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Editors' Note / Note de la rédaction
It was with more than a little trepidation that I deposited my doctoral thesis at the University of Ottawa this past September.
These feelings were, of course, accompanied by a great deal of satisfaction and relief at having (nearly) completed a project that I
have been working on for years now, and for which I remain passionate. My trepidation stems from the usual feelings of
anxiety that come with submitting a thesis: I pray to Clio for a speedy administrative process and for a timely and successful
defence. I hope the readers are satisfied with the work, but I also look forward to discussing it in detail with five different people who have actually read it! /Jâavais envoyĂ© cet hiver les Ă©tudiants de mon cours de mĂ©thode historique Ă BibliothĂšque et archives Canada. JâĂ©tais alors enthousiasmĂ© par le succĂšs de lâactivitĂ© : les Ă©tudiants sâĂ©taient lancĂ©s Ă la recherche de documents couvrant une pĂ©riode historique et un thĂšme spĂ©cifique choisi en classe. La phase de recherche et de consultation des
documents archivistiques avait ravi les étudiants, qui étaient fascinés par le processus de recherche documentaire et la facilité relative avec laquelle ils
avaient eu accĂšs aux documents
Geometry of logarithmic strain measures in solid mechanics
We consider the two logarithmic strain measureswhich are isotropic invariants of the
Hencky strain tensor , and show that they can be uniquely characterized
by purely geometric methods based on the geodesic distance on the general
linear group . Here, is the deformation gradient,
is the right Biot-stretch tensor, denotes the principal
matrix logarithm, is the Frobenius matrix norm, is the
trace operator and is the -dimensional deviator of
. This characterization identifies the Hencky (or
true) strain tensor as the natural nonlinear extension of the linear
(infinitesimal) strain tensor , which is the
symmetric part of the displacement gradient , and reveals a close
geometric relation between the classical quadratic isotropic energy potential
in
linear elasticity and the geometrically nonlinear quadratic isotropic Hencky
energywhere
is the shear modulus and denotes the bulk modulus. Our deduction
involves a new fundamental logarithmic minimization property of the orthogonal
polar factor , where is the polar decomposition of . We also
contrast our approach with prior attempts to establish the logarithmic Hencky
strain tensor directly as the preferred strain tensor in nonlinear isotropic
elasticity
The Productivity Slowdown, Measurement Issues, and the Explosion of Computer Power
macroeconomics, Productivity Slowdown, Measurement Issues, Computer Power
An ellipticity domain for the distortional Hencky-logarithmic strain energy
We describe ellipticity domains for the isochoric elastic energy for ,
where for . Here, is the deviatoric part of the
logarithmic strain tensor . For we identify the maximal
ellipticity domain, while for we show that the energy is
Legendre-Hadamard elliptic in the set , which is similar to the
von-Mises-Huber-Hencky maximum distortion strain energy criterion.
Our results complement the characterization of ellipticity domains for the
quadratic Hencky energy , with and
, previously obtained by Bruhns et al
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