Engineering 227: Advanced Elasticity

Problem Set 5

Due Wednesday, November 19, 2003

 

1. For an incompressible material, .

a. Confirm that

the function

                  

allows the principal Cauchy stress components to be written as . (no sum on i.)

b. Since the function  is evaluated using stretches that are constrained so that  we could just as easily have expressed that strain-energy density as a function  of the principal stretches  through

           

note that 

Show that the principal stress components can be written as  where  is a kinematically indeterminate pressure term. Relate  and p, and show that  is the true pressure field.

 

2.  The Ogden elastic potential has the form  with m and aconstant. The power-law material has the form  with m, b, and n constant. Find the stress-stretch relation (Nominal and Cauchy) for these materials in uniaxial stress. What is Young’s Modulus for each one? Plot the response for a range of the material parameters.

 

4. A neoHookean packman is in a state of plane strain. It is a disk of radius a  with a wedge-shaped gap of angle d=2p-g. The gap is closed by gluing its edges together. The outer edge of the disk is traction free.

 

a. Assume a deformation of the form

Here, a is a constant. Find  and a. The incompressibility constraint determines ; consider the volume of the material in the shaded region in the reference configuration (radius r) and its volume in the deformed consideration (radius  ). Find a using the displacement boundary condition.

b. For this deformation, F is symmetric; in a cylindrical coordinate system F has a diagonal matrix; the on-diagonal in-plane components are . These may be obtained through geometrical considerations. For example, you can use the fact that  are the stretches in the respective coordinate directions. Find these stretches in terms of the problem parameters.

 

c. Find the components Cauchy stress in cylindrical coordinates  and express the components as function of R,f. (Include the out-of-plane stress components.) 

d. Assume the pressure p depends only on the radial coordinate R then the relevant in-plane Eulerian equibilrium equation in cylindrical coordinates is

                                             

Determine the Cauchy stress components.

5. A neo-Hookean rubber cylinder (radius a height h) contains a screw dislocation. The lateral boundary are traction free. The end faces have no normal tractions applied, although shear tractions provide a twisting moment of magnitude M₃=T  to generate the dislocation. There is no net bending moment applied to the end faces M₁= M₂=0.   In terms of cylindrical coordinates  the slip plane is along the positive x₃ axis, and so the displacement conditions are that when  . The tractions across the slip plane are continuous. The goal is to relate T  to the slip u₀ and determine the stress fields in the material.

 

Assume an anti-plane shear deformation of the form .  There are no body forces. In what follows, p is the usual pressure field. .   The elastic potential is .

 

a. Show that the Cartesian components of P in this basis are as follows: .
 

b. Express the boundary conditions on r=a and q=0 in terms of  p, u₃ and its derivatives. Suggestion: use the cylindrical coordinates .

c. You may assume that .  (This may be shown to be a consequence of the equilibrium conditions and boundary conditions. Use the equilibrium equations and boundary conditions to find the function u in terms of the problem parameters.

d. Confirm that the end conditions are satisfied, and find a relation between the twisting moment T and the slip u₀.

e. Determine the Cauchy stress field. Compare with the linear stress field for a screw dislocation. (look up this solution if you are not familiar with it)