Condition de préservation de l’orientation
Application en hyperélasticité compressible
Keywords:
orientation preservation, compressible hyperelasticity, Blatz-Ko model, Newton- Raphson with arc length controlAbstract
For the Blatz-Ko compressible hyperelastic model, it was established that an orientation preservation defect leds to the divergence of Newton-Raphson algorithm. It is thus significant to have criterion in order to detect such a defect during a numerical calculation. The usual criterion, which relates to the sign of the deformation gradient matrix F determinant, is not always sufficient. Indeed, a surface reversal can appear with a positive determinant of F. One then proposes a more effective criterion, based on the change of sign of F eigenvalues between the reference and the deformed configurations. The application of this criterion is illustrated with the Blatz-Ko model, in the case of a specimen subjected to a tensile or compressive loading.
Downloads
References
[ARO 98] ARON M., CHRISTOPHER C. WANG Y., “On the straightening of compressible,
nonlinearly elastic, annular cylindrical sectors”, Mathematics and Mechanics of Solids,
vol. 3, p. 131-145, 1998.
[BLA 62] BLATZ P.J. AND KO W.L., “Application of finite elastic theory to the deformation
of rubbery materials”, Transactions of the Society of Rheology, vol. 6, p. 223-251, 1962.
[CIA 85] CIARLET P.G., Elasticité tridimensionnelle, Masson, Collection RMA, 1985.
[CRIS 81] CRISFIELD M.A., “A fast incremental/iterative solution procedure that handles
snap-through”, Comput. Struct., 13, 55-62, 1981.
[FEN 02] FENG Z.Q., CROS J.-M., “FER/SUBDomain: an integrated environment for finite
element analysis using object-oriented approach”, Mathematical Modelling and
Numerical Analysis, to appear, 2002.
[FOR 87] FORDE W.R.B. and STIEMER S.F., “Improved arc length orthogonality methods for
nonlinear finite element analysis”, Computer & Structures, vol. 27, n° 5, p. 625-630,
[HOR 95] HORGAN C.O. and POLIGNONE D.A., “A note on the pure torsion of a circular
cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy”,
Journal of Elasticity, vol. 37, p. 167-178, 1995.
[HOR 96] HORGAN C.O., “Remarks on ellipticity for the generalized Blatz-Ko constitutive
model for a compressible nonlinearly elastic solid”, Journal of Elasticity, vol. 42, p. 165-
, 1996.
[PEY 01] PEYRAUT F., LABED N., « Préservation de l’orientation et convergence de Newton-
Raphson avec le modèle hyperélastique compressible de Blatz-Ko », Revue Européenne
des Eléments Finis, n° 5, vol. 10, 2001.
[RAM 80] RAMM E., “Strategies for tracing the nonlinear response near limit points”, In
Europe-U.S. Workshop, nonlinear finite element analysis in structural mechanics, Ruhr-
Universität Bochum, Germany (Edited by W. Wunderlich, E. Stein and K.J. Bathe),
p. 63-89, 1980.
[WIN 95] WINEMAN A.S. and WALDRON JR W.K., “Normal stress effects induced during
circular shear of a compressible non-linear elastic cylinder”, Int. J. Non-Linear
Mechanics, vol. 30, n° 3, p. 323-339, 1995.
