188
4 Problems in Soft Tissue Biomechanics
To understand the physics behind the Poynting effect, consider a bar with
longitudinal fibers. If the length of the bar is fixed (λ = 1), twisting causes the fibers
to stretch as they spiral around the bar like a barber pole (see Fig. 4.11a). Since the
circumferential displacement increases with r, the fiber strain and the accompanying
fiber stress increase with radius. Moreover, the spiral geometry causes the fiber
stress to have components in both the longitudinal and circumferential directions.
If the bar is now allowed to change length freely as it twists, i.e., only twisting
moments are applied at the ends, the longitudinally directed fiber tension would
tend to shorten the bar. In contrast, circumferential tension in the outer region would
compress the inner region, forcing it to elongate axially to satisfy incompressibility.
The solution predicts that elongation dominates the response, leading to the
Poynting effect, which also has been observed experimentally (Poynting 1909). If
the bar has no fibers, the same argument applies if we replace “fibers” by “line
elements.”
Finally, it is important to emphasize that the above solution is “exact” only if the
stresses σ zz and σ zθ = σ θz on the ends of the cylinder are distributed according
to the calculations, e.g., see Fig. 4.12a. Otherwise, the solution provides a good
approximation sufficiently far from the ends (say more than a diameter away), while
end effects can produce significant error near the ends.
The take-away message here is that nonlinearities can lead to behavior that may
not be intuitively obvious. Such interesting behavior may not be captured by a linear
solution.
4.6 Inflation of a Spherical Shell
Another common biological shape is the sphere. Examples include the eye, bladder,
and white blood cell. After the first series of cell divisions, the embryo takes
the form of a hollow ball called the blastula. The shape of the cell nucleus is
generally ellipsoidal, but it can become spherical under some circumstances. To
a first approximation, these structures can be treated as pressurized fluid-filled
membranes or shells. 5 Here, we consider inflation of a thick-walled spherical shell.
4.6.1 Problem Statement
In the undeformed configuration, a spherical shell has inner and outer radii a 0 and
b 0 , respectively. The wall is composed of isotropic Mooney–Rivlin material with
W = c 1 (I 1 − 3) + c 2 (I 2 − 3).
(4.89)
5 In mechanics, a membrane is a very thin shell with negligible bending stiffness.
4 Problems in Soft Tissue Biomechanics
To understand the physics behind the Poynting effect, consider a bar with
longitudinal fibers. If the length of the bar is fixed (λ = 1), twisting causes the fibers
to stretch as they spiral around the bar like a barber pole (see Fig. 4.11a). Since the
circumferential displacement increases with r, the fiber strain and the accompanying
fiber stress increase with radius. Moreover, the spiral geometry causes the fiber
stress to have components in both the longitudinal and circumferential directions.
If the bar is now allowed to change length freely as it twists, i.e., only twisting
moments are applied at the ends, the longitudinally directed fiber tension would
tend to shorten the bar. In contrast, circumferential tension in the outer region would
compress the inner region, forcing it to elongate axially to satisfy incompressibility.
The solution predicts that elongation dominates the response, leading to the
Poynting effect, which also has been observed experimentally (Poynting 1909). If
the bar has no fibers, the same argument applies if we replace “fibers” by “line
elements.”
Finally, it is important to emphasize that the above solution is “exact” only if the
stresses σ zz and σ zθ = σ θz on the ends of the cylinder are distributed according
to the calculations, e.g., see Fig. 4.12a. Otherwise, the solution provides a good
approximation sufficiently far from the ends (say more than a diameter away), while
end effects can produce significant error near the ends.
The take-away message here is that nonlinearities can lead to behavior that may
not be intuitively obvious. Such interesting behavior may not be captured by a linear
solution.
4.6 Inflation of a Spherical Shell
Another common biological shape is the sphere. Examples include the eye, bladder,
and white blood cell. After the first series of cell divisions, the embryo takes
the form of a hollow ball called the blastula. The shape of the cell nucleus is
generally ellipsoidal, but it can become spherical under some circumstances. To
a first approximation, these structures can be treated as pressurized fluid-filled
membranes or shells. 5 Here, we consider inflation of a thick-walled spherical shell.
4.6.1 Problem Statement
In the undeformed configuration, a spherical shell has inner and outer radii a 0 and
b 0 , respectively. The wall is composed of isotropic Mooney–Rivlin material with
W = c 1 (I 1 − 3) + c 2 (I 2 − 3).
(4.89)
5 In mechanics, a membrane is a very thin shell with negligible bending stiffness.