Search

In this note, we address several issues, including some raised in recent works and commentary, related to bending measures and energies for plates and shells, and certain of their invariance properties. We discuss overlaps and distinctions in results arising from two different definitions of stretching, correct an error and citation oversights in our prior work, reiterate some of the early history of dilation-invariant bending measures, and provide additional brief observations regarding the relative size of energetic terms and the symmetrization of bending measures. A particular point of emphasis is the distinction between dilation-invariant measures and a recently introduced non-dilation-invariant measure for shells and curved rods. In the course of this discussion, we provide a simpler presentation of the elementary, but much neglected, fact that the through-thickness derivative of tangential stretch of material near the mid-surface of a thin body is the product of the mid-surface stretch and change in curvature., Comment: significant rewriting, errors corrected, references added, author order changed, acknowledgments updated

Experiments reveal that structural transitions in thin sheets are mediated by the passage of transient and stable mobile localized elastic excitations. These ``crumples'' or ``d-cones'' nucleate, propagate, interact, annihilate, and escape. Much of the dynamics occurs on millisecond time scales. Nucleation sites correspond to regions where generators of the ideal unstretched surface converge. Additional stable intermediate states illustrate two forms of quasistatic inter-crumple interaction through ridges or valleys. These interactions create pairs from which extended patterns may be constructed in larger specimens. The onset of localized transient deformation with increasing sheet size is correlated with a characteristic stable crumple size, whose measured scaling with thickness is consistent with prior theory and experiment for localized elastic features in thin sheets. We offer a new theoretical justification of this scaling., Comment: expanded data set, changes to text including rearrangements to suit journal format. contains link to videos

We derive stretching and bending energies for isotropic elastic plates and shells. Through the dimensional reduction of a bulk elastic energy quadratic in Biot strains, we obtain two-dimensional bending energies quadratic in bending measures featuring a bilinear coupling of stretches and geometric curvatures. For plates, the bending measure is invariant under spatial dilations and naturally extends primitive bending strains for straight rods. For shells or naturally-curved rods, the measure is not dilation invariant, and contrasts with previous \emph{ad hoc} postulated forms. The corresponding field equations and boundary conditions feature moments linear in the bending measures, and a decoupling of stretching and bending such that application of a pure moment results in isometric deformation of a unique neutral surface, primitive behaviors in agreement with classical linear response but not displayed by commonly used analytical models. We briefly comment on relations between our energies, those derived from a neo-Hookean bulk energy, and a commonly used discrete model for flat membranes. Although the derivation requires consideration of stretch and rotation fields, the resulting energy and field equations can be expressed entirely in terms of metric and curvature components of deformed and reference surfaces., Comment: correction appended to end of document

We propose bending energies for isotropic elastic plates and shells. For a plate, we define and employ a surface tensor that symmetrically couples stretch and curvature such that any elastic energy density constructed from its invariants is invariant under spatial dilations. This kinematic measure and its corresponding isotropic quadratic energy resolve outstanding issues in thin structure elasticity, including the natural extension of primitive bending strains for straight rods to plates, the assurance of a moment linear in the bending measure, and the avoidance of induced mid-plane strains in response to pure moments as found in some commonly used analytical plate models. Our analysis also reveals that some other commonly used numerical models have the right invariance properties, although they lack full generality at quadratic order in stretch. We further extend our result to naturally-curved rods and shells, for which the pure stretching of a curved rest configuration breaks dilation invariance; the new shell bending measure we provide contrasts with previous \emph{ad hoc} postulated forms. The concept that unifies these theories is not dilation invariance, but rather through-thickness uniformity of strain as a definition of pure stretching deformations. Our results provide a clean basis for simple models of low-dimensional elastic systems, and should enable more accurate analytical probing of the structure of singularities in sheets and membranes., Comment: final minor changes

Nonlinear models for pattern evolution by ion beam sputtering on a material surface present an ongoing opportunity for new numerical simulations. A numerical analysis of the evolution of preexisting patterns is proposed to investigate surface dynamics, based on a 2D anisotropic damped Kuramoto-Sivashinsky equation, with periodic boundary conditions. A finite-difference semi-implicit time splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is analyzed with time step and grid spacing tests for the pattern evolution, and the Method of Manufactured Solutions has been used to verify the proposed scheme. Ripples and hexagonal patterns were obtained from a monomodal initial condition for certain values of the damping coefficient, while spatiotemporal chaos appeared for lower values. The anisotropy effects on pattern formation were studied, varying the angle of incidence of the ion beam with respect to the irradiated surface. Analytical discussions are based on linear and weakly nonlinear analysis.

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed., Comment: Minor edits

Computational modeling of pattern formation in nonequilibrium systems is a fundamental tool for studying complex phenomena in biology, chemistry, materials science and engineering. The pursuit for theoretical descriptions of some among those physical problems led to the Swift-Hohenberg equation (SH3) which describes pattern selection in the vicinity of instabilities. A finite differences scheme, known as Stabilizing Correction (Christov & Pontes; 2001 DOI: 10.1016/S0895-7177(01)00151-0), developed to integrate the cubic Swift-Hohenberg equation in two dimensions, is reviewed and extended in the present paper. The original scheme features Generalized Dirichlet boundary conditions (GDBC), forcings with a spatial ramp of the control parameter, strict implementation of the associated Lyapunov functional, and second order representation of all derivatives. We now extend these results by including periodic boundary conditions (PBC), forcings with gaussian distributions of the control parameter and the quintic Swift-Hohenberg (SH35) model. The present scheme also features a strict implementation of the functional for all test cases. A code verification was accomplished, showing unconditional stability, along with second order accuracy in both time and space. Test cases confirmed the monotonic decay of the Lyapunov functional and all numerical experiments exhibit the main physical features: highly nonlinear behaviour, wavelength filter and competition between bulk and boundary effects.