26 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
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
Estimating the Effective Elasticity Properties of a Diamond/-SiC Composite Thin Film by 3D Reconstruction and Numerical Homogenization
The main aim of the present work is to estimate the effective elastic
stiffnesses of a two-phase diamond/-SiC composite thin film that is
fabricated by chemical vapor deposition. The parameters of linear elasticity
are determined by numerical homogenization. The database is sparse since for
the 3D volume of interest only two micrographs displaying the phase
distributions in perpendicular planes are available; micrographs each of a
cross-section and the surface of the thin film. A representative volume element
(RVE) is reconstructed by an optimization software and by means of identified
material symmetries in 2D of the specimen. The elastic homogenization results
indicate that the two-phase diamond/-SiC composite exhibits the behavior
of transverse isotropy, for which the set of six independent material
parameters is identified