of the classical energy minimization. At finite temperatures the usual statistical averaging produces the desired hardenning as a result of strain dependence of the configurational entropy. This effect is intimately related to the non-convexity of the energy and is absent if the energy is quadratic.
Professor Ecole Polytechnique Dipartimento di Ingeneria Civile Universita di Roma "Tor Vergata" ABSTRACT: TBA
Professor Ecole Polytechnique Palaiseau, France ABSTRACT: TBA |