zbMATH — the first resource for mathematics

Orientation preservation and Newton-Raphson convergence in the case of an hyperelastic sphere subjected to an hydrostatic pressure. (English) Zbl 1026.74074
Summary: We show that the incremental Newton-Raphson algorithm diverges if the orientation is not preserved. To illustrate this point, we treat the example of a compressible hyperelastic sphere subjected to hydrostatic pressure. The selected model is Blatz-Ko material. We also show that it is possible to choose an optimal step loading. This step guarantees at the same time the orientation preservation, the Newton-Raphson convergence and a minimum computing time.

74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text: DOI
[1] Aron, M.; Christopher, C.; Wang, Y., On the straightening of compressible, nonlinearly elastic, annular cylindrical sectors, Math. mech. solids, 3, 131-145, (1998) · Zbl 1001.74531
[2] Blatz, P.J.; Ko, W.L., Application of finite elastic theory to the deformation of rubbery materials, Trans. soc. rheol., 6, 223-251, (1962)
[3] Ciarlet, P.G., Elasticité tridimensionnelle, (1985), Collection RMA Masson · Zbl 0572.73027
[4] Chung, D.T.; Horgan, C.O.; Abeyaratne, R., The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials, Int. J. solids struct., 22, 12, 1557-1570, (1986) · Zbl 0603.73038
[5] Horgan, C.O.; Polignone, D.A., A note on the pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy, J. elasticity, 37, 167-178, (1995) · Zbl 0817.73008
[6] Horgan, C.O., Remarks on ellipticity for the generalized blatz – ko constitutive model for a compressible nonlinearly elastic solid, J. elasticity, 42, 165-176, (1996) · Zbl 0852.73019
[7] Peyraut, F.; Labed, N., Préservation de l’orientation et convergence de newton – raphson avec le modèle hyperélastique compressible de blatz – ko, Rev. europ. eléments finis, 10, 5, (2001) · Zbl 1054.74073
[8] Wineman, A.S.; Waldron, W.K., Normal stress effects induced during circular shear of a compressible non-linear elastic cylinder, Int. J. non-linear mech., 30, 3, 323-339, (1995) · Zbl 0853.73014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.