Brittle Thin Films

A two-dimensional model for brittle thin films is obtained from a three-dimensional fracture model for elastic material. Γ -convergence techniques are used to identify the limiting effective energy as the thickness of the sample approaches zero. Accepted 27 March 2001. Online publication 9 August 2001.     

On Ginzburg-Landau Vortices of Superconducting Thin Films

In this paper, we discuss the vortex structure of the superconducting thin films placed in a magnetic field. We show that the global minimizer of the functional modelling the superconducting thin films has a bounded number of vortices when the applied magnetic field hex 〈 Hc1 ＋ K log ｜ logε｜ where Hc1 is the lower critical field of the film obtained by Ding and Du in SIAM J. Math. Anal., 2002. The locations of the vortices are also given.     

Generalized Harmonic Functions and the Dewetting of Thin Films

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H¹Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.     

The Concertina Pattern: A Bifurcation in Ferromagnetic Thin Films

The concertina pattern is a metastable stage in the switching process of elongated thin-film elements. It is an approximately periodic structure of domains, separated by walls perpendicular to the long axis of the element. In this paper, we give arguments in favor of our claim that the period is frozen-in at nucleation, i.e., at the critical external field. In prior work, R. Cantero-Alvarez and F. Otto, Journal of Nonlinear Science (2006), we argued that there are four qualitatively different regimes for nucleation. In one of these asymptotic regimes, the unstable mode displays an oscillatory behavior in the direction of the long axis. In this work, we derive a scaling limit of the micromagnetic energy near the bifurcation point in the above regime. We also prove that the scaling limit is coercive for all values of the reduced external field. Because of this coercivity, there exists a branch of nontrivial local minimizers. Numerical minimization of the scaling limit reveals that this branch is indeed a continuous branch of concertina pattern. The scaling limit is derived by Γ-convergence of the suitably rescaled micromagnetic energy. This robust procedure combines the limit of an asymptotic parameter regime with a zoom-in in configuration space. The coercivity of the scaling limit is derived by suitable nonlinear interpolation estimates.     

Reduced Models for Ferromagnetic Thin Films with Periodic Surface Roughness

We investigate the influence of periodic surface roughness in thin ferromagnetic films on shape anisotropy and magnetization behavior inside the ferromagnet. Starting from the full micromagnetic energy and using methods of homogenization and \(\Gamma \)-convergence, we derive a two-dimensional local reduced model. Investigation of this model provides an insight into the formation mechanism of perpendicular magnetic anisotropy and uniaxial anisotropy with an arbitrary preferred direction of magnetization.     

Critical Fields in Ferromagnetic Thin Films: Identification of Four Regimes

We are interested in critical fields for ferromagnetic elements: At which strength of the external field does a branch of stationary magnetizations become unstable, and what is the unstable mode? We consider samples which are infinite in the direction of the external field and have a rectangular cross-section, of much smaller thickness than width, as an idealization of a thin film element. For this geometry Aharoni [Phys. Stat. Sol. 16, 3-42 (1966)] claims that there are only three different regimes: The unstable mode is either of coherent rotation type, of buckling type, or of curling type. We discover a large fourth parameter regime with an unstable mode displaying an oscillation in the infinite direction. We prove the existence of exactly four regimes by rigorously analyzing the scaling of the Rayleigh quotient of the Hessian of the energy functional. The parameters are the film width, the film thickness, and the exchange length.     

Effect of Time Delay of Support Propagation in Equations of Thin Films

We prove the existence of the effect of time delay of propagation of the support of strong solutions of the Cauchy problem for an equation of thin films and establish exact conditions on the behavior of an initial function near the free boundary that guarantee the appearance of this effect.     

Analysis of a Nonlinear Diffusive Amplitude Equation for Waves on Thin Films

This paper presents analytical and numerical solutions of a new amplitude equation governing long waves on thin films. At lowest order in the long-wave parameter, the equation is nondispersive and represents a balance between nonlinearity and cross-stream diffusion. Numerical solutions tracing the temporal evolution of an initially localized disturbance indicate that the aforementioned diffusion partly mitigates the tendency of the wave to break. We have also obtained a closed-form solution resembling an undular bore propagating in an oblique direction.     

Dimension Reduction for the Landau-de Gennes Model on Curved Nematic Thin Films

We use the method of \(\Gamma \)-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach in Golovaty et al. (J Nonlinear Sci 25(6):1431–1451, 2015) where we considered a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustum.     

Pinning of Vortices for a Variational Problem Related to the Superconducting Thin Films Having Variable Thickness

This paper is concerned with the minimization problem related to the superconducting thin films having variable thickness. The asymptotic behavior of the minimizers is discussed. The singularities are found to be located at the thinnest positions of the film and the vortices of u_ε are proved to be pinned near these regions.     

Thin Indicator Films to Assess the Residual Strength of a GFRP after a Local Contact Action

Mechanics of Composite Materials - Two types of thin impact indicator films were developed: made of an epoxy composite with hollow glass microspheres and of a selfadhesive thermoplastic with...     

Pseudoquaternion Geometry

We describe complex holomorphic transformations of a quaternion vector space taking left quaternion lines to left quaternion lines and real linear transformations of the quaternion plane simultaneously preserving the sets of left and right quaternion lines.     

Symplectic Geometry

Book Review

Symplectic GeometryA. T. Fomenko: Advanced Studies in Contemporary Mathematics, Volume 5, Gordon and Breach Science Publishers, 1988, 387 pp