Download Considerations on Incompressibility in Linear Elasticity

Considerations on Incompressibility in Linear Elasticity
Salvatore Federico 1 and Alfio Grillo 2
1
Dept of Mechanical and Manufacturing Engineering, The University of Calgary, Canada
tel: +1-403-220-5790; fax: +1-403-282-8406; email: [email protected]
2
IWR, Universität Heidelberg, Germany
tel: TEL; fax: FAX; email: [email protected]
Materials for which the stiffness under hydrostatic loads (bulk modulus) is very high
compared to that under distortional loads are often assumed to be incompressible. With this
assumption, the bulk elastic modulus κ is thought to diverge, which causes the elasticity
tensor L to be not defined. Therefore, “incompressible” materials are typically represented
by means of the compliance tensor A = L−1 (e.g. Destrade et al., 2002), so that the diverging
elastic modulus is reduced to a compliance approaching zero.
However, the most rigorous way to treat “incompressible” materials is to consider the motion
isochoric. In the realm of linear elasticity, this kinematical constraint is expressed by
prescribing the infinitesimal strain tensor ε to be purely deviatoric, i.e., tr( ε ) = 0 , and by
adding to the stress tensor the term − pI , where the hydrostatic pressure p is a Lagrange
multiplier (e.g., Ogden, 1984). In this picture, there is no diverging term in the elasticity
tensor L , but only some terms that are much larger than the others.
Based on the consideration that the “very large” components in L are completely “shielded”
by the constraint of isochoric motion ( tr( ε ) = 0 ), the purpose of this work is to show that it is
always possible to extract a component of the elasticity tensor which is purely deviatoric,
expressing resistance to distortional deformations only.
A given elasticity tensor L is represented as the orthogonal sum of four components. One
component is purely volumetric, and depends solely on the bulk modulus κ, one is purely
deviatoric, and two are mixed.
Since the volumetric and mixed components are related to tr( ε ) , these have to be dropped for
the case of “incompressible” materials. Therefore, the elasticity tensor for the case of
“incompressible” materials can be represented by the deviatoric part alone.
Dropping the volumetric component gives rise to one condition on the elastic constants
(always expressible by κ = 0 ), whereas dropping the mixed components gives a number of
conditions depending on the material symmetry considered.
If N is the number of independent parameters characteristic of the symmetry of the tensor (2
for isotropy, 5 for transverse isotropy, 9 for orthotropy, etc), then the independent parameters
for the deviatoric component are N − C , C being the number of conditions imposed by
dropping the volumetric and mixed components.
For the case of isotropy, it is straightforward to prove that the volumetric and distortional
parts are fully decoupled in the canonical representation given by Walpole (1984), and C = 1 .
For the case of anisotropic tensors, this ceases to be true. As an example, for the case of
transverse isotropy, the number of conditions is shown to be C = 2 , leaving N − C = 5 − 2 = 3
independent parameters for the deviatoric component of L .
References
Destrade, M., Martin, P.A., Ting, T.C.T., 2002, Journal of the Mechanics and Physics of Solids, 50, 1453-1468.
Ogden, R.W., 1984, Non-Linear Elastic Deformations. Dover Publications, Inc., Mineola, New York, USA.
Walpole, L.J., 1984, Proceedings of the Royal Society of London, Series A, 391, 149-179.