294
6 Growth
(R, ,, ,) and (r, θ, φ), being principal coordinates. Both regions are composed
of incompressible tissue characterized by the neo-Hookean strain-energy density
function
W = c (I 1 − 3)
I 1 = λ
2
r + λ
2
θ + λ
2
φ ,
(6.77)
and both undergo uniform isotropic growth with
G r = G θ = G φ =
G 1 for R < a 0
G 2 for R > a 0
,
(6.78)
where the growth ratios G 1 and G 2 are specified constants.
The sphere is subjected to no external loads or constraints. Our objective is to
compute stress distributions in both regions for a brain tumor (only the inner region
grows) and a tumor with a necrotic core (only the outer region grows).
6.8.2 Analysis
Section 6.6.5 lists the governing equations in spherical coordinates. For r = r(R),
the total stretch ratios are given by
λ r =
∂r
∂R
,
λ θ = λ φ =
r
R
,
(6.79)
which provide the total volume ratio
J = λ r λ θ λ φ =
∂r
∂R
r
R
2 = J G J
∗ ,
where J G = G r G θ G φ . For an incompressible material, setting J ∗ = 1 and
rearranging this equation give
r
2 dr = J G R
2 dR.
Integrating both sides and using the condition r(0) = 0 (no central cavity) yield
r
3
= 3
R
0
J G R
2 dR.
(6.80)
If growth differs between the inner and outer regions, the above integral must be
split for region 2 (R > a 0 ), i.e.,