Engineering 227: Advanced Elasticity

Problem Set 2

Due Wednesday, October 8, 2003


1.  Which of the following sets are groups?

  1. The set of all nonsingular tensors H with H¹I. 
  2. The set of tensors that are symmetric and positive definite.
  3. The set of nonsingular tensors that are symmetric and have the same principal basis  .
  4. The set of nonsingular tensors with eigenvector n.

2. Here is the general form for a transversely isotropic simple material without memory:

or

 

in which the scalar functions  depend on  . These are given by

 

,

 

and  is the material direction vector. Confirm that this representation satisfies objectivity and that the corresponding material symmetry group is set of all proper orthogonal tensors Q with Qn=n.

 

3.  Here is a constitutive law for an orthotropic simple material without memory. (This is not the most general form of the stress response function for an orthotropic material.)

 

 ,              (*)

or

,

 

in which the scalar functions  depend on  . These are given by

 

,

 

and  are the material directions. Confirm that this representation satisfies objectivity and that the corresponding material symmetry group is the one advertised in class.

 

4. A homogenous orthotropic rubber is described by the stress response function (*). A block of it is used as a bumper at the bottom of a rigid, rectangular container. The bumper has height h, width w, and depth d. The container walls are lubricated, and when filled, its contents exert a uniform pressure p on the top bumper surface. The base vectors  are fixed relative to the box. 

a.        What are the boundary conditions on each side of the bumper? Displacement conditions will be conditions on the components of , and traction boundary conditions should be expressed using components of the nominal stress . 

b.       If the bumper is aligned so that the material directions  are , show that a homogenous deformation  may be consistent with the boundary conditions.  

c.        Is such a deformation possible if the material directions  are not aligned with the base vectors? 

 

5. The strain-energy density of a homogeneous, hyperelastic solid is given by

,

with  constant.

a.        A bar (initial length and cross-sectional area , respectively) made of this material is subjected to  a state of uniaxial stress. A constant force F is applied to the ends; the other surfaces remain traction-free. The bar undergoes a homogeneous deformation of the form

 

i.                     Find a relation between the longitudinal stretch  and the transverse stretch . Suggestion: Express   in terms of the principal stretches .

ii.                    Find the longitudinal Cauchy and nominal stress components in terms of the longitudinal stretch . Relate  to  F and . Make a graph of the results.

iii.                  Find Young's modulus E and Poisson's ratio  to describe the small-strain elastic behavior of this material (let e =l1-1; |e|<<1).

 

b.       A rubber sheet made of this material is subjected to a state of biaxial tension in plane stress. q is the force per unit area in the reference (undeformed) configuration . A deformation of the form  is induced. Find a relation between the in-plane stretch  and the out-of-plane stretch . . Relate  to q.

c.        A glob of this material is subjected to a state of uniform pressure: on the boundary of the deformed configuration, , . Here,  p is a constant,  is the Cauchy traction vector, and  is the outer unit normal vector to . This might occur if the object were placed in a vacuum (p<0) or submerged under water (p>0).  A pure homogeneous deformation is induced:

Relate  to p; make a graph.

 

6.  The strain energy density described above is modified to represent a transversely isotropic solid (symmetry axis  ):

.

a.   As in the previous problem a bar (initial length and cross-sectional area , respectively) made of this material is subjected to a state of uniaxial stress. A constant force F is applied to the ends; the other surfaces remain traction-free. The bar undergoes a homogeneous deformation of the form

.             

i.                     Assume first that the longitudinal axis of the bar is lined up with the material symmetry axis (  ). Show that , and find a relation between the longitudinal stretch  and the transverse stretch . Then find the longitudinal Cauchy and nominal stress components in terms of the longitudinal stretch . Relate  to  and F . Graph the results, and compare with the results for the isotropic case.

ii.                    Next, assume that the longitudinal axis of the bar is perpendicular to the material symmetry axis; aligned with .  Do you expect  in this case as well? Explain.

 

  1. As in problem 5b, a  rubber sheet made of this material is subjected to a state of biaxial tension in plane stress. q is the force per unit area in the reference (undeformed) configuration . A pure homogenous deformation  is induced.  Relate   to q for the cases in which  and   and
  2. As in problem 5c, a piece of this transversely isotropic material is subjected to a state of uniform pressure: on the boundary of the deformed configuration, , one has . In a basis chosen such that , a pure homogeneous deformation of the following form is induced:

.

Find a single equation which relates p and l.  Again, graph the results. (Suggestion: use

Mathematica or Maple to help with the algebra on this one!)