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4 Problems in Soft Tissue Biomechanics
where E is the elastic (Young’s) modulus, ν is Poisson’s ratio, and
D =
Eh 3
12(1 − ν 2 )
is the flexural rigidity. In addition, ρ =
1
2 (r 1 + r 2 ) is the mean radius of curvature,
and κ = 1/ρ is the mean curvature. Thus, r − ρ represents the distance of a point
from the deformed middle surface.
Results are given in terms of the nondimensional quantities
(r
∗ , ρ
∗ ) = (r, ρ)/ h,
κ
∗
= hκ,
M
∗
=
M
2ch 2 c 1
,
σ
∗
=
σ
c 1
.
For an incompressible plate (ν = 0.5) composed of neo-Hookean material (c 2 = 0),
it can be shown that c 1 = μ/2 = E/6. Thus, Eqs. (4.128) take the form
M
∗
p = 2κ
∗ /3
σ
∗
p = 8 κ
∗ (r
∗
− ρ
∗ ).
(4.129)
Figure 4.17 shows plots of bending moment (M ∗ ) versus mean curvature, as well
as bending stress (σ ∗
θ ) across the beam thickness. As bending becomes stronger
Fig. 4.17 Results for bending of a block. (a, b) Comparison of elasticity theory and plate theory
for moment-curvature relation (a) and bending stress distribution (b). (c, d) Similar comparisons
for increasing material nonlinearity, as characterized by c 3 with W defined by Eq. (4.109)