Numerical studies on the identification of the material parameters of Rivlin's hyperelasticity using tension-torsion tests

SummaryThis paper deals with the identification of material parameters of elasticity relations based on Rivlin's hyperelasticity for incompressible material response, where the free energy evolves as a polynomial in the first and second invariant of the right Cauchy-Green tensor. This elasticity relation has the advantage of incorporating the material parameters linearily. The numerical studies are applied to tension, torsion and combined tension-torsion tests with cylindrical carbon black-filled rubber specimens represented in Haupt and Sedlan [1] and [2]. In the identification process the analytical solution of the resulting boundary value problem leads to a linear least square solution. In this article attention is focused on the numerical solution of several models proposed in the literature and their behavior for both a large and a small number of test data.

[1]  A. G. James,et al.  Strain energy functions of rubber. I. Characterization of gum vulcanizates , 1975 .

[2]  B. G. Kao,et al.  On the Determination of Strain Energy Functions of Rubbers , 1986 .

[3]  Ray W. Ogden,et al.  Nonlinear Elastic Deformations , 1985 .

[4]  S. Reese,et al.  A theory of finite viscoelasticity and numerical aspects , 1998 .

[5]  P. Haupt,et al.  Viscoplasticity of elastomeric materials: experimental facts and constitutive modelling , 2001 .

[6]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[7]  D. W. Saunders,et al.  Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber , 1951, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[8]  D. J. Montgomery,et al.  The physics of rubber elasticity , 1949 .

[9]  R. Rivlin.  LARGE ELASTIC DEFORMATIONS OF ISOTROPIC MATERIALS. I. FUNDAMENTAL CONCEPTS , 1997 .

[10]  M. Mooney.  A Theory of Large Elastic Deformation , 1940 .

[11]  R. Ogden.  Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  Stephen Wolfram,et al.  Mathematica: a system for doing mathematics by computer (2nd ed.) , 1991 .

[13]  Gene H. Golub,et al.  Matrix computations , 1983 .

[14]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[15]  A. Benjeddou,et al.  Determination of the Parameters of Ogden's Law Using Biaxial Data and Levenberg-Marquardt- Fletcher Algorithm , 1993 .

[16]  Nicholas W. Tschoegl.  Constitutive equations for elastomers , 1971 .

[17]  A. Bunse-Gerstner,et al.  Numerische lineare Algebra , 1985 .

[18]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[19]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[20]  N. Hashitsume,et al.  Statistical Theory of Rubber‐Like Elasticity. IV. (Two‐Dimensional Stretching) , 1951 .

[21]  Peter Haupt,et al.  Continuum Mechanics and Theory of Materials , 1999 .

[22]  Alexander Lion,et al.  On the large deformation behaviour of reinforced rubber at different temperatures , 1997 .

[23]  R. Fosdick.  Dynamically possible motions of incompressible, isotropic, simple materials , 1968 .