Engineering 227: Advanced Elasticity

Problem Set 3

Due Wednesday, October 22, 2003

 

1. A Neoprene rubber bar is subjected to a uniaxial tensile test. The bar’s initial cross sectional area was 1cm². Initially, the deformations are kept small and the bar is found to have Young’s modulus of 4.25 MPa. The extension is continued until the bar breaks, which occurs when the bar is stretched to 4.2 times its original length. Assume that the elastic potential has the form described in class: .
1. Find the value of the material modulus m in MPa.
2. Find the tensile strength of the material (maximum sustainable stress in tension) in MPa.
3. What was the maximum force applied to the bar during the test? Give the answer in Newtons and pounds.
4. What was the cross sectional area just before the bar broke?
2. Consider an infinitesimal homogeneous deformation with displacement gradient tensor b:  Assume in what follows that the reference configuration is stress free.
1. Show that the nominal stress is related to the displacement gradient b through the linear relations

                                                        

where  is called the elasticity four tensor.

2. Show that the nominal and Cauchy stress are related through . Show further that the balance of angular momentum requires that the elasticity tensor have the first minor symmetry  c_ijkl= c_jikl.
3. Suppose the material is hyperelastic, so that  Show further that the elasticity tensor has major symmetry  c_ijkl= c_klij. If the material is not hyperelastic, is it still possible for the elasticity tensor to have major symmetry? Why or why not?
4. Assume that the elasticity tensor does have major symmetry. Use this and the result of part (b) to show that the elasticity tensor have the second minor symmetry  c_ijkl= c_ijlk.
5. Next, show with the aid of part (d) that  the stress-strain relations can be written as  with the error of order e². The tensor  is the infinitesimal strain tensor.
6. Show that  
7. Now comes objectivity and material symmetry: recall that for every proper orthogonal Q and tensor F with positive determinant, objectivity demands  For solids, with the material symmetry group is a subset of the proper orthogonal group: . Then,  Putting these together gives  By differentiating this identity with respect to F and evaluating the result at F=I,  show that this implies  for every Q in the material symmetry group.
   
   

 

3. When subjected to small deformations, a material has a stress-strain law given by  in which  are material constants and n is a unit vector.
1. Show that the material is elastic and find the elastic potential
2. What type symmetry does this solid have?
3. The solid is in a state of uniaxial stress parallel to the e­₁-direction; the material is oriented such that n=e₂. One finds that the non-zero stress and strain components are related through .  Relate  to . Check the results by setting .
4. Again, the solid is in a state of uniaxial stress parallel to the e­₁-direction. This time, the material is oriented such that n=e₁. What are the resulting strain components, and how do they relate to , and ? Again, check the results by setting .
4. A nonlinear spring is made from a rubber cylindrical tube (inner radius a, outer radius b, height h) as shown below. The outside of this tube is cemented to a fixed, rigid container and the inside to a rigid shaft; the end faces of cylinder are traction free. In order to induce a net shaft displacement d, a force F is applied to the shaft end. The elastic potential for the material is the one you already know and love, from Problem Set 2, .
   
   The goal is to find F given d, and sketch a force-displacement curve for the spring. The boundary value problem consists of satisfying the usual equations of equilibrium subject to the boundary conditions:  on ,  on  , and  on . Consider an axially symmetric deformation of the form .  There are no body forces.
1. Show that the Cartesian components of P in this basis are as follows: .
    Here, , and  is the material constant.
2. Before you attempt to satisfy the equilibrium equations, see if this stress field consistent with the traction-free end conditions. If you can’t satisfy these end conditions exactly at every point on , see if it is possible to satisfy the end conditions in an integral sense, regardless of the actual form taken by u(r). Thus, show that the tractions generated by any deformation of the above form produce no net force (in any of the three directions) on the end-faces, and no net moment (about any of the three axes).
3. Now use the balance of linear momentum (equilibrium equations) and boundary conditions to find the function u(r) in terms of , and d. This function u(r) gives an approximate solution to the original boundary value problem.
4. Using the function u(r) determined above, find the net force F required to displace the inner bar, and sketch a curve of F versus d.
5. Find the components of Cauchy stress.