Mécanique non linéaire - LMEE

Comput Mech (2007) 40:1053–1060
DOI 10.1007/s00466-007-0163-0
ORIGINAL PAPER
A material-independent algorithm for preserving of the orientation
of the spatial basis attached to deforming medium
François Peyraut · Zhi-Qiang Feng · Nadia Labed
Received: 6 October 2006 / Accepted: 18 January 2007 / Published online: 15 February 2007
© Springer-Verlag 2007
Abstract The aim of the paper is to propose an
algorithm to satisfy the orientation-preserving condition
with hyperelastic materials. This algorithm is shown to
be applicable for many material models. Its efficiency is
assessed by numerical examples involving the Blatz–Ko,
the Ogden and the Gent models.
Keywords Orientation-preserving condition · Finite
element analysis · Compressible hyperelasticity
1 Introduction
Many industrial applications are concerned by foamlike
or rubber-like materials. The tire technology is for
example one of the main application fields of rubber-like
shells. Such materials are also used to improve the performance
of safety glass by using interlayers to join the
splinters in the case of a crash [4]. Another application
for the automotive industry is the use of polyurethane
foams for a better comfort of car seats [18]. All these
F. Peyraut
LERMPS, Université de Technologie de Belfort-Montbéliard,
90010 Belfort, France
e-mail: francois.peyraut@utbm.fr
Z.-Q. Feng (B)
Laboratoire de Mécanique d’Evry,
Université d’Evry-Val d’Essonne,
40 rue du Pelvoux, 91020 Evry, France
e-mail: feng@iup.univ-evry.fr
N. Labed
Laboratoire Mécatronique3M,
Université de Technologie de Belfort-Montbéliard,
90010 Belfort, France
e-mail: nadia.labed@utbm.fr
materials are highly non linear and require specific and
robust techniques to perform numerical computation.
In the last decades, many attempts have been made to
solve non linear problems, including hyperelastic materials,
with the finite element method. A very detailed
review on the finite element formulation for non linear
analysis, including two and three dimensional problems
and involving isotropic, orthotropic, rubber-like
and elasto-plastic materials, has been provided by Sussman
and Bathe [21]. But some numerical topics related
to the compressible hyperelasticity are still questionable.
For example, in the case of the Blatz–Ko material,
the loss of ellipticity leads to numerical problems near
the breaking loading values [14]. More generally, existence
and uniqueness of the solution can be discussed
on the issue of ellipticity, polyconvexity and convexity
of the energy density required to model the material
behavior [3,10,20]. Moreover, it has been proven that
the Newton–Raphson algorithm diverges if the orientation-preserving
condition is not satisfied [15]. To prevent
divergence in case of an orientation-preserving loss, an
algorithm based on the eigenvalues of the deformation
gradient has been proposed by the authors [16]. By using
eigenvalues instead of the determinant, this algorithm
can deal with any geometrical case (volume, area or
line) while the classical condition based on the determinant
only works for preserving volumes. By considering
the Blatz–Ko hyperelastic model [2], this algorithm
has already been found efficient for various geometrical
cases, boundary conditions and applied loads [13,15].
The aim of this paper is to establish the fact that this
algorithm, previously applied to the single case of the
Blatz–Ko hyperelastic model, is indeed efficient for any
kind of behavior laws. We actually prove its independency
on material models because it is found to be free of