4.7 Bending of a Block
195
more compliant. These geometric effects can be offset by increased material
nonlinearity, e.g., for c 2 > 0.
Lastly, Fig. 4.14b compares results given by the “exact” elasticity solution
and the membrane approximation (see Problem 4.7). As expected, the pressureradius curves come into closer agreement as the shell becomes thinner, i.e., a 0 /h 0
increases.
4.7 Bending of a Block
For bones, bending is a crucial deformation mode that can cause fracture or
stimulate growth. In soft-tissue biomechanics, bending of blood vessels can impede
flow and trigger the development of atherosclerosis at bifurcations (Fung 1997). In
mechanobiology, bending is especially important during embryonic development,
where examples include bending of cell sheets (epithelia) to create tubes and
bending of tubes to create organs, including the heart, brain, and gut (Davies 2005;
Savin et al. 2011; Taber 2014). This section considers bending of a rectangular
block.
4.7.1 Problem Statement
In the undeformed configuration, a rectangular block is bounded by the planes X =
a 1 , a 2 , Y = ±b, and Z = ±c (Fig. 4.16a). The block is composed of isotropic,
incompressible, hyperelastic material with
W = c 1 (I 1 − 3) +
c 2
c 3
e
c 3 (I 1 −3)
− 1
.
(4.109)
Fig. 4.16 Bending of an
isotropic block. (a)
Undeformed configuration.
(b) Deformed configuration.
Blue region represents a
differential element
X
Y
a 1
a 2
(a)
(b)
Y
X
b
b
r
T
r 2
r 1
N
M
195
more compliant. These geometric effects can be offset by increased material
nonlinearity, e.g., for c 2 > 0.
Lastly, Fig. 4.14b compares results given by the “exact” elasticity solution
and the membrane approximation (see Problem 4.7). As expected, the pressureradius curves come into closer agreement as the shell becomes thinner, i.e., a 0 /h 0
increases.
4.7 Bending of a Block
For bones, bending is a crucial deformation mode that can cause fracture or
stimulate growth. In soft-tissue biomechanics, bending of blood vessels can impede
flow and trigger the development of atherosclerosis at bifurcations (Fung 1997). In
mechanobiology, bending is especially important during embryonic development,
where examples include bending of cell sheets (epithelia) to create tubes and
bending of tubes to create organs, including the heart, brain, and gut (Davies 2005;
Savin et al. 2011; Taber 2014). This section considers bending of a rectangular
block.
4.7.1 Problem Statement
In the undeformed configuration, a rectangular block is bounded by the planes X =
a 1 , a 2 , Y = ±b, and Z = ±c (Fig. 4.16a). The block is composed of isotropic,
incompressible, hyperelastic material with
W = c 1 (I 1 − 3) +
c 2
c 3
e
c 3 (I 1 −3)
− 1
.
(4.109)
Fig. 4.16 Bending of an
isotropic block. (a)
Undeformed configuration.
(b) Deformed configuration.
Blue region represents a
differential element
X
Y
a 1
a 2
(a)
(b)
Y
X
b
b
r
T
r 2
r 1
N
M