Ot. Bruhns et al., Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky's logarithmic strain tensor, P ROY SOC A, 457(2013), 2001, pp. 2207-2226
Citations number
43
Categorie Soggetti
Multidisciplinary
Journal title
PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Hencky's strain-energy function for finite isotropic elasticity is obtained
by the replacement of the infinitesimal strain measure occurring in the cl
assical strain-energy function of infinitesimal isotropic elasticity with t
he Hencky or logarithmic strain measure. It has been shown recently by Anan
d that this simple strain-energy function, with two classical Lame elastic
constants, is in good agreement with a wide class of materials for moderate
ly large deformations. Very recently, it has been shown by these authors th
at the hyperelastic relation with the foregoing Hencky strain-energy functi
on may enter as a basic constituent into the Eulerian rate formulation of f
inite elastoplasticity for metals, etc. Now it is commonly used in finite-e
lement method (FEM) computations and in commercial packets of FEM codes, et
c. For this useful strain-energy function, there is a need to study the res
trictions and consequences resulting from certain well-founded constitutive
inequality conditions. Here, we consider the well-known Legendre-Hadamard,
or the ellipticity, condition. We first derive simple, explicit necessary
and sufficient conditions for ellipticity in terms of the principal stretch
es. Then we determine the largest common region for ellipticity in the prin
cipal stretch space, which applies to all Hencky strain-energy functions wi
th non-negative Lame constants. In particular, we find out the largest cube
contained in the common region just mentioned. We prove that the Hencky st
rain-energy function fulfils the Legendre-Hadamard condition whenever every
principal stretch falls within the range [alpha, (3)roote], where the lowe
r bound alpha = 0.21162... is the unique root of a certain transcendental e
quation involving the natural logarithmic function, and e = 2.718 28..., in
the upper bound, is the base of the natural logarithm. The range mentioned
above, i.e. [0.21162, 1.395 61], covers the range [0.7,1.3] set by Anand f
or moderately large deformations. Moreover, it is shown that Hencky's strai
n-energy function obeys the well-known Baker-Ericksen inequality and Hill's
inequality over the whole range of deformations.