Problems for Chapter 4
Solution to Simple Problems
4.3. Axially and Spherically Symmetric Solutions for QuasiStatic Large Strain Elasticity Problems
4.3.1. Consider the pressurized hyperelastic spherical shell described in Section 4.3.3. For simplicity, assume that the shell is made from an incompressible neoHookean material (recall that the NeoHookean constitutive equation is the special case ${\mu}_{2}=0$ in the MooneyRivlin material). Calculate the total strain energy of the sphere, in terms of relevant geometric and material parameters. Hence, derive an expression for the total potential energy of the system (assume that the interior and exterior are subjected to constant pressure). Show that the relationship between the internal pressure and the geometrical parameters $\alpha =a/A$, $\beta =b/B$ can be obtained by minimizing the potential energy of the system.

4.3.2. Consider an internally pressurized hollow rubber cylinder, as shown in the picture. Assume that
Before
deformation, the cylinder has inner radius A
and outer radius B
After
deformation, the cylinder has inner radius a
and outer radius b
The solid is
made from an incompressible MooneyRivlin solid, with strain energy potential
$U=\frac{{\mu}_{1}}{2}({I}_{1}3)+\frac{{\mu}_{2}}{2}({I}_{2}3)$
No body forces
act on the cylinder; the inner surface r=a
is subjected to pressure ${p}_{a}$;
while the outer surface r=b is free
of stress.
Assume plane
strain deformation.
Assume that a material particle that has radial position R before deformation moves to a position r=f(R) after the cylinder is loaded.
4.3.2.1. Express the incompressibility condition det(F)=1 in terms of f(R)
4.3.2.2. Integrate the incompressibility condition to calculate r in terms of R¸A and a, and also calculate the inverse expression that relates R to A, a and r.
4.3.2.3.
Calculate
the components of the left CauchyGreen deformation tensor ${B}_{rr},{B}_{\theta \theta}$
4.3.2.5.
Use
3.4 and the equilibrium equation to derive an expression for the radial stress ${\sigma}_{rr}$ in the cylinder. Use the boundary conditions to find a
relationship between the applied pressure and $\alpha =a/A$,
$\beta =b/B$.
4.3.2.6.
Plot
a graph showing the variation of normalized pressure ${p}_{a}{B}^{2}/({\mu}_{1}+{\mu}_{2})({B}^{2}{A}^{2})$ as a function of the normalized displacement
of the inner bore of the cylinder $a/A1$. Compare the nonlinear elastic solution with
the equivalent linear elastic solution.
4.3.3. A long rubber tube has internal radius A and external radius B.
The tube can be idealized as an incompressible neoHookean material with
material constant ${\mu}_{1}$. The tube is turned insideout, so that the
surface that lies at R=A in the
undeformed configuration moves to r=a
in the deformed solid, while the surface that lies at R=B moves to r=b. Note that B>A, and a>b. To approximate the deformation, assume that
planes that lie
perpendicular to ${e}_{z}$ in the undeformed solid remain perpendicular
to ${e}_{z}$ after deformation
The axial stretch ${\lambda}_{zz}$ in the tube is constant
It
is straightforward to show that the deformation mapping can be described as
$\begin{array}{l}r=\sqrt{{a}^{2}+({B}^{2}{R}^{2})/{\lambda}_{zz}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}R=\sqrt{{B}^{2}+({a}^{2}{r}^{2}){\lambda}_{zz}}\\ z={\lambda}_{zz}Z\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Z=z/{\lambda}_{zz}\end{array}$
4.3.3.1.
Calculate the
deformation gradient, expressing your answer in terms of R, as components in the $\{{e}_{r},{e}_{\theta},{e}_{z}\}$ basis.
Verify that the deformation preserves volume.
4.3.3.2.
Calculate the
components of the left CauchyGreen deformation tensor ${B}_{rr},{B}_{\theta \theta},{B}_{zz}$
4.3.3.3.
Find an
expression for the Cauchy stress components ${\sigma}_{rr},{\sigma}_{\theta \theta},{\sigma}_{zz}$ in the cylinder in terms of ${B}_{rr},{B}_{\theta \theta}$ and an indeterminate hydrostatic stress p.
4.3.3.4.
Use 4.3 and the
equilibrium equation and boundary conditions to calculate an expression for the
Cauchy stress components.
4.3.3.5.
Finally, use the
condition that the resultant force acting on any crosssection of the tube must
vanish to obtain an equation for the axial stretch ${\lambda}_{zz}$. Does the tube get longer or shorter when it
is inverted?

