The Linear Elasticity Tensor of Incompressible Materials
Salvatore Federico (The University of Calgary), Alfio Grillo (Polytechnic of Torino), Shoji Imatani (Kyoto University, Japan)
SES Medal Symposium in honor of D.J. Steigmann
Wed 9:00 - 10:30
MacMillan 115
The approximation of isochoric (volume-preserving) motion is often employed to describe the behaviour of materials having stiffness under volumetric compression several orders of magnitude higher than the stiffness in shear (e.g., Ogden, 1984). With a universally accepted abuse of terminology, these materials are called incompressible, although no real incompressible material exists in nature. This work proposes two approaches to the evaluation of the correct form of the linear elasticity tensor of the so-called incompressible materials, both stemming from the non-linear theory. In the first approach, one imposes the kinematical constraint of isochoric motion. In the second approach, which is often employed to enforce incompressibility in numerical applications such as the Finite Element Method, one instead assumes a decoupled form of the elastic strain energy potential, which is written as the sum of a function of the volumetric deformation only and a function of the distortional deformation only, and then imposes that the bulk modulus be much larger than all other moduli. We had previously (Federico et al., 2009) worked out the calculations for the case of isotropic quasi-incompressible materials and now aim at giving the general expression of the elasticity tensor for the strictly incompressible and the quasi-incompressible cases, regardless of material symmetry, and then study the important particular cases of isotropy, transverse isotropy and orthotropy. For the case of strict incompressibility, we show that the number of independent elastic constants decreases from 5 to 3 for transverse isotropy and from 9 to 6 for orthotropy. For the case of quasi-incompressibility, the bulk modulus is an additional independent elastic constant in all cases. References S. Federico, A. Grillo, and G. Wittum, Nuovo Cimento C, 32C:81-87, 2009. R.W. Ogden, Non-Linear Elastic Deformations, Dover Publications, 1984.