Extremal Mappings of Finite Distortion

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135425/1/plms0655.pd

A neohookean model of plates

This article is about hyperelastic deformations of plates (planar domains) which minimize a neohookean-type energy. Particularly, we investigate a stored energy functional introduced by J. M. Ball [Proc. Roy. Soc. Edinb. Sect. A, 88 (1981), pp. 315-328]. The mappings under consideration are Sobolev homeomorphisms and their weak limits. They are monotone in the sense of C. B. Morrey. One major advantage of adopting monotone Sobolev mappings lies in the existence of the energy-minimal deformations. However, injectivity is inevitably lost, so an obvious question to ask is, what are the largest subsets of the reference configuration on which minimal deformations remain injective? The fact that such subsets have full measure should be compared with the notion of global invertibility, which deals with subsets of the deformed configuration instead. In this connection we present a Cantor-type construction to show that both the branch set and its image may have positive area. Another novelty of our approach lies in allowing the elastic deformations to be free along the boundary, known as frictionless problems

Self-reported reasons for on-duty sleepiness among commercial airline pilots

Experimental and epidemiological research has shown that human sleepiness is determined especially by the circadian and homeostatic processes. The present field study examined which work-related factors airline pilots perceive as causing on-duty sleepiness during short-haul and long-haul flights. In addition, the association between the perceived reasons for sleepiness and actual sleepiness levels was examined, as well as the association between reporting inadequate sleep causing sleepiness and actual sleep-wake history. The study sample consisted of 29 long-haul (LH) pilots, 28 short-haul (SH) pilots, and 29 mixed fleet pilots (flying both SH and LH flights), each of whom participated in a 2-month field measurement period, yielding a total of 765 SH and 494 LH flight duty periods (FDPs) for analyses (FDP, a period between the start of a duty and the end of the last flight of that duty). The self-reports of sleepiness inducers were collected at the end of each FDP by an electronic select menu. On-duty sleepiness was rated at each flight phase by the Karolinska Sleepiness Scale (KSS). The sleep-wake data was collected by a diary and actigraph. The results showed that "FDP timing" and "inadequate sleep" were the most frequently reported reasons for on-duty sleepiness out of the seven options provided, regardless of FDP type (SH, LH). Reporting these reasons significantly increased the odds of increased on-duty sleepiness (KSS >= 7), except for reporting "inadequate sleep" during LH FDPs. Reporting "inadequate sleep" was also associated with increased odds of a reduced sleep-wake ratio (total sleep time/amount of wakefulnessPeer reviewe

Diffeomorphic approximation of Sobolev homeomorphisms

Every homeomorphism h : X -> Y between planar open sets that belongs to the Sobolev class W^{1,p}(X,Y), 1<p<\infty, can be approximated in the Sobolev norm by diffeomorphisms.Comment: 21 pages, 5 figure

Quasisymmetric graphs and Zygmund functions

A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class $\Lambda_*$. This relation makes it possible to use the tools of harmonic analysis to construct nontrivial examples of quasisymmetric graphs and of quasiconformal maps.Comment: 21 pages, no figure

Doubly connected minimal surfaces and extremal harmonic mappings

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.Comment: 35 pages, 2 figures. Minor edits, references adde

Quasiconvexity at the boundary and the nucleation of austenite

Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut

Mappings of least Dirichlet energy and their Hopf differentials

The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class $\bar{\mathscr H}_2(X, Y)$ of strong limits of homeomorphisms in the Sobolev space $W^{1,2}(X, Y)$, a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential $h_z \bar{h_{\bar{z}}} dz \otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h \in \bar{\mathscr H}_2(X, Y)$ is energy-minimal if and only if its Hopf-differential is analytic in $X$ and real along the boundary of $X$. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in $X$. Nevertheless, cracks are triggered only by the points in the boundary of $Y$ where $Y$ fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of $X$ toward the interior of $X$ where they eventually terminate before making a crosscut.Comment: 51 pages, 4 figure

Supplementary data for Onninen et al. Self-reported stress and stressors in tram and long-haul truck drivers

Supplementary data and figure for the research article: Self-reported stress and stressors in tram and long-haul truck drivers (Appendix A, Figure S1

Supplementary data for Onninen et al. The self-reported stress and stressors in tram and long-haul truck drivers

Supplementary data and figure for the research article: The self-reported stress and stressors in tram and long-haul truck drivers (Appendix A, Figure S1