Introduction to Functional Analysis.

A Glimpse at Set Theory. The Real Numbers. The Topology of Cartesian Spaces. Convergence. Continuous Functions. Functions of One Variable. Infinite Series. Differentiation in RP Integration in RP.

The elements of real analysis (2nd edition), by Robert G. Bartle. Pp xv, 480. £10. 1976. SBM 0 471 05464 X (Wiley)

We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two-dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions. (C) 2004 Wiley Periodicals, Inc.

Exact steady periodic water waves with vorticity

The effect of growth in the stability of elastic materials is studied. From a stability perspective, growth and resorption have two main effects. First a change of mass modifies the geometry of the system and possibly the critical lengths involved in stability thresholds. Second, growth may depend on stress but also it may induce residual stresses in the material. These stresses change the effective loads and they may both stabilize or destabilize the material. To discuss the stability of growing elastic materials, the theory of finite elasticity is used as a general framework for the mechanical description of elastic properties and growth is taken into account through the multiplicative decomposition of the deformation gradient. The formalism of incremental deformation is adapted to include growth effects. As an application of the formalism, the stability of a growing neo-Hookean incompressible spherical shell under external pressure is analyzed. Numerical and analytical methods are combined to obtain explicit stability results and to identify the role of mechanical and geometric effects. The importance of residual stress is established by showing that under large anisotropic growth a spherical shell can become spontaneously unstable without any external loading.

http://www.phys.ens.fr/~benamar/paper/2005-JMPS(growth).pdf

Growth and instability in elastic tissues

I. Fundamental concepts and examples.- 1. Hyperbolicity, genuine nonlinearity, and entropies.- 2. Shock formation and weak solutions.- 3. Singular limits and the entropy inequality.- 4. Examples of diffusive-dispersive models.- 5. Kinetic relations and traveling waves.- 1. Scalar Conservation Laws.- II. The Riemann problem.- 1. Entropy conditions.- 2. Classical Riemann solver.- 3. Entropy dissipation function.- 4. Nonclassical Riemann solver for concave-convex flux.- 5. Nonclassical Riemann solver for convex-concave flux.- III. Diffusive-dispersive traveling waves.- 1. Diffusive traveling waves.- 2. Kinetic functions for the cubic flux.- 3. Kinetic functions for general flux.- 4. Traveling waves for a given speed.- 5. Traveling waves for a given diffusion-dispersion ratio.- IV. Existence theory for the Cauchy problem.- 1. Classical entropy solutions for convex flux.- 2. Classical entropy solutions for general flux.- 3. Nonclassical entropy solutions.- 4. Refined estimates.- V. Continuous dependence of solutions.- 1. A class of linear hyperbolic equations.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- 2. Systems of Conservation Laws.- VI. The Riemann problem.- 1. Shock and rarefaction waves.- 2. Classical Riemann solver.- 3. Entropy dissipation and wave sets.- 4. Kinetic relation and nonclassical Riemann solver.- VII. Classical entropy solutions of the Cauchy problem.- 1. Glimm interaction estimates.- 2. Existence theory.- 3. Uniform estimates.- 4. Pointwise regularity properties.- VIII. Nonclassical entropy solutions of the Cauchy problem.- 1. A generalized total variation functional.- 2. A generalized weighted interaction potential.- 3. Existence theory.- 4. Pointwise regularity properties.- IX. Continuous dependence of solutions.- 1. A class of linear hyperbolic systems.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- X. Uniqueness of entropy solutions.- 1. Admissible entropy solutions.- 2. Tangency property.- 3. Uniqueness theory.- 4. Applications.

Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves

Bifurcation of solution branches in static or steady problems of nonlinear mechanics is often associated with the underlying symmetry of the physical system. The use of group-theoretic methods in local bifurcation theory for problems with symmetry is well known. In this paper it is shown that the exploitation of symmetry via group invariance also yields an efficient computational approach to global bifurcation problems. These techniques are illustrated in the analysis of a lattice-dome structure with hexagonal symmetry. The methodology leads to a drastic reduction in numerical effort in the determination of several global solution branches, and enables the accurate computation of numerous singular points.

A group-theoretic approach to computational bifurcation problems with symmetry

We consider the wrinkling of highly stretched, thin rectangular sheets—a problem that has attracted the attention of several investigators in recent years, nearly all of which employ the classical Foppl–von Karman (F–K) theory of plates. We first propose a rational model that correctly accounts for large mid-plane strain. We then carefully perform a numerical bifurcation/continuation analysis, identifying stable solutions (local energy minimizers). Our results in comparison to those from the F–K theory (also obtained herewith) show: (i) For a given fine thickness, only a certain range of aspect ratios admit stable wrinkling; for a fixed length (in the highly stretched direction), wrinkling does not occur if the width is too large or too small. In contrast, the F–K model erroneously predicts wrinkling in those very same regimes for sufficiently large applied macroscopic strain. (ii) When stable wrinkling emerges as the applied macroscopic strain is steadily increased, the amplitude first increases, reaches a maximum, decreases, and then returns to zero again. In contrast, the F–K model predicts an ever-increasing wrinkling amplitude as the macroscopic strain is increased. We identify (i) and (ii) as global isola-center bifurcations—in terms of both the macroscopic-strain parameter and an aspect-ratio parameter. (iii) When stable wrinkling occurs, for fixed parameters, the transverse pattern admits an entire orbit of neutrally stable (equally likely) possibilities: These include reflection symmetric solutions about the mid-plane, anti-symmetric solutions about the mid-line (a rotation by π radians about the mid-line leaves the wrinkled shape unchanged) and a continuously evolving family of shapes “in-between”, say, parametrized by an arbitrary phase angle, each profile of which is neither reflection symmetric nor anti-symmetric.

Wrinkling Behavior of Highly Stretched Rectangular Elastic Films via Parametric Global Bifurcation

We treat certain classes of material symmetry in straight nonlinearly elastic rods in the presence of a uniform helical microstructure. In particular, we consider rods with chirality or “handedness...

Material Symmetry and Chirality in Nonlinearly Elastic Rods

We consider large eigenvalue problems for skeletal structures with symmetry. We present an algorithm, based upon a novel combination of group-theoretic ideas and substructuring techniques, that block-diagonalizes such systems exactly and efficiently. The procedure requires only the structural matrices of a repeating substructure, together with the symmetry modes, which are obtained from symmetry considerations alone. We first present a simple paradigmatic example and then follow with several non-trivial applications involving large lattice structures.

Exact block diagonalization of large eigenvalue problems for structures with symmetry

Basic characteristics associated with snapthrough, bifurcation and postbuckling behavior of structures are illuminated and clarified. This is accomplished by presenting and examining in detail exac...

Timothy J. Healey

Papers

A group-theoretic approach to computational bifurcation problems with symmetry

Wrinkling Behavior of Highly Stretched Rectangular Elastic Films via Parametric Global Bifurcation

Material Symmetry and Chirality in Nonlinearly Elastic Rods

Exact block diagonalization of large eigenvalue problems for structures with symmetry

Snap‐Through and Bifurcation in a Simple Structure