A two-point boundary-value problem for the axial shear of hardening isotropic incompressible nonlinearly elastic materials

SIAM Journal on Applied Mathematics

Volumn 62, Issue 5, 2002, Pages 1712-1727

  Horgan, Cornelius O ^a   Saccomandi, Giuseppe ^a   Sgura, Ivonne ^a  

^a UNIVERSITY OF VIRGINIA   (United States)

Author keywords

Axial shear; Incompressible hyperelastic materials; Limiting chain extensibility; Material hardening; Power law materials; Rubber like and biological materials

Indexed keywords

BOUNDARY LAYERS; BOUNDARY VALUE PROBLEMS; COMPRESSIBILITY OF SOLIDS; COMPUTER SIMULATION; ELASTICITY; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PRESSURE EFFECTS; SHEAR STRESS; STRAIN HARDENING; TENSORS;

ISOTROPIC INCOMPRESSIBLE HYPERELASTIC MATERIALS; POWER-LAW MATERIALS;

PROBLEM SOLVING;

EID: 0036588812     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139901391963     Document Type: Article

References (27)

1. E.M. Arruda and M.C. Boyce, A three dimensional constitutive model for the large deformation stretch behavior of rubber elastic materials, J. Mech. Phys. Solids, 41 (1993), pp. 389-412.
2. U.M. Ascher, R.M.M. Mattheij, and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Appl. Math. 13, SIAM, Philadelphia, 1995.
3. M.F. Beatty, Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues-with examples, Appl. Mech. Rev., 40 (1987), pp. 1699-1733.
4. M.C. Boyce, Direct comparison of the Gent and the Arruda-Boyce constitutive models of rubber elasticity, Rubber Chem. Tech., 69 (1996), pp. 781-785.
5. L. Brugnano and D. Trigiante, A new mesh selection strategy for ODEs, Appl. Numer. Math., 24 (1997), pp. 1-21.
6. L. Brugnano and D. Trigiante, Solving ODEs by Linear Multistep Formulae: Initial and Boundary Value Methods, Gordon and Breach, London, 1998.
7. J.R. Cash and M.H. Wright, A deferred correction method for nonlinear two-point boundary value problems: Implementation and numerical evaluation, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 971-989.
8. E.J. Dean, An inexact Newton method for nonlinear two-point boundary-value problems, J. Optim. Theory Appl., 75 (1992), pp. 471-486.
9. W.H. Enright and P.H. Muir, Runge-Kutta software with defect control for boundary value ODEs, SIAM J. Sci. Comput., 17 (1996), pp. 479-497.
10. R.L. Fosdick and B.G. Kao, Transverse deformations associated with rectilinear shear in elastic solids, J. Elasticity, 8 (1978), pp. 117-142.
11. A.N. Gent, A new constitutive relation for rubber, Rubber Chem. Tech., 69 (1996), pp. 59-61.
12. C.O. Horgan, Anti-plane shear deformations in linear and nonlinear solid mechanics, SIAM Rev., 37 (1995), pp. 53-81.
13. C.O. Horgan and C. Saccomandi, Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility, J. Elasticity, 56 (1999), pp. 159-170.
14. C.O. Horgan and G. Saccomandi, Pure axial shear of isotropic incompressible nonlinearly elastic materials with limiting chain extensibility, J. Elasticity, 57 (1999), pp. 307-319.
15. C.O. Horgan and G. Saccomandi, Large deformations of a rotating solid cylinder for non-Gaussian isotropic, incompressible hyperelastic materials, J. Appl. Mech., 68 (2001), pp. 115-117.
16. C.O. Horgan and O. Saccomandi, Pure azimuthal shear of isotropic, incompressible hyperelastic materials with limiting chain extensibility, Internat. J. Nonlinear Mech., 36 (2001), pp. 465-475.
17. C.O. Horgan and G. Saccomandi, Anti-plane shear deformations for non-Gaussian isotropic, incompressible hyperelastic materials, Proc. Roy. Soc. London Set. A, 457 (2001), pp. 307-319.
18. C.O. Horgan and G. Saccomandi, Helical shear for hardening generalized neo-Hookean elastic materials, Z. Angew. Math. Phys., 53 (2002), in press.
19. X. Jiang and R.W. Ogden, Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material, Internat. J. Non-Linear Mech., 35 (2000), pp. 361-369.
20. J.K. Knowles, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids, Internat. J. Fracture, 13 (1977), pp. 611-639.
21. F. Mazzia and I. Soura, Numerical approximation of nonlinear BVPs by means of BVMs, Appl. Numer. Math., to appear.
22. R.W. Ogden, Non-Linear Elastic Deformations, Ellis Horwood, Chichester, 1984; reprinted by Dover, New York, 1997.
23. E. Pucci and G. Saccomandi, Some remarks on the Cent model of rubber elasticity, in Proceedings of the 1st Canadian Conference on Nonlinear Solid Mechanics, E.M. Croitoro, ed., University of Victoria Press, Victoria, BC, Canada, 1999, pp. 163-172.
24. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 1992.
25. R.S. Rivlin, Large elastic deformations of isotropic materials VI: Further results in the theory of torsion, shear and flexure, Philos. Trans. Roy. Soc. London Sar. A, 242 (1949), pp. 173-195.
26. I. Sgura, Approssimazione numerica di ODE con metodi BVM: Proprietà, Applicazioni e Confronti, Ph.D. thesis, Dipartimento di Matematica, Università di Pisa, Pisa, Italy, 1996.
27. J.P. Zhang and K.R. Rajagopal, Some inhomogeneous motions and deformations within the context of a nonlinear elastic solid, Internat. J. Engrg. Sci., 30 (1992), pp. 919-938.