The analytical expressions for the elements of the Jacobian matrix of the tensor-matrix system of FEM equations that describes the large deformations of an incompressible elastic body are derived using derivatives with respect to a tensor argument. The results are obtained for the general three-dimensional case, including the case of plane strain. The stress–strain state of a hollow square prism turned inside out is determined with a numerical method using the Jacobian matrix
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
D. V. Berezhnoi, V. N. Paimushin, and V. I. Shalashilin, “Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements,” Mech. Solids, 44, No. 6, 837–851 (2009).
V. V. Galishnikova, “An expansion method for the continuation of a solution at singular points,” Vest. RUDN., Ser. Mat. Inform. Fiz., No. 2, 123–132 (2011).
A. N. Guz, Stability of Elastic Bodies Subject to Finite Deformations [in Russian], Naukova Dumka, Kyiv (1973).
J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey (1983).
A. A. Zelenina and L. M. Zubov, “One-dimensional deformations of nonlinearly elastic micropolar bodies,” Mech. Solids, 45, No. 4, 575–582 (2010).
L. M. Zubov and S. I. Moiseenko, “Stability of the equilibrium of an inverted elastic sphere,” Izv. RAN, Mekh. Tverd. Tela, No. 5, 148–155 (1983).
A. I. Lurie, Nonlinear Theory of Elasticity, North-Holland, Amsterdam (1990).
J. T. Oden, Finite Elements of Nonlinear Continua, McGraw-Hill, New York (1971).
V. P. Sukhorukov, “Aerodynamic drag of a pipeline laid by turning inside out,” Nauk. Visn. Nats. Girn. Univ., No. 9, 72–74 (2009).
J. Aernouts, I. Couckuyt, K. Crombecq, and J. J. J. Dirckx, “Elastic characterization of membranes with a complex shape using point indentation measurements and inverse modelling,” Int. J. Eng. Sci., 48, No. 6, 599–611 (2010).
E. Chamberland, A. Fortin, and M. Fortin, “Comparison of the performance of some finite element discretizations for large deformation elasticity problems,” Compos. Struct., 88, No. 11–12, 664–673 (2010).
V. V. Chekhov, “Matrix FEM equation describing the large-strain deformation of an incompressible material,” Int. Appl. Mech., 46, No. 10, 1147–1153 (2010).
F. Conrad, K. Ehrmann, J. D. Choo, and B. A. Holden, “Finite element modeling of inverted (inside out) soft contact lenses,” Trans. ASME, J. Medic. Dev., 4, No. 2 (2010).
P. B. Goncalves, D. Pamplona, and S. R. X. Lopes, “Finite deformations of an initially stressed cylindrical shell under internal pressure,” Int. J. Mech. Sci., 50, No. 1, 92–103 (2008).
GSL Reference Manual. http://www.gnu.org/software/gsl/manual/gsl-ref.html.
Y.-M. Huang, “Finite element analysis of tube inversion process with radiused dies,” Int. J. Adv. Manuf. Tech., 26, No. 9–10, 991–998 (2005).
G. Karami, N. Grundman, N. Abolfathi, A. Naik, and M. Ziejewski, “A micromechanical hyperelastic modeling of brain white matter under large deformation,” J. Mech. Behavior Biomed. Mater., 2, 243–254 (2009).
V. A. Maksimyuk, E. A. Storozhuk, and I. S. Chernyshenko, “Using mesh-based methods to solve nonlinear problems of statics for thin shells,” Int. Appl. Mech., 45, No. 1, 32–56 (2009).
N. Promma, B. Raka, M. Grediac, E. Toussaint, J.-B. Le Cam, X. Balandraud, and F. Hild, “Application of the virtual fields method to mechanical characterization of elastomeric materials,” Int. J. Solids Struct., 46, No. 3–4, 698–715 (2009).
M. Sasso, G. Palmieri, G. Chiappini, and D. Amodio, “Characterization of hyperelastic rubber-like materials by biaxial and uniaxial stretching tests based on optical methods,” Polymer Testing, 27, 995–1004 (2008).
N. P. Semenyuk, V. M. Trach, and V. V. Ostapchuk, “Nonlinear axisymmetric deformation of anisotropic spherical shells,” Int. Appl. Mech., 45, No. 10, 1101–1111 (2009).
C. A. C. Silva and M. L. Bittencourt, “Structural shape optimization of 3D nearly-incompressible hyperelasticity problems,” Latin Amer. J. Solids Struct., 5, No. 2, 129–156 (2008).
Y. Zhu, X. Y. Luo, and R. W. Ogden, “Nonlinear axisymmetric deformations of an elastic tube under external pressure,” Europ. J. Mech., A/Solids, 29, No. 2, 216–229 (2010).
Th. Zisis, V. I. Zafiropoulou, and A. E. Giannakopoulos, “The adhesive contact of a flat punch on a hyperelastic substrate subject to a pull-out force or a bending moment,” Mech. Mater., 43, No. 1, 1–24 (2011).
Translated from Prikladnaya Mekhanika, Vol. 49, No. 6, pp. 37–43, November–December 2013.
About this article
Cite this article
Chekhov, V.V. Modification of the Finite-Element Method to Apply to Problems of the Equilibrium of Bodies Subject to Large Deformations. Int Appl Mech 49, 658–664 (2013). https://doi.org/10.1007/s10778-013-0599-1
- finite-element method
- tensor-matrix system of equations
- large deformations
- Finger strain measure
- incompressible elastic body
- tensor derivative
- Jacobian matrix
- hollow square prism turned inside out