Biomedical Engineering Reference
In-Depth Information
6.3.1
Linear Elastic Behavior: Generalized Hooke's Law for Isotropic Materials
In general Cauchy elasticity, the stress tensor
σ
ij
is a nonlinear tensor function of
the strain tensor
ε
kl
:
kl
) (6.15)
whereas in linear elasticity of homogeneous materials the Cauchy stress tensor is,
in extension to Hooke's law (
σ
=
f
ij
(
ε
ij
) of the year 1678, proportional to the Cauchy
strain tensor through the linear transformation:
σ
=
E
ε
σ
=
C
ijkl
ε
(6.16)
ij
kl
It may be noted here that Eq. (6.16) is the simplest generalization of the linear
dependence of stress and strain observed by Hooke in a simple tension test, and
consequently Eq. (6.16) is referred to as the
generalized Hooke's law
.Thefourth-order
tensor
C
ijkl
has 81 independent components in total and is called the
elasticity tensor
.
In the case of linear elasticity, the components of the elasticity tensor are constant
values. Owing to symmetry of
σ
kl
, the pairs of indices
ij
and
kl
in
C
ijkl
can be
permutated. Hence, the number of independent constants of the elasticity tensor
is reduced to 36. A further reduction of constants is obtained if we assume the
existence of a strain energy potential
w
(
ij
and
ε
ε
ij
) from which the stresses are derived by
differentiation as
w
∂ε
∂
σ
=
(6.17)
ij
kl
After partial differentiation of Eq. (6.16) with respect to
ε
ij
and consideration of
Eq. (6.17), one obtains the components of the elasticity tensor as
2
w
∂σ
ij
∂ε
∂
C
ijkl
=
kl
=
(6.18)
∂ε
∂ε
ij
kl
There is an additional symmetry due to the commutation of the order of the partial
derivative with respect to
ε
ε
kl
, and the number of independent constants
is reduced to 21. A material for which such a constitutive relation is assumed is
also called
Green elastic material
[9]. By considering the material symmetries, (e.g.,
crystal systems), a further reduction in the number of constants can be achieved.
For an isotropic material, the mechanical properties are the same for all directions
and it can be shown [10] that the components of the elasticity tensor are determined
by two independent constants, the so-called Lam
ij
and
´
e's constants
λ
and
µ
and the
stress-strain relationship is given for this case by
σ
ij
=
2
µε
ij
+
λε
kk
δ
ij
(6.19)
The characterization of metallic materials is in general based on the engineering
constants, Young's modulus
E
and Poisson's ratio
ν
. On the other hand, the
description of isotropic materials in soil or rock mechanics is based on the shear
modulus
G
and the bulk modulus
K
rather than
E
and
. The conversion of the
elastic constants can be done with the aid of the relationships given in Table 6.3.
ν