4.3.4. Two spherical, hyperelastic shells
are connected by a thin tube, as shown in the picture. When stress free, both spheres have internal
radius A and external radius B.
The material in each sphere can be idealized as an incompressible,
neoHookean solid, with material constant ${\mu}_{1}$.
Suppose that the two spheres together contain a volume $V\ge 8\pi {A}^{3}/3$ of
an incompressible fluid. As a result,
the two spheres have deformed internal and external radii $({a}_{1},{b}_{1})$,
$({a}_{2},{b}_{2})$ as shown in the picture. Investigate the
possible equilibrium configurations for the system, as functions of the
dimensionless fluid volume $\omega =V/(8\pi {A}^{3}/3)1$ and B/A. To display your results, plot a graph showing
the equilibrium values of ${\alpha}_{1}={a}_{1}/A$ as a function of $\omega $,
for various values of B/A. You should find that for small values of $\omega $ there is only a single stable equilibrium
configuration. For $\omega $ exceeding a critical value, there are three
possible equilibrium configurations: two in which one sphere is larger than the
other (these are stable), and a third in which the two spheres have the same
size (this is unstable).

4.3.5. In a model experiment intended to
duplicate the propulsion mechanism of the lysteria bacterium, a spherical bead
with radius a is coated with an
enzyme known as an “Arp2/3 activator.”
When suspended in a solution of actin, the enzyme causes the actin to polymerize
at the surface of the bead. The polymerization reaction causes a spherical gel
of a dense actin network to form around the bead. New gel is continuously formed at the
bead/gel interface, forcing the rest of the gel to expand radially around the
bead. The actin gel is a longchain polymer and consequently can be idealized
as a rubberlike incompressible neoHookean material. Experiments show that
after reaching a critical radius the actin gel loses spherical symmetry and
occasionally will fracture. Stresses in
the actin network are believed to drive both processes. In this problem you will calculate the
stress state in the growing, spherical, actin gel.
4.3.5.1.
Note
that this is an unusual boundary value problem in solid mechanics, because a
compatible reference configuration cannot be identified for the solid. Nevertheless, it is possible to write down a
deformation gradient field that characterizes the change in shape of
infinitesimal volume elements in the gel.
To this end: (a) write down the length of a circumferential line at the
surface of the bead; (b) write down the length of a circumferential line at
radius r in the gel; (c) use these
results, together with the incompressibility condition, to write down the
deformation gradient characterizing the shape change of a material element that
has been displaced from r=a to a
general position r. Assume that the bead is rigid, and that the
deformation is spherically symmetric.
4.3.5.2.
Suppose
that new actin polymer is generated at volumetric rate $\dot{V}$. Use the incompressibility condition to write
down the velocity field in the actin gel in terms of $\dot{V}$,
a and r (think about the volume of material crossing a radial line per
unit time)
4.3.5.3.
Calculate
the velocity gradient $v\otimes \nabla $ in the gel (a) by direct differentiation of
5.2 and (b) by using the results of 5.1.
Show that the results are consistent.
4.3.5.4.
Calculate
the components of the left CauchyGreen deformation tensor field and hence
write down an expression for the Cauchy stress field in the solid, in terms of
an indeterminate hydrostatic pressure.
4.3.5.5.
Use
the equilibrium equations and boundary condition to calculate the full Cauchy
stress distribution in the bead. Assume
that the outer surface of the gel (at r=b)
is traction free.

4.3.6. A rubber sheet is wrapped around a
rigid cylindrical shaft with radius a. The sheet has thickness t, and can be idealized as an incompressible neoHookean
solid. A constant tension $T$ per unit outofplane distance is applied to
the sheet during the wrapping process.
Calculate the full stress field in the solid rubber, and find an
expression for the radial pressure acting on the shaft. Assume that $t/b\to 0$ and neglect the shear stress component ${\sigma}_{r\theta}$.