Engineering 227: Advanced Elasticity

 

Problem Set 1

Due Wednesday, September 17, 2003

 

 

1.        Consider the map

defined on the strip

.

 

a.        Compute the Jacobian determinant for this map. Is it positive on R?

b.       What is the image R_* of R under the map? Is the mapping globally one-to-one? Locally?

c.        Show that the mapping is globally one-to-one on any a subset  of R defined by 

.

Here, k is a real number. Find the inverse map on this region.

 

 

 

2.  A finite deformation is rigid it is distance preserving. That is, the distance between any two points x and z in the reference configuration is equal to the distance between their image points in the deformed configuration:

                   (*)

 

IShow that distance preservation (*) implies that the deformation admits the representation

with Q, c a constant proper orthogonal tensor and vector, respectively. (Suggestion: square equation (*) and differentiate the result first with respect x and then with respect to y.) We showed in class that (**) implies (*).

 

3. A Cauchy stress-response function for a material in a reference configuration is given by

where M is a symmetric tensor field, m is a scalar field, and b is a constant.

 

a.        Show that this is an objective stress response function.

b.       Find the nominal stress response function.

c.        If  m>0 is constant and  M is constant and positive definite, and let  Can you find a stress-free configuration of the body? What is the gradient of the mapping to this stress-free reference configuration? Suggestion: Set  and solve for F. You might try something like F=cB^-1  with c a constant scalar.

 

For the rest of the problem, suppose b is a   positive constant and

.

d.        Find the material symmetry group for this material in this reference configuration. Would you describe it as isotropic? Transversely isotropic? Orthotropic? None of the above?

 

e.        The reference configuration for this material is a hollow sphere with inner radius a and outer radius b.  Show that the reference configuration is traction free on its outer boundary and is in static equilibrium in absence of body forces. Describe the tractions on the inside of the sphere.