1,457 research outputs found
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
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
A Riemannian approach to strain measures in nonlinear elasticity
The isotropic Hencky strain energy appears naturally as a distance measure of
the deformation gradient to the set SO(n) of rigid rotations in the canonical
left-invariant Riemannian metric on the general linear group GL(n). Objectivity
requires the Riemannian metric to be left-GL(n)-invariant, isotropy requires
the Riemannian metric to be right-O(n)-invariant. The latter two conditions are
satisfied for a three-parameter family of Riemannian metrics on the tangent
space of GL(n). Surprisingly, the final result is basically independent of the
chosen parameters. In deriving the result, geodesics on GL(n) have to be
parametrized and a novel minimization problem, involving the matrix logarithm
for non-symmetric arguments, has to be solved
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