Engineering 227: Advanced Elasticity

Problem Set 4

Due Wednesday, November 5, 2003

 

1. In class, we found that neoprene rubber in simple shear for neoprene rubber, the only non-zero Cauchy stress components are:

                                                      

Note that s₂₂ is compressive, even though the fibers aligned with the e₂ axis in the reference configuration are elongated.

a. Find the components of the surface traction on the top surface of the block in the directions e’₁, and e’₂, perpendicuar and parallel to the sloping side of the parallelogram. Comment on the directions of these traction components.

b. Caculate the components of the Cauchy stresses in the basis  aligned with the sloping edge of the parallelogram in the deformed configuration_.. Is s_2’2’ tensile or compressive?

 

c. Calculate the nominal stress components in the basis .

 

2. For an isotropic elastic solid with elastic potential , show that for a homogeneous plane stress deformation of the type  the  in-plane stress-deformation relations reduce to  where  is the elastic potential, , and the out of plane stretch function  is determined from .

 

3. A rubber balloon has a radius r₀ and thickness t₀ when empty. It is filled with an ideal gas with a pressure-volume relation given by

                                                      .

As in standard pressure vessel theory, you may assume that the stresses in the plane of the inflated balloon surface far exceed the pressure p, so that the balloon may be considered to be in a state of equi-biaxial plane stress. When inflated, the balloon has radius r and thickness t.

 

a) By considering static equilibrium of a hemisphere of the deformed balloon, show that the in-plane nominal stress component satisfies

                                                      

where  is the plane stress elastic potential.

 

b) Find and plot the radius R/r of the balloon as a function of   for the following strain energy densities:

i) Neoprene!

ii)  (from problem set 2) .

4.  A rubber shaft of length L and radius a is made of a neo-Hookean material. Each end is welded to a rigid plate; the plate at the end x₃=0 is held fixed and the end x₃=L is rotated in the (x₁,x₂)-plane through an angle f. The sides of the shaft are traction-free.

a) List the boundary conditions for this problem. Can this problem be treated as a plane strain problem?

 

b) Assume a deformation of the form

                                          

in which  is the specific angle of twist (radians per unit length). Interpret this deformation. Show that the deformed configuration is also a cylinder of radius a and length L. Is the deformation consistent with the displacement boundary conditions?

c) Show that the deformation gradient component matrix may be written as

Is the deformation consistent with the constraint of incompressibility?

 

d) Find the components of  G=FF^T as a function of  y, and use these to give the Cauchy stress as a function y.

e)  Apply the equilibrium condition  and from this, conclude that the pressure field p is independent of y₃.

f) Use the remaining two equilibrium equations,  to find the pressure field p as a function of  (y₁, y₂) and a constant of integration.  Determine the constant of integration using the boundary conditions on the lateral boundary of the shaft.

 

g) Using the Cauchy stress components, find the resultant normal force N(f) and torque T(f).  Is the normal force tensile or compressive? Compare these with the results from the linear theory of elastic torsion for a circular shaft.