Problems for Chapter 4

Solution to Simple Problems

4.3.  Axially and Spherically Symmetric Solutions for Quasi-Static 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 neo-Hookean material (recall that the Neo-Hookean constitutive equation is the special case ${\mu }_{2}=0$ in the Mooney-Rivlin 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 Mooney-Rivlin solid, with strain energy potential

$U=\frac{{\mu }_{1}}{2}\left({I}_{1}-3\right)+\frac{{\mu }_{2}}{2}\left({I}_{2}-3\right)$

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 Cauchy-Green deformation tensor ${B}_{rr},{B}_{\theta \theta }$

4.3.2.4.            Find an expression for the Cauchy stress components ${\sigma }_{rr},{\sigma }_{\theta \theta }$ in the cylinder in terms of  ${B}_{rr},{B}_{\theta \theta }$ and an indeterminate hydrostatic stress p.

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 ${p}_{a}{B}^{2}/\left({\mu }_{1}+{\mu }_{2}\right)\left({B}^{2}-{A}^{2}\right)$ as a function of the normalized displacement of the inner bore of the cylinder $a/A-1$.  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 neo-Hookean material with material constant ${\mu }_{1}$.  The tube is turned inside-out, 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

$\begin{array}{l}r=\sqrt{{a}^{2}+\left({B}^{2}-{R}^{2}\right)/{\lambda }_{zz}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}R=\sqrt{{B}^{2}+\left({a}^{2}-{r}^{2}\right){\lambda }_{zz}}\\ z=-{\lambda }_{zz}Z\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Z=-z/{\lambda }_{zz}\end{array}$

4.3.3.1.            Calculate the deformation gradient, expressing your answer in terms of R, as components in the $\left\{{e}_{r},{e}_{\theta },{e}_{z}\right\}$ basis.  Verify that the deformation preserves volume.

4.3.3.2.            Calculate the components of the left Cauchy-Green 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 cross-section 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, neo-Hookean 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 $\left({a}_{1},{b}_{1}\right)$, $\left({a}_{2},{b}_{2}\right)$ as shown in the picture. Investigate the possible equilibrium configurations for the system, as functions of the dimensionless fluid volume $\omega =V/\left(8\pi {A}^{3}/3\right)-1$ 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 long-chain polymer and consequently can be idealized as a rubber-like incompressible neo-Hookean 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 $\stackrel{˙}{V}$.  Use the incompressibility condition to write down the velocity field in the actin gel in terms of $\stackrel{˙}{V}$, 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 Cauchy-Green 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 neo-Hookean solid.    A constant tension $T$ per unit out-of-plane 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 ${\sigma }_{r\theta }$.