3.6 Constitutive Relations
137
(a)
(d)
(b)
(c)
isotropic
transversely isotropic
orthotropic
transversely isotropic
cross section
Fig. 3.25 Types of material symmetry. Fibers are shown in red
shown in the next chapter, the stress-strain curve for extension of a neo-Hookean
bar is relatively linear, limiting its scope. Examples where material nonlinearity is
relatively modest include tissues in the early embryo, where a neo-Hookean model
can be a useful first approximation.
Slightly modifying Eq. (3.217) leads to the Mooney-Rivlin strain-energy density
function (Mooney 1940; Rivlin 1947)
W = c 1 (I 1 − 3) + c 2 (I 2 − 3),
(3.218)
which involves two material constants. These two material models can be considered special cases of the general series representation (Rivlin 1956)
W =
n
i=0
n
j =0
c ij (I 1 − 3)
i (I 2 − 3)
j .
(3.219)
Another functional form for rubber-like materials was proposed by Ogden
(1972), who expressed W in terms of principal stretch ratios λ i = 1 + 2E i , which
themselves are invariant with respect to a change in coordinates. The proposed
form is
W =
n
i=1
c n
λ
b n
1 + λ
b n
2 + λ
b n
3 − 3
.
(3.220)
Except for problems that involve no shear in the chosen coordinate system, this
expression requires solving the eigenvalue problem for principal strains at each point
during the solution procedure.
The above relations are possible forms for incompressible materials. Because
soft tissues are mostly water, they often are treated as incompressible. However, if
fluid flow accompanies deformation, the effects of material compressibility may be
important. For compressible materials, W must include changes in volume through
the third strain invariant I 3 = J 2 . Generalizing Eq. (3.219) yields
137
(a)
(d)
(b)
(c)
isotropic
transversely isotropic
orthotropic
transversely isotropic
cross section
Fig. 3.25 Types of material symmetry. Fibers are shown in red
shown in the next chapter, the stress-strain curve for extension of a neo-Hookean
bar is relatively linear, limiting its scope. Examples where material nonlinearity is
relatively modest include tissues in the early embryo, where a neo-Hookean model
can be a useful first approximation.
Slightly modifying Eq. (3.217) leads to the Mooney-Rivlin strain-energy density
function (Mooney 1940; Rivlin 1947)
W = c 1 (I 1 − 3) + c 2 (I 2 − 3),
(3.218)
which involves two material constants. These two material models can be considered special cases of the general series representation (Rivlin 1956)
W =
n
i=0
n
j =0
c ij (I 1 − 3)
i (I 2 − 3)
j .
(3.219)
Another functional form for rubber-like materials was proposed by Ogden
(1972), who expressed W in terms of principal stretch ratios λ i = 1 + 2E i , which
themselves are invariant with respect to a change in coordinates. The proposed
form is
W =
n
i=1
c n
λ
b n
1 + λ
b n
2 + λ
b n
3 − 3
.
(3.220)
Except for problems that involve no shear in the chosen coordinate system, this
expression requires solving the eigenvalue problem for principal strains at each point
during the solution procedure.
The above relations are possible forms for incompressible materials. Because
soft tissues are mostly water, they often are treated as incompressible. However, if
fluid flow accompanies deformation, the effects of material compressibility may be
important. For compressible materials, W must include changes in volume through
the third strain invariant I 3 = J 2 . Generalizing Eq. (3.219) yields