Materials: Rubber

Description

Rubbers and rubber-like materials are generally isotropic but highly deformable, highly elastic (hyperelastic) and nearly incompressible. More specifically, they remain elastic even under very large strain and their volume change is often very small compared with the total deformation. The Materials: Isotropic is therefore not applicable to rubber materials when large deformation is expected.

The behaviour of rubber (hyperelastic) materials is described by certain assumed elastic potential functions, or strain energy density functions, for the strain energy stored in a unit volume of the undeformed material. Once the strain energy density function or the corresponding complementary energy density function is known, the material constitutive equation is determined. The four rubber material models available in Strand7 are defined by their respective energy density functions:

The neo-Hookean model is suitable for only vulcanised rubbers highly swollen with organic solvents. Both of the neo-Hookean and Mooney-Rivlin models are experimentally observed to be well-suited to most natural (unfilled) rubbers with accurate correlation up to about 500% strain. However, filled rubbers, as well as certain natural rubbers show poor correlation at strains in excess of about 100%. The generalised Mooney-Rivlin model should be used to accurately simulate the very large strain (100-700%) behaviour.

The Ogden model is different from the other three models in that the principal stretches are used instead of the strain invariants.

See the Strand7 Theoretical Manual for more information.

Determination of Material Parameters

The nominal stress and stretch relationship and inherently the strain energy density function of the rubber material is typically determined from experimental testing.

The nominal stress and stretch curve can then be used to determine the rubber material constants. See Properties: Rubber Coefficient Calculator.

Relationship between Bulk Modulus and Poisson's Ratio

As most rubbers have very small compressibility, they are often considered to be incompressible or nearly incompressible. The Poisson’s ratio for natural rubber is typically 0.499, whereas industrial elastomers, which are usually filled rubbers, exhibit Poisson’s ratio in the range 0.4985 - 0.4995. As the displacement in the thickness direction is not restrained in the membrane and plate/shell analyses, it is acceptable to assume that the material is incompressible and the strain in the thickness direction can be calculated from the in-plane components based on the assumption that there is no volume change. In the other analyses, however, it is very important to consider the compressibility even when it is small. Considering the small strain situation where an isotropic material model is interchangeable with a rubber material model, the following relationship between bulk modulus , Poisson's ratio and shear modulus is generally upheld:

To avoid numerical instability, values for the material constants and the bulk modulus should be checked to ensure that these yield a Poisson's ratio satisfying .

Literature

Oden, J.T. 1972, Finite Elements of Nonlinear Continua, McGraw-Hill, New York.

Hooke, R. "Lectures de Potentia Restitutiva", or "Of Spring, Explaining the Power of Spring Bodies", a pamphlet reproduced more recently by R.T. Gunther, Early Science in Oxford, vol. 8, pp. A169-A175.

Mooney, M. 1940, "A theory of large elastic deformation", Journal of Applied Physics, vol. 11, pp. 482-592.

Ogden, R.W. 1972, "Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubber-like materials", Proc. R. Soc. London, A326/328, pp. 567-583.

Iding, R.H. 1973, "Identification of Nonlinear Materials by Finite Element Methods", SESM Report, no. 73-4, Department of Civil Engineering, University of California, Berkeley, January.

Chen, J.S., Satyamurthy, K. & Hirschfelt, L.R. 1994, "Consistent finite element procedures for nonlinear rubber elasticity with a higher-order strain energy function", Comp. and Struct., vol. 50, pp. 715-727.

Chang, T.Y.P., Saleeb, A.F. & Li, G. 1991, "Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle", Comp. Mech., vol. 8, pp. 221-233.

Gadala, M.S. 1992, "Alternative methods for the solution of hyperelastic problems with incompressibility", Computers and Structures, vol. 42, pp. 1-10.

Peng, S.H. & Chang, W.V. 1997, "A compressible approach in finite element analysis of rubber-elastic materials", Computers and Structures, vol. 62, pp. 573-593.

Anand, L. 1996, "A constitutive model for compressible elastometric solids", Computational Mechanics, vol. 18, pp. 339-355.

See Also