Biomedical Engineering Reference
In-Depth Information
on anisotropic elasticity mentioned in the literature notes at the end of Chapter
6
.
In the first section of this chapter it was stated that one of the purposes of this
chapter was to obtain and record representations of
A
and
C
that represent the
effects of material symmetry. This has been done and the results are recorded in
Tables
4.3
,
4.4
and
4.5
for
A
and
C
, respectively. The representations in these tables
were developed with the minimum algebraic manipulation invoked and with the
minimum rigor but, we think, with the most concise method. The method employed
to obtain these representations is new. Cowin and Mehrabadi (
1987
) first pursued
using reflections to characterize the elastic symmetries. Later Cowin and
Mehrabadi (
1995
) developed all the linear elastic symmetries from this viewpoint.
Here we have not generally bothered to demonstrate invariance of these
representations for all the reflective symmetries that they enjoy. In this presentation
we have been content to use the minimum number of reflective symmetries
necessary to obtain the symmetry representations. So we do not show the reader
that further admissible reflective symmetries will not alter the representation
obtained, but it happens to be true.
The established and more general method to obtain these representations is
to use the symmetry groups associated with each of these symmetries.
The disadvantage of that path in the present text is that it takes too much text
space and distracts the reader from the main topic because it requires the introduc-
tion of the group concept and then the concept of the material symmetry group, etc.
In the method of presentation employed, only the concept of the plane of mirror or
reflective symmetry is necessary. The reflective symmetry approach is equivalent to
the group theory approach for
A
and
C
tensors (Chadwick et al.
2001
), but it may
not be so for nonlinear relationships between these tensors. It does, however, not
provide all the representations for the second rank
A
if
A
is not symmetric (Cowin
2003
).
References
Chadwick P, Vianello M, Cowin SC (2001) A proof that the number of linear anisotropic elastic
symmetries is eight. J Mech Phys Solids 49:2471-2492
Cowin SC (1995) On the number of distinct elastic constants associated with certain anisotropic
elastic symmetries. In Casey J, Crochet MJ (ed) Theoretical, experimental, and numerical
contributions to the mechanics of fluids and solids—a collection of papers in honor of Paul M.
Naghdi, Z Angew Math Phys (Special Issue) 46:S210-S214
Cowin SC (1999) Bone poroelasticity. J Biomech 32:218-238
Cowin SC (2002) Elastic symmetry restrictions from structural gradients. In: Podio-Guidugli P,
Brocato M (eds) Rational continua, classical and new—a collection of papers dedicated
to Gianfranco Capriz on the occasion of his 75th birthday. Springer Verlag, New York.
ISBN 88-470-0157-9
Cowin SC (2003) Symmetry plane classification criteria and the symmetry group classification
criteria are not equivalent in the case of asymmetric second order tensors. Chinese J Mech
19:9-14
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